# FeenoX manual

A free no-fee no-X uniX-like finite-element(ish) computational engineering tool

Jeremy Theler

# 1 Overview

FeenoX is a computational tool that can solve engineering problems which are usually casted as differential-algebraic equations (DAEs) or partial differential equations (PDEs). It is to finite elements programs and libraries what Markdown is to Word and TeX, respectively. In particular, it can solve

• dynamical systems defined by a set of user-provided DAEs (such as plant control dynamics for example)
• mechanical elasticity
• heat conduction
• structural modal analysis
• neutron diffusion
• neutron transport

FeenoX reads a plain-text input file which contains the problem definition and writes 100%-user defined results in ASCII (through PRINT or other user-defined output instructions within the input file). For PDE problems, it needs a reference to at least one Gmsh mesh file for the discretization of the domain. It can write post-processing views in either .msh or .vtk formats.

Keep in mind that FeenoX is just a back end reading a set of input files and writing a set of output files following the design philosophy of UNIX (separation, composition, representation, economy, extensibility, etc). Think of it as a transfer function (or a filter in computer-science jargon) between input files and output files:

                             +------------+
mesh (*.msh)  }             |            |             { terminal
data (*.dat)  } input ----> |   FeenoX   |----> output { data files
input (*.fee) }             |            |             { post (vtk/msh)
+------------+

Following the UNIX programming philosophy, there are no graphical interfaces attached to the FeenoX core, although a wide variety of pre and post-processors can be used with FeenoX. To illustrate the transfer-function approach, consider the following input file that solves Laplace’s equation \nabla^2 \phi = 0 on a square with some space-dependent boundary conditions:

\begin{cases} \phi(x,y) = +y & \text{for $x=-1$ (left)} \\ \phi(x,y) = -y & \text{for $x=+1$ (right)} \\ \nabla \phi \cdot \hat{\vec{n}} = \sin\left(\frac{\pi}{2} \cdot x\right) & \text{for $y=-1$ (bottom)} \\ \nabla \phi \cdot \hat{\vec{n}} =0 & \text{for $y=+1$ (top)} \\ \end{cases}

PROBLEM laplace 2d

# boundary conditions
BC left    phi=+y
BC right   phi=-y
BC bottom  dphidn=sin(pi/2*x)
BC top     dphidn=0

SOLVE_PROBLEM

# same output in .msh and in .vtk formats
WRITE_MESH laplace-square.msh phi VECTOR dphidx dphidy 0
WRITE_MESH laplace-square.vtk phi VECTOR dphidx dphidy 0

Figure 1: Laplace’s equation solved with FeenoX. a — Post-processed with Gmsh, b — Post-processed with Paraview

The .msh file can be post-processed with Gmsh, and the .vtk file can be post-processed with Paraview. See https://www.caeplex.com for a mobile-friendly web-based interface for solving finite elements in the cloud directly from the browser.

# 2 Introduction

FeenoX can be seen either as

• a syntactically-sweetened way of asking the computer to solve engineering-related mathematical problems, and/or
• a finite-element(ish) tool with a particular design basis.

Note that some of the problems solved with FeenoX might not actually rely on the finite element method, but on general mathematical models and even on the finite volumes method. That is why we say it is a finite-element(ish) tool.

In other words, FeenoX is a computational tool to solve

• dynamical systems written as sets of ODEs/DAEs, or
• steady or quasi-static thermo-mechanical problems, or
• steady or transient heat conduction problems, or
• modal analysis problems, or
• neutron diffusion or transport problems, or
• community-contributed problems

in such a way that the input is a near-English text file that defines the problem to be solved.

One of the main features of this allegedly particular design basis is that simple problems ought to have simple inputs (rule of simplicity) or, quoting Alan Kay, “simple things should be simple, complex things should be possible.”

For instance, to solve one-dimensional heat conduction over the domain x\in[0,1] (which is indeed one of the most simple engineering problems we can find) the following input file is enough:

PROBLEM thermal 1D               # tell FeenoX what we want to solve
k = 1                            # set uniform conductivity
BC left  T=0                     # set fixed temperatures as BCs
BC right T=1                     # "left" and "right" are defined in the mesh
SOLVE_PROBLEM                    # tell FeenoX we are ready to solve the problem
PRINT T(0.5)                     # ask for the temperature at x=0.5
$feenox thermal-1d-dirichlet-constant-k.fee 0.5$ 

The mesh is assumed to have been already created with Gmsh (or any other pre-processing tool and converted to .msh format with Meshio for example). This assumption follows the rule of composition and prevents the actual input file to be polluted with mesh-dependent data (such as node coordinates and/or nodal loads) so as to keep it simple and make it Git-friendly (rule of generation). The only link between the mesh and the FeenoX input file is through physical groups (in the case above left and right) used to set boundary conditions and/or material properties.

Another design-basis decision is that similar problems ought to have similar inputs (rule of least surprise). So in order to have a space-dependent conductivity, we only have to replace one line in the input above: instead of defining a scalar k we define a function of x (we also update the output to show the analytical solution as well):

PROBLEM thermal 1D
k(x) = 1+x                       # space-dependent conductivity
BC left  T=0
BC right T=1
SOLVE_PROBLEM
PRINT T(1/2) log(1+1/2)/log(2)   # print numerical and analytical solutions
$feenox thermal-1d-dirichlet-space-k.fee 0.584959 0.584963$

The other main decision in FeenoX design is an everything is an expression design principle, meaning that any numerical input can be an algebraic expression (e.g. T(1/2) is the same as T(0.5)). If we want to have a temperature-dependent conductivity (which renders the problem non-linear) we can take advantage of the fact that T(x) is available not only as an argument to PRINT but also for the definition of algebraic functions:

PROBLEM thermal 1D
k(x) = 1+T(x)                    # temperature-dependent conductivity
BC left  T=0
BC right T=1
SOLVE_PROBLEM
PRINT T(1/2) sqrt(1+(3*0.5))-1   # print numerical and analytical solutions
$feenox thermal-1d-dirichlet-temperature-k.fee 0.581139 0.581139$

For example, let us consider the famous chaotic Lorenz’ dynamical system. Here is one way of getting an image of the butterfly-shaped attractor using FeenoX to compute it and Gnuplot to draw it. Solve

\begin{cases} \dot{x} &= \sigma \cdot (y - x) \\ \dot{y} &= x \cdot (r - z) - y \\ \dot{z} &= x y - b z \\ \end{cases}

for 0 < t < 40 with initial conditions

\begin{cases} x(0) = -11 \\ y(0) = -16 \\ z(0) = 22.5 \\ \end{cases}

and \sigma=10, r=28 and b=8/3, which are the classical parameters that generate the butterfly as presented by Edward Lorenz back in his seminal 1963 paper Deterministic non-periodic flow.

The following ASCII input file ressembles the parameters, inital conditions and differential equations of the problem as naturally as possible:

PHASE_SPACE x y z     # Lorenz attractor’s phase space is x-y-z
end_time = 40         # we go from t=0 to 40 non-dimensional units

sigma = 10            # the original parameters from the 1963 paper
r = 28
b = 8/3

x_0 = -11             # initial conditions
y_0 = -16
z_0 = 22.5

# the dynamical system's equations written as naturally as possible
x_dot = sigma*(y - x)
y_dot = x*(r - z) - y
z_dot = x*y - b*z

PRINT t x y z        # four-column plain-ASCII output

Indeed, when executing FeenoX with this input file, we get four ASCII columns (t, x, y and z) which we can then redirect to a file and plot it with a standard tool such as Gnuplot. Note the importance of relying on plain ASCII text formats both for input and output, as recommended by the UNIX philosophy and the rule of composition: other programs can easily create inputs for FeenoX and other programs can easily understand FeenoX’ outputs. This is essentially how UNIX filters and pipes work.

Let us solve the linear elasticity benchmark problem NAFEMS LE10 “Thick plate pressure.” Assuming a proper mesh has already been created in Gmsh, note how well the FeenoX input file matches the problem statement from fig. 3:

# NAFEMS Benchmark LE-10: thick plate pressure
PROBLEM mechanical DIMENSIONS 3
READ_MESH nafems-le10.msh   # mesh in millimeters

BC upper    p=1      # 1 Mpa

# BOUNDARY CONDITIONS:
BC DCD'C'   v=0      # Face DCD'C' zero y-displacement
BC ABA'B'   u=0      # Face ABA'B' zero x-displacement
BC BCB'C'   u=0 v=0  # Face BCB'C' x and y displ. fixed
BC midplane w=0      #  z displacements fixed along mid-plane

# MATERIAL PROPERTIES: isotropic single-material properties
E = 210e3   # Young modulus in MPa
nu = 0.3    # Poisson's ratio

SOLVE_PROBLEM   # solve!

# print the direct stress y at D (and nothing more)
PRINT "σ_y @ D = " sigmay(2000,0,300) "MPa"

The problem asks for the normal stress in the y direction \sigma_y at point “D,” which is what FeenoX writes (and nothing else, rule of economy):

$feenox nafems-le10.fee sigma_y @ D = -5.38016 MPa$ 

Also note that since there is only one material there is no need to do an explicit link between material properties and physical volumes in the mesh (rule of simplicity). And since the properties are uniform and isotropic, a single global scalar for E and a global single scalar for \nu are enough.

For the sake of visual completeness, post-processing data with the scalar distribution of \sigma_y and the vector field of displacements [u,v,w] can be created by adding one line to the input file:

WRITE_MESH nafems-le10.vtk sigmay VECTOR u v w

This VTK file can then be post-processed to create interactive 3D views, still screenshots, browser and mobile-friendly webGL models, etc. In particular, using Paraview one can get a colorful bitmapped PNG (the displacements are far more interesting than the stresses in this problem).

1. The input files are very similar to the statements of each problem in plain English words (rule of clarity). Those with some experience may want to compare them to the inputs decks (sic) needed for other common FEA programs.
2. By design, 100% of FeenoX’ output is controlled by the user. Had there not been any PRINT or WRITE_MESH instructions, the output would have been empty, following the rule of silence. This is a significant change with respect to traditional engineering codes that date back from times when one CPU hour was worth dozens (or even hundreds) of engineering hours. At that time, cognizant engineers had to dig into thousands of lines of data to search for a single individual result. Nowadays, following the rule of economy, it is actually far easier to ask the code to write only what is needed in the particular format that suits the user.

Some basic rules are

• FeenoX is just a solver working as a transfer function between input and output files.

                             +------------+
mesh (*.msh)  }             |            |             { terminal
data (*.dat)  } input ----> |   FeenoX   |----> output { data files
input (*.fee) }             |            |             { post (vtk/msh)
+------------+

Following the rules of separation, parsimony and diversity, there is no embedded graphical interface but means of using generic pre and post processing tools—in particular, Gmsh and Paraview respectively. See also CAEplex for a web-based interface.

• The input files should be syntactically sugared so as to be as self-describing as possible.

• Simple problems ought to need simple input files.

• Similar problems ought to need similar input files.

• Everything is an expression. Whenever a number is expected, an algebraic expression can be entered as well. Variables, vectors, matrices and functions are supported. Here is how to replace the boundary condition on the right side of the slab above with a radiation condition:

sigma = 1       # non-dimensional stefan-boltzmann constant
e = 0.8         # emissivity
Tinf=1          # non-dimensional reference temperature
BC right q=sigma*e*(Tinf^4-T(x)^4)

This “everything is an expression” principle directly allows the application of the Method of Manufactured Solutions for code verification.

• FeenoX should run natively in the cloud and be able to massively scale in parallel. See the Software Requirements Specification and the Software Development Specification for details.

Since it is free (as in freedom) and open source, contributions to add features (and to fix bugs) are welcome. In particular, each kind of problem supported by FeenoX (thermal, mechanical, modal, etc.) has a subdirectory of source files which can be used as a template to add new problems, as implied in the “community-contributed problems” bullet above (rules of modularity and extensibility). See the documentation for details about how to contribute.

# 3 Running feenox

## 3.1 Invocation

The format for running the feenox program is:

feenox [options] inputfile [optional_extra_arguments] ...

The feenox executable supports the following options:

feenox [options] inputfile [replacement arguments] [petsc options]  
-h, --help

display options and detailed explanations of commmand-line usage

-v, --version

display brief version information and exit

-V, --versions

display detailed version information

--pdes

list the types of PROBLEMs that FeenoX can solve, one per line

--progress

print ASCII progress bars when solving PDEs

--mumps

ask PETSc to use the direct linear solver MUMPS

--linear

force FeenoX to solve the PDE problem as linear

--non-linear

force FeenoX to solve the PDE problem as non-linear

Instructions will be read from standard input if “-” is passed as inputfile, i.e.

$echo 'PRINT 2+2' | feenox - 4 The optional [replacement arguments] part of the command line mean that each argument after the input file that does not start with an hyphen will be expanded verbatim in the input file in each occurrence of $1, $2, etc. For example $ echo 'PRINT $1+$2' | feenox - 3 4
7

PETSc and SLEPc options can be passed in [petsc options] as well, with the difference that two hyphens have to be used instead of only once. For example, to pass the PETSc option -ksp_view the actual FeenoX invocation should be

$feenox input.fee --ksp_view For PETSc options that take values, en equal sign has to be used: $ feenox input.fee --mg_levels_pc_type=sor

See https://www.seamplex.com/feenox/examples for annotated examples.

## 3.2 Compilation

These detailed compilation instructions are aimed at amd64 Debian-based GNU/Linux distributions. The compilation procedure follows the POSIX standard, so it should work in other operating systems and architectures as well. Distributions not using apt for packages (i.e. yum) should change the package installation commands (and possibly the package names). The instructions should also work for in MacOS, although the apt-get commands should be replaced by brew or similar. Same for Windows under Cygwin, the packages should be installed through the Cygwin installer. WSL was not tested, but should work as well.

### 3.2.1 Quickstart

Note that the quickest way to get started is to download an already-compiled statically-linked binary executable. Note that getting a binary is the quickest and easiest way to go but it is the less flexible one. Mind the following instructions if a binary-only option is not suitable for your workflow and/or you do need to compile the source code from scratch.

On a GNU/Linux box (preferably Debian-based), follow these quick steps. See sec. 3.2.2 for the actual detailed explanations.

To compile the Git repository, proceed as follows. This procedure does need git and autoconf but new versions can be pulled and recompiled easily. If something goes wrong and you get an error, do not hesitate to ask in FeenoX’ discussion page.

1. Install mandatory dependencies

sudo apt-get install gcc make git automake autoconf libgsl-dev

If you cannot install libgsl-dev but still have git and the build toolchain, you can have the configure script to download and compile it for you. See point 4 below.

2. Install optional dependencies (of course these are optional but recommended)

sudo apt-get install libsundials-dev petsc-dev slepc-dev
3. Clone Github repository

git clone https://github.com/seamplex/feenox
4. Boostrap, configure, compile & make

cd feenox
./autogen.sh
./configure
make -j4

If you cannot (or do not want) to use libgsl-dev from a package repository, call configure with --enable-download-gsl:

./configure --enable-download-gsl

If you do not have Internet access, get the tarball manually, copy it to the same directory as configure and run again. See the detailed compilation instructions for an explanation.

5. Run test suite (optional)

make check
6. Install the binary system wide (optional)

sudo make install

To stay up to date, pull and then autogen, configure and make (and optionally install):

git pull
./autogen.sh; ./configure; make -j4
sudo make install

### 3.2.2 Detailed configuration and compilation

The main target and development environment is Debian GNU/Linux, although it should be possible to compile FeenoX in any free GNU/Linux variant (and even the in non-free MacOS and/or Windows platforms) running in virtually any hardware platform. FeenoX can run be run either in HPC cloud servers or a Raspberry Pi, and almost everything that sits in the middle.

Following the UNIX philosophy discussed in the SDS, FeenoX re-uses a lot of already-existing high-quality free and open source libraries that implement a wide variety of mathematical operations. This leads to a number of dependencies that FeenoX needs in order to implement certain features.

There is only one dependency that is mandatory, namely GNU GSL (see sec. 3.2.2.1.1), which if it not found then FeenoX cannot be compiled. All other dependencies are optional, meaning that FeenoX can be compiled but its capabilities will be partially reduced.

As per the SRS, all dependencies have to be available on mainstream GNU/Linux distributions and have to be free and open source software. But they can also be compiled from source in case the package repositories are not available or customized compilation flags are needed (i.e. optimization or debugging settings).

In particular, PETSc (and SLEPc) also depend on other mathematical libraries to perform particular operations such as low-level linear algebra operations. These extra dependencies can be either free (such as LAPACK) or non-free (such as Intel’s MKL), but there is always at least one combination of a working setup that involves only free and open source software which is compatible with FeenoX licensing terms (GPLv3+). See the documentation of each package for licensing details.

#### 3.2.2.1 Mandatory dependencies

FeenoX has one mandatory dependency for run-time execution and the standard build toolchain for compilation. It is written in C99 so only a C compiler is needed, although make is also required. Free and open source compilers are favored. The usual C compiler is gcc but clang can also be used. Nevertheless, the non-free icc has also been tested.

Note that there is no need to have a Fortran nor a C++ compiler to build FeenoX. They might be needed to build other dependencies (such as PETSc and its dependencies), but not to compile FeenoX if all the dependencies are installed from the oeprating system’s package repositories. In case the build toolchain is not already installed, do so with

sudo apt-get install gcc make

If the source is to be fetched from the Git repository then not only is git needed but also autoconf and automake since the configure script is not stored in the Git repository but the autogen.sh script that bootstraps the tree and creates it. So if instead of compiling a source tarball one wants to clone from GitHub, these packages are also mandatory:

sudo apt-get install git automake autoconf

Again, chances are that any existing GNU/Linux box has all these tools already installed.

##### 3.2.2.1.1 The GNU Scientific Library

The only run-time dependency is GNU GSL (not to be confused with Microsoft GSL). It can be installed with

sudo apt-get install libgsl-dev

In case this package is not available or you do not have enough permissions to install system-wide packages, there are two options.

1. Pass the option --enable-download-gsl to the configure script below.

If the configure script cannot find both the headers and the actual library, it will refuse to proceed. Note that the FeenoX binaries already contain a static version of the GSL so it is not needed to have it installed in order to run the statically-linked binaries.

#### 3.2.2.2 Optional dependencies

FeenoX has three optional run-time dependencies. It can be compiled without any of these, but functionality will be reduced:

• SUNDIALS provides support for solving systems of ordinary differential equations (ODEs) or differential-algebraic equations (DAEs). This dependency is needed when running inputs with the PHASE_SPACE keyword.

• PETSc provides support for solving partial differential equations (PDEs). This dependency is needed when running inputs with the PROBLEM keyword.

• SLEPc provides support for solving eigen-value problems in partial differential equations (PDEs). This dependency is needed for inputs with PROBLEM types with eigen-value formulations such as modal and neutron_transport.

In absence of all these, FeenoX can still be used to

These optional dependencies have to be installed separately. There is no option to have configure to download them as with --enable-download-gsl. When running the test suite (sec. 3.2.2.6), those tests that need an optional dependency which was not found at compile time will be skipped.

##### 3.2.2.2.1 SUNDIALS

SUNDIALS is a SUite of Nonlinear and DIfferential/ALgebraic equation Solvers. It is used by FeenoX to solve dynamical systems casted as DAEs with the keyword PHASE_SPACE, like the Lorenz system.

Install either by doing

sudo apt-get install libsundials-dev

or by following the instructions in the documentation.

##### 3.2.2.2.2 PETSc

The Portable, Extensible Toolkit for Scientific Computation, pronounced PET-see (/ˈpɛt-siː/), is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations. It is used by FeenoX to solve PDEs with the keyword PROBLEM, like the NAFEMS LE10 benchmark problem.

Install either by doing

sudo apt-get install petsc-dev

or by following the instructions in the documentation.

Note that

• Configuring and compiling PETSc from scratch might be difficult the first time. It has a lot of dependencies and options. Read the official documentation for a detailed explanation.
• There is a huge difference in efficiency between using PETSc compiled with debugging symbols and with optimization flags. Make sure to configure --with-debugging=0 for FeenoX production runs and leave the debugging symbols (which is the default) for development and debugging only.
• FeenoX needs PETSc to be configured with real double-precision scalars. It will compile but will complain at run-time when using complex and/or single or quad-precision scalars.
• FeenoX honors the PETSC_DIR and PETSC_ARCH environment variables when executing configure. If these two do not exist or are empty, it will try to use the default system-wide locations (i.e. the petsc-dev package).
##### 3.2.2.2.3 SLEPc

The Scalable Library for Eigenvalue Problem Computations, is a software library for the solution of large scale sparse eigenvalue problems on parallel computers. It is used by FeenoX to solve PDEs with the keyword PROBLEM that need eigen-value computations, such as modal analysis of a cantilevered beam.

Install either by doing

sudo apt-get install slepc-dev

or by following the instructions in the documentation.

Note that

• SLEPc is an extension of PETSc so the latter has to be already installed and configured.
• FeenoX honors the SLEPC_DIR environment variable when executing configure. If it does not exist or is empty it will try to use the default system-wide locations (i.e. the slepc-dev package).
• If PETSc was configured with --download-slepc then the SLEPC_DIR variable has to be set to the directory inside PETSC_DIR where SLEPc was cloned and compiled.

#### 3.2.2.3 FeenoX source code

There are two ways of getting FeenoX’ source code:

1. Cloning the GitHub repository at https://github.com/seamplex/feenox
##### 3.2.2.3.1 Git repository

The main Git repository is hosted on GitHub at https://github.com/seamplex/feenox. It is public so it can be cloned either through HTTPS or SSH without needing any particular credentials. It can also be forked freely. See the Programming Guide for details about pull requests and/or write access to the main repository.

Ideally, the main branch should have a usable snapshot. All other branches can contain code that might not compile or might not run or might not be tested. If you find a commit in the main branch that does not pass the tests, please report it in the issue tracker ASAP.

After cloning the repository

git clone https://github.com/seamplex/feenox

the autogen.sh script has to be called to bootstrap the working tree, since the configure script is not stored in the repository but created from configure.ac (which is in the repository) by autogen.sh.

Similarly, after updating the working tree with

git pull

it is recommended to re-run the autogen.sh script. It will do a make clean and re-compute the version string.

##### 3.2.2.3.2 Source tarballs

When downloading a source tarball, there is no need to run autogen.sh since the configure script is already included in the tarball. This method cannot update the working tree. For each new FeenoX release, the whole source tarball has to be downloaded again.

#### 3.2.2.4 Configuration

To create a proper Makefile for the particular architecture, dependencies and compilation options, the script configure has to be executed. This procedure follows the GNU Coding Standards.

./configure

Without any particular options, configure will check if the mandatory GNU Scientific Library is available (both its headers and run-time library). If it is not, then the option --enable-download-gsl can be used. This option will try to use wget (which should be installed) to download a source tarball, uncompress, configure and compile it. If these steps are successful, this GSL will be statically linked into the resulting FeenoX executable. If there is no internet connection, the configure script will say that the download failed. In that case, get the indicated tarball file manually, copy it into the current directory and re-run ./configure.

The script will also check for the availability of optional dependencies. At the end of the execution, a summary of what was found (or not) is printed in the standard output:

$./configure [...] ## ----------------------- ## ## Summary of dependencies ## ## ----------------------- ## GNU Scientific Library from system SUNDIALS IDA yes PETSc yes /usr/lib/petsc SLEPc no [...]  If for some reason one of the optional dependencies is available but FeenoX should not use it, then pass --without-sundials, --without-petsc and/or --without-slepc as arguments. For example $ ./configure --without-sundials --without-petsc
[...]
## ----------------------- ##
## Summary of dependencies ##
## ----------------------- ##
GNU Scientific Library  from system
SUNDIALS                no
PETSc                   no
SLEPc                   no
[...]  

If configure complains about contradicting values from the cached ones, run autogen.sh again before configure and/or clone/uncompress the source tarball in a fresh location.

To see all the available options run

./configure --help

#### 3.2.2.5 Source code compilation

After the successful execution of configure, a Makefile is created. To compile FeenoX, just execute

make

Compilation should take a dozen of seconds. It can be even sped up by using the -j option

make -j8

The binary executable will be located in the src directory but a copy will be made in the base directory as well. Test it by running without any arguments

$./feenox FeenoX v0.2.14-gbbf48c9-dirty a free no-fee no-X uniX-like finite-element(ish) computational engineering tool usage: feenox [options] inputfile [replacement arguments] [petsc options] -h, --help display options and detailed explanations of commmand-line usage -v, --version display brief version information and exit -V, --versions display detailed version information Run with --help for further explanations.$

The -v (or --version) option shows the version and a copyright notice:1

$./feenox -v FeenoX v0.2.14-gbbf48c9-dirty a free no-fee no-X uniX-like finite-element(ish) computational engineering tool Copyright © 2009--2022 Seamplex, https://seamplex.com/feenox GNU General Public License v3+, https://www.gnu.org/licenses/gpl.html. FeenoX is free software: you are free to change and redistribute it. There is NO WARRANTY, to the extent permitted by law.$

The -V (or --versions) option shows the dates of the last commits, the compiler options and the versions of the linked libraries:

$./feenox -V FeenoX v0.1.24-g6cfe063 a free no-fee no-X uniX-like finite-element(ish) computational engineering tool Last commit date : Sun Aug 29 11:34:04 2021 -0300 Build date : Sun Aug 29 11:44:50 2021 -0300 Build architecture : linux-gnu x86_64 Compiler version : gcc (Debian 10.2.1-6) 10.2.1 20210110 Compiler expansion : gcc -Wl,-z,relro -I/usr/include/x86_64-linux-gnu/mpich -L/usr/lib/x86_64-linux-gnu -lmpich Compiler flags : -O3 Builder : gtheler@chalmers GSL version : 2.6 SUNDIALS version : 4.1.0 PETSc version : Petsc Release Version 3.14.5, Mar 03, 2021 PETSc arch : PETSc options : --build=x86_64-linux-gnu --prefix=/usr --includedir=${prefix}/include --mandir=${prefix}/share/man --infodir=${prefix}/share/info --sysconfdir=/etc --localstatedir=/var --with-option-checking=0 --with-silent-rules=0 --libdir=${prefix}/lib/x86_64-linux-gnu --runstatedir=/run --with-maintainer-mode=0 --with-dependency-tracking=0 --with-debugging=0 --shared-library-extension=_real --with-shared-libraries --with-pic=1 --with-cc=mpicc --with-cxx=mpicxx --with-fc=mpif90 --with-cxx-dialect=C++11 --with-opencl=1 --with-blas-lib=-lblas --with-lapack-lib=-llapack --with-scalapack=1 --with-scalapack-lib=-lscalapack-openmpi --with-ptscotch=1 --with-ptscotch-include=/usr/include/scotch --with-ptscotch-lib="-lptesmumps -lptscotch -lptscotcherr" --with-fftw=1 --with-fftw-include="[]" --with-fftw-lib="-lfftw3 -lfftw3_mpi" --with-superlu_dist=1 --with-superlu_dist-include=/usr/include/superlu-dist --with-superlu_dist-lib=-lsuperlu_dist --with-hdf5-include=/usr/include/hdf5/openmpi --with-hdf5-lib="-L/usr/lib/x86_64-linux-gnu/hdf5/openmpi -L/usr/lib/x86_64-linux-gnu/openmpi/lib -lhdf5 -lmpi" --CXX_LINKER_FLAGS=-Wl,--no-as-needed --with-hypre=1 --with-hypre-include=/usr/include/hypre --with-hypre-lib=-lHYPRE_core --with-mumps=1 --with-mumps-include="[]" --with-mumps-lib="-ldmumps -lzmumps -lsmumps -lcmumps -lmumps_common -lpord" --with-suitesparse=1 --with-suitesparse-include=/usr/include/suitesparse --with-suitesparse-lib="-lumfpack -lamd -lcholmod -lklu" --with-superlu=1 --with-superlu-include=/usr/include/superlu --with-superlu-lib=-lsuperlu --prefix=/usr/lib/petscdir/petsc3.14/x86_64-linux-gnu-real --PETSC_ARCH=x86_64-linux-gnu-real CFLAGS="-g -O2 -ffile-prefix-map=/build/petsc-pVufYp/petsc-3.14.5+dfsg1=. -flto=auto -ffat-lto-objects -fstack-protector-strong -Wformat -Werror=format-security -fPIC" CXXFLAGS="-g -O2 -ffile-prefix-map=/build/petsc-pVufYp/petsc-3.14.5+dfsg1=. -flto=auto -ffat-lto-objects -fstack-protector-strong -Wformat -Werror=format-security -fPIC" FCFLAGS="-g -O2 -ffile-prefix-map=/build/petsc-pVufYp/petsc-3.14.5+dfsg1=. -flto=auto -ffat-lto-objects -fstack-protector-strong -fPIC -ffree-line-length-0" FFLAGS="-g -O2 -ffile-prefix-map=/build/petsc-pVufYp/petsc-3.14.5+dfsg1=. -flto=auto -ffat-lto-objects -fstack-protector-strong -fPIC -ffree-line-length-0" CPPFLAGS="-Wdate-time -D_FORTIFY_SOURCE=2" LDFLAGS="-Wl,-Bsymbolic-functions -flto=auto -Wl,-z,relro -fPIC" MAKEFLAGS=w SLEPc version : SLEPc Release Version 3.14.2, Feb 01, 2021$

#### 3.2.2.6 Test suite

The test directory contains a set of test cases whose output is known so that unintended regressions can be detected quickly (see the programming guide for more information). The test suite ought to be run after each modification in FeenoX’ source code. It consists of a set of scripts and input files needed to solve dozens of cases. The output of each execution is compared to a reference solution. In case the output does not match the reference, the test suite fails.

After compiling FeenoX as explained in sec. 3.2.2.5, the test suite can be run with make check. Ideally everything should be green meaning the tests passed:

$make check Making check in src make[1]: Entering directory '/home/gtheler/codigos/feenox/src' make[1]: Nothing to be done for 'check'. make[1]: Leaving directory '/home/gtheler/codigos/feenox/src' make[1]: Entering directory '/home/gtheler/codigos/feenox' cp -r src/feenox . make check-TESTS make[2]: Entering directory '/home/gtheler/codigos/feenox' make[3]: Entering directory '/home/gtheler/codigos/feenox' XFAIL: tests/abort.sh PASS: tests/algebraic_expr.sh PASS: tests/beam-modal.sh PASS: tests/beam-ortho.sh PASS: tests/builtin.sh PASS: tests/cylinder-traction-force.sh PASS: tests/default_argument_value.sh PASS: tests/expressions_constants.sh PASS: tests/expressions_variables.sh PASS: tests/expressions_functions.sh PASS: tests/exp.sh PASS: tests/i-beam-euler-bernoulli.sh PASS: tests/iaea-pwr.sh PASS: tests/iterative.sh PASS: tests/fit.sh PASS: tests/function_algebraic.sh PASS: tests/function_data.sh PASS: tests/function_file.sh PASS: tests/function_vectors.sh PASS: tests/integral.sh PASS: tests/laplace2d.sh PASS: tests/materials.sh PASS: tests/mesh.sh PASS: tests/moment-of-inertia.sh PASS: tests/nafems-le1.sh PASS: tests/nafems-le10.sh PASS: tests/nafems-le11.sh PASS: tests/nafems-t1-4.sh PASS: tests/nafems-t2-3.sh PASS: tests/neutron_diffusion_src.sh PASS: tests/neutron_diffusion_keff.sh PASS: tests/parallelepiped.sh PASS: tests/point-kinetics.sh PASS: tests/print.sh PASS: tests/thermal-1d.sh PASS: tests/thermal-2d.sh PASS: tests/trig.sh PASS: tests/two-cubes-isotropic.sh PASS: tests/two-cubes-orthotropic.sh PASS: tests/vector.sh XFAIL: tests/xfail-few-properties-ortho-young.sh XFAIL: tests/xfail-few-properties-ortho-poisson.sh XFAIL: tests/xfail-few-properties-ortho-shear.sh ============================================================================ Testsuite summary for feenox v0.2.6-g3237ce9 ============================================================================ # TOTAL: 43 # PASS: 39 # SKIP: 0 # XFAIL: 4 # FAIL: 0 # XPASS: 0 # ERROR: 0 ============================================================================ make[3]: Leaving directory '/home/gtheler/codigos/feenox' make[2]: Leaving directory '/home/gtheler/codigos/feenox' make[1]: Leaving directory '/home/gtheler/codigos/feenox'$

The XFAIL result means that those cases are expected to fail (they are there to test if FeenoX can handle errors). Failure would mean they passed. In case FeenoX was not compiled with any optional dependency, the corresponding tests will be skipped. Skipped tests do not mean any failure, but that the compiled FeenoX executable does not have the full capabilities. For example, when configuring with ./configure --without-petsc (but with SUNDIALS), the test suite output should be a mixture of green and blue:

$./configure --without-petsc [...] configure: creating ./src/version.h ## ----------------------- ## ## Summary of dependencies ## ## ----------------------- ## GNU Scientific Library from system SUNDIALS yes PETSc no SLEPc no Compiler gcc checking that generated files are newer than configure... done configure: creating ./config.status config.status: creating Makefile config.status: creating src/Makefile config.status: creating doc/Makefile config.status: executing depfiles commands$ make
[...]
$make check Making check in src make[1]: Entering directory '/home/gtheler/codigos/feenox/src' make[1]: Nothing to be done for 'check'. make[1]: Leaving directory '/home/gtheler/codigos/feenox/src' make[1]: Entering directory '/home/gtheler/codigos/feenox' cp -r src/feenox . make check-TESTS make[2]: Entering directory '/home/gtheler/codigos/feenox' make[3]: Entering directory '/home/gtheler/codigos/feenox' XFAIL: tests/abort.sh PASS: tests/algebraic_expr.sh SKIP: tests/beam-modal.sh SKIP: tests/beam-ortho.sh PASS: tests/builtin.sh SKIP: tests/cylinder-traction-force.sh PASS: tests/default_argument_value.sh PASS: tests/expressions_constants.sh PASS: tests/expressions_variables.sh PASS: tests/expressions_functions.sh PASS: tests/exp.sh SKIP: tests/i-beam-euler-bernoulli.sh SKIP: tests/iaea-pwr.sh PASS: tests/iterative.sh PASS: tests/fit.sh PASS: tests/function_algebraic.sh PASS: tests/function_data.sh PASS: tests/function_file.sh PASS: tests/function_vectors.sh PASS: tests/integral.sh SKIP: tests/laplace2d.sh PASS: tests/materials.sh PASS: tests/mesh.sh PASS: tests/moment-of-inertia.sh SKIP: tests/nafems-le1.sh SKIP: tests/nafems-le10.sh SKIP: tests/nafems-le11.sh SKIP: tests/nafems-t1-4.sh SKIP: tests/nafems-t2-3.sh SKIP: tests/neutron_diffusion_src.sh SKIP: tests/neutron_diffusion_keff.sh SKIP: tests/parallelepiped.sh PASS: tests/point-kinetics.sh PASS: tests/print.sh SKIP: tests/thermal-1d.sh SKIP: tests/thermal-2d.sh PASS: tests/trig.sh SKIP: tests/two-cubes-isotropic.sh SKIP: tests/two-cubes-orthotropic.sh PASS: tests/vector.sh SKIP: tests/xfail-few-properties-ortho-young.sh SKIP: tests/xfail-few-properties-ortho-poisson.sh SKIP: tests/xfail-few-properties-ortho-shear.sh ============================================================================ Testsuite summary for feenox v0.2.6-g3237ce9 ============================================================================ # TOTAL: 43 # PASS: 21 # SKIP: 21 # XFAIL: 1 # FAIL: 0 # XPASS: 0 # ERROR: 0 ============================================================================ make[3]: Leaving directory '/home/gtheler/codigos/feenox' make[2]: Leaving directory '/home/gtheler/codigos/feenox' make[1]: Leaving directory '/home/gtheler/codigos/feenox'$

To illustrate how regressions can be detected, let us add a bug deliberately and re-run the test suite.

Edit the source file that contains the shape functions of the second-order tetrahedra src/mesh/tet10.c, find the function feenox_mesh_tet10_h() and randomly change a sign, i.e. replace

      return t*(2*t-1);

with

      return t*(2*t+1);

Save, recompile, and re-run the test suite to obtain some red:

$git diff src/mesh/ diff --git a/src/mesh/tet10.c b/src/mesh/tet10.c index 72bc838..293c290 100644 --- a/src/mesh/tet10.c +++ b/src/mesh/tet10.c @@ -227,7 +227,7 @@ double feenox_mesh_tet10_h(int j, double *vec_r) { return s*(2*s-1); break; case 3: - return t*(2*t-1); + return t*(2*t+1); break; case 4:$ make
[...]
$make check Making check in src make[1]: Entering directory '/home/gtheler/codigos/feenox/src' make[1]: Nothing to be done for 'check'. make[1]: Leaving directory '/home/gtheler/codigos/feenox/src' make[1]: Entering directory '/home/gtheler/codigos/feenox' cp -r src/feenox . make check-TESTS make[2]: Entering directory '/home/gtheler/codigos/feenox' make[3]: Entering directory '/home/gtheler/codigos/feenox' XFAIL: tests/abort.sh PASS: tests/algebraic_expr.sh FAIL: tests/beam-modal.sh PASS: tests/beam-ortho.sh PASS: tests/builtin.sh PASS: tests/cylinder-traction-force.sh PASS: tests/default_argument_value.sh PASS: tests/expressions_constants.sh PASS: tests/expressions_variables.sh PASS: tests/expressions_functions.sh PASS: tests/exp.sh PASS: tests/i-beam-euler-bernoulli.sh PASS: tests/iaea-pwr.sh PASS: tests/iterative.sh PASS: tests/fit.sh PASS: tests/function_algebraic.sh PASS: tests/function_data.sh PASS: tests/function_file.sh PASS: tests/function_vectors.sh PASS: tests/integral.sh PASS: tests/laplace2d.sh PASS: tests/materials.sh PASS: tests/mesh.sh PASS: tests/moment-of-inertia.sh PASS: tests/nafems-le1.sh FAIL: tests/nafems-le10.sh FAIL: tests/nafems-le11.sh PASS: tests/nafems-t1-4.sh PASS: tests/nafems-t2-3.sh PASS: tests/neutron_diffusion_src.sh PASS: tests/neutron_diffusion_keff.sh FAIL: tests/parallelepiped.sh PASS: tests/point-kinetics.sh PASS: tests/print.sh PASS: tests/thermal-1d.sh PASS: tests/thermal-2d.sh PASS: tests/trig.sh PASS: tests/two-cubes-isotropic.sh PASS: tests/two-cubes-orthotropic.sh PASS: tests/vector.sh XFAIL: tests/xfail-few-properties-ortho-young.sh XFAIL: tests/xfail-few-properties-ortho-poisson.sh XFAIL: tests/xfail-few-properties-ortho-shear.sh ============================================================================ Testsuite summary for feenox v0.2.6-g3237ce9 ============================================================================ # TOTAL: 43 # PASS: 35 # SKIP: 0 # XFAIL: 4 # FAIL: 4 # XPASS: 0 # ERROR: 0 ============================================================================ See ./test-suite.log Please report to jeremy@seamplex.com ============================================================================ make[3]: *** [Makefile:1152: test-suite.log] Error 1 make[3]: Leaving directory '/home/gtheler/codigos/feenox' make[2]: *** [Makefile:1260: check-TESTS] Error 2 make[2]: Leaving directory '/home/gtheler/codigos/feenox' make[1]: *** [Makefile:1791: check-am] Error 2 make[1]: Leaving directory '/home/gtheler/codigos/feenox' make: *** [Makefile:1037: check-recursive] Error 1$

#### 3.2.2.7 Installation

To be able to execute FeenoX from any directory, the binary has to be copied to a directory available in the PATH environment variable. If you have root access, the easiest and cleanest way of doing this is by calling make install with sudo or su:

$sudo make install Making install in src make[1]: Entering directory '/home/gtheler/codigos/feenox/src' gmake[2]: Entering directory '/home/gtheler/codigos/feenox/src' /usr/bin/mkdir -p '/usr/local/bin' /usr/bin/install -c feenox '/usr/local/bin' gmake[2]: Nothing to be done for 'install-data-am'. gmake[2]: Leaving directory '/home/gtheler/codigos/feenox/src' make[1]: Leaving directory '/home/gtheler/codigos/feenox/src' make[1]: Entering directory '/home/gtheler/codigos/feenox' cp -r src/feenox . make[2]: Entering directory '/home/gtheler/codigos/feenox' make[2]: Nothing to be done for 'install-exec-am'. make[2]: Nothing to be done for 'install-data-am'. make[2]: Leaving directory '/home/gtheler/codigos/feenox' make[1]: Leaving directory '/home/gtheler/codigos/feenox'$

If you do not have root access or do not want to populate /usr/local/bin, you can either

• Configure with a different prefix (not covered here), or

• Copy (or symlink) the feenox executable to $HOME/bin: mkdir -p${HOME}/bin
cp feenox ${HOME}/bin If you plan to regularly update FeenoX (which you should), you might want to symlink instead of copy so you do not need to update the binary in $HOME/bin each time you recompile:

mkdir -p ${HOME}/bin ln -sf feenox${HOME}/bin

Check that FeenoX is now available from any directory (note the command is feenox and not ./feenox):

$cd$ feenox -v
FeenoX v0.2.14-gbbf48c9-dirty
a free no-fee no-X uniX-like finite-element(ish) computational engineering tool

FeenoX is free software: you are free to change and redistribute it.
There is NO WARRANTY, to the extent permitted by law.
$ If it is not and you went through the $HOME/bin path, make sure it is in the PATH (pun). Add

export PATH=${PATH}:${HOME}/bin

to your .bashrc in your home directory and re-login.

#### 3.2.3.1 Compiling with debug symbols

By default the C flags are -O3, without debugging. To add the -g flag, just use CFLAGS when configuring:

./configure CFLAGS="-g -O0"

#### 3.2.3.2 Using a different compiler

Without PETSc, FeenoX uses the CC environment variable to set the compiler. So configure like

./configure CC=clang

When PETSc is detected FeenoX uses the mpicc executable, which is a wrapper to an actual C compiler with extra flags needed to find the headers and the MPI library. To change the wrapped compiler, you should set MPICH_CC or OMPI_CC, depending if you are using MPICH or OpenMPI. For example, to force MPICH to use clang do

./configure MPICH_CC=clang CC=clang

To know which is the default MPI implementation, just run ./configure without arguments and pay attention to the “Compiler” line in the “Summary of dependencies” section. For example, for OpenMPI a typical summary would be

## ----------------------- ##
## Summary of dependencies ##
## ----------------------- ##
GNU Scientific Library  from system
SUNDIALS                yes
PETSc                   yes /usr/lib/petsc
SLEPc                   yes /usr/lib/slepc
Compiler                gcc -I/usr/lib/x86_64-linux-gnu/openmpi/include/openmpi -I/usr/lib/x86_64-linux-gnu/openmpi/include -pthread -L/usr/lib/x86_64-linux-gnu/openmpi/lib -lmpi

For MPICH:

## ----------------------- ##
## Summary of dependencies ##
## ----------------------- ##
GNU Scientific Library  from system
SUNDIALS                yes
PETSc                   yes /home/gtheler/libs/petsc-3.15.0 arch-linux2-c-debug
SLEPc                   yes /home/gtheler/libs/slepc-3.15.1
Compiler                gcc -Wl,-z,relro -I/usr/include/x86_64-linux-gnu/mpich -L/usr/lib/x86_64-linux-gnu -lmpich

Other non-free implementations like Intel MPI might work but were not tested. However, it should be noted that the MPI implementation used to compile FeenoX has to match the one used to compile PETSc. Therefore, if you compiled PETSc on your own, it is up to you to ensure MPI compatibility. If you are using PETSc as provided by your distribution’s repositories, you will have to find out which one was used (it is usually OpenMPI) and use the same one when compiling FeenoX.

The FeenoX executable will show the configured compiler and flags when invoked with the --versions option:

$feenox --versions FeenoX v0.2.14-gbbf48c9-dirty a free no-fee no-X uniX-like finite-element(ish) computational engineering tool Last commit date : Sat Feb 12 15:35:05 2022 -0300 Build date : Sat Feb 12 15:35:44 2022 -0300 Build architecture : linux-gnu x86_64 Compiler version : gcc (Debian 10.2.1-6) 10.2.1 20210110 Compiler expansion : gcc -Wl,-z,relro -I/usr/include/x86_64-linux-gnu/mpich -L/usr/lib/x86_64-linux-gnu -lmpich Compiler flags : -O3 Builder : gtheler@tom GSL version : 2.6 SUNDIALS version : 5.7.0 PETSc version : Petsc Release Version 3.16.3, Jan 05, 2022 PETSc arch : arch-linux-c-debug PETSc options : --download-eigen --download-hdf5 --download-hypre --download-metis --download-mumps --download-parmetis --download-pragmatic --download-scalapack SLEPc version : SLEPc Release Version 3.16.1, Nov 17, 2021$

Note that the reported values are the ones used in configure and not in make. Thus, the recommended way to set flags is in configure and not in make.

#### 3.2.3.3 Compiling PETSc

Particular explanation for FeenoX is to be done. For now, follow the general explanation from PETSc’s website.

export PETSC_DIR=$PWD export PETSC_ARCH=arch-linux-c-opt ./configure --with-debugging=0 --download-mumps --download-scalapack --with-cxx=0 --COPTFLAGS=-O3 --FOPTFLAGS=-O3  export PETSC_DIR=$PWD
./configure --with-debugging=0 --with-openmp=0 --with-x=0 --with-cxx=0 --COPTFLAGS=-O3 --FOPTFLAGS=-O3
make PETSC_DIR=/home/ubuntu/reflex-deps/petsc-3.17.2 PETSC_ARCH=arch-linux-c-opt all

# 4 Examples

## 4.1 Hello World (and Universe)!

PRINT "Hello $1!" $ feenox hello.fee World
Hello World!
$feenox hello.fee Universe Hello Universe!$

## 4.2 Lorenz’ attractor—the one with the butterfly

Solve

\begin{cases} \dot{x} &= \sigma \cdot (y - x) \\ \dot{y} &= x \cdot (r - z) - y \\ \dot{z} &= x y - b z \\ \end{cases}

for 0 < t < 40 with initial conditions

\begin{cases} x(0) = -11 \\ y(0) = -16 \\ z(0) = 22.5 \\ \end{cases}

and \sigma=10, r=28 and b=8/3, which are the classical parameters that generate the butterfly as presented by Edward Lorenz back in his seminal 1963 paper Deterministic non-periodic flow. This example’s input file ressembles the parameters, inital conditions and differential equations of the problem as naturally as possible with an ASCII file.

PHASE_SPACE x y z     # Lorenz attractor’s phase space is x-y-z
end_time = 40         # we go from t=0 to 40 non-dimensional units

sigma = 10            # the original parameters from the 1963 paper
r = 28
b = 8/3

x_0 = -11             # initial conditions
y_0 = -16
z_0 = 22.5

# the dynamical system's equations written as naturally as possible
x_dot = sigma*(y - x)
y_dot = x*(r - z) - y
z_dot = x*y - b*z

PRINT t x y z        # four-column plain-ASCII output
$feenox lorenz.fee > lorenz.dat$ gnuplot lorenz.gp
$python3 lorenz.py$ sh lorenz2x3d.sh < lorenz.dat > lorenz.html

Figure 5: The Lorenz attractor computed with FeenoX plotted with two different tools. a — Gnuplot, b — Matplotlib

## 4.3 The logistic map

Plot the asymptotic behavior of the logistic map

x_{n+1} = r \cdot x \cdot (1-x)

for a range of the parameter r.

DEFAULT_ARGUMENT_VALUE 1 2.6   # by default show r in [2.6:4]
DEFAULT_ARGUMENT_VALUE 2 4

steps_per_r = 2^10
steps_asymptotic = 2^8
steps_for_r = 2^10

static_steps = steps_for_r*steps_per_r

# change r every steps_per_r steps
IF mod(step_static,steps_per_r)=1
r = quasi_random($1,$2)
ENDIF

x_init = 0.5           # start at x = 0.5
x = r*x*(1-x)          # apply the map

IF step_static-steps_per_r*floor(step_static/steps_per_r)>(steps_per_r-steps_asymptotic)
# write the asymptotic behavior only
PRINT r x
ENDIF
$gnuplot gnuplot> plot "< feenox logistic.fee" w p pt 50 ps 0.02 gnuplot> quit$

## 4.4 Thermal slabs

### 4.4.1 One-dimensional linear

Solve heat conduction on the slab x \in [0:1] with boundary conditions

\begin{cases} T(0) = 0 & \text{(left)} \\ T(1) = 1 & \text{(right)} \\ \end{cases}

and uniform conductivity. Compute T\left(\frac{1}{2}\right).

• The input written in a self-evident English-like dialect
• Syntactic sugared plain-text ASCII file
• Simple problems (like this one) need simple inputs
• FeenoX follows the UNIX rule of simplicity
• Output is 100% user-defined
• No PRINT no output
• Feenox follows the UNIX rule of silence
• There is no node at x=1/2=0.5!
• FeenoX knows how to interpolate
• Mesh separated from problem
• The geometry comes from a Git-friendly .geo
Point(1) = {0, 0, 0};          // geometry:
Point(2) = {1, 0, 0};          // two points
Line(1) = {1, 2};              // and a line connecting them!

Physical Point("left") = {1};  // groups for BCs and materials
Physical Point("right") = {2};
Physical Line("bulk") = {1};   // needed due to how Gmsh works

Mesh.MeshSizeMax = 1/3;        // mesh size, three line elements
Mesh.MeshSizeMin = Mesh.MeshSizeMax;
• UNIX rule of composition
• The actual input file is a Git-friendly .fee
PROBLEM thermal 1D    # tell FeenoX what we want to solve
k = 1                 # set uniform conductivity
BC left  T=0          # set fixed temperatures as BCs
BC right T=1          # "left" and "right" are defined in the mesh
SOLVE_PROBLEM         # we are ready to solve the problem
PRINT T(1/2)          # ask for the temperature at x=1/2
$gmsh -1 slab.geo [...] Info : 4 nodes 5 elements Info : Writing 'slab.msh'... [...]$ feenox thermal-1d-dirichlet-uniform-k.fee
0.5
$ ## 4.5 NAFEMS LE10 “Thick plate pressure” benchmark Assuming the CAD has already been created in STEP format (for instance using Gmsh with this geo file), create a tetrahedral locally-refined unstructured grid with Gmsh using the following .geo file: // NAFEMS LE10 benchmark unstructured locally-refined tetrahedral mesh Merge "nafems-le10.step"; // load the CAD // define physical names from the geometrical entity ids Physical Surface("upper") = {7}; Physical Surface("DCD'C'") = {1}; Physical Surface("ABA'B'") = {3}; Physical Surface("BCB'C'") = {4, 5}; Physical Curve("midplane") = {14}; Physical Volume("bulk") = {1}; // meshing settings, read Gmsh' manual for further reference Mesh.ElementOrder = 2; // use second-order tetrahedra Mesh.Algorithm = 6; // 2D mesh algorithm: 6: Frontal Delaunay Mesh.Algorithm3D = 10; // 3D mesh algorithm: 10: HXT Mesh.Optimize = 1; // Optimize the mesh Mesh.HighOrderOptimize = 1; // Optimize high-order meshes? 2: elastic+optimization Mesh.MeshSizeMax = 80; // main element size Mesh.MeshSizeMin = 20; // refined element size // local refinement around the point D (entity 4) Field[1] = Distance; Field[1].NodesList = {4}; Field[2] = Threshold; Field[2].IField = 1; Field[2].LcMin = Mesh.MeshSizeMin; Field[2].LcMax = Mesh.MeshSizeMax; Field[2].DistMin = 2 * Mesh.MeshSizeMax; Field[2].DistMax = 6 * Mesh.MeshSizeMax; Background Field = {2}; and then use this pretty-straightforward input file that has a one-to-one correspondence with the original problem formulation from 1990: # NAFEMS Benchmark LE-10: thick plate pressure PROBLEM mechanical 3D READ_MESH nafems-le10.msh # mesh in millimeters # LOADING: uniform normal pressure on the upper surface BC upper p=1 # 1 Mpa # BOUNDARY CONDITIONS: BC DCD'C' v=0 # Face DCD'C' zero y-displacement BC ABA'B' u=0 # Face ABA'B' zero x-displacement BC BCB'C' u=0 v=0 # Face BCB'C' x and y displ. fixed BC midplane w=0 # z displacements fixed along mid-plane # MATERIAL PROPERTIES: isotropic single-material properties E = 210e3 # Young modulus in MPa nu = 0.3 # Poisson's ratio SOLVE_PROBLEM # solve! # print the direct stress y at D (and nothing more) PRINT "sigma_y @ D = " sigmay(2000,0,300) "MPa" # write post-processing data for paraview WRITE_MESH nafems-le10.vtk sigmay VECTOR u v w $ gmsh -3 nafems-le10.geo
[...]
$feenox nafems-le10.fee sigma_y @ D = -5.38016 MPa$

## 4.6 NAFEMS LE11 “Solid Cylinder/Taper/Sphere-Temperature” benchmark

Figure 6: The NAFEMS LE11 problem formulation. a — Problem statement, b — Structured hex mesh

Following the spirit from LE10, note how easy it is to give a space-dependent temperature field in FeenoX. Just write \sqrt{x^2+y^2}+z like sqrt(x^2 + y^2) + z!

# NAFEMS Benchmark LE-11: solid cylinder/taper/sphere-temperature
PROBLEM mechanical 3D

# as an algebraic expression as human-friendly as it can be
T(x,y,z) := sqrt(x^2 + y^2) + z

BC xz     v=0       # displacement vector is [u,v,w]
BC yz     u=0       # u = displacement in x
BC xy     w=0       # v = displacement in y
BC HIH'I' w=0       # w = displacement in z

E = 210e3*1e6       # mesh is in meters, so E=210e3 MPa -> Pa
nu = 0.3            # dimensionless
alpha = 2.3e-4      # in 1/ºC as in the problem
SOLVE_PROBLEM

# for post-processing in Paraview
WRITE_MESH nafems-le11.vtk VECTOR u v w   T sigmax sigmay sigmaz

PRINT "sigma_z(A) =" %.2f sigmaz(1,0,0)/1e6 "MPa" SEP " "
PRINT "wall time  =" %.2f wall_time() "seconds"  SEP " "
$gmsh -3 nafems-le11.geo [...]$ feenox nafems-le11.fee
sigma_z(A) = -105.04 MPa
wall time  = wall time  = 1.91 seconds
$ Figure 7: The NAFEMS LE11 problem results. a — Problem statement, b — Structured hex mesh ## 4.7 NAFEMS LE1 “Elliptical membrane” plane-stress benchmark Tell FenooX the problem is plane_stress. Use the nafems-le1.geo file provided to create the mesh. Read it with READ_MESH, set material properties, BCs and SOLVE_PROBLEM! PROBLEM mechanical plane_stress READ_MESH nafems-le1.msh E = 210e3 nu = 0.3 BC AB u=0 BC CD v=0 BC BC tension=10 SOLVE_PROBLEM WRITE_MESH nafems-le1.vtk VECTOR u v 0 sigmax sigmay tauxy PRINT "σy at point D = " %.4f sigmay(2000,0) "(reference is 92.7)" SEP " " $ gmsh -2 nafems-le11.geo
[...]
$feenox nafems-le1.fee σy at point D = 92.7011 (reference is 92.7)$

## 4.8 How to solve a maze without AI

Other people’s maze-related posts:

Say you are Homer Simpson and you want to solve a maze drawn in a restaurant’s placemat, one where both the start and end are known beforehand. In order to avoid falling into the alligator’s mouth, you can exploit the ellipticity of the Laplacian operator to solve any maze (even a hand-drawn one) without needing any fancy AI or ML algorithm. Just FeenoX and a bunch of standard open source tools to convert a bitmapped picture of the maze into an unstructured mesh.

PROBLEM laplace 2D  # pretty self-descriptive, isn't it?

# boundary conditions (default is homogeneous Neumann)
BC start  phi=0
BC end    phi=1

SOLVE_PROBLEM

# write the norm of gradient as a scalar field
# and the gradient as a 2d vector into a .msh file
WRITE_MESH maze-solved.msh \
sqrt(dphidx(x,y)^2+dphidy(x,y)^2) \
VECTOR dphidx dphidy 0 
$gmsh -2 maze.geo [...]$ feenox maze.fee
$ ### 4.8.1 Transient top-down Instead of solving a steady-state en exploiting the ellipticity of Laplace’s operator, let us see what happens if we solve a transient instead. PROBLEM laplace 2D READ_MESH maze.msh phi_0(x,y) = 0 # inital condition end_time = 100 # some end time where we know we reached the steady-state alpha = 1e-6 # factor of the time derivative to make it advance faster BC start phi=if(t<1,t,1) # a ramp from zero to avoid discontinuities with the initial condition BC end phi=0 # homogeneous BC at the end (so we move from top to bottom) SOLVE_PROBLEM PRINT t WRITE_MESH maze-tran-td.msh phi sqrt(dphidx(x,y)^2+dphidy(x,y)^2) VECTOR -dphidx(x,y) -dphidy(x,y) 0  $ feenox maze-tran-td.fee
0
0.00433078
0.00949491
0.0170774
0.0268599
[...]
55.8631
64.0819
74.5784
87.2892
100
$gmsh maze-tran-td-anim.geo # all frames dumped, now run ffmpeg -y -framerate 20 -f image2 -i maze-tran-td-%03d.png maze-tran-td.mp4 ffmpeg -y -framerate 20 -f image2 -i maze-tran-td-%03d.png maze-tran-td.gif$ ffmpeg -y -framerate 20 -f image2 -i maze-tran-td-%03d.png maze-tran-td.mp4
[...]
$ffmpeg -y -framerate 20 -f image2 -i maze-tran-td-%03d.png maze-tran-td.gif [...]  ### 4.8.2 Transient bottom-up Now let us see what happens if we travel the maze from the exit up to the inlet. It looks like the solver tries a few different paths that lead nowhere until the actual solution is found. PROBLEM laplace 2D READ_MESH maze.msh phi_0(x,y) = 0 end_time = 100 alpha = 1e-6 BC end phi=if(t<1,t,1) BC start phi=0 SOLVE_PROBLEM PRINT t WRITE_MESH maze-tran-bu.msh phi sqrt(dphidx(x,y)^2+dphidy(x,y)^2) VECTOR -dphidx(x,y) -dphidy(x,y) 0  $ feenox maze-tran-bu.fee
0
0.00402961
0.00954806
0.0180156
0.0285787
[...]
65.3715
72.6894
81.8234
90.9117
100
$gmsh maze-tran-bu-anim.geo # all frames dumped, now run ffmpeg -y -framerate 20 -f image2 -i maze-tran-bu-%03d.png maze-tran-bu.mp4 ffmpeg -y -framerate 20 -f image2 -i maze-tran-bu-%03d.png maze-tran-bu.gif$ ffmpeg -y -framerate 20 -f image2 -i maze-tran-bu-%03d.png maze-tran-bu.mp4
[...]
1	1
2	1
3	2
4	3
5	5
6	8
7	13
8	21
9	34
10	55
11	89
12	144
13	233
14	377
15	610
16	987
17	1597
18	2584
19	4181
20	6765
$ ### 4.9.3 Solving an iterative problem Finally, we print the sequence as an iterative problem and check that the three outputs are the same. static_steps = 20 #static_iterations = 1476 # limit of doubles IF step_static=1|step_static=2 f_n = 1 f_nminus1 = 1 f_nminus2 = 1 ELSE f_n = f_nminus1 + f_nminus2 f_nminus2 = f_nminus1 f_nminus1 = f_n ENDIF PRINT step_static f_n $ feenox fibo_iterative.fee > three
$diff one two$ diff two three
$ ## 4.10 Computing the derivative of a function as a UNIX filter This example illustrates how well FeenoX integrates into the UNIX philosophy. Let’s say one has a function f(t) as an ASCII file with two columns and one wants to compute the derivative f'(t). Just pipe the function file into this example’s input file derivative.fee used as a filter. For example, this small input file f.fee writes the function of time provided in the first command-line argument from zero up to the second command-line argument: end_time =$2
PRINT t $1 $ feenox f.fee "sin(t)" 1
0       0
0.0625  0.0624593
0.125   0.124675
0.1875  0.186403
0.25    0.247404
0.3125  0.307439
0.375   0.366273
0.4375  0.423676
0.5     0.479426
0.5625  0.533303
0.625   0.585097
0.6875  0.634607
0.75    0.681639
0.8125  0.726009
0.875   0.767544
0.9375  0.806081
1       0.841471
$ Then we can pipe the output of this command to the derivative filter. Note that • The derivative.fee has the execution flag has on and a shebang line pointing to a global location of the FeenoX binary in /usr/local/bin e.g. after doing sudo make install. • The first argument of derivative.fee controls the time step. This is only important to control the number of output lines. It does not have anything to do with precision, since the derivative is computed using an adaptive centered numerical differentiation scheme using the GNU Scientific Library. • Before doing the actual differentiation, the input data is interpolated using a third-order monotonous scheme (also with GSL). • TL;DR: this is not just “current value minus last value divided time increment.” #!/usr/local/bin/feenox # read the function from stdin FUNCTION f(t) FILE - INTERPOLATION steffen # detect the domain range a = vecmin(vec_f_t) b = vecmax(vec_f_t) # time step from arguments (or default 10 steps) DEFAULT_ARGUMENT_VALUE 1 (b-a)/10 h =$1

# compute the derivative with a wrapper for gsl_deriv_central()
VAR t'
f'(t) = derivative(f(t'),t',t)

# write the result
PRINT_FUNCTION f' MIN a+0.5*h MAX b-0.5*h STEP h
$chmod +x derivative.sh$ feenox f.fee "sin(t)" 1 | ./derivative.fee 0.1 | tee f_prime.dat
0.05    0.998725
0.15    0.989041
0.25    0.968288
0.35    0.939643
0.45    0.900427
0.55    0.852504
0.65    0.796311
0.75    0.731216
0.85    0.66018
0.95    0.574296
$ ## 4.11 Parametric study on a cantilevered beam If an external loop successively calls FeenoX with extra command-line arguments, a parametric run is obtained. This file cantilever.fee fixes the face called “left” and sets a load in the negative z direction of a mesh called cantilever-$1-$2.msh, where $1 is the first argument after the inpt file and $2 the second one. The output is a single line containing the number of nodes of the mesh and the displacement in the vertical direction w(500,0,0) at the center of the cantilever’s free face. The following Bash script first calls Gmsh to create the meshes. To do so, it first starts with a base cantilever.geo file that creates the CAD: // https://autofem.com/examples/determining_natural_frequencie.html SetFactory("OpenCASCADE"); L = 0.5; b = 0.05; h = 0.02; Box(1) = {0,-b/2,-h/2, L, b, h}; Physical Surface("left") = {1}; Physical Surface("right") = {2}; Physical Surface("top") = {4}; Physical Volume("bulk") = {1}; Transfinite Curve {1, 3, 5, 7} = 1/(Mesh.MeshSizeFactor*Mesh.ElementOrder) + 1; Transfinite Curve {2, 4, 6, 8} = 2/(Mesh.MeshSizeFactor*Mesh.ElementOrder) + 1; Transfinite Curve {9, 10, 11, 12} = 16/(Mesh.MeshSizeFactor*Mesh.ElementOrder) + 1; Transfinite Surface "*"; Transfinite Volume "*"; Then another .geo file is merged to build cantilever-${element}-${c}.msh where Figure 8: Cantilevered beam meshed with structured tetrahedra and hexahedra. a — Tetrahedra, b — Hexahedra It then calls FeenoX with the input cantilever.fee and passes ${element} and ${c} as extra arguments, which then are expanded as $1 and $2 respectively. #!/bin/bash rm -f *.dat for element in tet4 tet10 hex8 hex20 hex27; do for c in$(seq 1 10); do

# create mesh if not alreay cached
mesh=cantilever-${element}-${c}
if [ ! -e ${mesh}.msh ]; then scale=$(echo "PRINT 1/${c}" | feenox -) gmsh -3 -v 0 cantilever-${element}.geo -clscale ${scale} -o${mesh}.msh
fi

# call FeenoX
feenox cantilever.fee ${element}${c} | tee -a cantilever-${element}.dat done done After the execution of the Bash script, thanks to the design decision that output is 100% defined by the user (in this case with the PRINT instruction), one has several files cantilever-${element}.dat files. When plotted, these show the shear locking effect of fully-integrated first-order elements. The theoretical Euler-Bernoulli result is just a reference as, among other things, it does not take into account the effect of the material’s Poisson’s ratio. Note that the abscissa shows the number of nodes, which are proportional to the number of degrees of freedom (i.e. the size of the problem matrix) and not the number of elements, which is irrelevant here and in most problems.

PROBLEM elastic 3D
READ_MESH cantilever-$1-$2.msh   # in meters

E = 2.1e11         # Young modulus in Pascals
nu = 0.3           # Poisson's ratio

BC left   fixed
BC right  tz=-1e5  # traction in Pascals, negative z

SOLVE_PROBLEM

# z-displacement (components are u,v,w) at the tip vs. number of nodes
PRINT nodes %e w(500,0,0) "\# $1$2"
$./cantilever.sh 102 -7.641572e-05 # tet4 1 495 -2.047389e-04 # tet4 2 1372 -3.149658e-04 # tet4 3 [...] 19737 -5.916234e-04 # hex27 8 24795 -5.916724e-04 # hex27 9 37191 -5.917163e-04 # hex27 10$ pyxplot cantilever.ppl

## 4.12 Optimizing the length of a tuning fork

To illustrate how to use FeenoX in an optimization loop, let us consider the problem of finding the length \ell_1 of a tuning fork (fig. 9) such that the fundamental frequency on a free-free oscillation is equal to the base A frequency at 440 Hz.

The FeenoX input is extremely simple input file, since it has to solve the free-free mechanical modal problem (i.e. without any Dirichlet boundary condition). All it has to do is to print the fundamental frequency.

To find the length \ell_1, FeenoX is sucessively called from a Python driving script called fork.py. This script uses Gmsh’s Python API to create the CAD and the mesh of the tuning fork given the geometrical arguments r, w, \ell_1 and \ell_2. The parameter n controls the number of elements through the fork’s thickness. Here is the driving script without the CAD & mesh details (the full implementation of the function is available in the examples directory of the FeenoX distribution):

import math
import gmsh
import subprocess  # to call FeenoX and read back

def create_mesh(r, w, l1, l2, n):
gmsh.initialize()
...
gmsh.write("fork.msh")
gmsh.finalize()
return len(nodes)

def main():
target = 440    # target frequency
eps = 1e-2      # tolerance
r = 4.2e-3      # geometric parameters
w = 3e-3
l1 = 30e-3
l2 = 60e-3

for n in range(1,7):   # mesh refinement level
l1 = 60e-3              # restart l1 & error
error = 60
while abs(error) > eps:   # loop
l1 = l1 - 1e-4*error
# mesh with Gmsh Python API
nodes = create_mesh(r, w, l1, l2, n)
# call FeenoX and read scalar back
# TODO: FeenoX Python API (like Gmsh)
result = subprocess.run(['feenox', 'fork.fee'], stdout=subprocess.PIPE)
freq = float(result.stdout.decode('utf-8'))
error = target - freq

print(nodes, l1, freq)

Note that in this particular case, the FeenoX input files does not expand any command-line argument. The trick is that the mesh file fork.msh is overwritten in each call of the optimization loop. The detailed steps between gmsh.initialize() and gmsh.finalize() are not shown here,

Since the computed frequency depends both on the length \ell_1 and on the mesh refinement level n, there are actually two nested loops: one parametric over n=1,2\dots,7 and the optimization loop itself that tries to find \ell_1 so as to obtain a frequency equal to 440 Hz within 0.01% of error.

PROBLEM modal 3D MODES 1  # only one mode needed
E = 2.07e11         # in [Pa]
nu = 0.33
rho = 7829          # in [kg/m^2]

# no BCs! It is a free-free vibration problem
SOLVE_PROBLEM

# write back the fundamental frequency to stdout
PRINT f(1)
$python fork.py > fork.dat$ pyxplot fork.ppl
$ ## 4.13 IAEA 2D PWR Benchmark # BENCHMARK PROBLEM # # Identification: 11-A2 Source Situation ID.11 # Date Submitted: June 1976 By: R. R. Lee (CE) # D. A. Menely (Ontario Hydro) # B. Micheelsen (Riso-Denmark) # D. R. Vondy (ORNL) # M. R. Wagner (KWU) # W. Werner (GRS-Munich) # # Date Accepted: June 1977 By: H. L. Dodds, Jr. (U. of Tenn.) # M. V. Gregory (SRL) # # Descriptive Title: Two-dimensional LWR Problem, # also 2D IAEA Benchmark Problem # # Reduction of Source Situation # 1. Two-groupo diffusion theory # 2. Two-dimensional (x,y)-geometry # PROBLEM neutron_diffusion 2D GROUPS 2 DEFAULT_ARGUMENT_VALUE 1 quarter # either quarter or eigth READ_MESH iaea-2dpwr-$1.msh

# define materials and cross sections according to the two-group constants
# each material corresponds to a physical entity in the geometry file
Bg2 = 0.8e-4  # axial geometric buckling in the z direction
MATERIAL fuel1 {
D1=1.5    Sigma_a1=0.010+D1(x,y)*Bg2    Sigma_s1.2=0.02
D2=0.4    Sigma_a2=0.080+D2(x,y)*Bg2    nuSigma_f2=0.135   }#eSigmaF_2 nuSigmaF_2(x,y) }

MATERIAL fuel2 {
D1=1.5    Sigma_a1=0.010+D1(x,y)*Bg2    Sigma_s1.2=0.02
D2=0.4    Sigma_a2=0.085+D2(x,y)*Bg2    nuSigma_f2=0.135   }#eSigmaF_2 nuSigmaF_2(x,y) }

MATERIAL fuel2rod {
D1=1.5    Sigma_a1=0.010+D1(x,y)*Bg2    Sigma_s1.2=0.02
D2=0.4    Sigma_a2=0.130+D2(x,y)*Bg2    nuSigma_f2=0.135   }#eSigmaF_2 nuSigmaF_2(x,y) }

MATERIAL reflector {
D1=2.0    Sigma_a1=0.000+D1(x,y)*Bg2    Sigma_s1.2=0.04
D2=0.3    Sigma_a2=0.010+D2(x,y)*Bg2 }

# define boundary conditions as requested by the problem
BC external vacuum=0.4692  # "external" is the name of the entity in the .geo
BC mirror   mirror         # the first mirror is the name, the second is the BC type

# # set the power setpoint equal to the volume of the core
# # (and set eSigmaF_2 = nuSigmaF_2 as above)
# power = 17700

SOLVE_PROBLEM   # solve!
PRINT %.5f "keff = " keff
WRITE_MESH iaea-2dpwr-$1.vtk phi1 phi2 $ gmsh -2 iaea-2dpwr-quarter.geo
$[...]$ gmsh -2 iaea-2dpwr-eighth.geo
$[...]$ feenox iaea-2dpwr.fee quarter
keff =  1.02986
$feenox iaea-2dpwr.fee eighth$keff =  1.02975
$ ## 4.14 Cube-spherical bare reactor It is easy to compute the effective multiplication factor of a one-group bare cubical reactor. Or a spherical reactor. And we know that for the same mass, the k_\text{eff} for the former is smaller than for the latter. Figure 11: One eight of two bare reactors. a — Cubical reactor, b — Spherical reactor But what happens “in the middle”? That is to say, how does k_\text{eff} changes when we morph the cube into a sphere? Enter Gmsh & Feenox. Figure 12: Continuous morph between a cube and a sphere. a — 75% cube/25% sphere, b — 50% cube/50% sphere, c — 25% cube/75% sphere import os import math import gmsh def create_mesh(vol, F): gmsh.initialize() gmsh.option.setNumber("General.Terminal", 0) f = 0.01*F a = (vol / (1/8*4/3*math.pi*f**3 + 3*1/4*math.pi*f**2*(1-f) + 3*f*(1-f)**2 + (1-f)**3))**(1.0/3.0) internal = [] gmsh.model.add("cubesphere") if (F < 1): # a cube gmsh.model.occ.addBox(0, 0, 0, a, a, a, 1) internal = [1,3,5] external = [2,4,6] elif (F > 99): # a sphere gmsh.model.occ.addSphere(0, 0, 0, a, 1, 0, math.pi/2, math.pi/2) internal = [2,3,4] external = [1] else: gmsh.model.occ.addBox(0, 0, 0, a, a, a, 1) gmsh.model.occ.fillet([1], [12, 7, 6], [f*a], True) internal = [1,4,6] external = [2,3,5,7,8,9,10] gmsh.model.occ.synchronize() gmsh.model.addPhysicalGroup(3, [1], 1) gmsh.model.setPhysicalName(3, 1, "fuel") gmsh.model.addPhysicalGroup(2, internal, 2) gmsh.model.setPhysicalName(2, 2, "internal") gmsh.model.addPhysicalGroup(2, external, 3) gmsh.model.setPhysicalName(2, 3, "external") gmsh.model.occ.synchronize() gmsh.option.setNumber("Mesh.ElementOrder", 2) gmsh.option.setNumber("Mesh.Optimize", 1) gmsh.option.setNumber("Mesh.OptimizeNetgen", 1) gmsh.option.setNumber("Mesh.HighOrderOptimize", 1) gmsh.option.setNumber("Mesh.CharacteristicLengthMin", a/10); gmsh.option.setNumber("Mesh.CharacteristicLengthMax", a/10); gmsh.model.mesh.generate(3) gmsh.write("cubesphere-%g.msh"%(F)) gmsh.model.remove() #gmsh.fltk.run() gmsh.finalize() return def main(): vol0 = 100**3 for F in range(0,101,5): # mesh refinement level create_mesh(vol0, F) # TODO: FeenoX Python API! os.system("feenox cubesphere.fee %g"%(F)) if __name__ == "__main__": main() PROBLEM neutron_diffusion DIMENSIONS 3 READ_MESH cubesphere-$1.msh DIMENSIONS 3

# MATERIAL fuel
D1 = 1.03453E+00
Sigma_a1 = 5.59352E-03
nuSigma_f1 = 6.68462E-03
Sigma_s1.1 = 3.94389E-01

PHYSICAL_GROUP fuel DIM 3
BC internal    mirror
BC external    vacuum

SOLVE_PROBLEM

PRINT HEADER $1 keff 1e5*(keff-1)/keff fuel_volume $ python cubesphere.py | tee cubesphere.dat
0       1.05626 5326.13 1e+06
5       1.05638 5337.54 999980
10      1.05675 5370.58 999980
15      1.05734 5423.19 999992
20      1.05812 5492.93 999995
25      1.05906 5576.95 999995
30      1.06013 5672.15 999996
35      1.06129 5775.31 999997
40      1.06251 5883.41 999998
45      1.06376 5993.39 999998
50      1.06499 6102.55 999998
55      1.06619 6208.37 999998
60      1.06733 6308.65 999998
65      1.06839 6401.41 999999
70      1.06935 6485.03 999998
75      1.07018 6557.96 999998
80      1.07088 6618.95 999998
85      1.07143 6666.98 999999
90      1.07183 6701.24 999999
95      1.07206 6721.33 999998
100     1.07213 6727.64 999999
 ## 4.15 Parallelepiped whose Young’s modulus is a function of the temperature The problem consists of finding the non-dimensional temperature T and displacements u, v and w distributions within a solid parallelepiped of length l whose base is a square of h\times h. The solid is subject to heat fluxes and to a traction pressure at the same time. The non-dimensional Young’s modulus E of the material depends on the temperature T in a know algebraically way, whilst both the Poisson coefficient \nu and the thermal conductivity k are uniform and do not depend on the spatial coordinates: \begin{aligned} E(T) &= \frac{1000}{800-T} \\ \nu &= 0.3 \\ k &= 1 \\ \end{aligned} References: This thermo-mechanical problem is solved in two stages. First, the heat conduction equation is solved over a coarse first-order mesh to find the non-dimensional temperature distribution. Then, a mechanical problem is solved where T(x,y,z) is read from the first mesh and interpolated into a finer second-order mesh so to as evaluate the non-dimensional Young’s modulus as E\Big(T(x,y,z)\Big) = \frac{1000}{800-T(x,y,z)} Note that there are not thermal expansion effects (i.e. the thermal expansion coefficient is \alpha=0). Yet, suprinsingly enough, the problem has analytical solutions for both the temperature and the displacement fields. ### 4.15.1 Thermal problem The following input solves the thermal problem over a coarse first-order mesh, writes the resulting temperature distribution into parallelepiped-temperature.msh, and prints the L_2 error of the numerical result with respect to the analytical solution T(x,y,z) = 40 - 2x - 3y - 4z. PROBLEM thermal 3D READ_MESH parallelepiped-coarse.msh k = 1 # unitary non-dimensional thermal conductivity # boundary conditions BC left q=+2 BC right q=-2 BC front q=+3 BC back q=-3 BC bottom q=+4 BC top q=-4 BC A T=0 SOLVE_PROBLEM WRITE_MESH parallelepiped-temperature.msh T # compute the L-2 norm of the error in the displacement field Te(x,y,z) = 40 - 2*x - 3*y - 4*z # analytical solution, "e" means exact INTEGRATE (T(x,y,z)-Te(x,y,z))^2 RESULT num PHYSICAL_GROUP bulk DIM 3 # this is just to compute the volume PRINT num/bulk_volume  gmsh -3 parallelepiped.geo -order 1 -clscale 2 -o parallelepiped-coarse.msh
[...]
Info    : 117 nodes 567 elements
Info    : Writing 'parallelepiped-coarse.msh'...
Info    : Done writing 'parallelepiped-coarse.msh'
Info    : Stopped on Fri Dec 10 10:32:30 2021 (From start: Wall 0.0386516s, CPU 0.183052s)
$feenox parallelepiped-thermal.fee 6.18981e-12$

### 4.15.2 Mechanical problem

Now this input file reads the scalar function T stored in the coarse first-order mesh file parallelepiped-temperature.msh and uses it to solve the mechanical problem in the finer second-order mesh parallelepiped.msh. The numerical solution for the displacements over the fine mesh is written in a VTK file (along with the temperature as interpolated from the coarse mesh) and compared to the analytical solution using the L_2 norm.

PROBLEM mechanical 3D

# this is where we solve the mechanical problem

# this is where we read the temperature from

# mechanical properties
E(x,y,z) = 1000/(800-T(x,y,z))   # young's modulus
nu = 0.3                         # poisson's ratio

# boundary conditions
BC O fixed
BC B u=0 w=0
BC C u=0

# here "load" is a fantasy name applied to both "left" and "right"
BC load tension=1 PHYSICAL_GROUP left PHYSICAL_GROUP right

SOLVE_PROBLEM
WRITE_MESH parallelepiped-mechanical.vtk T VECTOR u v w

# analytical solutions
h = 10
A = 0.002
B = 0.003
C = 0.004
D = 0.76

# the "e" means exact
ue(x,y,z) := A/2*(x^2 + nu*(y^2+z^2)) + B*x*y + C*x*z + D*x - nu*A*h/4*(y+z)
ve(x,y,z) := -nu*(A*x*y + B/2*(y^2-z^2+x^2/nu) + C*y*z + D*y -A*h/4*x - C*h/4*z)
we(x,y,z) := -nu*(A*x*z + B*y*z + C/2*(z^2-y^2+x^2/nu) + D*z + C*h/4*y - A*h/4*x)

# compute the L-2 norm of the error in the displacement field
INTEGRATE (u(x,y,z)-ue(x,y,z))^2+(v(x,y,z)-ve(x,y,z))^2+(w(x,y,z)-we(x,y,z))^2 RESULT num MESH parallelepiped.msh
INTEGRATE 1 RESULT den MESH parallelepiped.msh
PRINT num/den
$gmsh -3 parallelepiped.geo -order 2 [...] Info : 2564 nodes 2162 elements Info : Writing 'parallelepiped.msh'... Info : Done writing 'parallelepiped.msh' Info : Stopped on Fri Dec 10 10:39:27 2021 (From start: Wall 0.165707s, CPU 0.258751s)$ feenox parallelepiped-mechanical.fee
0.000345839
$ ## 4.16 Non-dimensional transient heat conduction on a cylinder Let us solve a dimensionless transient problem over a cylinder. Conductivity and heat capacity are unity. Initial condition is a linear temperature profile along the x axis: T(x,y,z,0) = x The base of the cylinder has a prescribed time and space-dependent temperature T(0,y,z,t) = \sin( 2\pi \cdot t) \cdot \sin( 2\pi \cdot y) The other faces have a convection conditions with (non-dimensional) heat transfer coefficient h=0.1 and T_\text{ref} = 1. PROBLEM thermal 3D READ_MESH cylinder.msh end_time = 2 # final time [ non-dimensional units ] # the time step is automatically computed # initial condition (if not given, stead-state is computed) T_0(x,y,z) = x # dimensionless uniform and constant material properties k = 1 kappa = 1 # BCs BC hot T=sin(2*pi*t)*sin(2*pi*y) BC cool h=0.1 Tref=1 SOLVE_PROBLEM # print the temperature at the center of the base vs time PRINT %e t T(0,0,0) T(0.5,0,0) T(1,0,0) WRITE_MESH temp-cylinder.msh T IF done PRINT "\# open temp-anim-cylinder.geo in Gmsh to create a quick rough video" PRINT "\# run temp-anim-cylinder.py to get a nicer and smoother video" ENDIF $ gmsh -3 cylinder.geo
[...]
Info    : Done optimizing mesh (Wall 0.624941s, CPU 0.624932s)
Info    : 1986 nodes 10705 elements
Info    : Writing 'cylinder.msh'...
Info    : Done writing 'cylinder.msh'
Info    : Stopped on Fri Dec 24 10:35:32 2021 (From start: Wall 0.800542s, CPU 0.896698s)
$feenox temp-cylinder-tran.fee 0.000000e+00 0.000000e+00 5.000000e-01 1.000000e+00 1.451938e-04 4.406425e-07 5.000094e-01 9.960851e-01 3.016938e-04 9.155974e-07 5.000171e-01 9.921274e-01 5.566768e-04 1.689432e-06 5.000251e-01 9.862244e-01 8.565589e-04 2.599523e-06 5.000292e-01 9.800113e-01 1.245867e-03 3.780993e-06 5.000280e-01 9.728705e-01 1.780756e-03 5.404230e-06 5.000176e-01 9.643259e-01 2.492280e-03 7.563410e-06 4.999932e-01 9.545723e-01 3.428621e-03 1.040457e-05 4.999538e-01 9.436480e-01 [...] 1.978669e+00 -6.454358e-05 1.500891e-01 2.286112e-01 1.989334e+00 -3.234439e-05 1.500723e-01 2.285660e-01 2.000000e+00 1.001730e-14 1.500572e-01 2.285223e-01 # open temp-anim-cylinder.geo in Gmsh to create a quick rough video # run temp-anim-cylinder.py to get a nicer and smoother video$ python3 temp-anim-cylinder.py
Info    : 1986 nodes
Info    : 10612 elements
0 1 0.0
0.01 12 0.8208905327853042
0.02 15 0.8187351216040447
0.03 17 0.7902629708599855
[...]
Info    : Writing 'temp-cylinder-smooth-198.png'...
Info    : Done writing 'temp-cylinder-smooth-198.png'
199
Info    : Writing 'temp-cylinder-smooth-199.png'...
Info    : Done writing 'temp-cylinder-smooth-199.png'
all frames dumped, now run
ffmpeg -framerate 20 -f image2 -i temp-cylinder-smooth-%03d.png temp-cylinder-smooth.mp4
to get a video
$ffmpeg -y -f image2 -i temp-cylinder-smooth-%03d.png -framerate 20 -pix_fmt yuv420p -c:v libx264 -filter:v crop='floor(in_w/2)*2:floor(in_h/2)*2' temp-cylinder-smooth.mp4 [...]$

## 4.17 Five natural modes of a cantilevered wire

Back in College, we had this subject Experimental Physics 101. I had to measure the natural modes of two cantilevered wires and determine the Young modulus of of those measurements. The report is here. Two comments: 1. It is in Spanish 2. There was a systematic error and a factor of two sneaked in into the measured values

Here is a finite-element version of the experimental setup with a comparison to then theoretical values written directly as Markdown tables. The material (either aluminum or copper) and the mesh type (either tet or hex) and be chosen at runtime through command line arguments.

#
DEFAULT_ARGUMENT_VALUE 1 hex       # mesh, either hex or unstruct
DEFAULT_ARGUMENT_VALUE 2 copper    # material, either copper or aluminum

l = 0.5*303e-3   # cantilever wire length [ m ]
d = 1.948e-3     # wire diameter [ m ]

# material properties for copper
m_copper = 0.5*8.02e-3  # total mass (half the measured because of the experimental disposition) [ kg ]
E_copper = 2*66.2e9     # [ Pa ] Young modulus (twice because the factor-two error)

# material properties for aluminum
m_aluminum = 0.5*2.67e-3
E_aluminum = 2*40.2e9

# problem’s properties
E = E_$2 # [ MPa ] rho = m_$2/(pi*(0.5*d)^2*l)  # [ kg / m^3 ] density = mass (measured) / volume
nu = 0                       # Poisson’s ratio (does not appear in Euler-Bernoulli)

# analytical solution
VECTOR kl[5]
VECTOR f_euler[5]

# first compute the first five roots ok cosh(kl)*cos(kl)+1
kl[i] = root(cosh(t)*cos(t)+1, t, 3*i-2,3*i+1)

# then compute the frequencies according to Euler-Bernoulli
# note that we need to use SI inside the square root
A = pi * (d/2)^2
I = pi/4 * (d/2)^4
f_euler[i] = 1/(2*pi) * kl(i)^2 * sqrt((E * I)/(rho * A * l^4))

# now compute the modes numerically with FEM
# note that each mode is duplicated as it is degenerated
READ_MESH wire-$1.msh DIMENSIONS 3 PROBLEM modal MODES 10 BC fixed fixed SOLVE_PROBLEM # github-formatted markdown table # compare the frequencies PRINT " \$n\$| FEM [Hz] | Euler [Hz] | Relative difference [%]" PRINT ":------:|:-------------:|:-------------:|:--------------:" PRINT_VECTOR SEP "\t|\t" i %.4g f(2*i-1) f_euler %.2f 100*(f_euler(i)-f(2*i-1))/f_euler(i) PRINT PRINT ":$2 wire over $1 mesh" # commonmark table PRINT PRINT " \$n\$| \$L\$| \$\\Gamma\$| \$\\mu\$| \$M\$" PRINT ":------:+:---------------------:+:---------------------:+:-------------:+:--------------:" PRINT_VECTOR SEP "\t|\t" i "%+.1e" L Gamma "%.4f" mu Mu PRINT PRINT ":$2 wire over $1 mesh, participation and excitation factors \$L\$and \$\\Gamma\$, effective per-mode and cummulative mass fractions \$\\mu\$and \$M\$" # write the modes into a vtk file WRITE_MESH wire-$1-$2.vtk \ VECTOR u1 v1 w1 VECTOR u2 v2 w2 VECTOR u3 v3 w3 \ VECTOR u4 v4 w4 VECTOR u5 v5 w5 VECTOR u6 v6 w6 \ VECTOR u7 v7 w7 VECTOR u8 v8 w8 VECTOR u9 v9 w9 VECTOR u10 v10 w10 # and into a msh file WRITE_MESH wire-$1-$2.msh { u1 v1 w1 u2 v2 w2 u3 v3 w3 u4 v4 w4 u5 v5 w5 u6 v6 w6 u7 v7 w7 u8 v8 w8 u9 v9 w9 u10 v10 w10 } $ gmsh -3 wire-hex.geo
[...]
Info    : Done meshing order 2 (Wall 0.0169025s, CPU 0.016804s)
Info    : 8398 nodes 4676 elements
Info    : Writing 'wire-hex.msh'...
Info    : Done writing 'wire-hex.msh'
Info    : Stopped on Fri Dec 24 17:07:19 2021 (From start: Wall 0.0464517s, CPU 0.133498s)
$gmsh -3 wire-tet.geo [...] Info : Done optimizing mesh (Wall 0.0229018s, CPU 0.022892s) Info : 16579 nodes 13610 elements Info : Writing 'wire-tet.msh'... Info : Done writing 'wire-tet.msh' Info : Stopped on Fri Dec 24 17:07:59 2021 (From start: Wall 2.5798s, CPU 2.64745s)$ feenox wire.fee
$n$   |    FEM [Hz]   |   Euler [Hz]  |  Relative difference [%]
:------:|:-------------:|:-------------:|:--------------:
1       |       45.84   |       45.84   |       0.02
2       |       287.1   |       287.3   |       0.06
3       |       803.4   |       804.5   |       0.13
4       |       1573    |       1576    |       0.24
5       |       2596    |       2606    |       0.38

: copper wire over hex mesh

$n$   |          $L$          |       $\Gamma$        |      $\mu$    |       $M$
:------:+:---------------------:+:---------------------:+:-------------:+:--------------:
1       |       +1.3e-03        |       +4.2e-01        |       0.1371  |       0.1371
2       |       -1.8e-03        |       -5.9e-01        |       0.2716  |       0.4087
3       |       +9.1e-05        |       +1.7e-02        |       0.0004  |       0.4091
4       |       -1.7e-03        |       -3.0e-01        |       0.1252  |       0.5343
5       |       -3.3e-05        |       -5.9e-03        |       0.0000  |       0.5343
6       |       -9.9e-04        |       -1.8e-01        |       0.0431  |       0.5775
7       |       +7.3e-04        |       +1.2e-01        |       0.0221  |       0.5995
8       |       +4.5e-06        |       +7.5e-04        |       0.0000  |       0.5995
9       |       +5.4e-04        |       +9.9e-02        |       0.0134  |       0.6129
10      |       +2.7e-05        |       +4.9e-03        |       0.0000  |       0.6129

: copper wire over hex mesh, participation and excitation factors $L$ and $\Gamma$, effective per-mode and cummulative mass fractions $\mu$ and $M$
$feenox wire.fee hex copper | pandoc -o wire-hex-copper.html$ feenox wire.fee tet copper   | pandoc -o wire-tet-copper.html
$feenox wire.fee hex aluminum | pandoc -o wire-hex-aluminum.html$ feenox wire.fee tet aluminum | pandoc -o wire-tet-aluminum.html
copper wire over hex mesh
n FEM [Hz] Euler [Hz] Relative difference [%]
1 45.84 45.84 0.02
2 287.1 287.3 0.06
3 803.4 804.5 0.13
4 1573 1576 0.24
5 2596 2606 0.38
copper wire over hex mesh, participation and excitation factors L and \Gamma, effective per-mode and cummulative mass fractions \mu and M
n L \Gamma \mu M
1 -1.8e-03 -5.9e-01 0.2713 0.2713
2 +1.3e-03 +4.2e-01 0.1374 0.4087
3 +9.7e-05 +1.8e-02 0.0004 0.4091
4 -1.6e-03 -3.1e-01 0.1251 0.5343
5 -3.5e-05 -6.3e-03 0.0001 0.5343
6 -9.9e-04 -1.8e-01 0.0431 0.5774
7 +7.2e-04 +1.2e-01 0.0221 0.5995
8 -8.6e-06 -1.5e-03 0.0000 0.5995
9 -2.6e-05 -4.7e-03 0.0000 0.5996
10 +5.4e-04 +9.9e-02 0.0134 0.6130
copper wire over tet mesh
n FEM [Hz] Euler [Hz] Relative difference [%]
1 45.84 45.84 0.00
2 287.2 287.3 0.05
3 803.4 804.5 0.13
4 1573 1576 0.24
5 2596 2606 0.38
copper wire over tet mesh, participation and excitation factors L and \Gamma, effective per-mode and cummulative mass fractions \mu and M
n L \Gamma \mu M
1 -1.9e-03 -6.1e-01 0.2925 0.2925
2 +1.2e-03 +3.8e-01 0.1163 0.4087
3 -1.0e-03 -3.3e-01 0.0861 0.4948
4 +7.0e-04 +2.3e-01 0.0395 0.5343
5 -6.0e-04 -1.9e-01 0.0292 0.5634
6 +4.2e-04 +1.3e-01 0.0140 0.5774
7 -4.0e-04 -1.3e-01 0.0133 0.5908
8 +3.3e-04 +1.1e-01 0.0087 0.5995
9 +3.5e-04 +1.1e-01 0.0096 0.6091
10 -2.2e-04 -6.9e-02 0.0038 0.6129
aluminum wire over hex mesh
n FEM [Hz] Euler [Hz] Relative difference [%]
1 61.91 61.92 0.02
2 387.8 388 0.06
3 1085 1086 0.13
4 2124 2129 0.24
5 3506 3519 0.38
aluminum wire over hex mesh, participation and excitation factors L and \Gamma, effective per-mode and cummulative mass fractions \mu and M
n L \Gamma \mu M
1 -6.9e-04 -6.2e-01 0.3211 0.3211
2 +3.6e-04 +3.3e-01 0.0876 0.4087
3 +4.2e-05 +2.4e-02 0.0008 0.4095
4 -5.4e-04 -3.1e-01 0.1248 0.5343
5 +3.7e-05 +2.3e-02 0.0006 0.5349
6 -3.0e-04 -1.9e-01 0.0425 0.5774
7 +2.4e-04 +1.2e-01 0.0221 0.5995
8 -3.2e-06 -1.6e-03 0.0000 0.5995
9 +1.8e-04 +9.8e-02 0.0132 0.6127
10 -9.5e-06 -5.2e-03 0.0000 0.6128
aluminum wire over tet mesh
n FEM [Hz] Euler [Hz] Relative difference [%]
1 61.91 61.92 0.00
2 387.8 388 0.05
3 1085 1086 0.13
4 2124 2129 0.24
5 3506 3519 0.38
aluminum wire over tet mesh, participation and excitation factors L and \Gamma, effective per-mode and cummulative mass fractions \mu and M
n L \Gamma \mu M
1 -6.4e-04 -6.1e-01 0.2923 0.2923
2 +4.0e-04 +3.8e-01 0.1164 0.4087
3 -3.5e-04 -3.3e-01 0.0861 0.4948
4 +2.3e-04 +2.3e-01 0.0395 0.5343
5 -2.0e-04 -1.9e-01 0.0292 0.5634
6 +1.4e-04 +1.3e-01 0.0140 0.5774
7 -1.3e-04 -1.3e-01 0.0133 0.5908
8 +1.1e-04 +1.1e-01 0.0087 0.5995
9 +1.2e-04 +1.1e-01 0.0096 0.6091
10 -7.3e-05 -6.9e-02 0.0038 0.6129

## 4.18 On the evaluation of thermal expansion coefficients

When solving thermal-mechanical problems it is customary to use thermal expansion coefficients in order to take into account the mechanical strains induced by changes in the material temperature with respect to a reference temperature where the deformation is identically zero. These coefficients \alpha are defined as the partial derivative of the strain \epsilon with respect to temperature T such that differential relationships like

d\epsilon = \frac{\partial \epsilon}{\partial T} \, dT = \alpha \cdot dT

hold. This derivative \alpha is called the instantaneous thermal expansion coefficient. For finite temperature increments, one would like to be able to write

\Delta \epsilon = \alpha \cdot \Delta T

But if the strain is not linear with respect to the temperature—which is the most common case—then \alpha depends on T. Therefore, when dealing with finite temperature increments \Delta T = T-T_0 where the thermal expansion coefficient \alpha(T) depends on the temperature itself then mean values for the thermal expansion ought to be used:

\Delta \epsilon = \int_{\epsilon_0}^{\epsilon} d\epsilon^{\prime} = \int_{T_0}^{T} \frac{\partial \epsilon}{\partial T^\prime} \, dT^\prime = \int_{T_0}^{T} \alpha(T^\prime) \, dT^\prime

We can multiply and divide by \Delta T to obtain

\int_{T_0}^{T} \alpha(T^\prime) \, dT^\prime \cdot \frac{\Delta T}{\Delta T} = \bar{\alpha}(T) \cdot \Delta T

where the mean expansion coefficient for the temperature range [T_0,T] is

\bar{\alpha}(T) = \frac{\displaystyle \int_{T_0}^{T} \alpha(T^\prime) \, dT^\prime}{T-T_0}

This is of course the classical calculus result of the mean value of a continuous one-dimensional function in a certain range.

Let \epsilon(T) be the linear thermal expansion of a given material in a certain direction when heating a piece of such material from an initial temperature T_0 up to T so that \epsilon(T_0)=0.

From our previous analysis, we can see that in fig. 14:

\begin{aligned} A(T) &= \alpha(T) = \frac{\partial \epsilon}{\partial T} \\ B(T) &= \bar{\alpha}(T) = \frac{\epsilon(T)}{T-T_0} = \frac{\displaystyle \int_{T_0}^{T} \alpha(T^\prime) \, dT^\prime}{T - T_0} \\ C(T) &= \epsilon(T) = \int_{T_0}^T \alpha(T^\prime) \, dT^\prime \end{aligned}

Therefore,

1. A(T) can be computed out of C(T)
2. B(T) can be computed either out of A(T) or C(T)
3. C(T) can be computed out of A(T)
# just in case we wanted to interpolate with another method (linear, splines, etc.)
DEFAULT_ARGUMENT_VALUE 1 steffen

# read columns from data file and interpolate
# A is the instantaenous coefficient of thermal expansion x 10^-6 (mm/mm/ºC)
FUNCTION A(T) FILE asme-expansion-table.dat COLUMNS 1 2 INTERPOLATION $1 # B is the mean coefficient of thermal expansion x 10^-6 (mm/mm/ºC) in going # from 20ºC to indicated temperature FUNCTION B(T) FILE asme-expansion-table.dat COLUMNS 1 3 INTERPOLATION$1
# C is the linear thermal expansion (mm/m) in going from 20ºC
# to indicated temperature
FUNCTION C(T) FILE asme-expansion-table.dat COLUMNS 1 4 INTERPOLATION $1 VAR T' # dummy variable for integration T0 = 20 # reference temperature T_min = vecmin(vec_A_T) # smallest argument of function A(T) T_max = vecmax(vec_A_T) # largest argument of function A(T) # compute one column from another one A_fromC(T) := 1e3*derivative(C(T'), T', T) B_fromA(T) := integral(A(T'), T', T0, T)/(T-T0) B_fromC(T) := 1e3*C(T)/(T-T0) # C is in mm/m, hence the 1e3 C_fromA(T) := 1e-3*integral(A(T'), T', T0, T) # write interpolated results PRINT_FUNCTION A A_fromC B B_fromA B_fromC C C_fromA MIN T_min+1 MAX T_max-1 STEP 1 $ cat asme-expansion-table.dat
# temp  A       B        C
20      21.7    21.7     0
50      23.3    22.6     0.7
75      23.9    23.1     1.3
100     24.3    23.4     1.9
125     24.7    23.7     2.5
150     25.2    23.9     3.1
175     25.7    24.2     3.7
200     26.4    24.4     4.4
225     27.0    24.7     5.1
250     27.5    25.0     5.7
275     27.7    25.2     6.4
300     27.6    25.5     7.1
325     27.1    25.6     7.8
$feenox asme-expansion.fee > asme-expansion-interpolation.dat$ pyxplot asme-expansion.ppl
$ The conclusion (see this, this and this reports) is that values rounded to only one decimal value as presented in the ASME code section II subsection D tables are not enough to satisfy the mathematical relationships between the physical magnitudes related to thermal expansion properties of the materials listed. Therefore, care has to be taken as which of the three columns is chosen when using the data for actual thermo-mechanical numerical computations. As an exercise, the reader is encouraged to try different interpolation algorithms to see how the results change. Spoiler alert: they are also highly sensible to the interpolation method used to “fill in” the gaps between the table values. ### 4.18.1 Orthotropic free expansion of a cube To illustrate the point of the previous discussion, let us solve the thermal expansion of an unrestrained unitary cube [0,1~\text{mm}]\times[0,1~\text{mm}]\times[0,1~\text{mm}] subject to a linear radially-symmetric temperature field T(x,y,z) = 30 \text{ºC} + 150 \frac{\text{ºC}}{\text{mm}} \sqrt{x^2+y^2+z^2} with a mean thermal expansion coefficient for each of the three directions x, y and z computed from each of the three columns of the ASME table TE-2, respectively. If the data was consistent, the displacement at any point with the same coordinates x=y=z would be exactly equal. DEFAULT_ARGUMENT_VALUE 1 steffen DEFAULT_ARGUMENT_VALUE 2 hex PROBLEM mechanical READ_MESH cube-hex.msh # aluminum-like linear isotropic material properties E = 69e3 nu = 0.28 # free expansion BC left u=0 BC front v=0 BC bottom w=0 # reference temperature is 20ºC T0 = 20 # spatial temperature distribution symmetric wrt x,y & z T(x,y,z) = 30+150*sqrt(x^2+y^2+z^2) # read ASME data FUNCTION A(T') FILE asme-expansion-table.dat COLUMNS 1 2 INTERPOLATION$1
FUNCTION B(T') FILE asme-expansion-table.dat COLUMNS 1 3 INTERPOLATION $1 FUNCTION C(T') FILE asme-expansion-table.dat COLUMNS 1 4 INTERPOLATION$1

# remember that the thermal expansion coefficients have to be
#  1. the mean value between T0 and T
#  2. functions of space, so temperature has to be written as T(x,y,z)

# in the x direction, we use column B directly
alpha_x(x,y,z) = 1e-6*B(T(x,y,z))

# in the y direction, we convert column A to mean
alpha_y(x,y,z) = 1e-6*integral(A(T'), T', T0, T(x,y,z))/(T(x,y,z)-T0)

# in the z direction, we convert column C to mean
alpha_z(x,y,z) = 1e-3*C(T(x,y,z))/(T(x,y,z)-T0)

SOLVE_PROBLEM

WRITE_MESH cube-orthotropic-expansion-$1-$2.vtk T VECTOR u v w
PRINT %.3e "displacement in x at (1,1,1) = " u(1,1,1)
PRINT %.3e "displacement in y at (1,1,1) = " v(1,1,1)
PRINT %.3e "displacement in z at (1,1,1) = " w(1,1,1)
$gmsh -3 cube-hex.geo [...]$ gmsh -3 cube-tet.geo
[...]
$feenox cube-orthotropic-expansion.fee displacement in x at (1,1,1) = 4.451e-03 displacement in y at (1,1,1) = 4.449e-03 displacement in z at (1,1,1) = 4.437e-03$ feenox cube-orthotropic-expansion.fee linear tet
displacement in x at (1,1,1) =  4.451e-03
displacement in y at (1,1,1) =  4.447e-03
displacement in z at (1,1,1) =  4.438e-03
$feenox cube-orthotropic-expansion.fee akima hex displacement in x at (1,1,1) = 4.451e-03 displacement in y at (1,1,1) = 4.451e-03 displacement in z at (1,1,1) = 4.437e-03$ feenox cube-orthotropic-expansion.fee splines tet
displacement in x at (1,1,1) =  4.451e-03
displacement in y at (1,1,1) =  4.450e-03
displacement in z at (1,1,1) =  4.438e-03
$ Differences cannot be seen graphically, but they are there as the terminal mimic illustrates. Yet, they are not as large nor as sensible to meshing and interpolation settings as one would have expected after seeing the plots from the previous section. ## 4.19 Thermo-elastic expansion of finite cylinders Let us solve the following problem introduced by J. Veeder in his technical report AECL-2660 from 1967. Consider a finite solid cylinder (see insert) of radius b and length 2h, with the origin of coordinates at the centre. It may be shown that the temperature distribution in a cylindrical fuel pellet operating in a reactor is given approximately by T(r) = T_0 + T_1 \cdot \left[ 1 - \left(\frac{r}{b} \right)^2 \right] where T_0 is the pellet surface temperature and T_1 is the temperature difference between the centre and surface. The thermal expansion is thus seen to be the sum of two terms, the first of which produces uniform expansion (zero stress) at constant temperature T_0, and is therefore computationally trivial. Tho second term introduces non-uniform body forces which distort the pellet from its original cylindrical shape. The problem is axisymmetric on the azimutal angle and axially-symmetric along the mid-plane. The FeenoX input uses the tangential and radial boundary conditions applied to the base of the upper half of a 3D cylinder. The geometry is meshed using 27-noded hexahedra. Two one-dimensional profiles for the non-dimensional range [0:1] at the external surfaces are written into an ASCII file ready to be plotted: 1. axial dependency of the displacement v(z') = v(0,v,z'h) in the y direction at fixed x=0 and y=b, and 2. radial dependency of the displacement w(r') = w(0,r'b, h) in the z direction at fixed x=0 and z=h These two profiles are compared to the power expansion series given in the original report from 1967. Note that the authors expect a 5% difference between the reference solution and the real one. PROBLEM mechanical READ_MESH veeder.msh b = 1 # cylinder radius h = 0.5 # cylinder height E = 1 # young modulus (does not matter for the displacement, only for stresses) nu = 1/3 # poisson ratio alpha = 1e-5 # temperature expansion coefficient # temperature distribution as in the original paper T1 = 1 # maximum temperature T0 = 0 # reference temperature (where expansion is zero) T(x,y,z) := T0 + T1*(1-(x^2+y^2)/(b^2)) # boundary conditions (note that the cylinder can still expand on the x-y plane) BC inf tangential radial # solve! SOLVE_PROBLEM # write vtk output WRITE_MESH veeder.vtk T sigma dudx dudy dudz dvdx dvdy dvdz dwdx dwdy dwdz sigma1 sigma2 sigma3 VECTOR u v w # non-dimensional numerical displacement profiles v_num(z') = v(0, b, z'*h)/(alpha*T1*b) w_num(r') = w(0, r'*b, h)/(alpha*T1*b) ######## # reference solution # coefficients of displacement functions for h/b = 0.5 a00 = 0.66056 a01 = -0.44037 a10 = 0.23356 a02 = -0.06945 a11 = -0.10417 a20 = 0.00288 b00 = -0.01773 b01 = -0.46713 b10 = -0.04618 b02 = +0.10417 b11 = -0.01152 b20 = -0.00086 # coefficients of displacement functions for h/b = 1.0 # a00 = 0.73197 # a01 = -0.48798 # a10 = 0.45680 # a02 = -0.01140 # a11 = -0.06841 # a20 = 0.13611 # # b00 = 0.26941 # b01 = -0.45680 # b10 = -0.25670 # b02 = 0.03420 # b11 = -0.27222 # b20 = -0.08167 R(r') = r'^2 - 1 Z(z') = z'^2 - 1 v_ref(r',z') = r' * (a00 + a01*R(r') + a10*Z(z') + a02* R(r')^2 + a11 * R(r')*Z(z') + a20 * Z(z')^2) w_ref(r',z') = z' * (b00 + b01*R(r') + b10*Z(z') + b02* R(r')^2 + b11 * R(r')*Z(z') + b20 * Z(z')^2) PRINT_FUNCTION FILE veeder_v.dat v_num v_ref(1,z') MIN 0 MAX 1 NSTEPS 50 HEADER PRINT_FUNCTION FILE veeder_w.dat w_num w_ref(r',1) MIN 0 MAX 1 NSTEPS 50 HEADER $ gmsh -3 veeder.geo
[...]
$feenox veeder.fee$ pyxplot veeder.ppl
$ ## 4.20 Temperature-dependent material properties Let us solve a plane-strain square fixed on the left, with an horizontal traction on the right and free on the other two sides. The Young modulus depends on the temperature E(T) as given in the ASME II part D tables of material properties, interpolated using a monotonic cubic scheme. Actually, this example shows three cases: 1. Uniform temperature indentically equal to 200ºC 2. Linear temperature profile on the vertical direction given by an algebraic expression T(x,y) = 200 + 350\cdot y 3. The same linear profile but read from the output of a thermal conduction problem over a non-conformal mesh using this FeenoX input: PROBLEM thermal 2D READ_MESH square-centered-unstruct.msh # [-1:+1]x[-1:+1] BC bottom T=-150 BC top T=+550 k = 1 SOLVE_PROBLEM WRITE_MESH thermal-square-temperature.msh T Which of the three cases is executed is given by the first argument provided in the command line after the main input file. Depending on this argument, which is expanded as $1 in the main input file, either one of three secondary input files are included:

1. uniform

# uniform
T(x,y) := 200
2. linear

# algebraic expression
T(x,y) := 200 + 350*y
3. mesh

# read the temperature from a previous result
READ_MESH thermal-square-temperature.msh DIM 2 READ_FUNCTION T
# 2d plane strain mechanical problem over the [-1:+1]x[-1:+1] square
PROBLEM mechanical plane_strain

# fixed at left, uniform traction in the x direction at right
BC left    fixed
BC right   tx=50

# ASME II Part D pag. 785 Carbon steels with C<=0.30%
FUNCTION E_carbon(temp) INTERPOLATION steffen DATA {
-200  216
-125  212
-75   209
25    202
100   198
150   195
200   192
250   189
300   185
350   179
400   171
450   162
500   151
550   137
}

# read the temperature according to the run-time argument $1 INCLUDE mechanical-square-temperature-$1.fee

# Young modulus is the function above evaluated at the local temperature
E(x,y) := E_carbon(T(x,y))

# uniform Poisson's ratio
nu = 0.3

SOLVE_PROBLEM
PRINT u(1,1) v(1,1)
WRITE_MESH mechanical-square-temperature-$1.vtk E T VECTOR u v 0  $ gmsh -2 square-centered.geo
[...]
Info    : Done meshing 2D (Wall 0.00117144s, CPU 0.00373s)
Info    : 1089 nodes 1156 elements
Info    : Writing 'square-centered.msh'...
Info    : Done writing 'square-centered.msh'
Info    : Stopped on Thu Aug  4 09:40:09 2022 (From start: Wall 0.00818854s, CPU 0.031239s)
$feenox mechanical-square-temperature.fee uniform 0.465632 -0.105128$ feenox mechanical-square-temperature.fee linear
0.589859        -0.216061
$gmsh -2 square-centered-unstruct.geo [...] Info : Done meshing 2D (Wall 0.0274833s, CPU 0.061072s) Info : 65 nodes 132 elements Info : Writing 'square-centered-unstruct.msh'... Info : Done writing 'square-centered-unstruct.msh' Info : Stopped on Sun Aug 7 18:33:41 2022 (From start: Wall 0.0401667s, CPU 0.107659s)$ feenox thermal-square.fee
$feenox mechanical-square-temperature.fee mesh 0.589859 -0.216061$ 

To be done.

# 6 Description

FeenoX solves a problem defined in an plain-text input file and writes user-defined outputs to the standard output and/or files, either also plain-text or with a particular format for further post-processing. The syntax of this input file is designed to be as self-describing as possible, using English keywords that explains FeenoX what problem it has to solve in a way is understandable by both humans and computers. Keywords can work either as

1. Definitions, for instance ”define function f(x) and read its data from file f.dat”), or as
2. Instructions, such as “write the stress at point D into the standard output”.

A person can tell if a keyword is a definition or an instruction because the former are nouns (FUNCTION) and the latter verbs (PRINT). The equal sign = is a special keyword that is neither a verb nor a noun, and its meaning changes depending on what is on the left hand side of the assignment.

1. If there is a function, then it is a definition: define an algebraic function to be equal to the expression on the right-hand side, e.g.:

f(x,y) = exp(-x^2)*cos(pi*y)
2. If there is a variable, vector or matrix, it is an instruction: evaluate the expression on the right-hand side and assign it to the varible or vector (or matrix) element indicated in the left-hand side. Strictly speaking, if the variable has not already been defined (and implicit declaration is allowed), then the variable is also defined as well, e.g:

VAR a
VECTOR b[3]
a = sqrt(2)
b[i] = a*i^2

There is no need to explicitly define the scalar variable a with VAR since the first assigment also defines it implicitly (if this is allowed by the keyword IMPLICIT).

An input file can define its own variables as needed, such as my_var or flag. But there are some reserved names that are special in the sense that they either

1. can be set to modify the behavior of FeenoX, such as max_dt or dae_tol
2. can be read to get the internal status or results back from FeenoX, such as nodes or keff
3. can be either set or read, such as dt or done

The problem being solved can be static or transient, depending on whether the special variable end_time is zero (default) or not. If it is zero and static_steps is equal to one (default), the instructions in the input file are executed once and then FeenoX quits. For example

VAR x
PRINT %.7f func_min(cos(x)+1,x,0,6)

If static_steps is larger than one, the special variable step_static is increased and they are repeated the number of time indicated by static_steps:

static_steps = 10
f(n) = n^2 - n + 41
PRINT f(step_static^2-1)

If the special variable end_time is set to a non-zero value, after computing the static part a transient problem is solved. There are three kinds of transient problems:

1. Plain “standalone” transients
2. Differential-Algebraic equations (DAE) transients
3. Partial Differential equations (PDE) transients

In the first case, after all the instruction in the input file were executed, the special variable t is increased by the value of dt and then the instructions are executed all over again, until t reaches end_time:

end_time = 2*pi
dt = 1/10

y = lag(heaviside(t-1), 1)
z = random_gauss(0, sqrt(2)/10)

PRINT t sin(t) cos(t) y z HEADER

In the second case, the keyword PHASE_SPACE sets up DAE system. Then, one initial condition and one differential-algebraic equation has to be given for each element in the phase space. The instructions before the DAE block executed, then the DAE timestep is advanced and finally the instructions after DAE block are executed (there cannot be any instruction between the first and the last DAE):

PHASE_SPACE x
end_time = 1
x_0 = 1
x_dot = -x
PRINT t x exp(-t) HEADER

The timestep is chosen by the SUNDIALS library in order to keep an estimate of the residual error below dae_tol (default is 10^{-6}), although min_dt and max_dt can be used to control it. See the section of the [Differential-Algebraic Equations subsystem] for more information.

In the third cae, the type of PDE being solved is given by the keyword PROBLEM. Some types of PDEs do support transient problems (such as thermal) but some others do not (such as modal). See the detailed explanation of each problem type for details. Now the transient problem is handled by the TS framework of the PETSc library. In general transient PDEs involve a mesh, material properties, inital conditions, transient boundary conditions, etc. And they create a lot of data since results mean spatial and temporal distributions of one or more scalar fields:

# example of a 1D heat transient problem
# from https://www.mcs.anl.gov/petsc/petsc-current/src/ts/tutorials/ex3.c.html
# a non-dimensional slab 0 < x < 1 is kept at T(0) = T(1) = 0
# there is an initial non-trivial T(x)
# the steady-state is T(x) = 0
PROBLEM thermal 1d

end_time = 1e-1

# initial condition
T_0(x) := sin(6*pi*x) + 3*sin(2*pi*x)
# analytical solution
T_a(x,t) := exp(-36*pi^2*t)*sin(6*pi*x) + 3*exp(-4*pi^2*t)*sin(2*pi*x)

# unitary non-dimensional properties
k = 1
rho = 1
cp = 1

# boundary conditions
BC left  T=0
BC right T=0

SOLVE_PROBLEM

PRINT %e t dt T(0.1) T_a(0.1,t) T(0.7) T_a(0.7,t)
WRITE_MESH temp-slab.msh T

IF done
PRINT "\# open temp-anim-slab.geo in Gmsh to see the result!"
ENDIF

PETSc’s TS also honors the min_dt and max_dt variables, but the time step is controled by the allowed relative error with the special variable ts_rtol. Again, see the section of the [Partial Differential Equations subsystem] for more information.

## 6.1 Algebraic expressions

To be done.

• Everything is an expression.

# 7 Reference

This chapter contains a detailed reference of keywords, variables, functions and functionals available in FeenoX. These are used essentially to define the problem that FeenoX needs to solve and to define what the output should be. It should be noted that this chapter is to be used, indeed, as a reference and not as a tutorial.

## 7.1 Differential-Algebraic Equations subsystem

### 7.1.1 DAE keywords

#### 7.1.1.1INITIAL_CONDITIONS

Define how initial conditions of DAE problems are computed.

INITIAL_CONDITIONS { AS_PROVIDED | FROM_VARIABLES | FROM_DERIVATIVES }  

In DAE problems, initial conditions may be either:

• equal to the provided expressions (AS_PROVIDED)
• the derivatives computed from the provided phase-space variables (FROM_VARIABLES)
• the phase-space variables computed from the provided derivatives (FROM_DERIVATIVES)

In the first case, it is up to the user to fulfill the DAE system at t = 0. If the residuals are not small enough, a convergence error will occur. The FROM_VARIABLES option means calling IDA’s IDACalcIC routine with the parameter IDA_YA_YDP_INIT. The FROM_DERIVATIVES option means calling IDA’s IDACalcIC routine with the parameter IDA_Y_INIT. Wasora should be able to automatically detect which variables in phase-space are differential and which are purely algebraic. However, the [DIFFERENTIAL] keyword may be used to explicitly define them. See the (SUNDIALS documentation)[https://computation.llnl.gov/casc/sundials/documentation/ida_guide.pdf] for further information.

#### 7.1.1.2PHASE_SPACE

Asks FeenoX to solve a set of algebraic-differntial equations and define the variables, vectors and/or matrices that span the phase space.

PHASE_SPACE PHASE_SPACE <vars> ... <vectors> ... <matrices> ...   

#### 7.1.1.3TIME_PATH

Force time-dependent problems to pass through specific instants of time.

TIME_PATH <expr_1> [ <expr_2>  [ ... <expr_n> ] ]  

The time step dt will be reduced whenever the distance between the current time t and the next expression in the list is greater than dt so as to force t to coincide with the expressions given. The list of expresssions should evaluate to a sorted list of values for all times.

### 7.1.2 DAE variables

#### 7.1.2.1dae_rtol

Maximum allowed relative error for the solution of DAE systems.

Default value is is 1 \times 10^{-6}. If a fine per-variable error control is needed, special vector abs_error should be used.

## 7.2 Partial Differential Equations subsytem

### 7.2.1 PDE keywords

#### 7.2.1.1BC

Define a boundary condition to be applied to faces, edges and/or vertices.

BC <name> [ MESH <name> ] [ PHYSICAL_GROUP <name_1>  PHYSICAL_GROUP <name_2> ... ] [ <bc_data1> <bc_data2> ... ]  

If the name of the boundary condition matches a physical group in the mesh, it is automatically linked to that physical group. If there are many meshes, the mesh this keyword refers to has to be given with MESH. If the boundary condition applies to more than one physical group in the mesh, they can be added using as many PHYSICAL_GROUP keywords as needed. If at least one PHYSICAL_GROUP is given explicitly, then the BC name is not used to try to implicitly link it to a physical group in the mesh. Each <bc_data> argument is a single string whose meaning depends on the type of problem being solved. For instance T=150*sin(x/pi) prescribes the temperature to depend on space as the provided expression in a thermal problem and fixed fixes the displacements in all the directions in a mechanical or modal problem. See the particular section on boundary conditions for further details.

#### 7.2.1.2COMPUTE_REACTION

Compute the reaction (force, moment, power, etc.) at selected face, edge or vertex.

COMPUTE_REACTION <physical_group> [ MOMENT [ X0 <expr> ] [ Y0 <expr> ] [ Z0 <expr> ] ] RESULT { <variable> | <vector> }  

If the MOMENT keyword is not given, the zero-th order reaction is computed, i.e. force in elasticity and power in thermal. If the MOMENT keyword is given, then the coordinates of the center can be given with X0, Y0 and Z0. If they are not, the moment is computed about the barycenter of the physical group. The resulting reaction will be stored in the variable (thermal) or vector (elasticity) provided. If the variable or vector does not exist, it will be created.

#### 7.2.1.3DUMP

Dump raw PETSc objects used to solve PDEs into files.

DUMP [ FORMAT { binary | ascii | octave } ] [ K |   K_bc |   b |   b_bc |   M |   M_bc |  

#### 7.2.1.4FIND_EXTREMA

Find and/or compute the absolute extrema of a function or expression over a mesh (or a subset of it).

FIND_EXTREMA { <expression> | <function> } [ OVER <physical_group> ] [ MESH <mesh_identifier> ] [ NODES | CELLS | GAUSS ]
[ MIN <variable> ] [ MAX <variable> ] [ X_MIN <variable> ] [ X_MAX <variable> ] [ Y_MIN <variable> ] [ Y_MAX <variable> ] [ Z_MIN <variable> ] [ Z_MAX <variable> ] [ I_MIN <variable> ] [ I_MAX <variable> ]  

Either an expression or a function of space x, y and/or z should be given. By default the search is performed over the highest-dimensional elements of the mesh, i.e. over the whole volume, area or length for three, two and one-dimensional meshes, respectively. If the search is to be carried out over just a physical group, it has to be given in OVER. If there are more than one mesh defined, an explicit one has to be given with MESH. If neither NODES, CELLS or GAUSS is given then the search is performed over the three of them. With NODES only the function or expression is evalauted at the mesh nodes. With CELLS only the function or expression is evalauted at the element centers. With GAUSS only the function or expression is evalauted at the Gauss points. The value of the absolute minimum (maximum) is stored in the variable indicated by MIN (MAX). If the variable does not exist, it is created. The value of the x-y-z coordinate of the absolute minimum (maximum) is stored in the variable indicated by X_MIN-Y_MIN-Z_MIN (X_MAX-Y_MAX-Z_MAX). If the variable does not exist, it is created. The index (either node or cell) where the absolute minimum (maximum) is found is stored in the variable indicated by I_MIN (I_MAX).

#### 7.2.1.5INTEGRATE

Spatially integrate a function or expression over a mesh (or a subset of it).

INTEGRATE { <expression> | <function> } [ OVER <physical_group> ] [ MESH <mesh_identifier> ] [ NODES | CELLS ]
RESULT <variable>


Either an expression or a function of space x, y and/or z should be given. If the integrand is a function, do not include the arguments, i.e. instead of f(x,y,z) just write f. The results should be the same but efficiency will be different (faster for pure functions). By default the integration is performed over the highest-dimensional elements of the mesh, i.e. over the whole volume, area or length for three, two and one-dimensional meshes, respectively. If the integration is to be carried out over just a physical group, it has to be given in OVER. If there are more than one mesh defined, an explicit one has to be given with MESH. Either NODES or CELLS define how the integration is to be performed. With NODES the integration is performed using the Gauss points and weights associated to each element type. With CELLS the integral is computed as the sum of the product of the integrand at the center of each cell (element) and the cell’s volume. Do expect differences in the results and efficiency between these two approaches depending on the nature of the integrand. The scalar result of the integration is stored in the variable given by the mandatory keyword RESULT. If the variable does not exist, it is created.

#### 7.2.1.6LINEARIZE_STRESS

Compute linearized membrane and/or bending stresses according to ASME VIII Div 2 Sec 5.

LINEARIZE_STRESS  

#### 7.2.1.7MATERIAL

Define a material its and properties to be used in volumes.

MATERIAL <name> [ MESH <name> ] [ PHYSICAL_GROUP <name_1>  [ PHYSICAL_GROUP <name_2> [ ... ] ] ] [ <property_name_1>=<expr_1> [ <property_name_2>=<expr_2> [ ... ] ] ]  

If the name of the material matches a physical group in the mesh, it is automatically linked to that physical group. If there are many meshes, the mesh this keyword refers to has to be given with MESH. If the material applies to more than one physical group in the mesh, they can be added using as many PHYSICAL_GROUP keywords as needed. The names of the properties in principle can be arbitrary, but each problem type needs a minimum set of properties defined with particular names. For example, steady-state thermal problems need at least the conductivity which should be named k. If the problem is transient, it will also need heat capacity rhocp or diffusivity alpha. Mechanical problems need Young modulus E and Poisson’s ratio nu. Modal also needs density rho. Check the particular documentation for each problem type. Besides these mandatory properties, any other one can be defined. For instance, if one mandatory property dependend on the concentration of boron in the material, a new per-material property can be added named boron and then the function boron(x,y,z) can be used in the expression that defines the mandatory property.

#### 7.2.1.8PETSC_OPTIONS

Pass verbatim options to PETSc.

PETSC_OPTIONS  

Options for PETSc can be passed either in at run time in the command line (run with -h to see how) or they can be set in the input file with PETSC_OPTIONS. This is handy when a particular problem is best suited to be solved using a particular set of options which can the be embedded into the problem definition. @
The string is passed verbatim to PETSc as if the options were set in the command line. Note that in this case, the string is passed verbatim to PETSc. This means that they are non-POSIX options but they have to be in the native PETSc format. That is to say, while in the command line one would give --ksp_view, here one has to give -ksp_view. Conversely, instead of --mg_levels_pc_type=sor one has to give -mg_levels_pc_type sor.

#### 7.2.1.9PHYSICAL_GROUP

Explicitly defines a physical group of elements on a mesh.

PHYSICAL_GROUP <name> [ MESH <name> ] [ DIMENSION <expr> ] [ ID <expr> ]
[ MATERIAL <name> | | BC <name> [ BC ... ] ]


This keyword should seldom be needed. Most of the times, a combination of MATERIAL and BC ought to be enough for most purposes. The name of the PHYSICAL_GROUP keyword should match the name of the physical group defined within the input file. If there is no physical group with the provided name in the mesh, this instruction has no effect. If there are many meshes, an explicit mesh can be given with MESH. Otherwise, the physical group is defined on the main mesh. An explicit dimension of the physical group can be provided with DIMENSION. An explicit id can be given with ID. Both dimension and id should match the values in the mesh. For volumetric elements, physical groups can be linked to materials using MATERIAL. Note that if a material is created with the same name as a physical group in the mesh, they will be linked automatically, so there is no need to use PHYSCAL_GROUP for this. The MATERIAL keyword in PHYSICAL_GROUP is used to link a physical group in a mesh file and a material in the feenox input file with different names.

Likewise, for non-volumetric elements, physical groups can be linked to boundary using BC. As in the preceeding case, if a boundary condition is created with the same name as a physical group in the mesh, they will be linked automatically, so there is no need to use PHYSCAL_GROUP for this. The BC keyword in PHYSICAL_GROUP is used to link a physical group in a mesh file and a boundary condition in the feenox input file with different names. Note that while there can be only one MATERIAL associated to a physical group, there can be many BCs associated to a physical group.

#### 7.2.1.10PROBLEM

Ask FeenoX to solve a partial differential equation problem.

PROBLEM { laplace | mechanical | modal | neutron_diffusion | thermal }
[ 1D | 2D | 3D | DIM <expr> ] [ AXISYMMETRIC { x | y } ]
[ MESH <identifier> ] [ PROGRESS ] [ DETECT_HANGING_NODES | HANDLE_HANGING_NODES ]
[ TRANSIENT | QUASISTATIC ] [ LINEAR | NON_LINEAR ]
[ MODES <expr> ]
[ PRECONDITIONER { gamg | mumps | lu | hypre | sor | bjacobi | cholesky | ... } ]
[ LINEAR_SOLVER { gmres | mumps | bcgs | bicg | richardson | chebyshev | ... } ]
[ NONLINEAR_SOLVER { newtonls | newtontr | nrichardson | ngmres | qn | ngs | ... } ]
[ TRANSIENT_SOLVER { bdf | beuler | arkimex | rosw | glee | ... } ]
[ TIME_ADAPTATION { basic | none | dsp | cfl | glee | ... } ]
[ EIGEN_SOLVER { krylovschur | lanczos | arnoldi | power | gd | ... } ]
[ SPECTRAL_TRANSFORMATION { shift | sinvert | cayley | ... } ]
[ EIGEN_FORMULATION { omega | lambda } ]
[ DIRICHLET_SCALING { absolute <expr> | relative <expr> } ]


Currently, FeenoX can solve the following types of PDE-casted problems:

• laplace solves the Laplace (or Poisson) equation.
• modal computes the natural mechanical frequencies and oscillation modes.
• neutron_diffusion multi-group core-level neutron diffusion with a FEM formulation
• thermal solves the heat conduction problem.

If you are a programmer and want to contribute with another problem type, please do so! Check out the programming guide in the FeenoX repository.

The number of spatial dimensions of the problem needs to be given either as 1d, 2d, 3d or after the keyword DIM. Alternatively, one can define a MESH with an explicit DIMENSIONS keyword before PROBLEM. If the AXISYMMETRIC keyword is given, the mesh is expected to be two-dimensional in the x-y plane and the problem is assumed to be axi-symmetric around the given axis. If there are more than one MESHes defined, the one over which the problem is to be solved can be defined by giving the explicit mesh name with MESH. By default, the first mesh to be defined in the input file with READ_MESH (which can be defined after the PROBLEM keyword) is the one over which the problem is solved. If the keyword PROGRESS is given, three ASCII lines will show in the terminal the progress of the ensamble of the stiffness matrix (or matrices), the solution of the system of equations and the computation of gradients (stresses, heat fluxes, etc.), if applicable. If either DETECT_HANGING_NODES or HANDLE_HANGING_NODES are given, an intermediate check for nodes without any associated elements will be performed. For well-behaved meshes this check is redundant so by detault it is not done. With DETECT_HANGING_NODES, FeenoX will report the tag of the hanging nodes and stop. With HANDLE_HANGING_NODES, FeenoX will fix those nodes and try to solve the problem anyway. If the special variable end_time is zero, FeenoX solves a static problem—although the variable static_steps is still honored. If end_time is non-zero, FeenoX solves a transient or quasistatic problem. This can be controlled by TRANSIENT or QUASISTATIC. By default FeenoX tries to detect wheter the computation should be linear or non-linear. An explicit mode can be set with either LINEAR on NON_LINEAR. The number of modes to be computed when solving eigenvalue problems is given by MODES. The default value is problem dependent. The preconditioner (PC), linear (KSP), non-linear (SNES) and time-stepper (TS) solver types be any of those available in PETSc (first option is the default):

If the EIGEN_FORMULATION is omega then K \phi = \omega^2 M \phi is solved, and M \phi = \lambda K \phi if it is lambda. The DIRICHLET_SCALING keyword controls the way Dirichlet boundary conditions are scaled when computing the residual. Roughly, it defines how to compute the parameter \alpha.2 If absolute, then \alpha is equal to the given expression. If relative, then \alpha is equal to the given fraction of the average diagonal entries in the stiffness matrix. Default is \alpha = 1.

#### 7.2.1.11READ_MESH

Read an unstructured mesh and (optionally) functions of space-time from a file.

READ_MESH { <file_path> | <file_id> } [ DIM <num_expr> ]
[ SCALE <expr> ] [ OFFSET <expr_x> <expr_y> <expr_z> ]
[ INTEGRATION { full | reduced } ]
[ MAIN ] [ UPDATE_EACH_STEP ]


Either a file identifier (defined previously with a FILE keyword) or a file path should be given. The format is read from the extension, which should be either

Note than only MSH is suitable for defining PDE domains, as it is the only one that provides physical groups (a.k.a labels) which are needed in order to define materials and boundary conditions. The other formats are primarily supported to read function data contained in the file and eventually, to operate over these functions (i.e. take differences with other functions contained in other files to compare results). The file path or file id can be used to refer to a particular mesh when reading more than one, for instance in a WRITE_MESH or INTEGRATE keyword. If a file path is given such as cool_mesh.msh, it can be later referred to as either cool_mesh.msh or just cool_mesh.
The spatial dimensions can be given with DIM. If material properties are uniform and given with variables, the number of dimensions are not needed and will be read from the file at runtime. But if either properties are given by spatial functions or if functions are to be read from the mesh with READ_DATA or READ_FUNCTION, then the number of dimensions ought to be given explicitly because FeenoX needs to know how many arguments these functions take. If either OFFSET and/or SCALE are given, the node locations are first shifted and then scaled by the provided values. When defining several meshes and solving a PDE problem, the mesh used as the PDE domain is the one marked with MAIN. If none of the meshes is explicitly marked as main, the first one is used. If UPDATE_EACH_STEP is given, then the mesh data is re-read from the file at each time step. Default is to read the mesh once, except if the file path changes with time. For each READ_FIELD keyword, a point-wise defined scalar function of space named <function_name> is defined and filled with the scalar data named <name_in_mesh> contained in the mesh file. The READ_FUNCTION keyword is a shortcut when the scalar name and the to-be-defined function are the same. If no mesh is marked as MAIN, the first one is the main one.

#### 7.2.1.12SOLVE_PROBLEM

Explicitly solve the PDE problem.

SOLVE_PROBLEM  

Whenever the instruction SOLVE_PROBLEM is executed, FeenoX solves the PDE problem. For static problems, that means solving the equations and filling in the result functions. For transient or quasisstatic problems, that means advancing one time step.

#### 7.2.1.13WRITE_MESH

Write a mesh and functions of space-time to a file for post-processing.

WRITE_MESH <file> [ MESH <mesh_identifier> ] [ NO_MESH ] [ FILE_FORMAT { gmsh | vtk } ]
[ NO_PHYSICAL_NAMES ]  [ NODE | CELL ] [ SCALAR_FORMAT <printf_specification> ]
[ <scalar_field_1> ] [ <scalar_field_2> ] [...]
[ VECTOR <field_x> <field_y> <field_z> ] [...]
[ SYMMETRIC_TENSOR <field_xx> <field_yy> <field_zz> <field_xy> <field_yz> <field_zx> ] [...]


The format is automatically detected from the extension, which should be either msh (version 2.2 ASCII) or vtk (legacy ASCII). Otherwise, the keyword FILE_FORMAT has to be given to set the format explicitly. If there are several meshes defined by READ_MESH, the mesh used to write the data has be given explicitly with MESH. If the NO_MESH keyword is given, only the results are written into the output file without any mesh data. Depending on the output format, this can be used to avoid repeating data and/or creating partial output files which can the be latter assembled by post-processing scripts. When targetting the .msh output format, if NO_PHYSICAL_NAMES is given then the section that sets the actual names of the physical entities is not written.
This might be needed in some cases to avoid name clashes when dealing with multiple .msh files. The output is node-based by default. This can be controlled with both the NODE and CELL keywords. All fields that come after a NODE (CELL) keyword will be written at the node (cells). These keywords can be used several times and mixed with fields. For example CELL k(x,y,z) NODE T sqrt(x^2+y^2) CELL 1+z will write the conductivity and the expression 1+z as cell-based and the temperature T(x,y,z) and the expression \sqrt{x^2+y^2} as a node-based fields. The SCALAR_FORMAT keyword can be used to define the precision of the ASCII representation of the fields that follow. Default is %g. The data to be written has to be given as a list of fields, i.e. distributions (such as k or E), functions of space (such as T) and/or expressions (such as T(x,y,z)*sqrt(x^2+y^2+z^2)). Each field is written as a scalar, unless either the keywords VECTOR or SYMMETRIC_TENSOR are given. In the first case, the next three fields following the VECTOR keyword are taken as the vector elements. In the latter, the next six fields following the SYMMETRIC_TENSOR keyword are taken as the tensor elements.

## 7.3 Laplace’s equation

Set PROBLEM to laplace to solve Laplace’s equation

\nabla^2 \phi = 0

If end_time is set, then the transient problem is solved

\alpha(\vec{x}) \frac{\partial \phi}{\partial t} + \nabla^2 \phi = 0

### 7.3.1 Laplace results

#### 7.3.1.1phi

The scalar field \phi(\vec{x}) whose Laplacian is equal to zero or to f(\vec{x}).

### 7.3.2 Laplace properties

#### 7.3.2.1alpha

The coefficient of the temporal derivative for the transient equation  \alpha \frac{\partial \phi}{\partial t} + \nabla^2 \phi=f(\vec{x}). If not given, default is one.

#### 7.3.2.2f

The right hand side of the equation \nabla^2 \phi=f(\vec{x}). If not given, default is zero (i.e. Laplace).

### 7.3.3 Laplace boundary conditions

#### 7.3.3.1dphidn

Alias for phi'.

dphidn=<expr>  

#### 7.3.3.2phi

Dirichlet essential boundary condition in which the value of \phi is prescribed.

phi=<expr>  

#### 7.3.3.3phi'

Neumann natural boundary condition in which the value of the normal outward derivative \frac{\partial \phi}{\partial n} is prescribed.

phi'=<expr>  

## 7.4 The heat conduction equation

Set PROBLEM to thermal to solve thermal conduction:

\rho \c_p \frac{\partial T}{\partial t} + \text{div} \left[ k(\vec{x, T} \cdot \text{grad}{T} \right] = q'''(\vec{x}, T)

If end_time is zero, only the steady-state problem is solved. If k, q''' or any Neumann boundary condition depends on T, the problem is set to non-linear automatically.

### 7.4.1 Thermal results

#### 7.4.1.1qx

The heat flux field q_x(\vec{x}) = -k(\vec{x}) \cdot \frac{\partial T}{\partial x} in the x direction. This is a secondary unknown of the problem.

#### 7.4.1.2qy

The heat flux field q_y(\vec{x}) = -k(\vec{x}) \cdot \frac{\partial T}{\partial y} in the x direction. This is a secondary unknown of the problem. Only available for two and three-dimensional problems.

#### 7.4.1.3qz

The heat flux field q_z(\vec{x}) = -k(\vec{x}) \cdot \frac{\partial T}{\partial z} in the x direction. This is a secondary unknown of the problem.
Only available for three-dimensional problems.

#### 7.4.1.4T

The temperature field T(\vec{x}). This is the primary unknown of the problem.

### 7.4.2 Thermal properties

#### 7.4.2.1cp

Specific heat in units of energy per unit of mass per degree of temperature. Either kappa, rhocp or both rho and cp are needed for transient

cp  

#### 7.4.2.2k

The thermal conductivity in units of power per length per degree of temperature. This property is mandatory.

k  

#### 7.4.2.3kappa

Thermal diffusivity in units of area per unit of time. Equal to the thermal conductivity k divided by the density rho and specific heat capacity cp. Either kappa, rhocp or both rho and cp are needed for transient

kappa  

#### 7.4.2.4q

Alias for q'''

q  

#### 7.4.2.5q'''

The volumetric power dissipated in the material in units of power per unit of volume. Default is zero (i.e. no power).

q'''  

#### 7.4.2.6rho

Density in units of mass per unit of volume. Either kappa, rhocp or both rho and cp are needed for transient

rho  

#### 7.4.2.7rhocp

Product of the density rho times the specific heat capacity cp, in units of energy per unit of volume per degree of temperature. Either kappa, rhocp or both rho and cp are needed for transient

rhocp  

#### 7.4.2.8T_0

The initial condition for the temperature in transient problems. If not given, a steady-steady computation at t=0 is performed.
The initial guess for the temperature in steady-state problems. If not given, a uniform distribution equal to the the average of all the temperature appearing in boundary conditions is used.

### 7.4.5 Thermal variables

#### 7.4.5.1T_max

The maximum temperature T_\text{max}.

#### 7.4.5.2T_min

The minimum temperature T_\text{min}.

## 7.5 General & “standalone” mathematics

### 7.5.1 Keywords

#### 7.5.1.1ABORT

Catastrophically abort the execution and quit FeenoX.

ABORT  

Whenever the instruction ABORT is executed, FeenoX quits with a non-zero error leve. It does not close files nor unlock shared memory objects. The objective of this instruction is to either debug complex input files by using only parts of them or to conditionally abort the execution using IF clauses.

#### 7.5.1.2ALIAS

Define a scalar alias of an already-defined indentifier.

ALIAS { <new_var_name> IS <existing_object> | <existing_object> AS <new_name> }  

The existing object can be a variable, a vector element or a matrix element. In the first case, the name of the variable should be given as the existing object. In the second case, to alias the second element of vector v to the new name new, v(2) should be given as the existing object. In the third case, to alias second element (2,3) of matrix M to the new name new, M(2,3) should be given as the existing object.

#### 7.5.1.3CLOSE

Explicitly close a file after input/output.

CLOSE <name>  

The given <name> can be either a fixed-string path or an already-defined FILE.

#### 7.5.1.4DEFAULT_ARGUMENT_VALUE

Give a default value for an optional commandline argument.

DEFAULT_ARGUMENT_VALUE <constant> <string>  

If a $n construction is found in the input file but the commandline argument was not given, the default behavior is to fail complaining that an extra argument has to be given in the commandline. With this keyword, a default value can be assigned if no argument is given, thus avoiding the failure and making the argument optional. The <constant> should be 1, 2, 3, etc. and <string> will be expanded character-by-character where the $n construction is. Whether the resulting expression is to be interpreted as a string or as a numerical expression will depend on the context.

#### 7.5.1.5FILE

Define a file with a particularly formatted name to be used either as input or as output.

< FILE | OUTPUT_FILE | INPUT_FILE > <name> PATH <format> expr_1 expr_2 ... expr_n [ INPUT | OUTPUT | MODE <fopen_mode> ]  

For reading or writing into files with a fixed path, this instruction is usually not needed as the FILE keyword of other instructions (such as PRINT or MESH) can take a fixed-string path as an argument. However, if the file name changes as the execution progresses (say because one file for each step is needed), then an explicit FILE needs to be defined with this keyword and later referenced by the given name.
The path should be given as a printf-like format string followed by the expressions which shuold be evaluated in order to obtain the actual file path. The expressions will always be floating-point expressions, but the particular integer specifier %d is allowed and internally transformed to %.0f. The file can be explicitly defined and INPUT, OUTPUT or a certain fopen() mode can be given (i.e. “a”). If not explicitly given, the nature of the file will be taken from context, i.e. FILEs in PRINT will be OUTPUT and FILEs in FUNCTION will be INPUT. This keyword justs defines the FILE, it does not open it. The file will be actually openened (and eventually closed) automatically. In the rare case where the automated opening and closing does not fit the expected workflow, the file can be explicitly opened or closed with the instructions FILE_OPEN and FILE_CLOSE.

#### 7.5.1.6FIT

Find parameters to fit an analytical function to a pointwise-defined function.

FIT <function_to_be_fitted>  TO <function_with_data> VIA <var_1> <var_2> ... <var_n>
[ GRADIENT <expr_1> <expr_2> ... <expr_n> ]
[ RANGE_MIN <expr_1> <expr_2> ... <expr_j> ]
[ RANGE_MAX <expr_1> <expr_2> ... <expr_n> ]
[ TOL_REL <expr> ] [ TOL_ABS <expr> ] [ MAX_ITER <expr> ]
[ VERBOSE ]  

The function with the data has to be point-wise defined (i.e. a FUNCTION read from a file, with inline DATA or defined over a mesh). The function to be fitted has to be parametrized with at least one of the variables provided after the USING keyword. For example to fit f(x,y)=a x^2 + b sqrt(y) to a pointwise-defined function g(x,y) one gives FIT f TO g VIA a b. Only the names of the functions have to be given, not the arguments. Both functions have to have the same number of arguments. The initial guess of the solution is given by the initial value of the variables after the VIA keyword. Analytical expressions for the gradient of the function to be fitted with respect to the parameters to be fitted can be optionally given with the GRADIENT keyword. If none is provided, the gradient will be computed numerically using finite differences. A range over which the residuals are to be minimized can be given with RANGE_MIN and RANGE_MAX. The expressions give the range of the arguments of the functions, not of the parameters. For multidimensional fits, the range is an hypercube. If no range is given, all the definition points of the function with the data are used for the fit. Convergence can be controlled by giving the relative and absolute tolreances with TOL_REL (default DEFAULT_NLIN_FIT_EPSREL) and TOL_ABS (default DEFAULT_NLIN_FIT_EPSABS), and with the maximum number of iterations MAX_ITER (default DEFAULT_NLIN_FIT_MAX_ITER). If the optional keyword VERBOSE is given, some data of the intermediate steps is written in the standard output.

#### 7.5.1.7FUNCTION

Define a scalar function of one or more variables.

FUNCTION <function_name>(<var_1>[,var2,...,var_n]) {
= <expr> |
FILE { <file> } |
VECTORS <vector_1> <vector_2> ... <vector_n> <vector_data> |
MESH <mesh> |
DATA <num_1> <num_2> ... <num_N>
}
[ COLUMNS <expr_1> <expr_2> ... <expr_n> <expr_n+1> ]
[ INTERPOLATION { linear | polynomial | spline | spline_periodic | akima | akima_periodic | steffen |
nearest | shepard | shepard_kd | bilinear } ]
[ INTERPOLATION_THRESHOLD <expr> ] [ SHEPARD_RADIUS <expr> ] [ SHEPARD_EXPONENT <expr> ]  

The number of variables n is given by the number of arguments given between parenthesis after the function name. The arguments are defined as new variables if they had not been already defined explictly as scalar variables. If the function is given as an algebraic expression, the short-hand operator = (or := for compatiblity with Maxima) can be used. That is to say, FUNCTION f(x) = x^2 is equivalent to f(x) = x^2 (or f(x) := x^2). If a FILE is given, an ASCII file containing at least n+1 columns is expected. By default, the first n columns are the values of the arguments and the last column is the value of the function at those points. The order of the columns can be changed with the keyword COLUMNS, which expects n+1 expressions corresponding to the column numbers. If VECTORS is given, a set of n+1 vectors of the same size is expected. The first n correspond to the arguments and the last one to the function values. If MESH is given, the function is point-wise defined over the mesh topology. That is to say, the independent variables (i.e. the spatial coordinates) coincide with the mesh nodes. The dependent variable (i.e. the function value) is set by “filling” a vector named vec_f (where f has to be replaced with the function name) of size equal to the number of nodes.
The function can be pointwise-defined inline in the input using DATA. This should be the last keyword of the line, followed by N=k \cdot (n+1) expresions giving k definition points: n arguments and the value of the function. Multiline continuation using brackets { and } can be used for a clean data organization. Interpolation schemes can be given for either one or multi-dimensional functions with INTERPOLATION. Available schemes for n=1 are:

• linear
• polynomial, the grade is equal to the number of data minus one
• spline, cubic (needs at least 3 points)
• spline_periodic
• akima (needs at least 5 points)
• akima_periodic (needs at least 5 points)
• steffen, always-monotonic splines-like interpolator

Default interpolation scheme for one-dimensional functions is DEFAULT_INTERPOLATION.

Available schemes for n>1 are:

For n>1, if the euclidean distance between the arguments and the definition points is smaller than INTERPOLATION_THRESHOLD, the definition point is returned and no interpolation is performed. Default value is square root of DEFAULT_MULTIDIM_INTERPOLATION_THRESHOLD.

The initial radius of points to take into account in shepard_kd is given by SHEPARD_RADIUS. If no points are found, the radius is double until at least one definition point is found. The radius is doubled until at least one point is found. Default is DEFAULT_SHEPARD_RADIUS. The exponent of the shepard method is given by SHEPARD_EXPONENT. Default is DEFAULT_SHEPARD_EXPONENT.

#### 7.5.1.8IF

Execute a set of instructions if a condition is met.

IF expr
<block_of_instructions_if_expr_is_true>
[ ELSE
<block_of_instructions_if_expr_is_false> ]
ENDIF  

#### 7.5.1.9IMPLICIT

Define whether implicit definition of variables is allowed or not.

IMPLICIT { NONE | ALLOWED }  

By default, FeenoX allows variables (but not vectors nor matrices) to be implicitly declared. To avoid introducing errors due to typos, explicit declaration of variables can be forced by giving IMPLICIT NONE. Whether implicit declaration is allowed or explicit declaration is required depends on the last IMPLICIT keyword given, which by default is ALLOWED.

#### 7.5.1.10INCLUDE

Include another FeenoX input file.

INCLUDE <file_path> [ FROM <num_expr> ] [ TO <num_expr> ]  

Includes the input file located in the string file_path at the current location. The effect is the same as copying and pasting the contents of the included file at the location of the INCLUDE keyword. The path can be relative or absolute. Note, however, that when including files inside IF blocks that instructions are conditionally-executed but all definitions (such as function definitions) are processed at parse-time independently from the evaluation of the conditional. The included file has to be an actual file path (i.e. it cannot be a FeenoX FILE) because it needs to be resolved at parse time. Yet, the name can contain a commandline replacement argument such as $1 so INCLUDE$1.fee will include the file specified after the main input file in the command line. The optional FROM and TO keywords can be used to include only portions of a file.

#### 7.5.1.11MATRIX

Define a matrix.

MATRIX <name> ROWS <expr> COLS <expr> [ DATA <expr_1> <expr_2> ... <expr_n> |  

A new matrix of the prescribed size is defined. The number of rows and columns can be an expression which will be evaluated the very first time the matrix is used and then kept at those constant values. All elements will be initialized to zero unless DATA is given (which should be the last keyword of the line), in which case the expressions will be evaluated the very first time the matrix is used and row-major-assigned to each of the elements. If there are less elements than the matrix size, the remaining values will be zero. If there are more elements than the matrix size, the values will be ignored.

#### 7.5.1.12OPEN

Explicitly open a file for input/output.

OPEN <name> [ MODE <fopen_mode> ]  

The given <name> can be either a fixed-string path or an already-defined FILE. The mode is only taken into account if the file is not already defined. Default is write w.

#### 7.5.1.13PRINT

Write plain-text and/or formatted data to the standard output or into an output file.

PRINT [ <object_1> <object_2> ... <object_n> ] [ TEXT <string_1> ... TEXT <string_n> ]
[ FILE { <file_path> | <file_id> } ] [ HEADER ] [ NONEWLINE ] [ SEP <string> ]
[ SKIP_STEP <expr> ] [ SKIP_STATIC_STEP <expr> ] [ SKIP_TIME <expr> ] [ SKIP_HEADER_STEP <expr> ]


Each argument object which is not a keyword of the PRINT instruction will be part of the output. Objects can be either a matrix, a vector or any valid scalar algebraic expression. If the given object cannot be solved into a valid matrix, vector or expression, it is treated as a string literal if IMPLICIT is ALLOWED, otherwise a parser error is raised. To explicitly interpret an object as a literal string even if it resolves to a valid numerical expression, it should be prefixed with the TEXT keyword such as PRINT TEXT 1+1 that would print 1+1 instead of 2. Objects and string literals can be mixed and given in any order. Hashes # appearing literal in text strings have to be quoted to prevent the parser to treat them as comments within the FeenoX input file and thus ignoring the rest of the line, like PRINT "\# this is a printed comment". Whenever an argument starts with a porcentage sign %, it is treated as a C printf-compatible format specifier and all the objects that follow it are printed using the given format until a new format definition is found. The objects are treated as double-precision floating point numbers, so only floating point formats should be given. See the printf(3) man page for further details. The default format is DEFAULT_PRINT_FORMAT. Matrices, vectors, scalar expressions, format modifiers and string literals can be given in any desired order, and are processed from left to right. Vectors are printed element-by-element in a single row. See PRINT_VECTOR to print one or more vectors with one element per line (i.e. vertically). Matrices are printed element-by-element in a single line using row-major ordering if mixed with other objects but in the natural row and column fashion if it is the only given object in the PRINT instruction. If the FILE keyword is not provided, default is to write to stdout. If the HEADER keyword is given, a single line containing the literal text given for each object is printed at the very first time the PRINT instruction is processed, starting with a hash # character.
If the NONEWLINE keyword is not provided, default is to write a newline \n character after all the objects are processed. Otherwise, if the last token to be printed is a numerical value, a separator string will be printed but not the newline \n character. If the last token is a string, neither the separator nor the newline will be printed. The SEP keyword expects a string used to separate printed objects. To print objects without any separation in between give an empty string like SEP "". The default is a tabulator character ‘DEFAULT_PRINT_SEPARATOR’ character. To print an empty line write PRINT without arguments. By default the PRINT instruction is evaluated every step. If the SKIP_STEP (SKIP_STATIC_STEP) keyword is given, the instruction is processed only every the number of transient (static) steps that results in evaluating the expression, which may not be constant. The SKIP_HEADER_STEP keyword works similarly for the optional HEADER but by default it is only printed once. The SKIP_TIME keyword use time advancements to choose how to skip printing and may be useful for non-constant time-step problems.

Print one or more functions as a table of values of dependent and independent variables.

PRINT_FUNCTION <function_1> [ { function | expr } ... { function | expr } ]
[ FILE { <file_path> | <file_id> } ] [ HEADER ]
[ MIN <expr_1> <expr_2> ... <expr_k> ] [ MAX <expr_1> <expr_2> ... <expr_k> ]
[ STEP <expr_1> <expr_2> ... <expr_k> ] [ NSTEPs <expr_1> <expr_2> ... <expr_k> ]
[ FORMAT <print_format> ] <vector_1> [ { vector | expr } ... { vector | expr } ]  

Each argument should be either a function or an expression. The output of this instruction consists of n+k columns, where n is the number of arguments of the first function of the list and k is the number of functions and expressions given. The first n columns are the arguments (independent variables) and the last k one has the evaluated functions and expressions. The columns are separated by a tabulator, which is the format that most plotting tools understand. Only function names without arguments are expected. All functions should have the same number of arguments. Expressions can involve the arguments of the first function. If the FILE keyword is not provided, default is to write to stdout. If HEADER is given, the output is prepended with a single line containing the names of the arguments and the names of the functions, separated by tabs. The header starts with a hash # that usually acts as a comment and is ignored by most plotting tools. If there is no explicit range where to evaluate the functions and the first function is point-wise defined, they are evalauted at the points of definition of the first one. The range can be explicitly given as a product of n ranges [x_{i,\min},x_{i,\max}] for i=1,\dots,n.
The values x_{i,\min} and x_{i,\max} are given with the MIN and MAX keywords. The discretization steps of the ranges are given by either STEP that gives \delta x or NSTEPS that gives the number of steps. If the first function is not point-wise defined, the ranges are mandatory.

Print the elements of one or more vectors, one element per line.

PRINT_VECTOR
[ FILE { <file_path> | <file_id> } ] [ HEADER ]
[ SEP <string> ]  

Each argument should be either a vector or an expression of the integer i. If the FILE keyword is not provided, default is to write to stdout. If HEADER is given, the output is prepended with a single line containing the names of the arguments and the names of the functions, separated by tabs. The header starts with a hash # that usually acts as a comment and is ignored by most plotting tools. The SEP keyword expects a string used to separate printed objects. To print objects without any separation in between give an empty string like SEP "". The default is a tabulator character ‘DEFAULT_PRINT_SEPARATOR’ character.

#### 7.5.1.16SOLVE

Solve a (small) system of non-linear equations.

SOLVE FOR <n> UNKNOWNS <var_1> <var_2> ... <var_n> [ METHOD { dnewton | hybrid | hybrids | broyden } ]
[ EPSABS <expr> ] [ EPSREL <expr> ] [ MAX_ITER <expr> ]  

#### 7.5.1.17SORT_VECTOR

Sort the elements of a vector, optionally making the same rearrangement in another vector.

SORT_VECTOR <vector> [ ASCENDING | DESCENDING ] [ <other_vector> ]  

This instruction sorts the elements of <vector> into either ascending or descending numerical order. If <other_vector> is given, the same rearrangement is made on it. Default is ascending order.

#### 7.5.1.18VAR

Explicitly define one or more scalar variables.

VAR <name_1> [ <name_2> ] ... [ <name_n> ]  

When implicit definition is allowed (see IMPLICIT), scalar variables need not to be defined before being used if from the context FeenoX can tell that an scalar variable is needed. For instance, when defining a function like f(x) = x^2 it is not needed to declare x explictly as a scalar variable. But if one wants to define a function like g(x) = integral(f(x'), x', 0, x) then the variable x' needs to be explicitly defined as VAR x' before the integral.

#### 7.5.1.19VECTOR

Define a vector.

VECTOR <name> SIZE <expr> [ FUNCTION_DATA <function> ] [ DATA <expr_1> <expr_2> ... <expr_n> |  

A new vector of the prescribed size is defined. The size can be an expression which will be evaluated the very first time the vector is used and then kept at that constant value. If the keyword FUNCTION_DATA is given, the elements of the vector will be synchronized with the inpedendent values of the function, which should be point-wise defined. The sizes of both the function and the vector should match. All elements will be initialized to zero unless DATA is given (which should be the last keyword of the line), in which case the expressions will be evaluated the very first time the vector is used and assigned to each of the elements. If there are less elements than the vector size, the remaining values will be zero. If there are more elements than the vector size, the values will be ignored.

### 7.5.2 Variables

#### 7.5.2.1done

Flag that indicates whether the overall calculation is over.

This variable is set to true by FeenoX when the computation finished so it can be checked in an IF block to do something only in the last step. But this variable can also be set to true from the input file, indicating that the current step should also be the last one. For example, one can set end_time = infinite and then finish the computation at t=10 by setting done = t > 10. This done variable can also come from (and sent to) other sources, like a shared memory object for coupled calculations.

#### 7.5.2.2done_static

Flag that indicates whether the static calculation is over or not.

It is set to true (i.e. \neq 0) by feenox if step_static \ge static_steps. If the user sets it to true, the current step is marked as the last static step and the static calculation ends after finishing the step. It can be used in IF blocks to check if the static step is finished or not.

#### 7.5.2.3done_transient

Flag that indicates whether the transient calculation is over or not.

It is set to true (i.e. \neq 0) by feenox if t \ge end_time. If the user sets it to true, the current step is marked as the last transient step and the transient calculation ends after finishing the step. It can be used in IF blocks to check if the transient steps are finished or not.

#### 7.5.2.4dt

Actual value of the time step for transient calculations.

When solving DAE systems, this variable is set by feenox. It can be written by the user for example by importing it from another transient code by means of shared-memory objects. Care should be taken when solving DAE systems and overwriting t. Default value is DEFAULT_DT, which is a power of two and roundoff errors are thus reduced.

#### 7.5.2.5end_time

Final time of the transient calculation, to be set by the user.

The default value is zero, meaning no transient calculation.

#### 7.5.2.6i

Dummy index, used mainly in vector and matrix row subindex expressions.

#### 7.5.2.7infinite

A very big positive number.

It can be used as end_time = infinite or to define improper integrals with infinite limits. Default is 2^{50} \approx 1 \times 10^{15}.

#### 7.5.2.8in_static

Flag that indicates if FeenoX is solving the iterative static calculation.

This is a read-only variable that is non zero if the static calculation.

#### 7.5.2.9in_static_first

Flag that indicates if feenox is in the first step of the iterative static calculation.

#### 7.5.2.10in_static_last

Flag that indicates if feenox is in the last step of the iterative static calculation.

#### 7.5.2.11in_transient

Flag that indicates if feenox is solving transient calculation.

#### 7.5.2.12in_transient_first

Flag that indicates if feenox is in the first step of the transient calculation.

#### 7.5.2.13in_transient_last

Flag that indicates if feenox is in the last step of the transient calculation.

#### 7.5.2.14j

Dummy index, used mainly in matrix column subindex expressions.

#### 7.5.2.15max_dt

Maximum bound for the time step that feenox should take when solving DAE systems.

#### 7.5.2.16min_dt

Minimum bound for the time step that feenox should take when solving DAE systems.

#### 7.5.2.17ncores

The number of online available cores, as returned by sysconf(_SC_NPROCESSORS_ONLN).

This value can be used in the MAX_DAUGHTERS expression of the PARAMETRIC keyword (i.e ncores/2).

#### 7.5.2.18on_gsl_error

This should be set to a mask that indicates how to proceed if an error ir raised in any routine of the GNU Scientific Library.

#### 7.5.2.19on_ida_error

This should be set to a mask that indicates how to proceed if an error ir raised in any routine of the SUNDIALS Library.

#### 7.5.2.20on_nan

This should be set to a mask that indicates how to proceed if Not-A-Number signal (such as a division by zero) is generated when evaluating any expression within feenox.

#### 7.5.2.21pi

A double-precision floating point representaion of the number \pi

It is equal to the M_PI constant in math.h .

#### 7.5.2.22pid

The UNIX process id of the FeenoX instance.

#### 7.5.2.23static_steps

Number of steps that ought to be taken during the static calculation, to be set by the user.

The default value is one, meaning only one static step.

#### 7.5.2.24step_static

Indicates the current step number of the iterative static calculation.

This is a read-only variable that contains the current step of the static calculation.

#### 7.5.2.25step_transient

Indicates the current step number of the transient static calculation.

This is a read-only variable that contains the current step of the transient calculation.

#### 7.5.2.26t

Actual value of the time for transient calculations.

This variable is set by FeenoX, but can be written by the user for example by importing it from another transient code by means of shared-memory objects. Care should be taken when solving DAE systems and overwriting t.

#### 7.5.2.27zero

A very small positive number.

It is taken to avoid roundoff errors when comparing floating point numbers such as replacing a \leq a_\text{max} with a < a_\text{max} + zero. Default is (1/2)^{-50} \approx 9\times 10^{-16} .

## 7.6 Functions

### 7.6.1abs

Returns the absolute value of the argument x.

abs(x)  

|x|

### 7.6.2acos

Computes the arc in radians whose cosine is equal to the argument x. A NaN error is raised if |x|>1.

acos(x)  

\arccos(x)

### 7.6.3asin

Computes the arc in radians whose sine is equal to the argument x. A NaN error is raised if |x|>1.

asin(x)  

\arcsin(x)

### 7.6.4atan

Computes, in radians, the arc tangent of the argument x.

atan(x)  

\arctan(x)

### 7.6.5atan2

Computes, in radians, the arc tangent of quotient y/x, using the signs of the two arguments to determine the quadrant of the result, which is in the range [-\pi,\pi].

atan2(y,x)  

\arctan(y/x)

### 7.6.6ceil

Returns the smallest integral value not less than the argument x.

ceil(x)  

\lceil x \rceil

### 7.6.7clock

Returns the value of a certain clock in seconds measured from a certain (but specific) milestone. The kind of clock and the initial milestone depend on the optional integer argument f. It defaults to one, meaning CLOCK_MONOTONIC. The list and the meanings of the other available values for f can be checked in the clock_gettime (2) system call manual page.

clock([f])  

### 7.6.8cos

Computes the cosine of the argument x, where x is in radians. A cosine wave can be generated by passing as the argument x a linear function of time such as \omega t+\phi, where \omega controls the frequency of the wave and \phi controls its phase.

cos(x)  

\cos(x)

### 7.6.9cosh

Computes the hyperbolic cosine of the argument x, where x is in radians.

cosh(x)  

\cosh(x)

### 7.6.10cpu_time

Returns the CPU time used by FeenoX, in seconds. If the optional argument f is not provided or it is zero (default), the sum of times for both user-space and kernel-space usage is returned. For f=1 only user time is returned. For f=2 only system time is returned.

cpu_time([f])  

### 7.6.11d_dt

Computes the time derivative of the expression given in the argument x during a transient problem using the difference between the value of the signal in the previous time step and the actual value divided by the time step \delta t stored in dt. The argument x does not neet to be a variable, it can be an expression involving one or more variables that change in time. For t=0, the return value is zero. Unlike the functional derivative, the full dependence of these variables with time does not need to be known beforehand, i.e. the expression x might involve variables read from a shared-memory object at each time step.

d_dt(x)  

\frac{x(t) - x(t-\Delta t)}{\Delta t} \approx \frac{d}{dt} \Big( x (t) \Big)

### 7.6.12deadband

Filters the first argument x with a deadband centered at zero with an amplitude given by the second argument a.

deadband(x, a)  

\begin{cases} 0 & \text{if $| x | \leq a$} \\ x + a & \text{if $x < a$} \\ x - a & \text{if $x > a$} \end{cases}

### 7.6.13equal

Checks if the two first expressions a and b are equal, up to the tolerance given by the third optional argument \epsilon. If either |a|>1 or |b|>1, the arguments are compared using GSL’s gsl_fcmp, otherwise the absolute value of their difference is compared against \epsilon. This function returns zero if the arguments are not equal and one otherwise. Default value for \epsilon = 10^{-9}.

equal(a, b, [eps])  

\begin{cases} 1 & \text{if $a = b$} \\ 0 & \text{if $a \neq b$} \end{cases}

### 7.6.14exp

Computes the exponential function the argument x, i.e. the base of the natural logarithm e raised to the x-th power.

exp(x)  

e^x

### 7.6.15expint1

Computes the first exponential integral function of the argument x. If x is zero, a NaN error is issued.

expint1(x)  

\text{Re} \left[ \int_1^{\infty}\! \frac{\exp(-xt)}{t} \, dt \right]

### 7.6.16expint2

Computes the second exponential integral function of the argument x.

expint2(x)  

\text{Re} \left[ \int_1^{\infty}\! \frac{\exp(-xt)}{t^2} \, dt \right]

### 7.6.17expint3

Computes the third exponential integral function of the argument x.

expint3(x)  

\text{Re} \left[ \int_1^{\infty}\! \frac{\exp(-xt)}{t^3} \, dt \right]

### 7.6.18expintn

Computes the n-th exponential integral function of the argument x. If n is zero or one and x is zero, a NaN error is issued.

expintn(n,x)  

\text{Re} \left[ \int_1^{\infty}\! \frac{\exp(-xt)}{t^n} \, dt \right]

### 7.6.19floor

Returns the largest integral value not greater than the argument x.

floor(x)  

\lfloor x \rfloor

### 7.6.20heaviside

Computes the zero-centered Heaviside step function of the argument x. If the optional second argument \delta is provided, the discontinuous step at x=0 is replaced by a ramp starting at x=0 and finishing at x=\delta.

heaviside(x, [delta])  

\begin{cases} 0 & \text{if $x < 0$} \\ x / \delta & \text{if $0 < x < \delta$} \\ 1 & \text{if $x > \delta$} \end{cases}

### 7.6.21if

Performs a conditional testing of the first argument a, and returns either the second optional argument b if a is different from zero or the third optional argument c if a evaluates to zero. The comparison of the condition a with zero is performed within the precision given by the optional fourth argument \epsilon. If the second argument c is not given and a is not zero, the function returns one. If the third argument c is not given and a is zero, the function returns zero. The default precision is \epsilon = 10^{-9}. Even though if is a logical operation, all the arguments and the returned value are double-precision floating point numbers.

if(a, [b], [c], [eps])  

\begin{cases} b & \text{if $|a|<\epsilon$} \\ c & \text{otherwise} \end{cases}

### 7.6.22integral_dt

Computes the time integral of the expression given in the argument x during a transient problem with the trapezoidal rule using the value of the signal in the previous time step and the current value. At t = 0 the integral is initialized to zero. Unlike the functional integral, the full dependence of these variables with time does not need to be known beforehand, i.e. the expression x might involve variables read from a shared-memory object at each time step.

integral_dt(x)  

z^{-1}\left[ \int_0^{t-\Delta t} x(t') \, dt' \right] + \frac{x(t) + x(t-\Delta t)}{2} \, \Delta t \approx \int_0^{t} x(t') \, dt'

### 7.6.23integral_euler_dt

Idem as integral_dt but uses the backward Euler rule to update the instantaenous integral value. This function is provided in case this particular way of approximating time integrals is needed, for instance to compare FeenoX solutions with other computer codes. In general, it is recommended to use integral_dt.

integral_euler_dt(x)  

z^{-1}\left[ \int_0^{t-\Delta t} x(t') \, dt' \right] + x(t) \, \Delta t \approx \int_0^{t} x(t') \, dt'

### 7.6.24is_even

Returns one if the argument x rounded to the nearest integer is even.

is_even(x)  

\begin{cases}1 &\text{if $x$ is even} \\ 0 &\text{if $x$ is odd} \end{cases}

### 7.6.25is_in_interval

Returns true if the argument x is in the interval [a,b), i.e. including a but excluding b.

is_in_interval(x, a, b)  

\begin{cases} 1 & \text{if $a \leq x < b$} \\ 0 & \text{otherwise} \end{cases}

### 7.6.26is_odd

Returns one if the argument x rounded to the nearest integer is odd.

is_odd(x)  

\begin{cases}1 &\text{if $x$ is odd} \\ 0 &\text{if $x$ is even} \end{cases}

### 7.6.27j0

Computes the regular cylindrical Bessel function of zeroth order evaluated at the argument x.

j0(x)  

J_0(x)

### 7.6.28lag

Filters the first argument x(t) with a first-order lag of characteristic time \tau, i.e. this function applies the transfer function G(s) = \frac{1}{1 + s\tau} to the time-dependent signal x(t) to obtain a filtered signal y(t), by assuming that it is constant during the time interval [t-\Delta t,t] and using the analytical solution of the differential equation for that case at t = \Delta t with the initial condition y(0) = y(t-\Delta t).

lag(x, tau)  

x(t) - \Big[ x(t) - y(t-\Delta t) \Big] \cdot \exp\left(-\frac{\Delta t}{\tau}\right)

### 7.6.29lag_bilinear

Filters the first argument x(t) with a first-order lag of characteristic time \tau to the time-dependent signal x(t) by using the bilinear transformation formula.

lag_bilinear(x, tau)  

x(t-\Delta t) \cdot \left[ 1 - \frac{\Delta t}{2\tau} \right] + \left[ \frac{x(t) + x(t - \Delta t)}{1 + \frac{\Delta t}{2\tau}}\right] \cdot \frac{\Delta t}{2\tau}

### 7.6.30lag_euler

Filters the first argument x(t) with a first-order lag of characteristic time \tau to the time-dependent signal x(t) by using the Euler forward rule.

lag_euler(x, tau)  

x(t-\Delta t) + \Big[ x(t) - x(t - \Delta t) \Big] \cdot \frac{\Delta t}{\tau}

### 7.6.31last

Returns the value the variable x had in the previous time step. This function is equivalent to the Z-transform operator “delay” denoted by z^{-1}\left[ x\right]. For t=0 the function returns the actual value of x. The optional flag p should be set to one if the reference to last is done in an assignment over a variable that already appears inside expression x such as x = last(x). See example number 2.

last(x,[p])  

z^{-1}\left[ x \right] = x(t-\Delta t)

### 7.6.32limit

Limits the first argument x to the interval [a,b]. The second argument a should be less than the third argument b.

limit(x, a, b)  

\begin{cases} a & \text{if $x < a$} \\ x & \text{if $a \leq x \leq b$} \\ b & \text{if $x > b$} \end{cases}

### 7.6.33limit_dt

Limits the value of the first argument x(t) so to that its time derivative is bounded to the interval [a,b]. The second argument a should be less than the third argument b.

limit_dt(x, a, b)  

\begin{cases} x(t) & \text{if $a \leq dx/dt \leq b$} \\ x(t-\Delta t) + a \cdot \Delta t & \text{if $dx/dt < a$} \\ x(t-\Delta t) + b \cdot \Delta t & \text{if $dx/dt > b$} \end{cases}

### 7.6.34log

Computes the natural logarithm of the argument x. If x is zero or negative, a NaN error is issued.

log(x)  

\ln(x)

### 7.6.35mark_max

Returns the integer index i of the maximum of the arguments x_i provided. Currently only maximum of ten arguments can be provided.

mark_max(x1, x2, [...], [x10])  

i / \max \Big (x_1, x_2, \dots, x_{10} \Big) = x_i

### 7.6.36mark_min

Returns the integer index i of the minimum of the arguments x_i provided. Currently only maximum of ten arguments can be provided.

mark_max(x1, x2, [...], [x10])  

i / \min \Big (x_1, x_2, \dots, x_{10} \Big) = x_i

### 7.6.37max

Returns the maximum of the arguments x_i provided. Currently only maximum of ten arguments can be given.

max(x1, x2, [...], [x10])  

\max \Big (x_1, x_2, \dots, x_{10} \Big)

### 7.6.38memory

Returns the maximum memory (resident set size) used by FeenoX, in Gigabytes.

memory()  

### 7.6.39min

Returns the minimum of the arguments x_i provided. Currently only maximum of ten arguments can be given.

min(x1, x2, [...], [x10])  

\min \Big (x_1, x_2, \dots, x_{10} \Big)

### 7.6.40mod

Returns the remainder of the division between the first argument a and the second one b. Both arguments may be non-integral.

mod(a, b)  

a - \left\lfloor \frac{a}{b} \right\rfloor \cdot b

### 7.6.41not

Returns one if the first argument x is zero and zero otherwise. The second optional argument \epsilon gives the precision of the “zero” evaluation. If not given, default is \epsilon = 10^{-9}.

not(x, [eps])  

\begin{cases}1 &\text{if $|x| < \epsilon$} \\ 0 &\text{otherwise} \end{cases}

### 7.6.42random

Returns a random real number uniformly distributed between the first real argument x_1 and the second one x_2. If the third integer argument s is given, it is used as the seed and thus repetitive sequences can be obtained. If no seed is provided, the current time (in seconds) plus the internal address of the expression is used. Therefore, two successive calls to the function without seed (hopefully) do not give the same result. This function uses a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., Section 3.6.

random(x1, x2, [s])  

x_1 + r \cdot (x_2-x_1) \quad \quad 0 \leq r < 1

### 7.6.43random_gauss

Returns a random real number with a Gaussian distribution with a mean equal to the first argument x_1 and a standard deviation equatl to the second one x_2. If the third integer argument s is given, it is used as the seed and thus repetitive sequences can be obtained. If no seed is provided, the current time (in seconds) plus the internal address of the expression is used. Therefore, two successive calls to the function without seed (hopefully) do not give the same result. This function uses a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., Section 3.6.

random_gauss(x1, x2, [s])  

### 7.6.44round

Rounds the argument x to the nearest integer. Halfway cases are rounded away from zero.

round(x)  

\begin{cases} \lceil x \rceil & \text{if $\lceil x \rceil - x < 0.5$} \\ \lceil x \rceil & \text{if $\lceil x \rceil - x = 0.5 \wedge x > 0$} \\ \lfloor x \rfloor & \text{if $x-\lfloor x \rfloor < 0.5$} \\ \lfloor x \rfloor & \text{if $x-\lfloor x \rfloor = 0.5 \wedge x < 0$} \end{cases}

### 7.6.45sawtooth_wave

Computes a sawtooth wave between zero and one with a period equal to one. As with the sine wave, a sawtooh wave can be generated by passing as the argument x a linear function of time such as \omega t+\phi, where \omega controls the frequency of the wave and \phi controls its phase.

sawtooth_wave(x)  

x - \lfloor x \rfloor

### 7.6.46sgn

Returns minus one, zero or plus one depending on the sign of the first argument x. The second optional argument \epsilon gives the precision of the “zero” evaluation. If not given, default is \epsilon = 10^{-9}.

sgn(x, [eps])  

\begin{cases}-1 &\text{if $x \le -\epsilon$} \\ 0 &\text{if $|x| < \epsilon$} \\ +1 &\text{if $x \ge +\epsilon$} \end{cases}

### 7.6.47sin

Computes the sine of the argument x, where x is in radians. A sine wave can be generated by passing as the argument x a linear function of time such as \omega t+\phi, where \omega controls the frequency of the wave and \phi controls its phase.

sin(x)  

\sin(x)

### 7.6.48sinh

Computes the hyperbolic sine of the argument x, where x is in radians.

sinh(x)  

\sinh(x)

### 7.6.49sqrt

Computes the positive square root of the argument x. If x is negative, a NaN error is issued.

sqrt(x)  

+\sqrt{x}

### 7.6.50square_wave

Computes a square function between zero and one with a period equal to one. The output is one for 0 < x < 1/2 and zero for 1/2 \leq x < 1. As with the sine wave, a square wave can be generated by passing as the argument x a linear function of time such as \omega t+\phi, where \omega controls the frequency of the wave and \phi controls its phase.

square_wave(x)  

\begin{cases} 1 & \text{if $x - \lfloor x \rfloor < 0.5$} \\ 0 & \text{otherwise} \end{cases}

### 7.6.51tan

Computes the tangent of the argument x, where x is in radians.

tan(x)  

\tan(x)

### 7.6.52tanh

Computes the hyperbolic tangent of the argument x, where x is in radians.

tanh(x)  

\tanh(x)

### 7.6.53threshold_max

Returns one if the first argument x is greater than the threshold given by the second argument a, and zero otherwise. If the optional third argument b is provided, an hysteresis of width b is needed in order to reset the function value. Default is no hysteresis, i.e. b=0.

threshold_max(x, a, [b])  

\begin{cases} 1 & \text{if $x > a$} \\ 0 & \text{if $x < a-b$} \\ \text{last value of $y$} & \text{otherwise} \end{cases}

### 7.6.54threshold_min

Returns one if the first argument x is less than the threshold given by the second argument a, and zero otherwise. If the optional third argument b is provided, an hysteresis of width b is needed in order to reset the function value. Default is no hysteresis, i.e. b=0.

threshold_min(x, a, [b])  

\begin{cases} 1 & \text{if $x < a$} \\ 0 & \text{if $x > a+b$} \\ \text{last value of $y$} & \text{otherwise} \end{cases}

### 7.6.55triangular_wave

Computes a triangular wave between zero and one with a period equal to one. As with the sine wave, a triangular wave can be generated by passing as the argument x a linear function of time such as \omega t+\phi, where \omega controls the frequency of the wave and \phi controls its phase.

triangular_wave(x)  

\begin{cases} 2 (x - \lfloor x \rfloor) & \text{if $x - \lfloor x \rfloor < 0.5$} \\ 2 [1-(x - \lfloor x \rfloor)] & \text{otherwise} \end{cases}

### 7.6.56wall_time

Returns the time ellapsed since the invocation of FeenoX, in seconds.

wall_time()  

## 7.7 Functionals

### 7.7.1derivative

Computes the derivative of the expression f(x) given in the first argument with respect to the variable x given in the second argument at the point x=a given in the third argument using an adaptive scheme. The fourth optional argument h is the initial width of the range the adaptive derivation method starts with. The fifth optional argument p is a flag that indicates whether a backward (p < 0), centered (p = 0) or forward (p > 0) stencil is to be used. This functional calls the GSL functions gsl_deriv_backward, gsl_deriv_central or gsl_deriv_forward according to the indicated flag p. Defaults are h = (1/2)^{-10} \approx 9.8 \times 10^{-4} and p = 0.

derivative(f(x), x, a, [h], [p])  

\left. \frac{d}{dx} \Big[ f(x) \Big] \right|_{x = a}

### 7.7.2func_min

Finds the value of the variable x given in the second argument which makes the expression f(x) given in the first argument to take local a minimum in the in the range [a,b] given by the third and fourth arguments. If there are many local minima, the one that is closest to (a+b)/2 is returned. The optional fifth argument \epsilon gives a relative tolerance for testing convergence, corresponding to GSL epsrel (note that epsabs is set also to \epsilon). The sixth optional argument is an integer which indicates the algorithm to use: 0 (default) is quad_golden, 1 is brent and 2 is goldensection. See the GSL documentation for further information on the algorithms. The seventh optional argument p is a flag that indicates how to proceed if there is no local minimum in the range [a,b]. If p = 0 (default), a is returned if f(a) < f(b) and b otherwise. If p = 1 then the local minimum algorimth is tried nevertheless. Default is \epsilon = (1/2)^{-20} \approx 9.6\times 10^{-7}.

y = func_min(f(x), x, a, b, [eps], [alg], [p])  

y = \left\{ x \in [a,b] / f(x) = \min_{[a,b]} f(x) \right \}

### 7.7.3gauss_kronrod

Computes the integral of the expression f(x) given in the first argument with respect to variable x given in the second argument over the interval [a,b] given in the third and fourth arguments respectively using a non-adaptive procedure which uses fixed Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions. The algorithm applies the Gauss-Kronrod 10-point, 21-point, 43-point and 87-point integration rules in succession until an estimate of the integral is achieved within the relative tolerance given in the fifth optional argument \epsilon It correspondes to GSL’s epsrel parameter (epsabs is set to zero).
The rules are designed in such a way that each rule uses all the results of its predecessors, in order to minimize the total number of function evaluations. Defaults are \epsilon = (1/2)^{-10} \approx 10^{-3}. See GSL reference for further information.

gauss_kronrod(f(x), x, a, b, [eps])  

\int_a^b f(x) \, dx

### 7.7.4gauss_legendre

Computes the integral of the expression f(x) given in the first argument with respect to variable x given in the second argument over the interval [a,b] given in the third and fourth arguments respectively using the n-point Gauss-Legendre rule, where n is given in the optional fourth argument. It is provided for fast integration of smooth functions with known polynomic order (it is exact for polynomials of order 2n-1). This functional calls GSL function gsl_integration_glfixedp. Default is n = 12. See GSL reference for further information.

gauss_legendre(f(x), x, a, b, [n])  

\int_a^b f(x) \, dx

### 7.7.5integral

Computes the integral of the expression f(x) given in the first argument with respect to variable x given in the second argument over the interval [a,b] given in the third and fourth arguments respectively using an adaptive scheme, in which the domain is divided into a number of maximum number of subintervals and a fixed-point Gauss-Kronrod-Patterson scheme is applied to each quadrature subinterval. Based on an estimation of the error commited, one or more of these subintervals may be split to repeat the numerical integration alogorithm with a refined division. The fifth optional argument \epsilon is is a relative tolerance used to check for convergence. It correspondes to GSL’s epsrel parameter (epsabs is set to zero). The sixth optional argument 1\leq k \le 6 is an integer key that indicates the integration rule to apply in each interval. It corresponds to GSL’s parameter key. The seventh optional argument gives the maximum number of subdivisions, which defaults to 1024. If the integration interval [a,b] if finite, this functional calls the GSL function gsl_integration_qag. If a is less that minus the internal variable infinite, b is greater that infinite or both conditions hold, GSL functions gsl_integration_qagil, gsl_integration_qagiu or gsl_integration_qagi are called. The condition of finiteness of a fixed range [a,b] can thus be changed by modifying the internal variable infinite. Defaults are \epsilon = (1/2)^{-10} \approx 10^{-3} and k=3. The maximum numbers of subintervals is limited to 1024. Due to the adaptivity nature of the integration method, this function gives good results with arbitrary integrands, even for infinite and semi-infinite integration ranges. However, for certain integrands, the adaptive algorithm may be too expensive or even fail to converge. In these cases, non-adaptive quadrature functionals ought to be used instead. See GSL reference for further information.

integral(f(x), x, a, b, [eps], [k], [max_subdivisions])  

\int_a^b f(x) \, dx

### 7.7.6prod

Computes product of the N=b-a expressions f(i) given in the first argument by varying the variable~i given in the second argument between~a given in the third argument and~b given in the fourth argument,~i = a, a+1, \dots ,b-1,b.

prod(f(i), i, a, b)  

\prod_{i=a}^b f_i

### 7.7.7root

Computes the value of the variable x given in the second argument which makes the expression f(x) given in the first argument to be equal to zero by using a root bracketing algorithm. The root should be in the range [a,b] given by the third and fourth arguments. The optional fifth argument \epsilon gives a relative tolerance for testing convergence, corresponding to GSL epsrel (note that epsabs is set also to \epsilon). The sixth optional argument is an integer which indicates the algorithm to use: 0 (default) is brent, 1 is falsepos and 2 is bisection. See the GSL documentation for further information on the algorithms. The seventh optional argument p is a flag that indicates how to proceed if the sign of f(a) is equal to the sign of f(b). If p=0 (default) an error is raised, otherwise it is not. If more than one root is contained in the specified range, the first one to be found is returned. The initial guess is x_0 = (a+b)/2. If no roots are contained in the range and p \neq 0, the returned value can be any value. Default is \epsilon = (1/2)^{-10} \approx 10^{3}.

root(f(x), x, a, b, [eps], [alg], [p])  

\left\{ x \in [a,b] / f(x) = 0 \right \}

### 7.7.8sum

Computes sum of the N=b-a expressions f_i given in the first argument by varying the variable i given in the second argument between a given in the third argument and b given in the fourth argument, i=a,a+1,\dots,b-1,b.

sum(f_i, i, a, b)  

\sum_{i=a}^b f_i

## 7.8 Vector functions

### 7.8.1derivative

Computes the derivative of the expression f(x) given in the first argument with respect to the variable x given in the second argument at the point x=a given in the third argument using an adaptive scheme. The fourth optional argument h is the initial width of the range the adaptive derivation method starts with. The fifth optional argument p is a flag that indicates whether a backward (p < 0), centered (p = 0) or forward (p > 0) stencil is to be used. This functional calls the GSL functions gsl_deriv_backward, gsl_deriv_central or gsl_deriv_forward according to the indicated flag p. Defaults are h = (1/2)^{-10} \approx 9.8 \times 10^{-4} and p = 0.

derivative(f(x), x, a, [h], [p])  

\left. \frac{d}{dx} \Big[ f(x) \Big] \right|_{x = a}

### 7.8.2func_min

Finds the value of the variable x given in the second argument which makes the expression f(x) given in the first argument to take local a minimum in the in the range [a,b] given by the third and fourth arguments. If there are many local minima, the one that is closest to (a+b)/2 is returned. The optional fifth argument \epsilon gives a relative tolerance for testing convergence, corresponding to GSL epsrel (note that epsabs is set also to \epsilon). The sixth optional argument is an integer which indicates the algorithm to use: 0 (default) is quad_golden, 1 is brent and 2 is goldensection. See the GSL documentation for further information on the algorithms. The seventh optional argument p is a flag that indicates how to proceed if there is no local minimum in the range [a,b]. If p = 0 (default), a is returned if f(a) < f(b) and b otherwise. If p = 1 then the local minimum algorimth is tried nevertheless. Default is \epsilon = (1/2)^{-20} \approx 9.6\times 10^{-7}.

y = func_min(f(x), x, a, b, [eps], [alg], [p])  

y = \left\{ x \in [a,b] / f(x) = \min_{[a,b]} f(x) \right \}

### 7.8.3gauss_kronrod

Computes the integral of the expression f(x) given in the first argument with respect to variable x given in the second argument over the interval [a,b] given in the third and fourth arguments respectively using a non-adaptive procedure which uses fixed Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions. The algorithm applies the Gauss-Kronrod 10-point, 21-point, 43-point and 87-point integration rules in succession until an estimate of the integral is achieved within the relative tolerance given in the fifth optional argument \epsilon It correspondes to GSL’s epsrel parameter (epsabs is set to zero).
The rules are designed in such a way that each rule uses all the results of its predecessors, in order to minimize the total number of function evaluations. Defaults are \epsilon = (1/2)^{-10} \approx 10^{-3}. See GSL reference for further information.

gauss_kronrod(f(x), x, a, b, [eps])  

\int_a^b f(x) \, dx

### 7.8.4gauss_legendre

Computes the integral of the expression f(x) given in the first argument with respect to variable x given in the second argument over the interval [a,b] given in the third and fourth arguments respectively using the n-point Gauss-Legendre rule, where n is given in the optional fourth argument. It is provided for fast integration of smooth functions with known polynomic order (it is exact for polynomials of order 2n-1). This functional calls GSL function gsl_integration_glfixedp. Default is n = 12. See GSL reference for further information.

gauss_legendre(f(x), x, a, b, [n])  

\int_a^b f(x) \, dx

### 7.8.5integral

Computes the integral of the expression f(x) given in the first argument with respect to variable x given in the second argument over the interval [a,b] given in the third and fourth arguments respectively using an adaptive scheme, in which the domain is divided into a number of maximum number of subintervals and a fixed-point Gauss-Kronrod-Patterson scheme is applied to each quadrature subinterval. Based on an estimation of the error commited, one or more of these subintervals may be split to repeat the numerical integration alogorithm with a refined division. The fifth optional argument \epsilon is is a relative tolerance used to check for convergence. It correspondes to GSL’s epsrel parameter (epsabs is set to zero). The sixth optional argument 1\leq k \le 6 is an integer key that indicates the integration rule to apply in each interval. It corresponds to GSL’s parameter key. The seventh optional argument gives the maximum number of subdivisions, which defaults to 1024. If the integration interval [a,b] if finite, this functional calls the GSL function gsl_integration_qag. If a is less that minus the internal variable infinite, b is greater that infinite or both conditions hold, GSL functions gsl_integration_qagil, gsl_integration_qagiu or gsl_integration_qagi are called. The condition of finiteness of a fixed range [a,b] can thus be changed by modifying the internal variable infinite. Defaults are \epsilon = (1/2)^{-10} \approx 10^{-3} and k=3. The maximum numbers of subintervals is limited to 1024. Due to the adaptivity nature of the integration method, this function gives good results with arbitrary integrands, even for infinite and semi-infinite integration ranges. However, for certain integrands, the adaptive algorithm may be too expensive or even fail to converge. In these cases, non-adaptive quadrature functionals ought to be used instead. See GSL reference for further information.

integral(f(x), x, a, b, [eps], [k], [max_subdivisions])  

\int_a^b f(x) \, dx

### 7.8.6prod

Computes product of the N=b-a expressions f(i) given in the first argument by varying the variable~i given in the second argument between~a given in the third argument and~b given in the fourth argument,~i = a, a+1, \dots ,b-1,b.

prod(f(i), i, a, b)  

\prod_{i=a}^b f_i

### 7.8.7root

Computes the value of the variable x given in the second argument which makes the expression f(x) given in the first argument to be equal to zero by using a root bracketing algorithm. The root should be in the range [a,b] given by the third and fourth arguments. The optional fifth argument \epsilon gives a relative tolerance for testing convergence, corresponding to GSL epsrel (note that epsabs is set also to \epsilon). The sixth optional argument is an integer which indicates the algorithm to use: 0 (default) is brent, 1 is falsepos and 2 is bisection. See the GSL documentation for further information on the algorithms. The seventh optional argument p is a flag that indicates how to proceed if the sign of f(a) is equal to the sign of f(b). If p=0 (default) an error is raised, otherwise it is not. If more than one root is contained in the specified range, the first one to be found is returned. The initial guess is x_0 = (a+b)/2. If no roots are contained in the range and p \neq 0, the returned value can be any value. Default is \epsilon = (1/2)^{-10} \approx 10^{3}.

root(f(x), x, a, b, [eps], [alg], [p])  

\left\{ x \in [a,b] / f(x) = 0 \right \}

### 7.8.8sum

Computes sum of the N=b-a expressions f_i given in the first argument by varying the variable i given in the second argument between a given in the third argument and b given in the fourth argument, i=a,a+1,\dots,b-1,b.

sum(f_i, i, a, b)  

\sum_{i=a}^b f_i

# 8 FeenoX & the UNIX Philospohy

## 8.1 Rule of Modularity

Developers should build a program out of simple parts connected by well defined interfaces, so problems are local, and parts of the program can be replaced in future versions to support new features. This rule aims to save time on debugging code that is complex, long, and unreadable.

• FeenoX uses third-party high-quality libraries
• GNU Scientific Library
• SUNDIALS
• PETSc
• SLEPc

## 8.2 Rule of Clarity

Developers should write programs as if the most important communication is to the developer who will read and maintain the program, rather than the computer. This rule aims to make code as readable and comprehensible as possible for whoever works on the code in the future.

• Example two squares in thermal contact.
• LE10 & LE11: a one-to-one correspondence between the problem text and the FeenoX input.

## 8.3 Rule of Composition

Developers should write programs that can communicate easily with other programs. This rule aims to allow developers to break down projects into small, simple programs rather than overly complex monolithic programs.

• FeenoX uses meshes created by a separate mesher (i.e. Gmsh).
• FeenoX writes data that has to be plotted or post-processed by other tools (Gnuplot, Gmsh, Paraview, etc.).
• ASCII output is 100% controlled by the user so it can be tailored to suit any other programs’ input needs such as AWK filters to create LaTeX tables.

## 8.4 Rule of Separation

Developers should separate the mechanisms of the programs from the policies of the programs; one method is to divide a program into a front-end interface and a back-end engine with which that interface communicates. This rule aims to prevent bug introduction by allowing policies to be changed with minimum likelihood of destabilizing operational mechanisms.

• FeenoX does not include a GUI, but it is GUI-friendly.

## 8.5 Rule of Simplicity

Developers should design for simplicity by looking for ways to break up program systems into small, straightforward cooperating pieces. This rule aims to discourage developers’ affection for writing “intricate and beautiful complexities” that are in reality bug prone programs.

• Simple problems need simple input.
• Similar problems need similar inputs.
• English-like self-evident input files matching as close as possible the problem text.
• If there is a single material there is no need to link volumes to properties.

## 8.6 Rule of Parsimony

Developers should avoid writing big programs. This rule aims to prevent overinvestment of development time in failed or suboptimal approaches caused by the owners of the program’s reluctance to throw away visibly large pieces of work. Smaller programs are not only easier to write, optimize, and maintain; they are easier to delete when deprecated.

• Parametric and/or optimization runs have to be driven from an outer script (Bash, Python, etc.)

## 8.7 Rule of Transparency

Developers should design for visibility and discoverability by writing in a way that their thought process can lucidly be seen by future developers working on the project and using input and output formats that make it easy to identify valid input and correct output. This rule aims to reduce debugging time and extend the lifespan of programs.

• Written in C99
• Makes use of structures and function pointers to give the same functionality as C++’s virtual methods without needing to introduce other complexities that make the code base harder to maintain and to debug.

## 8.8 Rule of Robustness

Developers should design robust programs by designing for transparency and discoverability, because code that is easy to understand is easier to stress test for unexpected conditions that may not be foreseeable in complex programs. This rule aims to help developers build robust, reliable products.

## 8.9 Rule of Representation

Developers should choose to make data more complicated rather than the procedural logic of the program when faced with the choice, because it is easier for humans to understand complex data compared with complex logic. This rule aims to make programs more readable for any developer working on the project, which allows the program to be maintained.

## 8.10 Rule of Least Surprise

Developers should design programs that build on top of the potential users’ expected knowledge; for example, ‘+’ in a calculator program should always mean ‘addition’. This rule aims to encourage developers to build intuitive products that are easy to use.

• If one needs a problem where the conductivity depends on x as k(x)=1+x then the input is

k(x) = 1+x
• If a problem needs a temperature distribution given by an algebraic expression T(x,y,z)=\sqrt{x^2+y^2}+z then do

T(x,y,z) = sqrt(x^2+y^2) + z

## 8.11 Rule of Silence

Developers should design programs so that they do not print unnecessary output. This rule aims to allow other programs and developers to pick out the information they need from a program’s output without having to parse verbosity.

• No PRINT (or WRITE_MESH), no output.

## 8.12 Rule of Repair

Developers should design programs that fail in a manner that is easy to localize and diagnose or in other words “fail noisily”. This rule aims to prevent incorrect output from a program from becoming an input and corrupting the output of other code undetected.

• Input errors are detected before the computation is started:

$feenox thermal-error.fee error: undefined thermal conductivity 'k'$ 
• Run-time errors can be user controled, they can be fatal or ignored.

## 8.13 Rule of Economy

Developers should value developer time over machine time, because machine cycles today are relatively inexpensive compared to prices in the 1970s. This rule aims to reduce development costs of projects.

• Output is 100% user-defined so the desired results is directly obtained instead of needing further digging into tons of undesired data.The approach of “compute and write everything you can in one single run” made sense in 1970 where CPU time was more expensive than human time, but not anymore.
• Example: LE10 & LE11.

## 8.14 Rule of Generation

Developers should avoid writing code by hand and instead write abstract high-level programs that generate code. This rule aims to reduce human errors and save time.

• Inputs are M4-like-macro friendly.
• Parametric runs can be done from scripts through command line arguments expansion.
• Documentation is created out of simple Markdown sources and assembled as needed.

## 8.15 Rule of Optimization

Developers should prototype software before polishing it. This rule aims to prevent developers from spending too much time for marginal gains.

• Premature optimization is the root of all evil
• We are still building. We will optimize later.
• Code optimization: TODO
• Parallelization: TODO
• Comparison with other tools: TODO

## 8.16 Rule of Diversity

Developers should design their programs to be flexible and open. This rule aims to make programs flexible, allowing them to be used in ways other than those their developers intended.

• Either Gmsh or Paraview can be used to post-process results.
• Other formats can be added.

## 8.17 Rule of Extensibility

Developers should design for the future by making their protocols extensible, allowing for easy plugins without modification to the program’s architecture by other developers, noting the version of the program, and more. This rule aims to extend the lifespan and enhance the utility of the code the developer writes.

• FeenoX is GPLv3+. The ‘+’ is for the future.
• Each PDE has a separate source directory. Any of them can be used as a template for new PDEs, especially laplace for elliptic operators.

# 9 History

Very much like UNIX in the late 1960s, FeenoX is a third-system effect: I wrote a first hack that seemed to work better than I had expected. Then I tried to add a lot of features and complexities which I felt the code needed. After ten years of actual usage, I then realized

1. what was worth keeping,
2. what needed to be rewrittenm and

The first version was called wasora, the second was “The wasora suite” (i.e. a generic framework plus a bunch of “plugins”, including a thermo-mechanical one named Fino) and then finally FeenoX. The story that follows explains why I wrote the first hack to begin with.

It was at the movies when I first heard about dynamical systems, non-linear equations and chaos theory. The year was 1993, I was ten years old and the movie was Jurassic Park. Dr. Ian Malcolm (the character portrayed by Jeff Goldblum) explained sensitivity to initial conditions in a memorable scene, which is worth watching again and again. Since then, the fact that tiny variations may lead to unexpected results has always fascinated me. During high school I attended a very interesting course on fractals and chaos that made me think further about complexity and its mathematical description. Nevertheless, it was not not until college that I was able to really model and solve the differential equations that give rise to chaotic behavior.

In fact, initial-value ordinary differential equations arise in a great variety of subjects in science and engineering. Classical mechanics, chemical kinetics, structural dynamics, heat transfer analysis and dynamical systems, among other disciplines, heavily rely on equations of the form

\dot{\mathbf{x}} = F(\mathbf{x},t)

During my years of undergraduate student (circa 2004–2007), whenever I had to solve these kind of equations I had to choose one of the following three options:

1. to program an ad-hoc numerical method such as Euler or Runge-Kutta, matching the requirements of the system of equations to solve, or
2. to use a standard numerical library such as the GNU Scientific Library and code the equations to solve into a C program (or maybe in Python), or
3. to use a high-level system such as Octave, Maxima, or some non-free (and worse, see below) programs.

Of course, each option had its pros and its cons. But none provided the combination of advantages I was looking for, namely flexibility (option one), efficiency (option two) and reduced input work (partially given by option three). Back in those days I ended up wandering between options one and two, depending on the type of problem I had to solve. However, even though one can, with some effort, make the code read some parameters from a text file, any other drastic change usually requires a modification in the source code—some times involving a substantial amount of work—and a further recompilation of the code. This was what I most disliked about this way of working, but I could nevertheless live with it.

Regardless of this situation, during my last year of Nuclear Engineering, the tipping point came along. Here’s a slightly-fictionalized of a dialog between myself and the teacher at the computer lab, as it might have happened (or not):

— (Prof.) Open MATLAB.™
— (Me) It’s not installed here. I type mathlab and it does not work.
— (Prof.) It’s spelled matlab.
— (Me) Ok, working. (A screen with blocks and lines connecting them appears)
— (Me) What’s this?
— (Prof.) The point reactor equations.
— (Me) It’s not. These are the point reactor equations:

\begin{cases} \dot{\phi}(t) = \displaystyle \frac{\rho(t) - \beta}{\Lambda} \cdot \phi(t) + \sum_{i=1}^{N} \lambda_i \cdot c_i \\ \dot{c}_i(t) = \displaystyle \frac{\beta_i}{\Lambda} \cdot \phi(t) - \lambda_i \cdot c_i \end{cases}

— (Me) And in any case, I’d write them like this in a computer:

phi_dot = (rho-Beta)/Lambda * phi + sum(lambda[i], c[i], i, 1, N)
c_dot[i] = beta[i]/Lambda * phi - lambda[i]*c[i]

This conversation forced me to re-think the ODE-solving issue. I could not (and still cannot) understand why somebody would prefer to solve a very simple set of differential equations by drawing blocks and connecting them with a mouse with no mathematical sense whatsoever. Fast forward fifteen years, and what I wrote above is essentially how one would solve the point kinetics equations with FeenoX.

1. The word dirty means the current Git worktree used to compile the binary had some changes that were not commited yet.↩︎