# 1 Keywords

## 1.1 .=

Add an equation to the DAE system to be solved in the phase space spanned by PHASE_SPACE.

{ 0[(i[,j]][<imin:imax[;jmin:jmax]>] | <expr1> } .= <expr2>

## 1.2 =

Assign an expression to a variable, a vector or a matrix.

<var>[ [<expr_tmin>, <expr_tmax>] |
<expr_t> ] = <expr> <vector>(<expr_i>)[<expr_i_min, expr_i_max>] [ [<expr_tmin>, <expr_tmax>] |
<expr_t> ] = <expr> <matrix>(<expr_i>,<expr_j>)[<expr_i_min, expr_i_max; expr_j_min, expr_j_max>] [ [<expr_tmin>, <expr_tmax>] |
<expr_t> ] = <expr>

## 1.3 ABORT

Catastrophically abort the execution and quit wasora.

ABORT

Whenever the instruction ABORT is executed, wasora quits without closing files or unlocking shared memory objects. The objective of this instruction is, as illustrated in the examples, either to debug complex input files and check the values of certain variables or to conditionally abort the execution using IF clauses.

## 1.4 ALIAS

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.

## 1.5 CALL

Call a previously dynamically-loaded user-provided routine.

CALL <name> [ expr_1 expr_2 ... expr_n ]

## 1.6 CLOSE

Explicitly close an already-OPENed file.

CLOSE

## 1.7 CONST

Mark a scalar variable, vector or matrix as a constant.

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

## 1.8 DEFAULT_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. ## 1.9 DIFFERENTIAL Explicitly mark variables, vectors or matrices as “differential” to compute intial conditions of DAE systems. DIFFERENTIAL { <var_1> <var_2> ... | <vector_1> <vector_2> ... | <matrix_1> <matrix_2> ... } ## 1.10 DO_NOT_EVALUATE_AT_PARSE_TIME Ask wasora not to evaluate assignments at parse time. DO_NOT_EVALUATE_AT_PARSE_TIME ## 1.11 FILE Define a file, either as input or as output, for further usage. < FILE | OUTPUT_FILE | INPUT_FILE > <name> <printf_format> [ expr_1 expr_2 ... expr_n ] [ INPUT | OUTPUT | MODE <fopen_mode> ] [ OPEN | DO_NOT_OPEN ] ## 1.12 FIT Fit a function of one or more arguments to a set of data. 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_n> ] [ RANGE_MAX <expr_1> <expr_2> ... <expr_n> ] [ DELTAEPSREL <expr> ] [ DELTAEPSABS <expr> ] [ MAX_ITER <expr> ] [ VERBOSE ] [ RERUN | DO_NOT_RERUN ] The function with the data has to be point-wise defined. The function to be fitted hast to be parametrized with at least one of the variables provided after the VIA keyword. Only the names of the functions have to be given. 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 listed in 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. For multidimensional fits, the range is an hypercube. If no range is given, all the definition points of the function witht the data are used for the fit. Convergence can be controlled by given the relative and absolute tolreances with DELTAEPSREL (default 1e-4) and DELTAEPSABS (default 1e-6), and with the maximum number of iterations MAX_ITER (default 100). If the optional keyword VERBOSE is given, some data of the intermediate steps is written in the standard output. ## 1.13 FUNCTION Define a function of one or more variables. FUNCTION <name>(<var_1>[,var2,...,var_n]) { [ = <expr> | FILE_PATH <file_path> | ROUTINE <name> | | MESH <name> { DATA <new_vector_name> | VECTOR <existing_vector_name> } { NODES | CELLS } | [ VECTOR_DATA <vector_1> <vector_2> ... <vector_n> <vector_n+1> ] } [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> ] [ SIZES <expr_1> <expr_2> ... <expr_n> ] [ X_INCREASES_FIRST <expr> ] [ DATA <num_1> <num_2> ... <num_N> ] 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 as variables. If the function is given as an algebraic expression, the short-hand operator := can be used. That is to say, FUNCTION f(x) = x^2 is equivalent to f(x) := x^2. If a FILE_PATH 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. A function of type ROUTINE calls an already-defined user-provided routine using the CALL keyword and passes the values of the variables in each required evaluation as a double * argument. If MESH is given, the definition points are the nodes or the cells of the mesh. The function arguments should be (x), (x,y) or (x,y,z) matching the dimension the mesh. If the keyword DATA is used, a new empty vector of the appropriate size is defined. The elements of this new vector can be assigned to the values of the function at the i-th node or cell. If the keyword VECTOR is used, the values of the dependent variable are taken to be the values of the already-existing vector. Note that this vector should have the size of the number of nodes or cells the mesh has, depending on whether NODES or CELLS is given. If VECTOR_DATA is given, a set of n+1 vectors of the same size is expected. The first n+1 correspond to the arguments and the last one is the function value. 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 (available only with GSL >= 2.0) Default interpolation scheme for one-dimensional functions is (*gsl_interp_linear). 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 9.5367431640625e-07. 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 1.0. The exponent of the shepard method is given by SHEPARD_EXPONENT. Default is 2. When requesting bilinear interpolation for n>3, the number of definition points for each argument variable has to be given with SIZES, and wether the definition data is sorted with the first argument changing first (X_INCREASES_FIRST evaluating to non-zero) or with the last argument changing first (zero). 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. See the examples. ## 1.14 HISTORY Record the time history of a variable as a function of time. HISTORY <variable> <function> ## 1.15 IF Begin a conditional block. IF expr <block_of_instructions_if_expr_is_true> [ ELSE ] [block_of_instructions_if_expr_is_false] ENDIF ## 1.16 IMPLICIT Define whether implicit declaration of variables is allowed or not. IMPLICIT { NONE | ALLOWED } By default, wasora 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. ## 1.17 INCLUDE Include another wasora 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 optional FROM and TO keywords can be used to include only portions of a file. ## 1.18 INITIAL_CONDITIONS_MODE Define how initial conditions of DAE problems are computed. INITIAL_CONDITIONS_MODE { 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. ## 1.19 LOAD_PLUGIN Load a wasora plug-in from a dynamic shared object. LOAD_PLUGIN { <file_path> | <plugin_name> } A wasora plugin in the form of a dynamic shared object (i.e. .so) can be loaded either with the LOAD_PLUGIN keyword or from the command line with the -p option. Either a file path or a plugin name can be given. The following locations are tried: • the current directory ./ • the parent directory ../ • the user’s LD_LIBRARY_PATH • the cache file /etc/ld.so.cache • the directories /lib, /usr/lib, /usr/local/lib If a wasora plugin was compiled and installed following the make install procedure, the plugin should be loaded by just passing the name to LOAD_PLUGIN. ## 1.20 LOAD_ROUTINE Load one or more routines from a dynamic shared object. LOAD_ROUTINE <file_path> <routine_1> [ <routine_2> ... <routine_n> ] ## 1.21 M4 Call the m4 macro processor with definitions from wasora variables or expressions. M4 { INPUT_FILE <file_id> | FILE_PATH <file_path> } { OUTPUT_FILE <file_id> | OUTPUT_FILE_PATH <file_path> } [ EXPAND <name> ] ... } [ MACRO <name> [ <format> ] <definition> ] ... } ## 1.22 MATRIX Define a matrix. MATRIX <name> ROWS <expr> COLS <expr> [ DATA num_expr_1 num_expr_2 ... num_expr_n ] ## 1.23 MINIMIZE Find the combination of arguments that give a (relative) minimum of a function, i.e. run an optimization problem. MINIMIZE <function> <function> [ METHOD { conjugate_fr | conjugate_pr | vector_bfgs2 | vector_bfgs | steepest_descent | nmsimplex2 | nmsimplex | nmsimplex2rand } [ GRADIENT <expr_1> <expr_2> ... <expr_n> ] [ GUESS <expr_1> <expr_2> ... <expr_n> ] [ MIN <expr_1> <expr_2> ... <expr_n> ] [ MAX <expr_1> <expr_2> ... <expr_n> ] [ STEP <expr_1> <expr_2> ... <expr_n> ] [ VERBOSE ] [ NORERUN ] [ MAX_ITER <expr> ] [ TOL <expr> ] [ GRADTOL <expr> ] ## 1.24 PARAMETRIC Systematically sweep a zone of the parameter space, i.e. perform a parametric run. PARAMETRIC <var_1> [ ... <var_n> ] [ TYPE { linear | logarithmic | random | gaussianrandom | sobol | niederreiter | halton | reversehalton } ] [ MIN <num_expr_1> ... <num_expr_n> ] [ MAX <num_expr_1> ... <num_expr_n> ] [ STEP <num_expr_1> ... <num_expr_n> ] [ NSTEPS <num_expr_1> ... <num_expr_n> ] [ OUTER_STEPS <num_expr> ] [ MAX_DAUGHTERS <num_expr> ] [ OFFSET <num_expr> ] [ ADIABATIC ] ## 1.25 PHASE_SPACE Define which variables, vectors and/or matrices belong to the phase space of the DAE system to be solved. PHASE_SPACE { <vars> | <vectors> | <matrices> } ## 1.26 PRINT Print plain-text and/or formatted data to the standard output or into an output file. PRINT [ FILE <file_id> | FILE_PATH <file_path> ] [ NONEWLINE ] [ SEP <string> ] [ NOSEP ] [ HEADER ] [ SKIP_STEP <expr> ] [ SKIP_STATIC_STEP <expr> ] [ SKIP_TIME <expr> ] [ SKIP_HEADER_STEP <expr> ] [ <object_1> <object_2> ... <object_n> ] [ TEXT <string_1> ... TEXT <string_n> ] Each argument object that is not a keyword is expected to be part of the output, can be either a matrix, a vector, an 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. Hashes # appearing literal in text strings have to be quoted to prevent the parser to treat them as comments within the wasora input file and thus ignoring the rest of the line. Whenever an argument starts with a porcentage sign %, it is treated as a C printf-compatible format definition 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. The default format is "%g". 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 vectors column-wise. 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. If the FILE keyword is not provided, default is to write to stdout. If the NONEWLINE keyword is not provided, default is to write a newline ‘’ character after all the objects are processed. The SEP keywords expects a string used to separate printed objects, the default is a tab ‘DEFAULT_PRINT_SEPARATOR’ character. Use the NOSEP keyword to define an empty string as object separator. 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 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. By default the PRINT instruction is evaluated every step. 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_2 | expr_1 } ... { function_n | expr_n-1 } ] [ FILE <file_id> | FILE_PATH <file_path> ] [ HEADER ] [ MIN <expr_1> <expr_2> ... <expr_m> ] [ MAX <expr_1> <expr_2> ... <expr_m> ] [ STEP <expr_1> <expr_2> ... <expr_m> ] [ NSTEPs <expr_1> <expr_2> ... <expr_m> ] [ FORMAT <print_format> ] [ PHYSICAL_ENTITY <name> ] Print the elements of one or more vectors. PRINT_VECTOR [ FILE <file_id> ] FILE_PATH <file_path> ] [ { VERTICAL | HORIZONTAL } ] [ ELEMS_PER_LINE <expr> ] [ FORMAT <print_format> ] <vector_1> [ vector_2 ... vector_n ] ## 1.29 READ Read data (variables, vectors o matrices) from files or shared-memory segments. [ READ | WRITE ] [ SHM <name> ] [ { ASCII_FILE_PATH | BINARY_FILE_PATH } <file_path> ] [ { ASCII_FILE | BINARY_FILE } <identifier> ] [ IGNORE_NULL ] [ object_1 object_2 ... object_n ] ## 1.30 SEMAPHORE Perform either a wait or a post operation on a named shared semaphore. [ SEMAPHORE | SEM ] <name> { WAIT | POST } ## 1.31 SHELL Execute a shell command. SHELL <print_format> [ expr_1 expr_2 ... expr_n ] ## 1.32 SOLVE Solve a non-linear system of n equations with n unknowns. SOLVE <n> UNKNOWNS <var_1> <var_2> ... <var_n> RESIDUALS <expr_1> <expr_2> ... <expr_n> ] GUESS <expr_1> <expr_2> ... <expr_n> ] [ METHOD { dnewton | hybrid | hybrids | broyden } ] [ EPSABS <expr> ] [ EPSREL <expr> ] [ MAX_ITER <expr> ] [ VERBOSE ] ## 1.33 TIME_PATH Force transient 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. ## 1.34 VAR Define one or more scalar variables. VAR <name_1> [ <name_2> ] ... [ <name_n> ] ## 1.35 VECTOR Define a vector. VECTOR <name> SIZE <expr> [ DATA <expr_1> <expr_2> ... <expr_n> | FUNCTION_DATA <function> ] ## 1.36 VECTOR_SORT Sort the elements of a vector using a specific numerical order, potentially making the same rearrangement of another vector. VECTOR_SORT <vector> [ ASCENDING_ORDER | DESCENDING_ORDER ] [ <vector> ] ## 1.37 WRITE Write data (variables, vectors o matrices) to files or shared-memory segments. See the READ keyword for usage details. # 2 Mesh-related keywords ## 2.1 MATERIAL MATERIAL <name> [ MESH <name> ] [ PHYSICAL_GROUP <name_1> [ PHYSICAL_GROUP <name_2> [ ... ] ] ] [ <property_name_1> <expr_1> [ <property_name_2> <expr_2> [ ... ] ] ] ## 2.2 MESH Reads an unstructured mesh from an external file in MSH, VTK or FRD format. MESH [ NAME <name> ] { FILE <file_id> | FILE_PATH <file_path> } [ DIMENSIONS <num_expr> ] [ SCALE <expr> ] [ OFFSET <expr_x> <expr_y> <expr_z> ] [ READ_SCALAR <name_in_mesh> AS <function_name> ] [...] [ READ_FUNCTION <function_name> ] [...] If there will be only one mesh in the input file, the NAME is optional. Yet it might be needed in cases where there are many meshes and one needs to refer to a particular mesh, such as in MESH_POST or MESH_INTEGRATE. When solving PDEs (such as in Fino or milonga), the first mesh is the problem mesh. 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 information about physical groups. The spatial dimensions should be given with DIMENSION. If material properties are uniform and given with variables, the dimensions are not needed and will be read from the file. But if spatial functions are needed (either for properties or read from the mesh file), an explicit value for the mesh dimensions is needed. If either SCALE or OFFSET are given, the node position if first shifted and then scaled by the provided amounts. For each READ_SCALAR keyword, a point-wise defined 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. ## 2.3 MESH_FILL_VECTOR Fills the elements of a vector with data evaluated at the nodes or the cells of a mesh. MESH_FILL_VECTOR VECTOR <vector> { FUNCTION <function> | EXPRESSION <expr> } [ MESH <name> ] [ NODES | CELLS ] The vector to be filled needs to be already defined and to have the appropriate size, either the number of nodes or cells of the mesh depending on NODES or CELLS (default is nodes). The elements of the vectors will be either the FUNCTION or the EXPRESSION of x, y and z evaluated at the nodes or cells of the provided mesh. If there is more than one mesh, the name has to be given. ## 2.4 MESH_FIND_MINMAX Finds absolute extrema of a function or expression within a mesh-based domain. MESH_FIND_MINMAX { FUNCTION <function> | EXPRESSION <expr> } [ MESH <name> ] [ NODES | CELLS ] [ MIN <variable> ] [ I_MIN <variable> ] [ X_MIN <variable> ] [ Y_MIN <variable> ] [Z_MIN <variable> ] [ MAX <variable> ] [ I_MAX <variable> ] [ X_MAX <variable> ] [ Y_MAX <variable> ] [Z_MAX <variable> ] Either a FUNCTION or an EXPRESSION should be given. In the first case, just the function name is expected (i.e. not its arguments). ## 2.5 MESH_INTEGRATE Performs a spatial integration of a function or expression over a mesh. MESH_INTEGRATE { FUNCTION <function> | EXPRESSION <expr> } [ MESH <mesh_identifier> ] [ OVER <physical_group> ] [ NODES | CELLS ] RESULT <variable> The integrand may be either a FUNCTION or an EXPRESSION. In the first case, just the function name is expected (i.e. not its arguments). In the second case, a full algebraic expression including the arguments is expected. If the expression is just 1 then the volume (or area or length) of the domain is computed. Note that arguments ought to be x, y and/or z. If there are more than one mesh defined, an explicit one has to be given with MESH. By default the integration is performed over the highest-dimensional elements of the mesh. If the integration is to be carried out over just a physical group, it has to be given in OVER. Either NODES or CELLS define how the integration is to be performed. In the first case a the integration is performed using the Gauss points and weights associated to each element type. In the second case, the integral is computed as the sum of the product of the function evaluated at the center of each cell (element) and the cell’s volume. The scalar result of the integration is stored in the variable given by RESULT. If the variable does not exist, it is created. In the second case, a full algebraic expression including the arguments is expected. ## 2.6 MESH_MAIN MESH_MAIN [ <name> ] ## 2.7 MESH_POST MESH_POST [ MESH <mesh_identifier> ] { FILE <name> | FILE_PATH <file_path> } [ NO_MESH ] [ FORMAT { gmsh | vtk } ] [ CELLS | ] NODES ] [ NO_PHYSICAL_NAMES ] [ VECTOR <function1_x> <function1_y> <function1_z> ] [...] [ <scalar_function_1> ] [ <scalar_function_2> ] ... ## 2.8 PHYSICAL_GROUP Defines a physical group of elements within a mesh file. PHYSICAL_GROUP <name> [ MESH <name> ] [ DIMENSION <expr> ] [ MATERIAL <name> ] [ BC <bc_1> <bc_2> ... ] A name is mandatory for each physical group defined within the input file. If there is no physical group with the provided name in the mesh, this instruction makes 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. 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. The MATERIAL keyword in PHYSICAL_GROUP is used to link a physical group in a mesh file and a material in the wasora input file with different names. For non-volumetric elements, boundary conditions can be assigned by using the BC keyword. This should be the last keyword of the line, and any token afterwards is treated specially by the underlying solver (i.e. Fino or milonga). ## 2.9 PHYSICAL_PROPERTY PHYSICAL_PROPERTY <name> [ <material_name1> <expr1> [ <material_name2> <expr2> ] ... ] # 3 Variables ## 3.1 done Flag that indicates whether the overall calculation is over. ## 3.2 done_outer Flag that indicates whether the parametric, optimization of fit calculation is over or not. It is set to true (i.e. \neq 0) by wasora whenever the outer calculation is considered to be finished, which can be that the parametric calculation swept the desired parameter space or that the optimization algorithm reached the desired convergence criteria. If the user sets it to true, the current step is marked as the last outer step and the transient calculation ends after finishing the step. ## 3.3 done_static Flag that indicates whether the static calculation is over or not. It is set to true (i.e. \neq 0) by wasora 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. ## 3.4 done_transient Flag that indicates whether the transient calculation is over or not. It is set to true (i.e. \neq 0) by wasora 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. ## 3.5 dt Actual value of the time step for transient calculations. When solving DAE systems, this variable is set by wasora. 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 1/16, which is a power of two and roundoff errors are thus reduced. ## 3.6 end_time Final time of the transient calculation, to be set by the user. The default value is zero, meaning no transient calculation. ## 3.7 i Dummy index, used mainly in vector and matrix row subindex expressions. ## 3.8 infinite A very big positive number, which can be used as end_time = infinite or to define improper integrals with infinite limits. Default is 2^{50} \approx 1 \times 10^{15}. ## 3.9 in_outer_initial Flag that indicates if the current step is the initial step of an optimization of fit run. ## 3.10 in_static Flag that indicates if wasora is solving the iterative static calculation. Flag that indicates if wasora is in the first step of the iterative static calculation. Flag that indicates if wasora is in the last step of the iterative static calculation. ## 3.11 in_transient Flag that indicates if wasora is solving transient calculation. ## 3.12 in_transient_first Flag that indicates if wasora is in the first step of the transient calculation. ## 3.13 in_transient_last Flag that indicates if wasora is in the last step of the transient calculation. ## 3.14 j Dummy index, used mainly in matrix column subindex expressions. ## 3.15 max_dt Maximum bound for the time step that wasora should take when solving DAE systems. ## 3.16 min_dt Minimum bound for the time step that wasora should take when solving DAE systems. ## 3.17 ncores 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). ## 3.18 on_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. ## 3.19 on_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 IDA Library. ## 3.20 on_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 wasora. ## 3.21 pi A double-precision floating point representaion of the number \pi, equal to math.h ’s M_PI constant. ## 3.22 pid The UNIX process id of wasora (or the plugin). ## 3.23 realtime_scale If this variable is not zero, then the transient problem is run trying to syncrhonize the problem time with realtime, up to a scale given. For example, if the scale is set to one, then wasora will advance the problem time at the same pace that the real wall time advances. If set to two, wasora’s time wil advance twice as fast as real time, and so on. If the calculation time is slower than real time modified by the scale, this variable has no effect on the overall behavior and execution will proceed as quick as possible with no delays. ## 3.24 rel_error 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. ## 3.25 static_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. ## 3.26 step_outer Indicates the current step number of the iterative outer calculation (parametric, optimization or fit). Indicates the current step number of the iterative inner calculation (optimization or fit). ## 3.27 step_static Indicates the current step number of the iterative static calculation. ## 3.28 step_transient Indicates the current step number of the transient static calculation. ## 3.29 t Actual value of the time for transient calculations. This variable is set by wasora, 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. ## 3.30 zero A very small positive number, which 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} . # 4 Mesh-related variables ## 4.1 bbox_min Minimum values of the mesh’s bounding box (vector of size 3) Maximum values of the mesh’s bounding box (vector of size 3) ## 4.2 cells Number of cells of the unstructured grid. This number is the actual quantity of volumetric elements in which the domain was discretized. ## 4.3 elements Number of total elements of the unstructured grid. This number include those surface elements that belong to boundary physical entities. ## 4.4 nodes Number of nodes of the unstructured grid. # 5 Functions ## 5.1 abs Returns the absolute value of the argument x. y = abs(x) ## 5.2 acos Computes arc in radians whose cosine is equal to the argument x. A NaN error is raised if |x|>1. y = acos(x) ## 5.3 asin Computes arc in radians whose sine is equal to the argument x. A NaN error is raised if |x|>1. y = asin(x) ## 5.4 atan Computes, in radians, the arc tangent of the argument x. atan(x) ## 5.5 atan2 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]. atan(y,x) ## 5.6 ceil Returns the smallest integral value not less than the argument x. ceil(x) ## 5.7 clock Returns the value of a certain clock in seconds measured from a certain (but specific) milestone. The kind of clock and the initial milestone depends on the optional flag f. It defaults to zero, meaning wall time since the UNIX Epoch. 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]) ## 5.8 cos 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) ## 5.9 cosh Computes the hyperbolic cosine of the argument x, where x is in radians. cosh(x) ## 5.10 d_dt Computes the time derivative of the signal x using the difference between the value of the signal in the previous time step and the actual value divided by the time step. For t=0, the return value is zero. Unlike the functional derivative, this function works with expressions and not with functions. Therefore the argument x may be for example an expression involving a variable that may be read from a shared-memory object, whose time derivative cannot be computed with derivative. d_dt(x) ## 5.11 deadband Filters the first argument x with a deadband centered at zero with an amplitude given by the second argument a. deadband(x, a) ## 5.12 equal 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^{-16}. equal(a, b, [eps]) ## 5.13 exp Computes the exponential function the argument x, i.e. the base of the natural logarithms raised to the x-th power. exp(x) ## 5.14 expint1 Computes the first exponential integral function of the argument x. If x equals zero, a NaN error is issued. expint1(x) ## 5.15 expint2 Computes the second exponential integral function of the argument x. expint2(x) ## 5.16 expint3 Computes the third exponential integral function of the argument x. expint3(x) ## 5.17 expintn Computes the n-th exponential integral function of the argument x. If n equals zero or one and x zero, a NaN error is issued. expintn(n,x) ## 5.18 floor Returns the largest integral value not greater than the argument x. floor(x) ## 5.19 heaviside Computes the zero-centered Heaviside step function of the argument x. If the optional second argument \epsilon is provided, the discontinuous step at x=0 is replaced by a ramp starting at x=0 and finishing at x=\epsilon. heaviside(x, [eps]) ## 5.20 if 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^{-16}. 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]) ## 5.21 integral_dt Computes the time integral of the signal x using 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, this function works with expressions and not with functions. Therefore the argument x may be for example an expression involving a variable that may be read from a shared-memory object, whose time integral cannot be computed with integral. integral_dt(x) ## 5.22 integral_euler_dt Idem as integral_dt but uses the backward Euler rule to update the integral value. This function is provided in case this particular way of approximating time integrals is needed. integral_euler_dt(x) ## 5.23 is_even Returns one if the argument x rounded to the nearest integer is even. y = is_even(x) ## 5.24 is_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) ## 5.25 is_odd Returns one if the argument x rounded to the nearest integer is odd. y = is_odd(x) ## 5.26 j0 Computes the regular cylindrical Bessel function of zeroth order evaluated at the argument x. j0(x) ## 5.27 lag Filters the first argument x(t) with a first-order lag of characteristic time \tau, i.e. this function applies the transfer function !bt [ G(s) = ] !et to the time-dependent signal x(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) ## 5.28 lag_bilinear Filters the first argument x(t) with a first-order lag of characteristic time \tau, i.e. this function applies the transfer function !bt [ G(s) = ] !et to the time-dependent signal x(t) by using the bilinear transformation formula. lag_bilinear(x, tau) ## 5.29 lag_euler Filters the first argument x(t) with a first-order lag of characteristic time \tau, i.e. this function applies the transfer function !bt [ G(s) = ] !et to the time-dependent signal x(t) by using the Euler forward rule. lag_euler(x, tau) ## 5.30 last Returns the value the signal 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 insi expression x. See example number 2. last(x,[p]) ## 5.31 limit 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) ## 5.32 limit_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) ## 5.33 log Computes the natural logarithm of the argument x. If x is zero or negative, a NaN error is issued. log(x) ## 5.34 mark_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]) ## 5.35 mark_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]) ## 5.36 max Returns the maximum of the arguments x_i provided. Currently only maximum of ten arguments can be provided. max(x1, x2, [...], [x10]) ## 5.37 min Returns the minimum of the arguments x_i provided. Currently only maximum of ten arguments can be provided. min(x1, x2, [...], [x10]) ## 5.38 mod Returns the remainder of the division between the first argument a and the second b. Both arguments may be non-integral. mod(a, b) ## 5.39 not 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^{-16}. not(x, [eps]) ## 5.40 random 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]) ## 5.41 random_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]) ## 5.42 round Rounds the argument x to the nearest integer. Halfway cases are rounded away from zero. round(x) ## 5.43 sawtooth_wave Computes a sawtooth wave betwen 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) ## 5.44 sgn 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^{-16}. sgn(x, [eps]) ## 5.45 sin 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) ## 5.46 sinh Computes the hyperbolic sine of the argument x, where x is in radians. sinh(x) ## 5.47 sqrt Computes the positive square root of the argument x. If x is negative, a NaN error is issued. sqrt(x) ## 5.48 square_wave Computes a square function betwen zero and one with a period equal to one. The output is one for 0 < x < 1/2 and goes to zero for 1/2 < 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) ## 5.49 tan Computes the tangent of the argument x, where x is in radians. tan(x) ## 5.50 tanh Computes the hyperbolic tangent of the argument x, where x is in radians. tanh(x) ## 5.51 threshold_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]) ## 5.52 threshold_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]) ## 5.53 triangular_wave Computes a triangular wave betwen 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$\$ controls the frequency of the wave and \phi controls its phase.

triangular_wave(x)

# 6 Functionals

## 6.1 derivative

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_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])

## 6.2 func_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])

## 6.3 gauss_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])

## 6.4 gauss_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])

## 6.5 integral

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])

## 6.6 prod

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)

## 6.7 root

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])

## 6.8 sum

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)

# 7 Vector functions

## 7.1 vecdot

Computes the dot product between vectors \vec{a} and \vec{b}, which should have the same size.

vecdot(a,b)

## 7.2 vecmax

Returns the biggest element of vector \vec{b}, taking into account its sign (i.e. 1 > -2).

vecmax(b)

## 7.3 vecmaxindex

Returns the index of the biggest element of vector \vec{b}, taking into account its sign (i.e. 2 > -1).

vecmaxindex(b)

## 7.4 vecmin

Returns the smallest element of vector \vec{b}, taking into account its sign (i.e. -2 < 1).

vecmin(b)

## 7.5 vecminindex

Returns the index of the smallest element of vector \vec{b}, taking into account its sign (i.e. -2 < 1).

vecminindex(b)

## 7.6 vecnorm

Computes euclidean norm of vector \vec{b}. Other norms can be computed explicitly using the sum functional, as illustrated in the example.

vecnorm(b)

## 7.7 vecsize

Returns the size of vector \vec{b}.

vecsize(b)

## 7.8 vecsum

Computes the sum of all the components of vector \vec{b}.

vecsum(b)