FEATool Multiphysics
v1.14
Finite Element Analysis Toolbox

This section describes the different physics solvers available with the FEATool Multiphysics toolbox. Physics solvers compute numerical approximations to equations and boundary conditions with discretization schemes such as FDM/FEM/FVM. This results in sparse matrix systems which are subsequently solved (inverted) to yield discrete and approximate numerical solutions to the equations.
FEATool features integrated GUI and CLI interfaces to the linear and nonlinear stationary (solvestat), timedependent (solvetime), and eigenvalue (solveeig) multiphysics solvers. In addition, interfaces to external solvers, such as the Computational Fluid Dynamics (CFD) solvers OpenFOAM and SU2, and the FEA solver FEniCS is also builtin.
The external solver interfaces allow for setting up and defining multiphysics problems directly in MATLAB with the FEATool GUI and CLI interfaces, then export and solve them with the external solvers. Afterwards, the solutions can automatically be imported into the FEATool GUI for further postprocessing. Moreover, the multisimulation solver approach employed by FEATool conveniently allows switching between several solvers for the same problem with oneclick, this can be very useful when comparing and validating results and solutions.
After geometry, grid, and equation and boundary conditions have been defined one can switch to solve mode by either pressing the mode button, or by selecting Solve Mode in the Solve menu. The solve mode toolbar buttons and menu actions are the following
The builtin solvers allow for solving coupled stationary, timedependent, and eigenvalue/frequency multiphysics problems, for which solver settings and parameters and are discussed in the following.
The Solver Settings control panel and dialog box shown below can be opened by pressing the corresponding toolbar button or by selecting Solver Settings... in the Solve menu. The solver settings dialog box allow selecting and configuring the builtin multiphysics solvers.
The Solver Type dropdown box allows for either of the
solvers to be selected. The builtin multiphysics solvers are monolithic, meaning that all equations and dependent variables are discretized and solved together in a large coupled system, rather than iteratively solving a segregated set of smaller decoupled systems. This typically allows for faster and more robust convergence behavior at the expense of higher memory usage.
The Initial Condition setting controls whether initial values should be computed from Expressions (from the equation settings initial value fields), or use a previously Computed solution. In the latter case the initial solution can be selected from the corresponding list box. Note that for this to work the geometry, grid, physics modes, and finite element shape functions must all remain the same.
The Linear solver used to solve sparse matrix systems can be selected from the following options
The default linear solver option is to use either the builtin direct solver (MATLAB's backslash operator) or, alternatively the direct solver mumps which can be noticeably faster and more efficient for many problems. Direct solvers are very robust and fast for a wide range of problems, but are not as memory efficient as iterative solvers such as gmres, bicgstab, or amg. The algebraic multigrid amg solver is experimental, but when it works it potentially can combine the best of solving speed with memory efficiency. Users can also incorporate their own or custom linear solvers by overloading or editing the solvelin function.
The Integration rule or numerical quadrature order used to numerically approximate equation integrals is automatically set to 1 + max(FEM shape function order) by default, but can also be set manually between 1st to 5th order if desired.
A fixed point/Newton nonlinear solution process is automatically used when dependent variables are detected in equations and/or boundary coefficients. The NonLinear Solver Settings allows control over the Maximum number of nonlinear iterations, and the Nonlinear relaxation parameter which is the fraction of the new to old solution to weigh in to the next nonlinear iteration. The nonlinear solver will terminate and stop when both the Defect stopping criteria and Solution changes stopping criteria are met which can be set to be measured in either absolute or relative error norms.
The Time Dependent Settings control timestepping parameters when using a timedependent solver. Time step sets the time step size for the timestepping schemes. Time dependent simulations will terminate when either the accumulated time has reached the specified Simulation time, or the normed changes in the dependent solution variables are smaller than the Time stopping criteria, which would indicate a stationary state.
The Time stepping scheme setting allows choosing between the following schemes
The default CrankNicolson scheme works well in most situations, for very nonlinear or sensitive timestepping problems one can use the backward Euler method which is more robust but less accurate (to compensate smaller time steps can be used).
Lastly, one can choose between using the Full mass matrix, Row rowsum lumped (default), Diagonal lumped, or consistent HRZlumped Mass matrix type representations.
In the Eigenvalue Settings frame one can control the Number of eigenvalues to be computed, as well as the Type of eigenvalues to find (smallest/largest or a search region around a specified eigenvalue/frequency).
Solver hooks allow modification of the finite element problem struct and matrices before and after the linear solver calls in the stationary (solvestat) or timedependent (solvetime) solution process. This enables implementation of nonstandard equations and boundary conditions, such as periodic boundary conditions. To define a solver hook a Neumann boundary condition must be assigned with the argument solve_hook_fname
where fname is the name of the custom function to be called (an mfile function must exist in the MATLAB paths or current working directory with this name). The function to be called must have a function signature of the form
function [ A, f, fea ] = fname( A, f, fea, i_dvar, j_bdr, solve_step )
where A
is the sparse matrix, f
the right hand side vector, fea
the finite element problem struct, i_dvar
dependent variable index, j_bdr
boundary number, and solve_step
equal to 1 after matrix assembly (but before the linear solver call) and 2 after the linear solver call.
For examples of using solver hooks see the following examples with periodic boundary condition examples
It is also possible to define custom solver monitoring functions by creating a mscript function with the name fea_solve_monitor.m
. If present and detected by MATLAB the function will be called by the builtin solvers after each iteration or timestep with the following function signatures
function [ halt ] = fea_solve_monitor( fea, it ) % for solvestat function [ halt ] = fea_solve_monitor( fea, tlist ) % for solvetime
where fea
is the finite element problem struct, it
the nonlinear iteration number, and tlist
the list of time steps corresponding to the number of columns in the solution vector stored in fea.sol.u. The output argument halt
is a logical scalar which indicates if the solution process should be terminated early. For example, the following function for the Shrink Fitting of an Assembly model example will monitor and plot the temperature, T, as a function of time at the point (0.06, 0.06), and stop the solution process when the temperature has reached 15 degrees or lower.
function [ halt ] = fea_solve_monitor( fea, tlist ); persistent h_fig if( isempty(h_fig) ) h_fig = figure( 'Name', 'Custom Solver Monitor' ); end for i=1:length(tlist) T(i) = evalexpr( 'T', [0.06;0.06], fea, i ); end clf plot( tlist, T ) grid on title( ['T@t = ',num2str(tlist(end))]) xlabel( 'Time [s]' ) ylabel( 'Temperature, T(0.06,0.06)' ) drawnow halt = T(end) <= 15;
Fluidstructure Interaction (FSI) problems feature a dedicated solver (fsisolve) which handles the physics coupling between the fluid and solid domains, and allows for movement of the grid by using a solid extension mesh motion technique with Jacobianbased stiffening [1].
The FSI solver supports timedependent problems with either fullyimplicit or semiimplicit discretizations (with explicit geometry treatment). The Initial Condition section allows for either initial conditions from a predefined constant or expression, or interpolated from a previously computed solution.
The Time Dependent Settings section allows for prescribing the Initial time, Time step size, and Final time. Specific Output times can also be supplied as a vector of specific times to save and output (default [t_{0}:t_{step}:t_{end}]). Moreover, the order and accuracy of the employed time discretization schemes can also be selected (1st to 5th for the fluid domain using a Backward Difference Formula (BDF) discretization, and Newmark scheme for the solid domain).
[1] A. Quarteroni, A. Manzoni, F. Negri. Reduced Basis Methods for Partial Differential Equations. An Introduction, Springer, 2016.
The following functions can be used on the MATLAB command line interface for to solve FEATool Multiphysics simulation problems.
Function  Description 

fenics  Solve problem with the external FEniCS solver 
fenics_data  Convert FEATool problem to FEniCS 
openfoam  Solve problem with the external OpenFOAM solver 
turbulence_inletbccalc  Calculate turbulence quantities at inlets 
su2  Solve problem with the external SU2 solver 
fsisolve  Solver for fluidstructure interaction problems 
solvelin  Wrapper around linear sparse Ax=b system solvers 
solveeig  Solve eigenvalue/frequency problem 
solvestat  Solve stationary problem 
solvetime  Solve timedependent/instationary problem 