Select one or multiple of the stopping criteria Tolerance, Residual, Maximal Iterations or Maximal Run Time.

The default stopping criterion of the EJ solver, Tolerance, detects if the iterative process becomes stationary. This occurs when the change in the Diffusivity value from iteration to iteration becomes sufficiently small. If the relative change is smaller than the value entered for Tolerance, the iteration is stopped:

Alternatively, the Residual stopping criterion can be used. In this case, the iteration is stopped, if the solution satisfies the equation up to the required accuracy.

When the solver stops because the Maximal Iterations value or Maximal Run Time has been reached, no guarantee on the quality of solution can be given. In this case:

• Check the corresponding .log file to see how large the residual values and conductivity increase are. If these values are already very close to the desired result, you may decide to use the current result.
• Double check the structure and parameter values. Unphysical parameters or too rough resolution of the structure (leading, e.g., to artificial unconnected components) can cause the iterative solver to fail.

Which stopping criterion has occurred, can be seen in the result file under the Results Report tab.

The calculations run by the solvers can be restarted from intermediate results, and the interval between auto-saves can be configured from the value entered in Restart Save Interval (h).

Depending on the purchased license, the simulation process can be parallelized. The Parallelization Options dialog box opens when clicking the Edit... button, to choose between Sequential, Parallel (Shared Memory), Automatic Number of Threads and Cluster. For details on how to set up und run parallel computations, consult the High Performance Computing handbook of the User Guide.

In some situations, it may be useful to re-use previously computed results and, thus, reduce the runtime of the diffusivity computation. Typical examples would be when trying another boundary condition (Dirichlet instead of Periodic or other way around) or non-sufficient accuracy of some computation, when it is suspected that more iterations may improve the quality of the result.

To use some previously computed result, Restart from .gdr File can be checked and Browse used to search for the file.

Note that the structure used for restarting for both the current and the restart result file, must be the same. If this is not the case, an error message is displayed.

Checking the Discard PDE Solver Files box causes the deletion of all intermediate computation files. While having the benefit of saving storage place, discarding solver files has also the side effect of disabling the 3D visualization of the results.

Of course, the contents of the result file (*.gdr) are not discarded even in this case.

With Write Diffusion Flux into Solution File, the flux in the three coordinate directions is saved, allowing a detailed analysis of the flux field. The size of the result files increases when selecting this option.

Expand Advanced Options to see Analyze Geometry for EJ. If Analyze Geometry is checked, GeoDict checks if the structure contains a through path before starting the computation. If no path for the diffusing species through the structure is found, the partial differential equation does not need to be solved, and a diffusivity of 0.0 is directly reported as the solution. For LIR, there are more advanced options.

The default stopping criterion of the LIR solver, Error Bound, uses the result of previous iterations and predicts the final solution based on linear and quadratic extrapolation. The solver stops if the relative difference regarding the prediction is smaller than the specified error bound.

The stopping criterion recognizes oscillations in the convergence behavior and prevents premature stopping at local minima or maxima. A damped convergence curve is fit through the oscillating curve and the solver stops then regarding the damped convergence curve.

The stopping criteria Tolerance, Maximal Iterations and Maximal Run Time work as described for the EJ solver.

The options are the same as for the EJ solver and a description can be found above.

Depending on the purchased license, the simulation process can be parallelized. The Parallelization Options dialog box opens when clicking the Edit... button, to choose between Sequential, Parallel (Shared Memory) and Automatic Number of Threads. For details on how to set up und run parallel computations, consult the High Performance Computations handbook of the User Guide.

Parallelization Benchmark Results As an example for the parallelization, the diffusivity in through-direction in a Gas Diffusion Layer (GDL) is computed with different number of processes. The computation is run on a server with 2 x Intel E5-2697A v4 processors with 16 cores each, running with a maximum of 3.60 GHz, and 1 TB RAM. The GPL structure has size 8,192 x 8,192 x 512 voxels and is created in GeoDict with the GeoApp Gas Diffusion Layer, available in FiberGeo. The computation for the structure with over 34 billion voxels, requires 302 GB of RAM. The runtimes for a different number of parallel processes show that even with 8 processes, the simulation for the huge structure is possible in 3:50 h. Due to the very good speed-up for increasing number of processes, the computation with 32 processes is possible in 1:15 h. Runtimes in GeoDict 2025 are improved compared to GeoDict 2024 due to improvements in the cell generation code of the LIR solver.

If one of the constituent materials has a transverse isotropic or orthotropic material law, a local orientation is needed to compute the diffusion. There are three different choices how to determine the local orientation. The standard case is to use the orientation defined by the local orientation of the GAD objects, e.g. the direction of a fiber.

However, if the current structure was not generated using one of the structure generation modules, but imported from a 3D image, GAD object information is not available. In such a case, the local orientation must be estimated from the image first, e.g. by using FiberFind or GrainFind. It is then possible to load the local orientation from a file generated by one of those modules:

Last, one can simply use the coordinate system:

In this case, the entered diffusivities are the diffusivities in the X, Y, Z space directions:

For LIR, more solver settings are hidden under Advanced Options. Expand it to make them visible.

If Analyze Geometry is checked, GeoDict checks if the structure contains a through path before starting the computation. If no path for the diffusing species through the structure is found, the partial differential equation does not need to be solved, and a diffusivity of 0.0 is directly reported as the solution.

If the option Write Compressed Volume Fields is checked for LIR solver then the adaptive grid structure is used as compression method for writing out HHT files. This option allows to save 80-90% space on hard drive.

The runtime for writing HHT files is also reduced significantly. If the option Write Compressed Volume Fields is not checked then a usual regular grid is used for writing out HHT files.

The Multigrid Method (see e.g. Wesseling, 2004) was introduced to speed-up the computation and reduce the runtime significantly. The main idea of multigrid is the usage of multiple coarser adaptive grids to speed up convergence behavior but requires only little more memory.

The method is available for solving the Stokes and Stokes-Brinkman equations in FlowDict as well as for solving diffusion, thermal and electrical conduction in DiffuDict and ConductoDict and is enabled by default.

Depending on the structure and the corresponding material parameters, a significant speedup of the LIR can be achieved by using the BiCGstab method to compute the solution. Using the BiCGstab method approximately doubles the amount of RAM needed for the computation.

When Use Krylov Subspace Method is set to Automatic, GeoDict decides based on structure, material parameters and boundary condition which method is expected to be faster and uses this method. In case that the Krylov subspace method (BICGstab) is used, the Relaxation is also chosen automatically.

Alternatively, the user may also explicitly enable or disable this method.

If the ratio of the largest and smallest diffusivity within the structure (i.e. high contrast) is large (approx. >), usage of the Krylov method is recommended. For structures without a high diffusivity contrast, the usage of this option is not recommended.

Depending on the material parameters and geometry of the structure, the underlying mathematical problem can vary in complexity, thus influencing the behavior of the solver. The iterative method uses the Relaxation number to adjust it from stable (with smaller number chosen, which results in higher number of iterations, slower time stepping, and longer solver run times), to fast with higher number chosen, which makes the solver run less iterations but implies the risk that the solver does not converge.

For the LIR solver, this balance is managed through the Relaxation. The value should be between 0 and 2. For relaxation values smaller than one (<1.0), the simulation is more stable. For relaxation values larger than one (>1.0), the simulation is faster.

The LIR solver can Optimize for speed or memory.

If Speed is chosen, the solver constructs additional optimization structures. The runtime is decreased by up to 30% but requires up to 50% more memory compared to the other option. If Memory is chosen, then the runtime is increased by up to 40% but the solver requires up to 50% less memory.

The Grid Type decides what kind of tree structure is used for the simulation.

The default option is LIR-Tree and should always be used. The solver uses an adaptive tree structure called LIR-tree and needs up to 10 times less runtime and memory compared the Regular Grid option.

The solver can analyze the velocity and pressure field during the computation and improves the adaptive grid in places where more accuracy is needed. The LIR solver splits cells where a high velocity-gradient or high pressure-gradient occurs. The analysis is enabled if the Grid Refinement Criterion option is set to Automatic or Manual.

If the Grid Refinement is set to Automatic, the solver chooses the Number of Grid Refinements and Threshold for Grid Refinement automatically.

If the Grid Refinement is set to Manual, the user can enter the parameters manually.

The Number of Grid Refinements controls how many velocity-based and pressure-based grid refinements are allowed during the simulation. The value should be between 0 and 10. Velocity-based and pressure-based grid refinements may increase the number of iterations, runtime and memory requirements.

The Number of Grid Refinements can be zero in most of the cases and should be greater than zero if a flow simulation is done on a structure with a very long inlet and outlet, for pleated filter structures, or for Navier-Stokes simulations.

Refinement is done at regions with high-velocity gradient or high-pressure gradient. Cells are split where the current velocity gradient (or pressure gradient) is greater than the Threshold for Grid Refinement multiplied by the maximal velocity gradient (or pressure gradient). This threshold must be between 0.0 and 1.0. The recommended value range is between 0.05 and 0.1.