Both LIR and EJ solve the Poisson equations by an iterative approach. The basic idea of an iterative method is to:

1. Start with some initial guess for the unknown values.
2. Improve the current values in each iterative step.
3. Repeat the iterative process until one of the stopping criteria (Error Bound, Tolerance, Residual, Maximal Iterations or Maximal Run Time) occurs.

For the EJ solver, select either Tolerance (recommended default) or Residual as stopping criterion. For the LIR solver, select Error Bound (recommended default) or Tolerance as stopping criterion.

Tolerance detects if the iterative process becomes stationary. This occurs when the change in the conductivity value from iteration to iteration becomes extremely small. If the relative change is smaller than the value entered for Tolerance the iteration is stopped.

When the Residual stopping criterion is used, the iteration is stopped if the solution satisfies the equation up to the required accuracy.

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.

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. Following possibilities might help:

• 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 an iterative solver to fail.

Which stopping criterion has occurred, can be seen in the Result Viewer of the GeoDict result file (*.gdr) under the Results Map tab.

The calculations run by the solvers can be restarted from saved intermediate result files 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 EJ, and Sequential, Parallel (Shared Memory), and Automatic Number of Threads for LIR.

For details on how to set up und run parallel computations, consult the High Performance Computing handbook.

Parallelization Benchmark Results Two examples of parallelization benchmark results for ConductoDict are shown here. Both benchmark computations were run on a server with 2xIntel E5-2697A v4 processors with 16 cores each, running with a maximum of 3.60 GHz, and 1024 GB RAM. The first example is the computation of the electrical conductivity through a synthetic X-ray tomography data of a sandstone structure (Mattila et al. 2016).The structure has a porosity of 13.6% and a size of 2048x2048x2048 voxels. The electrical conductivity of the structure, for pores filled with brine, is computed with the EJ and the LIR solver. For sandstone, an electrical conductivity of 1e-08 S/m and for the brine of 5 S/m is used as input values. The following figure shows the runtime for a different number of processes to compute the electrical conductivity of the structure of 0.1137 S/m with an error bound of 0.001 for LIR and of 0.1135 S/m with a tolerance of 0.001 with EJ solver. For both solvers, the ideal speedup, i.e. getting half of the runtime for twice the number of processes, is also shown. The speedup for LIR solver is close to the ideal one, which is usually not possible to reach for real life examples. The speedup of EJ differs more from the ideal one. For both EJ and LIR, the performance in GeoDict 2025 is comparable to the one in GeoDict 2024. The memory requirement for this example is 233 GB when LIR is used with the option of speed optimization, and is 239GB when EJ is used. The second benchmark is the computation of transverse isotropic thermal conductivity in a gas diffusion layer (GDL). The structure size is 4096x4096x512. A portion of the structure, of size 512x512x512 voxels, is shown in the figure below. The synthetic GDL, created with GeoDict, consists of 7% carbon fibers and 4% binder. The transverse isotropic thermal conductivity of 0.3 W/(mK) in both in-plane directions of the GDL and of 0.053 W/(mK) in through-plane direction is computed with the LIR solver with an error bound of 0.01. The following thermal conductivities for the different materials of the structure where used as input for the computation. Material Thermal Conductivity W/(mK) Fibers, in fiber direction 10 Fibers, orthogonal to fiber direction 2 Binder 1 Air 0.0257 Runtimes for a different number of processes are shown in the following figure. Again, the speedup of the LIR solver is close to the ideal one. For this example, LIR requires 259 GB memory.

In some situations, it may be useful to re-use previously computed results and save a great amount of time by restarting the process from an existing GeoDict result file (*.gdr). Typical examples would be

• non-sufficient accuracy of some computation, when it is suspected that more iterations may improve the quality of the result,
• trying another boundary condition (e.g., Dirichlet instead of Periodic or other way around),
• continuing an iteration process,
• or using similar parameter values for the computation.

To restart and initialize the solver from a result file (*.gdr), check Restart from .gdr File and enter the name of a file, or Browse for it in the project folder.

When using Restart From .gdr File only the volume field (*.hht file) associated to the loaded *.gdr file will be used to continue the simulation.

No other parameters from the options dialog will be loaded from the *.gdr file, but it is possible to load them by clicking Load Parameters in the Result Viewer. This way the simulation can be continued while using the same settings as in the initial simulation.

Checking Discard PDE Solver Files causes the deletion of all intermediate computation files, such as log files and flow field files.

Only the content of the final GeoDict result file (*.gdr) is stored.

While having the benefit of saving hard disk storage place, discarding intermediate solver files has also the effect of disabling the 3D visualization of the results.

If one of the constituent materials has a non-isotropic material law, a local orientation is needed to compute the permeability. 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 (Use Orientation from Analytic Objects (gad)).

However, if the current structure was not generated using one of the structure generation modules, but imported from a 3D scan, 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, by selecting Load Orientation Information from File (*.gof).

The last option is to simply Use the Global XYZ-Coordinate System.

In this case, the permeabilities entered in the Constituent Materials tab are the permeabilities in the X-, Y- and Z-directions.

If contact resistances between objects of the same material have been set in the Constituent Materials tab, GeoDict must be able to distinguish between different objects of the same type. This can be done in two different ways.

The standard approach is to use the GAD information available for the current structure:

Such information is available if a green dot is shown in front of GAD Objects in the Project Status section (left side of the GUI).

A red dot might be shown if the current structure was not generated using one of the structure generation modules but imported from a 3D image. Then, GAD object information is not available, and the structure must be segmented into separate objects first, e.g. by using FiberFind or GrainFind. It is then possible to load the segmentation results from a file generated by one of those modules:

Additional 3D data can be added to the solution *.hht files for visualization or later analysis. With Write Heat Flux into Solution File the Heat Flux in the three coordinate directions is saved, allowing a detailed analysis of the field. The size of the result files and the memory requirements increase when selecting this option.

Checking Analyze Geometry performs an analysis of the geometry before the solver computations.

If no percolation path through the structure is found, the partial differential equation does not need to be solved, and a permeability of 0.0 is directly reported as the solution.

Furthermore, the solver operates on the whole structure, regardless of which parts are connected or unconnected. However, unconnected components are not transport relevant, thus the effort of solving in these parts is not necessary.

For these reasons, a geometrical analysis is routinely run to determine whether a through path exists. After the analysis is finished, unconnected components are removed from the computational grid.

This may speed up the computations but requires some time for the geometrical analysis, especially for very large structures. So if you know beforehand that there are no unrelevant parts or that the structure has been processed to eliminate them, the geometry analysis can be switched off by unchecking Analyze Geometry.

The LIR solver uses a very memory efficient adaptive grid structure for the simulations.

If the option Write Compressed Volume Fields is checked, then the adaptive grid 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. But the runtime for loading and uncompressing of compressed *.hht is increased by the amount of runtime that was saved for writing out compressed *.hht files.

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 to solve the Stokes and Stokes–Brinkman equations as well as for solving mechanics, diffusion, thermal, and electrical conduction and is enabled by default.

Another speed-up option to accelerate the convergence behavior of the LIR solver is called Krylov Subspace Method.

The runtime of the LIR solver depends on many different properties of the structure and the simulation parameters. The BiCGStab algorithm is used, which can reduce the runtime for challenging simulation very drastically.

Unfortunately, the Krylov method is not always faster than a simulation without the Krylov method and therefore we introduced an Automatic mode which uses some heuristics to choose the most efficient method based on structure, material parameters, and boundary conditions automatically. Of course, it is possible to explicitly enable (Enabled) or disable (Disabled) the method.

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 more complex the problem is, the more stable the solver settings should be.

With the Relaxation number, the solver is adjusted from Stable (which results in higher number of iterations, slower time stepping, and longer solver run times), to Fast, which makes the solver run less iterations but implies the risk that the solver does not converge. The Relaxation is a parameter of the SOR method and must be between 0 and 2 to ensure convergence. For relaxation values smaller than one (<1.0), the simulation is more stable. For relaxation values larger than one (>1.0), the simulation converges 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, 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 grid structure called LIR-tree and needs up to 10 times less runtime and memory compared to the Regular Grid option.

The solver can analyze the result field during the computation and improves the adaptive grid in places where more accuracy is needed. The LIR solver splits cells where a high gradient occurs.

The solver refines the adaptive grid based on the computed fields. New cells are introduced at locations where higher accuracy is needed. Three settings are available here:

• Automatic: The computational cells are split when the difference in values between neighboring cells exceeds an automatically determined threshold. The solver automatically chooses all internal parameters based on the structure and simulation settings.
• Manual: The computational cells are split when the difference in values between neighboring cells exceeds the specified threshold. With this option you can specify the parameters Threshold for Grid Refinement and the Number of Grid Refinements manually. The Threshold for Grid Refinement must be between 0.0 and 1.0, while the recommended value range is between 0.05 and 0.1. The solver refines the grid at regions with high gradients and thus, the accuracy of the grid is enhanced in areas where it is necessary. Cells are split where the current gradient exceeds the Threshold multiplied by the maximum gradient. The Number of Grid Refinements is the maximum number of grid refinements that the solver can perform and should be between 1 and 10. The gradient-based grid refinement can increase the number of iterations, runtime, and memory requirements.
• Disabled: The grid refinement is turned off.