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Academy of Sciences CRCzech Geo

Finite element software GEM

A proprietary software called GEM is being developed and maintained for mathematical modelling and simulations of the thermo-hydro-mechanical processes in the rock mass that are associated with the construction, operation and safety of underground structures, e.g. mines or waste deposits.

GEM can be characterized as non-commercial 3D finite-element (FE) package oriented on the solution of problems arising in geosciences. GEM serves both research purposes and practical modelling. Its development is mainly problem-driven, reflecting the requirements of the current research and applications on increasing complexity of the models and methods.

Further, we address mainly GEM’s solvers, i.e. the modules responsible for numerical processing of the systems of equations arising from the FE analysis. It is computationally the most demanding phase in the FE method simulation chain preprocessing – assembling the FE system – solution of the system – postprocessing. GEM considers modelling of both the mechanical response, being in the centre of our interest from the very beginning, and the heat and flow phenomena included in the software recently.

Building blocks

Finite elements. For the solution of the problem of both elastic deformation and nonstationary temperature distribution, the finite element method is employed. The FE discretization of these boundary value problems, based on linear tetrahedral finite elements in GEM, leads to the linear system with a symmetric positive definite stiffness matrix, the right-hand side given by the loading and the vector of unknown nodal displacements. In the case of nonstationary heat equations, the linear system is solved in each time step.

Structured meshes. Since its very beginning, GEM uses structured meshes for the discretization of the modeled domain, which can be viewed as adaptation (deformation) of a regular rectangular (reference) grid of nodes to the solved problem.

Conjugate gradients and preconditioning. The symmetry and positive definiteness of the stiffness matrix permit to solve the linear system by the standard (iterative) preconditioned conjugate gradient method. In the PCG algorithm, the preconditioning should be efficient and parallelizable.

Parallel processing. The increasing demands on the size and complexity of the modelling and the growing availability of the multiprocessor systems promote parallel processing in the numerical solution. However, the parallelization of the solution is not straightforward, due to the irreducible global character of the solved systems.

Selected modules

  • Parallel iterative solver ISOL 1.45a for 3D boundary value problems of elasticity. The code follows the algorithm of the preconditioned conjugate gradient method and its parallelization is based on Schwarz methods and on a decomposition of the original domain into overlapping subdomains. The communication of parallel processes is realized by message passing given by the MPI standard.
  • Parallel iterative solver TERMO 1a for 3D solution of nonstationary heat equations. The solution of the linear system in each time step is realized by the conjugate gradient method with the preconditioning given by the additive Schwarz method with overlapping. The program code is parallelized with the aid of OpenMP paradigm.

Solved problems