Department of applied mathematics and computer science
Main research topics
The topics correspond to the research plan Effective Methods of Numerical Modeling of Physical Phenomena in the Geological Environment. This orientation includes several research fields. The first of them is numerical modelling of heat (T), hydraulic (H), and mechanical (M) processes in rocks, which is done both separately, neglecting the mutual influences, and jointly as coupled multiphysics modelling. The research in this area includes development of numerical methods and high performance means of solving large-scale 3D problems, considering more complicated nonlinear behaviour of geomaterials as well as analyzing and reducing the influence of uncertainty in input data. The described field of modelling brings a lot of motivation to applied computer science. More precisely, it motivates the research in code optimization, in parallel and distributed computing or computer graphics and the work in effective for methods of preparing input data and the visualization of the computed physical fields. The methods for multi-scale modelling of problems with high heterogeneity or microstructures and methods for the identification of material parameters are newly opened research topics. A specific field is the modelling of special reinforcement and geotechnical elements (bolts, geosynthetic materials) for geomechanics and civil engineering and modelling of behaviour of cable stayed and suspension bridges.
Important scientific achievements
- Modelling of THM processes in rocks
The in-house finite element software GEM3 is developed for the modelling of thermo-mechanical processes and the testing of numerical methods. The code has been applied to solving large scale problems with millions of nodal points. Originally the code was used for the solution of geomechanical problems, it is currently used also for thermo-mechanical analysis with application to the assessment of repositories of the spent nuclear fuel. The application of iterative methods and parallel programming techniques developed in the department allows us to solve large scale problems of huge dimensions.
- Schwarz domain decomposition methods
A series of new results came from the analysis and application of the Schwarz method for parallel solution of boundary value problems of heat conduction and elasticity. This method uses decomposition of the original domain into overlapping subdomains. We showed that for efficient solution of elasticity problems, it is necessary to complete the domain decomposition with some coarse approximate solution of the original problem, e.g. a solution of a coarser discretization of the problem. We also demonstrated that such an approximate solution can be obtained by the inexact solution of an algebraically constructed aggregated problem using inner iterations. Moreover, it is possible to use correctly nonsymmetric variants of the preconditioning. For evolutionary heat conduction problem, we showed that the coarse solution is not required. Recently, we have proposed a variant of the Schwarz methods for the solution of underground water flow problems by the mixed finite element method.
- Aggregation and algebraic multigrid
Many useful numerical methods can be based on a nested coarse and fine division of the problem domain into finite elements. But sometimes, it is difficult to use such nested divisions and there are many reasons for thinking about an algebraic construction of the coarse discretization. One possible algebraic construction of a coarse grid discretization is the aggregation which we have been dealing with for long time. New results are connected with a combination of aggregation and hierarchical preconditioning, the use of aggregation in the framework of nonconforming finite element methods and with Schwarz aggregation methods for solving elliptic problems.
- Parallel computations
The development of parallel computations at the Institute was started in 1995 already, in the context of two EC projects called High Performance Computing in Geosciences I, II. At present, we focus on the efficient implementation of the Schwarz methods for the solution of T, H, M problems on parallel computers. The parallel codes themselves are based on current standards for parallel processing, namely on MPI for platforms with distributed memory, e.g. clusters, and OpenMP for platforms with shared memory, symmetric multiprocessors. The Institute owns smaller parallel systems of both types. Thanks to the involvement of the Institute in the fast academic network infrastructure, larger parallel computations, which make use of tens of processors, can be carried out externally. For example, a great number of computations were completed in the UPPMAX computing centre at the Uppsala University, supported by the joint project Parallel Computing in Geosciences.
- Hierarchical methods
Standard hierarchical finite element methods use an approximation on nested grids and the corresponding decomposition of the stiffness matrix. Our results are connected with the analysis of efficiency of hierarchical decompositions, robustness with respect to anisotropy, application to nonconforming finite element methods and construction of aposteriori estimates of the discretization error.
- Homogenization techniques
Homogenization techniques are used in continuum mechanics problems to obtain coefficients describing the behaviour of environments with some microstructure. Our results are mainly connected with the modelling of rock masses reinforced with bolt anchoring and heat conduction together with mechanics of porous media in the case of nonlinear behaviour of material. The technique connected with the homogenization of bolt anchoring was implemented in the code GEM2 and successfully tested in practical problems.
- Analysis of cable stayed and suspension bridges
Cable stayed and suspension bridges are modelled as a one dimensional beam suspended by a cable system. For cable stayed bridges the main attention was paid to asymptotic stability in lateral wind. The influence of eigenvalues corresponding to vertical and torsion oscillations on the stability of bridges was also studied. For suspension bridges the main focus was on the nonlinearity of cable systems, where cable stays are attached to the main cable, which is freely flexible. This nonlinearity is the source of instability influencing the behaviour of such bridges.