Two classes of problems in Geosciences, namely, target practical problems originating from mine exploration and foundation engineering have been considered. The problems are formulated in mathematical form as two- and three-dimensional partial differential equations as used in elasticity and plasticity. What concerns the models, the major aim is to consider sufficiently accurate 3D mathematical models. This brings requirements for use of various material models (elasticity, plasticity, rheology) as well as for modelling of contacts of bodies and/or thermo-mechanical influences in some cases. The accuracy and reliability may require also to perform more accurate local analysis (submodelling) or to consider a number of variants for determining the influence of some uncertain input parameters.
Once the mathematical models have been defined, the second major task becomes the development of efficient numerical solution methods. Emphasis is given both to the finite element (discretization) methodology and solution techniques of the arising linear and nonlinear algebraic systems. The discretization strategies include adaptive refinement, based of patched (overlaid) meshes, where the solution gradients are large. This captures the solution peculiarities more efficiently and accurately than if a uniformly sized mesh is used.
The numerical solution requires handling of linear and nonlinear algebraic systems of equations. Nonlinear equations solvers are studied in detail, namely, certain nonlinear equation solvers based on inexact Newton methods for nonlinear elliptic and plastic flow problems are analysed and implemented. The nonlinearity studied is due to material nonlinearity and can involve non-differentiable functions. A rate of convergence of the outer (nonlinear) iterations independent of meshsize and certain problem parameters is striven for. The treatment on nonlinear problems include two-grid methods and local refinement as well. The solution of linear systems is an inseparable ingredient of the numerical solution process both because the linear systems describe completely some model problems of interest and also they have to be solved as a part of the nonlinear solution procedures. A very important issue, however, is the size of the systems, which easily reaches ranges of several hundreds of thousands to several millions of degrees of freedom for real-life problems, to which the project is oriented. For instance, the ability to make more reliable prediction of safety of constructions sensitive to rock deformations, requires much more accurate modelling, and, hence, more degrees of freedom. Therefore, it is necessary to solve these systems by iteration using methods which are highly efficient on certain parallel computer architectures, having a fast rate of convergence (independent of meshsize and certain problem parameters) and at the same time, requiring little communication overhead. The aim is to use the parallel computer facility up to its full capacity achieving high Megaflop rates and high parallel efficiency which will result in total execution times at least 100 fold smaller than with presently used methods and computers for the type of problems considered. This turns out to be possible by combining accurate modelling and fast solution times in order to permit performing many simulation runs to support the decision taking process.
The computer implementation of the above numerical methods is based on some available codes for three-dimensional problems of elasticity and plasticity which are further enriched with possibilities to work with various finite element methods (including non-conforming ones) and various linear and nonlinear solution methods. The code is structured and open for future upgrading. It includes the new efficient solution techniques related to the local refinement and nonlinearity. Also, some linear equation solutions methods are implemented on various parallel computer platforms, namely, a parallel cluster of workstations, symmetric shared memory multiprocessors and a massively parallel distributed memory computer. These cover the commonly accessible computer architectures which can be expected to be available when solving the considered type of problems in the future. A two-dimensional more research oriented code using mixed finite element methods for some model elasticity and plasticity problems has also been developed.
The developed software is tested on real sized problems and the obtained results have been subject to analysis from experts in the corresponding application field and also compared with in situ observations.
Last changed on Mar 26 2001, mn