- Title
- Efficient iterative sparse linear solvers for large-scale computational geomechanics
- Creator
- Kardani, Omid
- Relation
- University of Newcastle Research Higher Degree Thesis
- Resource Type
- thesis
- Date
- 2015
- Description
- Research Doctorate - Doctor of Philosophy (PhD)
- Description
- It can be argued that a substantial part of engineering design problems can be formulated and solved using mathematical programming or simply optimisation techniques. Therefore, there is no surprise that during last two decades, the use of mathematical programming has been intensively applied to problems in geomechanics. Recently, significant advances have been made in a special branch of optimisation known as conic programming, which has attracted much attention as an efficient tool in dealing with difficult problems in geomechanics. Some of its important applications in geomechanics include elastoplastic analysis, upper- and lower-bound finite element limit analysis (FELA) and most recently granular contact dynamics. Upon formulating the original engineering problem as second-order cone program (SOCP), it can be solved by the well-known primal-dual interior point method (IPM). However, in each step of this method, a symmetric indefinite augmented linear system of equations needs to be solved. Traditionally, the augmented system is reduced to its Schur complement form and then solved by a direct solver. Being symmetric positive definite (SPD), the Schur complement system can be efficiently solved by the Cholesky decomposition method. While being robust and accurate, direct solvers require polynomially growing memory requirements as well as computational time for large three-dimensional problems. In fact, with problem size increasing beyond a few million variables, the use of direct solvers becomes more and more prohibitive (both in storage and computational times) due to the increased bandwidth of 3D problems, even on modern computers. This motivates the feasibility study of preconditioned iterative methods to solve systems of equations resulting from interior point method (IPM) in the context of computational geomechanics. In this thesis, preconditioned iterative solution schemes are developed that can efficiently deal with extremely large problems in the framework of IPM algorithm. To this end, it investigates potential preconditioned iterative solution methods for this context and their efficiency for solving large-scale problems in computational Geomechanics. A major part of the present research is devoted to the design, development and implementation of efficient iterative solvers in two major stages based on target computational environments: sequential and parallel. The first stage aims to develop efficient solvers for sequential computational environments, targeting implementation on the central processing unit (CPU) of a standard computer. A number of potential preconditioning techniques as well as iterative solution methods are considered. Most suitable methods are identified through an extensive comparative study. Finally, a novel preconditioned iterative solution method is designed and implemented in the IPM framework and its robustness and efficiency are verified by numerical tests. Particularly, it is shown that the developed solver effectively overcomes the memory barrier encountered by the direct solvers. In the second stage, the possible improvement of the computational time of the preconditioned iterative solution method is investigated by implementation on massively parallel graphic processing units (GPUs). For this purpose, potential preconditioning methods suitable for parallel implementation are compared and the most efficient solution scheme is identified based on the corresponding improvement of the computational time resulting from different methods. Finally, a novel GPU-accelerated preconditioned iterative solution method is designed and implemented in the IPM framework and its efficiency in improving storage and computational requirements is verified by means of large-scale numerical tests.
- Subject
- computational geomechanics; sparse linear solvers; preconditioning; finite element limit analysis; conic optimisation; interior point method; graphic processing units
- Identifier
- http://hdl.handle.net/1959.13/1310569
- Identifier
- uon:22053
- Rights
- Copyright 2015 Omid Kardani
- Language
- eng
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