Solving large scale engineering problems on parallel computers

Abstract

PhD thesis deals with three separate topics. The first topic of the thesis is the description of the parallel implementation of the conjugate gradient method into the SIFEL open source project. The SIFEL open source project has been developed at the Department of Mechanics, Faculty of Civil Engineering, CTU in Prague. The project is written in C/C++ programming language and is based on the finite element method. The conjugate gradient method is an iterative method for solving systems of linear algebraic equations and was published in 1952. The method uses the matrix-vector multiplication and the vector product. Therefore, the method is suitable for parallel implementation and for using on clusters and massive parallel computers. The thesis describes the implementation which is based not only on the parallelisation of the matrix-vector multiplication but also on the parallelisation of the vector product. An incomplete LU factorisation is used for preconditioning of the conjugate gradient method. The PETSc library contains the incomplete LU factorisation which is used for preconditioning. Numerical experiments show that the implementation is effective up to 100 processors. The preconditioned method can be effective used for fourth-order problems. The second separate topic of the thesis is the description of the preconditioning into the FETI method. The Finite Element Tearing and Interconnecting (FETI) method is one of the nonoverlapping domain decomposition method which was published by Farhat et al. in 1994. The method was implemented earlier but without a preconditioning. The preconditioning is needed for achieving the scalability. The scalability of the FETI method was theoretically and numerically proved in the articles by Farhat et al. Two preconditionings was published in the articles, lumped and Dirichlet. The Dirichlet preconditioning is mathematically optimal and is based on the evaluation of the Schur complement of subdomain matrices. However, it has a large demands for time. The lumped preconditionig is computationally optimal and is based on the matrix-vector multiplication. Both of these preconditionings were implemented into the SIFEL code. The description of the implementation is in the thesis. Numerical experiments show that results are in agreement with published results in the articles by Farhat et al. The third and main topic of the thesis is development of the algorithm for the selection of the fixing nodes in the FETI-DP method. The Finite Element Tearing and Interconnecting Dual-Primal (FETI-DP) method is one of the nonoverlapping domain decomposition method which was published by Farthat et al. in 2001. The method was developed due to problems with singular matrices in the original FETI method. The method is a combination of the original FETI method and Schur complement method. The method divides unknowns in the problems into three groups. The first group contains of fixing unknowns, which are selected form interface unknowns, and enforce the non-singularity of subdomain matrices. The second group contains of remaining interface unknowns. Lagrange multipliers, which are defined on remaining interface unknowns, enforce the continuity conditions among subdomains. The last group includes internal unknowns. The selection of fixing unknowns deserve special attention. Two algorithms are described in the thesis. These algorithms use graphs, which are defined with the help of interface finite element nodes. Numerical experiments show that the minimal needed number of fixing nodes does not lead to the best time of the solution. The increasing number of fixing unknowns leads to the decreasing the number of iterations in the coarse problem, which is obtained after elimination of internal and remaining interface unknowns. However, a large number of fixing unknowns prolong the whole time of solution. The optimal number of fixing unknowns, which leads to the shortest time of the solution, exists for every .

PhD thesis deals with three separate topics. The first topic of the thesis is the description of the parallel implementation of the conjugate gradient method into the SIFEL open source project. The SIFEL open source project has been developed at the Department of Mechanics, Faculty of Civil Engineering, CTU in Prague. The project is written in C/C++ programming language and is based on the finite element method. The conjugate gradient method is an iterative method for solving systems of linear algebraic equations and was published in 1952. The method uses the matrix-vector multiplication and the vector product. Therefore, the method is suitable for parallel implementation and for using on clusters and massive parallel computers. The thesis describes the implementation which is based not only on the parallelisation of the matrix-vector multiplication but also on the parallelisation of the vector product. An incomplete LU factorisation is used for preconditioning of the conjugate gradient method. The PETSc library contains the incomplete LU factorisation which is used for preconditioning. Numerical experiments show that the implementation is effective up to 100 processors. The preconditioned method can be effective used for fourth-order problems. The second separate topic of the thesis is the description of the preconditioning into the FETI method. The Finite Element Tearing and Interconnecting (FETI) method is one of the nonoverlapping domain decomposition method which was published by Farhat et al. in 1994. The method was implemented earlier but without a preconditioning. The preconditioning is needed for achieving the scalability. The scalability of the FETI method was theoretically and numerically proved in the articles by Farhat et al. Two preconditionings was published in the articles, lumped and Dirichlet. The Dirichlet preconditioning is mathematically optimal and is based on the evaluation of the Schur complement of subdomain matrices. However, it has a large demands for time. The lumped preconditionig is computationally optimal and is based on the matrix-vector multiplication. Both of these preconditionings were implemented into the SIFEL code. The description of the implementation is in the thesis. Numerical experiments show that results are in agreement with published results in the articles by Farhat et al. The third and main topic of the thesis is development of the algorithm for the selection of the fixing nodes in the FETI-DP method. The Finite Element Tearing and Interconnecting Dual-Primal (FETI-DP) method is one of the nonoverlapping domain decomposition method which was published by Farthat et al. in 2001. The method was developed due to problems with singular matrices in the original FETI method. The method is a combination of the original FETI method and Schur complement method. The method divides unknowns in the problems into three groups. The first group contains of fixing unknowns, which are selected form interface unknowns, and enforce the non-singularity of subdomain matrices. The second group contains of remaining interface unknowns. Lagrange multipliers, which are defined on remaining interface unknowns, enforce the continuity conditions among subdomains. The last group includes internal unknowns. The selection of fixing unknowns deserve special attention. Two algorithms are described in the thesis. These algorithms use graphs, which are defined with the help of interface finite element nodes. Numerical experiments show that the minimal needed number of fixing nodes does not lead to the best time of the solution. The increasing number of fixing unknowns leads to the decreasing the number of iterations in the coarse problem, which is obtained after elimination of internal and remaining interface unknowns. However, a large number of fixing unknowns prolong the whole time of solution. The optimal number of fixing unknowns, which leads to the shortest time of the solution, exists for every .