Academia.edu no longer supports Internet Explorer.
To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser.
2006, Proceedings Int. Conf. on Boundary Element Techniques
…
28 pages
1 file
IEEE Transactions on Magnetics, 2000
The main problems for applying boundary element methods (BEM) in computational electromagnetism are related to the large memory requirements of the matrices and the convergence of the iterative solver. In this paper, we solve a Laplace problem with mixed boundary conditions by making use of a variational symmetric direct boundary integral equation. The Galerkin discretization results in densely populated matrices that are here compressed by adaptive cross approximation. This leads to an approximation of the underlying BEMoperator by means of so-called hierarchical matrices (-Matrices). These matrices are then used to construct an effective preconditioner for the iterative solver. Numerical experiments demonstrate the application of the method.
In this paper, a new parallel acoustic simulation package has been created, using the boundary element method (BEM). The package is built on top of SPECFEM3D, which is parallel software for doing seismic simulations, e.g. earthquake simulations of the globe. The acoustical simulation relies on a Fourier transform of the seismic elastodynamic data, resulting from SPECFEM3D_GLOBE, which are then postprocessed by a sequence of solutions to Helmholtz equations, in the exterior of the globe. For the acoustic simulations BEM has been employed, which reduces computation to the sphere; however, its naive implementation suffers from quadratic time and memory complexity, with respect to the number of unknowns. To overcome the latter, the method was accelerated by using hierarchical matrices and adaptive cross approximation techniques, which is referred to as Fast BEM. First, a hierarchical clustering of the globe surface triangulation is performed. The arising cluster pairs decompose the fully populated BEM matrices into a hierarchy of blocks, which are classified as far-field or near-field. While the near-field blocks are kept as full matrices, the far-field blocks are approximated by low-rank matrices. This reduces the quadratic complexity of the serial code to almost linear complexity, i.e. O(n log(n)), where n denotes the number of triangles. Furthermore, a parallel implementation was done, so that the blocks are assigned to concurrent MPI processes with an optimal load balance. The novelty of our approach is based on a nontrivial and theoretically supported memory distribution of the hierarchical matrices and right-hand side vectors so that the overall memory consumption leads to O(n log(n)/N+n/sqrt(N)), which is the theoretical limit at the same time.
International Journal for Numerical Methods in Engineering, 2006
In the past two decades, considerable improvements concerning integration algorithms and solvers involved in boundary-element formulations have been obtained. First, a great deal of efficient techniques for evaluating singular and quasi-singular boundary-element integrals have been, definitely, established, and second, iterative Krylov solvers have proven to be advantageous when compared to direct ones also including non-Hermitian matrices. The former fact has implied in CPU-time reduction during the assembling of the system of equations and the latter fact in its faster solution. In this paper, a triangle-polar-coordinate transformation and the Telles coordinate transformation, applied in previous works independently for evaluating singular and quasi-singular integrals, are combined to increase the efficiency of the integration algorithms, and so, to improve the performance of the matrixassembly routines. In addition, the Jacobi-preconditioned biconjugate gradient (J-BiCG) solver is used to develop a generic substructuring boundary-element algorithm. In this way, it is not only the system solution accelerated but also the computer memory optimized. Discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions. Several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving. In the discussion of results, expressed in terms of accuracy and CPU time, advantages and potential applications of the BE code developed are highlighted.
Proceedings of the …
The boundary element method has become a popular tool for the solution of Maxwell's equations in electromagnetism. It discretizes only the surface of the radiating object and gives rise to linear systems that are smaller in size compared to those arising from finite element or finite difference discretizations. However, these systems are prohibitevely demanding in terms of memory for direct methods and challenging to solve by iterative methods. In this paper we address the iterative solution via preconditioned Krylov methods of electromagnetic scattering problems expressed in an integral formulation, with main focus on the design of the preconditioner. We consider an approximate inverse method based on the Frobeniusnorm minimization with a pattern prescribed in advance. The preconditioner is constructed from a sparse approximation of the dense coefficient matrix, and the patterns both for the preconditioner and for the coefficient matrix are computed a priori using geometric information from the mesh. We describe the implementation of the approximate inverse in an out-of-core parallel code that uses multipole techniques for the matrix-vector products, and show results on the numerical scalability of our method on systems of size up to one million unknowns. We propose an embedded iterative scheme based on the GMRES method and combined with multipole techniques, aimed at improving the robustness of the approximate inverse for large problems. We prove by numerical experiments that the proposed scheme enables the solution of very large and difficult problems efficiently at reduced computational and memory cost. Finally we perform a preliminary study on a spectral two-level preconditioner to enhance the robustness of our method. This numerical technique exploits spectral information of the preconditioned systems to build a low rank-update of the preconditioner.
2008
In this paper a fast solver for three-dimensional BEM and DBEM is developed. The technique is based on the use of hierarchical matrices for the representation of the collocation matrix and uses a preconditioned GMRES for the solution of the algebraic system of equations. The preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. Special algorithms are developed to deal with crack problems within the context of DBEM. The structure of DBEM matrices has been efficiently exploited and it has been demonstrated that, since the cracks form only small parts of the whole structure, the use of hierarchical matrices can be particularly advantageous. Test examples presented show that, with the proposed technique, substantial increase in number of elements over the crack surfaces leads only to moderate increases in memory storage and solution time.
Engineering Analysis With Boundary Elements, 2004
Engineering problems, such as those dealing with boundary element methods (BEM), usually involve high computational costs. Previous works have proposed effective parallel implementations of BEM. These works have proved the benefit of using parallel computers to solve large civil engineering problems. Despite the profits of the parallel implementation in the discretization step of these methods, the drawback remains the solution of the systems of equations obtained. Because of their fully populated coefficient matrices, direct methods are usually preferable to solve these linear systems. However, parallel implementations of iterative methods achieve higher efficiency, but need suitable preconditioners in order to speed up their convergence. The aim of this work is the research into parallel Sparse Approximate Inverse Preconditioners (SPAI) for the iterative solution of dense systems of equations arising from the boundary element codes. Different heuristics for the construction of SPAI preconditioners suitable for this kind of systems are proposed. Some experiments and numerical results, on distributed-memory computers, are presented in this paper. q
We consider the numerical solution of 3D linear elasticity equations. The investigated problem is described by a coupled system of second order elliptic partial differential equations. This system is then discretized by conforming or nonconforming finite elements. After applying the Finite Element Method (FEM) based discretization, a system of linear algebraic equations has to be solved. In this system the stiffness matrix is large, sparse and symmetric positive definite. In the solution process we utilize a well-known fact that the preconditioned conjugate gradient method is the best tool for efficient solution of large-scale symmetric systems with sparse positive definite matrices. In this context, the displacement decomposition (DD) technique is applied at the first step to construct a preconditioner that is based on a decoupled block diagonal part of the original matrix. Then two preconditioners, namely the Modified Incomplete Cholesky factorization MIC(0) and the Circulant Block-Factorization (CBF) preconditioning, are used to precondition thus obtained block diagonal matrix.
IEEE Transactions on Magnetics, 2003
Fast methods like the fast multipole method or the adaptive cross approximation technique reduce the memory requirements and the computational costs of the boundary-element method (BEM) to approximately (). In this paper, both fast methods are applied in combination with BEM-finite-element method coupling to nonlinear magnetostatic problems. Index Terms-Adaptive cross approximation (ACA) technique, boundary element methods (BEMs), fast multipole method, finite element methods (FEMs), iterative solution methods.
2022
We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions, using an extension of the recently introduced strong recursive skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an ${LU}$-like hierarchical factorization of the dense system matrix, permitting application of the inverse in $O(N)$ time, where $N$ is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted, adaptive octree data structure and therefore it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nyström quadrature methods to integrate the singular and weakly-singular Green's functions used in the integral representations. Our method has immediate application to a variety of problems in computational physics. We conce...
1999
In this work one aims to present an efficient algorithm for coupling a generic number of subregions discretized with boundary elements. The followed strategy does not base on condensing the field variables into certain variable blocks, as e.g. the interface variable ones, but instead considers directly the resulting sparse matrix of the coupled system. The main idea of the algorithm is that by using reliable iterative solvers it is then possible to develop multi-zone algorithms without condensing techniques such that, independently of the numbering scheme adopted, only the block matrices with non-zero coefficients are stored and manipulated during the analysis process and, furthermore, the CPU time for solving the system of equations is strongly decreased in comparison with necessary CPU times by using direct ones, as profit of the efficiency of iterative solvers, which become specially suitable for treatment of sparse matrix, is taken into account. Specifically, the Lanczos acceler...
Ochrona zdrowia i gospodarka. Pacjenci, świadczeniodawcy, turystyka medyczna, 2019
Anuario Colombiano de Historia Social y de la Cultura, 2011
Actividad de aprendizaje 13 Evidencia 3: Taller “Plan de integración y TIC, 2019
International journal of research - granthaalayah, 2021
The History of Distributed Cognition, 2019
Journal of Thoracic Oncology, 2017
Applicable Analysis and Discrete Mathematics, 2009
Scientific reports, 2017
International Journal of Plant and Soil Science, 2023
Linguistic Forum - A Journal of Linguistics, 2022