Inexact Data‐Sparse Boundary Element Tearing and Interconnecting Methods

The Boundary Element Tearing and Interconnecting (BETI) methods were recently introduced as boundary element counterparts of the well established Finite Element Tearing and Interconnecting (FETI) methods. Here we combine the BETI method preconditioned by the projector to the “natural coarse grid” with recently proposed optimal algorithms for the solution of bound and equality constrained quadratic programming problems in order to develop a theoretically supported scalable solver for elliptic multidomain boundary variational inequalities such as those describing the equilibrium of a system of bodies in mutual contact. The key observation is that the “natural coarse grid” defines a subspace that contains the solution, so that the preconditioning affects also the non-linear steps. The results are validated by numerical experiments.

This paper presents a novel approach, based on the domain decomposition method in conjunction with finite element tearing and interconnecting (FETI) algorithm, to model complex EMC problems. FETI algorithm results in the computation of numerical Green's function, which can be compressed efficiently via an efficient rank-revealing SVD algorithm. The numerical Green's function, once obtained, can be readily combined with other domains as a matrix-vector multiplication in the DDM solution process. Few examples are shown to validate the present approach.

A domain decomposition method with Lagrange multipliers for the Stokes problem is developed and analysed. A common approach to solve the Stokes problem, termed the Uzawa algorithm, is to decouple the velocity and the pressure. This approach yields the Schur complement system for the pressure Lagrange multiplier which is solved with an iterative solver. Each outer iteration of the Uzawa procedure involves the inversion of a Laplacian in each spatial direction. The objective of this paper is to effectively solve this inner system (the vector Laplacian system) by applying the finite‐element tearing and interconnecting (FETI) method. Previously calculated search directions for the FETI solver are reused in subsequent outer Uzawa iterations. The advantage of the approach proposed in this paper is that pressure is continuous across the entire computational domain. Numerical tests are performed by solving the driven cavity problem. An analysis of the number of outer Uzawa iterations and inner FETI iterations is reported. Results show that the total number of inner iterations is almost numerically scalable since it grows asymptotically with the mesh size and the number of subdomains. Copyright © 2003 John Wiley & Sons, Ltd.

A class of finite element tearing and interconnecting (FETI) methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed. First, an abstract framework is presented for the analysis of a class of FETI methods where a natural coarse problem, associated with the substructures, is lacking. Then, a family of FETI methods for edge element approximations is proposed. It is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures. The estimate is also independent of the jumps of both of the coefficients of the original problem. Numerical results validating our theoretical bounds are given. The method and its analysis can be easily generalized to Raviart--Thomas element approximations in two and three dimensions.

A new domain decomposition method with Lagrange multipliers for elliptic problems is introduced. It is based on a reformulation of the well-known finite element tearing and interconnecting (FETI) method as a saddle point problem with both primal and dual variables as unknowns. The resulting linear system is solved with block-structured preconditioners combined with a suitable Krylov subspace method. This approach allows the use of inexact subdomain solvers for the positive definite subproblems. It is shown that the condition number of the preconditioned saddle point problem is bounded independently of the number of subregions and depends only polylogarithmically on the number of degrees of freedom of individual local subproblems. Numerical results are presented for a plane stress cantilever membrane problem.

We focus on the analysis of fracture in quasi-brittle materials by exploiting the potential of domain decomposition techniques. More specifically, we restrict our attention to the FETI (Finite Element Tearing and Interconnecting) method which is used as a solver in our non-linear solution scheme. We develop criteria to anticipate the linear/non-linear character of different regions in the structure according to a damage constitutive model. A first application of our scheme focuses on the efficiency increase of a monoscale analysis by simplifying the computations in those areas that remain linear. The second application treats the problem in a multiscale fashion where the resolution of the non-linear domains is increased in order to describe failure phenomena with a higher degree of accuracy. Both applications represent a significant improvement from a computational standpoint when the main non-linear regions are small compared to the size of the whole specimen. This is often the case for brittle and quasi-brittle materials where strain localization is expected to occur upon failure.

Despite their merits, composites have weakness in transverse strength making them susceptible to failures such as delamination. In order to understand the failure mechanism an accurate evaluation of the interlaminar stresses is required. An e‐cient procedure for accurate analysis of multi-layer composite structures is presented, with particular interest in the interlaminar stresses. Parallelization of sequential simulations approach together with the use of domain decomposition methods signiflcantly broadens the range of applications of numerical simulations for computationally demanding problems. In the present paper, attention will be restricted to the Finite Element Tearing and Interconnecting (FETI) method, an e‐cient solution scheme based on the domain decomposition of the structure and enforcing the compatibility conditions between the difierent sub-domains by means of Lagrange multipliers. With this approach, connecting forces between the difierent sub-domains at the interfacial nodes can be obtained and, as a consequence, an accurate post-processing recovery procedure of the interlaminar stresses based on equilibrium has been formulated. In addition, a consistent variational recovery procedure has been implemented to completely recover stress flelds. The reliability of the approach has been tested with problems involving composite plates of various geometries subjected to difierent mechanical loadings.

In this paper, the transient electrical-thermal co-simulation is implemented with the finite-element method. The capability of the co-simulation is extended to solve large-scale problems by incorporating a domain decomposition scheme called the finite element tearing and interconnecting (FETI). With the high parallel efficiency of FETI for computing with multiple processors, a significant reduction in computation time is achieved. The transient electrical-thermal behaviors of largescale power distribution networks (PDNs), including on-chip power grids, solder bump arrays, and TSV-based PDN are simulated and analyzed with the proposed method. The impacts on the design of the PDNs from different types of input pulses, power maps, and via pitches are also investigated.

An electromagnetism module is under development in the commercial software LS-DYNA in order to perform coupled mechanical/thermal/electromagnetic simulations. The main part of the module is a so-called eddy current solver. This module allows us to introduce source electrical currents into solid conductors, and to compute the associated magnetic fields, electric fields, induced currents, Lorentz forces, Joule heating and so forth. The Maxwell equations are solved using a Finite Element Method (FEM) for the solid conductors coupled with a Boundary Element Method (BEM) for the surrounding air (or insulators). Both the FEM and the BEM are based on discrete differential forms (Nedelec-like elements). The use of the BEM for the air allows to handle complex 3D geometries with multiply connected conductors, very small air gaps, and motion of the conductors, due, in particular, to the electromagnetic forces. The variable of the BEM is a so-called surface current allowing the connection of the model to external current sources by simple dirichlet constraints, as well as direct computations of the self and mutual inductances of the system. In order to handle the large and dense BEM matrices, a domain decomposition is performed and low rank approximations are done on the off-diagonal blocks of the resulting block matrices. The diagonal blocks are used as an efficient preconditioner when solving the BEM part of the system. The singularities arising in the computation of the self and neighbor matrix elements of the Galerkin BEM are taken into account using a method based on Duffy transforms. The BEM method will be presented as well as benchmarks and real life application examples.

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

We present a distributed-memory library for computations with dense structured matrices. A matrix is considered structured if its off-diagonal blocks can be approximated by a rank-deficient matrix with low numerical rank. Here, we use Hierarchically Semi-Separable (HSS) representations. Such matrices appear in many applications, for example, finite-element methods, boundary element methods, and so on. Exploiting this structure allows for fast solution of linear systems and/or fast computation of matrix-vector products, which are the two main building blocks of matrix computations. The compression algorithm that we use, that computes the HSS form of an input dense matrix, relies on randomized sampling with a novel adaptive sampling mechanism. We discuss the parallelization of this algorithm and also present the parallelization of structured matrix-vector product, structured factorization, and solution routines. The efficiency of the approach is demonstrated on large problems from different academic and industrial applications, on up to 8,000 cores. This work is part of a more global effort, the STRUctured Matrices PACKage (STRUMPACK) software package for computations with sparse and dense structured matrices. Hence, although useful on their own right, the routines also represent a step in the direction of a distributed-memory sparse solver.

The phase field approach to fracture becomes popular for complicated fracture problems in recent years. However, its widespread application is hindered by its high computational cost. In this article, we propose an efficient parallel explicit‐implicit solution scheme for the phase field approach to dynamic fracture based on a domain decomposition method, specifically, the dual‐primal finite element tearing and interconnecting (FETI‐DP) method. In this scheme, the displacement field is updated by an explicit algorithm in parallel, and the phase field is implicitly solved by the FETI‐DP method. In particular, Lagrange multipliers are introduced to ensure the interface continuity of the phase field. In the computational process, the information exchange among subdomains merely exists in a few substeps, which renders the cost for communication very small. Moreover, the size of equations to be solved is proportional to the total area of subdomain interfaces, which is significantly reduced compared with a typical single‐domain solution procedure. The solution scheme is able to perform phase field simulations with a million of degrees of freedom using only 0.034 core hours per load step, and has flexible extensibility for existing phase field codes. Several numerical examples demonstrate the accuracy and efficiency of the proposed scheme.

The boundary element method (BEM) can be effectively used to solve mixed boundary value problems, as well as fluid-structure interaction problems. Among the advantages of BEM are the reduction of the problem dimension by one, high accuracy and the efficient treatment of problems with infinite domains. The direct BEM leads to a system of equations with dense and unsymmetric matrices. Thus, coupling with symmetric domain discretization techniques, such as finite element methods, is numerically inefficient. For this reason, alternative approaches are use herein derived from variational principles leading to symmetry by construction. By relaxing continuity between the fields on the boundary and those in the domain two multi-field variational principles have been formulated. Emerging from these variational principles, two Hybrid Boundary Element Methods (HBEM) were developed, the Hybrid Displacement Method and the Hybrid Stress Method, respectively. The choice of weighted fundamental solutions as domain approximations selected independently from polynomial approximations of boundary variables allow to map domain integrals in boundary integrals. Symmetric sytems of equations for frequency and time domain are obtained.

This paper discusses a class of two-stage iterations whose outer iterative methods are the block SOR methods for parallel solution of linear systems. Convergence is showed for symmetric positive definite linear systems, and an approximate optimal relaxation factor is defined for block tridiagonal matrices. The numerical results for Poisson model problem are presented.

Some results on convergence of two-stage iterative methods for symmetric positive definite linear systems