Ulrich Langer

Boundary Element Tearing and Interconnecting Methods

AbstractIn this paper we introduce the Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established Finite Element Tearing and Interconnecting (FETI) methods. In some practical important applications such as far field computations, handling of singularities and moving parts etc., BETI methods have certainly some advantages over their finite element counterparts. This claim is especially true for the sparse versions of the BETI preconditioners resp. methods. Moreover, there is an unified framework for coupling, handling, and analyzing both methods. In particular, the FETI methods can benefit from preconditioning components constructed by boundary element techniques. The first numerical results confirm the efficiency and the robustness predicted by our analysis.

Finite-element simulation of wave propagation in periodic piezoelectric SAW structures

Many surface acoustic wave (SAW) devices consist of quasiperiodic structures that are designed by successive repetition of a base cell. The precise numerical simulation of such devices, including all physical effects, is currently beyond the capacity of high-end computation. Therefore, we have to restrict the numerical analysis to the periodic substructure. By using the finite-element method (FEM), this can be done by introducing periodic boundary conditions (PBCs) at special artificial boundaries. To be able to describe the complete dispersion behavior of waves, including damping effects, the PBC has to be able to model each mode that can be excited within the periodic structure. Therefore, the condition used for the PBCs must hold for each phase and amplitude difference existing at periodic boundaries. Based on the Floquet theorem, our two newly developed PBC algorithms allow the calculation of both, the phase and the amplitude coefficients of the wave. In the first part of this paper we describe the basic theory of the PBCs. Based on the FEM, we develop two different methods that deliver the same results but have totally different numerical properties and, therefore, allow the use of problem-adapted solvers. Further on, we show how to compute the charge distribution of periodic SAW structures with the aid of the new PBCs. In the second part, we compare the measured and simulated dispersion behavior of waves propagating on periodic SAW structures for two different piezoelectric substrates. Then we compare measured and simulated input admittances of structures similar to SAW resonators.

Numerical analysis of nonlinear multiharmonic eddy current problems

SummaryThis work is devoted to non-linear eddy current problems and their numerical treatment by the so-called multiharmonic approach. Since the sources are usually alternating currents, we propose a truncated Fourier series expansion instead of a costly time-stepping scheme. Moreover, we suggest to introduce some regularization parameter that ensures unique solvability not only in the factor space of divergence-free functions, but also in the whole space H(curl). Finally, we provide a rigorous estimate for the total error that is due to the use of truncated Fourier series, the regularization technique and the spatial finite element discretization.

The approximate Dirichlet domain decomposition method. Part I: an algebraic approach

We present a new approach to the construction of Domain Decomposition (DD) preconditioners for the conjugate gradient method applied to the solution of symmetric and positive definite finite element equations. The DD technique is based on a non-overlapping decomposition of the domain Ω intop subdomains connected later with thep processors of a MIMD computer. The DD preconditioner derived contains three block matrices which must be specified for the specific problem considered. One of the matrices is used for the transformation of the nodal finite element basis into the approximate discrete harmonic basis. The other two matrices are block preconditioners for the Dirichlet problems arising on the subdomains and for a modified Schur complement defined over all nodes on the coupling boundaries between the subdomains. The relative spectral condition number is estimated. Relations to the additive Schwarz method are discussed. In the second part of this paper, we will apply the results of this paper to two-dimensional, symmetric, second-order, elliptic boundary value problems and present numerical results performed on a transputer-network.ZusammenfassungIn der vorliegenden Arbeit wird ein neuer Zugang zur Konstruktion von Vorkonditionierungsoperatoren auf der Basis von Gebietsdekompositionstechniken (DD Techniken) beschrieben. Anwendungen finden diese DD Vorkonditionierungen im Verfahren der konjugierten Gradienten zur iterativen Lösung von symmetrischen und positiv definiten Finiten-Elemente Gleichungen. Die DD Technik basiert auf einer Zerlegung des Gebietes Ω inp sich nicht überlappende Teilgebiete, die später denp Prozessoren eines MIMD Rechners zugeordnet sind. Die DD Vorkonditionierung enthält drei Blockmatrizen, die für ein konkretes Anwendungsproblem jeweils zu spezifizieren sind. Eine dieser Matrizen wird genutzt, um die Knotenbasis in eine näherungsweise diskret harmonische Basis zu transformieren. Die anderen beiden Matrizen können als Blockvorkonditionierungen für die in jedem Teilgebiet entstehenden Dirichlet-Probleme und für ein modifiziertes Schurkomplement auf den Knoten der Koppelränder zwischen den Teilgebieten interpretiert werden. Die relative spektrale Konditionszahl wird abgeschätzt. Eine direkte Verbindung der vorgeschlagenen DD Vorkonditionierung zu einer Additiven Schwarzschen Methode kann gezeigt werden. Im zweiten Teil dieser Artikelserie werden die Resultate dieser Arbeit auf ebene, symmetrische Randwertprobleme für partielle Differentialgleichungen zweiter Ordnung angewandt und die numerischen Resultate, die auf einem Transputer-Hypercube erzeugt wurden, diskutiert.

Tutorial on Elliptic PDE Solvers and Their Parallelization

From the Publisher: This compact yet thorough tutorial is the perfect introduction to the basic concepts of solving partial differential equations (PDEs) using parallel numerical methods. In just eight short chapters, the authors provide readers with enough basic knowledge of PDEs, discretization methods, solution techniques, parallel computers, parallel programming, and the run-time behavior of parallel algorithms to allow them to understand, develop, and implement parallel PDE solvers. Examples throughout the book are intentionally kept simple so that the parallelization strategies are not dominated by technical details. A Tutorial on Elliptic PDE Solvers and Their Parallelization is a valuable aid for learning about the possible errors and bottlenecks in parallel computing. One of the highlights of the tutorial is that the course material can run on a laptop, not just on a parallel computer or cluster of PCs, thus allowing readers to experience their first successes in parallel computing in a relatively short amount of time. This tutorial is intended for advanced undergraduate and graduate students in computational sciences and engineering; however, it may also be helpful to professionals who use PDE-based parallel computer simulations in the field.

Inexact Data-Sparse Boundary Element Tearing and Interconnecting Methods

The boundary element tearing and interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well-established finite element tearing and interconnecting (FETI) methods. In this paper we present inexact data-sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices. However, instead of symmetric and positive definite systems, we finally have to solve twofold saddle point problems. The proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor. Moreover, the solvers are robust with respect to large coefficient jumps.

Multipatch Discontinuous Galerkin Isogeometric Analysis

Isogeometric Analysis (IgA) uses the same class of basis functions for both representing the geometry of the computational domain and approximating the solution of the boundary value problem under consideration. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This multi-patch representation corresponds to a decomposition of the computational domain into non-overlapping subdomains also called patches in the geometrical framework. We will present discontinuous Galerkin (dG) methods that allow for discontinuities across the subdomain (patch) boundaries. The required interface conditions are weakly imposed by the dG terms associated with the boundary of the subdomains. The construction and the corresponding discretization error analysis of such dG multi-patch IgA schemes is given for heterogeneous diffusion model problems in volumetric 2d and 3d domains as well as on open and closed surfaces. The theoretical results are confirmed by numerous numerical experiments which have been performed in G+SMO. The concept and the main features of the IgA library G +SMO are also described.

Coupled Boundary and Finite Element Tearing and Interconnecting Methods

We have recently introduced the Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established Finite Element Tearing and Interconnecting (FETI) methods. Since Finite Element Methods (FEM) and Boundary Element Methods (BEM) have certain complementary properties, it is sometimes very useful to couple these discretization techniques and to benefit from both worlds. Combining our BETI techniques with the FETI methods gives new, quite attractive tearing and interconnecting parallel solvers for large scale coupled boundary and finite element equations. There is an unified framework for coupling, handling, and analyzing both methods. In particular, the FETI methods can benefit from preconditioning components constructed by boundary element techniques. This is especially true for sparse versions of the boundary element method such as the fast multipole method which avoid fully populated matrices arising in classical boundary element methods.

The approximate Dirichlet domain decomposition method. Part II: applications to 2nd-order elliptic B.V.P.s

In the first part of this article series, we had derived Domain Decomposition (DD) preconditioners containing three block matrices which must be specified for specific applications. In the present paper, we consider finite element equations arising from the DD discretization of plane, symmetric, 2nd-order, elliptic b.v.p.s and specify the matrices involved in the preconditioner via multigrid and hierarchical techniques. The resulting DD-PCCG methods are asymptotically almost optimal with respect to the operation count and well suited for parallel computations on MIMD computers with local memory and message passing. The numerical experiments performed on a transputer hypercube confirm the efficiency of the DD preconditioners proposed.ZusammenfassungIm ersten Teil dieser Artikelserie haben wir auf Basis von Gebietsdekompositionstechniken (DD Techniken) Vorkonditionierungsoperatoren konstruiert. Diese DD Vorkonditionierungen enthalten drei Blockmatrizen, die für spezifische Anwendungsfälle zu konkretisieren sind. In der vorliegenden Arbeit betrachten wir Finite-Elemente-Gleichungen, die bei der DD Diskretisierung von ebenen, symmetrischen, elliptischen Randwertproblemen für partielle Differentialgleichungen zweiter Ordnung entstehen. Zur Definition der oben genannten Blockmatrizen werden Mehrgitter-und hierarchische Techniken herangezogen. Die entstehenden DD-PCCCG Verfahren sind bezüglich des arithmetischen Aufwands asymptotisch fast optimal und bestens zur Parallelrechnung auf MIMD-Computern mit lokalem Speicher und Botschaftenaustausch geeignet. Die auf einem Transputer-Hypercube durchgeführten numerischen Experimente belegen nachhaltig die Effektivität der vorgeschlagenen DD Vorkonditionierungen.

Adaptive Domain Decomposition Methods for Finite and Boundary Element Equations

The use of the FEM and BEM in different subdomains of a non-overlapping Domain Decomposition (DD) and their coupling over the coupling boundaries (interfaces) brings about several advantages in many practical applications. The paper presents parallel solvers for large-scaled coupled FE-BE-DD equations approximating linear and nonlinear plane magnetic field problems as well as plane linear elasticity problems. The parallel algorithms presented are of asymptotically optimal, or, at least, almost optimal complexity and of high parallel efficiency.