Finite element methods for Maxwell's equations.

This paper presents the adaptive cross approximation (ACA) algorithm to reduce memory and CPU time overhead in the method of moments (MoM) solution of surface integral equations. The present algorithm is purely algebraic; hence, its formulation and implementation are integral equation kernel (Green's function) independent. The algorithm starts with a multilevel partitioning of the computational domain. The interactions of well-separated partitioning clusters are accounted through a rank-revealing LU decomposition. The acceleration and memory savings of ACA come from the partial assembly of the rank-deficient interaction submatrices. It has been demonstrated that the ACA algorithm results in O(NlogN) complexity (where N is the number of unknowns) when applied to static and electrically small electromagnetic problems. In this paper the ACA algorithm is extended to electromagnetic compatibility-related problems of moderate electrical size. Specifically, the ACA algorithm is used to study compact-range ground planes and electromagnetic interference and shielding in vehicles. Through numerical experiments, it is concluded that for moderate electrical size problems the memory and CPU time requirements for the ACA algorithm scale as N/sup 4/3/logN.

The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems

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Nonlinear Functional Analysis And Its Applications

Coupling methods for continuum models with molecular models are developed. Two methods are studied here: an overlapping domain decomposition method, which has overlapping domain, and an edge-to-edge decomposition method, which has an interface between two models. These two methods enforce the compatibility on the overlapping domain or interface nodes/atoms by the Lagrange multiplier method or the augmented Lagrangian method.

Coupling Methods for Continuum Model with Molecular Model

It is well known that the classical boundary-element method (BEM) yields fully populated matrices. Their manipulation is cumbersome with respect to memory consumption and computational costs. This paper describes a novel approach where the matrices are split into collections of blocks of various sizes. Those blocks which describe remote interactions are adaptively approximated by low rank submatrices. This procedure reduces the algorithmic complexity for matrix setup and matrix-by-vector products to approximately O(N). The proposed method has been examined in a testing environment and implemented into an existing BEM-finite-element method (FEM) code for electromagnetic and electromechanical problems. The advantages of the new method are demonstrated by means of several examples.

The adaptive cross-approximation technique for the 3D boundary-element method

This paper summarizes the theoretical background of 3D eddy current problems with moving bodies. A novel 3D formulation combining a Lagrangian description and BEM-FEM coupling is subsequently developed. The conducting and permeable bodies are described by the FEM in their respective rest frame, whereas the surrounding space is treated by the BEM in the laboratory frame. The FEM and BEM descriptions are coupled together taking into account the transformation between the different frames. The proposed formulation contains no explicit velocity terms.

A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling

A fast isogeometric BEM for the three dimensional Laplace- and Helmholtz problems

Discrete electromagnetism (DEM)-in the authors' view-should be a self-consistent theory, mirroring the properties of the continuous electromagnetic theory in a discrete setting. Any recursion to continuous techniques can be interpreted as an inconsistency in the discrete theory. Recently, discrete Hodge operators on tetrahedra and triangles have been introduced that avoid the concepts of interpolation and integration of fields. In this paper we introduce a geometrical definition of a discrete Hodge operator for general dimension n and degree p,0lesnles3,0lesplesn. The definition generalizes previously published definitions. The increased level of abstraction allows for a short definition and a concise discussion of the properties of this operator

A geometrically defined discrete hodge operator on simplicial cells

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 O(N). In this paper, both fast methods are applied in combination with BEM-finite-element method coupling to nonlinear magnetostatic problems.

Stefan Kurz

Papers

The adaptive cross-approximation technique for the 3D boundary-element method

A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling

A fast isogeometric BEM for the three dimensional Laplace- and Helmholtz problems

A geometrically defined discrete hodge operator on simplicial cells

Comparison between different approaches for fast and efficient 3-D BEM computations