Christian Lage - Academia.edu (original) (raw)

Papers by Christian Lage

Research report / Seminar für Angewandte Mathematik, 1996

In this paper we present the design of a class library to support the development of software for... more In this paper we present the design of a class library to support the development of software for boundary elements. We discuss the extensibility and reusability of the library and give an example of its application.

IEEE Transactions on Magnetics, Mar 1, 2004

In the extraction and simulation of integrated circuit (IC) chips using electromagnetic field sol... more In the extraction and simulation of integrated circuit (IC) chips using electromagnetic field solvers, one of the difficulties is the modeling of the multilayered dielectric structures. The consideration of these thin layers in the field solvers dramatically increases the memory and the computation time. This paper presents a numerical homogenization strategy of multilayered dielectric media that allows removing the dielectric layers from the field solver models. A technique that transforms the homogenized anisotropic materials to isotropic ones is also developed. The homogenization strategy has been applied in both the finite-element method and a fast multipole expansion accelerated boundary-element method. The validation is accomplished through examples of capacitance extraction on a real digital IC chip.

Journal of Applied Mathematics and Mechanics, 1996

We will present efficient techniques to approximate singular and nearly singular surface integral... more We will present efficient techniques to approximate singular and nearly singular surface integrals arising massively in Galerkin boundary element discretizations of Fredholm integral equations on two-dimensional surfaces. The technique is based on a new representation of the functionals which arise by computing the so-called local element matrices, i.e., integrals over pairs of panels. The remaining integrals are approximated by introducing relative co-ordinates which fix the location of the singularity. These integrals can be integrated using polar co-ordinates. It turns out that the number of kernel evaluations which is needed to compute the integrals up to the required accuracy is independent of the order of the singularity. This enables us to use the hypersingular formulation of integral equations which is the method of choice from the theoretical point of view, i.e., stability, robustness with respect to non-smooth surfaces, etc.

Springer eBooks, 1997

The boundary element method (BEM) is an elegant tool for solving elliptic boundary value problems... more The boundary element method (BEM) is an elegant tool for solving elliptic boundary value problems numerically. First, the method of integral equations is applied transforming the PDE on a domain Ω into an integral equation defined on the boundary of Ω. This integral equation can be discretized by Petrov-Galerkin methods defined on the surface of Ω. Instead of the discretization of the whole domain, only the lower dimensional boundary of Ω has to be partitioned into a FE-grid. This is one of the major advantages of the BEM. Especially for 3-d problems, grid generation of the whole domain Ω in many cases is still an extremely time consuming step. As a consequence, the dimension of the stiffness matrix is much smaller as for the corresponding FE-discretization. Furthermore, the matrix condition number is smaller compared to the FEM-system governing the convergence speed of iterative solvers applied to the linear system. On the other hand, the major drawback of the BEM is that the arising system matrix is full and, in addition, the computation of the matrix entries requires the evaluation of complicated surface integrals. Recently, many attempts have been made to overcome these two difficulties. The full matrix can avoided by representing the integral operator on the discrete level in an alternative form, which is based on the approximation of the kernel function of the integral operator.

2022 23rd International Conference on the Computation of Electromagnetic Fields (COMPUMAG)

2022 IEEE 20th Biennial Conference on Electromagnetic Field Computation (CEFC)

The continuously growing computing power of modern comput ers admits to tackle numerical problems... more The continuously growing computing power of modern comput ers admits to tackle numerical problems of extreme complexity This complexity carries over to the numerical methods applied to solve the problems Whereas the mathematical formulation of these methods does not raise any diculties their implementation turns out to be the bottleneck in the realization of numerical applications In the last years in order to aord relief object oriented methods were applied to promote reusable and extensible numerical software since this kind of exibility is the key to manage complexity It be came evident that a carefully chosen modularization of the considered methods is a necessary requirement to provide exible software com ponents In this paper we give a brief review of object oriented methods to identify the key issues that support a exible software design and discuss a modularization technique based on mathematical concepts Finally the application of this concept oriented approach to boundary ...

The continuously growing computing power of modern comput ers admits to tackle numerical problems... more The continuously growing computing power of modern comput ers admits to tackle numerical problems of extreme complexity This complexity carries over to the numerical methods applied to solve the problems Whereas the mathematical formulation of these methods does not raise any di culties their implementation turns out to be the bottleneck in the realization of numerical applications In the last years in order to a ord relief object oriented methods were applied to promote reusable and extensible numerical software since this kind of exibility is the key to manage complexity It be came evident that a carefully chosen modularization of the considered methods is a necessary requirement to provide exible software com ponents In this paper we give a brief review of object oriented methods to identify the key issues that support a exible software design and discuss a modularization technique based on mathematical concepts Finally the application of this concept oriented approach to boundar...

Engineering Analysis with Boundary Elements, 2003

Abstract Weakly singular boundary integral equations (BIEs) of the first kind on polyhedral surfa... more Abstract Weakly singular boundary integral equations (BIEs) of the first kind on polyhedral surfaces Γ in R 3 are discretized by Galerkin BEM on shape-regular, but otherwise unstructured meshes of meshwidth h. Strong ellipticity of the integral operator is shown to give nonsingular stiffness matrices and, for piecewise constant approximations, up to O(h3) convergence of the farfield. The condition number of the stiffness matrix behaves like O(h−1) in the standard basis. An O(N) agglomeration algorithm for the construction of a multilevel wavelet basis on Γ is introduced resulting in a preconditioner which reduces the condition number to O (| log h|). A class of kernel-independent clustering algorithms (containing the fast multipole method as special case) is introduced for approximate matrix–vector multiplication in O (N( log N) 3 ) memory and operations. Iterative approximate solution of the linear system by CG or GMRES with wavelet preconditioning and clustering-acceleration of matrix–vector multiplication is shown to yield an approximate solution in log-linear complexity which preserves the O(h3) convergence of the potentials. Numerical experiments are given which confirm the theory.

IEEE Transactions on Magnetics

The Mathematics of Finite Elements and Applications X, 2000

The implementation of a wavelet-based Galerkin discretization of the double layer potential opera... more The implementation of a wavelet-based Galerkin discretization of the double layer potential operator on polyhedral surfaces Γ⊂ℝ 3 is described. The algorithm generates an approximate stiffness matrix with O(N(logN) 2 ) entries in O(N(logN) 3 ) operations where N is the number of degrees of freedom on the boundary. The condition number of the compressed stiffness matrix is bounded uniformly with respect to N. A C++ realization of the data structure containing the compressed stiffness matrix is described. It can be set up in O(N(logN) 2 ) operations and requires O(N(logN) 2 ) memory. Numerical experiments show the asymptotic complexity estimates and convergence rates to be accurate already for moderate N. Problems with N>10 5 were computed in core on a workstation.

International Association of Geodesy Symposia, 2001

Research report / Seminar für Angewandte Mathematik, 1996

In this paper we present the design of a class library to support the development of software for... more In this paper we present the design of a class library to support the development of software for boundary elements. We discuss the extensibility and reusability of the library and give an example of its application.

IEEE Transactions on Magnetics, Mar 1, 2004

In the extraction and simulation of integrated circuit (IC) chips using electromagnetic field sol... more In the extraction and simulation of integrated circuit (IC) chips using electromagnetic field solvers, one of the difficulties is the modeling of the multilayered dielectric structures. The consideration of these thin layers in the field solvers dramatically increases the memory and the computation time. This paper presents a numerical homogenization strategy of multilayered dielectric media that allows removing the dielectric layers from the field solver models. A technique that transforms the homogenized anisotropic materials to isotropic ones is also developed. The homogenization strategy has been applied in both the finite-element method and a fast multipole expansion accelerated boundary-element method. The validation is accomplished through examples of capacitance extraction on a real digital IC chip.

Journal of Applied Mathematics and Mechanics, 1996

We will present efficient techniques to approximate singular and nearly singular surface integral... more We will present efficient techniques to approximate singular and nearly singular surface integrals arising massively in Galerkin boundary element discretizations of Fredholm integral equations on two-dimensional surfaces. The technique is based on a new representation of the functionals which arise by computing the so-called local element matrices, i.e., integrals over pairs of panels. The remaining integrals are approximated by introducing relative co-ordinates which fix the location of the singularity. These integrals can be integrated using polar co-ordinates. It turns out that the number of kernel evaluations which is needed to compute the integrals up to the required accuracy is independent of the order of the singularity. This enables us to use the hypersingular formulation of integral equations which is the method of choice from the theoretical point of view, i.e., stability, robustness with respect to non-smooth surfaces, etc.

Springer eBooks, 1997

The boundary element method (BEM) is an elegant tool for solving elliptic boundary value problems... more The boundary element method (BEM) is an elegant tool for solving elliptic boundary value problems numerically. First, the method of integral equations is applied transforming the PDE on a domain Ω into an integral equation defined on the boundary of Ω. This integral equation can be discretized by Petrov-Galerkin methods defined on the surface of Ω. Instead of the discretization of the whole domain, only the lower dimensional boundary of Ω has to be partitioned into a FE-grid. This is one of the major advantages of the BEM. Especially for 3-d problems, grid generation of the whole domain Ω in many cases is still an extremely time consuming step. As a consequence, the dimension of the stiffness matrix is much smaller as for the corresponding FE-discretization. Furthermore, the matrix condition number is smaller compared to the FEM-system governing the convergence speed of iterative solvers applied to the linear system. On the other hand, the major drawback of the BEM is that the arising system matrix is full and, in addition, the computation of the matrix entries requires the evaluation of complicated surface integrals. Recently, many attempts have been made to overcome these two difficulties. The full matrix can avoided by representing the integral operator on the discrete level in an alternative form, which is based on the approximation of the kernel function of the integral operator.

2022 23rd International Conference on the Computation of Electromagnetic Fields (COMPUMAG)

2022 IEEE 20th Biennial Conference on Electromagnetic Field Computation (CEFC)

The continuously growing computing power of modern comput ers admits to tackle numerical problems... more The continuously growing computing power of modern comput ers admits to tackle numerical problems of extreme complexity This complexity carries over to the numerical methods applied to solve the problems Whereas the mathematical formulation of these methods does not raise any diculties their implementation turns out to be the bottleneck in the realization of numerical applications In the last years in order to aord relief object oriented methods were applied to promote reusable and extensible numerical software since this kind of exibility is the key to manage complexity It be came evident that a carefully chosen modularization of the considered methods is a necessary requirement to provide exible software com ponents In this paper we give a brief review of object oriented methods to identify the key issues that support a exible software design and discuss a modularization technique based on mathematical concepts Finally the application of this concept oriented approach to boundary ...

The continuously growing computing power of modern comput ers admits to tackle numerical problems... more The continuously growing computing power of modern comput ers admits to tackle numerical problems of extreme complexity This complexity carries over to the numerical methods applied to solve the problems Whereas the mathematical formulation of these methods does not raise any di culties their implementation turns out to be the bottleneck in the realization of numerical applications In the last years in order to a ord relief object oriented methods were applied to promote reusable and extensible numerical software since this kind of exibility is the key to manage complexity It be came evident that a carefully chosen modularization of the considered methods is a necessary requirement to provide exible software com ponents In this paper we give a brief review of object oriented methods to identify the key issues that support a exible software design and discuss a modularization technique based on mathematical concepts Finally the application of this concept oriented approach to boundar...

Engineering Analysis with Boundary Elements, 2003

Abstract Weakly singular boundary integral equations (BIEs) of the first kind on polyhedral surfa... more Abstract Weakly singular boundary integral equations (BIEs) of the first kind on polyhedral surfaces Γ in R 3 are discretized by Galerkin BEM on shape-regular, but otherwise unstructured meshes of meshwidth h. Strong ellipticity of the integral operator is shown to give nonsingular stiffness matrices and, for piecewise constant approximations, up to O(h3) convergence of the farfield. The condition number of the stiffness matrix behaves like O(h−1) in the standard basis. An O(N) agglomeration algorithm for the construction of a multilevel wavelet basis on Γ is introduced resulting in a preconditioner which reduces the condition number to O (| log h|). A class of kernel-independent clustering algorithms (containing the fast multipole method as special case) is introduced for approximate matrix–vector multiplication in O (N( log N) 3 ) memory and operations. Iterative approximate solution of the linear system by CG or GMRES with wavelet preconditioning and clustering-acceleration of matrix–vector multiplication is shown to yield an approximate solution in log-linear complexity which preserves the O(h3) convergence of the potentials. Numerical experiments are given which confirm the theory.

IEEE Transactions on Magnetics

The Mathematics of Finite Elements and Applications X, 2000

The implementation of a wavelet-based Galerkin discretization of the double layer potential opera... more The implementation of a wavelet-based Galerkin discretization of the double layer potential operator on polyhedral surfaces Γ⊂ℝ 3 is described. The algorithm generates an approximate stiffness matrix with O(N(logN) 2 ) entries in O(N(logN) 3 ) operations where N is the number of degrees of freedom on the boundary. The condition number of the compressed stiffness matrix is bounded uniformly with respect to N. A C++ realization of the data structure containing the compressed stiffness matrix is described. It can be set up in O(N(logN) 2 ) operations and requires O(N(logN) 2 ) memory. Numerical experiments show the asymptotic complexity estimates and convergence rates to be accurate already for moderate N. Problems with N>10 5 were computed in core on a workstation.

International Association of Geodesy Symposia, 2001