Johannes Tausch

A multiscale method for fast capacitance extraction

The many levels of metal used in aggressive deep submicron process technologies has made fast and accurate capacitance extraction of complicated 3D geometries of conductors essential, and many novel approaches have been recently developed. In this paper we present an accelerated boundary-element method, like the well-known FASTCAP program, but instead of using an adaptive fast multipole algorithm we use a numerically generated multiscale basis for constructing a sparse representation of the dense boundary-element matrix. Results are presented to demonstrate that the multiscale method can be applied to complicated geometries, generates a sparser boundary-element matrix than the adaptive fast multipole method and provides an inexpensive but effective preconditioner. Examples are used to show that the better sparsification and the effective preconditioner yield a method that can be 25 times faster than FASTCAP while still maintain accuracy in the smallest coupling capacitances.

Multiscale Bases for the Sparse Representation of Boundary Integral Operators on Complex Geometry

A multilevel transform is introduced to represent discretizations of integral operators from potential theory by nearly sparse matrices. The new feature presented here is to construct the basis in a hierarchical decomposition of the three-space and not, as in previous approaches, in a parameter space of the boundary manifold. This construction leads to sparse representations of the operator even for geometrically complicated, multiply connected domains. We will demonstrate that the numerical cost to apply a vector to the operator using the nonstandard form is essentially equal to performing the same operation with the fast multipole method. With a second compression scheme the multiscale approach can be further optimized. The diagonal blocks of the transformed matrix can be used as an inexpensive preconditioner which is empirically shown to reduce the condition number of discretizations of the single layer operator so as to be independent of mesh size.

A fast method for solving the heat equation by layer potentials

Boundary integral formulations of the heat equation involve time convolutions in addition to surface potentials. If M is the number of time steps and N is the number of degrees of freedom of the spatial discretization then the direct computation of a heat potential involves order N2M2 operations. This article describes a fast method to compute three-dimensional heat potentials which is based on Chebyshev interpolation of the heat kernel in both space and time. The computational complexity is order p4q2NM operations, where p and q are the orders of the polynomial approximation in space and time.

Capacitance extraction of 3-D conductor systems in dielectric media with high-permittivity ratios

We describe a perturbation formulation for the problem of computing electrostatic capacitances of multiple conductors embedded in multiple dielectric materials. Unlike the commonly used equivalent-charge formulation (ECF), this new approach insures that the capacitances are computed accurately even when the permittivity ratios of the dielectric materials are very large. Computational results from a three-dimensional multipole-accelerated algorithm based on this approach are presented. The results show that the accuracy of this new approach is nearly independent of the permittivity ratios and superior to the ECF for realistic interconnect structures.

Second-kind integral formulations of the capacitance problem

The standard approach to calculating electrostatic forces and capacitances involves solving a surface integral equation of the first kind. However, discretizations of this problem lead to ill-conditioned linear systems and second-kind integral equations usually solve for the dipole density, which can not be directly related to electrostatic forces. This paper describes a second-kind equation for the monopole or charge density and investigates different discretization schemes for this integral formulation. Numerical experiments, using multipole accelerated matrix–vector multiplications, demonstrate the efficiency and accuracy of the new approach.

On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

The present paper is concerned with the numerical solution of a shape identification problem for the heat equation. The goal is to determine of the shape of a void or an inclusion of zero temperature from measurements of the temperature and the heat flux at the exterior boundary. This nonlinear and ill-posed shape identification problem is reformulated in terms of three different shape optimization problems: (a) minimization of a least-squares energy variational functional, (b) tracking of the Dirichlet data, and (c) tracking of the Neumann data. The states and their adjoint equations are expressed as parabolic boundary integral equations and solved using a Nystrom discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. Special quadrature rules are derived to handle singularities of the kernel and the solution. Numerical experiments are carried out to demonstrate and compare the different formulations.

Improved integral formulations for fast 3-D method-of-moments solvers

This paper introduces a new integral formulation to calculate charge densities of conductor systems that may include multiple dielectric materials. We show that the conditioning of our formulation is much better than that of the standard equivalent charge formulation. When combined with a nonstandard discretization scheme, results can be obtained with higher accuracy at reduced numerical cost. We present a multipole accelerated implementation of our formulation. The results demonstrate that the new approach can cut the iteration count by a factor between two and four. Moreover, we will demonstrate that in the presence of sparsification errors and multiple dielectric materials second-kind formulations are much more accurate than the standard first-kind formulations.

A variable order wavelet method for the sparse representation of layer potentials in the non-standard form

We discuss a variable order wavelet method for boundary integral formulations of elliptic boundary value problems. The wavelet basis functions are transformations of standard nodal basis functions and have a variable number of vanishing moments. For integral equations of the second kind we will show that the non-standard form can be compressed to contain only O(N) non-vanishing entries while retaining the asymptotic converge of the full Galerkin scheme, where N is the number of degrees of freedom in the discretization.

The variable order fast multipole method for boundary integral equations of the second kind

We discuss the variable order Fast Multipole Method (FMM) applied to piecewise constant Galerkin discretizations of boundary integral equations. In this version of the FMM low-order expansions are employed in the finest level and orders are increased in the coarser levels. Two versions will be discussed, the first version computes exact moments, the second is based on approximated moments. When applied to integral equations of the second kind, both versions retain the asymptotic error of the direct method. The complexity estimate of the first version contains a logarithmic term while the second version is O(N) where N is the number of panels.

An efficient numerical method for a shape-identification problem arising from the heat equation

This paper is dedicated to the determination of the shape of a compactly supported constant source in the heat equation from measurements of the heat flux through the boundary. This shape-identification problem is formulated as the minimization of a least-squares cost functional for the desired heat flux at the boundary. The shape gradient of the shape functional under consideration is computed by means of the adjoint method. A gradient-based nonlinear Ritz–Galerkin scheme is applied to discretize the shape optimization problem. The state equation and its adjoint are computed by a fast space-time multipole method for the heat equation. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach.