Marcus Wagner

Numerical treatment of acoustic problems with the hybrid boundary element method

Abstract The symmetric hybrid boundary element method in the frequency and time domain is introduced for the computation of acoustic radiation and scattering in closed and infinite domains. The hybrid stress boundary element method in a frequency domain formulation is based on the dynamical Hellinger–Reissner potential and leads to a Hermitian, frequency-dependent stiffness equation. As compared to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix are communicated. On the other hand, the hybrid displacement boundary element method for time domain starts out from Hamiltons principle formulated with the velocity potential. The field variables in both formulations are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions, generated by Dirac distributions, and generalized loads, that are time dependent in the transient case. The domain is modified such that small spheres centered at the nodes are subtracted. Then the property of the Dirac distribution, now acting outside the domain, cancels the remaining domain integral in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. In the time domain formulation, an analytical transformation is employed to transform the remaining domain integral into a boundary one. This approach results in a linear system of equations with a symmetric stiffness and mass matrix. Earlier 2D results are generalized in the present paper by a 3D implementation. Numerical results of transient pressure wave propagation in a closed domain are presented.

Efficient field point evaluation by combined direct and hybrid boundary element methods

A new procedure for calculating field points with the boundary element method (BEM) is outlined. In the conventional BEM (CBEM) field points are computed as post-processing by recurrently solving the boundary integral equation with known boundary data for every field point of interest. This procedure is very time-consuming. On the other hand, newly developed hybrid boundary element formulations (HBEMs) based on variational principles compute field points from the known boundary solution by simply transforming it to the domain using fundamental solutions as field approximation. The advantage of the HBEM is that the related system matrices are symmetric by construction but the computational effort to obtain the boundary solution is higher than in CBEMs. Thus, if symmetry of the system matrices has no priority, it is advantageous to compute the boundary solution with the CBEM and the field point solution with the HBEM. The additional implementation effort is relatively low, since only one extra matrix has to be computed, which is sparse and has analytical entries.

Boundary Element Method for Potential Problems

In Chapter 1, we introduced the Boundary Element Method by means of a simple one-dimensional example for which we sketched the basic steps from the differential equation to the boundary integral equation. However, since the boundary in a 1-D continuum degenerates to two points, we have neither encountered the special problems arising from the singularity of the fundamental solutions, nor could we describe the discretisation process with boundary elements. With the mathematical and physical background given in Chapters 2 and 3, we can now direct our attention to the more interesting and practically relevant two- and three-dimensional problems. In the present chapter, we will deal with simple potential problems in 2-D, so that we can concentrate on the basic features of the Boundary Element formulation. In Chapter 5, we then apply the method to 3-D problems of elastomechanics and static piezoelectricity.

Efficient variation-aware statistical dynamic timing analysis for delay test applications

Increasing parameter variations, caused by variations in process, temperature, power supply, and wear-out, have emerged as one of the most important challenges in semiconductor manufacturing and test. As a consequence for gate delay testing, a single test vector pair is no longer sufficient to provide the required low test escape probabilities for a single delay fault. Recently proposed statistical test generation methods are therefore guided by a metric, which defines the probability of detecting a delay fault with a given test set. However, since runtime and accuracy are dominated by the large number of required metric evaluations, more efficient approximation methods are mandatory for any practical application. In this work, a new statistical dynamic timing analysis algorithm is introduced to tackle this problem. The associated approximation error is very small and predominantly caused by the impact of delay variations on path sensitization and hazards. The experimental results show a large speedup compared to classical Monte Carlo simulations.

Incremental computation of delay fault detection probability for variation-aware test generation

Large process variations in recent technology nodes present a major challenge for the timing analysis of digital integrated circuits. The optimization decisions of a statistical delay test generation method must therefore rely on the probability of detecting a target delay fault with the currently chosen test vector pairs. However, the huge number of probability evaluations in practical applications creates a large computational overhead. To address this issue, this paper presents the first incremental delay fault detection probability computation algorithm in the literature, which is suitable for the inner loop of automatic test pattern generation methods. Compared to Monte Carlo simulations of NXP benchmark circuits, the new method consistently shows a very large speedup and only a small approximation error.

A Symmetric Hybrid Boundary Element Method for Acoustical Problems

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.

Probabilistic sensitization analysis for variation-aware path delay fault test evaluation

With the ever increasing process variability in recent technology nodes, path delay fault testing of digital integrated circuits has become a major challenge. A randomly chosen long path often has no robust test and many of the existing non-robust tests are likely invalidated by process variations. To generate path delay fault tests that are more tolerant towards process variations, the delay test generation must evaluate different non-robust tests and only those tests that sensitize the target path with a sufficiently high probability in presence of process variations must be selected. This requires a huge number of probability computations for a large number of target paths and makes the development of very efficient approximation algorithms mandatory for any practical application. In this paper, a novel and efficient probabilistic sensitization analysis is presented which is used to extract a small subcircuit for a given test vector-pair. The probability that a target path is sensitized by the vector-pair is computed efficiently and without significant error by a Monte-Carlo simulation of the subcircuit.

Solution of the Equations of Motion

When using static or stationary fundamental solutions in conjunction with the Dual Reciprocity Boundary Element Method, the resulting systems of equations contain time-independent system matrices. These systems are solved using different algorithms adapted to the particular kind of analysis to be performed, for example, transient analysis, solution of the eigenproblem, etc. The algorithms are similar to those employed in Finite Element analysis, but the system matrices in the Boundary Element Method are neither symmetric nor positive-definite; this can lead to difficulties in the solution of the eigenproblem or in transient analyses if the algorithms are not employed judiciously.

Properties of Elastic Materials

The classification, mass densities, and elasticity matrices C of four elastic materials are given in Table A.1 and A.2

The Hybrid Boundary Element Method in Time Domain

In this chapter, we describe the Hybrid Displacement Boundary Element Method for the time domain and derive formulations for elastodynamics and acoustics. These formulations lead to symmetric systems of equations with time-invariant mass and stiffness matrices. The symmetry is exploited to formulate a symmetric coupled system of equations for fluid-structure interaction in time domain.