Space-time Boundary Integral Equation Research Articles

Adaptive time stepping for generalized convolution quadrature

AbstractThrough the current work, we aim to improve the boundary element method (BEM) in acoustics. The proposed adaptivity is based on a time‐domain boundary element formulation. In this setting, an elegant method for solving the acoustic problem is through the transformation of the underlying partial differential equation on to the boundary of the domain as a space–time boundary integral equation. However, accurate modeling of temporal behavior is crucial, requiring careful consideration of the time step size to ensure numerical stability and capture the dynamics of the system accurately. The boundary integral equation in the time domain has a convolution in time and the generalized convolution quadrature method (gCQ) is introduced, providing a framework for numerically evaluating the convolution integral within the boundary integral equation. The gCQ method under consideration uses a higher order time stepping method from the Runge–Kutta family. The gCQ method allows for adaptive time stepping, enabling the refinement of the time step size to focus computational resources where they are most needed. The current work shows that an approach similar to that of the ordinary differential equations can be included in the gCQ to introduce an adaptive control of the time step size in the 3D boundary element formulation. Numerical experiments are conducted to validate the effectiveness of the proposed approach. The results demonstrate improved solution resolution, particularly in capturing steep changes and localized variations through the adaptive time stepping scheme. The proposed method shows promising performance, dynamically adjusting the time step size based on the evolving solution, thereby enabling efficient and accurate computations in BEM simulations of radiation phenomena.

On a time-discrete convolution—space collocation BEM for the numerical solution of two-dimensional wave propagation problems in unbounded domains

Abstract For the numerical solution of a classical two-dimensional Dirichlet exterior problem for the wave equation in the time domain, we consider a space-time boundary integral equation approach, based on its discretization by means of the coupling of a time discrete convolution quadrature of order 2, with the classical continuous piecewise linear boundary element method. The latter is either the Galerkin method or the collocation one, this being computationally much cheaper. For the efficient evaluation of most of the integrals generated by these two boundary element methods, but also for the subsequent analysis, we first define a new Gaussian quadrature. After recalling that stability and convergence have been proved for the Galerkin method, while for the collocation one we only have numerical evidences of these properties, to (partially) fill this gap, we first note that collocation can be obtained directly from the Galerkin system, after discretizing its inner product integral by using the composite trapezoidal rule, which in this case turns out to be the Gaussian rule mentioned above with $1$ node. Then, we show that for all (positive) time step-sizes $\varDelta _t$, when we let $h\rightarrow 0$, independently from $\varDelta _t$, the matrix of the linear system produced by the collocation method, properly normalized, converges, in the $\infty $-norm, to the corresponding Galerkin matrix. The behavior of the associated absolute and relative errors in the $\infty $-norm, as $h\rightarrow 0$, are $O\big (h^{\frac {5}{3}}+\varDelta _t^{-(1+\varepsilon )}h^{\frac {5+\varepsilon }{2}}[|\!\ln \varDelta _t|+|\!\ln h|]\big )$ and $O\big (h^{\frac {2}{3}}/|\!\ln \varDelta _t|+\varDelta _t^{-(1+\varepsilon )}h^{\frac {3+\varepsilon }{2}}[1+|\!\ln h|/|\!\ln \varDelta _t|]\big )$, respectively, with $\varepsilon>0$ as close as one likes to $0$ and the $O(\cdot )$ constant independent from $\varDelta _t, h$. Some possible extensions of the investigation we carried out are also mentioned.

Partially pivoted ACA based acceleration of the energetic BEM for time-domain acoustic and elastic waves exterior problems

We consider acoustic and elastic wave propagation problems in 2D unbounded domains, reformulated in terms of space-time Boundary Integral Equations (BIEs). For their numerical solution, we employ a weak formulation related to the energy of the system and we discretize the weak problems by a Galerkin-type Boundary Element Method (BEM): this approach, called Energetic BEM, has revealed accurate and stable even on large time intervals of analysis. In particular, it results that, when standard Lagrangian basis functions are considered, the BEM matrices have Toeplitz lower triangular block structure, where blocks for growing time become fully populated; hence the overall memory cost of the energetic BEM is O(M2N), M and N being the number of the space degrees of freedom and the total number of performed time steps, respectively. This drawback prevents the application of such method to large scale problems. As a possible remedy, we propose a fast technique based on the Adaptive Cross Approximation (ACA). The core of this procedure is the approximation of sufficiently large time blocks of the energetic BEM matrix through the partially pivoted ACA algorithm, which allows to compute only few of the original entries. This leads to reduced assembly time, which for the energetic BEM is generally relevant, coupled with reduced memory storage requirements. Additionally, the consequent acceleration of the matrix/vector multiplication together with a marching on time procedure, leads to remarkable reduction of the computational solution time. The effectiveness of the proposed method is theoretically proved and several numerical results are presented and discussed.

Two FEM-BEM methods for the numerical solution of 2D transient elastodynamics problems in unbounded domains

We consider wave propagation problems in 2D unbounded isotropic homogeneous elastic media, with rigid boundary conditions. For their solution, we propose and compare two alternative numerical approaches, both obtained by coupling the differential equation with the associated space-time boundary integral equation. The latter is defined on an artificial boundary, chosen to surround the (bounded) exterior computational domain of interest. The integral equation defines a boundary condition which is non-reflecting for incoming and also for outgoing waves.In both approaches, the differential equations are discretized by applying a finite element method combined with the Crank Nicolson time marching scheme, while the discretization of the integral equation is obtained by coupling a time convolution quadrature with a space collocation boundary element method. The construction of the two approaches is described and discussed. Some numerical tests are also presented.

A New Approach to Space-Time Boundary Integral Equations for the Wave Equation

We present a new approach for boundary integral equations for the wave equation with zero initial conditions. Unlike previous attempts, our mathematical formulation allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise efficient adaptive space-time boundary element methods, which will be tackled in future work. On the other hand, the proposed approach is compatible with current time dependent boundary element method's implementations and we predict that it explains many of the behaviours observed in practice but that were not understood with the existing theory.

Highly accurate quadrature schemes for singular integrals in energetic BEM applied to elastodynamics

We consider an exterior linear elastodynamics problem with vanishing initial conditions and Dirichlet datum on the scatterer. We convert the Navier Equation, governing the wave behaviour, into two space–time Boundary Integral Equations (BIEs) whose solution is approximated by the energetic Boundary Element Method (BEM). To apply this technique, we have to set the BIEs in a weak form related to the energy of the differential problem solution at the final time instant of analysis. After the space–time discretization of the weak formulation, we have to deal with double space–time integrals, with a weakly singular kernel depending on primary and secondary wave speeds and multiplied by Heaviside functions. The main purpose of this work is the analysis of these peculiar integrals and the study of suitable quadrature schemes for their approximation.

Fully Dynamic Earthquake Cycle Simulations on a Nonplanar Fault Using the Spectral Boundary Integral Element Method (sBIEM)

ABSTRACT One of the most suitable methods for modeling fully dynamic earthquake cycle simulations is the spectral boundary integral element method (sBIEM), which takes advantage of the fast Fourier transform (FFT) to make a complex numerical dynamic rupture tractable. However, this method has the serious drawback of requiring a flat fault geometry due to the FFT approach. Here, we present an analytical formulation that extends the sBIEM to a mildly nonplanar fault. We start from a regularized boundary element method and apply a small-slope approximation of the fault geometry. Making this assumption, it is possible to show that the main effect of nonplanar fault geometry is to change the normal traction along the fault, which is controlled by the local curvature along the fault. We then convert this space–time boundary integral equation of the normal traction into a spectral-time formulation and incorporate this change in normal traction into the existing sBIEM methodology. This approach allows us to model fully dynamic seismic cycle simulations on nonplanar faults in a particularly efficient way. We then test this method against a regular BIEM for both rough-fault and seamount-fault geometries and demonstrate that this sBIEM maintains the scaling between the fault geometry and slip distribution.

Energetic boundary element method for accurate solution of damped waves hard scattering problems

The paper deals with the numerical solution of 2D wave propagation exterior problems including viscous and material damping coefficients and equipped by Neumann boundary condition, hence modeling the hard scattering of damped waves. The differential problem, which includes, besides diffusion, advection and reaction terms, is written as a space–time boundary integral equation (BIE) whose kernel is given by the hypersingular fundamental solution of the 2D damped waves operator. The resulting BIE is solved by a modified Energetic Boundary Element Method, where a suitable kernel treatment is introduced for the evaluation of the discretization linear system matrix entries represented by space–time quadruple integrals with hypersingular kernel in space variables. A wide variety of numerical results, obtained varying both damping coefficients and discretization parameters, is presented and shows accuracy and stability of the proposed technique, confirming what was theoretically proved for the simpler undamped case. Post-processing phase is also taken into account, giving the approximate solution of the exterior differential problem involving damped waves propagation around disconnected obstacles and bounded domains.

Efficient Solution of Two-Dimensional Wave Propagation Problems by CQ-Wavelet BEM: Algorithm and Applications

In this paper we consider wave propagation problems in two-dimensional unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equations. For their s...

Two boundary integral equation methods for linear elastodynamics problems on unbounded domains

We consider (transient) 3D elastic wave propagation problems in unbounded isotropic homogeneous media, which can be reduced to corresponding 2D ones. For their solution, we propose and compare two boundary integral equation approaches, both based on the coupling of a discrete time convolution quadrature with a classical space collocation discretization. In the first approach, the PDE problem is preliminarily replaced by the equivalent well known (vector) space–time boundary integral equation formulation, while in the second, the same PDE is replaced by a system of two (coupled) wave equations, each one of which is then represented by the associated boundary integral equation. The construction of these two approaches is described and discussed. Some numerical testing are also presented.

Restrictions matrices for Platonic solids invariance and applications to space–time energetic BEM

Geometrical symmetry is found in many linear field problems. Group representation theory is the only valuable tool for exploiting this property from a computational point of view. In this context, here we apply a technique for taking into account equivariance in the numerical treatment of space–time boundary integral equations, which are invariant under a finite group G of congruences of R 3 , each related to one of the so-called Platonic solids. This technique is based upon suitable restriction matrices strictly related to a system of unitary, irreducible, pairwise not-equivalent matrix representations of G . The main development is expounded in the framework of space–time energetic boundary element method applied to Neumann exterior wave propagation problems, where the discretization matrices have a block lower triangular Toeplitz structure, and the diagonal block, to be inverted at each time step, is typically dense. Several numerical results, related to computational saving, are presented.

A parallel space–time boundary element method for the heat equation

In this paper we introduce a new parallel solver for the weakly singular space–time boundary integral equation for the heat equation. The space–time boundary mesh is decomposed into a given number of submeshes. Pairs of the submeshes represent dense blocks in the system matrices, which are distributed among computational nodes by an algorithm based on a cyclic decomposition of complete graphs ensuring load balance. In addition, we employ vectorization and threading in shared memory to ensure intra-node efficiency. We present scalability experiments on different CPU architectures to evaluate the performance of the proposed parallelization techniques. All levels of parallelism allow us to tackle large problems and lead to an almost optimal speedup.

On the discretization and application of two space–time boundary integral equations for 3D wave propagation problems in unbounded domains

In this paper, we consider 3D wave propagation problems in unbounded domains, such as those of acoustic waves in non viscous fluids, or of seismic waves in (infinite) homogeneous isotropic materials, where the propagation velocity c is much higher than 1. For example, in the case of air and water c≈343m/s and c≈1500m/s respectively, while for seismic P-waves in linear solids we may have c≈6000m/s or higher. These waves can be generated by sources, possible away from the obstacles. We further assume that the dimensions of the obstacles are much smaller than that of the wave velocity, and that the problem transients are not excessively short.For their solution we consider two different approaches. The first directly uses a known space–time boundary integral equation to determine the problem solution. In the second one, after having defined an artificial boundary delimiting the region of computational interest, the above mentioned integral equation is interpreted as a non reflecting boundary condition to be coupled with a classical finite element method.For such problems, we show that in some cases the computational cost and storage, required by the above numerical approaches, can be significantly reduced by taking into account a property that till now has not been considered. To show the effectiveness of this reduction, the proposed approach is applied to several problems, including multiple scattering.

A space–time BIE method for 2D wave equation problems with mixed boundary conditions

In this paper we consider the 2D exterior problem for the non homogeneous wave equation, with mixed type boundary conditions and non homogeneous initial conditions, these latter having local supports. First we derive a space–time boundary integral equation formulation to solve the problem. Then we propose a numerical approach for its solution, which couples a Lubich discrete convolution quadrature with a classical collocation method. For the computation of the extra “volume” integrals generated by the initial data, having local supports, we propose a new numerical approach. Finally, we solve some test problems and present some of the numerical results we have obtained.

A fast Galerkin method for parabolic space-time boundary integral equations

An efficient scheme for solving boundary integral equations of the heat equation based on the Galerkin method is introduced. The parabolic fast multipole method (pFMM) is applied to accelerate the ...

A fast Galerkin method for parabolic space–time boundary integral equations

An efficient scheme for solving boundary integral equations of the heat equation based on the Galerkin method is introduced. The parabolic fast multipole method (pFMM) is applied to accelerate the evaluation of the thermal layer potentials. In order to remain attractive for a wide range of applications, a key issue is to ensure efficiency for a big variety of temporal to spatial mesh ratios. Within the parabolic Galerkin FMM (pGFMM) it turns out that the temporal nearfield can become very costly. To that end, a modified fast Gauss transform (FGT) is developed. The complexity and convergence behavior of the method are analyzed and numerically investigated on a range of model problems. The results demonstrate that the complexity is nearly optimal in the number of discretization parameters while the convergence rate of the Galerkin method is preserved.

Lubich convolution quadratures and their application to problems described by space-time BIEs

In the last years several authors have used Lubich convolution quadrature formulas to discretize space-time boundary integral equations representing time dependent problems. These rules have the fundamental property of not using explicitly the expression of the kernel of the integral equation they are applied to, which is instead replaced by that of its Laplace transform, usually given by a simple analytic function. In this paper, a review of these rules, which includes their main properties, several new remarks and some conjectures, will be presented when they are applied to the heat and wave space-time boundary integral equation formulations. The construction and behavior of the corresponding coefficients are analyzed and tested numerically. When the quadrature is defined by a BDF method, a new approach for the representation of its coefficients is presented.

Analysis of Boundary Element Methods

The workshop \\emph{Analysis of Boundary Element Methods}, organized by Martin Costabel (Rennes) and Ernst P. Stephan (Hannover). This meeting brought together 46 experts in numerical analysis. Boundary element methods (BEM) are well established numerical methods with a wide range of applications. There are still many challenging problems, the aim of this workshop was to coordinate the efforts to tackle these problems. A central theme of many talks was the error analysis. In the following we focus on some central aspects. \\begin{enumerate} \\item\[1.] \\textbf{\\textit{BEM for time dependent problems:}} Space-time boundary integral equations using integral representations with the fundamental solution of hyperbolic problems have become increasingly important in elastodynamics and electrodynamics. New results were presented concerning error analysis, stability and performance of numerical algorithms. Special attention was given to efficient quadrature of retarded potentials and also to the method of convolution quadrature for the computation of transient acoustic waves. \\pagebreak \\item\[2.]\\textit{\\textbf{Stability at high frequencies:}} The numerical modeling of scattering problems at higher frequencies poses practical challenges due to the need to approximate highly oscillating functions. In addition, the analysis of many of the standard BEM loses usefulness in the high frequency range because the constants in the stability and error estimates either grow rapidly with the frequency of have unspecified dependency on the frequency. New boundary element methods have been presented whose stability deteriorates only slowly with frequency or are even unconditionally stable for all frequencies. \\item\[3.] \\textbf{\\textit{Fast algorithms:}} Multilevel approaches have produced several families of highly efficient algorithms. Several talks reported on the applications of the fast multipole method and on iterative techniques based on multigrid methods, H-matrices or on Schwarz preconditioners. \\item\[4.] \\textit{\\textbf{New concepts for BEM:}} Topics have been here: sparse p-version BEM with random loading, meshless methods for pseudodifferential equations, adaptive procedures, least squares FEM-BEM coupling, p-version applied to EFIE, BEM for eigenvalue problems. \\item\[5.] \\textbf{\\textit{Specific applications:}} Here the following have been addressed: Source identification and source reconstructions in acoustics with the help of inverse boundary elements, BEM for Maxwell's equations (currents and charges boundary integral equations, shape optimization of integrated lens antennas, electromagnetic scattering by a dielectric body), BEM for option pricing. \\end{enumerate} The list of participants included several of the founders of the discipline of mathematical analysis of BEM, but also some PhD students. This mix of both established and younger researchers created a particularly stimulating research atmosphere.

Transient Response of Nonlinearly Loaded Wires in a Two Media Configuration

The transient behavior of thin wires with nonlinear resistive load in the presence of a dielectric half-space is analyzed directly in the time domain. The nonlinear wire problem is formulated by the space-time Hallen integral equation. The effect of a two-media configuration is taken into account via the space-time reflection coefficient appearing within the Green function. The resulting integral equation is handled by the space-time boundary integral equation method. The transient response for the case of a thin wire isolated in free space computed by this direct time-domain approach is compared with results obtained by another method solved using data derived from frequency-domain analysis. Results for various other configurations are also presented.