Adaptive Boundary Element Method Research Articles

Intelligent Inversion of Coastal Earth Resistivity

Coastal grounding electrodes are currently an important means to alleviate land grounding electrode land constraints. In order to better invert the terrestrial geodesic resistivity in the coastal region, this paper proposes a complete set of inversion technology schemes. First, this paper proposes a layered land model for the coastal region, and a composite geodetic model is modeled by the fold junction of the land model and the ocean. Based on this, an adaptive subdivision boundary element method is proposed for solving the composite soil grounding calculation problem, and the accuracy and advantages of the method are demonstrated by examples. Finally, the paper uses the differential evolutionary algorithm to invert the exploration data of the four-point method in the coastal area, and obtains the parameters of the terrestrial layered geodetic model that meet the engineering requirements. The comparison with the grounding software CDEGS illustrates the effectiveness of the method. This paper carries out the research on the modeling and inversion methods of composite layered soil model, combining advanced numerical calculation methods and artificial intelligence algorithms to provide the support of computational tools for coastal resistivity inversion.

Benchmark computations for the polarization tensor characterization of small conducting objects

The characterisation of small low conducting inclusions in an otherwise uniform background from low-frequency electrical field measurements has important applications in medical imaging using electrical impedance tomography as well as in geological imaging using electrical resistivity tomography. It is known that such objects can be characterised by a Póyla-Szegö (polarizability) tensor. Such characterisations have attracted interest as they can provide object features in a machine learning classification algorithm and provide an alternative imaging solution. However, to be able train machine learning algorithms, large dictionaries are required and it is essential that the characterisations are accurate. In this work, we obtain accurate numerical approximations to the tensor coefficients, by applying an adaptive boundary element method. The goal being to provide a sequence of benchmark computations for the tensor coefficients to allow other software developers check the accuracy of their codes.

Stable Implementation of Adaptive IGABEM in 2D in MATLAB

Abstract We report on the Matlab program package IGABEM2D which provides an easily accessible implementation of adaptive Galerkin boundary element methods in the frame of isogeometric analysis and which is available on the web for free download. Numerical experiments with IGABEM2D underline the particular importance of adaptive mesh refinement for high accuracy in isogeometric analysis.

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.

Adaptive space-time BEM for the heat equation

We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Dirichlet data. We propose a residual-type a posteriori error estimator that is a lower bound and, up to weighted L2-norms of the residual, also an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be prismatic, i.e., their elements are tensor-products J×K of elements in time J and space K. While the results do not depend on the local aspect ratio between time and space, assuming the scaling |J|≂diam(K)2 for all elements and using Galerkin BEM, the estimator is shown to be efficient and reliable without the additional L2-terms. In the considered numerical experiments on two-dimensional domains in space, the estimator seems to be equivalent to the error, independently of these assumptions. In particular for adaptive anisotropic refinement, both converge with the best possible convergence rate.

An adaptive boundary element method for the transmission problem with hyperbolic metamaterials

In this work we present an adaptive boundary-integral equation method for computing the electromagnetic response of wave interactions in hyperbolic metamaterials. The indefiniteness of the permittivity tensor gives rise to preferential wave radiation within the propagating cone for the hyperbolic media, and this induces sharp transition for the solution of the integral equation across the cone boundary when waves start to decay or grow exponentially. In order to avoid a global refined mesh over the whole boundary, we employ a two-level a posteriori error estimator and an adaptive mesh refinement procedure to resolve the singularity locally for the solution of the integral equation. Such an adaptive procedure allows for the reduction of the number of the degrees of freedom significantly for the integral equation solver while achieving desired accuracy for the solution. In addition, to resolve the fast transition of the fundamental solution and its derivatives accurately across the propagation cone boundary, adaptive numerical quadrature rules are applied to evaluate the integrals for the stiffness matrices. Finally, to formulate the integral equations over the boundary we also derive the limits of layer potentials and their derivatives in the hyperbolic media when the target points approach the boundary.

Adaptive isogeometric boundary element methods with local smoothness control

In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the boundary partition and, moreover, steers the smoothness of the NURBS ansatz functions across elements. In particular and unlike prior work, the algorithm can increase and decrease the local smoothness properties and hence exploits the full potential of isogeometric analysis. We prove that the new adaptive strategy leads to linear convergence with optimal algebraic rates. Numerical experiments confirm the theoretical results. A short appendix comments on analogous results for the weakly-singular integral equation.

Adaptive boundary element methods for the computation of the electrostatic capacity on complex polyhedra

The accurate computation of the electrostatic capacity of three dimensional objects is a fascinating benchmark problem with a long and rich history. In particular, the capacity of the unit cube has widely been studied, and recent advances allow to compute its capacity to more than ten digits of accuracy. However, the accurate computation of the capacity for general three dimensional polyhedra is still an open problem. In this paper, we propose a new algorithm based on a combination of ZZ-type a posteriori error estimation and effective operator preconditioned boundary integral formulations to easily compute the capacity of complex three dimensional polyhedra to 5 digits and more. While this paper focuses on the capacity as a benchmark problem, it also discusses implementational issues of adaptive boundary element solvers, and we provide codes based on the boundary element package Bempp to make the underlying techniques accessible to a wide range of practical problems.

Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods

We define and analyze (local) multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the refinement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results.

Adaptive BEM with optimal convergence rates for the Helmholtz equation

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.

On Adaptive Wavelet Boundary Element Methods

The present article is concerned with the numerical solution of boundary integral equa- tions by an adaptive wavelet boundary element method. This method approximates the solution with a computational complexity that is proportional to the solution’s best N -term approximation. The focus of this article is on algorithmic issues which includes the crucial building blocks and details about the efficient implementation. By numerical examples for the Laplace equation and the Helmholtz equation, solved for different geometries and right-hand sides, we validate the feasibility and efficiency of the adaptive wavelet boundary element method.

π-BEM: A flexible parallel implementation for adaptive, geometry aware, and high order boundary element methods

Many physical phenomena can be modelled using boundary integral equations, and discretised using the boundary element method (BEM). Such models only require the discretisation of the boundary of the domain, making the setup of the simulation straightforward and lowering the number of degrees of freedom. However, while many parallel efficient libraries are available for the Finite Element Method (FEM), the implementation of scalable BEM solvers still poses many challenges. We present the open source framework π-BEM (where π stands for parallel): a novel boundary element method solver, combining distributed and shared memory paradigms to achieve high scalability. π-BEM exploits high performance libraries and graph partitioning tools to deliver a parallel solver employing automatic domain decomposition, high order elements, local refinement capabilities, and exact geometry-adaptivity (using CAD files). A preliminary fast multipole accelerator is included in the implementation. Every aspect of the library is modular and easily extendible by the community. We discuss the internal structure of the code, and present some examples to demonstrate the reliability and scalability of our implementation.

An adaptive isogeometric boundary element method for predicting the effective thermal conductivity of steady state heterogeneity

This work presents an adaptive isogeometric boundary element method (IGBEM) for the calculation of effective thermal conductivity of steady state heterogeneities. Based on the generalized self-consistent scheme (GSCS), some integral equation formulations which only contain the unknown temperatures on the interface are used to calculate the effective thermal conductivity of steady state composites. In our approach, the geometries of the inclusion and original matrix are described using NURBS basis functions. The advantage over currently used methods is that no geometrical errors exist in the analysis process. And the geometry data in the isogeometric GSCS model can be taken directly from CAD programs. In addition, based on the upper bound of the relative error of the Gaussian quadrature formula, an adaptive integration method is used to compute the boundary integrals, which makes the computation of the integrals easier and more efficient at optimal computational cost. The comparisons between the results obtained by the present method and the existing counterparts are carried out and the good agreement can be observed.

Adaptive Wavelet BEM for Boundary Integral Equations: Theory and Numerical Experiments

ABSTRACTWe are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ℝ3. The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution.

Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver

We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in d≥1 dimensions. Our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and, where necessary, modifying them so that they can be applied to the fractional PDE case. Improved a priori error estimates are derived for the case of quasi-uniform meshes which are seen to deliver sub-optimal rates of convergence owing to the presence of singularities. Attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure. We assume that the resulting refined meshes are locally quasi-uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods, including the multigrid method. The storage of the dense matrices along with efficient techniques for computing the dense matrix–vector products needed for the iterative solution is also considered. The performance and efficiency of the resulting algorithm is illustrated for a variety of examples.

Adaptive time domain boundary element methods with engineering applications

Parts of this work were funded by BMWi under the project SPERoN 2020, part II, Leiser StraB enverkehr, grant number 19 U 10016 F. H. G. acknowledges support by ERC Advanced Grant HARG 268105.

Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations

In a recent work (Feischl et al. in Eng Anal Bound Elem 62:141–153, 2016), we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates. Technical contributions include a novel mesh-size function which also monitors the knot multiplicity as well as inverse estimates for NURBS in fractional-order Sobolev norms.

Investigation on the dynamics of air-gun array bubbles based on the dual fast multipole boundary element method

The air-gun array has important applications in ocean engineering. In this work, new dynamic behaviors of the air-gun array bubbles were investigated. The new adaptive fast multipole boundary element method (FMBEM) was implemented to accelerate the calculation of large scale moving boundary problems. The dual boundary integral equation (BIE) with an optimal linear combination of the regular BIE and the gradient BIE was used to facilitate a faster convergent FMBEM. It has been verified that the accuracy of the present numerical model was satisfactory for the dynamic simulation and the overall computational cost was scaled to O(N1.1) with N being the number of unknowns, which surpassed the conventional BEM largely. The behavior of air-gun array bubbles was studied for different distribution forms of the air-guns with the gravity effect considered. By increasing the dimensions of the bubble array, it can be seen that the “shield effect” of outer bubbles on inner bubbles was more evident. Pressure gradient was the dominant factor that determined the re-entrant jet's direction. The period of bubble array was longer if array scale increased. The distance between bubbles versus the intensity of interaction effect has been studied. In addition, the evolutions and the movements of the bubbles are quite different in this kind of large scale problem.

Adaptive 2D IGA boundary element methods

We derive and discuss a posteriori error estimators for Galerkin and collocation IGA boundary element methods for weakly singular integral equations of the first-kind in 2D. While recent own work considered the Faermann residual error estimator for Galerkin IGA boundary element methods, the present work focuses more on collocation and weighted-residual error estimators, which provide reliable upper bounds for the energy error. Our analysis allows piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. We formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments show that the proposed adaptive strategy leads to optimal convergence, and related IGA boundary element methods are superior to standard boundary element methods with piecewise polynomials.

Adaptive boundary element methods for optimal convergence of point errors

One particular strength of the boundary element method is that it allows for a high-order pointwise approximation of the solution of the related partial differential equation via the representation formula. However, the high-order convergence and hence accuracy usually suffers from singularities of the Cauchy data. We propose two adaptive mesh-refining algorithms and prove their quasi-optimal convergence behavior with respect to an a posteriori computable bound for the point error in the representation formula. Numerical examples for the weakly-singular integral equations for the 2D and 3D Laplacian underline our theoretical findings.