Galerkin Boundary Element Method Research Articles

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

SummaryAcoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space‐time stochastic Galerkin boundary element method which is applied to sound‐hard, sound‐soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

Thermally and mechanically induced strain gradient fields in architected 2D materials and beam structures

The current contribution investigates the strain-gradient thermomechanical performance of architected materials and structures with uniform and graded inner designs. To that scope, an integral representation of strain-gradients in thermoelasticity, along with its Galerkin Boundary Element Method (GBEM) implementation are elaborated. The formulation accounts for both mechanical and thermal strain-gradients for the first time. Thereupon, the complete strain-gradient response upon uniaxial tensile (UT) and thermal loading (Th) is analyzed, performing direct comparisons among the strain-gradient fields induced in each case and providing summarizing statistics that associate higher-order thermal and mechanical effects. The numerical framework is used as a basis for the quantification of the impact of the underlying structural patterns on the equivalent internal length parameters of architected beam-type structures under thermomechanical loads in the context of simple gradient theory. It is found that thermal loads relate to comparable, yet lower, internal length parameters with respect to the ones obtained for uniaxially tensioned structures with uniform inner cellular designs. Both internal length and temperature variation contributions determine the strain-gradient thermomechanical response of beam-type architected structures, for which, exact, higher-order equivalent 1D displacement field solutions are first derived. Thermally-induced, higher-gradient displacements are found to be comparable with the ones obtained in UT-loaded structures with uniform inner cellular topologies. Moreover, inner material gradings are found to be able to considerably mitigate higher-order effects, a sensitivity that is not reproduced in the UT loading case. The results provided, along with the numerical and analytical methodologies elaborated, set the basis for the thermomechanical strain-gradient analysis of advanced architected media well-beyond the designs here investigated.

An efficient computational framework for height-contained growing and intersecting hydraulic fracturing simulation via SGBEM–FEM

An efficient computational framework to model the growth of intersecting height-contained hydraulic fractures is proposed. The fracture in solids is modeled by the classical theory of linear elastic fracture mechanics, whereas the injection flow in hydraulic fractures is treated as channel flow. The governing equations of fracture mechanics are formulated in terms of weakly-singular weak-form boundary integral equations, with the knowledge of the fracture surface geometry exploited to reduce the integral dimension. The fluid flow within the fracture is treated using finite element method as one dimensional flow in an arbitrarily curved channel. The symmetric Galerkin boundary element method and finite element method are coupled with the traction continuity condition. The model is then formulated in a variational form and augmented with Lagrange multipliers to prescribe displacement constraints on intersections of hydraulic fractures, while fluid constraints are enforced by a consolidation of associated pressure degrees of freedom. Pertinent numerical implementation techniques, including spatial and temporal discretizations, special crack tip element, crack propagation criteria and remeshing strategy, are presented. The proposed framework is verified by comparing to the numerical results from a three dimensional hydraulic fracturing simulator and then verified by comparing the stress intensity factors to the XFEM and Murakami’s results for both star shaped cracks and symmetric branched cracks. The verification shows the utility properly enforces the physical constraints on the intersections of fractures. Three numerical examples are shown to demonstrate the capability of the proposed computational framework modeling the intersecting hydraulic fracturing process in a single or separated height-contained pay zones.

B-Spline Surface-Based Reduced-Order Modeling of Nonplanar Crack Growth in Structural Digital Twins

Nonplanar crack growth holds a critical role in aeronautical structures, necessitating effective analysis under mixed fatigue loading to assess structural integrity. This study introduces a reduced-order modeling (ROM) method for predicting nonplanar crack growth in structural digital twins. The method’s advantage lies in its representation of the entire crack surface morphology using a B-spline surface, which better captures its impact on crack growth. The symmetric Galerkin boundary element method–finite element method coupling method is adopted as a full-order method to generate the crack database. Isoparametric coordinates of the crack surface and stress intensity factor serve as input and output, respectively, for training the ROM, which integrates -means clustering, principal component analysis, and Gaussian process regression. The proposed approach is demonstrated using a rotorcraft-mast-like component. Results reveal superior fracture mechanics parameter prediction compared to the crack-front-based ROM. Furthermore, the method boasts three orders of magnitude greater efficiency than full-order simulation, enabling its coupling with approaches like Monte Carlo for probabilistic crack growth analysis. Future work entails integrating our method into the probabilistic framework of digital twins.

Distributed ℋ 2 -Matrices for Boundary Element Methods

Standard discretization techniques for boundary integral equations, e.g., the Galerkin boundary element method, lead to large densely populated matrices that require fast and efficient compression techniques like the fast multipole method or hierarchical matrices. If the underlying mesh is very large, running the corresponding algorithms on a distributed computer is attractive, e.g., since distributed computers frequently are cost-effective and offer a high accumulated memory bandwidth. Compared to the closely related particle methods, for which distributed algorithms are well-established, the Galerkin discretization poses a challenge, since the supports of the basis functions influence the block structure of the matrix and therefore the flow of data in the corresponding algorithms. This article introduces distributed ℋ 2 -matrices, a class of hierarchical matrices that is closely related to fast multipole methods and particularly well-suited for distributed computing. While earlier efforts required the global tree structure of the ℋ 2 -matrix to be stored in every node of the distributed system, the new approach needs only local multilevel information that can be obtained via a simple distributed algorithm, allowing us to scale to significantly larger systems. Experiments show that this approach can handle very large meshes with more than 130 million triangles efficiently.

Shear deformable plate by SGBEM: Indirect “progenitor matrix” approach

The Mindlin-Reissner's formulation of shear deformable plate is addressed through the Symmetric Galerkin Boundary Element Method (SGBEM). The thick plate's elastic problem is resolved and fundamental solutions matrix is obtained. The application of Betti's theorem in the unlimited domain allows obtaining the Somigliana's Identities (S.I.s). The solving system is obtained by an indirect approach, i.e., through the “progenitor matrix”. The “progenitor matrix” allows simulating all possible plate's load conditions and is the starting point for an automatic calculation code. The coefficients are calculated through the theory of distributions, using double integrals without resorting to regularization techniques but by exploiting the expansion in power series of the Bessel functions present in the fundamental solutions.Particular attention is paid to the presence of domain loads, whose domain integral is transformed into a boundary one using the Radial Integral Method technique (RIM).These strategies lead to a robust procedure that allows obtaining good results even in the presence of boundary sparse discretization. This is demonstrated by the results of the examples carried out which, compared with the analytical solutions present in the literature, show a very good convergence.

Analysis of near-interface cracks in three-dimensional anisotropic multi-materials by efficient BIEM

This paper presents the full analysis of near-interface cracks in three-dimensional linear elastic multimaterial bodies by a weakly singular boundary integral equation method (BIEM). The proposed technique was implemented in a general framework that allowed the treatment of finite cracked bodies with general configurations, material anisotropy, and mode-mixity. A system of integral equations governing unknown data on the boundary, crack surface, and material interface was established using a pair of weakly singular weak-form displacement and traction integral equations together with continuity along the material interface. A symmetric Galerkin boundary element method along with the finite element technique were implemented to solve the governing integral equations. In addition, the special near-front interpolation was employed to enhance the approximation of the relative crack-face displacement in the neighborhood of the crack front. The solved crack-face data were used directly to post-process the stress intensity factors and T-stresses along the crack front. An extensive numerical study was carried out not only to demonstrate the accuracy, convergence, and capability of the proposed technique, but also to explore the influence of material stiffness and distance to the material interface on fracture data along the crack front.

ReLU Neural Network Galerkin BEM

We introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation-based method, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built by using NNs that, in turn, employ the so-called Rectified Linear Units (ReLU) as the underlying activation function. The ReLU-NNs used to approximate the solutions to the BIEs depend nonlinearly on the parameters characterizing the NNs themselves. Consequently, the computation of a numerical solution to a BIE by means of ReLU-NNs boils down to a fine tuning of these parameters, in network training. We argue that ReLU-NNs of fixed depth and with a variable width allow us to recover well-known approximation rate results for the standard Galerkin Boundary Element Method (BEM). This observation hinges on existing well-known properties concerning the regularity of the solution of the BIEs on Lipschitz, polygonal boundaries, i.e. accounting for the effect of corner singularities, and the expressive power of ReLU-NNs over different classes of functions. We prove that shallow ReLU-NNs, i.e. networks having a fixed, moderate depth but with increasing width, can achieve optimal order algebraic convergence rates. We propose novel loss functions for NN training which are obtained using computable, local residual a posteriori error estimators with ReLU-NNs for the numerical approximation of BIEs. We find that weighted residual estimators, which are reliable without further assumptions on the quasi-uniformity of the underlying mesh, can be employed for the construction of computationally efficient loss functions for ReLU-NN training. The proposed framework allows us to leverage on state-of-the-art computational deep learning technologies such as TENSORFLOW and TPUs for the numerical solution of BIEs using ReLU-NNs. Exploratory numerical experiments validate our theoretical findings and indicate the viability of the proposed ReLU-NN Galerkin BEM approach.

Pixel-based boundary element method for computing effective thermal conductivity of heterogeneous materials

In the present article, we propose an efficient 2D Pixel-based Boundary Element formulation to compute the effective thermal conductivity of heterogeneous materials. In the proposed formulation, each pixel of the digital image is represented by a subdomain with four boundary elements. We develop the formulation within the context of the Symmetric Galerkin Boundary Element Method and, apart from the Boundary Element Method itself, the other main ingredients are the Domain Decomposition technique, to treat each pixel as a subdomain, and the Element-by-Element technique used in conjunction with the Preconditioned Conjugate Gradient method to solve the resulting symmetric positive definite large linear system. In addition, Periodic Boundary Conditions are employed. We validate our formulation and implementation with the aid of analytical results for the case of a plate with periodic circular inclusions. Further, we illustrate the application of the proposed formulation by computing the effective thermal conductivity of a plate with periodic square inclusions, a woven-like material, and a cast iron sample, and compare the obtained results with those obtained via finite elements. Our results indicate that the Pixel-based Boundary Element formulation performs, at least, as good as the Pixel-based Finite Element formulation.

Raytracing in Galerkin Boundary Integral form

Raytracing is an established method for computing the late-time part of room impulse responses. But it has the drawback that only very crude Monte Carlo models of boundary scattering and diffraction are possible to include without losing its attractive computational cost scaling. This happens because higher-resolution models of these processes output multiple child rays for every parent ray received, making the number of rays grow with reflection order. An emerging solution is so-called 'Surface-Based' Geometrical Acoustics. Here the distribution of rays arriving at a boundary is mapped onto a predefined set of spatial elements and angular interpolation functions, producing a vector of coefficients. Re-radiation of subsequent reflections is then a matrix multiplication, with the steady state solution being solvable via a Neumann series. Since rays only every propagate one reflection order before being collected, diffraction and scattering process that cause them to multiply can be included without issue. Here we present a new formulation based on a Galerkin Boundary Element Method (BEM). A unique feature is its ability to readily change interpolation functions, so their effect on accuracy and convergence can be assessed. In this preliminary work, it is verified against an Image Source Model for a rectangular room.

Numerical quadrature for singular integrals on fractals

We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Gamma subset mathbb {R}^n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on Gamma, including in particular the Hausdorff measure mathcal {H}^d restricted to Gamma, where d is the Hausdorff dimension of Gamma. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Gamma are decomposed into sums of integrals over suitable partitions of Gamma into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens.

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.

Time domain boundary integral equations and convolution quadrature for scattering by composite media

We consider acoustic scattering in heterogeneous media with piecewise constant wave number. The discretization is carried out using a Galerkin boundary element method in space and Runge-Kutta convolution quadrature in time. We prove well-posedness of the scheme and providea prioriestimates for the convergence in space and time.

Weakly-singular symmetric Galerkin boundary element method in thermoelasticity for the fracture analysis of three-dimensional solids

The Symmetric Galerkin Boundary Element Method has demonstrated significant advantages for the fracture mechanics analysis of three-dimensional solid structures. However, when dealing with structures in thermal environments, strongly-singular domain integrals resulting from the temperature field require meshing of the interior of the domain. In this paper, formulation and implementation of the weakly-singular SGBEM of three-dimensional thermoelastic intact and cracked solids are presented where temperature-induced domain integrals are transformed into boundary integrals. The strong singularity of these temperature-induced integrals is reduced to the weak singularity by exploring the feature of double-layer surface integrals in the SGBEM. Obtained temperature-induced boundary integrals affect only the right-hand-side vector of the SGBEM equation, and therefore the advantages of the SGBEM, such as the symmetry of the stiffness matrix, are retained. Several examples of three-dimensional solids with or without cracks subjected to thermal loadings are presented to validate the developed methods. Advantages and disadvantages of SGBEMs with different temperature-induced boundary integrals are also discussed.

Adaptive quadrature rules for Galerkin BEM

The singular integrals in the Galerkin Boundary Element Method are usually treated using singularity removing transformations. The regularized integrals are approximated by tensor product Gaussian quadrature rules. Although these integrals are smooth, the convergence rate depends strongly on the type of singularity and the aspect ratio of the triangulation. Here an adaptive quadrature scheme is presented that ensures a convergence at a user-specified rate. The effectiveness of the resulting quadrature scheme is compared with the non-adaptive approach.

Spectral Galerkin boundary element methods for high-frequency sound-hard scattering problems

This paper is concerned with the design of two different classes of Galerkin boundary element methods for the solution of high-frequency sound-hard scattering problems in the exterior of two-dimensional smooth convex scatterers. We prove in this paper that both methods require a small increase (in the order of \(k^\epsilon \) for any \(\epsilon > 0\)) in the number of degrees of freedom to guarantee frequency independent precisions with increasing wavenumber k. In addition, the accuracy of the numerical solutions are independent of frequency provided sufficiently many terms in the asymptotic expansion are incorporated into the integral equation formulation. Numerical results validating \(\mathcal {O}(k^\epsilon )\) algorithms are presented.

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.

Real-Time Prediction of Probabilistic Crack Growth with a Helicopter Component Digital Twin

To deploy the airframe digital twin or to conduct probabilistic evaluations of the remaining life of a structural component, a (near) real-time crack-growth simulation method is critical. In this paper, a reduced-order simulation approach is developed to achieve this goal by leveraging two methods. On the one hand, the symmetric Galerkin boundary element method–finite element method (SGBEM-FEM) coupling method is combined with parametric modeling to generate the database of computed stress intensity factors for cracks with various sizes/shapes in a complex structural component, by which hundreds of samples are automatically simulated within a day. On the other hand, machine learning methods are applied to establish the relation between crack sizes/shapes and crack-front stress intensity factors. By combining the reduced-order computational model with load inputs and fatigue growth laws, a real-time prediction of probabilistic crack growth in complex structures with minimum computational burden is realized. In an example of a round-robin helicopter component, even though the fatigue crack growth is simulated cycle by cycle, the simulation is faster than real-time (as compared with the physical test). The proposed approach is a key simulation technology toward realizing the digital twin of complex structures, which further requires fusion of model predictions with flight/inspection/monitoring data.

Weakly Singular Symmetric Galerkin Boundary Element Method for Fracture Analysis of Three-Dimensional Structures Considering Rotational Inertia and Gravitational Forces

The Symmetric Galerkin Boundary Element Method is advantageous for the linear elastic fracture and crackgrowth analysis of solid structures, because only boundary and crack-surface elements are needed. However, for engineering structures subjected to body forces such as rotational inertia and gravitational loads, additional domain integral terms in the Galerkin boundary integral equation will necessitate meshing of the interior of the domain. In this study, weakly-singular SGBEM for fracture analysis of three-dimensional structures considering rotational inertia and gravitational forces are developed. By using divergence theorem or alternatively the radial integration method, the domain integral terms caused by body forces are transformed into boundary integrals. And due to the weak singularity of the formulated boundary integral equations, a simple Gauss-Legendre quadrature with a few integral points is suffcient for numerically evaluating the SGBEM equations. Some numerical examples are presented to verify this approach and results are compared with benchmark solutions.

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm <1/2 as an operator on the natural trace space H^{1/2}(Gamma ) whenever Gamma is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in H^{1/2}(Gamma ) for any sequence of finite-dimensional subspaces ({{mathcal {H}}}_N)_{N=1}^infty that is asymptotically dense in H^{1/2}(Gamma ). Long-standing open questions are whether the essential norm is also <1/2 for D as an operator on L^2(Gamma ) for all Lipschitz Gamma in 2-d; or whether, for all Lipschitz Gamma in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators pm frac{1}{2}I+D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces ({{mathcal {H}}}_N)_{N=1}^infty that is asymptotically dense in L^2(Gamma ). We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is ge 1/2, and examples with Lipschitz constant two for which the operators pm frac{1}{2}I +D are not coercive plus compact. We also give, for every C>0, examples of Lipschitz polyhedra for which the essential norm is ge C and for which lambda I+D is not a compact perturbation of a coercive operator for any real or complex lambda with |lambda |le C. We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the L^2(Gamma ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on C(Gamma ), equivalent to the standard supremum norm, for which the essential norm of D on C(Gamma ) is <1/2.