Boundary Integral Representation Research Articles

Goal-oriented model adaptivity for viscous incompressible flows

In van Opstal et al. (Comput Mech 50:779---788, 2012) airbag inflation simulations were performed where the flow was approximated by Stokes flow. Inside the intricately folded initial geometry the Stokes assumption is argued to hold. This linearity assumption leads to a boundary-integral representation, the key to bypassing mesh generation and remeshing. It therefore enables very large displacements with near-contact. However, such a coarse assumption cannot hold throughout the domain, where it breaks down one needs to revert to the original model. The present work formalizes this idea. A model adaptive approach is proposed, in which the coarse model (a Stokes boundary-integral equation) is locally replaced by the original high-fidelity model (Navier---Stokes) based on a-posteriori estimates of the error in a quantity of interest. This adaptive modeling framework aims at taking away the burden and heuristics of manually partitioning the domain while providing new insight into the physics. We elucidate how challenges pertaining to model disparity can be addressed. Essentially, the solution in the interior of the coarse model domain is reconstructed as a post-processing step. We furthermore present a two-dimensional numerical experiments to show that the error estimator is reliable.

Boundary Integral Solvers for an Evolutionary Exterior Stokes Problem

This paper proposes and analyzes a full discretization of the exterior transient Stokes problem with Dirichlet boundary conditions. The method is based on a single layer boundary integral representation, using Galerkin semidiscretization in the space variables and multistep convolution quadrature in time. Convergence estimates are based on a Laplace domain analysis, which translates into a detailed study of the exterior Brinkman problem. Some numerical experiments are provided.

An Efficient Galerkin Boundary Element Method for the Transient Heat Equation

We present boundary integral representations of several initial boundary value problems related to the heat equation. A Galerkin discretization with piecewise constant functions in time and piecewise linear functions in space leads to optimal a priori error estimates, provided that the meshwidths in space and time satisfy $h_t=\mathcal{O}(h_x^2)$. Each time step involves the solution of a linear system, whose spectral condition number is independent of the refinement under the same assumption on the mesh. We show that if the parabolic multipole method is used to apply parabolic boundary integral operators, the overall complexity of the scheme is log-linear while preserving the convergence of the Galerkin discretization method. The theoretical estimates are confirmed numerically at the end of the paper.

Complex variables-based approach for analytical evaluation of boundary integral representations of three-dimensional acoustic scattering

The paper presents the complex variables-based approach for analytical evaluation of three-dimensional integrals involved in boundary integral representations (potentials) for the Helmholtz equation. The boundary element is assumed to be planar bounded by an arbitrary number of straight lines and/or circular arcs. The integrals are re-written in local (element) coordinates, while in-plane components of the fields are described in terms of certain complex combinations. The use of Cauchy–Pompeiu formula (a particular case of Bochner–Martinelli formula) allows for the reduction of surface integrals over the element to the line integrals over its boundary. By considering the requirement of the minimum number of elements per wavelength and using an asymptotic analysis, analytical expressions for the line integrals are obtained for various density functions. A comparative study of numerical and analytical integration for particular integrals over two types of elements is performed.

A Poisson boundary for topological semigroups

We give a non-probabilistic proof for the boundary integral representation of μ-harmonic functions on topological semigroups.

BEM numerical simulation of coupled heat and moisture flow through a porous solid

Abstract The problem of unsteady coupled moisture and heat energy transport through a porous solid is studied numerically using singular boundary integral representation of the governing equations. The integral equations are discretized using mixed-boundary elements and a multidomain method also known as the macro-elements technique. Two discretization models over time are presented, i.e. constant and linear interpolation in time. Numerical simulations were performed for the benchmark problems of moisture uptake within a semi-infinite region and for the drying out of a layer. Non-uniform grids of increasing mesh density were employed to ensure accurate solutions. Satisfactory agreement is obtained with benchmark results, especially when using the linear in time numerical model.

Integral Equation-Wavelet Collocation Method for Geometric Transformation and Application to Image Processing

Geometric (or shape) distortion may occur in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by a certain geometric transformation model, we can apply its inverse transformation to remove the distortion for the geometric restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. In this paper, a novel wavelet-based method is presented, which consists of three phases. In phase 1, the partial differential equation is converted into boundary integral equation and representation by an indirect method. In phase 2, the boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. In phase 3, the plane integral equation and representation are then solved by a method we call wavelet collocation. The performance of our method is evaluated by numerical experiments.

A biomolecular electrostatics solver using Python, GPUs and boundary elements that can handle solvent-filled cavities and Stern layers

The continuum theory applied to biomolecular electrostatics leads to an implicit-solvent model governed by the Poisson–Boltzmann equation. Solvers relying on a boundary integral representation typically do not consider features like solvent-filled cavities or ion-exclusion (Stern) layers, due to the added difficulty of treating multiple boundary surfaces. This has hindered meaningful comparisons with volume-based methods, and the effects on accuracy of including these features has remained unknown. This work presents a solver called PyGBe that uses a boundary-element formulation and can handle multiple interacting surfaces. It was used to study the effects of solvent-filled cavities and Stern layers on the accuracy of calculating solvation energy and binding energy of proteins, using the well-known apbsfinite-difference code for comparison. The results suggest that if required accuracy for an application allows errors larger than about 2% in solvation energy, then the simpler, single-surface model can be used. When calculating binding energies, the need for a multi-surface model is problem-dependent, becoming more critical when ligand and receptor are of comparable size. Comparing with the apbssolver, the boundary-element solver is faster when the accuracy requirements are higher. The cross-over point for the PyGBe code is on the order of 1–2% error, when running on one gpu card (nvidiaTesla C2075), compared with apbsrunning on six Intel Xeon cpu cores. PyGBe achieves algorithmic acceleration of the boundary element method using a treecode, and hardware acceleration using gpus via PyCuda from a user-visible code that is all Python. The code is open-source under MIT license.

A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks

On a finite network (connected weighted undirected graph), the relationship between the natural Dirichlet form E and the discrete Laplace operator Δ is given by \(\varepsilon (u,v) = {\langle u,\Delta v\rangle _{{\ell ^2}}}\), where the latter is the usual ℓ 2 inner product. This formula is not generally true for infinite networks; earlier authors have given various conditions under which this formula remains valid. Instead, we extend this formula to arbitrary infinite networks (including the case when Δ is unbounded) by including a new (boundary) term, in parallel with the classical Gauss-Green identity. This tool allows for detailed study of the boundary of the network. We construct a reproducing kernel for the space of functions of finite energy which allows us to specify a dense domain for Δ and give several criteria for the transience of the random walk on the network. The extended Gauss-Green identity and the reproducing kernel also yield a boundary integral representation for harmonic functions of finite energy. The boundary representation is developed further in [24].

Modelado numérico para estudiar interfases fluido-sólidas ante excitaciones dinámicas

This work shows the wave propagation in fluid-solid interfaces due to dynamic excitations, such interface waves are known as Scholte's waves. We studied a wide range of elastic solid materials used in engineering. The interface connects an acoustic medium (fluid) and another solid. It has been shown that by means of an analysis of diffracted waves in a fluid, it is possible to deduce the mechanical characteristics of the solid medium, specifically, its propagation velocities. For this purpose, the diffracted field of pressures and displacements, due to an initial pressure in the fluid, are expressed using boundary integral representations, which satisfy the equation of motion. The initial pressure in the fluid is represented by a Hankel's function of second kind and zero order. The solution to this problem of wave propagation is obtained by means of the Indirect Boundary Element Method, which is equivalent to the well-known Somigliana's representation theorem. The validation of the results was performed by means of the Discrete Wave Number Method. Firstly, spectra of pressures to illustrate the behavior of the fluid for each solid material considered are included, then, the Fast Fourier Transform algorithm to display the results in the time domain is applied, where the emergence of Scholte's waves and the amount of energy that they carry are highlighted.

A three-dimensional cohesive-frictional grain-boundary micromechanical model for intergranular degradation and failure in polycrystalline materials

In this study, a novel three-dimensional micro-mechanical crystal-level model for the analysis of intergranular degradation and failure in polycrystalline materials is presented. The polycrystalline microstructures are generated as Voronoi tessellations, that are able to retain the main statistical features of polycrystalline aggregates. The formulation is based on a grain-boundary integral representation of the elastic problem for the aggregate crystals, that are modeled as three-dimensional anisotropic elastic domains with random orientation in the three-dimensional space. The boundary integral representation involves only intergranular variables, namely interface displacement discontinuities and interface tractions, that play an important role in the micromechanics of polycrystals. The integrity of the aggregate is restored by enforcing suitable interface conditions, at the interface between adjacent grains. The onset and evolution of damage at the grain boundaries is modeled using an extrinsic non-potential irreversible cohesive linear law, able to address mixed-mode failure conditions. The derivation of the traction-separation law and its relation with potential-based laws is discussed. Upon interface failure, a non-linear frictional contact analysis is used, to address separation, sliding or sticking between micro-crack surfaces. To avoid a sudden transition between cohesive and contact laws, when interface failure happens under compressive loading conditions, the concept of cohesive-frictional law is introduced, to model the smooth onset of friction during the mode II decohesion process. The incremental-iterative algorithm for tracking the degradation and micro-cracking evolution is presented and discussed. Several numerical tests on pseudo- and fully three-dimensional polycrystalline microstructures have been performed. The influence of several intergranular parameters, such as cohesive strength, fracture toughness and friction, on the microcracking patterns and on the aggregate response of the polycrystals has been analyzed. The tests have demonstrated the capability of the formulation to track the nucleation, evolution and coalescence of multiple damage and cracks, under either tensile or compressive loads.

Consistent skew-symmetric couple stress theory for size-dependent creeping flow

Within the framework of continuum fluid dynamics, couple-stresses appear as an inevitable consequence of non-central forces and the discrete character of matter at the finest scales. As a result, the force-stress tensor becomes non-symmetric and classical theory may not accurately predict the behavior. Recent theoretical work has shown that the couple-stress tensor must be skew-symmetric and that mean curvature rate is the energy-conjugate kinematical measure. The resulting fully consistent couple stress theory incorporates a characteristic material length scale for the fluid that becomes increasingly important, as the characteristic geometric dimension of the problem approaches that level. This size-dependent non-Newtonian theory is essential to understand a range of behavior at micro-scales with potential applications to blood flow and lubrication, among others. Here we concentrate on steady creeping flow within this newly developed fully determinate linear couple stress theory and formulate a boundary integral representation for two-dimensional problems. These boundary integral equations are written in terms of velocities, angular velocities, force-tractions and moment-tractions as primary variables. Details are provided on the derivation of fundamental solutions and on the corresponding boundary element implementation. Afterwards, the new boundary element method is applied for the solution of two basic problems to explore some consequences of this size-dependent couple stress fluid mechanics.

A Boundary Element Formulation for the Pricing of Barrier Options

In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a boundary value problem of heat equation with a moving boundary. The boundary integral representation and integral equation are derived. A boundary element method is designed to solve the integral equation. Special quadrature rules for the singular integral are used. A numerical example is also demonstrated. This boundary element formulation is correct.

Scholte waves on fluid-solid interfaces by means of an integral formulation

The present work shows the propagation of Scholte interface waves at the boundary of a fluid in contact with an elastic solid, for a broad range of solid materials. It has been demonstrated that by an analysis of diffracted waves in a fluid it is possible to infer the mechanical properties of the elastic solid medium, specifically, its propagation velocities. For this purpose, the diffracted wave field of pressures and displacements, due to an initial wave of pressure in the fluid, are expressed using boundary integral representations, which satisfy the equation of motion. The source in the fluid is represented by a Hankel’s function of second kind and zero order. The solution to this wave propagation problem is obtained by means of the Indirect Boundary Element Method, which is equivalent to the well-known Somigliana representation theorem. The validation of the results is carried out by using the Discrete Wave Number Method and the Spectral Element Method. Firstly, we show spectra of pressures that illustrate the behavior of the fluid for each solid material considered, then, we apply the Fast Fourier Transform to show results in time domain. Snapshots to exemplify the emergence of Scholte’s waves are also included.

TOWARDS WAVE DISTURBANCE IN PORTS COMPUTED BY A DETERMINISTIC CONVOLUTION-TYPE MODEL

Among the wide range of potential applications of the convolution-type approach to deterministic wave modeling, this paper looks into the challenge of complex shaped domains. The canonical case of diffraction around a semiinfinite vertical barrier, the 'Sommerfeld diffraction' case, is first studied. Focusing on locally constant water depth, the convolution method is related to a boundary integral representation by which the impulse response function representing the convolution kernel is related to a Green's function for the Laplace equation. This provides a framework for determining the impulse response function by solving a local, three-dimensional Laplace problem prior to the time-stepping of the wave transformation problem. For the Sommerfeld case, numerical results for the impulse response function near the barrier are computed numerically and compared with an analytical solution. For complex-shaped domains, numerical determination of the impulse response functions is the only solution. A very preliminary example of application to wave disturbance in a real port is given.

A three-dimensional grain boundary formulation for microstructural modeling of polycrystalline materials

A three-dimensional grain boundary formulation is presented for the analysis of polycrystalline microstructures. The formulation is based on a boundary integral representation of the elastic problem for the single grains of the polycrystalline aggregate and it is expressed in terms of the intergranular fields, namely displacements and tractions, that play an important role in polycrystalline micromechanics. The artificial polycrystalline morphology is represented using the Hardcore Voronoi tessellation, which is simple to generate and able to embody the main statistical features of polycrystalline microstructures. The details of the microstructure generation and meshing, which involve only the discretization of the grains surface, and not their volume, thus resulting in a remarkable simplification of data preparation, are discussed. The single crystals are represented as anisotropic elastic regions. The integrity of the aggregate is restored by enforcing both continuity and equilibrium at the interface between contiguous grains. The developed technique has been applied to the numerical homogenization of cubic polycrystals and the obtained results agree well with available data, thus confirming the reliability of the method. Some numerical issues and directions of further investigations are highlighted and discussed.

COMPUTATIONAL HOMOGENIZATION OF POLYCRYSTALLINE MATERIALS WITH PORES: A THREE-DIMENSIONAL GRAIN BOUNDARY FORMULATION

In this study, the influence of porosity on the elastic effective properties of polycrystalline materials is investigated using a 3D grain boundary micro mechanical model. The volume fraction of pores, their size and distribution can be varied to better simulate the response of real porous materials. The formulation is built on a boundary integral representation of the elastic problem for the grains, which are modeled as 3D linearly elastic orthotropic domains with arbitrary spatial orientation. The artificial polycrystalline morphology is represented using 3D Voronoi Tessellations. The formulation is expressed in terms of intergranular fields, namely displacements and tractions that play an important role in polycrystalline micromechanics. The continuity of the aggregate is enforced through suitable intergranular conditions. The effective material properties are obtained through material homogenization, computing the volume averages of micro-strains and stresses and taking the ensemble average over a certain number of microstructural samples. The obtained results show the capability of the model to assess the macroscopic effects of porosity.

Frequency-domain method for discrete frequency noise prediction of rotors in arbitrary steady motion

A novel frequency-domain formulation for the prediction of the tonal noise emitted by rotors in arbitrary steady motion is presented. It is derived from Farassat's ‘Formulation 1A’, that is a time-domain boundary integral representation for the solution of the Ffowcs-Williams and Hawkings equation, and represents noise as harmonic response to body kinematics and aerodynamic loads via frequency-response-function matrices. The proposed frequency-domain solver is applicable to rotor configurations for which sound pressure levels of discrete tones are much higher than those of broadband noise. The numerical investigation concerns the analysis of noise produced by an advancing helicopter rotor in blade–vortex interaction conditions, as well as the examination of pressure disturbances radiated by the interaction of a marine propeller with a non-uniform inflow.

Multi-domain boundary element method with dissipation

The wave diffraction and radiation around a floating body is considered within the framework of the linear potential theory in a fairly perfect fluid. The fluid domain extended infinitely in the horizontal directions but is limited by the sea bed, the body hull, and the part of the free surface excluding the body waterplane, and is subdivided into two subdomains according to the body geometry. The two subdomains are connected by a control surface in fluid. In each subdomain, the velocity potential is described by using the usual boundary integral representation involving Green functions. The boundary integral equations are then established by satisfying the boundary conditions and the continuous condition of the potential and the normal derivation across the control surface. This multi-domain boundary element method (MDBEM) is particularly interesting for bodies with a hull form including moonpools to which the usual BEM presents singularities and slow convergence of numerical results. The application of the MDBEM to study the resonant motion of a water column in moonpools shows that the MDBEM provides an efficient and reliable prediction method.

Simulation of spontaneous rupture based on a combined boundary integral equation method and finite element method approach: SH and P-SV cases

SUMMARY We present a hybrid approach for solving the dynamic rupture problem in an anelastic medium. It combines the advantages of the boundary integral equation method (BIEM), which is capable of representing accurately the solution near a crack tip, with those of the domain finite element method (FEM), which can handle conveniently heterogeneous materials, as well as the traction-free condition on a free surface and the continuity of traction across interfaces. When applied jointly, the proposed BIEM and the FEM can be used to solve efficiently spontaneous rupture problems in heterogeneous media, provided the rupture surface is contained within a homogeneous portion of the domain. The proposed method, BIEM–FEM approach (BDM, for Boundary/Domain Method), is verified for several antiplane (SH) and in-plane (P-SV) 2-D cases, in which the slip is prescribed along the fault (kinematic rupture). For spontaneous rupture, we examine the performance of the BDM by computing its convergence rate for an example in a homogeneous half-space with a slip-weakening friction law on the fault. By including the boundary integral representation on the fault, without using any refinement near the crack tips, the solution converges with the same rate as problems that do not exhibit any stress concentrations; that is, for the piecewise linear elements used here, the rms of the slip-rate distribution and of the traction distribution on the fault converge as O(Δx), whereas the slip distribution converges as O(Δx2), in which Δx is the mesh size of the finite element mesh. Additional 2-D spontaneous rupture simulations are performed for an inclined fault in a half-space for different values of the dip angle and for a folded layer system in a half-space, focusing only the P-SV case. The results show that the waves generated at the free surface and within the layers contribute to the propagation of the fault rupture and have a visible and sometimes strong effect on the ensuing ground motion.