Implementation Of Boundary Element Method Research Articles

Numerical modeling of acoustic visco-vhermal losses: simplified implementations and their limitations

Several numerical adaptations have been proposed to account for viscous and thermal losses of acoustic waves. This is a physical effect that is mostly relevant in a region very close to boundaries, the so-called boundary layers, in the micrometer range. It is therefore relevant when modeling small devices such as acoustic transducers and acoustic metamaterials. An often used modeling technique collapses the viscous and thermal losses into a boundary layer impedance (BLI), which is used in a calculation where the regular wave equation with no losses is discretized. However, this method has two shortcomings: i) the boundary layers may not overlap, and ii) it assumes flat surfaces. The BLI can be used either with the Finite Element Method (FEM) or the Boundary Element Method (BEM). BEM and FEM implementations with visco-thermal losses having no such restrictions also exist, at the cost of a heavier computational burden. This contribution discusses the shortcomings and advantages of the simulations using BLI as compared with full implementations and points to possible ways to overcome them. The BEM is employed to illustrate the analysis.

Fast and accurate boundary element methods for large-scale computational acoustics

The boundary element method (BEM) is a powerful algorithm to solve the Helmholtz equation for harmonic acoustic waves. The explicit use of Green’s functions avoids domain truncation of unbounded regions and accurately models wave propagation through homogeneous materials. Furthermore, fast multipole and hierarchical compression techniques provide efficient computations for dense matrix multiplications. However, the convergence of the iterative linear solvers deteriorates significantly when frequencies are high or materials have large contrasts in density or speed of sound. This talk presents several algorithmic improvements of the BEM. First, a preconditioner based on on-surface radiation conditions drastically reduces the iteration count of linear solvers at high frequencies. Second, anovel boundary integral formulation remains well-conditioned for high-contrast transmission problems. We used our fast and accurate BEM implementation to simulate focused ultrasound propagation in the human body, which can be translated to important biomedical applications such as the non-invasive treatment of liver cancer and neuromodulation of the brain. We validated the methodology within the benchmarking exercise of the International Transcranial Ultrasonic Stimulation Safety and Standards (ITRUSST) consortium. As a second application, we simulated the collective resonances of water-entrained arrays of air bubbles. Finally, we implemented all functionality in our open-source Python library, OptimUS.

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.

A high-order BEM for acoustic problems in a subsonic uniform flow

The present work is to develop a boundary element method (BEM) with curved surface elements for 3D acoustic problems in a subsonic uniform flow. For exterior sound propagation problems in moving flows, the convected conventional boundary integral equation (CBIE) suffers from the irregular frequency issue which becomes more severe in the presence of moving flows. In this paper, a hybrid convected boundary integral formula based on the Burton-Miller method is implemented to avoid the fictitious eigenfrequency problem. However, the hypersingular integral involved in the convected normal derivative boundary integral equation (NDBIE) is one of the crucial troublesomeness in the implementation of BEM with curved continuous elements for 3D acoustic problems. Moreover, the presence of the Mach number further complicates the evaluation of the singular integrals. To resolve this dilemma, an effective method based on polar coordinate transformation (PCT) is developed to compute the singular integrals. By mapping physical coordinates to the intrinsic local coordinates, this method can effectually remove various kinds of singularities appearing in the CBIE and NDBIE. Additionally, a complete scheme on the high-order convected BEM including the scheme of collocation points is also constructed for acoustic problems in a subsonic uniform flow. Several numerical examples are elaborately designed to verify the accuracy of the developed method.

A fundamental solution for the harmonic vibration of asymmetrically laminated composite plates described by a higher-order theory of shear strains

We derive here a fundamental solution for the harmonic vibration of thick asymmetrically laminated composite plates. The kinematics of the plate is based on Reddy’s third-order shear deformation theory (J Appl Mech 51:745–752, 1984). The fundamental solution is derived via the Fourier transform, and its final form is given in terms of definite integrals, which are evaluated numerically. All spatial singularities can be obtained explicitly, which makes the fundamental solution suitable for a boundary element method implementation.

A dual interpolation boundary face method for 3D elasticity

The dual interpolation boundary face method (DiBFM) proposed recently has been successfully applied to solve various problems in two dimensions. Compared with the conventional boundary element method (BEM), it has been proved that the DiBFM has the advantages of higher accuracy, convergence rate and computational efficiency. In addition, the DiBFM is suitable to unify the conforming and nonconforming elements in the BEM implementation, as well as to approximate both continuous and discontinuous fields. Moreover, there are no geometric errors by the DiBFM in the computational process. In this paper, the DiBFM is extended successfully to solve the elasticity problems in three-dimensions (3D) with formulations derived in details. A number of numerical examples are presented in order to validate the accuracy and convergence rate of the proposed method.

A fundamental solution for the harmonic vibration of laminated composite plates with coupled dynamic bending and quasistatic extension

We obtain here a new fundamental solution for the harmonic vibration of asymmetric, laminated, anisotropic plates. The fundamental solution is derived via the Fourier transform and its final form is given in terms of definite integrals, which are evaluated numerically. Moreover, we present some of the higher order derivatives of the solution and their explicit spatial singularities, which are necessary for a boundary element method implementation.

Numerical implementation of boundary element method for determining aerodynamic characteristics around airfoil system by viscous media

In this paper, the boundary element method is applied to determine the characteristics of distributed flow around the airfoil group. Because the boundary integral equation can describe various initial boundary value problems in mechanics and has many advantages such as good convergence, it can be used as a tool to establish appropriate mathematical models and numerical realization methods, so as to comprehensively study the distribution kinematics and dynamics of the airfoil group motion system in the viscous fluid flow co-flow process. In this paper, some numerical results are presented to determine the aerodynamic distribution characteristics of the viscous flow field around the airfoil group in the control area.

FEM and BEM Implementations of a High Order Surface Impedance Boundary Condition for Three-Dimensional Eddy Current Problems

We implement a three-dimensional formulation for eddy current problems based on the reduced magnetic scalar potential enforcing a high order surface impedance boundary condition (SIBC) which takes into account the curvatures of the conductor surface. Based on perturbation theory, the formulation reduces to three Laplace boundary value problems with Neumann boundary conditions and hence can be easily implemented in any existing Finite Element Method (FEM) or Boundary Element Method (BEM) code for the Laplace equation. The appropriate choice of the small parameter in the perturbation approach correctly represents the order of accuracy of the SIBC. The validation is carried out by comparison with full FEM solutions of a canonical test problem and of a more realistic example of a non-destructive testing probe. The validity of the extension of a high order SIBC to lower frequencies is verified and the fields can be obtained at any frequency in the range of interest once the formulation is solved only once.

Boundary element method analysis for mode III linear fracture mechanics in anisotropic and nonhomogeneous media

AbstractWe present here a Boundary Element Method implementation for anti‐plane fracture mechanics in nonhomogeneous and anisotropic media. The results are obtained using two different formulations based on displacement and traction Boundary Integral Equations (BIEs). The crack tips are discretized with special quarter‐point boundary elements. The main crack configurations are studied to determine the stress intensity factors (SIFs). Results for homogeneous and isotropic/anisotropic cases are compared with available solutions in the literature, showing good results. For the case of exponential nonhomogeneity, numerical results are obtained for moderately stiffer media and for softer media.

Helmholtz Decomposition and Boundary Element Method Applied to Dynamic Linear Elastic Problems

The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the longitudinal and transverse fields satisfy scalar Helmholtz equations that can be solved using a desingularized boundary element method (BEM) framework. The curl free longitudinal and divergence free transversal conditions can also be cast as additional scalar Helmholtz equations. When compared to other BEM implementations, the current framework leads to smaller matrix dimensions and a simpler conceptual approach. The numerical implementation of this approach is benchmarked against the 3D elastic wave field generated by a rigid vibrating sphere embedded in an infinite linear elastic medium for which the analytical solution has been derived. Examples of focussed 3D elastic waves generated by a vibrating bowl-shaped rigid object with convex and concave surfaces are also considered. In the static zero frequency limit, the Helmholtz decomposition becomes non-unique, and both the longitudinal and transverse components contain divergent terms that are proportional to the inverse square of the frequency. However, these divergences are equal and opposite so that their sum, that is the displacement field that reflects the physics of the problem, remains finite in the zero frequency limit.

A new solution technique for cathodic protection systems with homogeneous region by the boundary element method

The purpose of this work is to efficiently evaluate the design of cathodic protection (CP) systems of tank bottoms using concentric ring or linear anodes. As customary in current CP systems, the outer surface of the tank bottom is usually in electrical contact with a slender homogeneous layer of conductive concrete (or something similar) which in turn is in direct contact with the homogeneous deep soil region. The boundary element method (BEM) together with a subregion technique has been widely adopted to analyse such CP systems where the domain consists of two (or even more) homogeneous zones. However, the numerical solution of the final matrix system of equations can be quite time-consuming, especially if the slender intermediate layer is to be discretised, requiring a considerable number of elements, due to its somewhat reduced thickness. To overcome this problem, the present work proposes a new methodology in which the slender subregion is indirectly introduced, as a theoretically created polarisation curve, acting as a new boundary condition at the boundary of the soil domain (original common interface). Numerical simulations have been carried out using BEM implementations and results are discussed, including CP studies of practical axisymmetric and three-dimensional engineering problems.

Sensitivity analysis of 3D frictional contact with BEM using complex-step differentiation

Abstract This paper presents a study of the complex step differentiation method applied to a parameter sensitivity analysis for 3D elastic contact problem. The analysis is performed with the Boundary Element Method (BEM) using discontinuous elements and the Generalized Newton Method with line search (GNMls). A standard BEM implementation is used and the contact restrictions are fulfilled through the augmented Lagrangian method. This methodology in conjunction with the BEM avoids the calculation of the nonlinear derivatives during the solution process, allowing a fast and reliable solution procedure. The parameter sensitivity is evaluated using complex-step differentiation. This well-known method approximates the derivative of a function analogously to the standard finite differences method, with the advantages of being numerically exact and nearly insensitive to the step-size. As an example, a Hertz-type problem is solved and the sensitivity of the contact pressures with respect to the Young Modulus variation is evaluated. The obtained results are compared with analytical and numerical solutions found in the literature.

A Three-Dimensional Acoustic Boundary Element Method Formulation with Viscous and Thermal Losses Based on Shape Function Derivatives

Sound waves in fluids are subject to viscous and thermal losses, which are particularly relevant in the so-called viscous and thermal boundary layers at the boundaries, with thicknesses in the micrometer range at audible frequencies. Small devices such as acoustic transducers or hearing aids must then be modeled with numerical methods that include losses. In recent years, versions of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) including viscous and thermal losses have been developed. This paper deals with an improved formulation in three dimensions of the BEM with losses which avoids the calculation of tangential derivatives on the surface by finite differences used in a previous BEM implementation. Instead, the tangential derivatives are obtained from the element shape functions. The improved implementation is demonstrated using an oscillating sphere, where an analytical solution exists, and a condenser microphone as test cases.

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.

A new solution technique for cathodic protection systems with homogeneous region by the boundary element method

The purpose of this work is to efficiently evaluate the design of cathodic protection (CP) systems of tank bottoms using concentric ring or linear anodes. As customary in current CP systems, the outer surface of the tank bottom is usually in electrical contact with a slender homogeneous layer of conductive concrete (or something similar) which in turn is in direct contact with the homogeneous deep soil region. The boundary element method (BEM) together with a subregion technique has been widely adopted to analyse such CP systems where the domain consists of two (or even more) homogeneous zones. However, the numerical solution of the final matrix system of equations can be quite time-consuming, especially if the slender intermediate layer is to be discretised, requiring a considerable number of elements, due to its somewhat reduced thickness. To overcome this problem, the present work proposes a new methodology in which the slender subregion is indirectly introduced, as a theoretically created polarisation curve, acting as a new boundary condition at the boundary of the soil domain (original common interface). Numerical simulations have been carried out using BEM implementations and results are discussed, including CP studies of practical axisymmetric and three-dimensional engineering problems.

A new implementation of BEM by an expanding element interpolation method

A new implementation of boundary element method (BEM) by an expanding element interpolation method is presented in this paper. The expanding element is achieved by collocating virtual nodes along the perimeter of the traditional discontinuous element. With the virtual nodes, both continuous and discontinuous fields on the domain boundary can be accurately approximated, and the interpolation accuracy increases by two orders compared with the original discontinuous element. The boundary integral equations are built up for the inner nodes of the traditional discontinuous elements, only (taking these nodes as source points), while the virtual nodes are used for connecting the shape functions at the source points, thus the size of the final system of linear equations will not increase. The expanding element inherits the advantages of both the continuous and discontinuous elements while overcomes their disadvantages. Successful numerical examples with different boundary conditions have demonstrated that our new implementation is very encouraging and promising.

OBTAINING AND INVESTIGATION OF THE INTEGRAL REPRESENTATION OF SOLUTION AND BOUNDARY INTEGRAL EQUATION FOR THE NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY

Technological processes in the energy sector and engineering require the calculation of temperature regime of functioning of different constructions. Mathematical model of thermal loading of constructions is reduced to a non-stationary initial-boundary value problem of thermal conductivity. The article examines the formulation of the non-stationary initial-boundary value problem of thermal conductivity in the form of a boundary integral equation, analyzes the singular equation and builds the fundamental solution. To build the integral representation of the solution the method of weighted residuals is used. The correctness of the obtained integral representation of the solution in Minkowski space is confirmed. Singularity of the fundamental solution is investigated. The boundary integral equation and fundamental solution for axially symmetric domain for internal problem is built. The results of the article can be useful for numerical implementation of boundary element method.

Boundary Element Analysis of Interior Thermoelastic Stresses in Three-Dimensional Generally Anisotropic Bodies

AbstractThis paper is to present the treatment of internal thermoelastic stress analysis in 3D anisotropic bodies by the boundary element method (BEM). Fundamentally, thermal effects will give rise to an additional volume integral in the boundary integral equation (BIE). By applying the fundamental solutions represented by Fourier series, the volume integral has been analytically transformed to the boundary. For the present work, spatial differentiations of the integral equation are performed to give displacement gradients at internal points of interest. This differentiated integral equation is further implemented to perform thermoelastic stress analysis inside 3D anisotropic bodies. This analysis is particularly important in engineering applications when thermoelastic stresses concentrations are present inside the bodies. The present work is the first BEM implementation for this study by the transformed BIE. In the end, two benchmark examples are tested to demonstrate the applicability of the present BEM treatment.

Boundary element method application in wall dampness tomography

The boundary element method implementation used to find tomographic image of damped wall humidity distribution is presented. Damped wall represents non-homogeneous media where humidity has a spatial distribution that varies with two co-ordinates. Real measurements were taken and function coefficients where found using boundary element method, verified again by finite element method.