Conforming Boundary Elements Research Articles

Conforming boundary element methods in acoustics

Conforming boundary element methods are developed for three-dimensional acoustic wave scattering by sound hard (Neumann problem) and sound soft (Dirichlet problem) objects. Theoretically, these methods guarantee convergence of the solution in the norm of an appropriate function space as the number of degrees of freedom is increased. For the integral equations of the first kind conforming boundary elements can be obtained with conventional Galerkin׳s method and finite element spaces (FE). However, for the integral equations of the second kind this standard technique does not lead to a conforming method, rather Petrov–Galerkin׳s methods, where the basis and testing functions are not identical, are required. In the numerical implementation of the Petrov–Galerkin methods special “dual” FE spaces defined on the barycentrically refined mesh are used to test the equations. These dual FE spaces also play an important role in the Calderon matrix multiplicative preconditioners applied to regularize ill-conditioned integral equations of the first kind.

Radial basis functions for the solution of hypersingular operators on open surfaces

We analyze the approximation by radial basis functions of a hypersingular integral equation on an open surface. In order to accommodate the homogeneous essential boundary condition along the surface boundary, scaled radial basis functions on an extended surface and Lagrangian multipliers on the extension are used. We prove that our method converges quasi-optimally. Approximation results for scaled radial basis functions indicate that, for highly regular radial basis functions, the achieved convergence rates are close to the one of low-order conforming boundary element schemes. Numerical experiments confirm our conclusions.

Stable multilevel splittings of boundary edge element spaces

We establish the stability of nodal multilevel decompositions of lowest-order conforming boundary element subspaces of the trace space \({\boldsymbol{H}}^{-\frac {1}{2}}(\operatorname {div}_{\varGamma },{\varGamma })\) of \({\boldsymbol{H}}(\operatorname {\bf curl},{\varOmega })\) on boundaries of triangulated Lipschitz polyhedra. The decompositions are based on nested triangular meshes created by uniform refinement and the stability bounds are uniform in the number of refinement levels.The main tool is the general theory of P. Oswald (Interface preconditioners and multilevel extension operators, in Proc. 11th Intern. Conf. on Domain Decomposition Methods, London, 1998, pp. 96–103) that teaches, when stability of decompositions of boundary element spaces with respect to trace norms can be inferred from corresponding stability results for finite element spaces. \({\boldsymbol{H}}(\operatorname {\bf curl},{\varOmega })\)-stable discrete extension operators are instrumental in this.Stable multilevel decompositions immediately spawn subspace correction preconditioners whose performance will not degrade on very fine surface meshes. Thus, the results of this article demonstrate how to construct optimal iterative solvers for the linear systems of equations arising from the Galerkin edge element discretization of boundary integral equations for eddy current problems.

Multiple traces boundary integral formulation for Helmholtz transmission problems

We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in subdomains) that directly lends itself to operator preconditioning via Calderon projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise $\widetilde{H}^{-1/2}$ -distributions over each associated interface. Through the use of interior Calderon projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions' projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning.

Dynamic element discretization method for solving 2D traction boundary integral equations

A sufficient condition for the existence of element singular integral of the traction boundary integral equation for elastic problems requires that the tangential derivatives of the boundary displacements are Hölder continuous at collocation points. This condition is violated if a collocation point is at the junction between two standard conforming boundary elements even if the field variables themselves are Hölder continuous there. Various methods are proposed to overcome this difficulty. Some of them are rather complicated and others are too different from the conventional boundary element method. A dynamic element discretization method to overcome this difficulty is proposed in this work. This method is novel and very simple: the form of the standard traction boundary integral equation remains the same; the standard conforming isoparametric elements are still used and all collocation points are located in the interior of elements where the continuity requirements are satisfied. For boundary elements with boundary points where the field variables themselves are singular, such as crack tips, corners and other boundary points where the stress tensors are not unique, a general procedure to construct special elements has been developed in this paper. Highly accurate numerical results for various typical examples have been obtained.

A Coercive Combined Field Integral Equation for Electromagnetic Scattering

Many boundary integral equation methods used in the simulation of direct electromagnetic scattering of a time-harmonic wave at a perfectly conducting obstacle break down when applied at frequencies close to a resonant frequency of the obstacle. A remedy is offered by special indirect boundary element methods based on the so-called combined field integral equation. However, hitherto no theoretical results about the convergence of discretized combined field integral equations have been available. In this paper we propose a new combined field integral equation, convert it into variational form, establish its coercivity in the natural trace spaces for electromagnetic fields, and conclude existence and uniqueness of solutions for any frequency. Moreover, a conforming Galerkin discretization of the variational equations by means of ${\rm div}_\Gamma$-conforming boundary elements can be shown to be asymptotically quasi-optimal. This permits us to derive quantitative convergence rates on sufficiently fine, uniformly shape-regular sequences of surface triangulations.

Coupling of Finite Elements and Boundary Elements in Electromagnetic Scattering

We consider the scattering of monochromatic electromagnetic waves at a dielectric object with a rough surface. We investigate the coupling of a weak formulation of Maxwell's equations inside the scatterer with boundary integral equations that arise from the homogeneous problem in the unbounded region outside the scatterer. The symmetric coupling approach based on the full Calderon projector for Maxwell's equations is employed. By splitting both the electric field inside the scatterer and the surface currents into components of predominantly electric and magnetic nature, we can establish coercivity of the coupled variational problem, provided that the frequency is away from resonant frequencies. Discretization relies on both curl-conforming edge elements inside the scatterer and $\bDiv$-\break conforming boundary elements for the surface currents. The splitting idea, adjusted to the discrete setting, permits us to show uniform stability of the discretized problem. We exploit it to come up with a priori convergence estimates.

Boundary Element Analysis of Raft Foundations on Piles

The boundary element method is used in the formulation of models for the analysis of raft foundations on piles. Two models are considered: a Kirchhoff plate on a layered elastic half-space and a Kirchhoff plate on a Winkler soil. The plates are modelled using conforming boundary elements and the piles by using linear finite elements. Mindlin's solution is used as influence function within the half-space while Boussinesq's solution, a precursor although a particular case of Mindlin's solution, is used to derive the deflections of the soil surface. The models are used in the analysis of some raft foundations on piles and the results and relative merits are discussed.

An integral equation method for the electromagnetic scattering from cavities

Consider a time-harmonic electromagnetic plane wave incident on a cavity in a ground plane. The physical process is modelled by Maxwell's equations. In this paper, integral representations of the solutions to the model problem in both fundamental polarizations are derived and studied. Existence and uniqueness of the solutions for the integral equations are established. The integral equations approach forms a basis for numerical solution of the model problem. In particular, for each fundamental polarization, an integral formulation with Gårding-type estimates is derived. These formulations provide a basis for variational boundary element methods for solving the cavity problem. The Gårding-type estimates imply convergence results for conforming boundary element methods. Copyright © 2000 John Wiley & Sons, Ltd

STEADY-STATE RESPONSE OF ACOUSTIC CAVITIES BOUNDED BY PIEZOELECTRIC COMPOSITE SHELL STRUCTURES

A formulation to calculate the coupled response of composite shells with embedded piezoelectric layers and an enclosed acoustic fluid is presented in this paper. The methodology consists of three parts: (1) a formulation for the electro-mechanical response of piezoelectric shells; (2) a formulation for the three-dimensional acoustic response of the enclosed fluid; and (3) the combination of the formulations (1) and (2) to calculate the coupled smart structure-acoustic fluid response. A recently developed mixed field laminate theory is adapted for the analysis of piezoelectric shells. The theory combines the first order shear theory kinematic assumptions with a layer-wise approximation for the electric potential. Shell geometry is described in an orthogonal curvilinear co-ordinate system and general piezoelectric material descriptions and laminate configurations are considered. A boundary element formulation is developed to calculate the acoustic response of the enclosed fluid. Quadratic conforming boundary elements are used to discretize the fluid boundary. Advanced numerical integration techniques are employed to calculate singular elements in boundary element matrices. The treatment of distributed acoustic sources is also presented. A formulation to calculate the coupled fluid-structure response is also developed. Relations between the structural and acoustic variable on the structure-fluid interface are utilized to generate the coupled system of equations in terms of the kinematic shell variables and acoustic pressures on the fluid boundary. The convergence of the present developments is established by studying a circular cylindrical shell with an attached piezoelectric layer. The coupled response is investigated for various types of mechanical loads and active voltage patterns.

Influence of the location of the contact point in node-to-point contact approaches using non conforming boundary element discretizations

The study of contact between deformable bodies with non conforming discretizations inherently involves the possibility of a node having contact with an element at any position along it. This paper presents a study of the influence that the position along the element of this contacting point has on the results The numerical technique used is the Boundary Element Method and the classic problem of the compression of a deformable cylinder on a deformable foundation, which belongs to the class of advancing contact problems, will be used as a reference to evaluate the errors induced by the position of the contacting point.

Stochastic BEM—Random Excitations and Time-Domain Analysis

Boundary element formulations for the treatment of boundary value problems in 2-D elasticity with random boundary conditions are presented. It is assumed that random boundary conditions may be described as second order random fields that possess finite moments up to the second order. This permits the use of the mean square calculus for which the operations of integration and mathematical expectations commute. Spatially correlated, time-independent and time-dependent boundary conditions are considered. For time-independent boundary conditions, the stochastic equivalent of Somigliana's identity is used to obtain deterministic integral equations for mathematical expectations and covariances of the response variables, and crosscovariances of the response variables with respect to prescribed boundary conditions. The random field used to describe the boundary conditions is discretized into a finite set of random variables defined at the element nodes. Quadratic, conforming boundary elements are used to arrive a...

HYPERSINGULAR INTEGRALS: HOW SMOOTH MUST THE DENSITY BE?

Hypersingular integrals are guaranteed to exist at a point x only if the density function f in the integrand satises certain conditions in a neighbourhood of x. It is well known that a su- cient condition is that f has a Holder-continuous rst derivative. This is a stringent condition, especially when it is incorporated into boundary-element methods for solving hypersingular in- tegral equations. This paper is concerned with nding weaker conditions for the existence of one-dimensional Hadamard nite-part integrals: it is shown that it is sucien t for the even part of f (with respect to x) to have a Holder-continuous rst derivative { the odd part is al- lowed to be discontinuous. A similar condition is obtained for Cauchy principal-value integrals. These simple results have non-trivial consequences. They are applied to the calculation of the tangential derivative of a single-layer potential and to the normal derivative of a double-layer potential. Particular attention is paid to discontinuous densities and to discontinuous boundary conditions. Also, despite the weaker sucien t conditions, it is rearmed that, for hypersingular integral equations, collocation at a point x at the junction between two standard conforming boundary elements is not permissible, theoretically. Various modications to the denition of nite-part integral are explored.

Stochastic boundary elements for two-dimensional potential flow in non-homogeneous media

A stochastic boundary element formulation is presented for the treatment of two-dimensional problems of steady-state potential flow in non-homogeneous media that involve a random operator in the governing differential equation. The randomness is introduced through the material parameter of the domain which is described as a non-homogeneous random field. The random field is discretized into a set of correlated random variables and a perturbation is applied to the differential equation of the problem. This leads to differential equations for the unknown potential and its first- and second-order derivatives, respectively, evaluated at the mathematical expectations of the random variables resulting from the discretization of the random field. An approximate method is applied for the solution of these equations by expressing the potential and its derivatives as a sum of functions of descending order. These solutions are introduced into the differential equations and upon equating similar order terms, a sequence of Poisson's equations is obtained for each order. A transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations by retaining a small number of transformed random variables. The resulting equations are solved using the boundary element method to obtain the unknown boundary values of the potentials and their respective first- and second-order derivatives which are then used to compute the desired response statistics. Quadratic, conforming boundary elements are used in the boundary integration and four-node quadrilateral cells are used in the domain integration. Strongly singular terms of the boundary element matrices are obtained indirectly by applying a state of uniform unit potential over the entire contour of the object. The singular domain integrals are calculated analytically. Direct solution techniques are used to calculate the response variables and their derivatives, respectively. A number of example problems are presented and the results are compared with those obtained from Monte Carlo simulations. A good agreement of the results is observed.

Stochastic boundary elements in elastostatics

A stochastic boundary element formulation for the treatment of boundary value problems in two-dimensional elastostatics that involve a random operator is presented. A general perturbation procedure is formulated for the set of correlated random variables governing the response of the solid. This procedure is then specialized for the cases of (a) random geometry and (b) random material properties. The problems involving a random configuration are analyzed using the random variable model, and those with a random material property are analyzed using the random field model. The random field is first discretized into a set of correlated random variables, which are then transformed into an uncorrelated set to simplify the analysis. The derivatives of the boundary element matrices appearing in the systems of equations from the perturbation of the random variables are derived analytically. The direct solution methods are used to obtain the response variables and their first- and second-order derivatives, respectively. Quadratic, conforming boundary elements are employed in the boundary element discretization and the strongly singular terms of the boundary element matrices and their first- and second-order derivatives are obtained using the conditions associated with rigid body motions of the solid. The present formulation has been evaluated for a number of example problems through comparisons with the solutions obtained by Monte Carlo simulation. A good agreement of the results is observed.

A multiple‐node method to resolve the difficulties in the boundary integral equation method caused by corners and discontinuous boundary conditions

AbstractIn the boundary integral equation method (BIEM), use of Lagrangian shape functions together with conforming boundary elements requires continuity of functions at the interelement boundary. When the flux or the traction is discontinuous due to the presence of corners or discontinuous boundary conditions, conforming elements can be a source of error. In this paper, we detail a multiple‐node method in which this error can either be eliminated or substantially reduced. The paper is limited to the Laplace problem and the problem of elastostatics in two dimensions. However, the method can be easily extended to problems of other types and to higher dimensions.

Conforming boundary elements in plane elasticity for shape design sensitivity

AbstractThe structural design sensitivity analysis of a two‐dimensional continuum using conforming (continuous) boundary elements is investigated. Implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices. A singular formulation of the boundary element method is used. Implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis. The behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated. A rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices, thus avoiding the problems associated with their numerical integration. A formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented. The design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results. Numerical data comparing the performance of conforming and non‐conforming formulations in the calculation of design sensitivities are also presented. The accuracy of the present results is demonstrated through comparisons with existing analytical results.