Efficient solution of time‐domain boundary integral equations arising in sound‐hard scattering

We consider the efficient numerical solution of the three‐dimensional wave equation with Neumann boundary conditions via time‐domain boundary integral equations. A space‐time Galerkin method with C^∞‐smooth, compactly supported basis functions in time and piecewise polynomial basis functions in space is employed. We discuss the structure of the system matrix and its efficient parallel assembly. Different preconditioning strategies for the solution of the arising systems with block Hessenberg matrices are proposed and investigated numerically. Furthermore, a C++ implementation parallelized by OpenMP and MPI in shared and distributed memory, respectively, is presented. The code is part of the boundary element library BEM4I. Results of numerical experiments including convergence and scalability tests up to a thousand cores on a cluster are provided. The presented implementation shows good parallel scalability of the system matrix assembly. Moreover, the proposed algebraic preconditioner in combination with the FGMRES solver leads to a significant reduction of the computational time. Copyright © 2015 John Wiley & Sons, Ltd.     

Efficient preconditioners for boundary element matrices based on grey‐box algebraic multigrid methods

This paper is concerned with the iterative solution of the boundary element equations arising from standard Galerkin boundary element discretizations of first‐kind boundary integral operators of positive and negative order. We construct efficient preconditioners on the basis of so‐called grey‐box algebraic multigrid methods that are well adapted to the treatment of boundary element matrices. In particular, the coarsening is based on an auxiliary matrix that represents the underlying topology in a certain sense. This auxiliary matrix is additionally used for the construction of the smoothers and the transfer operators. Finally, we present the results of some numerical studies that show the efficiency of the proposed algebraic multigrid preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

Free surfaces: shape sensitivity analysis and numerical methods

In this paper we consider numerical methods for stationary free boundary problems. We start by analysing systematically different shape optimization formulations of a model problem and show how the optimality conditions relate to construction of trial type methods. Shape sensitivity analysis of the free boundary leads also to the so‐called total linearization method which combines the good properties of Newton method and trial methods, i.e. fast convergence and relative simplicity of implementation. Detailed implementation for a model problem together with numerical tests is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

A different view of ridge analysis from numerical optimization

This research falls into the gap between applied statistics and numerical optimization in a specific topic—Ridge Analysis (RA). This article proposes using the trust-region (TR) methods in numerical optimization to solve the RA problem, arising from the literature of response surface methodology (RSM) in applied statistics, where its goal is to help engineers for ‘process improvement’ to find the better response value of the predicted response function within the boundary of experimentation. In the field of numerical optimization, as the family of TR approaches always exhibits excellent mathematical properties during optimization steps, thus the algorithm presented in this study guarantees global optima for the RA problem. Two examples found in the RSM literature are included to illustrate the algorithm, demonstrating its capability of locating better operating conditions than existing computing methods and pointing out particular circumstances (termed the ‘hard case’) where the classical RA procedure fails. An important application to the response modeling problem arising from the philosophy of Taguchi's quality engineering illustrates the hard case. Finally, the utility of the presented TR algorithm is demonstrated through a sequential framework with iterative updates of the TR model under local approximation provided that the predicted response model is a high-order or even non-polynomial function.     

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L² inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.     

A meshfree method for incompressible fluid dynamics problems

We show that meshfree variational methods may be used for the solution of incompressible fluid dynamics problems using the R‐function method (RFM). The proposed approach constructs an approximate solution that satisfies all prescribed boundary conditions exactly using approximate distance fields for portions of the boundary, transfinite interpolation, and computations on a non‐conforming spatial grid. We give detailed implementation of the method for two common formulations of the incompressible fluid dynamics problem: first using scalar stream function formulation and then using vector formulation of the Navier–Stokes problem with artificial compressibility approach. Extensive numerical comparisons with commercial solvers and experimental data for the benchmark back‐facing step channel problem reveal strengths and weaknesses of the proposed meshfree method. Copyright © 2003 John Wiley & Sons, Ltd.     

Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

We present new iterative solvers for large‐scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second‐order accuracy can be obtained at very small overcost with respect to first‐order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2 × 2 block symmetric indefinite linear system arising from mixed (displacement‐pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

Free vibration of generally supported rectangular Kirchhoff plates: State‐space‐based differential quadrature method

The title problem is investigated using the differential quadrature method based on the state‐space formalism. The plates, with mixed boundary conditions, may cross over one‐way internal rigid line supports that impose zero transverse displacement constraints. Differential quadrature procedure is applied in the direction of line supports, while exact solution is sought in the transfer domain perpendicular to the line supports using the state space method. To avoid numerical instability in the transfer matrix method, joint coupling matrices are introduced, mainly according to the continuity conditions at line joints. Natural frequencies of rectangular Kirchhoff plates with general boundary conditions are calculated and compared with the results from other methods. Effects of location of internal line supports and the mixed boundary conditions on the frequency parameters are discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

Element‐wise a posteriori estimates based on hierarchical bases for non‐linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non‐linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right‐hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right‐hand side the residual of the p‐FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non‐linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non‐linear parabolic cases are provided. Copyright © 2005 John Wiley & Sons, Ltd.     

Direct solution of ill‐posed boundary value problems by radial basis function collocation method

Numerical solution of ill‐posed boundary value problems normally requires iterative procedures. In a typical solution, the ill‐posed problem is first converted to a well‐posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill‐posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method. Copyright © 2005 John Wiley & Sons, Ltd.     

A solution via generalised intergral transform technique for the simultaneous transport processes during combustion of wood cylinders

A non‐linear moving boundary diffusion problem is proposed as a simple model for the heat transfer during combustion of wood cylinders. Such a problem is solved here by applying the generalized integral transform technique. A new filtering strategy, denoted as local‐instantaneous filter, is used in order to accelerate the convergence of the series‐solution obtained with the present hybrid numerical–analytical technique. We show that the use of such filtering approach reduces the stiffness of the system of ordinary differential equations, resultant from the integral transformation of the original problem. Hence, subroutines based on simpler and faster methods can be used for the solution of such systems. Results are presented in the paper for the combustion of cylinders of different sizes and involving different initial moisture contents and densities. The effects on the solution of different models available in the literature for the evaluation of thermal conductivity and specific heat are also addressed on the paper. Copyright © 2000 John Wiley & Sons, Ltd.     

Low‐frequency assessment of the in situ acoustic absorption of materials in rooms: an inverse problem approach using evolutionary optimization

The in situ assessment of the acoustic absorption of materials is often a necessity. The need to cover the whole frequency range of interest for the building engineer has led the authors to an approach involving two frequency‐complementary measurement methods. This paper deals with the part dedicated to low frequencies. The measurement is defined here as a boundary inverse interior problem. A numerical model of the room under investigation, allowing for the computation of the pressure field in the volume, given impedance boundary conditions and a point source, is combined to a global optimization algorithm. The algorithm explores the set of possible boundary conditions in order to minimize the difference between the computed pressure values and the one observed at a few measurement points, leading to the determination of all the boundary conditions at a time. In practice, the finite element method (FEM) or the finite difference method (FDM) is used here to model the room and an Evolution Strategy as the optimization tool. After describing the ES operators, a numerical study is carried out on simulated measurements, both on problem‐ and algorithm‐specific parameters, in the case of an academic two‐dimensional room geometry. The method is then applied to a three‐dimensional room with promising results. Copyright © 2002 John Wiley & Sons, Ltd.     

Regularized meshless method for antiplane shear problems with multiple inclusions

In this paper, we employ the regularized meshless method to solve antiplane shear problems with multiple inclusions. The solution is represented by a distribution of double‐layer potentials. The singularities of kernels are regularized by using a subtracting and adding‐back technique. Therefore, the troublesome singularity in the method of fundamental solutions (MFS) is avoided and the diagonal terms of influence matrices are determined. An inclusion problem is decomposed into two parts: one is the exterior problem for a matrix with holes subjected to remote shear, the other is the interior problem for each inclusion. The two boundary densities, essential and natural data, along the interface between the inclusion and matrix satisfy the continuity and equilibrium conditions. A linear algebraic system is obtained by matching boundary conditions and interface conditions. Finally, numerical results demonstrate the accuracy of the present solution. Good agreements are obtained and compare well with analytical solutions and Gong's results. Copyright © 2007 John Wiley & Sons, Ltd.     

The Steklov‐Poincaré technique for data completion: Preconditioning and filtering

This paper presents a study of primal and dual Steklov‐Poincaré approaches for the identification of unknown boundary conditions of elliptic problems. After giving elementary properties of the discretized operators, we investigate the numerical solution with Krylov solvers. Different preconditioning and acceleration strategies are evaluated. We show that costless filtering of the solution is possible by postprocessing Ritz elements. Assessments are provided on a 3D mechanical problem.     

Efficient computation of order and mode of corner singularities in 3D‐elasticity

A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non‐smooth domains in three‐dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin–Petrov finite element method. A quadratic eigenvalue problem ( P +λ Q +λ² R ) u = 0 is obtained, with explicitly analytically defined matrices P , Q , R . Moreover, the three matrices are found to have optimal structure, so that P , R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ?e(λ)∈(?0.5,1.0) (no eigenpairs can be ‘lost’) as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill‐in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd.     

Iterative algorithms for impressed cathodic protection systems

A non‐linear cathodic protection problem arising from corrosion engineering is considered. The objective is to present some iterative methods and study the convergence. We show both numerically and theoretically that the Newton–Raphson iteration is monotonically lower‐convergent and a proposed combined iteration is alternative‐convergent. In the combined case, the exact solution locates between two subsequent iterative solutions. Boundary element methods are employed for the numerical calculation. Copyright © 2000 John Wiley & Sons, Ltd.     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of Laplace's equation in three dimensions and Poisson's equation in two dimensions. Both are two‐dimensional weakly singular integrals and are evaluated using (in a product fashion) methods which have recently been used for evaluating one‐dimensional weakly singular integrals arising in the boundary element method. The methods used are based on various polynomial transformations of conventional Gaussian quadrature points where the transformation polynomial has zero Jacobian at the singular point. Methods which split the region of integration into sub‐regions are considered as well as non‐splitting methods. In particular, the newly introduced and highly accurate generalized composite subtraction of singularity and non‐linear transformation approach (GSSNT) is applied to various two‐dimensional weakly singular integrals. A study of the different methods reveals complex relationships between transformation orders, position of the singular point, integration kernel and basis function. It is concluded that the GSSNT method gives the best overall results for the two‐dimensional weakly singular integrals studied. Copyright © 2002 John Wiley & Sons, Ltd.     

Solution of Inverse Boundary Optimization Problem by Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm

The inverse boundary optimization problem, governed by the Helmholtz equation, is analyzed by the Trefftz method (TM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the TM, one kind of boundary-type meshless methods, is adopted in this study, since it can avoid the generation of mesh grid and numerical integration. In the boundary optimization problem governed by the Helmholtz equation, the numerical solution of TM is expressed as linear combination of the T-complete functions. When this problem is considered by TM, a system of nonlinear algebraic equations will be formed and solved by ECSHA which will converge exponentially. The evolutionary process of ECSHA can acquire the unknown coefficients in TM and the spatial position of the unknown boundary simultaneously. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. Besides, the stability of the proposed meshless method will be validated by adding some noise into the boundary conditions.     

Surrogate Duality Based Method for Contact Problems

We present a numerical method based on surrogate duality to solve contact problems in elasticity. The primal optimization problem arising from the contact problem is converted to a corresponding dual problem using surrogate duality. An explicit dual formulation for the surrogate dual problem is obtained on some certain conditions, and a Karmarkar's interior point based method is introduced to solve the explicit surrogate dual problem. Numerical examples including some quadratic programming problems with dense matrices and one elastic contact problem are solved using the present method to show its feasibility and efficiency.