Shape sensitivity analysis of natural frequencies of structures using boundary elements

The boundary element approach to shape sensitivity analysis of eigenvalues of free vibrating elastic structures is presented. The eigenvalue problem is described in terms of the boundary integral equation method. Using the variational approach for variable regions, first-order sensitivities of simple frequencies are derived. Dependence of eigenvalues with respect to the stochastic shape of the boundary is considered. The numerical procedure of discretization of the problem is characterized. Numerical examples for two-dimensional problems are presented.     

Shape sensitivity analysis using a fixed basis function finite element approach

An approach is presented for the determination of solution sensitivity to changes in problem domain or shape. A finite element displacement formulation is adopted and the point of view is taken that the finite element basis functions and grid are fixed during the sensitivity analysis; therefore, the method is referred to as a “fixed basis function” finite element shape sensitivity analysis. This approach avoids the requirement of explicit or approximate differentiation of finite element matrices and vectors and the difficulty or errors resulting from such calculations. Effectively, the sensitivity to boundary shape change is determined exactly; thus, the accuracy of the solution sensitivity is dictated only by the finite element mesh used. The evaluation of sensitivity matrices and force vectors requires only modest calculations beyond those of the reference problem finite element analysis; that is, certain boundary integrals and reaction forces on the reference location of the moving boundary are required. In addition, the formulation provides the unique family of element domain changes which completely eliminates the inclusion of grid sensitivity from the shape sensitivity calculation. The work is illustrated for some one-dimensional beam problems and is outlined for a two-dimensional C⁰ problem; the extension to three-dimensional problems is straight-forward. Received December 5, 1999?Revised mansucript received July 6, 2000     

GPU-accelerated indirect boundary element method for voxel model analyses with fast multipole method

An indirect boundary element method (BEM) that uses the fast multipole method (FMM) was accelerated using graphics processing units (GPUs) to reduce the time required to calculate a three-dimensional electrostatic field. The BEM is designed to handle cubic voxel models and is specialized to consider square voxel walls as boundary surface elements. The FMM handles the interactions among the surface charge elements and directly outputs surface integrals of the fields over each individual element. The CPU code was originally developed for field analysis in human voxel models derived from anatomical images. FMM processes are programmed using the NVIDIA Compute Unified Device Architecture (CUDA) with double-precision floating-point arithmetic on the basis of a shared pseudocode template. The electric field induced by DC-current application between two electrodes is calculated for two models with 499,629 (model 1) and 1,458,813 (model 2) surface elements. The calculation times were measured with a four-GPU configuration (two NVIDIA GTX295 cards) with four CPU cores (an Intel Core i7-975 processor). The times required by a linear system solver are 31 s and 186 s for models 1 and 2, respectively. The speed-up ratios of the FMM range from 5.9 to 8.2 for model 1 and from 5.0 to 5.6 for model 2. The calculation speed for element-interaction in this BEM analysis was comparable to that of particle-interaction using FMM on a GPU.     

Reconstruction of heat transfer coefficients using the boundary element method

This paper describes the reconstruction of the heat transfer coefficient (space, Problem I, or time dependent, Problem II) in one-dimensional transient inverse heat conduction problems from surface temperature or average temperature measurements. Since the inverse problem posed does not involve internal temperature measurements, this means that non-destructive testing of materials can be performed. In the formulation, convective boundary conditions relate the boundary temperature to the heat flux. Numerical results obtained using the boundary element method are presented and discussed.     

Error analysis of a Galerkin method to solve the forward problem in MEG using the boundary element method

Sources of brain activity, e.g. epileptic foci, can be localized with Magnetoencephalography (MEG) measurements by recording the magnetic field outside the head. For a successful surgery a very high localization accuracy is needed. The most often used conductor model in the source localization is an analytic sphere, which is not always adequate, and thus a realistically shaped conductor model is needed. In this paper we examine a Galerkin method with linear basis functions to solve the forward problem in MEG using the boundary element method. Its accuracy is compared to the collocation method with constant and linear basis functions. The accuracies are determined for a unit sphere for which analytic solutions are available. The Galerkin method gives a clear improvement in the accuracy of the forward problem especially for the tangential component of the magnetic field. At realistic MEG measurement distances from the brain the Galerkin method reaches a given accuracy with lower computational costs than the collocation methods starting from a few hundreds of unknowns. With larger meshes the difference for the Galerkin method increases significantly.     

Shape sensitivity analysis using low- and high-order finite elements

A general procedure to perform the sensitivity analysis for the shape optimal design of elastic structures is proposed. The method is based on the implicit differentiation of the discretized equilibrium equations used in the finite element method (FEM). The so-called semianalytical approach is followed, that is, finite differences are used to differentiate the finite element matrices. The technique takes advantage of the geometric modeling concepts typical of the computer-aided design (CAD) technology used in the creation of a compact design model. This procedure is largely independent of the types of finite elements used in the analysis and has been implemented in ah-version andp-version finite element program. Very accurate and stable shape sensitivity derivatives were obtained from both programs over a wide range of finite difference step sizes. It is shown that the method is computationally efficient, general, and relatively easy to implement. Some classical shape optimal design problems have been solved using the CONLIN optimizer supplied with these gradients.     

Second-order shape design sensitivity using P-version finite element analysis

Thep-version finite element analysis (FEA) approach is attractive for design sensitivity analysis (DSA) and optimization due to its high accuracy of analysis results, even with coarse mesh; insensitivity to finite element mesh distortion and aspect ratio; and tolerance for large shape design changes during design iterations. A continuum second-order shape DSA formulation is derived and implemented usingp-version FEA. The second-order shape design sensitivity can be used for reliability based analysis and design optimization by incorporating it with the second-order reliability analysis method (SORM). Both the second-order shape DSA formulations with respect to the single and mixed shape design parameters are derived for elastic solids using the material derivative concept. Both the direct differentiation and hybrid methods are presented in this paper. A shape DSA is implemented by using an establishedp-version FEA code, STRESS CHECK. Two numerical examples, a connecting rod and bracket, are presented to demonstrate the feasibility and accuracy of the proposed seond-order shape DSA approach.     

A new numerical algorithm for 2D moving boundary problems using a boundary element method

Phase change problems are of practical importance and can be found in a wide range of engineering applications. In the present paper, two proposed numerical algorithms are developed; the first one is general for phase change problems, while the second one is for ablation problems. The boundary elements method is used as a mathematical tool in conjunction with the proposed algorithms. Two test examples were solved and the results agree with the physics of the problems.     

Efficient error estimation and adaptive meshing method for boundary element analysis

This paper describes a new and efficient error estimator by using the Direct Regular Method and h or h-r adaptive meshing for BEM analysis. This posteriori error estimator correctly indicates the discretization errors on each element. Based on the error distribution, and the adaptive meshing is generated automatically. The accuracy and convergence of this method are demonstrated by the numerical results on the stress concentration problem and the crack problem.     

A thermoelastic analysis of shrink-fit type constructions by boundary element method

In this paper a boundary element method for the analysis of shrink fits is presented. The contact stresses created at the interference layer of the mating bodies and all over boundaries can be accurately evaluated. The shrinkage is usually generated or relieved by thermal expansions or by inertia forces and thus a thermoelastic bodyforce analysis is performed. The method is straightforward. Only the boundaries of the mating bodies are required to be discretized. Examples are shown to verify the accuracy of the analysis.     

Direct and indirect boundary element methods for solving the heat conduction problem

The boundary element method is used to solve the stationary heat conduction problem as a Dirichlet, a Neumann or as a mixed boundary value problem. Using singularities which are interpreted physically, a number of Fredholm integral equations of the first or second kind is derived by the indirect method. With the aid of Green's third identity and Kupradze's functional equation further direct integral equations are obtained for the given problem. Finally a numerical method is described for solving the integral equations using Hermitian polynomials for the boundary elements and constant, linear, quadratic or cubic polynomials for the unknown functions.     

Partitioning technique for boundary element method

The partitioning technique and its Fortran subroutines for the solution of a non-symmetrical fully populated matrix system encountered in the boundary element method are presented. The technique is such that the computer backing store is fully utilized and therefore a large-scale problem can be solved with a small computer. The derivation of formulae for this technique is given in detail. A corresponding computer program is developed and verified through two two-dimensional elastostatic problems. The influence of submatrix size on cost of computation time is discussed.     

Alternative technique for the solution of plate bending problems using the boundary element method

The objective of this work consists of presenting a boundary element technique for analysis of bending plates where all algebraic representations of rotations for boundary nodes are avoided. The technique requires outside load points whose definition is extensively studied in this article.     

Boundary element sensitivity evaluation for elasticity problems using complex variable method

The complex variable method is used to evaluate sensitivities in two-dimensional elasticity problems, using the BEM as numerical method. The method shows negligible dependency on the perturbation magnitude, and can be easily incorporated to existing codes which handle complex algebra. Numerical results for elasticity problems show that the numerical accuracy of the sensitivities obtained with complex variable differentiation is competitive with other methods, and can be easily implemented in any other approximation methods as well.     

Three-dimensional eigenvalue analysis of the Helmholtz equation by multiple reciprocity boundary element method

The eigenvalue of the three-dimensional Helmholtz equation is determined efficiently by extending the previously developed method for the two-dimensional problem. Boundary integral equation is formulated in the realm of the multiple reciprocity method, using higher order fundamental solutions for the Laplace equation; yielding polynomial coefficient matrices in terms of unknown wavenumber (eigenvalue). The Newton iteration method with the help of LU decomposition is employed to search eigenvalue, which can reduce the computational task significantly.     

Geometrically nonlinear analysis of a Reissner type plate by the boundary element method

The boundary element method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates taking into account the effects of transverse shear deformation. This paper presents the general equations for finite deformation of a Reissner type plate, and also gives an integral formulation of the Von Kármán type nonlinear governing equations which involve coupling between in-plane and out-of-plane deformation. The boundary and the domain of the plate are discretized to solve nonlinear plate bending problems. All unknown variables are at the boundary. An iterative procedure is applied to achieve linearization of the nonlinear equations. Some numerical results of the computation are compared with the analytical solutions and other numerical techniques, and good agreement is obtained.     

Postbuckling analysis of plates on an elastic foundation by the boundary element method

The postbuckling behavior of plates on an elastic foundation is studied by using the boundary element method (BEM). A new fundamental solution of lateral deflection is derived through the resolution theory of a differential operator, and a set of boundary element formulae in incremental form is presented. By using these formulae, the BEM solution procedure becomes relatively simple. The results of a number of numerical examples are compared with existing solutions and good agreement is observed. It shows that the proposed method is effective for solving the postbuckling problems of plates with arbitrary shape and various boundary conditions.     

The boundary element method on the ICL DAP

In this paper we consider the implementation, on the ICL DAP, of a boundary element solution for two-dimensional potential problems. The DAP is shown to be particularly suited to this problem. In particular the numerical quadratures in the contributions to the overall system matrices are calculated in parallel and the rearrangement of the resulting system of equations is conveniently performed using the logical mask facility. A speedup by a factor of 100 is obtained compared with a corresponding sequential code on a DEC 1091.     

A Matlab library for solving quasi-static volume conduction problems using the boundary element method

The boundary element method (BEM) is commonly used in the modeling of bioelectromagnetic phenomena. The Matlab language is increasingly popular among students and researchers, but there is no free, easy-to-use Matlab library for boundary element computations. We present a hands-on, freely available Matlab BEM source code for solving bioelectromagnetic volume conduction problems and any (quasi-)static potential problems that obey the Laplace equation. The basic principle of the BEM is presented and discretization of the surface integral equation for electric potential is worked through in detail. Contents and design of the library are described, and results of example computations in spherical volume conductors are validated against analytical solutions. Three application examples are also presented. Further information, source code for application examples, and information on obtaining the library are available in the WWW-page of the library: (http://biomed.tkk.fi/BEM).