Boundary elements in shape design sensitivity analysis and optimal design of vibrating structures

A general approach to shape design sensitivity analysis and optimal design for dynamic transient and free vibrations problems using boundary elements is presented. The material derivatives and the adjoint system method are applied to obtain first-order sensitivities for the effect of boundary shape variations. A numerical example of shape sensitivity analysis and optimal design for free vibrations of an elastic body is presented.     

Applications of BEM in sensitivity analysis and optimization

A general approach to shape design sensitivity analysis and optimal design for static and vibration problems using boundary elements is presented. The adjoint variable method is applied to obtain first-order sensitivities for the effect of boundary shape variations. The boundary element procedure for numerical calculations of sensitivities are used. Typical objective and constraints functionals are described for shape optimal design. Several numerical examples of applications of boundary elements in shape optimal design are presented.It is a part of the paper New trends and applications of BEM in sensitivity analysis and optimization—a survey, presented during International IABEM-92 Symposium on Boundary Element Methods at University of Colorado, Boulder, 3–6 August 1992     

A new approach to 2-D eigenvalue shape design

A new approach to solve problems in the optimal shape design of eigenvalues is presented. The idea behind the approach is based on a combination of the known material derivative method^4,7 and a specially adapted Green's functions formulation.¹⁴ In the current study, numerical testing has been limited to two-dimensional shape optimization. The two test cases involve the first eigenvalues of boundary value problems which model (i) free vibrations of a fixed membrane and (ii) natural frequencies of clamped thin plates. The basic algorithm can easily be adapted to serve other types of boundary value problems dealing with the highest eigenvalues that occur in applied mechanics. Computational aspects of the proposed approach, such as accuracy of the obtained sensitivities, cost of computation and so on are discussed in References 16 (in preparation).     

THREE-DIMENSIONAL DESIGN SENSITIVITY ANALYSIS USING A BOUNDARY INTEGRAL APPROACH

A general shape design sensitivity analysis approach, different from traditional sensitivity methods is developed for three-dimensional elastostatic problems. The boundary integral design sensitivity formulation is given in order to obtain traction, displacement and equivalent stress sensitivities which are required for design optimization. Those integral equations are derived analytically by differentiation with respect to the normal to the surface at design variable points. Subdivision of boundary elements into sub-elements and rigid body translation methods are employed to deal with singularities that occur during the numerical discretization of the domain. Four different examples are demonstrated to show the accuracy of the method. The boundary integral sensitivity results are compared with the finite difference sensitivity results. Excellent agreement is achieved between the two methods. © 1997 by John Wiley & Sons, Ltd.     

Conforming boundary elements in plane elasticity for shape design sensitivity

The 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.     

On evaluation of shape sensitivities of non‐linear critical loads

The present paper focuses on the evaluation of the shape sensitivities of the limit and bifurcation loads of geometrically non‐linear structures. The analytical approach is applied for isoparametric elements, leading to exact results for a given mesh. Since this approach is difficult to apply to other element types, the semi‐analytical method has been widely used for shape sensitivity computation. This method combines ease of implementation with computational efficiency, but presents severe accuracy problems. Thus, a general procedure to improve the semi‐analytical sensitivities of the non‐linear critical loads is presented. The numerical examples show that this procedure leads to sensitivities with sufficient accuracy for shape optimization applications. Copyright © 2002 John Wiley & Sons, Ltd.     

Boundary integral equation method for shape optimization of elastic structures

A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.     

Dynamic analysis of 3-D structures by a transformed boundary element method

In this work, an advanced implementation of the direct boundary element method applicable to periodic (steadystate) and transient dynamic problems involving three-dimensional structures of arbitrary shape and connectivity is presented. Interior, exterior and halfspace type of problems can all be solved by the present method. The discussion first focuses on the formulation of the method, followed by material pertaining to the fundamental singular solutions and to the isoparametric boundary elements used for discretizing the surface of the problem. Subsequently, numerical integration techniques and the solution algorithm are introduced. This methodology has been incorporated in a versatile, general purpose computer program. Finally, the stability and high accuracy of this dynamic analysis technique are established through comparisons with available analytical and numerical results.     

Material constant sensitivity boundary integral equation for anisotropic solids

In this paper, the material constant sensitivity boundary integral equation is presented, and its numerical solution proposed, based on boundary element techniques. The formulation deals with plane problems with general rectilinear anisotropy. Expressions for the computation of sensitivities for displacements, tractions, strains and stresses are derived, both for boundary and interior points. The sensitivities can be computed with respect to the bulk material properties or to the properties of part of the domain (inclusions, coatings, etc.). To assess the accuracy of the proposed approach, the computed results are compared to analytical ones derived from exact solutions obtained by complex potential theory, when possible, or finite difference derivatives otherwise. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary formulations for shape sensitivity of temperature dependent conductivity problems

Used in concert with the Kirchhoff transformation, implicit differentiation of the discretized boundary integral equations governing the conduction of heat in solids with temperature dependent thermal conductivity is shown to generate an accurate and economical approach for computation of shape sensitivities. For problems with specified temperature and heat flux boundary conditions, a linear problem results for both the analysis and sensitivity analysis. In problems with either convection or radiation boundary conditions, a non-linear problem is generated. Several iterative strategies are presented for the solution of the resulting sets of non-linear equations and the computational performances examined in detail. Multi-zone analysis and zone condensation strategies are demonstrated to provide substantive computational economies in this process for models with either localized non-iinear boundary conditions or regions of geometric insensitivity to design variables. A series of non-linear example problems is presented that have closed form solutions. Exact anaytical expressions tor the shape sensitivities associated with these problems are developed and these are compared with the sensitivities computed using the boundary element formulation.     

EVALUATION OF THE STRESS TENSOR IN 3-D ELASTOPLASTICITY BY DIRECT SOLVING OF HYPERSINGULAR INTEGRALS

A 3-D hypersingular Boundary Integral Equation (BIE) of elastoplasticity is derived. Using this formulation the displacement rate gradients and the complete stress tensor on the boundary can be evaluated directly as opposed to the classical approach, where the shape functions derivatives are to be calculated. The regularization of strongly singular and hypersingular boundary integrals, as well as strongly singular domain integrals for a source point positioned on the boundary is carried out in a general manner. Arbitrary types of elements and arbitrary positions of the source point with respect to continuity requirements can be used. Numerical 3-D elastoplastic examples (notch and crack problems) illustrate the advantages of the proposed method.     

Shape design sensitivity analysis in elasticity using the boundary element method

This paper deals with sensitivity analysis of the different functionals appearing in optimum shape design in elasticity using boundary element method (BEM). First, a general review concerning sensitivity analysis of the most usual functionals in elasticity is presented, based on the continuum approach. The accuracy in sensitivity analysis depends on the accuracy in evaluating strains and stresses on the boundary. A general procedure for strain calculation based upon some results of differential geometry of surfaces is shown. Another essential aspect in sensitivity analysis is the definition of the design velocity on the boundary, which defines the change in the geometry of the elastic solid. A computational treatment independent of the design variables used is presented, defining nodal values of the design velocity and taking advantage of the boundary element approximation. Finally, the feasibility and accuracy of the proposed procedures are assessed through several example problems.     

A boundary element formulation for design sensitivities in materially nonlinear problems

Summary This paper presents a formulation for the determination of design sensitivities for shape optimization in materially nonlinear problems. This approach is based on direct differentiation (DDA) of the relevant boundary element method (BEM) formulation of the problem. It combines the accuracy advantages of the BEM without the difficulty of dealing with strongly singular kernels. This approach provides a new avenue towards efficient shape optimization of small strain elastic-viscoplastic and elastic-plastic problems.With 1 Figure     

Shape design sensitivities improvement using convected unstructured meshes

This paper focuses on various forms of direct differentiation methods for design sensitivity computation in the shape optimisation of continuum structures and the role of convected meshes on the accuracy of the sensitivities. A Pseudo-Analytical Sensitivity Analysis (P-ASA) method is presented and tested. In this method the response analysis component uses unstructured finite element meshes and the sensitivity algorithm entails shape-perturbation for each design variable. A material point is convected during a change of shape and the design sensitivities are therefore intrinsically associated with the mesh-sensitivities of the finite element discretization. Such mesh sensitivities are obtained using a very efficient boundary element point-tracking analysis of an affine notional underlying elastic domain. All of the differentiation, with respect to shape variables, is done exactly except for the case of mesh-sensitivities: hence the method is almost analytical. In contrast to many other competing methods, the P-ASA method is, by definition, independent of perturbation step-size, making it particularly robust. Furthermore, the sensitivity accuracy improves with mesh refinement. The boundary element point-tracking method is also combined with two popular methods of sensitivity computation, namely the global finite difference method and the semi-analytical method. Increases in accuracy and perturbation range are observed for both methods.     

Adaptive mesh refinement of the boundary element method for potential problems by using mesh sensitivities as error indicators

This paper presents a novel method for error estimation and h-version adaptive mesh refinement for potential problems which are solved by the boundary element method (BEM). Special sensitivities, denoted as mesh sensitivities, are used to evaluate a posteriori error indicators for each element, and a global error estimator. A mesh sensitivity is the sensitivity of a physical quantity at a boundary node with respect to perturbation of the mesh. The element error indicators for all the elements can be evaluated from these mesh sensitivities. Mesh refinement can then be performed by using these element error indicators as guides.The method presented here is suitable for both potential and elastostatics problems, and can be applied for adaptive mesh refinement with either linear or quadratic boundary elements. For potential problems, the physical quantities are potential and/or flux; for elastostatics problems, the physical quantities are tractions/displacements (or tangential derivatives of displacements). In this paper, the focus is on potential problems with linear elements, and the proposed method is validated with two illustrative examples. However, it is easy to extend these ideas to elastostatics problems and to quadratic elements.The computing for this research has been supported by the Cornell National Supercomputer Facility.     

Automatic monitoring of element shape quality in 2-D and 3-D computational mesh dynamics

One of the major problems in fluid–structure interaction using the arbitrary Lagrangian Eulerian approach lies in the area of dynamic mesh generation. For accurate fluid-dynamic computations, meshes must be generated at each time step. The fluid mesh must be regenerated in the deformed fluid domain in order to account for the displacements of the elastic body computed by the structural dynamics solver. In the elasticity-based computational dynamic mesh procedure, the fluid mesh is modeled as a pseudo-elastic solid the deformation of which is based on the displacement boundary conditions, resulting from the solution of the computational structural dynamics problem. This approach has a distinct advantage over other mesh-movement algorithms in that it is a very general, physically based approach that can be applied to both structured and unstructured meshes. The major drawback of the linear elastostatic solver is that it does not guarantee the absence of severe element distortion. This paper describes a novel mesh-movement procedure for mesh quality control of 2-D and 3-D dynamic meshes based on solving a pseudo-nonlinear elastostatic problem. An inexpensive distortion measure for different types of elements is introduced and used for controlling the element shape quality. The mesh-movement procedure is illustrated with several examples (large-displacement and free-boundary problems) that highlight its advantages in terms of performance, mesh quality, and robustness. It is believed that the resulting scheme will result in a more economical simulation of the motion of complex geometry, 3-D elastic bodies immersed in temporally and spatially evolving flows. Received 20 April 2000     

Sensitivity analysis and shape optimization in nonlinear solid mechanics

The objective of this paper is twofold. First, it presents a boundary element formulation for sensitivity analysis for solid mechanics problems involving both material and geometric nonlinearities. The second focus is on the use of such sensitivities to obtain optimal design for problems of this class. Numerical examples include sensitivity analysis for small (material nonlinearities only) and large deformation problems. These numerical results are in good agreement with direct integration results. Further, by using these sensitivities, a shape optimization problem has been solved for a plate with a cutout involving only material nonlinearities. The difference between the optimal shapes of solids, undergoing purely elastic or elasto-viscoplastic deformation is shown clearly in this example.     

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.     

求解二维Helmholtz外问题的一种快速算法

本基于虚拟边界积分法，通过将虚拟积分曲线选为多(单)条圆形曲线，并在这些圆形积分曲线上将未知源强密度函数用Fourier级数展开，同时借助快速数值Fourier逆变换(IFFT)计算程序，提出了一种求解二维Helmholtz外问题的快速算等。该方法由于不需要将分布在虚拟边界上的未知函数进行单元分散，不仅克服了边界元法或虚拟边界元法中由于单元形函数是由低阶多项式函数构成导致其结果只适用于较低频率范围的不足，而且具有很高的计算精度和效率。中给出的数值算例表明了这种快速算法的计算效率是虚拟边界元法的20-80倍。     

Three dimensional boundary formulations for nonlinear thermal shape sensitivities

Implicit differentiation of the discretized boundary integral equations governing the conduction of heat in three dimensional (3D) solid objects, subjected to nonlinear boundary conditions, and with temperature dependent material properties, is shown to generate an accurate and economical approach for the computation of shape sensitivities. The theoretical formulation for primary response (surface temperature and normal heat flux) sensitivities and secondary response (surface tangential heat flux components and internal temperature and heat flux components) sensitivities is given. Iterative strategies are described for the solution of the resulting sets of nonlinear equations and computational performances examined. Multi-zone analysis and zone condensation strategies are demonstrated to provide substantial computational economies in this process for models with either localized nonlinear boundary conditions or regions of geometric insensitivity to design variables. A series of nonlinear sensitivity example problems are presented that have closed form solutions. Sensitivities computed using the boundary formulation are shown to be in excellent agreement with these exact expressions.