Showing papers on "Shape optimization published in 1991"

Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions

Abstract: We study an optimal design problem for the domain of an elliptic equation with Dirichlet boundary conditions. We introduce a relaxed formulation of the problem which always admits a solution, and we prove some necessary conditions for optimality both for the relaxed and for the original problem.

Integrated topology and shape optimization in structural design

Abstract: Structural optimization procedures usually start from a given design topology and vary proportions or boundary shapes to achieve optimality under various constraints. This article presents an approach for initiating formal structural optimization at an earlier stage, where the design topology is rigorously generated in addition to selecting shape and size dimensions. A three-phase design process is discussed. An optimal initial topology is created by a homogenization method as a gray level image. This topology is then transformed to a realizable design using computer vision techniques, parameterized, and treated in detail by size and shape optimization. A fully automated process is described for trusses. Examples for two-dimensional solid structures are also discussed.

Applications of the COC Algorithm in Layout Optimization

Abstract: Whilst the advantages of iterative continuum-based optimality criteria (COC) methods in cross-section optimization (“sizing”) were discussed elsewhere (e.g. [4, 5]), this paper is devoted to applications of the above technique in layout optimization. Earlier studies of the latter field by others used a two-stage process consisting of separate topological and geometrical optimization and considered only a very small number of members. It will be shown here that the COC algorithm achieves a simultaneous optimization of the topology and geometry in layout problems with many thousand potential members. Moreover, some observations on shape optimization by the “homogenization” method are offered and an alternative approach to global shape optimization is suggested.

Velocity method and Lagrangian formulation for the computation of the shape Hessian

Abstract: The object of this paper is to study the shape Hessian of a shape functional by the velocity (speed) method. It contains a review and an extension of the velocity method and its connections with methods using first- or second-order perturbations of the identity. The key point is that all these methods yield the same shape gradient but different and unequal shape Hessian since each method depends on a choice of “connection.” However, for autonomous velocity fields the velocity method yields a canonical bilinear Hessian. Expressions obtained by other methods can be recovered by adding to that canonical term the shape gradient acting on the acceleration of the velocity field associated with the choice of perturbation of the identity. The second part of the paper is an application of the Lagrangian method with function space embedding to compute the shape gradient and Hessian of a simple cost function associated with the nonhomogeneous Dirichlet problem.

Integrated topology and boundary shape optimization of 2-D solids

Abstract: This study is concerned with the development of an integrated procedure for the computation of the optimal topology as well as the optimal boundary shape of a two-dimensional, linear elastic body. The topology is computed by regarding the body as a domain of the plane with a high density of material and the objective is to maximize the overall stiffness, subject to a constraint on the material volume of the body. This optimal topology is then used as the basis for a shape optimal design method that regards the body as given by boundary curves. For this case the objective is to minimize the maximum value of the Von Mises equivalent stress in the body, subject to an isoperimetric constraint on the area as well as a constraint on the stiffness. The solution procedures for the shape design are based on variational formulations for the problems and the results of a variational analysis are implemented via finite element discretizations. The discretization grids are generated automatically by an elliptical method for general two-dimensional domains. Computational results are presented for the design of a fillet, a beam and a portal frame.

On Reliability-Based Structural Optimization

Abstract: In this paper a brief presentation of the state-of-the-art of reliability-based structural optimization (RBSO) is given. Special emphasis is put on problems related to application of RBSO on real (large) structures. Shape optimization, knowledge-based optimization and optimal inspection strategies are briefly discussed. A list of 125 references is included in the appendix.

Sensitivity analysis based on analytic approach for shape optimization of electromagnetic devices: interface problem of iron and air

Abstract: The material derivative concept of continuum mechanics and the adjoint variable method of design sensitivity analysis are used to express the relation between the change of shape and that of the objective function. The explicit sensitivity formula for the interface variation between iron and air is derived in a two-dimensional electromagnetic system. Implementation of the sensitivity analysis is carried out with existing finite element codes. The sensitivity analysis is applied to the pole optimization of a quadrupole. >

Shape Design Sensitivity Analysis and Optimization Using the Boundary Element Method

Abstract: 1 Introduction.- 1.1 Introduction.- 1.2 Review of the Shape Optimization.- 1.2.1 Shape Optimization using the Finite Element Method.- 1.2.2 Shape Optimization using the Boundary Element Method.- 1.2.3 Shape Design Sensitivity Analysis.- 1.3 References.- 2 Basic Numerical Optimization Techniques.- 2.1 Introduction.- 2.2 Basic Concepts and Terminology.- 2.3 Mathematical Programming Method.- 2.4 References.- 3 The Boundary Element Method in Elastostatics.- 3.1 Introduction.- 3.2 Review of the Boundary Element Method in Elastostatics.- 3.3 The Boundary Element Method in Elastostatics.- 3.3.1 Basic Equations of Linear Elasticity.- 3.3.2 The Boundary Integral Formulation of Elasticity.- 3.3.3 Numerical Implementation of the Boundary Element Method.- 3.4 Conclusion Remarks.- 3.5 References.- 4 Shape Design Sensitivity Analysis using the Boundary Element Method.- 4.1 Introduction.- 4.2 Two Basic Approaches for Design Sensitivity Analysis.- 4.2.1 The Discretized Approach (DA).- 4.2.2 The Continuum Approach (CA).- 4.2.3 Comparisons of the Two Approaches.- 4.3 The Implementation of the Material Derivative of Displacements.- 4.4 Stress Sensitivity Analysis by CA.- 4.4.1 A Simple Case.- 4.4.2 The Stress Sensitivity Formulation for the General Case.- 4.5 The Modelling of the Adjoint Problem.- 4.5.1 Numerical Approaches for Problems with singular Loads.- 4.5.2 Mesh Refinement and Special Elements Methods.- 4.5.3 Local Singular Function Method.- 4.5.4 Smooth Loading Method.- 4.5.5 Singularity Subtraction Method.- 4.5.6 Concluding Remarks.- 4.6 Implementation of the Singularity Subtraction Method.- 4.7 A New Finite Difference Based Approach to Shape Design Sensitivity Analysis.- 4.7.1 A Simple Example.- 4.7.2 Derivation of the Finite Difference Load Method.- 4.7.3 Further Discussions of FDLM.- 4.7.4 Concluding Remarks.- 4.8 Numerical Examples.- 4.8.1 A Cantilever Beam.- 4.8.2 A Circular Plate Under Internal Pressure.- 4.8.3 A Fillet Example.- 4.8.4 An Elastic Ring under a Concentrated Load.- 4.9 Concluding Remarks.- 4.10 References.- 5 Shape Optimization Using the Boundary Element Method.- 5.1 Introduction.- 5.2 The Design Model and the Analysis Model.- 5.2.1 The Design Model.- 5.2.2 The Analysis Model - Remeshing Problem.- 5.3 Shape Optimization Implementation.- 5.3.1 SOP - A Shape Optimization Program.- 5.4 Numerical Examples.- 5.4.1 A Beam Example.- 5.4.2 A Fillet Example.- 5.4.3 A Plate With a Hole.- 5.4.4 A Connecting Rod.- 5.5 References.- A Fundamental Solutions of the Semi-infinite Plane.- B Derivatives of Boundary Stresses on the Normal Direction.

Anatomy of the Shape Hessian

Abstract: The computation of the Shape Gradient with respect to domain perturbations plays a central role in the theory and numerical solution of Shape Optimization problems. In 1907 J. Hadamard introduced a method which has been and still is widely used to obtain many useful results for applications. The mathematical limitation of his method rests in the fact that the deformations of the domain are a function of the smoothness of the normal to the boundary (hence the smoothness of the boundary). New developments by the Nice School (J. Cea and J. P. Zolesio) using arbitrary velocity fields of deformation relaxed the condition that the deformation be carried by the normal to the boundary. Finally the use of «Shape Lagrangians» by Delfour and Zolesio made it possible to obtain Shape Gradients by a simple constructive method which does not require the derivative of the state with respect to the domain. In this paper we apply this last method to semi convex cost functions. This extension makes it possible to compute the «Shape Hessian» or «Shape directional second derivative». We give several expressions for the «Shape Hessian» and a set of equations characterizing its kernel.

Design optimization using the boundary element method

Abstract: Structural optimization is often confined to sizing simple design variables, such as plate thicknesses and bar cross-sectional areas, while the geometric shapes of components remain largely unchanged. Shape optimization is more complex, changing the shape of the boundary, subject to appropriate constraints. The boundary element method, being a boundary-oriented technique is very appropriate for this purpose. It can overcome a number of the difficulties associated with its main rival, the finite element method. Firstly, because of the continuously changing geometry. the accuracy of the finite element analysis using the initial mesh of elements may become inadequate during the optimization process. Secondly, if during this process, the finite element mesh has to be re-generated, the cost is relatively high. Finally, and most importantly, the sensitivity analysis in the calculation of the derivatives with respect to the design variables may be obtained directly in the boundary element approach rather...

Optimal shape design for systems governed by variational inequalities, part 2: existence theory for the evolution case

Abstract: Some general existence results for optimal shape design problems for systems governed by parabolic variational inequalities are established by the mapping method and variational convergence theory. Then, an existence theorem is given for the optimal shape for an electrochemical machining problem, in which the cost functional is not lower semicontinuous, by extending the general results to this case. Furthermore, this problem is approximated by a set of optimal shape design problems which have more smooth cost functionals and are easier to handle computationally.

A general methodological analysis for optimum design

Abstract: DAO, a computer aided optimum design system by the finite element method, has been developed. The system can solve efficiently 2D and 3D structural fixd-geometry and shape optimization problems. The power and viability of this methodology is illustrated by the solution to a structural optimization problem. The shape of the central section of an arch dam is optimized.

Shape sensitivity analysis of piezoelectric structures by the adjoint variable method

Abstract: Shape sensitivity expressions are derived for linear piezoelectric structures with coupled mechanical and elastic fields. By adopting the quasielectrostatic approximation for these inherently anisotropic materials, the adjoint variable method of optimization and the material derivative formulation of shape variations are used in a systematic procedure to evaluate the total variation of a general performance criterion with respect to shape variations. The material (total) derivative of the adopted integral functional is found in terms of primary and adjoint quantities, as well as the deformation velocity field. Since the structure is assumed to undergo dynamic response, domain integrations evaluated at the initial time are also needed in this formulation.

Transient thermoelasticity and other body force effects in boundary element shape sensitivity analysis

Abstract: Structural shape design sensitivity analysis (DSA) is presented for objects subjected to gravity, centrifugal, and general thermal loading. DSA employs implicit differentiation of the governing boundary integral equations. Both a surface integral and a particular integral approach are discussed for computation of these effects in response analysis and sensitivity analysis. It is shown that the particular integral approach is capable of computing the sensitivities of objects subjected to nonharmonic temperature distributions. Attention is also given to several techniques for the modeling of temperature distributions in the DSA process. In particular, an approach that eliminates the need for the incorporation of thermal response analysis and sensitivity analysis is described in the overall shape optimization process. A sharp corner formulation is also presented that can account for the jump in the surface traction components at a sharp corner in an interzone boundary.

Shape Optimization of Engineering Components by Adaptive Biological Growth

Abstract: Biological load carriers always grow into a shape whereby a constant stress can be found everywhere along the surface of the biological component for the most significant natural loading applied. This avoids local stress peaks and therefore pre-defined failure points in the design. This mechanism of adaptive growth is copied by the so called CAO-method (Computer Aided Optimization). The method is briefly described and the shape optimization of a tree fork illustrates the adaptive growth. Furthermore a rubber bearing, a bending bar with rectangular window as well as a joint of metal sheets are shape-optimized as engineering examples. In cases where the design proposal which the CAO-method starts from cannot be guessed easily, an oversized rough proposal can be analysed by FEM. After cutting off unloaded parts, the remaining structure can then be used as a starting design for CAO.

Shape sensitivities and optimal configurations for heat diffusion problems : a BEM approach

Abstract: Design sensitivity analysis, along with the shape optimization of heat diffusion problems using the boundary element method (BEM), is presented in this paper. The present approach utilizes the implicit differentiation of discretized boundary integral equations with respect to the design variables to yield the sensitivity equations. A technique based on the response of an object to a constant boundary temperature is presented for the evaluation of singular terms in the thermal sensitivity kernels. A procedure for the design sensitivity analysis of a reduced system of equations obtained via substructuring and condensation is also presented. The BEM formulations are implemented for both two-dimensional and axisymmetric objects. A number of sample problems are solved to demonstrate the accuracy of the present sensitivity formulation and to obtain optimal configurations of some mechanical components of practical interest, which are subjected to different thermal environments.

First- and second-order shape sensitivity analysis of structures

Abstract: The first- and second-order sensitivities with respect to varying structural shape are discussed for an arbitrary stress, strain and displacement functional. It is assumed that only the traction-free boundary of a structure can undergo the shape modification described by a set of shape design parameters. The first derivatives of a functional with respect to these parameters are derived using both the direct and adjoint approaches. Next the second derivatives are obtained using the mixed approach in which both the direct and adjoint first-order solutions are used. The general results are particularized for the case of complementary and potential energy of a structure. Some simple examples illustrate the theory presented.

Error Analysis of Semianalytic Displacement Derivatives for Shape and Sizing Variables

Abstract: For efficiency, the derivatives used in structural optimization are often calculated using a technique known as the semianalytic method. Here, we investigate the accuracy of this method for calculating static displacement derivatives with respect to both shape and sizing design variables. We show that the errors are entirely due to inaccuracies introduced during the finite-difference approximation of the stiffness matrix derivatives. Two types of errors are discussed. In the first case, the finite-difference error causes uniform scaling of the derivatives relative to their true values and the magnitude of the relative error depends only on the choice of the finite-difference parameter and the finite-difference formula. In the second case, errors in the finite-difference operation may lead to nonuniform errors in the displacement derivatives. The magnitudes of the relative errors depend not only on the finite-difference parameter and formula, but on the location of the element(s) associated with the design variable and the discretization of the structure. We demonstrate our error analysis using several example problems, including a representative automotive frame. We show for these problems that the relative errors can be adequately controlled through the choice of the finite-difference parameter.

Three-Dimensional Shape Optimization with Substructuring

Abstract: The substructure or superelement formulation used in the finite element technique was employed for threedimensional shape optimization problems. In a design process, one often encounters the situation that only a small part of a complex component is allowed to be modified due to considerations other than structural performance. In this case, the substructure formulation is more efficient than the full-structure finite element approach and is well suited for structural optimization, since the design process is iterative and the analysis has to be performed many times before the design is finalized. The substructure formulation is one in which only the stiffness matrix for the modified part is reformulated, reduced to the exterior degrees-of-freedom for the changed design, and can provide significant savings in computing resources. In shape optimization, the computational advantage is not only in analysis but also in the sensitivity calculations which provide in a further reduction of computing time. A front steering knuckle is used as an example to demonstrate the use and efficiency of the superelement formulation.