Structural Shape and Topology Optimization

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.

This paper deals with simultaneous topology and shape optimization of elastic contact problems. The structural optimization problem for an elastic contact problem is formulated. Shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization and to calculate a descent direction in numerical algorithm. Level set based numerical algorithm for the solution of this optimization problem is proposed. Numerical examples are provided and discussed.

Shape optimization (with topological changes) has become a de facto standard approach for synthesizing new designs. Current approaches heavily rely on voxelized modifications of the initial domain, followed by heuristic reconstruction procedures that result in a detailed shape design. The initial domain is almost always conservatively estimated so that the design space spans a sufficiently large set of possible solutions. We propose a novel approach to shape optimization that exploits the geometric and topologic properties of medial zones and a new shape modification paradigm to synthesize shapes with topological guarantees. The medial zone is a ‘thick skeleton’ of the domain induced by prescribed boundary conditions, and represents an initial solution that is topologically valid. We modify such an initial solution by adding or subtracting ‘bubbles’ of material defined via prescribed shape functions. The proposed approach has many advantages in terms of topological guarantees, shape control, and computational cost, supports parametric optimization of free-form geometry, and can be implemented in any geometric representation that supports distance computations.

3R25. Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, Vol 146. - G Allaire (Center of Appl Math, Ecole Polytechnic, Paliseau Cedex, 91128, France). Springer-Verlag, New York. 2002. 456 pp. ISBN 0-387-95298-5. $79.95.Reviewed by MS Qatu (Ford Motor Co, EVB Bldg, MD X8, 20800 Oakwood Blvd, Dearborn MI 48124).This is mainly a research monograph on the theory and application of the homogenization method in shape optimization problems. Applications given in the book are in the fields of composite materials, conductivity, and elasticity with emphasis on the minimization of compliance. The audience for this book includes researchers and advanced graduate students in applied mathematics and engineers interested in the design of shapes of structures for conductive and elastic purposes. It also fits very well for material scientists with inclination towards theoretical aspects of design and analysis of composite materials. Based on the theoretical content and examples in the book, the word shape in the title should rather be understood as topology. In some engineering fields, shape is more related to boundary perturbation, however, the methods presented in this book are designed to define the boundary, rather than to perturb it. This concept of defining the boundary is closer to the idea of topology optimization as a material distribution problem. The book is divided into five chapters. The first chapter covers the subject of homogenization in quite details. Periodicity is first assumed and used to derive homogenized operators in elasticity by means of the two-scale asymptotic expansion. Definitions and proofs of H- and G-convergence are then presented in a rigorous way. The second chapter is titled mathematical modeling of composite materials. In this chapter, Allaire explains carefully the G-closure problem and its interpretations in the mechanical setting. Finding bounds on elastic properties of bi-phase materials is an old problem in mechanics that can be explained and treated mathematically with the help of the G-closure theory. Rank laminated materials are used to demonstrate the bounds derived in the book. Optimal design of materials under electric conductivity considerations is addressed in the third chapter. With an extensive number of proofs, this chapter explains the theory behind the computation of optimal topology (shape) using homogenization theory as a mean to relax the optimization problem. Composite materials are used to relax the problem, and homogenization of conductivity tensors is then used to go from an ill-posed problem to a well-posed one. Structural optimization has been an important subject of research since the days of Lagrange and Saint Venant. This chapter is the counterpart of the previous one, this time, in the field of elastic structures. Composite materials are again introduced to relax the optimization problem and homogenized elastic properties are computed in order to attain a solution to the problem. The minimization of compliance (maximization of stiffness) is addressed intensively in this chapter. The fifth and last chapter is titled Numerical Algorithms. Academic two-dimensional rectangular domains using quadrilateral elements in regular meshes are used to exemplify the theory presented in the previous chapters. Examples include minimization of compliance and maximization of eigenvalues. Multiple loadings and optimum topologies (shapes) of complaint mechanisms were considered. In summary, Shape Optimization by the Homogenization Method is an excellent book that complements very well the extensive literature in the field, including Sanchez-Palencia and Zaoui [1], Bendsøe [2], Cherkaev [3], and Bendsøe and Sigmund [4].

In conventional level set methods, the slope of the level set function needs to be well controlled to maintain the numerical stability during the topology optimization process. One common solution is to regularize the level set function to be a signed distance function, which is usually achieved by periodically implementing the so called re-initialization scheme to force the level set function to gain the desired signed distance property. However, the re-initialization scheme will bring some unwanted drawbacks to the optimization process, such as zero level set drifting, time consuming etc. In addition, re-initialization is usually implemented outside the optimization loop, which will cause convergence issues. In this paper, a distance regularization functional is introduced to the structural topology optimization objective functional to ensure the signed distance property of the level set function near the structure boundaries. This functional can also keep the level set function to be constant-value at positions far away from the structural boundaries. The radial basis function (RBF) based parameterization technique together with the mathematical programming are utilized to improve the potential capability of handling multiple constraints for the topology optimization. The combination of these two techniques makes the level set based topology optimization be capable of handling complicated multi-constrained problems with higher numerical efficiency, leaving no compromise to multiple drawbacks. To demonstrate the validity of the proposed scheme, benchmark examples on minimum compliance structural optimization are employed. This type of problem is computed by the conventional level set method with the introduced distance regularization functional, the RBF based parametric level set and at last, the distance regularized RBF based parametric level set separately to demonstrate their differences.

Structural topology optimization.- On basic properties of Michell's structures.- Validation of numerical method by analytical benchmarks and verification of exact solutions by numerical methods.- Introduction to shape and topology optimization.- Homogenization method for shape and topology optimization.- Level set method for shape and topology optimization.- Compliance minimization of two-material elastic structures.- Some notes on topology optimization of vibrating continuum structures.- Topology optimization of vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps.- On optimum shape design and periodicity of band-gape beam structures.- Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration.- Topological design for minimum sound emission from bi-material structures subjected to forced vibration.- Discrete material optimization of vibrating laminated composite plates for minimum sound emission.- Optimization of diffusive transport problems.- Fluid topology optimization: stokes and Navier-stokes models.- Topology optimization of coupled multi-physics problems.- Topology optimization based on the extended finite element method.- Topology optimization of meso- and nano-scale problems.- Topology optimization under uncertainty.- A brief review of numerical methods of structural topology optimization.- The free material design in planar elasticity.

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending.

Weakly determinate problems of Boolean linear programming with a partially specified set of admissible solutions

Beyond Lack of Compactness and Lack of Stability of a Coupled Parabolic-Hyperbolic Fluid-Structure System.- A Continuous Adjoint Approach to Shape Optimization for Navier Stokes Flow.- Recent Advances in the Analysis of State-constrained Elliptic Optimal Control Problems.- Fast and Strongly Localized Observation for a perturbed Plate Equation.- Representations, Composition, and Decomposition of C 1,1-hypersurfaces.- On Some Nonlinear Optimal Control Problems with Vector-valued Affine Control Constraints.- Weak Solutions to a Model for Crystal Growth from the Melt in Changing Magnetic Fields.- Lavrentiev Prox-regularization Methods for Optimal Control Problems with Pointwise State Constraints.- Nonlinear Feedback Solutions for a Class of Quantum Control Problems.- Optimal Feedback Synthesis for Bolza Control Problem Arising in Linearized Fluid Structure Interaction.- Single-step One-shot Aerodynamic Shape Optimization.- Shape Differentiability of Drag Functional for Compressible Navier-Stokes Equations.- Null-controllability for a Coupled Heat-Finite-dimensional Beam System.- Feedback Modal Control of Partial Differential Equations.- Optimization Problems for Thin Elastic Structures.- A New Non-linear Semidefinite Programming Algorithm with an Application to Multidisciplinary Free Material Optimization.- How to Check Numerically the Sufficient Optimality Conditions for Infinite-dimensional Optimization Problems.- Hidden Boundary Shape Derivative for the Solution to Maxwell Equations and Non Cylindrical Wave Equations.

In order to achieve a low vibration design of engine block,verify the correctness of finite element model based on the test method,ensure the undertake of different levels of optimization model using the method of topology and shape optimization,modal frequency and vibration intensity of engine block are taken as optimization objectives,vibration analysis of engine block is made in the simulation,and the body vibration condition distribution are identified in the optimization design.The four cylinder engine block is set as the design objective,the optimal solution design variables are got through the topology optimization and shape optimization method,and then the final optimization scheme with whole layout and detailed design are obtained.The results of optimized structure show that frequencies of each mode are improved,and the vibration intensity value of engine block decreases by 34.8%,then vibration intensity of improved engine block is minimized in a safe area subject to rigidity and strength constraints.Efficiency and precision of the combination method of topology and shape optimization has an obvious enhance,and the feasibility and efficiency of the proposed methods are demonstrated by the experiment.

Integrated topology and shape optimization software for compliant MEMS mechanism design

Finite-element approximation of 2D elliptic optimal design

A level set solution to the stress-based structural shape and topology optimization

In this paper we present theoretical, computational, and practical aspects concerning 3-dimensional shape optimization governed by linear magnetostatics. The state solution is approximated by the finite element method using Nédélec elements on tetrahedra. Concerning optimization, the shape controls the interface between the air and the ferromagnetic parts while the whole domain is fixed. We prove the existence of an optimal shape. Then we state a finite element approximation to the optimization problem and prove the convergence of the approximated solutions. In the end, we solve the problem for the optimal shape of an electromagnet that arises in the research on magnetooptic effects and that was manufactured afterwards.

A novel approach to the three-dimensional optimal design of magnetic shields is presented, in which both shape optimization and topology optimization are achieved with the aid of numerical magnetic-field analysis. These numerical procedures are implemented by means of the hybrid finite element and boundary element method combined with the genetic algorithm. This results in an overall increase in the optimization speed with acceptable computational accuracy. Finally, some numerical results that demonstrate the validity of the proposed approach are presented.