An element-by-element strategy for non-linear shape optimization

A strategy for the efficient solution of non-linear shape optimization problems is developed. This strategy employs an integrated element-by-element approach to the solution of the governing partial differential equations, and, more particularly, to the computation of the necessary gradients of the objective function and constraints using an adjoint formulation. This proves to be a very efficient strategy and also is relatively easy to implement, because the local effect of design changes can be exploited. The method is tested with an application involving the design of the shape of electromagnet poles in order to obtain a desired field in the interpolar region.     

Sensitivity analysis and shape optimization for preform design in thermo‐mechanical coupled analysis

In complex forging processes, it is essential to find the optimal deformation path and the optimal preform shape that will lead to the desired final shape and service properties. A sensitivity analysis and optimization for preform billet shape in thermo‐mechanical coupled simulation is developed in this work. Non‐linear sensitivity analysis of temperatures, flow‐stresses, strains and strain‐rates are presented with respect to design variables. Both analytical and finite‐difference gradients are employed to validate the effectiveness of sensitivity analysis developed in this work. Numerous iterations of coupled thermo‐mechanical analysis are performed to determine an optimum preform shape based on a given criterion of minimizing the objective function on effective strain variance within the final forging. The design constraints are imposed on die underfill, material scrap, folding defects and temperatures. In addition, a method for material data processing is given in order to determine the flow stress and its derivatives. The shape optimization scheme is demonstrated with the preform designs of an axisymmetric disk and a plane strain problem. Copyright © 1999 John Wiley & Sons, Ltd.     

A unified parametric design approach to structural shape optimization

This paper presents a general parametric design approach for 2-D shape optimization problems. This approach has been achieved by integrating practical design methodologies into numerical procedures. It is characterized by three features: (i) automatic selection of a minimum number of shape design variables based on the CAD geometric model; (ii) integration of sequential convex programming algorithms to solve equality constrained optimization problems; (iii) efficient sensitivity analysis by means of the improved semi-analytical method. It is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities. Numerical solutions are performed to demonstrate the applicability of the proposed approach. A discussion of the results is also given:     

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.     

Structural shape and topology optimization based on level-set modelling and the element-propagating method

This article introduces the element-propagating method to structural shape and topology optimization. Structural optimization based on the conventional level-set method needs to solve several partial differential equations. By the insertion and deletion of basic material elements around the geometric boundary, the element-propagating method can avoid solving the partial differential equations and realize the dynamic updating of the material region. This approach also places no restrictions on the signed distance function and the Courant–Friedrichs–Lewy condition for numerical stability. At the same time, in order to suppress the dependence on the design initialization for the 2D structural optimization problem, the strain energy density is taken as a criterion to generate new holes in the material region. The coupled algorithm of the element-propagating method and the method for generating new holes makes the structural optimization more robust. Numerical examples demonstrate that the proposed approach greatly improves numerical efficiency, compared with the conventional level-set method for structural topology optimization.     

Compliant mechanism design with non-linear materials using topology optimization

In this paper, compliant mechanism design with non-linear materials using topology optimization is presented. A general displacement functional with non-linear material model is used in the topology optimization formulation. Sensitivity analysis of this displacement functional is derived from the adjoint method. Optimal compliant mechanism examples for maximizing the mechanical advantage are presented and the effect of non-linear material on the optimal design are considered.     

On structural shape optimization using an embedding domain discretization technique

This contribution presents a novel approach to structural shape optimization that relies on an embedding domain discretization technique. The evolving shape design is embedded within a uniform finite element background mesh which is then used for the solution of the physical state problem throughout the course of the optimization. We consider a boundary tracking procedure based on adaptive mesh refinement to separate between interior elements, exterior elements, and elements intersected by the physical domain boundary. A selective domain integration procedure is employed to account for the geometric mismatch between the uniform embedding domain discretization and the evolving structural component. Thereby, we avoid the need to provide a finite element mesh that conforms to the structural component for every design iteration, as it is the case for a standard Lagrangian approach to structural shape optimization. Still, we adopt an explicit shape parametrization that allows for a direct manipulation of boundary vertices for the design evolution process. In order to avoid irregular and impracticable design updates, we consider a geometric regularization technique to render feasible descent directions for the course of the optimization. Copyright © 2016 John Wiley & Sons, Ltd.     

Acoustic shape optimization using cut finite elements

Fictitious domain methods are attractive for shape optimization applications, since they do not require deformed or regenerated meshes. A recently developed such method is the CutFEM approach, which allows crisp boundary representations and for which uniformly well‐conditioned system matrices can be guaranteed. Here, we investigate the use of the CutFEM approach for acoustic shape optimization, using as test problem the design of an acoustic horn for favorable impedance‐matching properties. The CutFEM approach is used to solve the Helmholtz equation, and the geometry of the horn is implicitly described by a level‐set function. To promote smooth algorithmic updates of the geometry, we propose to use the nodal values of the Laplacian of the level‐set function as design variables. This strategy also improves the algorithm's convergence rate, counteracts mesh dependence, and, in combination with Tikhonov regularization, controls small details in the optimized designs. An advantage with the proposed method is that the exact derivatives of the discrete objective function can be expressed as boundary integrals, as opposed to when using a traditional method that uses mesh deformations. The resulting horns possess excellent impedance‐matching properties and exhibit surprising subwavelength structures, not previously seen, which are possible to capture due to the fixed mesh approach.     

A bispace parameterization method for shape optimization of thin‐walled curved shell structures with openings

Simultaneous shape optimization of thin‐walled curved shell structures and involved hole boundaries is studied in this paper. A novel bispace parameterization method is proposed for the first time to define global and local shape design variables both in the Cartesian coordinate system and the intrinsic coordinate system. This method has the advantage of achieving a simultaneous optimization of the global shape of the shell surface and the local shape of the openings attached automatically on the former. Inherent problems, for example, the effective parameterization of shape design variables, mapping operation between two spaces, and sensitivity analysis with respect to both kinds of design variables are highlighted. A design procedure is given to show how both kinds of design variables are managed together and how the whole design flowchart is carried out with relevant formulations. Numerical examples are presented and the effects of both kinds of design variables upon the optimal solutions are discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust shape and topology optimization considering geometric uncertainties with stochastic level set perturbation

When geometric uncertainties arising from manufacturing errors are comparable with the characteristic length or the product responses are sensitive to such uncertainties, the products of deterministic design cannot perform robustly. This paper presents a new level set‐based framework for robust shape and topology optimization against geometric uncertainties. We first propose a stochastic level set perturbation model of uncertain topology/shape to characterize manufacturing errors in conjunction with Karhunen–Loève (K–L) expansion. We then utilize polynomial chaos expansion to implement the stochastic response analysis. In this context, the mathematical formulation of the considered robust shape and topology optimization problem is developed, and the adjoint‐variable shape sensitivity scheme is derived. An advantage of this method is that relatively large shape variations and even topological changes can be accounted for with desired accuracy and efficiency. Numerical examples are given to demonstrate the validity of the present formulation and numerical techniques. In particular, this method is justified by the observations in minimum compliance problems, where slender bars vanish when the manufacturing errors become comparable with the characteristic length of the structures. Copyright © 2016 John Wiley & Sons, Ltd.     

Sensitivity and optimization for shape and non‐linear boundary conditions in thermal boundary elements

This paper is concerned with the minimization of functionals of the form ∫_Γ(b) f( h ,T( b, h )) dΓ( b ) where variation of the vector b modifies the shape of the domain Ω on which the potential problem, ?²T=0, is defined. The vector h is dependent on non‐linear boundary conditions that are defined on the boundary Γ. The method proposed is founded on the material derivative adjoint variable method traditionally used for shape optimization. Attention is restricted to problems where the shape of Γ is described by a boundary element mesh, where nodal co‐ordinates are used in the definition of b . Propositions are presented to show how design sensitivities for the modified functional ∫_Γ(b) f( h ,T ( b, h )) dΓ( b ) +∫_Ω(b) λ( b, h ) ?²T( b, h ) dΩ( b ) can be derived more readily with knowledge of the form of the adjoint function λ determined via non‐shape variations. The methods developed in the paper are applied to a problem in pressure die casting, where the objective is the determination of cooling channel shapes for optimum cooling. The results of the method are shown to be highly convergent. Copyright © 2002 John Wiley & Sons, Ltd.     

Automated structural optimization system for integrated topology and shape optimization

Abstract

This paper combines previously developed techniques for image‐preprocessing and characteristic image‐interpreting together with a newly proposed automated shape‐optimization modeling technique into an integrated topology‐optimization and shape‐optimization system. As a result, structure designers are provided with an efficient and reliable automated structural optimization system (ASOS). The automated shape‐optimization modeling technique, the key technique in ASOS, uses hole‐expanding strategy, interference analysis, and hole shape‐adjusting strategy to automatically define the design variables and side constraints needed for shape optimization. This technique not only eliminates the need to manually define design variables and side constraints for shape optimization, but during the process of shape optimization also prevents interference between the interior holes and the exterior boundary. The ASOS is tested in three different structural configuration design examples.     

Stresses,stress sensitivities and shape optimization in two-dimensional linear elasticity by the Boundary Contour Method

This paper presents new formulations for computing stresses as well as their sensitivities in two-dimensional (2-D) linear elasticity by the Boundary Contour Method (BCM). Contrary to previous work (e.g. Reference 1), the formulations presented here are established directly from the boundary contour version of the Hypersingular Boundary Integral Equation (HBIE) which can provide accurate numerical results and is very efficient with regard to numerical implementation as well as computational time. The Design Sensitivity Coefficients (DSCs) computed from the above formulations are then coupled with a mathematical programming method, here the Successive Quadratic Programming (SQP) algorithm, in order to solve shape optimization problems. Numerical examples are presented to demonstrate the validity of the new formulations for calculation of DSCs. Also, based on these formulations, shape optimization examples by the BCM are presented here for the first time. © 1998 John Wiley & Sons, Ltd.     

Shape optimization with topological changes and parametric control

Recent advances in shape optimization rely on free-form implicit representations, such as level sets, to support boundary deformations and topological changes. By contrast, parametric shape optimization is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. We propose a novel approach to shape optimization that combines and retains the advantages of the earlier optimization techniques. The shapes in the design space are represented implicitly as level sets of a higher-dimensional function that is constructed using B-splines (to allow free-form deformations), and parameterized primitives combined with R-functions (to support desired parametric changes). Our approach to shape design and optimization offers great flexibility because it provides explicit parametric control of geometry and topology within a large space of free-form shapes. The resulting method is also general in that it subsumes most other types of shape optimization as special cases. We describe an implementation of the proposed technique with attractive numerical properties. The explicit construction of an implicit representation supports straightforward sensitivity analysis that can be used with most gradient-based optimization methods. Furthermore, our implementation does not require any error-prone polygonization or approximation of level sets (isocurves and isosurfaces). The effectiveness of the method is demonstrated by several numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

An alternating optimization approach for mixed discrete non-linear programming

This article contributes to the development of the field of alternating optimization (AO) and general mixed discrete non-linear programming (MDNLP) by introducing a new decomposition algorithm (AO-MDNLP) based on the augmented Lagrangian multipliers method. In the proposed algorithm, an iterative solution strategy is proposed by transforming the constrained MDNLP problem into two unconstrained components or units; one solving for the discrete variables, and another for the continuous ones. Each unit focuses on minimizing a different set of variables while the other type is frozen. During optimizing each unit, the penalty parameters and multipliers are consecutively updated until the solution moves towards the feasible region. The two units take turns in evolving independently for a small number of cycles. The validity, robustness and effectiveness of the proposed algorithm are exemplified through some well known benchmark mixed discrete optimization problems.     

A continuum approach for second-order shape design sensitivity of elastic solids with loaded boundaries

A second-order shape design sensitivity analysis (DSA) method applicable to the shape change on the loaded boundaries is derived for three-dimensional linear elastic solids using a continuum method with the material derivative. The continuum method is also used to derive mixed second-order variations of stress and displacement performance measures with respect to shape design variables and distribution of non-conservative traction loads, and also with respect to shape design variables and material properties. A shape design acceleration field is defined for the second-order shape design sensitivity. Both direct differentiation and hybrid methods are presented in this paper. A numerical method, which can be implemented using established finite element analysis (FEA) codes, is developed. The feasibility and accuracy of the proposed second-order shape DSA method has been demonstrated by solving a structural example-doubly curved arch dam.     

Certified real-time shape optimization using isogeometric analysis,PGD model reduction,and a posteriori error estimation

In this paper, we address the effective and accurate solution of problems with parameterized geometry. Considering the attractive framework of isogeometric analysis, which enables a natural and flexible link between computer-aided design and simulation tools, the parameterization of the geometry is defined on the mapping from the isogeometric analysis parametric space to the physical space. From the subsequent multidimensional problem, model reduction based on the proper generalized decomposition technique with off-line/online steps is introduced in order to describe the resulting manifold of parametric solutions with reduced CPU cost. Eventually, a posteriori estimation of various error sources inheriting from discretization and model reduction is performed in order to control the quality of the approximate solution, for any geometry, and feed a robust adaptive algorithm that optimizes the computational effort for prescribed accuracy. The overall approach thus constitutes an effective and reliable numerical tool for shape optimization analyses. Its performance is illustrated on several two- and three-dimensional numerical experiments.