On the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem

We study the ‘classical’ topology optimization problem, in which minimum compliance is sought, subject to linear constraints. Using a dual statement, we propose two separable and strictly convex subproblems for use in sequential approximate optimization (SAO) algorithms. Respectively, the subproblems use reciprocal and exponential intermediate variables in approximating the non‐linear compliance objective function. Any number of linear constraints (or linearly approximated constraints) are provided for. The relationships between the primal variables and the dual variables are found in analytical form. For the special case when only a single linear constraint on volume is present, we note that application of the ever‐popular optimality criterion (OC) method to the topology optimization problem, combined with arbitrary values for the heuristic numerical damping factor η proposed by Bendsøe, results in an updating scheme for the design variables that is identical to the application of a rudimentary dual SAO algorithm, in which the subproblems are based on exponential intermediate variables. What is more, we show that the popular choice for the damping factor η=0.5 is identical to the use of SAO with reciprocal intervening variables. Finally, computational experiments reveal that subproblems based on exponential intervening variables result in improved efficiency and accuracy, when compared to SAO subproblems based on reciprocal intermediate variables (and hence, the heuristic topology OC method hitherto used). This is attributed to the fact that a different exponent is computed for each design variable in the two‐point exponential approximation we have used, using gradient information at the previously visited point. Copyright © 2007 John Wiley & Sons, Ltd.     

Non‐convex dual forms based on exponential intervening variables,with application to weight minimization

We study the weight minimization problem in a dual setting. We propose new dual formulations for non‐linear multipoint approximations with diagonal approximate Hessian matrices, which derive from separable series expansions in terms of exponential intervening variables. These, generally, nonconvex approximations are formulated in terms of intervening variables with negative exponents, and are therefore applicable to the solution of the weight minimization problem in a sequential approximate optimization (SAO) framework. Problems in structural optimization are traditionally solved using SAO algorithms, like the method of moving asymptotes, which require the approximate subproblems to be strictly convex. Hence, during solution, the nonconvex problems are approximated using convex functions, and this process may in general be inefficient. We argue, based on Falk's definition of the dual, that it is possible to base the dual formulation on nonconvex approximations. To this end we reintroduce a nonconvex approach to the weight minimization problem originally due to Fleury, and we explore certain convex and nonconvex forms for subproblems derived from the exponential approximations by the application of various methods of mixed variables. We show in each case that the dual is well defined for the form concerned, which may consequently be of use to the future code developers. Copyright © 2009 John Wiley & Sons, Ltd.     

Multipoint approximation development: thermal structural optimization case study

Function approximation is one of the most important fields of research in design optimization. Accurate function approximation reduces the repetitive cost of finite element analysis. Many local approximations, such as one‐ and two‐point local approximations, are already available. These are, however, only valid in a small domain and require stringent move‐limits on design variables. The objective of this research is to achieve an efficient and accurate multipoint approximation to constraints by integrating the most accurate segment of all local approximations previously constructed. The proposed multipoint approximation is constructed by combining weighting functions with local function approximations. With function and gradient information at a series of points, local approximations are established at those points. Once established, all local approximations are blended into a multipoint approximation by use of a weighting function. Function and gradient values of this multipoint approximation correspond directly with exact counterparts at the points where the local approximations were generated. Finally, the multipoint approximation is applied to a plate and wing box thermal‐structural optimization. Published in 2000 by John Wiley & Sons, Ltd.     

A novel minimum weight formulation of topology optimization implemented with reanalysis approach

In this paper, we develop an efficient diagonal quadratic optimization formulation for minimum weight design problem subject to multiple constraints. A high-efficiency computational approach of topology optimization is implemented within the framework of approximate reanalysis. The key point of the formulation is the introduction of the reciprocal-type variables. The topology optimization seeking for minimum weight can be transformed as a sequence of quadratic program with separable and strictly positive definite Hessian matrix, thus can be solved by a sequential quadratic programming approach. A modified sensitivity filtering scheme is suggested to remove undesirable checkerboard patterns and mesh dependence. Several typical examples are provided to validate the presented approach. It is observed that the optimized structure can achieve lighter weight than those from the established method by the demonstrative numerical test. Considerable computational savings can be achieved without loss of accuracy of the final design for 3D structure. Moreover, the effects of multiple constraints and upper bound of the allowable compliance upon the optimized designs are investigated by numerical examples.     

Hybrid approximation algorithm with Kriging and quadratic polynomial‐based approach for approximate optimization

This paper describes a new hybrid algorithm that uses a Kriging and quadratic polynomial‐based approach for approximate optimization. The Kriging method is used for generating a global approximation model, and the polynomial‐based approximation method is used for generating a local approximation model. The Kriging system is only used to construct a polynomial‐based locally approximate model by estimating some function values and Hessian components of an estimated surface. The number of Kriging estimations can be reduced in comparison with direct Kriging‐based optimization, and a local optimum solution on an approximated surface can be clearly estimated without use of an optimization procedure based on a local appropriate quadratic polynomial model. Numerical examples of engineering optimization using the proposed method illustrate validity and effectiveness of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

Structural optimization of uncertain dynamical systems considering mixed-design variables

In this paper attention is directed to the reliability-based optimization of uncertain structural systems under stochastic excitation involving discrete-continuous sizing type of design variables. The reliability-based optimization problem is formulated as the minimization of an objective function subject to multiple reliability constraints. The probability that design conditions are satisfied within a given time interval is used as a measure of system reliability. The problem is solved by a sequential approximate optimization strategy cast into the framework of conservative convex and separable approximations. To this end, the objective function and the reliability constraints are approximated by using a hybrid form of linear, reciprocal and quadratic approximations. The approximations are combined with an effective sensitivity analysis of the reliability constraints in order to generate explicit expressions of the constraints in terms of the design variables. The explicit approximate sub-optimization problems are solved by an appropriate discrete optimization technique. The optimization scheme exhibits monotonic convergence properties. Two numerical examples showing the effectiveness of the approach reported herein are presented.     

Parametric topology optimization with multiresolution finite element models

We present a methodical procedure for topology optimization under uncertainty with multiresolution finite element (FE) models. We use our framework in a bifidelity setting where a coarse and a fine mesh corresponding to low- and high-resolution models are available. The inexpensive low-resolution model is used to explore the parameter space and approximate the parameterized high-resolution model and its sensitivity, where parameters are considered in both structural load and stiffness. We provide error bounds for bifidelity FE approximations and their sensitivities and conduct numerical studies to verify these theoretical estimates. We demonstrate our approach on benchmark compliance minimization problems, where we show significant reduction in computational cost for expensive problems such as topology optimization under manufacturing variability, reliability-based topology optimization, and three-dimensional topology optimization while generating almost identical designs to those obtained with a single-resolution mesh. We also compute the parametric von Mises stress for the generated designs via our bifidelity FE approximation and compare them with standard Monte Carlo simulations. The implementation of our algorithm, which extends the well-known 88-line topology optimization code in MATLAB, is provided.     

Uniform luminance design of the direct‐type backlight unit of liquid crystal displays using neural networks

Abstract

Light efficiency optimization of a direct‐type backlight unit (BLU) of the liquid crystal display (LCD) is considered in this paper. The purpose of this research is to obtain the greatest uniformity in a direct‐type BLU, while the brightness is maintained in a satisfactory level.

It is difficult to find a numerical optimization algorithm that is readily applicable to this problem because of the existence of discrete variables, implicit objective functions and constraints, or even binary constraints. The sequential neural‐network approximation (SNA) method is designed specifically for this type of engineering design optimization problem. In this paper, a two variable LCD BLU design example is presented first to illustrate the design process using the SNA method. Then a four variable LCD BLU design optimization problem is solved.     

APPROXIMATE DISCRETE VARIABLE OPTIMIZATION OF FRAME STRUCTURES WITH DUAL METHODS

The purpose of this work is to present an efficient method for optimum design of frame structures, using approximation concepts. A dual strategy in which the design variables can be considered as discrete variables is used. A two-level approximation concept is used. In the first level, all the structural response quantities such as forces and displacements are approximated as functions of some intermediate variables. Then the second level approximation is employed to convert the first-level approximation problem into a series of problems of separable forms, which can be solved easily by dual methods with discrete variables. In the second-level approximation, the objective function and the approximate constraints are linearized. The objective of the first-level approximation is to reduce the number of structural analyses required in the optimization problem and the second level approximation reduces the computational cost of the optimization technique. A portal frame and a single layer grid are used as design examples to demonstrate the efficiency of the proposed method.     

Approximate reanalysis in topology optimization

In the nested approach to structural optimization, most of the computational effort is invested in the solution of finite element analysis equations. In this study, the integration of an approximate reanalysis procedure into the framework of topology optimization of continuum structures is investigated. The nested optimization problem is reformulated to accommodate the use of an approximate displacement vector and the design sensitivities are derived accordingly. It is shown that relatively rough approximations are acceptable since the errors are taken into account in the sensitivity analysis. The implementation is tested on several small and medium scale problems, including 2‐D and 3‐D minimum compliance problems and 2‐D compliant force inverter problems. Accurate results are obtained and the savings in computation time are promising. Copyright © 2008 John Wiley & Sons, Ltd.     

On sequential approximate simultaneous analysis and design in classical topology optimization

We study the simultaneous analysis and design (SAND) formulation of the ‘classical’ topology optimization problem subject to linear constraints on material density variables. Based on a dual method in theory, and a primal‐dual method in practice, we propose a separable and strictly convex quadratic Lagrange–Newton subproblem for use in sequential approximate optimization of the SAND‐formulated classical topology design problem. The SAND problem is characterized by a large number of nonlinear equality constraints (the equations of equilibrium) that are linearized in the approximate convex subproblems. The availability of cheap second‐order information is exploited in a Lagrange–Newton sequential quadratic programming‐like framework. In the spirit of efficient structural optimization methods, the quadratic terms are restricted to the diagonal of the Hessian matrix; the subproblems have minimal storage requirements, are easy to solve, and positive definiteness of the diagonal Hessian matrix is trivially enforced. Theoretical considerations reveal that the dual statement of the proposed subproblem for SAND minimum compliance design agrees with the ever‐popular optimality criterion method – which is a nested analysis and design formulation. This relates, in turn, to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal (and exponential) intervening variables and the optimality criterion method. Copyright © 2016 John Wiley & Sons, Ltd.     

Multifidelity Response Surface Approximations for the Optimum Design of Diffuser Flows

Response surface methods use least-squares regression analysis to fit low-order polynomials to a set of experimental data. It is becoming increasingly more popular to apply response surface approximations for the purpose of engineering design optimization based on computer simulations. However, the substantial expense involved in obtaining enough data to build quadratic response approximations seriously limits the practical size of problems. Multifidelity techniques, which combine cheap low-fidelity analyses with more accurate but expensive high-fidelity solutions, offer means by which the prohibitive computational cost can be reduced. Two optimum design problems are considered, both pertaining to the fluid flow in diffusers. In both cases, the high-fidelity analyses consist of solutions to the full Navier-Stokes equations, whereas the low-fidelity analyses are either simple empirical formulas or flow solutions to the Navier-Stokes equations achieved using coarse computational meshes. The multifidelity strategy includes the construction of two separate response surfaces: a quadratic approximation based on the low-fidelity data, and a linear correction response surface that approximates the ratio of high-and low-fidelity function evaluations. The paper demonstrates that this approach may yield major computational savings.     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

Efficient generation of pareto‐optimal topologies for compliance optimization

In multi‐objective optimization, a design is defined to beit pareto‐optimal if no other design exists that is better with respect to one objective, and as good with respect to other objectives. In this paper, we first show that if a topology is pareto‐optimal, then it must satisfy certain properties associated with the topological sensitivity field, i.e. no further comparison is necessary. This, in turn, leads to a deterministic, i.e. non‐stochastic, method for efficiently generating pareto‐optimal topologies using the classic fixed‐point iteration scheme. The proposed method is illustrated, and compared against SIMP‐based methods, through numerical examples. In this paper, the proposed method of generating pareto‐optimal topologies is limited to bi‐objective optimization, namely compliance–volume and compliance–compliance. The future work will focus on extending the method to non‐compliance and higher dimensional pareto optimization. Copyright © 2011 John Wiley & Sons, Ltd.     

Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm

Transient heat conduction analysis involves extensive computational cost. It becomes more serious for multi-material topology optimization, in which many design variables are involved and hundreds of iterations are usually required for convergence. This article aims to provide an efficient quadratic approximation for multi-material topology optimization of transient heat conduction problems. Reciprocal-type variables, instead of relative densities, are introduced as design variables. The sequential quadratic programming approach with explicit Hessians can be utilized as the optimizer for the computationally demanding optimization problem, by setting up a sequence of quadratic programs, in which the thermal compliance and weight can be explicitly approximated by the first and second order Taylor series expansion in terms of design variables. Numerical examples show clearly that the present approach can achieve better performance in terms of computational efficiency and iteration number than the solid isotropic material with penalization method solved by the commonly used method of moving asymptotes. In addition, a more lightweight design can be achieved by using multi-phase materials for the transient heat conductive problem, which demonstrates the necessity for multi-material topology optimization.     

General topology optimization method with continuous and discrete orientation design using isoparametric projection

A general topology optimization method, which is capable of simultaneous design of density and orientation of anisotropic material, is proposed by introducing orientation design variables in addition to the density design variable. In this work, the Cartesian components of the orientation vector are utilized as the orientation design variables. The proposed method supports continuous orientation design, which is out of the scope of discrete material optimization approaches, as well as design using discrete angle sets. The advantage of this approach is that vector element representation is less likely to fail into local optima because it depends less on designs of former steps, especially compared with using the angle as a design variable (Continuous Fiber Angle Optimization) by providing a flexible path from one angle to another with relaxation of orientation design space. An additional advantage is that it is compatible with various projection or filtering methods such as sensitivity filters and density filters because it is free from unphysical bound or discontinuity such as the one at θ = 2π and θ = 0 seen with direct angle representation. One complication of Cartesian component representation is the point‐wise quadratic bound of the design variables; that is, each pair of element values has to reside in a given circular bound. To overcome this issue, we propose an isoparametric projection method, which transforms box bounds into circular bounds by a coordinate transformation with isoparametric shape functions without having the singular point that is seen at the origin with polar coordinate representation. A new topology optimization method is built by taking advantage of the aforementioned features and modern topology optimization techniques. Several numerical examples are provided to demonstrate its capability. Copyright © 2014 John Wiley & Sons, Ltd.     

Local Pareto approximation for multi-objective optimization

The design process of complex systems often resorts to solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically, this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the objective function space. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution for sensitivity analysis. In this article, new linear and quadratic local approximations of the Pareto surface are derived and compared to existing formulas. The case of non-differentiable Pareto points (solutions) in the objective space is also analysed. The concept of a local quick Pareto analyser based on local sensitivity analysis is proposed. This Pareto analysis provides a quantitative insight into the relation between variations of the different objective functions under constraints. A few examples are considered to illustrate the concept and its advantages.     

Continuous approximation of material distribution for topology optimization

In this paper, we propose a checkerboard‐free topology optimization method without introducing any additional constraint parameter. This aim is accomplished by the introduction of finite element approximation for continuous material distribution in a fixed design domain. That is, the continuous distribution of microstructures, or equivalently design variables, is realized in the whole design domain in the context of the homogenization design method (HDM), by the discretization with finite element interpolations. By virtue of this continuous FE approximation of design variables, discontinuous distribution like checkerboard patterns disappear without any filtering schemes. We call this proposed method the method of continuous approximation of material distribution (CAMD) to emphasize the continuity imposed on the ‘material field’. Two representative numerical examples are presented to demonstrate the capability and the efficiency of the proposed approach against some classes of numerical instabilities. Copyright © 2004 John Wiley & Sons, Ltd.     

A sequential approximation method using neural networks for engineering design optimization problems

There are three characteristics in engineering design optimization problems: (1) the design variables are often discrete physical quantities; (2) the constraint functions often cannot be expressed analytically in terms of design variables; (3) in many engineering design applications, critical constraints are often ‘pass–fail’, ‘0–1’ type binary constraints. This paper presents a sequential approximation method specifically for engineering optimization problems with the three characteristics. In this method a back-propagation neural network is trained to simulate a rough map of the feasible domain formed by the constraints using a few representative training data. A training data point consists of a discrete design point and whether this design point is feasible or infeasible. Function values of the constraints are not required. A search algorithm then searches for the optimal point in the feasible domain simulated by the neural network. This new design point is checked against the true constraints to see whether it is feasible, and is then added to the training set. The neural network is trained again with this added information, in the hope that the network will better simulate the boundary of the feasible domain of the true optimization problem. Then a further search is made for the optimal point in this new approximated feasible domain. This process continues in an iterative manner until the approximate model locates the same optimal point in consecutive iterations. A restart strategy is also employed so that the method may have a better chance to reach a global optimum. Design examples with large discrete design spaces and implicit constraints are solved to demonstrate the practicality of this method.     

Response surface approximations for structural optimization

Response surface methodology can be used to construct global and midrange approximations to functions in structural optimization. Since structural optimization requires expensive function evaluations, it is important to construct accurate function approximations so that rapid convergence may be achieved. In this paper techniques to find the region of interest containing the optimal design, and techniques for finding more accurate approximations are reviewed and investigated. Aspects considered are experimental design techniques, the selection of the ‘best’ regression equation, intermediate response functions and the location and size of the region of interest. Standard examples in structural optimization are used to show that the accuracy is largely dependent on the choice of the approximating function with its associated subregion size, while the selection of a larger number of points is not necessarily cost-effective. In a further attempt to improve efficiency, different regression models were investigated. The results indicate that the use of the two methods investigated does not significantly improve the results. Finding an accurate global approximation is challenging, and sufficient accuracy could only be achieved in the example problems by considering a smaller region of the design space. © 1998 John Wiley & Sons, Ltd.