A review of homogenization and topology optimization III-topology optimization using optimality criteria

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.     

Fuzzy tolerance multilevel approach for structural topology optimization

This paper presents a novel methodology, fuzzy tolerance multilevel programming approach, for applying fuzzy set theory and sequence multilevel method to multi-objective topology optimization problems of continuum structures undergoing multiple loading cases. Ridge-type nonlinear membership functions in fuzzy set theory are applied to embody fuzzy and uncertain characteristics essentially involved by the objective and constraint functions. Sequence multilevel method is used to characterize the different priorities of loading cases at different levels making contribution to the final optimum solution, which is practically beneficial to reduce the subjective influence transferred by using weighted approaches. The solid isotropic material with penalization (SIMP) is adopted as the density-stiffness interpolation scheme to relax the original optimization problem and indicate the dependence of material properties with element pseudo-densities. Sequential linear programming (SLP) is used as the optimizer to solve the multi-objective optimization problem formulated using fuzzy tolerance multilevel programming scheme. Numerical instabilities, such as checkerboards and mesh dependencies are summarized and a duplicate sensitivity filtering method, in favor of contributing to the mesh-dependent optimum designs, is subsequently proposed to regularize the singularity of the optimization problem. The validation of the methodologies presented in this work has been demonstrated by detailed examples of numerical applications.     

A method for system identification using the optimality criteria optimization approach

An algorithm similar to the optimality criteria approach used in structural optimization is presented for identifying stiffnesses of structural members by using vibration test data. A set of equivalent static inertia forces are obtained from the vibration analysis using d'Alembert's principle and are used to solve the multiple displacement constraint problem. The displacement constraint values are specified based on the measured experimental modal displacement data at critical locations. The algorithm is used to find the changes needed in the stiffnesses of the elements and the distribution of nonstructural mass of the nominal analytical model to correlate the analytical and experimental data. The algorithm alternates between the vibration analysis and static analysis to find the equivalent load vector and modify the stiffnesses. The identified stiffness properties of the structural elements can be used to control and study the dynamic response of the structure.     

A pseudo-sensitivity based discrete-variable approach to structural topology optimization with multiple materials

An algorithm has been developed which uses material as a discrete variable in multi-material topology optimization and thus provides an alternative to traditional methods using material interpolation and level-set approaches. The algorithm computes ‘pseudo-sensitivities’ of the objective and constraint functions to discrete material changes. These are used to rank elements, based on which a fraction of elements are selected for material ID modification during the optimization iteration. The algorithm is of general applicability and avoids frequent matrix factorizations so that it is applicable to large structural problems. In addition to the conventionally used evolutionary and morphogenesis approaches for iteration, a new iteration scheme of ‘resubstitution’ which combines the two approaches is presented. The application and functioning of the algorithm is demonstrated through case studies and comparisons with a few benchmark problems, showing its capability in providing mass-optimal topologies under stiffness constraints for various structural problems where multiple materials are considered.     

Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques

Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.     

Traditional vs. extended optimality in topology optimization

Owing to its implications with respect to a critical examination of the SIMP and ESO methods in a Forum Article, extended optimality in topology optimization is revisited, with a view to clarifying certain issues and to illustrate this concept with a case study. It is concluded that extended optimality can result in a much lower structural volume than traditional optimality.     

An efficient optimality criteria approach to the minimum weight design of elastic structures

This paper is concerned with the optimality criteria approach to the minimum weight design of elastic structures analyzed by finite elements. It is first shown that the classical methods apply the lagrangian multiplier technique to an explicit problem. This one results from high quality, first order approximations of the displacement constraints and cruder, zero order approximations of the stress constraints. A generalized optimality criterion is then proposed as the explicit Kuhn-Tucker conditions of a first order approximate problem. Hence a hybrid optimality criterion is developed by using both zero and first order approximations of the stress constraints, according to their criticality. Efficient solution algorithms of the explicit approximate problem are suggested. Its dual statement generalizes the classical lagrangian approaches. Its primal statement leads to a rigorous definition of the optimality criteria approach, which appears to be closely related to the linearization methods of mathematical programming. Finally some numerical applications clearly illustrate the efficiency of the generalized and hybrid optimality criteria.     

Stochastic structural topology optimization: discretization and penalty function approach

We consider structural topology optimization problems, including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or non-compression conditions for elastic ropes. To construct more realistic models and to circumvent possible failures or inefficient behaviour of optimal structures, we allow parameters (for example, loads) defining the problem to be stochastic. The resulting non-smooth stochastic optimization problem is an instance of stochastic mathematical programs with equilibrium constraints (MPEC), or stochastic bilevel programs. We propose a solution scheme based first on the approximation of the given topology optimization problem by a sequence of simpler sizing optimization problems, and second on approximating the probability measure in the latter problems. For stress-constrained weight-minimization problems, an alternative to -perturbation based on a new penalty function is proposed.     

An efficient second-order SQP method for structural topology optimization

This article presents a Sequential Quadratic Programming (SQP) solver for structural topology optimization problems named TopSQP. The implementation is based on the general SQP method proposed in Morales et al. J Numer Anal 32(2):553–579 (2010) called SQP+. The topology optimization problem is modelled using a density approach and thus, is classified as a nonconvex problem. More specifically, the SQP method is designed for the classical minimum compliance problem with a constraint on the volume of the structure. The sub-problems are defined using second-order information. They are reformulated using the specific mathematical properties of the problem to significantly improve the efficiency of the solver. The performance of the TopSQP solver is compared to the special-purpose structural optimization method, the Globally Convergent Method of Moving Asymptotes (GCMMA) and the two general nonlinear solvers IPOPT and SNOPT. Numerical experiments on a large set of benchmark problems show good performance of TopSQP in terms of number of function evaluations. In addition, the use of second-order information helps to decrease the objective function value.     

An approach of topology optimization of multi-rigid-body mechanism

Topology synthesis of multi-rigid-body mechanisms has always been a very important stage in the mechanism design process. In most cases, the topology of the multi-rigid-body mechanism for particular task is obtained by designers’ experience and ingenuity, rather than automatic approach. In this work, an approach of topology optimization of multi-rigid-body mechanisms is investigated. The core process of the approach is an automatic optimization design process. In this approach, we construct kinematics mapping from truss structures to the joint-linked mechanisms, which transforms the topology optimization problem of multi-body system into the truss structure optimization problem. We also develop a new strategy for topology optimization of statically determinate truss, the advantage of which lies in the ability dealing with statically determinate truss topology optimization problem compared to the existing methods. By automatically optimizing the topology of the truss structure, the topology of the multi-rigid-body mechanism is optimized automatically, accordingly. Here, we utilize the investigated approach to design suitable layout for multi-rigid-body micro-displacement amplifying mechanisms (MMAMs) with a large amplification ratio (>50). The layout consists of not only the topology information of the mechanism, but also the dimension parameters of the mechanism. The procedure of the approach is carried out in steps, and a human–computer interaction program has been developed for it. Using the developed program, different MMAMs are achieved. Meanwhile, the direct kinematics analysis of the MMAMs is achieved automatically, the existence of dead point position in the mechanism within movement range is checked and the micro-displacement amplification ratio is calculated out. The computing results are validated by the ADAMS^® motion simulation, which proves that the achieved MMAMs fully fulfill the functional requirement. Along with two of the achieved MMAMs, the approach is explained, its functionality is shown, its advantages, limitations, some open problems and future works are discussed.     

An engineering approach: Hierarchical optimization criteria

An area of difficulty in the effective application of modern optimization techniques is often the choice of a performance criterion which adequately reflects the various factors of importance in the proper proportions. This short paper presents an approach appropriate to problems in which two or more performance measures can be ranked as "most important," "next most important," etc. It involves the successive application of the performance measures, with the constraint set at a given stage in the process being chosen on the basis of the results obtained in the previous stages. The resulting optimal control may be a much better compromise choice between the competing criteria than might be obtained by blind reliance on some preselected criterion.     

An integrated approach to topology,sizing, and shape optimization

Topology optimization has become very popular in industrial applications, and most FEM codes have implemented certain capabilities of topology optimization. However, most codes do not allow simultaneous treatment of sizing and shape optimization during the topology optimization phase. This poses a limitation on the design space and therefore prevents finding possible better designs since the interaction of sizing and shape variables with topology modification is excluded. In this paper, an integrated approach is developed to provide the user with the freedom of combining sizing, shape, and topology optimization in a single process.     

Benchmarking optimization solvers for structural topology optimization

The purpose of this article is to benchmark different optimization solvers when applied to various finite element based structural topology optimization problems. An extensive and representative library of minimum compliance, minimum volume, and mechanism design problem instances for different sizes is developed for this benchmarking. The problems are based on a material interpolation scheme combined with a density filter. Different optimization solvers including Optimality Criteria (OC), the Method of Moving Asymptotes (MMA) and its globally convergent version GCMMA, the interior point solvers in IPOPT and FMINCON, and the sequential quadratic programming method in SNOPT, are benchmarked on the library using performance profiles. Whenever possible the methods are applied to both the nested and the Simultaneous Analysis and Design (SAND) formulations of the problem. The performance profiles conclude that general solvers are as efficient and reliable as classical structural topology optimization solvers. Moreover, the use of the exact Hessians in SAND formulations, generally produce designs with better objective function values. However, with the benchmarked implementations solving SAND formulations consumes more computational time than solving the corresponding nested formulations.     

An efficient approach to reliability-based topology optimization for continua under material uncertainty

This contribution presents a computationally efficient method for reliability-based topology optimization for continuum domains under material properties uncertainty. Material Young’s modulus is assumed to be lognormally distributed and correlated within the domain. The computational efficiency is achieved through estimating the response statistics with stochastic perturbation of second order, using these statistics to fit an appropriate distribution that follows the empirical distribution of the response, and employing an efficient gradient-based optimizer. Two widely-studied topology optimization problems are examined and the changes in the optimized topology is discussed for various levels of target reliability and correlation strength. Accuracy of the proposed algorithm is verified using Monte Carlo simulation.     

Simulation approach to structural optimization

A unified approach to various problems of structural optimization is presented. It is based on a combination of mathematical models of different complexity. The models describe the behaviour of a designed structure. From the computational point of view, it is connected with the sequential approximation of design problem constraints and/or an objective function. In each step, a subregion of the initial search region in the space of design variables is chosen. In this subregion, various points (designs) are selected, for which response analyses are carried out using a numerical method (mostly FEM). Using the least-squares method, analytical expressions are formulated, which then replace the initial problem functions. They are used as functions of a particular mathematical programming problem. The size and location of sequential subregions may be changed according to the result of the search. The choice of one particular form of the analytical expressions is described. The application of the approach is shown by means of test examples and comparison with other optimization techniques is presented.Visiting scientist in the Department of Solid Mechanics, The Technical University of Denmark, Lyngby, Denmark, September 1987 – August 1988.     

A density-and-strain-based K-clustering approach to microstructural topology optimization

Structural and Multidisciplinary Optimization - Microstructural topology optimization (MTO) is the simultaneous optimization of macroscale topology and microscale structure. MTO holds the promise...     

POD-assisted strategies for structural topology optimization

We propose a new numerical tool for structural optimization design. To cut down the computational burden typical of the Solid Isotropic Material with Penalization (SIMP) method, we apply Proper Orthogonal Decomposition on SIMP snapshots computed on a fixed grid to construct a rough structure (predictor) which becomes the input of a SIMP procedure performed on an anisotropic adapted mesh (corrector). The benefit of the proposed design tool is to deliver smooth and sharp layouts which require a contained computational effort before moving to the 3D printing production phase.     

An optimality criteria based method for discrete design optimization taking into account buildability constraints

This paper presents a heuristic design optimization method specifically developed for practicing structural engineers. Practical design optimization problems are often governed by buildability constraints. The majority of optimization methods that have recently been proposed for design optimization under buildability constraints are based on evolutionary computing. While these methods are generally easy to implement, they require a large number of function evaluations (finite element analyses), and they involve algorithmic parameters that require careful tuning. As a consequence, both the computation time and the engineering time are high. The discrete design optimization algorithm presented in this paper is based on the optimality criteria method for continuous optimization. It is faster than an evolutionary algorithm and it is free of tuning parameters. The algorithm is successfully applied to two classical benchmark problems (the design of a ten-bar truss and an eight-story frame) and to a practical truss design optimization problem.