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.     

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.     

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.     

A SIMPLEX-BASED APPROACH TO A CLASS OF PROBLEMS ASSOCIATED WITH TRUSS DESIGN

Determination of the cross-sectional areas of the members of a structural truss in order to minimize structural weight, subject to performance constraints, gives rise to a discrete non-linear mathematical programming problem. A Lagrangian scheme for the solution of this type of problem is described and an effective simplex-based method developed for solving the Lagrangian dual subproblems that thus arise. Structure inherent in the problem permits the use of simplex tableaux which are very much reduced in size, thereby enabling subproblems to be solved quickly. Moreover, the ideas developed here have potential for more general application.     

Lower bound limit analysis of cohesive‐frictional materials using second‐order cone programming

The formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second‐order cone programming (SOCP), for which efficient primal‐dual interior‐point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr–Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker–Prager criterion in 2D or 3D, and Nielsen's criterion for plates). Copyright © 2005 John Wiley & Sons, Ltd.     

An engineering method for complex structural optimization involving both size and topology design variables

This work presents an engineering method for optimizing structures made of bars, beams, plates, or a combination of those components. Corresponding problems involve both continuous (size) and discrete (topology) variables. Using a branched multipoint approximate function, which involves such mixed variables, a series of sequential approximate problems are constructed to make the primal problem explicit. To solve the approximate problems, genetic algorithm (GA) is utilized to optimize discrete variables, and when calculating individual fitness values in GA, a second-level approximate problem only involving retained continuous variables is built to optimize continuous variables. The solution to the second-level approximate problem can be easily obtained with dual methods. Structural analyses are only needed before improving the branched approximate functions in the iteration cycles. The method aims at optimal design of discrete structures consisting of bars, beams, plates, or other components. Numerical examples are given to illustrate its effectiveness, including frame topology optimization, layout optimization of stiffeners modeled with beams or shells, concurrent layout optimization of beam and shell components, and an application in a microsatellite structure. Optimization results show that the number of structural analyses is dramatically decreased when compared with pure GA while even comparable to pure sizing optimization.     

Geometry and topology optimization of sheet metal profiles by using a branch‐and‐bound framework

In this paper well established procedures from partial differential equation (PDE)‐constrained and discrete optimization are combined in a new way to find an optimal design of a multi‐chambered profile. Given a starting profile design, a load case and corresponding design constraints (e.g. sheet thickness, chamber sizes), the aim is to find an optimal subdivision into a predefined number of chambers with optimal shape subject to structural stiffness. In the presented optimization scheme a branch‐and‐bound tree is generated with one additional chamber in each level. Before adding the next chamber, the geometry of the profile is optimized. Then a relaxation of a topology optimization problem is solved. Based on this relaxation, a best fitting feasible topology subject to manufacturability conditions is determined using a new mixed integer method employing shortest paths. To improve the running time, the finite element simulations for the geometry optimization and topology relaxation are performed with different levels of accuracy. Finally, numerical experiments are presented including different starting geometries, load scenarios and mesh sizes.     

Topology optimization with multiple materials via moving morphable component (MMC) method

With the fast development of additive manufacturing technology, topology optimization involving multiple materials has received ever increasing attention. Traditionally, this kind of optimization problem is solved within the implicit solution framework by using the Solid Isotropic Material with Penalization or level set method. This treatment, however, will inevitably lead to a large number of design variables especially when many types of materials are involved and 3‐dimensional (3D) problems are considered. This is because for each type of material, a corresponding density field/level function defined on the entire design domain must be introduced to describe its distribution. In the present paper, a novel approach for topology optimization with multiple materials is established based on the Moving Morphable Component framework. With use of this approach, topology optimization problems with multiple materials can be solved with much less numbers of design variables and degrees of freedom. Numerical examples provided demonstrate the effectiveness of the proposed approach.     

Mesh adaptive computation of upper and lower bounds in limit analysis

An efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented, with a formulation that explicitly considers the exact convex yield condition. The approach consists of two main steps. First, the continuous problem, under the form of the static principle of limit analysis, is discretized twice (one per bound) using particularly chosen finite element spaces for the stresses and velocities that guarantee the attainment of an upper or a lower bound. The second step consists of solving the resulting discrete non‐linear optimization problems. These are reformulated as second‐order cone programs, which allows for the use of primal–dual interior point methods that optimally exploit the convexity and duality properties of the limit analysis model. To benefit from the fact that collapse mechanisms are typically highly localized, a novel method for adaptive meshing is introduced. The method first decomposes the total bound gap as the sum of positive contributions from each element in the mesh and then refines those elements with higher contributions. The efficiency of the methodology is illustrated with applications in plane stress and plane strain problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Level set topology optimization of structural problems with interface cohesion

This paper presents a finite element topology optimization framework for the design of two‐phase structural systems considering contact and cohesion phenomena along the interface. The geometry of the material interface is described by an explicit level set method, and the structural response is predicted by the extended finite element method. In this work, the interface condition is described by a bilinear cohesive zone model on the basis of the traction‐separation constitutive relation. The non‐penetration condition in the presence of compressive interface forces is enforced by a stabilized Lagrange multiplier method. The mechanical model assumes a linear elastic isotropic material, infinitesimal strain theory, and a quasi‐static response. The optimization problem is solved by a nonlinear programming method, and the design sensitivities are computed by the adjoint method. The performance of the presented method is evaluated by 2D and 3D numerical examples. The results obtained from topology optimization reveal distinct design characteristics for the various interface phenomena considered. In addition, 3D examples demonstrate optimal geometries that cannot be fully captured by reduced dimensionality. The optimization framework presented is limited to two‐phase structural systems where the material interface is coincident in the undeformed configuration, and to structural responses that remain valid considering small strain kinematics. Copyright © 2017 John Wiley & Sons, Ltd.     

A mixed FEM approach to stress‐constrained topology optimization

We present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion. The main idea is to adopt a mixed finite‐element discretization scheme wherein not only displacements (as usual) but also stresses are the variables entering the formulation. By doing so, any stress constraint may be handled within the optimization procedure without resorting to post‐processing operation typical of displacement‐based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed. Two dual variational principles of Hellinger–Reissner type are presented in continuous and discrete form that, which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes (Int. J. Numer. Meth. Engng. 1984; 24 (3):359–373). Extensive numerical simulations are performed and ongoing extensions outlined, including the optimization of elastoplastic and incompressible media. Copyright © 2007 John Wiley & Sons, Ltd.     

Topology design of structures using a dual algorithm and a constraint on the perimeter

Dual optimization algorithms are well suited for the topology design of continuum structures in discrete variables, since in these problems the number of constraints is small in comparison to the number of design variables. The ‘raw’ dual algorithm, which was originally proposed for the minimum compliance design problem, worked well when a perimeter constraint was added in addition to the volume constraint. However, if the perimeter constraint was gradually relaxed by increasing the upper bound on the allowable perimeter, the algorithm tended to behave erratically. Recently, a simple strategy has been suggested which modifies the raw dual algorithm to make it more robust in the absence of the perimeter constraint; in particular the problem of checkerboarding which is frequently observed with the use of lower‐order finite elements is eliminated. In this work, we show how the perimeter constraint can be incorporated in this improved algorithm, so that it not only provides a designer with a control over the topology, but also generates good topologies irrespective of the value of the upper bound on the perimeter. Copyright © 2002 John Wiley & Sons, Ltd.     

A TWO-LEVEL DECOMPOSITION METHOD FOR SHAPE OPTIMIZATION OF STRUCTURES

The paper proposes a two-level decomposition method for shape optimization of structures. The optimization problem is divided into two subproblems on the basis of the different effects on structural behaviour of different kinds of design variables. A minimum mass subproblem is solved to determine the sizing variables and a constraint evaluation function based on norm optimization is minimized to determine the shape variables. An efficient coupling technique is used between the subproblems to ensure very rapid and steady convergence. Numerical results are presented to demonstrate the effectiveness of the method. © 1997 by John Wiley & Sons, Ltd.     

A consistent grayscale‐free topology optimization method using the level‐set method and zero‐level boundary tracking mesh

This paper proposes a level‐set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming. Because the boundary tracking mesh is always conformed to the structural boundary, good approximation to the boundary is maintained during optimization; therefore, structural design problems are solved completely without grayscale material. Previously, we introduced the boundary tracking mesh generating method into level‐set based topology optimization and updated the design variables by solving the level‐set equation. In order to adapt our previous method to general structural optimization frameworks, the incorporation of the method with nonlinear programming is investigated in this paper. To successfully incorporate nonlinear programming, the optimization problem is regularized using a double‐well potential. Furthermore, the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming. We expect the investigation to open up a new class of grayscale‐free topology optimization. The usefulness of the proposed method is demonstrated using several numerical examples targeting two‐dimensional compliant mechanism and metallic waveguide design problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Topology optimization of periodic cellular solids based on a superelement method

It is well known that the structural performance of lightweight cellular solids depends greatly on the design of the representative volume element (RVE). In this article, an integrated topology optimization procedure is developed for the global stiffness maximization of 2D periodic and cyclic-symmetry cellular solids. A design variable linking technique and a superelement method are applied to model the structural periodicity and to reduce the computing time. In order to prevent the numerical instabilities associated with checkerboards in the design process, the quadratic perimeter constraint is used. Finally, the topology optimization problem is solved by the dual optimization algorithm. Several numerical examples are used to test the efficiency of the optimization procedure. Results show that the optimal topology of the RVE is not unique. It greatly depends on the size of the RVE. The computing efficiency can be greatly improved by means of the superelement technique. Also, for the optimal solution, the equivalent torsional rigidity has been compared with what is in the literature, to check the structural efficiency of the obtained topology. It has been observed that the current topology solution has the strongest rigidity when the same volume fraction of solid-phase materials is used.     

An enhanced genetic algorithm for structural topology optimization

Genetic algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced GA is proposed for structural topology optimization by using problem‐specific knowledge. The original discrete black‐and‐white (0–1) problem is directly solved by using a bit‐array representation method. To address the related pronounced connectivity issue effectively, the four‐neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well‐posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge‐free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM‐based function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near‐optimum solutions and viable topologies with checkerboard free, mesh independent and hinge‐free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problem‐specific knowledge can be a powerful global search tool for structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Minimum cost design of framed structures by the mini-max dual method

Minimum cost design of a framed structure is considered by using the mini-mas dual method. Stress and/or displacement constraints are imposed as behavioural constraints. The minimum cost design problem has a discrete objective function and discrete design variables. A sequence of approximate optimization problems in created by using the first-order Taylor series expansion for displacements with respect to the reciprocals of cross-sectional areas and moments of inertia. Each approximate problem is solved in the dual space. Two simple structural examples are given to show the appropriateness and efficiency of the proposed procedure. Approximate solutions are obtained within five structural analysis.     

Structural optimization: A new dual method using mixed variables

A new and powerful mathematical programming method is described, which is capable of solving a broad class of structural optimization problems. The method employs mixed direct/reciprocal design variables in order to get conservative, first-order approximations to the objective function and to the constraints. By this approach the primary optimization problem is replaced with a sequence of explicit subproblems. Each subproblem being convex and separable, it can be efficiently solved by using a dual formulation. An attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs. Examples of application to real-life aerospace structures are offered to demonstrate the power and generality of the approach presented.     

A Combined Convex Approximation—Interior Point Approach for Large Scale Nonlinear Programming

The method of moving asymptotes (MMA) and its globally convergent extension SCP (sequential convex programming) are known to work well in the context of structural optimization. The two main reasons are that the approximation scheme used for the objective function and the constraints fits very well to these applications and that at an iteration point a local optimization model is used such that additional expensive function and gradient evaluations of the original problem are avoided. The subproblems that occur in both methods are special nonlinear convex programs and have traditionally been solved using a dual approach. This is now replaced by an interior point approach. The latter one is more suitable for large problems because sparsity properties of the original problem can be preserved and the separability property of the approximation functions is exploited. The effectiveness of the new method is demonstrated by a few examples dealing with problems of structural optimization.