Variable chromosome length genetic algorithm for progressive refinement in topology optimization

This article introduces variable chromosome lengths (VCL) in the context of a genetic algorithm (GA). This concept is applied to structural topology optimization but is also suitable to a broader class of design problems. In traditional genetic algorithms, the chromosome length is determined a priori when the phenotype is encoded into the corresponding genotype. Subsequently, the chromosome length does not change. This approach does not effectively solve problems with large numbers of design variables in complex design spaces such as those encountered in structural topology optimization. We propose an alternative approach based on a progressive refinement strategy, where a GA starts with a short chromosome and first finds an optimum solution in the simple design space. The optimum solutions are then transferred to the following stages with longer chromosomes, while maintaining diversity in the population. Progressively refined solutions are obtained in subsequent stages. A strain energy filter is used in order to filter out inefficiently used design cells such as protrusions or isolated islands. The variable chromosome length genetic algorithm (VCL-GA) is applied to two structural topology optimization problems: a short cantilever and a bridge problem. The performance of the method is compared to a brute-force approach GA, which operates ab initio at the highest level of resolution.     

ε-relaxed approach in structural topology optimization

This paper presents a so-called -relaxed approach for structural topology optimization problems of discrete structures. The distinctive feature of this new approach is that unlike the typical treatment of topology optimization problems based on the ground structure approach, we eliminate the singular optima from the problem formulation and thus unify the sizing and topology optimization within the same framework. As a result, numerical methods developed for sizing optimization problems can be applied directly to the solution of topology optimization problems without any further treatment. The application of the proposed approach and its effectiveness are illustrated with several numerical examples.     

A method for structural optimization which combines secondorder approximations and dual techniques

A structural optimization problem is usually solved iteratively as a sequence of approximate design problems. Traditionally, a variety of approximation concepts are used, but lately second-order approximation strategies have received most interest since high quality approximations can be obtained in this way. Furthermore, difficulties in choosing tuning parameters such as step-size restrictions may be avoided in these approaches. Methods that utilize second-order approximations can be divided into two groups; in the first, a Quadratic Programming (QP) subproblem including all available second-order information is stated, after which it is solved with a standard QP method, whereas the second approach uses only an approximate QP subproblem whose underlying structure can be efficiently exploited. In the latter case, only the diagonal terms of the second-order information are used, which makes it possible to adopt dual methods that require separability. An advantage of the first group of methods is that all available second-order information is used when stating the approximate problem, but a disadvantage is that a rather difficult QP subproblem must be solved in each iteration. The second group of methods benefits from the possibility of using efficient dual methods, but lacks in not using all available information. In this paper, we propose an efficient approach to solve the QP problems, based on the creation of a sequence of fully separable subproblems, each of which is efficiently solvable by dual methods. This procedure makes it possible to combine the advantages of each of the two former approaches. The numerical results show that the proposed solution procedure is a valid approach to solve the QP subproblems arising in second-order approximation schemes.Presented at NATO ASI Optimization of Large Structural Systems, Berchtesgaden, Germany, Sept. 23 – Oct. 4, 1991     

On the orthogonal similarity transformation (OST)-based sensitivity analysis method for robust topology optimization under loading uncertainty: a mathematical proof and its extension

The main purpose of this work is to provide a mathematical proof of our previously proposed orthogonal similarity transformation (OST)-based sensitivity analysis method (Zhao et al. Struct Multidisc Optim 50(3):517–522 2014a, Comput Methods Appl Mech Engrg 273:204–218 c); the proof is designed to show the method’s computational effectiveness. Theoretical study of computational efficiency for both robust topology optimization and robust concurrent topology optimization problems shows the necessity of the OST-based sensitivity analysis method for practical problems. Numerical studies were conducted to demonstrate the computational accuracy of the OST-based sensitivity analysis method and its efficiency over the conventional method. The research leads us to conclude that the OST-based sensitivity analysis method can bring considerable computational savings when used for large-scale robust topology optimization problems, as well as robust concurrent topology optimization problems.     

An application of relative difference quotient algorithm to topology optimization of truss structures with discrete variables

In this paper, the definitions of cross-sectional variable and topological variable are advanced, and a mathematical model of topology optimization of truss structures with discrete variables including two kinds of variables is developed. The model has considered the coupling relations between cross-sectional variables and topological variables, so that is reflects the innate characteristics of topology optimization as a combinatorial optimization problem. Moreover, problems such as limit stress and singular solution of structural optimization can be overcome by using this model. The model of topology optimization of truss structures with discrete variables including two kinds of variables is solved directly by using the relative difference quotient algorithm. The computational results are satisfactory and some new topologies and better solutions are obtained.     

On reducing computational effort in topology optimization: we can go at least this far!

In this work, we attempt to answer the question posed in Amir O., Sigmund O.: On reducing computational effort in topology optimization: how far can we go? (Struct. Multidiscip. Optim. 44(1):25–29 2011). Namely, we are interested in assessing how inaccurately we can solve the governing equations during the course of a topology optimization process while still obtaining accurate results. We consider this question from a “PDE-based” angle, using a posteriori residual estimates to gain insight into the behavior of the residuals over the course of Krylov solver iterations. Our main observation is that the residual estimates are dominated by discretization error after only a few iterations of an iterative solver. This provides us with a quantitative measure for early termination of iterative solvers. We illustrate this approach using benchmark examples from linear elasticity and demonstrate that the number of Krylov solver iterations can be significantly reduced, even when compared to previous heuristic recommendations, although each Krylov iteration becomes considerably more expensive.     

Structural topology optimization of multibody systems

Flexible multibody dynamics (FMD) has found many applications in control, analysis and design of mechanical systems. FMD together with the theory of structural optimization can be used for designing multibody systems with bodies which are lighter, but stronger. Topology optimization of static structures is an active research topic in structural mechanics. However, the extension to the dynamic case is less investigated as one has to face serious numerical difficulties. One way of extending static structural topology optimization to topology optimization of dynamic flexible multibody system with large rotational and transitional motion is investigated in this paper. The optimization can be performed simultaneously on all flexible bodies. The simulation part of optimization is based on an FEM approach together with modal reduction. The resulting nonlinear differential-algebraic systems are solved with the error controlled integrator IDA (Sundials) wrapped into Python environment by Assimulo (Andersson et al. in Math. Comput. Simul. 116(0):26–43, 2015). A modified formulation of solid isotropic material with penalization (SIMP) method is suggested to avoid numerical instabilities and convergence failures of the optimizer. Sensitivity analysis is central in structural optimization. The sensitivities are approximated to circumvent the expensive calculations. The provided examples show that the method is indeed suitable for optimizing a wide range of multibody systems. Standard SIMP method in structural topology optimization suggests stiffness penalization. To overcome the problem of instabilities and mesh distortion in the dynamic case we consider here additionally element mass penalization.     

Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes–Darcy problem

We propose and study a combination of two second-order implicit–explicit (IMEX) methods for the coupled Stokes–Darcy system that governs flows in karst aquifers. The first is a second-order explicit two-step MacCormack scheme and the second is a second-order implicit Crank–Nicolson method. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. This combination so called the MacCormack rapid solver method is very efficient (faster, at least of second order accuracy in time and space) and can be easily implemented using legacy codes. Under time step limitation of the form $\Delta t\le Ch$ (where $h,$$\Delta t$ are mesh size and time step, respectively, and $C$ is a physical parameter) we prove both long time stability and the rate of convergence of the method. Some numerical experiments are presented and discussed.     

A simple and compact Python code for complex 3D topology optimization

This paper presents a 100-line Python code for general 3D topology optimization. The code adopts the Abaqus Scripting Interface that provides convenient access to advanced finite element analysis (FEA). It is developed for the compliance minimization with a volume constraint using the Bi-directional Evolutionary Structural Optimization (BESO) method. The source code is composed of a main program controlling the iterative procedure and five independent functions realizing input model preparation, FEA, mesh-independent filter and BESO algorithm. The code reads the initial design from a model database (.cae file) that can be of arbitrary 3D geometries generated in Abaqus/CAE or converted from various widely used CAD modelling packages. This well-structured code can be conveniently extended to various other topology optimization problems. As examples of easy modifications to the code, extensions to multiple load cases and nonlinearities are presented. This code is useful for researchers in the topology optimization field and for practicing engineers seeking automated conceptual design tools. With further extensions, the code could solve sophisticated 3D conceptual design problems in structural engineering, mechanical engineering and architecture practice. The complete code is given in the appendix section and can also be downloaded from the website: www.rmit.edu.au/research/cism/.     

Growth method for size,topology, and geometry optimization of truss structures

The problem of optimally designing the topology of plane trusses has, in most cases, been dealt with as a size problem in which members are eliminated when their size tends to zero. This article presents a novel growth method for the optimal design in a sequential manner of size, geometry, and topology of plane trusses without the need of a ground structure. The method has been applied to single load case problems with stress and size constraints. It works sequentially by adding new joints and members optimally, requiring five basic steps: (1) domain specification, (2) topology and size optimization, (3) geometry optimization, (4) optimality verification, and (5) topology growth. To demonstrate the proposed growth method, three examples were carried out: Michell cantilever, Messerschmidt–Bölkow–Blohm beam, and Michell cantilever with fixed circular boundary. The results obtained with the proposed growth method agree perfectly with the analytical solutions. A Windows XP program, which demonstrates the method, can be downloaded from http://www.upct.es/~deyc/software/tto/.     

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.     

Bit duplication technique to generate hard quadratic unconstrained binary optimization problems with adjustable sizes

Quadratic unconstrained binary optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The main contribution of this article is to propose the bit duplication technique that can specify the number of duplicated bits, so that it can generate hard QUBO problem with adjustable sizes. The idea is to duplicate specified number of bits and then to give constraints so that the corresponding two bits take the same binary values. By this technique, any QUBO problem with $n$ bits is converted to a hard QUBO problem with $(m+n)$ bits . We use random QUBO problems, N-Queen problems, traveling salesman problem and maximum weight matching problems for experiments. The performance of QUBO solvers including Gurobi optimizer, Fixstars Amplify AE, OpenJij with SA, D-Wave samplers with SA, D-Wave hybrid and ABS2 QUBO solver are evaluated for solving these QUBO problems. The experimental results show that only a small scale of duplicated bits can make QUBO problems harder. Hence, the bit duplication technique is a potent method to generate hard QUBO problems and generated QUBO problems can be used as benchmark problems for evaluating the search performance of QUBO solvers.     

Large-scale parallel topology optimization using a dual-primal substructuring solver

Parallel computing is an integral part of many scientific disciplines. In this paper, we discuss issues and difficulties arising when a state-of-the-art parallel linear solver is applied to topology optimization problems. Within the topology optimization framework, we cannot readjust domain decomposition to align with material decomposition, which leads to the deterioration of performance of the substructuring solver. We illustrate the difficulties with detailed condition number estimates and numerical studies. We also report the practical performances of finite element tearing and interconnection/dual–primal solver for topology optimization problems and our attempts to improve it by applying additional scaling and/or preconditioning strategies. The performance of the method is finally illustrated with large-scale topology optimization problems coming from different optimal design fields: compliance minimization, design of compliant mechanisms, and design of elastic surface wave-guides. The authors acknowledge the support of the Air Force Office of Scientific Research (AFOSR) under grant FA9550-05-1-0046. The computational facility was obtained under the grant AFOSR-DURIP FA9550-05-1-0291.     

Complex weight control of array pattern nulling

An efficient method based on the sequential quadratic programming (SQP) algorithm for the linear antenna arrays pattern synthesis with prescribed nulls in the interference direction and minimum side lobe levels by the complex weights of each array element is presented. In general, the pattern synthesis technique that generates a desired pattern is a greatly nonlinear optimization problem. SQP method is a versatile method to solve the general nonlinear constrained optimization problems and is much simpler to implement. It transforms the nonlinear minimization problem to a sequence of quadratic subproblem that is easier to solve, based on a quadratic approximation of the Lagrangian function. Several numerical results of Chebyshev pattern with the imposed single, multiple, and broad nulls sectors are provided and compared with published results to illustrate the performance of the proposed method. © 2007 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2007.     

A new approach to optimization of viscoelastic beams: minimization of the input/output transfer function $\boldsymbol {H}_{\infty }$-norm

A new approach to structural optimization in dynamic regime is presented that is based on the minimization of the \(H_{\infty }\) norm of the transfer function between the external loads and the structural response. The method is successfully applied to the sizing optimization of viscoelastic beams as shown by extensive numerical investigations that are presented in much detail. The abstract nature of the proposed approach makes it applicable to a wide class of dynamical systems including 2D and 3D systems within general topology optimization frameworks that are object of ongoing analysis.     

A parallel multigrid solver for incompressible flows on computing architectures with accelerators

An efficient parallel multigrid pressure correction algorithm is proposed for the solution of the incompressible Navier–Stokes equations on computing architectures with acceleration devices. The pressure correction procedure is based on the numerical solution of a Poisson-type problem, which is discretized using a fourth-order finite difference compact scheme. Since this is the most time-consuming part of the solver, we propose a parallel pressure correction algorithm using an iterative method based on a block cyclic reduction solution method combined with a multigrid technique. The grid points are numbered with respect to the red–black ordering scheme for the parallel Gauss–Seidel smoother. These parallelization techniques allow the execution of the entire simulation computations on the acceleration device, minimizing memory communication costs. The realization is developed using the OpenACC API, and the numerical method is demonstrated for the solution of two classical incompressible flow test problems. The first is the two-dimensional lid-driven cavity problem over equal mesh sizes while the other is the Stokes boundary layer, which is a decent benchmark problem for unequal mesh spacing. The effect of several multigrid components on modern and legacy acceleration architectures is examined. Eventually the performance investigation demonstrates that the proposed parallel multigrid solver achieves an acceleration of more than 10\(\times \) over the sequential solver and more than 4\(\times \) over multi-core CPU only realizations for all tested accelerators.     

A variational inequality approach to optimal plastic design of structures via the Prager-Rozvany theory

The theory of optimal plastic design of structures via optimality criteria (W. Prager approach) transforms the optimal design problem into a certain nonlinear elastic structural analysis problem with appropriate stress-strain laws, which are derived by the adopted specific cost function for the members of the structure and which generally have complete vertical branches. Moreover, the concept of structural universe (introduced by G.I.N. Rozvany) permits us to tackle complicated optimal layout problems.On the other hand, a significant effort in the field of nonsmooth mechanics has recently been devoted to the solution of structural analysis problems with complete material and boundary laws, e.g. stress-strain laws or reaction-displacement laws with vertical branches.In this paper, the problem of optimal plastic design and layout of structures following the approach of Prager-Rozvany is revised within the framework of recent progress in the area of nonsmooth structural analysis and it is treated by means of techniques primarily developed for the solution of inequality mechanics problems. The problem of the optimal layout of trusses is used here as a model problem. The introduction of general convex, continuous and piecewise linear specific cost functions for the structural members leads to the formulation of linear variational inequalities or equivalent piecewise linear, convex but nonsmooth optimization problems. An algorithm exploiting the particular structure of the minimization problem is then described for the numerical solution. Thus, practical structural optimization problems of large size can be treated. Finally, numerical examples illustrate the applicability and the advantages of the method.On leave from the Institute of Applied Mechanics, Department of Engineering Sciences, Technical University of Crete, GR-73100 Chania, Greece     

Multiparameter structural optimization using FEM and multipoint explicit approximations

A unified approach to various problems of structural optimization, based on approximation concepts, is presented. The approach is concerned with the development of the iterative technique, which uses in each iteration the information gained at several previous design points (multipoint approximations) in order to better fit constraints and/or objective functions and to reduce the total number of FE analyses needed to solve the optimization problem. In each iteration, the subregion of the initial region in the space of design variables, defined by move limits, is chosen. In this subregion, several points (designs) are selected, for which response analyses and design sensitivity analyses are carried out using FEM. The explicit expressions are formulated using the weighted least-squares method. The explicit expressions obtained then replace initial problem functions. They are used as functions of a particular mathematical programming problem. Several particular forms of the explicit expressions are considered. The basic features of the presented approximations are shown by means of classical test examples, and the method is compared with other optimization techniques.Presented at NATO ASI Optimization of Large Strucutral Systems, held in Berchtesgarden, Germany, Sept. 23 — Oct. 4, 1991     

On the performance of MaxSAT and MinSAT solvers on 2SAT-MaxOnes

We analyze and compare two solvers for Boolean optimization problems: WMaxSatz, a solver for Partial MaxSAT, and MinSatz, a solver for Partial MinSAT. Both MaxSAT and MinSAT are similar, but previous results indicate that when solving optimization problems using both solvers, the performance is quite different on some cases. For getting insights about the differences in the performance of the two solvers, we analyze their behaviour when solving 2SAT-MaxOnes problem instances, given that 2SAT-MaxOnes is probably the most simple, but NP-hard, optimization problem we can solve with them. The analysis is based first on the study of the bounds computed by both algorithms on some particular 2SAT-MaxOnes instances, characterized by the presence of certain particular structures. We find that the fraction of positive literals in the clauses is an important factor regarding the quality of the bounds computed by the algorithms. Then, we also study the importance of this factor on the typical case complexity of Random-p 2SAT-MaxOnes, a variant of the problem where instances are randomly generated with a probability p of having positive literals in the clauses. For the case p=0, the performance results indicate a clear advantage of MinSatz with respect to WMaxSatz, but as we consider positive values of p WMaxSatz starts to show a better performance, although at the same time the typical complexity of Random-p 2SAT-MaxOnes decreases as p increases. We also study the typical value of the bound computed by the two algorithms on these sets of instances, showing that the behaviour is consistent with our analysis of the bounds computed on the particular instances we studied first.     

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.