Showing papers in "Structural and Multidisciplinary Optimization in 2001"

A 99 line topology optimization code written in Matlab

Abstract: The paper presents a compact Matlab implementation of a topology optimization code for compliance minimization of statically loaded structures. The total number of Matlab input lines is 99 including optimizer and Finite Element subroutine. The 99 lines are divided into 36 lines for the main program, 12 lines for the Optimality Criteria based optimizer, 16 lines for a mesh-independency filter and 35 lines for the finite element code. In fact, excluding comment lines and lines associated with output and finite element analysis, it is shown that only 49 Matlab input lines are required for solving a well-posed topology optimization problem. By adding three additional lines, the program can solve problems with multiple load cases. The code is intended for educational purposes. The complete Matlab code is given in the Appendix and can be down-loaded from the web-site http://www.topopt.dtu.dk.

Comparative studies of metamodelling techniques under multiple modelling criteria

Abstract: Despite advances in computer capacity, the enormous computational cost of running complex engineering simulations makes it impractical to rely exclusively on simulation for the purpose of design optimization. To cut down the cost, surrogate models, also known as metamodels, are constructed from and then used in place of the actual simulation models. In this paper, we systematically compare four popular metamodelling techniques – polynomial regression, multivariate adaptive regression splines, radial basis functions, and kriging – based on multiple performance criteria using fourteen test problems representing different classes of problems. Our objective in this study is to investigate the advantages and disadvantages of these four metamodelling techniques using multiple criteria and multiple test problems rather than a single measure of merit and a single test problem.

An alternative interpolation scheme for minimum compliance topology optimization

Abstract: We consider the discretized zero-one continuum topology optimization problem of finding the optimal distribution of two linearly elastic materials such that compliance is minimized. The geometric complexity of the design is limited using a constraint on the perimeter of the design. A common approach to solve these problems is to relax the zero-one constraints and model the material properties by a power law which gives noninteger solutions very little stiffness in comparison to the amount of material used. We propose a material interpolation model based on a certain rational function, parameterized by a positive scalar q such that the compliance is a convex function when q is zero and a concave function for a finite and a priori known value on q. This increases the probability to obtain a zero-one solution of the relaxed problem.

Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics

Abstract: Topology optimization of structures and composite continua has two main subfields: Layout Optimization (LO) deals with grid-like structures having very low volume fractions and Generalized Shape Optimization (GSO) is concerned with higher volume fractions, optimizing simultaneously the topology and shape of internal boundaries of porous or composite continua. The solutions for both problem classes can be exact/analytical or discretized/FE-based. This review article discusses FE-based generalized shape optimization, which can be classified with respect to the types of topologies involved, namely Isotropic-Solid/Empty (ISE), Anisotropic-Solid/Empty (ASE), and Isotropic-Solid/Empty/Porous (ISEP) topologies. Considering in detail the most important class of (i.e. ISE) topologies, the computational efficiency of various solution strategies, such as SIMP (Solid Isotropic Microstructure with Penalization), OMP (Optimal Microstructure with Penalization) and NOM (NonOptimal Microstructures) are compared. The SIMP method was proposed under the terms "direct approach" or "artificial density approach" by Bendsoe over a decade ago; it was derived independently, used extensively and promoted by the author's research group since 1990. The term "SIMP" was introducted by the author in 1992. After being out of favour with most other research schools until recently, SIMP is becoming generally accepted in topology optimization as a technique of considerable advantages. It seems, therefore, useful to review in greater detail the origins, theoretical background, history, range of validity and major advantages of this method.

Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme

Abstract: In the topology optimization of structures, compliant mechanisms or materials, a density-like func- tion is often used for material interpolation to overcome the computational difficulties encountered in the large "0-1" type integer programming problem. In this paper, we illustrate that a gradually formed continuous peak function can be used for material interpolation. One of the advantages of introducing the peak function is that multiple materials can easily be incorporated into the topology optimization without increasing the number of design variables. By using the peak function and the op- timality criteria method, we synthesize compliant mech- anisms with multiple materials with and without the ma- terial resource constraint. The numerical examples in- clude the two-phase, three-phase, and four-phase materi- als where void is treated as one material. This newdesign method enables us to optimally juxtapose stiff and flex- ible materials in compliant mechanisms, which can be built using modern manufacturing methods.

Checkerboard and minimum member size control in topology optimization

Abstract: Checkerboard-like material distributions are frequently encountered in topology optimization of continuum structures, especially when first order finite elements are used. It has been shown that this phenomenon is caused by errors in the finite element formulation. Minimum member size control is closely related to the problem of mesh dependency of solutions in topology optimization. With increasing mesh density, the solution of a broad class of problems tends to form an increasing number of members with decreasing size. Different approaches have been developed in recent years to overcome these numerical difficulties. However, limitations exist for those methods, either in generality or in efficiency. In this paper, a new algorithm for checkerboard and direct minimum member size control has been developed that is applicable to the general problem formulation involving multiple constraints. This method is implemented in the commercial software Altair OptiStruct.

Sufficiency of a finite exponent in SIMP (power law) methods

Abstract: A common way to perform discrete optimization in shape or topology optimization is to use a method called the artificial power law or SIMP. The focus of this paper is to show that this method gives a discrete solution under some conditions. Examples from topology optimization are included for illustrative purposes.

On the validity of ESO type methods in topology optimization

Abstract: It is shown on a simple test example that ESO's rejection criteria may result in a highly nonoptimal design. Reasons for this failure are also discussed.

A simple checkerboard suppression algorithm for evolutionary structural optimization

Abstract: Checkerboard patterns are quite common in various fixed grid finite element based structural optimization methods. In the evolutionary structural optimization procedure, such checkerboard patterns have been observed under various design criteria. The presence of checkerboard patterns makes the interpretation of optimal material distribution and subsequent geometric extraction for manufacturing difficult. To prevent checkerboarding, an effective smoothing algorithm in terms of the surrounding element’s reference factors is proposed in this paper. The approach does not alter the mesh of the finite element model, nor increase the degree of freedom of the structural system, therefore, it does not affect the computational efficiency. To demonstrate the capabilities of this algorithm, a wide range of illustrative examples are presented in this paper.

On the trajectories of penalization methods for topology optimization

Abstract: We consider the discretized zero-one minimum compliance topology optimization problem of elastic continuum structures under multiple load conditions. The binary design variables indicate presence or absence of material in the finite elements. A common approach to solve these problems is to relax the binary constraints, i.e. allow the design variables to attain values between zero and one, and penalize intermediate values to obtain a "black and white" (zero-one) design. To avoid convergence to a local minimum, it has been suggested that a continuation method should be used, where the penalized problems are solved with increasing penalization. In this paper, the trajectories associated with optimal solutions to the penalized problems, for continuously increasing penalization, are studied on some carefully chosen examples. Two different penalization techniques are used. The global trajectory is defined as the path followed by the global optimal solutions to the penalized problems, and we present examples for which the global trajectory is discontinuous even though the original zero-one problem has a unique solution. Furthermore, we present examples where the penalization method combined with a continuation approach fails to produce a black and white design, no matter how large the penalization becomes.

Pareto analysis in multiobjective optimization using the collinearity theorem and scaling method

Abstract: This paper presents a method to predict the relative objective weighting scheme necessary to cause arbitrary members of a Pareto solution set to become optimal. First, a polynomial description of the Pareto set is constructed utilizing simulation and high performance computing. Then, using geometric relationships between the member of the Pareto set in question, the location of the utopia point and the polynomial coefficients, the weighting of the performance metrics which causes a particular member of the Pareto set to become optimal is determined. The use of this technique, termed the scaling method, is examined via using a sample problem from the field of vehicle dynamics optimization. The scaling method is based on the collinearity theorem which is also presented in the paper.

Optimization of car body under constraints of noise, vibration, and harshness (NVH), and crash

Abstract: A car body structure was optimized for minimum weight under the constraints of noise, vibration, and harshness (NVH), and a crash event, using up to 254 concurrently operating processors. The crash analysis alone, if executed on a single processor and repeated the number of times this optimization required, would have taken 257 days of elapsed computing time. Parallel processing has compressed the elapsed time to one day demonstrating how a multiprocessor machine may be useful in solving engineering tasks that heretofore were regarded as intractable. The optimization procedure transformed the structure initially infeasible to one having its weight reduced and all the constraints satisfied. The experience gained in the reported application indicated it is important to tailor the solution method to the characteristics of the multiprocessor computer architecture and to understand the data handling options offered by that architecture. Another conclusion drawn from this case is that the coarse-grained parallelism whereby an existing code is being replicated over an array of processors should be regarded as an effective way of utilization of multiprocessor machines, immediately available in the interim before solutions are redeveloped from ground up specifically for that class of machines.

A genetic algorithm with real-value coding to optimize multimodal continuous functions

Abstract: In this paper a new Genetic Algorithm (GA) to optimize multimodal continuous functions is proposed. It is based on a splitting of the traditional GA into a sequence of three processes. The first process creates several appropriate sub-populations using the information entropy theory. The second process applies the genetic operators (selection, crossover and mutation) on every subpopulation that is so gradually enriched with better individuals. We then determine the best point s* among the best solutions issued from each of the preceding subpopulations. In the neighbourhood of this point s* is generated a population used to initialize a traditional GA in the third process. Inthis last process, the population is entirely renewed after each generation, the new population being generated in the neighborhood of the best point found. The neighborhood size is decreased after each generation. A detailedcomparison of performances with several stochastic global search methods is presented, using test functions of which local and global minima are known.

Topology optimization in structural mechanics

Abstract: G.I.N. Rozvany, Aims, Scope, Basic Concepts and Methods of Topology Optimization.- W. Achtziger, Topology Optimization of Discrete Structures: an Introduction in View of Computational and Nonsmooth Aspects.- G. Allaire, The Homogenization Method for Topology and Shape Optimization.- H.A. Eschenauer, A. Schumacher, Topology and Shape Optimization Procedures Using Hole Positioning Criteria - Theory and Applications.- U. Kirsch, Reduction and Expansion Processes in Topology Optimization.- U. Kirsch, Singular and Local Optima in Layout Optimization.- U. Kirsch, Reanalysis Models for Topology Optimization - Concepts.- U. Kirsch, Reanalysis Models for Topology Optimization - Applications.- L.A. Krog, N. Olhoff, Topology and Reinforcement Layout Optimization of Disk, Plate, and Shell Structures.

Sequential approximate optimization using variable fidelity response surface approximations

Abstract: The dimensionality and complexity of typical multidisciplinary systems hinders the use of formal optimization techniques in application to this class of problems. The use of approximations to represent the system design metrics and constraints has become vital for achieving good performance in many multidisciplinary design optimization (MDO) algorithms. This paper reports recent research efforts on the use of variable fidelity response surface approximations (RSA) to drive the convergence of MDO problems using a trust region model management algorithm. The present study focuses on a comparative study of different response sampling strategies based on design of experiment (DOE) approaches within the disciplines to generate the zero order data to build the RSAs. Two MDO test problems that have complex coupling between disciplines are used to benchmark the performance of each sampling strategy. The results show that these types of variable fidelity RSAs can be effectively managed by the trust region model management strategy to drive convergence of MDO problems. It is observed that the efficiency of the optimization algorithm depends on the sampling strategy used. A comparison of the DOE approaches with those obtained using a optimization based sampling strategy (i.e. concurrent subspace optimization --- CSSO) shows the DOE methodologies to be competitive with the CSSO based sampling methodology in some cases. However, the CSSO based sampling strategy was found to be, in general, more efficient in driving the optimization.

Optimization of a car body component subjected to side impact

Abstract: This paper explores structural optimization methods applied to a car side impact. The use of local and global approximation methods has been compared, resulting in a recommendation that global approximation methods should be used for this kind of transient loading problems. The numerical simulations have been carried out using the explicit finite element program LS-DYNA. The problem studied is the weight minimization of the B-pillar, situated between the front and the rear door of the car, without compromising the safety of the car occupants. All results are related to the original B-pillar in the SAAB 95 car. By using global approximations in the form of linear and quadratic response surfaces it is shown that the weight of the B-pillar can be reduced by 25% without the loss of safety.

On design-dependent constraints and singular topologies

Abstract: A historical perspective of design-dependent constraints and singular topologies is presented and their theoretical background as well as fundamental features discussed, together with methods for treating computational difficulties. This note contains some rather surprising new facts about singular topologies and it is hoped that it will provide both a comprehensive review and additional insights.

Integration of topology and shape optimization for design of structural components

Abstract: This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach.

A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints

Abstract: The present paper investigates problems of truss topology optimization under local buckling constraints. A new approach for the solution of singular problems caused by stress and local buckling constraints is proposed. At first, a second order smooth-extended technique is used to make the disjoint feasible domains connect, then the so-called e-relaxed method is applied to eliminate the singular optima from problem formulation. By means of this approach, the singular optimum of the original problem caused by stress and local buckling constraints can be searched approximately by employing the algorithms developed for sizing optimization problems with high accuracy. Therefore, the numerical problem resulting from stress and local buckling constraints can be solved in an elegant way. The applications of the proposed approach and its effectiveness are illustrated with several numerical examples.

On the trajectories of the epsilon-relaxation approach for stress-constrained truss topology optimization

Abstract: We consider the nonconvex problem of minimizing the weight of a linearly elastic truss structure subject to stress constraints under multiple load conditions. The design variables are the cross-sectional areas of the elements, and the stress constraints are imposed only on elements with strictly positive areas. To avoid degenerate feasible domains, it has been suggested that the stress constraints of the original problem should be relaxed by a positive scalar ?, leading to the so-called ?-relaxed problem. In this paper, the trajectories associated with optimal solutions of the ?-relaxed problems, for continuously decreasing values of ?, are studied in detail on some carefully chosen examples. The global trajectory is defined as the path followed by the global optimal solution to the ?-relaxed problem, and we present two parameterized examples for which the global trajectory is discontinuous for arbitrarily small values of ?>0. From that we conclude that, in practice, a sequence of solutions to the ?-relaxed problem for decreasing values on ? may not converge to the global optimal solution of the original problem, even if the starting point is on the global trajectory.

Stochastic bilevel programming in structural optimization

Abstract: We consider the mathematical modelling and solution of robust and cost-optimizing structural (topology) design problems. The setting is the optimal design of a linear-elastic structure, for example a truss topology, under unilateral frictionless contact, and under uncertainty in the data describing the load conditions, the material properties, and the rigid foundation. The resulting stochastic bilevel optimization model finds a structural design that responds the best to the given probability distribution in the data. This model is of special interest when a structural failure will lead to a reconstruction cost, rather than loss of life. For the mathematical model, we provide results on the existence of optimal solutions which allow for zero lower design bounds. We establish that the optimal solution is continuous in the lower design bounds, a result which validates the use of small but positive values of them, and for such bounds we also establish the locally Lipschitz continuity and directional differentiability of the implicit upper-level objective function. We also provide a heuristic algorithm for the solution of the problem, which makes use of its differentiability properties and parallelization strategies across the scenarios. A small set of numerical experiments illustrates the behaviour of the stochastic solution compared to an average-case deterministic one, establishing an increased robustness.

Stress ratio and compliance based methods in topology optimization --- a critical review

Abstract: Several currently popular methods of topology optimization are closely related to the classical Fully Stressed Design (FSD)/Stress Ratio (SR) or Minimum Compliance (MC)/Uniform Energy Distribution (UED) methods. The ranges of validity of the above techniques --- and of recent variations on the same themes --- are examined critically and possible extensions of their validity considered. Particular attention is paid to so-called "hard-kill" or Evolutionary Structural Optimization (ESO) or Adaptive Biological Growth (ABG) methods and to the Generalized Stress Design (GSD) technique.

Structural reanalysis for topological modifications - A unified approach

Abstract: A unified approach for structural reanalysis of all types of topological modifications is presented. The modifications considered include various cases of deletion and addition of members and joints. The most challenging problem where the structural model is itself allowed to vary is presented. The two cases, where the number of degrees of freedom is decreased and increased, are considered. Various types of modified topologies are discussed, including the common conditionally unstable structures. The solution procedure is based on the combined approximations approach and involves small computational effort. Numerical examples show that accurate results are achieved for significant topological modifications. Exact solutions are obtained efficiently for modifications in a small number of members.

Robust optimization of an automobile rearview mirror for vibration reduction

Abstract: An automobile outside rearview mirror system has been analysed and designed to reduce vibration with a finite element model. Modal analysis is conducted for the calculation of natural frequencies. Harmonic analysis is utilized to estimate the displacements of the glass surface under dynamic loads. The model is verified with the vibration experiment for the parts and the assembled body. The structure of the mirror system is optimized for robustness defined by the Taguchi concept. At first, many potential design variables are defined. Final design variables are selected based on the amount of contribution to the objective function. That is, sensitive variables are chosen. The signal-to-noise (S/N) ratio in the Taguchi method is replaced by an objective function with mean and standard deviations of the quality characteristics. The defined objective function is appropriate for structural design in that the vibration displacements are minimized while the robustness is improved.

Development of simplified models for design and optimization of automotive structures for crashworthiness

Abstract: Simplified models can be useful for up-front design of automotive structures for passenger safety during crash. Formulations based on the system identification approach are presented for development of simplified models for simulation and design for automotive crash environment. Numerical crash data available from experiments or simulations are used in the development of such models. Parametric as well as nonparametric formulations of the problem are investigated. Standard nonlinear programming optimality conditions and methods are used to solve the resulting nonlinear identification problem. Simple numerical examples are solved to illustrate the proposed formulations and methodologies. As a practical example, the front horn of an automotive structure is replaced by a single degree of freedom system (SDOF). Two basis functions that identify the given target data are studied: Hat functions (piecewise linear) and Chebyshev polynomials. Effects of the number of design variables on the final solution to the problem are investigated. In addition, using the identified SDOF model, redesign of the front horn to improve its performance is discussed.

Visual design steering for optimization solution improvement

Abstract: The paradigm of Visual Design Steering is applied to formal optimal design solution algorithms to improve efficiency, accuracy, and reliability. A formal procedure is presented that guides a designer to utilize the method of Graph Morphing with a formal optimization process. The procedure starts with reduction of the problem for visualization purposes through a ranking and reduction of design variables and constraints. The procedure then outlines how the reduced problem may be visualized using the method of Graph Morphing. Following this procedure an initial point is selected which can produce an improved final solution point in less time than if the solution were started from a random initial point. Several examples are presented which demonstrate this capability.

On the use of energy minimization for CA based analysis in elasticity

Abstract: There has been recent interest in exploring alternative computational models for structural analysis that are better suited for a design environment requiring repetitive analysis The need for such models is brought about by significant increases in computer processing speeds, realized primarily through parallel processing To take full advantage of such parallel machines, however, the computational approach itself must be revisited from a totally different perspective; parallelization of inherently serial paradigms is subject to limitations introduced by a requirement of information coordination The cellular automata (CA) model of decentralized computations provides one such approach which is ideally tailored for parallel computers The proposed paper examines the applicability of the cellular automata model in problems of 2-D elasticity The focus of the paper is in the use of a genetic algorithm based optimization process to derive the rules for local interaction required in evolving the cellular automata

Structural application of a shape optimization method based on a genetic algorithm

Abstract: This paper presents a structural application of a shape optimization method based on a Genetic Algorithm (GA). The method produces a sequence of fixed-distance step-wise movements of the boundary nodes of a finite element model to derive optimal shapes from an arbitrary initial design space. The GA is used to find the optimal or near-optimal combination of boundary nodes to be moved for a given step movement. The GA uses both basic and advanced operators. For illustrative purposes, the method has been applied to structural shape-optimization. The shape-optimization methodology presented allows local optimization, where only crucial parts of a structure are optimized as well as global shape-optimization which involves finding the optimal shape of the structure as a whole for a given environment as described by its loading and freedom conditions. Material can be removed or added to reach the optimal shape. Two examples of structural shape optimization are included showing local and global optimization through material removal and addition.

Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases

Abstract: The reliability-based design optimization (RBDO) can be described by the design potential concept in a unified system space, where the probabilistic constraint is identified by the design potential surface of the reliability target that is obtained analytically from the first-order reliability method (FORM). This paper extends the design potential concept to treat nonsmooth probabilistic constraints and extreme case design in RBDO. In addition, refinement of the design potential surface, which yields better optimum design, can be obtained using more accurate second-order reliability method (SORM). By integrating performance probability analysis into the iterative design optimization process, the design potential concept leads to a very effective design potential method (DPM) for robust system parameter design. It can also be applied effectively to extreme case design (ECD) by directly representing a probabilistic constraint in terms of the system performance function.

A web-based topology optimization program

Abstract: The paper presents a web-based interface for a topology optimization program. The program is accessible over the World Wide Web at the address http://www.topopt.dtu.dk. The paper discusses implementation issues and educational aspects as well as statistics and experience with the program.