Showing papers in "Structural and Multidisciplinary Optimization in 2005"

Adaptive weighted-sum method for bi-objective optimization: Pareto front generation

Abstract: This paper presents a new method that effectively determines a Pareto front for bi-objective optimization with potential application to multiple objectives. A traditional method for multiobjective optimization is the weighted-sum method, which seeks Pareto optimal solutions one by one by systematically changing the weights among the objective functions. Previous research has shown that this method often produces poorly distributed solutions along a Pareto front, and that it does not find Pareto optimal solutions in non-convex regions. The proposed adaptive weighted sum method focuses on unexplored regions by changing the weights adaptively rather than by using a priori weight selections and by specifying additional inequality constraints. It is demonstrated that the adaptive weighted sum method produces well-distributed solutions, finds Pareto optimal solutions in non-convex regions, and neglects non-Pareto optimal solutions. This last point can be a potential liability of Normal Boundary Intersection, an otherwise successful multiobjective method, which is mainly caused by its reliance on equality constraints. The promise of this robust algorithm is demonstrated with two numerical examples and a simple structural optimization problem.

Topology optimization of channel flow problems

Abstract: This paper describes a topology design method for simple two-dimensional flow problems. We consider steady, incompressible laminar viscous flows at low-to-moderate Reynolds numbers. This makes the flow problem nonlinear and hence a nontrivial extension of the work of Borrvall and Petersson (2003).Further, the inclusion of inertia effects significantly alters the physics, enabling solutions of new classes of optimization problems, such as velocity-driven switches, that are not addressed by the earlier method. Specifically, we determine optimal layouts of channel flows that extremize a cost function which measures either some local aspect of the velocity field or a global quantity, such as the rate of energy dissipation. We use the finite element method to model the flow, and we solve the optimization problem with a gradient-based math-programming algorithm that is driven by analytical sensitivities. Our target application is optimal layout design of channels in fluid network systems. Using concepts borrowed from topology optimization of compliant mechanisms in solid mechanics, we introduce a method for the synthesis of fluidic components, such as switches, diodes, etc.

The effect of normalization norms in multiple attribute decision making models: a case study in gear material selection

Abstract: The main objective of this paper is to evaluate the effect of different normalization norms within multiple attribute decision making (MADM) models. The application of the work is dedicated to gear material selection for power transmission. To this end, the general scheme of the decision model is first presented, with close attention to the context of material selection. Subsequently, the entropy method and technique for order preference by similarity to ideal solution (TOPSIS) are employed to weigh the selected failure criteria and to rank the selected material IDs, respectively. Finally, by the introduction of different norms to the solution algorithm, the effect of normalization formalism on the material selection using the MADM models is studied. A simple multiaxial strategy is also recommended from which safer engineering decisions may be attained.

Adaptive probability analysis using an enhanced hybrid mean value method

Abstract: This paper proposes an adaptive probability analysis method that can effectively generate the probability distribution of the output performance function by identifying the propagation of input uncertainty to output uncertainty. The method is based on an enhanced hybrid mean value (HMV+) analysis in the performance measure approach (PMA) for numerical stability and efficiency in search of the most probable point (MPP). The HMV+ method improves numerical stability and efficiency especially for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. The proposed adaptive probability analysis method approximates the MPP locus, and then adaptively refines this locus using an a posteriori error estimator. Using the fact that probability levels can be easily set a priori in PMA, the MPP locus is approximated using the interpolated moving least-squares method. For refinement of the approximated MPP locus, additional probability levels are adaptively determined through an a posteriori error estimator. The adaptive probability analysis method will determine the minimum number of necessary probability levels, while ensuring accuracy of the approximated MPP locus. Several examples are used to show the effectiveness of the proposed adaptive probability analysis method using the enhanced HMV+ method.

Multi-objective robust optimization using a sensitivity region concept

Abstract: In multi-objective design optimization, it is quite desirable to obtain solutions that are “multi-objectively” optimum and insensitive to uncontrollable (noisy) parameter variations. We call such solutions robust Pareto solutions. In this paper we present a method to measure the multi-objective sensitivity of a design alternative, and an approach to use such a measure to obtain multi-objectively robust Pareto optimum solutions. Our sensitivity measure does not require a presumed probability distribution of uncontrollable parameters and does not utilize gradient information; therefore, it is applicable to multi-objective optimization problems that have non-differentiable and/or discontinuous objective functions, and also to problems with large parameter variations. As a demonstration, we apply our robust optimization method to an engineering example, the design of a vibrating platform. We show that the solutions obtained for this example are indeed robust.

Note on topology optimization of continuum structures including self-weight

Abstract: This paper proposes to investigate topology optimization with density-dependent body forces and especially self-weight loading. Surprisingly the solution of such problems cannot be based on a direct extension of the solution procedure used for minimum-compliance topology optimization with fixed external loads. At first the particular difficulties arising in the considered topology problems are pointed out: non-monotonous behaviour of the compliance, possible unconstrained character of the optimum and the parasitic effect for low densities when using the power model (SIMP). To get rid of the last problem requires the modification of the power law model for low densities. The other problems require that the solution procedure and the selection of appropriate structural approximations be revisited. Numerical applications compare the efficiency of different approximation schemes of the MMA family. It is shown that important improvements are achieved when the solution is carried out using the gradient-based method of moving asymptotes (GBMMA) approximations. Criteria for selecting the approximations are suggested. In addition, the applications also provide the opportunity to illustrate the strong influence of the ratio between the applied loads and the structural weight on the optimal structural topology.

Review of formulations for structural and mechanical system optimization

Abstract: Alternative formulations for optimization and simulation of structural and mechanical systems and other related fields are reviewed. The material is divided roughly into two parts. Part 1 focuses on the developments in structural and mechanical systems, including configuration and topology optimization. Here the formulations are classified into three broad categories: (i) the conventional formulation where only the structural design variables are treated as optimization variables, (ii) simultaneous analysis and design (SAND) formulations where design and some of the state variables are treated as optimization variables, and (iii) a displacement-based two-phase approach where the displacements are treated as unknowns in the outer loop and the design variables as the unknowns in the inner loop. Part 2 covers more general formulations that are applicable to diverse fields, such as economics, optimal control, multidisciplinary problems and other engineering disciplines. In these fields, SAND-type formulations have been called mathematical programs with equilibrium constraints (MPEC), and partial differential equations (PDE)-constrained optimization problems. These formulations are viewed as generalizations of the SAND formulations developed in the structural optimization field. Based on the review, it is concluded that the basic ideas of the formulations presented in diverse fields can be integrated to conduct further research and develop alternative formulations and solution procedures for practical engineering applications. The paper lists 187 references on the subject.

Optimization of fiber orientations near a hole for increased load-carrying capacity of composite laminates

Abstract: In this paper, a procedure for optimizing the fiber orientations near a hole in a single layer of multilayer composite laminates for increased strength is presented. A symmetric six-layer [(C-0)/+45/-45]s laminate with central hole is considered. Within the ±45° layers, the fiber orientations are fixed. The C-0 layer is divided into two fields: a small near field around the hole and a relatively large far field away from the hole. In the far field, the fiber orientations are fixed at 0°, and in the near field, the fiber orientations are assumed to have continuous distribution represented by piecewise bilinear interpolation functions. The Tsai–Wu-criterion-based failure load magnitudes of [(C-0)/+45/-45]s and [C-0]6 designs are maximized by iteratively alternating between a gradient-based search and a genetic algorithm. The results show that the load-carrying capacity of composite laminates with hole can be greatly increased through the optimization of continuous fiber orientation distribution within a small area around the hole in the C-0 layer, and the optimum fiber orientation distribution pattern in the C-0 layer is similar to that of the corresponding principal stress orientation distribution. The [(C-0)/+45/-45]s design is only a few percent weaker than the [C-0]6 design, which is important for carrying off-design loadings.

Multiobjective structural optimization using a microgenetic algorithm

Abstract: In this paper, we present a genetic algorithm with a very small population and a reinitialization process (a microgenetic algorithm) for solving multiobjective optimization problems. Our approach uses three forms of elitism, including an external memory (or secondary population) to keep the nondominated solutions found along the evolutionary process. We validate our proposal using several engineering optimization problems taken from the specialized literature and compare our results with respect to two other algorithms (NSGA-II and PAES) using three different metrics. Our results indicate that our approach is very efficient (computationally speaking) and performs very well in problems with different degrees of complexity.

A reevaluation of the SIMP method with filtering and an alternative formulation for solid-void topology optimization

Abstract: The most popular way to introduce the notion of topology into the structural analysis of the topology optimization problem is through the Solid Isotropic Material with Penalization (SIMP) method. The fundamental principle behind its use requires a density design variable dependent material constitutive law that penalizes intermediate density material in combination with an active volume constraint. Here, the SIMP method with filtering is reevaluated, and an alternative topology optimization problem formulation, called the SINH (pronounced “cinch”) method, is developed that exploits this principle. The main advantages of the SINH method are that the optimization problem is consistently defined, the topology description is unambiguous, and the method leads to predominantly solid–void designs.

Design of band-gap grid structures

Abstract: This paper discusses issues related to designing band-gaps in periodic plane grid structures. Finite element analysis is used to solve the dynamic behavior of a representative unit cell and Bloch–Floquet theory is used to extend the results to the infinite structure. Particular attention is given to the addition of non-structural masses that are introduced as design variables. These are used to create desirable features in the dispersion diagram. Physical insight is presented into the optimal choice of locations where masses should be added and the results of several numerical examples are provided to highlight this and other features of how band-gaps can be created and located at desired frequency ranges. The effect of the skew angle of the underlying grid structure is also explored, as are mathematical refinements of the modelling of the beam elements and the rotational inertia of the added masses. A scaling feature between the size of the reducible and the irreducible reference cell is exploited and the manner in which this can simplify optimization approaches is discussed.

Compliant mechanism design using multi-objective topology optimization scheme of continuum structures

Abstract: Topology optimization problems for compliant mechanisms using a density interpolation scheme, the rational approximation of material properties (RAMP) method, and a globally convergent version of the method of moving asymptotes (GCMMA) are primarily discussed. First, a new multi-objective formulation is proposed for topology optimization of compliant mechanisms, in which the maximization of mutual energy (flexibility) and the minimization of mean compliance (stiffness) are considered simultaneously. The formulation of one-node connected hinges, as well as checkerboards and mesh-dependency, is typically encountered in the design of compliant mechanisms. A new hybrid-filtering scheme is proposed to solve numerical instabilities, which can not only eliminate checkerboards and mesh-dependency efficiently, but also prevent one-node connected hinges from occurring in the resulting mechanisms to some extent. Several numerical applications are performed to demonstrate the validity of the methods presented in this paper.

A review of optimization of cast parts using topology optimization

Abstract: In recent years, there has been considerable progress in the optimization of cast parts with respect to strength, stiffness, and frequency. Here, topology optimization has been the most important tool in finding the optimal features of a cast part, such as optimal cross-section or number and arrangement of ribs. An optimization process with integrated topology optimization has been used very successfully at Adam Opel AG in recent years, and many components have been optimized. This two-paper review gives an overview of the application and experience in this area. This is the first part of a two-paper review of optimization of cast parts.Here, we want to focus on the application of the original topology optimization codes, which do not take manufacturing constraints for cast parts into account. Additionally, the role of shape optimization as a fine-tuning tool will be briefly analyzed and discussed.

Combined topology and fiber path design of composite layers using cellular automata

Abstract: The popular Solid Isotropic Material Penalization (SIMP) technique of topology design is extended to simultaneous fiber-angle and topology design of composite laminae in a cellular automata (CA) framework. CA is a novel methodology to simulate a physical phenomenon based on iterative local updates of both field and design variables. Displacements are updated satisfying local equilibrium of CA cells. Fiber angles and density measures are updated based on the optimality criteria for the minimum compliance design. Numerical results for the design of 2D cantilever plates for single and multiple load cases are used to demonstrate the robustness of the proposed algorithm.

Variable chromosome length genetic algorithm for progressive refinement in topology optimization

Abstract: 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.

Structure topology optimization: fully coupled level set method via FEMLAB

Abstract: This paper presents a procedure which can easily implement the 2D compliance minimization structure topology optimization by the level set method using the FEMLAB package. Instead of a finite difference solver for the level set equation, as is usually the case, a finite element solver for the reaction–diffusion equation is used to evolve the material boundaries. All of the optimization procedures are implemented in a user-friendly manner. A FEMLAB code can be downloaded from the homepage www.imtek.de/simulation and is free for educational purposes.

Topology design of large displacement compliant mechanisms with multiple materials and multiple output ports

Abstract: Topology optimization of compliant mechanisms is presented in this paper wherein the layout design problem is addressed in its original binary or discrete (0-1) form. Design variables are modeled as discrete variables and allowed to assume values pertaining only to their void (0) or solid (1) states. Due to this discrete nature, a genetic algorithm is employed as an optimization routine. Using the barrier assignment approach, the search algorithm is extended to use with multiple materials. The layout design of compliant mechanisms is performed wherein displacements at multiple points (ports) in the design region are maximized along the respective prescribed directions. With multiple output ports and multiple materials, additional freedom in motion and force transduction can be achieved with compliant mechanisms. Geometrically large deformation analysis is employed to compute the displacement-based multiple objectives that are extremized using Nondominated Sorting in Genetic Algorithms (or NSGA). With genetic algorithms, buckling or snap through like issues with nonconvergent solutions in the population when computing nonlinear deformations can be implicitly circumvented.

On polynomial response surfaces and Kriging for use in structural optimization of crashworthiness

Abstract: The accuracy of different approximating response surfaces is investigated. In the classical response surface methodology (CRSM) the true response function is usually replaced with a low-order polynomial. In Kriging the true response function is replaced with a low-order polynomial and an error correcting function. In this paper the error part of the approximating response surface is obtained from “simple point Kriging” theory. The combined polynomial and error correcting function will be addressed as a Kriging surface approximation.

Large deformations and stability in topology optimization

Abstract: The present contribution focuses on the influence of geometrical nonlinearities on the structural behavior in the design process. The notion of the stiffest structure loses its clear definition in the case of nonlinear kinematics; here we will discuss this concept on the basis of different objectives. Apparently topology optimization is often a generator of slender struts, which tend to buckle before the structure is completely loaded. To include the instability phenomena into the design process, the critical load level will be determined directly; this condition will be included as an inequality constraint. Further on, to reduce the imperfection sensitivity, a geometrically modified structure including the imperfection shape is also introduced. The present optimization procedures are demonstrated by examples showing rather the principal effects of the enhancements than real practical design problems.

A review of optimization of cast parts using topology optimization: I - Topology optimization without manufacturing constraints

Abstract: In recent years, there has been considerable progress in the optimization of cast parts with respect to strength, stiffness, and frequency. Here, topology optimization has been the most important tool in finding the optimal features of a cast part, such as optimal cross-section or number and arrangement of ribs. An optimization process with integrated topology optimization has been used very successfully at Adam Opel AG in recent years, and many components have been optimized. This two-paper review gives an overview of the application and experience in this area. This is the first part of a two-paper review of optimization of cast parts.Here, we want to focus on the application of the original topology optimization codes, which do not take manufacturing constraints for cast parts into account. Additionally, the role of shape optimization as a fine-tuning tool will be briefly analyzed and discussed.

Parameter optimization of the sheet metal forming process using an iterative parallel Kriging algorithm

Abstract: Different numerical optimization strategies were used to find an optimized parameter setting for the sheet metal forming process. A parameterization of a time-dependent blank-holder force was used to control the deep-drawing simulation. Besides the already well-established gradient and direct search algorithms and the response surface method the novel Kriging approach was used as an optimization strategy. Results for two analytical and two sheet metal forming test problems reveal that the new Kriging approach leads to a fast and stable convergence of the optimization process. Parallel simulation is perfectly supported by this method.

Experience with approximate reliability-based optimization methods II: an exhaust system problem

Abstract: Traditional reliability-based design optimization (RBDO) requires a double-loop iteration process. The inner optimization loop is to find the reliability and the outer is the regular optimization loop to optimize the RBDO problem with reliability objectives or constraints. It is known that the computation can be prohibitive when the associated function evaluation is expensive. This situation is even worse when a large number of reliability constraints are present. As a result, many approximate RBDO methods, which convert the double loop to a single loop, have been developed. In this research, an engineering problem with a large number of constraints (144) is designed to test RBDO methods based on the first-order reliability method (FORM), including single- and double-loop methods. In addition to the number of constraints, this problem possesses many local minimums. Some original authors of the RBDO methods are also asked to solve the same problem. The results and the efficiencies for different methods are published and discussed.

Structural topology optimization using a genetic algorithm with a morphological geometric representation scheme

Abstract: This paper describes a versatile methodology for solving topology design optimization problems using a genetic algorithm (GA). The key to its effectiveness is a geometric representation scheme that works by specifying a skeleton which defines the underlying topology/connectivity of a structural continuum together with segments of material surrounding the skeleton. The required design variables are encoded in a chromosome which is in the form of a directed graph that embodies this underlying topology so that appropriate crossover and mutation operators can be devised to recombine and help preserve any desirable geometry characteristics of the design through succeeding generations in the evolutionary process. The overall methodology is first tested by solving ‘target matching’ problems—simulated topology optimization problems in each of which a ‘target’ geometry is first created and predefined as the optimum solution, and the objective of the optimization problem is to evolve design solutions to converge towards this target shape. The methodology is then applied to design two path-generating compliant mechanisms—large-displacement flexural structures that undergo some desired displacement paths at some point when given a straight line input displacement at some other point—by an actual process of topology/shape optimization.

Integrating linear physical programming within collaborative optimization for multiobjective multidisciplinary design optimization

Abstract: Multidisciplinary design optimization (MDO) is a concurrent engineering design tool for large-scale, complex systems design that can be affected through the optimal design of several smaller functional units or subsystems. Due to the multiobjective nature of most MDO problems, recent work has focused on formulating the MDO problem to resolve tradeoffs between multiple, conflicting objectives. In this paper, we describe the novel integration of linear physical programming within the collaborative optimization framework, which enables designers to formulate multiple system-level objectives in terms of physically meaningful parameters. The proposed formulation extends our previous multiobjective formulation of collaborative optimization, which uses goal programming at the system and subsystem levels to enable multiple objectives to be considered at both levels during optimization. The proposed framework is demonstrated using a racecar design example that consists of two subsystem level analyses — force and aerodynamics — and incorporates two system-level objectives: (1) minimize lap time and (2) maximize normalized weight distribution. The aerodynamics subsystem also seeks to minimize rearwheel downforce as a secondary objective. The racecar design example is presented in detail to provide a benchmark problem for other researchers. It is solved using the proposed formulation and compared against a traditional formulation without collaborative optimization or linear physical programming. The proposed framework capitalizes on the disciplinary organization encountered during large-scale systems design.

Nonparametric gradient-less shape optimization for real-world applications

Abstract: A nonparametric gradient-less shape optimization approach for finite element stress minimization problems is presented The shape optimization algorithm is based on optimality criteria, which leads to a robust and fast convergence independent of the number of design variables Sensitivity information of the objective function and constraints are not required, which results in superior performance and offers the possibility to solve the structural analysis task using fast and reliable industry standard finite element solvers such as ABAQUS, ANSYS, I-DEAS, MARC, NASTRAN or PERMAS The approach has been successfully extended to complex nonlinear problems including material, boundary and geometric nonlinear behavior The nonparametric geometry representation creates a complete design space for the optimization problem, which includes all possible solutions for the finite element discretization The approach is available within the optimization system TOSCA and has been used successfully for real-world optimization problems in industry for several years The approach is compared to other approaches and the benefits and restrictions are highlighted Several academic and real-world examples are presented

Minimization of the expected compliance as an alternative approach to multiload truss optimization

Abstract: We show that a problem of finding the truss of minimum expected compliance under stochastic loading conditions is equivalent to the dual of a special convex minimax problem, and therefore may be efficiently solved. This equivalence makes it possible to provide classic multiload compliance minimization problems with interpretations in a probabilistic setting. In fact, we prove that minimizing the expected compliance amounts to solving a multiload-like problem associated with a particular finite set of loading scenarios, which depend on the mean and the variance of the perturbations.

Topology optimization of trusses with stress and local constraints on nodal stability and member intersection

Abstract: A truss topology optimization problem under stress constraints is formulated as a Mixed Integer Programming (MIP) problem with variables indicating the existence of nodes and members. The local constraints on nodal stability and intersection of members are considered, and a moderately large lower bound is given for the cross-sectional area of an existing member. A lower-bound objective value is found by neglecting the compatibility conditions, where linear programming problems are successively solved based on a branch-and-bound method. An upper-bound solution is obtained as a solution of a Nonlinear Programming (NLP) problem for the topology satisfying the local constraints. It is shown in the examples that upper- and lower-bound solutions with a small gap in the objective value can be found by the branch-and-bound method, and the computational cost can be reduced by using the local constraints.

Shape design of pin-jointed multistable compliant mechanisms using snapthrough behavior

Abstract: A general approach is presented for generating pin-jointed multistable compliant mechanisms using snapthrough behavior. An optimization problem is formulated for minimizing the total structural volume under constraints on the displacements at the specified nodes, stiffnesses at initial and final states, and load factors to lead to snapthrough behavior. The design variables are cross-sectional areas and the nodal coordinates. It is shown in the numerical examples that several mechanisms can be naturally found as a result of optimization starting from randomly selected initial solutions. It is also shown that no local bifurcation point exists along the equilibrium path, and the obtained mechanism is not sensitive to initial imperfections.

Optimum Design of Truss Topology under Buckling Constraints

Abstract: The present paper studies the optimum design of truss topology under buckling constraints based on a new formulation of the problem. Through the incorporation of a global system stability constraint into the problem formulation, isolated compressive bars are eliminated from the final optimal topology. Furthermore, by including overlapping bars in the initial ground structure, the difficulty caused by hinge cancellation as pointed out by Rozvany (1996) can be overcome. Also, the importance of inclusion of compatibility conditions in the problem formulation is demonstrated. Finally, several numerical examples are presented for demonstration of the effectiveness of the proposed approach.

Vibration reduction optimum design of a steam-turbine rotor-bearing system using a hybrid genetic algorithm

Abstract: This paper describes the vibration optimum design for the low-pressure steam-turbine rotor of a 1007-MW nuclear power plant by using a hybrid genetic algorithm (HGA) that combines a genetic algorithm and a local concentration search algorithm using a modified simplex method. This algorithm not only calculates the optimum solution faster and more accurately than the standard genetic algorithm but can also find the global and local optimum solutions. The objective function is to minimize the resonance response (Q-factor) of the second occurring mode in the excessive vibration. Under the constraints of shaft diameter, bearing length and clearance, these factors play a very important role in the design of a rotor-bearing system. In the present work, the shaft diameter, bearing length and clearance are chosen as the design variables. The results show that the HGA can reduce the excessive response at the critical speed and improve the stability.