Optimal structural topologies with transmissible loads

In optimal topological design of structures one obtains the configuration of optimal structures when the design domain, the displacement boundary conditions and the applied loads are specified. In the optimal structure one often notices a marked difference between the main bearing structure and the load transfer zones. The latter are composed of relatively light elements the exact nature of which is not always very distinct. The main purpose of this paper is to allow the main bearing part of the structure to emerge. Moreover the actual location of the load along its line of action is not always a design requirement. In order to include this relaxed condition regarding the loading position the concept of transmissible or sliding forces is introduced in topological design of structures. A transmissible force is a force of given magnitude and direction which can be applied at any point along the line of action of the force. The optimization formulation is similar to standard topological design procedure in addition to the condition of transmissability of the forces. It is shown that this condition reduces to an equal displacement constraint along the line of action of the forces. The method is illustrated by typical structural examples. It is observed that this numerical method produces indeed crisp images of the main structural components, unblurred by the secondary load transfer elements. It is also indicated that many results are often replicas of Prager structures which were previously obtained by analytical methods. Received March 3, 1999     

Integration of topology and shape optimization for design of structural components

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. Received January 27, 2000 Communicated by J. Sobieski     

Evolving structural design solutions using an implicit redundant Genetic Algorithm

Performing synthesis during conceptual design provides substantial cost savings by selecting an efficient design topology and geometry, in addition to selecting the structural member properties. A new evolutionary-based representation, which combines redundancy and implicit fitness constraints, is introduced to represent and search for design solutions in an unstructured, multi-objective structural frame problem. The implicit redundant representation genetic algorithm, in tandem with the unstructured problem domain definition, allows the evaluation of diverse frame topologies and geometries. The IRR GA allows the representation of a variable number of location independent parameters, which overcomes the fixed parameter limitations of standard GAs. The novel frame designs evolved by the IRR GA synthesis design method compare favourably with traditional frame design solutions calculated by trial and error. Received May 27, 1999     

Issues of commercial optimization software development

Commercial optimization software development has a different set of goals and constraints than the development of academic or industrial research codes. Commercial codes must be all things to all people. They must contain a wide range of analysis options and be able to handle large, real world, industrial analysis models. As most of the users of the software in industry come from analysis, rather than design optimization backgrounds, the codes must perform in a robust manner. Inconsistent input data must be detected. Optimization methods must be automatically chosen by the program. Optimization parameters need to be adjusted automatically by the program. Another very important aspect is ease of use. A very intuitive and easy to use GUI (Graphical User Interface) should be developed. This work describes some of the development objectives and concerns that are essential to the development of commercial optimization software products. Received December 30, 2000     

Automotive door design with the ULSAB concept. using structural optimization

Weight reduction for an automobile body is sought to achieve fuel efficiency and energy conservation. Recently, the UltraLight Steel Auto Body (ULSAB) concept is suggested using a few methods. ULSAB pursues a lightweight automotive with steel structure. Tailor welded blank (TWB) is one of the ULSAB methods and TWB can be utilized for an automobile door. Optimization technology is applied to the inner panel of a door which is made by TWB. A design process is appropriately defined for the inner panel. The design starts from an existing component. At first, the inner reinforcements are removed to use TWB technology. In the conceptual design stage, topology optimization is conducted to find the distribution of the variable thickness. The number of parts and the welding lines are determined from the topology design. In the detailed design process, size optimization is carried out to find thickness while the stiffness constraints are satisfied. Size optimization is performed based on the welding lines determined from topology optimization. The final parting lines are tuned by shape optimization. The results from size optimization are considered constant in shape optimization. A commercial optimization software GENESIS is utilized for the optimization processes. Received November 10, 2000     

Multi-material structural topology optimization with decision making of stiffness design criteria

A new methodology for making design decisions of structures using multi-material optimum topology information is presented. Multi-material analysis contributes significant applications to enhance the bearing capacity and performance of structures. A method that chooses an appropriate material combination satisfying design stiffness requirement economically is currently needed. An alternative method of making design-decision is to utilize a multi-material topology optimization (MMTO) approach. This study provides a new computational design optimization procedure as a guideline to find the optimal multi-material design by considering structure strain energy and material cost. The MMTO problem is analyzed using an alternative active-phase approach. The procedure consists of three design steps. First, steel grid configurations and composite with material properties are defined as a given structure for automatic design decision-making (DDM). And then design criteria of the steel composites structure is given to be limited strain energy by designers and engineers. Second, topology changes in the automatic distribution of multi-steel materials combination and volume control of each material during optimization procedures are achieved and at the same time, their converged minimal strain energy is produced for each material combination. And third, the strain energy and material cost which is computed based on the material ratio in the combinations are used as design decision parameters. A study in constructional steel composites to produce optimal and economical multi-material designs demonstrates the efficiency of the present DDM methodology.     

Selection strategies in a structural design support system

This paper presents two selection strategies for the process of establishing a structural design support system. The distinct difference between the strategies is the point of time when the decision maker participates in the design process. The first strategy is called the pre-selection strategy in which the designer tries to grasp the intention or preference of the decision maker at an early stage of the design, and the second one is called the post-selection strategy in which the designer prepares all design information as much as possible, and then the decision maker selects the final design solution. Based on these two strategies, the shared and essential processes of the structural design support system will be explicated in advance, and then the characteristics of these two strategies will be compared. The crucial parts of the structural design support system are common design process that is shared in the two strategies and the alternative based design approach. The common design process means the basic sequences of the design process, such as problem specification, concept design, preliminary design and selection of the final design solution, that are used conventionally in the general design process. These design processes are carried out with the screened region of the design alternatives, which is efficient and effective for the selection of the final design solution. With these concepts, the structural design support system especially the grid-like type structures will be examined and the usefulness of our design methodology will be confirmed. Received October 14, 1999     

Optimal design for 2D wave equations

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. ?veràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.     

Study of a new local update scheme for cellular automata in structural design

This paper investigates an improved local update scheme for cellular automata (CA) applied to structural design. Local analysis and design rules are derived for equilibrium and minimum compliance design. The new update scheme consists of repeating analysis and optimality-based design rules locally. The benefits of this approach are demonstrated through a series of systematic experiments. Truss topology design problems of various sizes are used based on the Gauss–Seidel and the Jacobi iteration modes. Experiments show the robust convergence of the approach as compared to an earlier CA implementation. The approach is also extended to a plate problem.     

Optimal design of underwater stiffened shells

The optimal design parameters of stiffened shells are determined using a rational multicriteria optimization approach. The adopted approach aims at simultaneously minimizing the shell vibration, associated sound radiation, weight of the stiffening rings as well as the cost of the stiffened shell. A finite element model is developed to determine the vibration and noise radiation from cylindrical shells into the surrounding fluid domain. The production cost as well as the life cycle and maintenance costs of the stiffened shells are computed using the Parametric Review of Information for Costing and Evaluation (PRICE) model. A Pareto/min-max multicriteria optimization approach is then utilized to select the optimal dimensions and spacing of the stiffeners. Numerical examples are presented to compare the vibration and noise radiation characteristics of optimally designed stiffened shells with the corresponding characteristics of plain un-stiffened shells. The obtained results emphasis the importance of the adopted multicriteria optimization approach in the design of quiet, low weight and low cost underwater shells which are suitable for various critical applications. Received September 14, 2000 Communicated by J. Sobieski     

Truss topology optimization with discrete design variables—Guaranteed global optimality and benchmark examples

This paper considers the problem of optimal truss topology design subject to multiple loading conditions. We minimize a weighted average of the compliances subject to a volume constraint. Based on the ground structure approach, the cross-sectional areas are chosen as the design variables. While this problem is well-studied for continuous bar areas, we consider in this study the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single available bar area, i.e., a 0/1 problem. In contrast to the heuristic methods considered in many other approaches, our goal is to compute guaranteed globally optimal structures. This is done by a branch-and-bound method for which convergence can be proven. In this branch-and-bound framework, lower bounds of the optimal objective function values are calculated by treating a sequence of continuous but non-convex relaxations of the original mixed-integer problem. The main effect of using this approach lies in the fact that these relaxed problems can be equivalently reformulated as convex problems and, thus, can be solved to global optimality. In addition, these convex problems can be further relaxed to quadratic programs for which very efficient numerical solution procedures exist. By exploiting this special problem structure, much larger problem instances can be solved to global optimality compared to similar mixed-integer problems. The main intention of this paper is to provide optimal solutions for single and multiple load benchmark examples, which can be used for testing and validating other methods or heuristics for the treatment of this discrete topology design problem.     

Topology optimization for minimum stress design with the homogenization method

This paper is devoted to minimum stress design in structural optimization. The homogenization method is extended to such a framework and yields an efficient numerical algorithm for topology optimization. The main idea is to use a partial relaxation of the problem obtained by introducing special microstructures which are sequential laminated composites. Indeed, the so-called corrector terms of such microgeometries are explicitly known, which allows us to compute the relaxed objective function. These correctors can be interpreted as stress amplification factors, caused by the underlying microstructure.     

Shape feature control in structural topology optimization

A variational approach to shape feature control in topology optimization is presented in this paper. The method is based on a new class of surface energies known as higher-order energies as opposed to the conventional energies for problem regularization, which are linear. In employing a quadratic energy functional in the objective of the topology optimization, non-trivial interactions between different points on the structural boundary are introduced, thus favoring a family of shapes with strip-like (or beam) features. In addition, the quadratic energy functional can be seamlessly integrated into the level set framework that represents the geometry of the structure implicitly. The shape gradient of the quadratic energy functional is fully derived in the paper, and it is incorporated in the level set approach for topology optimization. The approach is demonstrated with benchmark examples of structure optimization and compliant mechanism design. The results presented show that this method is capable of generating strip-like (or beam) designs with specified feature width, which have highly desirable characteristics and practical benefits and uniquely distinguish the proposed method.     

A mutual energy formulation for optimal structural design

This paper presents a variational formulation for the design of elastic structures where the function to be minimized by the optimal design, i.e. the objective, is expressed in abstract form. The resulting statement of necessary conditions is uniformly applicable for all admissible objectives. Both state and adjoint state variables appear directly in the problem statement, and all objectives and the arguments of constraints are scalars. The adjoint pair of state variables appear in symmetric roles via the expression termed “mutual energy". Application of the generalized formulation is demonstrated by treatment of the following examples: design to minimize the maximum value of displacement or to minimize a global measure of stress, design for generalized compliance, design where self-weight is taken into account, and multicriterion design. Received April 28, 2000     

Optimal design of a welded I-section frame using four conceptually different optimization algorithms

The purpose of this study is to investigate the suitability of four conceptually different optimization algorithms for specifically the optimal design of welded I-section frames. The cost function to be minimized is the volume of the frame. Constraints on lateral-torsional buckling as well as local buckling of the beam and column webs and flanges are taken into consideration. The algorithms evaluated include a genetic algorithm, a novel leap-frog gradient method without line searches, as well as an orthogonal search method requiring no gradients and the differential evolution technique. Received September 26, 2001 RID="*" ID="*"Part of this paper was presented at the World Congress on Structural and Multidisciplinary Optimization, June 4–8 2001, Dalian, China     

Parallelized structural topology optimization and CFD coupling for design of aircraft wing structures

A set of structural optimization tools are presented for topology optimization of aircraft wing structures coupled with Computational Fluid Dynamics (CFD) analyses. The topology optimization tool used for design is the material distribution technique. Because reducing the weight requires numerous calculations, the CFD and structural optimization codes are parallelized and coupled via a code/mesh coupling scheme. In this study, the algorithms used and the results obtained are presented for topology design of a wing cross-section under a given critical aerodynamic loading and two different spar positions to determine the optimum rib topology.     

Optimal riser design in sand casting process with evolutionary topology optimization

The optimal design of a casting feeding system is considered. The riser topology is systematically modified to minimize the riser volume, while simultaneously ensuring that no defect appears in the product. In this approach, we combine finite-difference analysis of the solidification process with evolutionary topology optimization to systematically improve the feeding system design. The outstanding features of the presented method are: its efficiency, ease of implementation and simplifying definition of the initial design. The efficiency and capability of the presented method are supported by illustrative examples.     

Optimization under uncertainty with applications to design of truss structures

Many real-world engineering design problems are naturally cast in the form of optimization programs with uncertainty-contaminated data. In this context, a reliable design must be able to cope in some way with the presence of uncertainty. In this paper, we consider two standard philosophies for finding optimal solutions for uncertain convex optimization problems. In the first approach, classical in the stochastic optimization literature, the optimal design should minimize the expected value of the objective function with respect to uncertainty (average approach), while in the second one it should minimize the worst-case objective (worst-case or min–max approach). Both approaches are briefly reviewed in this paper and are shown to lead to exact and numerically efficient solution schemes when the uncertainty enters the data in simple form. For general uncertainty dependence however, the problems are numerically hard. In this paper, we present two techniques based on uncertainty randomization that permit to solve efficiently some suitable probabilistic relaxation of the indicated problems, with full generality with respect to the way in which the uncertainty enters the problem data. In the specific context of truss topology design, uncertainty in the problem arises, for instance, from imprecise knowledge of material characteristics and/or loading configurations. In this paper, we show how reliable structural design can be obtained using the proposed techniques based on the interplay of convex optimization and randomization.