Showing papers on "Shape optimization published in 1993"
TL;DR: In this paper, a modified optimization criterion method (MOC) was proposed to solve the frequency response optimization problem for both the optimal layout and the reinforcement of an elastic structure, which is based on a simpler idea that employs a shifted Lagrangian function to make a convex approximation.
Abstract: A topology and shape optimization technique using the homogenization method was developed for stiffness of a linearly elastic structure by Bendsoe and Kikuchi (1988), Suzuki and Kikuchi (1990, 1991), and others. This method has also been extended to deal with an optimal reinforcement problem for a free vibration structure by Diaz and Kikuchi (1992). In this paper, we consider a frequency response optimization problem for both the optimal layout and the reinforcement of an elastic structure. First, the structural optimization problem is transformed to an Optimal Material Distribution problem (OMD) introducing microscale voids, and then the homogenization method is employed to determine and equivalent “averaged” structural analysis model. A new optimization algorithm, which is derived from a Sequential Approximate Optimization approach (SAO) with the dual method, is presented to solve the present optimization problem. This optimization algorithm is different from the CONLIN (Fleury 1986) and MMA (Svanderg 1987), and it is based on a simpler idea that employs a shifted Lagrangian function to make a convex approximation. The new algorithm is called “Modified Optimality Criteria method (MOC)” because it can be reduced to the traditional OC method by using a zero value for the shift parameter. Two sensitivity analysis methods, the Direct Frequency Response method (DFR) and the Modal Frequency Response method (MFR), are employed to calculate the sensitivities of the object functions. Finally, three examples are given to show the feasibility of the present approach.
270 citations
TL;DR: In this article, a foam-like material, with variable density, is used for the evaluation of material properties, and the approach is implemented in a pre-optimizer for optimal shape finding.
263 citations
Dissertation•
1 Jan 1993
174 citations
1 Jan 1993
TL;DR: In this paper, the form finding and thickness variation are embedded in the concept of structural optimization which combines design modelling, structural and sensitivity analyses and mathematical optimization schemes to a general design tool.
Abstract: Shells are known to be optimal in many ways, provided certain basic shell oriented design rules are followed. The shape, thickness and material distribution play a dominant role. Minimum material, a specific frequency response, maximum load carrying capacity, a pure membrane stress state are typical design objectives. In the present contribution the form finding and thickness variation are embedded in the concept of structural optimization which combines design modelling, structural and sensitivity analyses and mathematical optimization schemes to a general design tool. The structural response may be based on linear elastic, eigenvalue and geometrically nonlinear analyses. In particular, the imperfection sensitivity with respect to buckling is discussed. A few selected examples demonstrate the versatility of optimization schemes in shell design, among these are the tuning of a bell and the form finding of a classical reinforced concrete dome shell.
83 citations
TL;DR: In this article, non-uniform rational B-splines (NURBs) are used to describe the shape of a structure and a sensitivity analysis using NURB defined geometries is developed.
80 citations
TL;DR: In this article, an aerodynamic shape optimization method has been developed by the authors using the Euler equations and has been applied to supersonic-hypersonic nozzle designs.
Abstract: An aerodynamic shape optimization method has previously been developed by the authors using the Euler equations and has been applied to supersonic-hypersonic nozzle designs. This method has also included a flowfield extrapolation (or flow prediction) method based on the Taylor series expansion of an existing CFD solution. The present paper reports on the extension of this method to the thin-layer Navier-Stokes equations in order to account for the viscous effects. Also, to test the method under highly nonlinear conditions, it has been applied to the transonic flows. Initially, the success of the flow prediction method is tested. Then, the overall method is demonstrated by optimizing the shapes of two supercritical transonic airfoils at zero angle of attack. The first one is shape optimized to achieve a minimum drag while obtaining a lift above a specified value. Whereas, the second one is shape optimized for a maximum lift while attaining a drag below a specified value. The results of these two cases indicate that the present method can produce successfully optimized aerodynamic shapes.
65 citations
1 Nov 1993
TL;DR: In this article, a numerical approach for the treatment of optimal shape problems governed by the Euler equations is discussed, in particular for flows with embedded shocks, and some results are given to illustrate the effectiveness of the method.
Abstract: In this paper we discuss a numerical approach for the treatment of optimal shape problems governed by the Euler equations. In particular, we focus on flows with embedded shocks. We consider a very simple problem: the design of a quasi-one-dimensional Laval nozzle. We introduce a cost function and a set of Lagrange multipliers to achieve the minimum. The nature of the resulting costate equations is discussed. A theoretical difficulty that arises for cases with embedded shocks is pointed out and solved. Finally, some results are given to illustrate the effectiveness of the method.
61 citations
Journal Article•
59 citations
TL;DR: This methodology is based on the use of the adaptive mesh refinement techniques in the context of 2D shape optimization problems and the analytical sensitivity analysis of the different items ruling the problem (B-splines, finite element mesh, structural behaviour and error estimator) is studied in detail.
Abstract: This methodology is based on the use of the adaptive mesh refinement techniques in the context of 2D shape optimization problems. A technique for the parametrization of the optimization problem, using B-splines to define the boundary, is first presented. Then mesh generation, using the advancing frontal method, the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems. In particular, the analytical sensitivity analysis of the different items ruling the problem (B-splines, finite element mesh, structural behaviour and error estimator) is studied in detail
55 citations
TL;DR: In this article, an optimization method is developed for the design of intermediate die shapes needed in the plane strain and axisymmetric forging operations, based on backward deformation simulation using nonlinear rigid viscoplastic finite element method and shape optimization techniques.
Abstract: An optimization method is developed for the design of intermediate die shapes needed in the plane strain and axisymmetric forging operations. The approach is based on backward deformation simulation using nonlinear rigid viscoplastic finite element method and shape optimization techniques. The advantage of this optimization approach is that it has the ability to determine the intermediate die shapes from the final product shape by applying constraints on the plastic deformation of the material. This paper presents axisymmetric disk and plane strain case studies to demonstrate the new design procedures for minimizing variations in deformation rates during a multistage forging operation
50 citations
TL;DR: In this paper, an aerodynamic optimization method with two design variables using sensitivity analysis on the first-order-accurate discretization of the Euler equations is presented, which is more efficient than the traditional design methods for a few reasons, which include the use of flow predictions and the elimination of a priori guessing of possible shapes from which the optimum is to be selected.
Abstract: Previously, the authors have shown an aerodynamic optimization method with two design variables using sensitivity analysis on the first-order-accurate discretization of the Euler equations. Two advancements of this method are reported in this article. First, nonlinear fluid dynamic phenomena including flow discontinuities are better predicted by an improved flow prediction method which uses the third-order accurate discretization of the Euler equations. Using this method, the flowfield of a modified shape which generates shocks and other large gradients is predicted based on the shock-free flowfield of the original shape and without solving the flowfield equations. Secondly, every surface grid point is used as a design variable, which virtually eliminates all geometrical restrictions on the shape as it is optimized for the specified objective. This improved algorithm is demonstrated by optimizing the ramp shape of a scramjet-afterbody configuration for maximum axial thrust. Starting with totally different initial designs, virtually identical shapes are obtained as the optimum. The method is more efficient than the traditional design methods for a few reasons, which include the use of flow predictions and the elimination of a priori guessing of possible shapes from which the optimum is to be selected.
TL;DR: In this article, a numerical method to compute free surfaces in electromagnetic shaping and levitation of liquid metals is described, where the surfaces are represented by piecewise linear finite elements and each step requires solving an elliptic boundary value problem in the exterior of the intermediate surfaces.
Abstract: We describe a numerical method to compute free surfaces in electromagnetic shaping and levitation of liquid metals. We use an energetic variational formulation and optimization techniques to compute, a critical point. The surfaces are represented by piecewise linear finite elements. Each step of the algorithm requires solving an elliptic boundary value problem in the exterior of the intermediate surfaces. This is done by using an integral representation on these surfaces.
TL;DR: In this paper, a homogenized plate model for shape optimization is presented, which accommodates without significant changes the modelling of ribbed, honeycomb and perforated plates.
Abstract: The development of a homogenized plate model suitable for shape optimization is presented. The development is based on a homogenization method for layered materials with a periodic microstructure. A particular advantage of the approach is that it accommodates without significant changes the modelling of ribbed, honeycomb and perforated plates. The model is compared with others that have appeared in the literature and that are also useful in the context of the optimization of the shape and layout of plates and plate-like structures. The results indicate that the model presented here is useful in the optimization of both thick and thin plates.
TL;DR: In this article, a geometric representation scheme is outlined which utilizes the natural design variable concept, which can be used for all geometric design studies; it is demonstrated here for structural optimization, and a similar optimization problem is presented for an automobile crankshaft section.
Abstract: A geometric representation scheme is outlined which utilizes the natural design variable concept. A base configuration with distinct topological features is created. This configuration is then deformed to define components with similar topology but different geometry. The values of the deforming loads are the geometric entities used in the shape representation. The representation can be used for all geometric design studies; it is demonstrated here for structural optimization. This technique can be used in parametric design studies, where the system response is defined as functions of geometric entities. It can also be used in shape optimization, where the geometric entities of an original design are modified to maximize performance and satisfy constraints. Two example problems are provided. A cantilever beam is elongated to meet new design specifications and then optimized to reduce volume and satisfy stress constraints. A similar optimization problem is presented for an automobile crankshaft section. The finite element method is used to perform the analyses.
1 Jan 1993
TL;DR: It is demonstrated that solid isotropic microstructures with penalty (SIMP) for intermediate densities are highly efficient in locating optimal SE topologies.
Abstract: Two algorithms for the iterative optimization of discretized systems are discussed in this lecture: one concerns layout optimization, the simultaneous optimization of topology, geometry and cross-sectional dimensions for grid-like structures; and the other one generalized shape optimization, the simultaneous optimization of boundary topology and boundary shape for continua. Both methods are based on new optimality criteria methods (COC, DCOC). Discretized layout optimization is illustrated with test examples involving trusses and grillages, and combinations of stress and displacement constraints. In generalized shape optimization, the emphasis is on solutions in which porous regions are suppressed and only solid and empty regions remain (SE topologies). It is demonstrated that solid isotropic microstructures with penalty (SIMP) for intermediate densities are highly efficient in locating optimal SE topologies.
TL;DR: In this article, structural shape optimization of shells and folded plates using two-noded Mindlin-Reissner C (0) finite strips was carried out by integrating finite strip analysis, cubic spline shape definition, automatic mesh generation, sensitivity analysis and mathematical programming methods in an efficient way.
TL;DR: In this article, it was shown that when the shape of an elastic body in contact with a flat rigid foundation is chosen on the same premises, the uniform pressure distribution is found only if displacement gradients can be considered small.
Abstract: This paper addresses the problem of finding shapes of contacting bodies avoiding undesirable stress concentrations It has previously been shown that designing the shape of a rigid body in contact with a fixed linear elastic body by minimizing the equilibrium potential energy under an isoparametric constraint results in a uniform contact pressure distribution As an extension of this result, it is shown here that when the shape of an elastic body in contact with a flat rigid foundation is chosen on the same premises, the uniform pressure distribution is found only if displacement gradients can be considered small From the point of view of applications, an important conclusion is that this smallness holds in a case when linear elasticity is physically valid
TL;DR: In this article, a method to optimize the cross-sectional shape of a prismatic bar under Saint-Venant's torsion using direct integration of the integral equations is developed.
TL;DR: In this article, a finite element-based shape optimization program was developed for three-dimensional shell structures which allows for large shape changes by linking together adaptive mesh generation, substructuring, and linear and nonlinear optimization techniques to a commercial finite element analysis program.
1 Jan 1993
TL;DR: On the basis of the optimal layout theory discussed in the preceding lecture, an iterative COC procedure is outlined and then applied to problems involving optimal plastic design and optimal elastic design with a given compliance.
Abstract: On the basis of the optimal layout theory discussed in the preceding lecture, an iterative COC procedure is outlined and then applied to problems involving optimal plastic design and optimal elastic design with a given compliance It is demonstrated that (i) the proposed method can handle up to several thousand potential members and (ii) the results show an excellent agreement with analytical solutions An additional example with two alternate loading conditions is presented and it is found that for that case, the optimal plastic design differs from the optimal elastic design Finally, the problem of generalized shape optimization is reviewed and an improved homogenization method is put forward
1 Jan 1993
TL;DR: In this article, different models that can be used to compute the optimum layout and shape of plate structures using homogenization techniques are presented, which differ in the plate equation used as the basis for the analysis, the microstructure assumed for the material, and the order in which the homogeneization and assembly of plies is performed.
Abstract: Different models that can be used to compute the optimum layout and shape of plate structures using homogenization techniques are presented. These models differ in the plate equation used as the basis for the analysis, the microstructure assumed for the material, and the order in which the homogenization and the assembly of plies is performed. Different ply stacking arrangements to assemble the plate are used to build plates with external ribs, internal ribs (honeycomb plates) and transversely perforated plates. The shape optimization problem is stated and the optimality conditions are derived. It is found that the optimality condition for the orientation of the ribs requires the solution of a fourth order polynomial equation, giving directions that are not necessarily aligned with the in-plane principal stress directions. Some shape optimization examples are solved and compared with results from the literature.
1 Jan 1993
TL;DR: In this article, the difference of optimal shapes of a cantilever disc made of ductile and brittle materials is presented and furthermore, the Bubble Method is used for finding a best possible initial design of an initial CPD made from ductile materials.
Abstract: The “Bubble Method” is one of the methods of topology optimization techniques. Its basic idea is to iteratively position new holes (bubbles) in a structure by means of a definite function and a hierachically secondary shape optimization. The expression of the definite function depends on the special optimization functionals and the material behaviour. In this paper the difference of optimal shapes of a cantilever disc made of ductile and brittle materials is presented and, furthermore, the Bubble Method is used for finding a best possible initial design of a cantilever disc made of ductile materials.
TL;DR: Improved design procedures are proposed to extend the capability of the approaches for shape optimization problems to simulate the behavior of adaptive shapes of biological structures by growth and atrophy.
Abstract: These approaches simulate the behavior of adaptive shapes of biological structures by growth and atrophy with respect to the natural loading applied Improved design procedures are proposed to extend the capability of the approaches for shape optimization problems
TL;DR: In this paper, an adaptation of the gradient projection algorithm for shape optimization problems is described along with a method to reduce the intermediate size of the optimization problem by allowing adaptive refinement of the shape.
Abstract: A large part of the computational effort in shape optimization problems is expended in the numerical computation of the gradients for sensitivity information. This effort increases dramatically with an increase in the number of variables used to represent the shape. An adaptation of the gradient projection algorithm for shape optimization problems is described here along with a method to reduce the intermediate size of the optimization problem by allowing adaptive refinement of the shape. The method is demonstrated with a simple representative test case.
TL;DR: This procedure is implemented as a module in a general system for shape optimal design of linear elastic structures and provides the mesh generation and its adaptive refinement in a wholly automatic way.
TL;DR: In this paper, a boundary element formulation for sensitivity analysis for solid mechanics problems involving both material and geometric nonlinearities is presented, and a shape optimization problem has been solved for a plate with a cutout involving only material non-linearities.
Abstract: The objective of this paper is twofold. First, it presents a boundary element formulation for sensitivity analysis for solid mechanics problems involving both material and geometric nonlinearities. The second focus is on the use of such sensitivities to obtain optimal design for problems of this class. Numerical examples include sensitivity analysis for small (material nonlinearities only) and large deformation problems. These numerical results are in good agreement with direct integration results. Further, by using these sensitivities, a shape optimization problem has been solved for a plate with a cutout involving only material nonlinearities. The difference between the optimal shapes of solids, undergoing purely elastic or elasto-viscoplastic deformation is shown clearly in this example.
TL;DR: In this article, an efficient approximation method to determine the optimum shape of the minimum weight of a body subjected to stress and displacement constraints is suggested by using the boundary element method, where the objective function of weight is approximated to an expansion of a second-order Taylor series and the stress and displacements constraints to expansions of a first-orderTaylor series, based on the sensitivity analysis at the current design point.
TL;DR: In this paper, a general approach to shape design sensitivity analysis and optimal design for static and vibration problems using boundary elements is presented, and the adjoint variable method is applied to obtain first-order sensitivities for the effect of boundary shape variations.
Abstract: A general approach to shape design sensitivity analysis and optimal design for static and vibration problems using boundary elements is presented. The adjoint variable method is applied to obtain first-order sensitivities for the effect of boundary shape variations. The boundary element procedure for numerical calculations of sensitivities are used. Typical objective and constraints functionals are described for shape optimal design. Several numerical examples of applications of boundary elements in shape optimal design are presented.
TL;DR: In this paper, a robust and versatile algorithm for shape optimization with adaptive finite element procedures is developed for the design of axisymmetric structures based on the use of boundary parameterization with cubic splines for describing shape changes and taking advantage of the utilities available in an advancing front type mesh generator.
Abstract: A robust and versatile algorithm for shape optimization with adaptive finite element procedures is developed for the design of axisymmetric structures. The algorithm is based on the use of boundary parameterization with cubic splines for describing shape changes and takes advantage of the utilities available in an advancing front type mesh generator. Six-noded triangular elements are adopted. Shape optimization examples involving solid axisymmetric structures are presented to illustrate the various features of the integrated approach.