Showing papers on "Shape optimization published in 2005"

Variational methods in shape optimization problems

Abstract: * Preface * Introduction to Shape Optimization Theory and Some Classical Problems > General formulation of a shape optimization problem > The isoperimetric problem and some of its variants > The Newton problem of minimal aerodynamical resistance > Optimal interfaces between two media > The optimal shape of a thin insulating layer * Optimization Problems Over Classes of Convex Domains > A general existence result for variational integrals > Some necessary conditions of optimality > Optimization for boundary integrals > Problems governed by PDE of higher order * Optimal Control Problems: A General Scheme > A topological framework for general optimization problems > A quick survey on 'gamma'-convergence theory > The topology of 'gamma'-convergence for control variables > A general definition of relaxed controls > Optimal control problems governed by ODE > Examples of relaxed shape optimization problems * Shape Optimization Problems with Dirichlet Condition on the Free Boundary > A short survey on capacities > Nonexistence of optimal solutions > The relaxed form of a Dirichlet problem > Necessary conditions of optimality > Boundary variation > Continuity under geometric constraints > Continuity under topological constraints: Sverak's result > Nonlinear operators: necessary and sufficient conditions for the 'gamma'-convergence > Stability in the sense of Keldysh > Further remarks and generalizations * Existence of Classical Solutions > Existence of optimal domains under geometrical constraints > A general abstract result for monotone costs > The weak'gamma'-convergence for quasi-open domains > Examples of monotone costs > The problem of optimal partitions > Optimal obstacles * Optimization Problems for Functions of Eigenvalues > Stability of eigenvalues under geometric domain perturbation > Setting the optimization problem > A short survey on continuous Steiner symmetrization > The case of the first two eigenvalues of the Laplace operator > Unbounded design regions > Some open questions * Shape Optimization Problems with Neumann Condition on the Free Boundary > Some examples > Boundary variation for Neumann problems > General facts in RN > Topological constraints for shape stability > The optimal cutting problem > Eigenvalues of the Neumann Laplacian * Bibliography * Index

Structural optimization using topological and shape sensitivity via a level set method

Abstract: A numerical coupling of two recent methods in shape and topology optimization of structures is proposed. On the one hand, the level set method, based on the classical shape derivative, is known to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes (at least in 2-d). On the other hand, the bubble or topological gradient method is precisely designed for introducing new holes in the optimization process. Therefore, the coupling of these two method yields an efficient algorithm which can escape from local minima in a given topological class of shapes. Both methods relies on a notion of gradient computed through an adjoint analysis, and have a low CPU cost since they capture a shape on a fixed Eulerian mesh. The main advantage of our coupled algorithm is to make the resulting optimal design largely independent of the initial guess.

Mesh generation for implicit geometries

Abstract: We present new techniques for generation of unstructured meshes for geometries specified by implicit functions An initial mesh is iteratively improved by solving for a force equilibrium in the element edges, and the boundary nodes are projected using the implicit geometry definition Our algorithm generalizes to any dimension and it typically produces meshes of very high quality We show a simplified version of the method in just one page of MATLAB code, and we describe how to improve and extend our implementation Prior to generating the mesh we compute a mesh size function to specify the desired size of the elements We have developed algorithms for automatic generation of size functions, adapted to the curvature and the feature size of the geometry We propose a new method for limiting the gradients in the size function by solving a non-linear partial differential equation We show that the solution to our gradient limiting equation is optimal for convex geometries, and we discuss efficient methods to solve it numerically The iterative nature of the algorithm makes it particularly useful for moving meshes, and we show how to combine it with the level set method for applications in fluid dynamics, shape optimization, and structural deformations It is also appropriate for numerical adaptation; where the previous mesh is used to represent the size function and as the initial mesh for the refinements Finally, we show how to generate meshes for regions in images by using implicit representations (Copies available exclusively from MIT Libraries, Rm 14-0551, Cambridge, MA 02139-4307 Ph 617-253-5668; Fax 617-253-1690)

Shape optimization in steady blood flow: a numerical study of non-newtonian effects

Abstract: We investigate the influence of the fluid constitutive model on the outcome of shape optimization tasks, motivated by optimal design problems in biomedical engineering. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. The generalized Newtonian treatment exhibits striking differences in the velocity field for smaller shear rates. We apply sensitivity-based optimization procedure to a flow through an idealized arterial graft. For this problem we study the influence of the inflow velocity, and thus the shear rate. Furthermore, we introduce an additional factor in the form of a geometric parameter, and study its effect on the optimal shape obtained.

Using an Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design

Abstract: An algorithm for efficiently incorporating the effects of mesh sensitivities in a computational design framework is introduced. The method is based on an adjoint approach and eliminates the need for explicit linearizations of the mesh movement scheme with respect to the geometric parameterization variables, an expense that has hindered practical large-scale design optimization using discrete adjoint methods. The effects of the mesh sensitivities can be accounted for through the solution of an adjoint problem equivalent in cost to a single mesh movement computation, followed by an explicit matrix-vector product scaling with the number of design variables and the resolution of the parameterized surface grid. The accuracy of the implementation is established and dramatic computational savings obtained using the new approach are demonstrated using several test cases. Sample design optimizations are also shown.

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.

Improvements in Fixed-Valve Micropump Performance Through Shape Optimization of Valves

Abstract: The fixed-geometry valve micropump is a seemingly simple device in which the interaction between mechanical, electrical, and fluidic components produces a maximum output near resonance. This type of pump offers advantages such as scalability, durability, and ease of fabrication in a variety of materials. Our past work focused on the development of a linear dynamic model for pump design based on maximizing resonance, while little has been done to improve valve shape. Here we present a method for optimizing valve shape using two-dimensional computational fluid dynamics in conjunction with an optimization procedure. A Tesla-type valve was optimized using a set of six independent, non-dimensional geometric design variables. The result was a 25% higher ratio of reverse to forward flow resistance (diodicity) averaged over the Reynolds number range 0

Optimal Design of Unitized Panels with Curvilinear Stiffeners

Abstract: This paper summarizes some numerical results of optimal design study of curvilinear stiffened panels where the reference axes of the stiffeners can be curvilinear. An integrated approach to use different capabilities in relevant fields, such as NURBS, DistMesh, and MSC.NASTRAN, etc., are adopted in the MATLAB environment. Numerical studies on optimal stiffened panel designs of buckling dominant problems are made to look into the effects of certain factors, such as, orientation, spacing, location, and curvature, etc., on the optimal designs. It is observed that curvilinear placement of stiffeners plays the role of orientation, spacing, location and intersection placement of infinitesimal straight stiffeners, and provides an enhanced design space. This enhanced design space may lead to better designs than using straight stiffeners though it is not always so. This work reveals the necessity to use global optimization techniques to perform topology/placement/shape optimization of curvilinear stiffened panels along with size optimization.

The topological asymptotic for the navier-stokes equations

Abstract: The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional and a numerical application is presented.

Nonlinear parametric optimization using cylindrical algebraic decomposition

Abstract: In this paper, a new method is presented for optimization of parametric families of polynomial functions subject to polynomial constraints. The method is based on cylindrical algebraic decomposition (CAD). Given the polynomial objective and constraints, the method constructs the corresponding CAD offline, extracting in advance all the relevant structural information. Then, given the parameter value, an online procedure uses the precomputed information to efficiently evaluate the optimal solution of the original optimization problem. The method is very general and can be applied to a broad range of problems.

Modelling of topological derivatives for contact problems

Abstract: The problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the topological derivatives of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimization.

Topological Shape Optimization of Heat Conduction Problems using Level Set Approach

Abstract: A topological shape optimization method for heat conduction problems is developed using a level set method. The level set function obtained from the “Hamilton-Jacobi type” equation is embedded into a fixed initial domain to implicitly represent thermal boundaries and obtain the finite-element response and adjoint sensitivity. The developed method minimizes the thermal compliance, satisfying the constraint of allowable volume by varying the implicit boundary. During optimization, the boundary velocity to integrate the Hamilton-Jacobi equation is obtained from the optimality condition. The newly developed method shows no numerical instability and makes it easy to represent topological shape variations.

Shape Optimization of a Dimpled Channel to Enhance Turbulent Heat Transfer

Abstract: Numerical optimization of a dimpled channel has been carried out to enhance the turbulent heat transfer. The response surface-based optimization is used as an optimization technique with three-dimensional Reynolds-averaged Navier–Stokes analysis. Computational results for heat transfer rate show good agreement with the experimental data. The objective function is defined as a linear combination of heat transfer- and friction loss-related terms with a weighting factor. Twenty-seven training points obtained by full factorial designs for three design variables construct a reliable response surface. In the sensitivity analysis, it is found that the objective function is most sensitive to the ratio of dimple depth to dimple print diameter. Optimal values of the design variables have been obtained in a range of the weighting factor.

Nondifferentiable Multiplier Rules for Optimization and Bilevel Optimization Problems

Abstract: In this paper we study optimization problems with equality and inequality constraints on a Banach space where the objective function and the binding constraints are either differentiable at the optimal solution or Lipschitz near the optimal solution. Necessary and sufficient optimality conditions and constraint qualifications in terms of the Michel--Penot subdifferential are given, and the results are applied to bilevel optimization problems.

Boundary Methods: Elements, Contours, and Nodes

Abstract: INTRODUCTION TO BOUNDARY METHODS I SELECTED TOPICS IN BOUNDARY ELEMENT METHODS BOUNDARY INTEGRAL EQUATIONS Potential Theory in Three Dimensions Linear Elasticity in Three Dimensions Nearly Singular Integrals in Linear Elasticity Finite Parts of Hypersingular Equations ERROR ESTIMATION Linear Operators Iterated HBIE and Error Estimation Element-Based Error Indicators Numerical Examples THIN FEATURES Exterior BIE for Potential Theory: MEMS BIE for Elasticity: Cracks and Thin Shells II THE BOUNDARY CONTOUR METHOD LINEAR ELASTICITY Surface and Boundary Contour Equations Hypersingular Boundary Integral Equations Internal Displacements and Stresses Numerical Results SHAPE SENSITIVITY ANALYSIS Sensitivities of Boundary Variables Sensitivities of Surface Stresses Sensitivities of Variables at Internal Points Numerical Results: Hollow Sphere Numerical Results: Block with a Hole SHAPE OPTIMIZATION Shape Optimization Problems Numerical Results ERROR ESTIMATION AND ADAPTIVITY Hypersingular Residuals as Local Error Estimators Adaptive Meshing Strategy Numerical Results III THE BOUNDARY NODE METHOD SURFACE APPROXIMANTS Moving Least Squares (MLS) Approximants Surface Derivatives Weight Functions Use of Cartesian Coordinates POTENTIAL THEORY AND ELASTICITY Potential Theory in Three Dimensions Linear Elasticity in Three Dimensions ADAPTIVITY FOR 3-D POTENTIAL THEORY Hypersingular and Singular Residuals Error Estimation and Adaptive Strategy Progressively Adaptive Solutions: Cube Problem One-Step Adaptive Cell Refinement ADAPTIVITY FOR 3-D LINEAR ELASTICITY Hypersingular and Singular Residuals Error Estimation and Adaptive Strategy Progressively Adaptive Solutions: Pulling a Rod One-Step Adaptive Cell Refinement Bibliography Index

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.

A regularized Newton method in electrical impedance tomography using shape Hessian information

Abstract: The present paper is concerned with the identifica- tion of an obstacle or void of different conductivity included in a two-dimensional domain by measurements of voltage and currents at the boundary. We employ a reformulation of the given identifica- tion problem as a shape optimization problem as proposed by Roche and Sokolowski (1996). It turns out that the shape Hessian degener- ates at the given hole which gives a further hint on the ill-posedness of the problem. For numerical methods, we propose a preprocessing for detecting the barycentre and a crude approximation of the void or hole. Then, we resolve the shape of the hole by a regularized Newton method.

Lipschitz continuity of state functions in some optimal shaping

Abstract: We prove local Lipschitz continuity of the solution to the state equation in two kinds of shape optimization problems with constraint on the volume: the minimal shaping for the Dirichlet energy, with no sign condition on the state function, and the minimal shaping for the first eigenvalue of the Laplacian. This is a main first step for proving regularity of the optimal shapes themselves.

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

A new inverse analysis approach for multi-region heat conduction BEM using complex-variable-differentiation method *

Abstract: This paper presents a new inverse analysis approach for identifying material properties and unknown geometries for multi-region problems using the Boundary Element Method (BEM). In this approach, the material properties and coordinates of an unknown region boundary are taken as the optimization variables, and the sensitivity coefficients are computed by the Complex-Variable-Differentiation Method (CVDM). Due to the use of CVDM, the sensitivity coefficients can be accurately determined in a way that is as simple to use as the Finite Difference Method (FDM) and an inverse analysis for a complex composite structure can be easily performed through a similar procedure to the direct computation. Although basic integral equations are presented for heat conduction problems, the application of the proposed algorithm to other problems, such as elastic problems, is straightforward. Two numerical examples are given to demonstrate the potential of the proposed approach.

Shape design by optimal flow control and reduced basis techniques

Abstract: The purpose of this thesis is to develop numerical methods for optimization, control and shape design in computational fluid dynamics, more precisely in haemodynamics. The application studied is related with the shape optimization of an aorto-coronaric bypass. The optimization process has to keep into account aspects which are very different and sometimes conflicting, for this reason the process has been organized in more levels dealing with a geometrical scale. Moreover we have chosen to use simplified low fidelity models during the application of the complex optimization tools and to verify in feed-back with higher fidelity models the configurations previously obtained. In our case we deal with fluid models based on Stokes and Navier-Stokes equations, for lower and higher fidelity approach respectively, also in the unsteady formulation. At an outer level of the optimization process, efficient numerical methods based on parametrized partial differential equations have been developed to get real-time and accurate information concerning the preliminary configurations, and to get a sensitivity analysis on geometrical quantities of interest and on functionals, related with fluid mechanics quantities. This approach is carried out by reduced basis methods which let us rebuild approximate solutions for parametrized equations by other solutions already computed and stored, allowing huge computational savings. At an inner level we have developed local shape optimization methods by optimal flow control theory based on adjoint approach. Two different approaches have been developed: the former is based on the local displacement of each node on the boundary, the latter is based on small perturbation theory into a reference domain. This approach is more complex but let us avoid mesh reconstruction at each iteration and study the problem into a deeper context from a theoretical point of view and do a generalization dealing with unsteady flows.

Applied High-Speed Plate Penetration Dynamics

Abstract: Localized Interaction Approach.- Cavity Expansion Approximations.- Power-Law Realationships Between Impact, Residual and Ballistic Limit Velocity.- Towards Shape Optimization of Impactors.- Shape Optimization of Impactors Penetrating into Ductile Shields.- Shape Optimization of Impactors Penetrating into Concrete Shields.- Optimum Shape of Impactors Against Fibre-Reinforced Plastic Laminates.- Area Rules for Penetrating Impactors.- Optimization of Multi-layered and Spaced Ductile Shields.- Optimization of Two-Component Ceramic Shields.