Topic

# Shape optimization

About: Shape optimization is a research topic. Over the lifetime, 7629 publications have been published within this topic receiving 134582 citations.

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TL;DR: In this article, various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable are described. But none of these methods can be used for shape optimization in a general setting.

Abstract: Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

3,434 citations

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TL;DR: A new approach to structural topology optimization that represents the structural boundary by a level set model that is embedded in a scalar function of a higher dimension that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation.

Abstract: This paper presents a new approach to structural topology optimization. We represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension. Such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure. Furthermore, a well-founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints. The result is a 3D topology optimization technique that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation. We have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods. The benefit and the advantages of the proposed method are illustrated with several 2D examples that are widely used in the recent literature of topology optimization, especially in the homogenization based methods.

2,404 citations

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TL;DR: A new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation, which can easily handle topology changes and is strongly dependent on the initial guess.

Abstract: In the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation. We implement this method in two and three space dimensions for a model of linear or nonlinear elasticity. We consider various objective functions with weight and perimeter constraints. The shape derivative is computed by an adjoint method. The cost of our numerical algorithm is moderate since the shape is captured on a fixed Eulerian mesh. Although this method is not specifically designed for topology optimization, it can easily handle topology changes. However, the resulting optimal shape is strongly dependent on the initial guess.

2,176 citations

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31 Dec 1984

TL;DR: In this article, the authors present an approach for the optimization of structural components of a ten-bar truss and a twenty-five-bar trestle in the context of structural optimization.

Abstract: 1. Introduction.- 1.1 Function Optimization and Parameter Optimization.- 1.2 Elements of Problem Formulation.- Design Variables.- Objective Function.- Constraints.- Standard Formulation.- 1.3 The Solution Process.- 1.4 Analysis and Design Formulations.- 1.5 Specific Versus General Methods.- 1.6 Exercises.- 1.7 References.- 2. Classical Tools in Structural Optimization.- 2.1 Optimization Using Differential Calculus.- 2.2 Optimization Using Variational Calculus.- to the Calculus of Variations.- 2.3 Classical Methods for Constrained Problems.- Method of Lagrange Multipliers.- Function Subjected to an Integral Constraint.- Finite Subsidiary Conditions.- 2.4 Local Constraints and the Minmax Approach.- 2.5 Necessary and Sufficient Conditions for Optimality.- Elastic Structures of Maximum Stiffness.- Optimal Design of Euler-Bernoulli Columns.- Optimum Vibrating Euler-Bernoulli Beams.- 2.6 Use of Series Solutions in Structural Optimization.- 2.7 Exercises.- 2.8 References.- 3. Linear Programming.- 3.1 Limit Analysis and Design of Structures Formulated as LP Problems.- 3.2 Prestressed Concrete Design by Linear Programming.- 3.3 Minimum Weight Design of Statically Determinate Trusses.- 3.4 Graphical Solutions of Simple LP Problems.- 3.5 A Linear Program in a Standard Form.- Basic Solution.- 3.6 The Simplex Method.- Changing the Basis.- Improving the Objective Function.- Generating a Basic Feasible Solution-Use of Artificial Variables.- 3.7 Duality in Linear Programming.- 3.8 An Interior Method-Karmarkar's Algorithm.- Direction of Move.- Transformation of Coordinates.- Move Distance.- 3.9 Integer Linear Programming.- Branch-and-Bound Algorithm.- 3.10 Exercises.- 3.11 References.- 4. Unconstrained Optimization.- 4.1 Minimization of Functions of One Variable.- Zeroth Order Methods.- First Order Methods.- Second Order Method.- Safeguarded Polynomial Interpolation.- 4.2 Minimization of Functions of Several Variables.- Zeroth Order Methods.- First Order Methods.- Second Order Methods.- Applications to Analysis.- 4.3 Specialized Quasi-Newton Methods.- Exploiting Sparsity.- Coercion of Hessians for Suitability with Quasi-Newton Methods.- Making Quasi-Newton Methods Globally Convergent.- 4.4 Probabilistic Search Algorithms.- Simulated Annealing.- Genetic Algorithms.- 4.5 Exercises.- 4.6 References.- 5. Constrained Optimization.- 5.1 The Kuhn-Tucker Conditions.- General Case.- Convex Problems.- 5.2 Quadratic Programming Problems.- 5.3 Computing the Lagrange Multipliers.- 5.4 Sensitivity of Optimum Solution to Problem Parameters.- 5.5 Gradient Projection and Reduced Gradient Methods.- 5.6 The Feasible Directions Method.- 5.7 Penalty Function Methods.- Exterior Penalty Function.- Interior and Extended Interior Penalty Functions.- Unconstrained Minimization with Penalty Functions.- Integer Programming with Penalty Functions.- 5.8 Multiplier Methods.- 5.9 Projected Lagrangian Methods (Sequential Quadratic Prog.).- 5.10 Exercises.- 5.11 References.- 6. Aspects of the Optimization Process in Practice.- 6.1 Generic Approximations.- Local Approximations.- Global and Midrange Approximations.- 6.2 Fast Reanalysis Techniques.- Linear Static Response.- Eigenvalue Problems.- 6.3 Sequential Linear Programming.- 6.4 Sequential Nonlinear Approximate Optimization.- 6.5 Special Problems Associated with Shape Optimization.- 6.6 Optimization Packages.- 6.7 Test Problems.- Ten-Bar Truss.- Twenty-Five-Bar Truss.- Seventy-Two-Bar Truss.- 6.8 Exercises.- 6.9 References.- 7. Sensitivity of Discrete Systems.- 7.1 Finite Difference Approximations.- Accuracy and Step Size Selection.- Iterative Methods.- Effect of Derivative Magnitude on Accuracy.- 7.2 Sensitivity Derivatives of Static Displacement and Stress Constraints.- Analytical First Derivatives.- Second Derivatives.- The Semi-Analytical Method.- Nonlinear Analysis.- Sensitivity of Limit Loads.- 7.3 Sensitivity Calculations for Eigenvalue Problems.- Sensitivity Derivatives of Vibration and Buckling Constraints.- Sensitivity Derivatives for Non-Hermitian Eigenvalue Problems.- Sensitivity Derivatives for Nonlinear Eigenvalue Problems.- 7.4 Sensitivity of Constraints on Transient Response.- Equivalent Constraints.- Derivatives of Constraints.- Linear Structural Dynamics.- 7.5 Exercises.- 7.6 References.- 8. Introduction to Variational Sensitivity Analysis.- 8.1 Linear Static Analysis.- The Direct Method.- The Adjoint Method.- Implementation Notes.- 8.2 Nonlinear Static Analysis and Limit Loads.- Static Analysis.- Limit Loads.- Implementation Notes.- 8.3 Vibration and Buckling.- The Direct Method.- The Adjoint Method.- 8.4 Static Shape Sensitivity.- The Material Derivative.- Domain Parametrization.- The Direct Method.- The Adjoint Method.- 8.5 Exercise.- 8.6 References.- 9. Dual and Optimality Criteria Methods.- 9.1 Intuitive Optimality Criteria Methods.- Fully Stressed Design.- Other Intuitive Methods.- 9.2 Dual Methods.- General Formulation.- Application to Separable Problems.- Discrete Design Variables.- Application with First Order Approximations.- 9.3 Optimality Criteria Methods for a Single Constraint.- The Reciprocal Approximation for a Displacement Constraint.- A Single Displacement Constraint.- Generalization for Other Constraints.- Scaling-based Resizing.- 9.4 Several Constraints.- Reciprocal-Approximation Based Approach.- Scaling-based Approach.- Other Formulations.- 9.5 Exercises.- 9.6 References.- 10. Decomposition and Multilevel Optimization.- 10.1 The Relation between Decomposition and Multilevel Formulation.- 10.2 Decomposition.- 10.3 Coordination and Multilevel Optimization.- 10.4 Penalty and Envelope Function Approaches.- 10.5 Narrow-Tree Multilevel Problems.- Simultaneous Analysis and Design.- Other Applications.- 10.6 Decomposition in Response and Sensitivity Calculations.- 10.7 Exercises.- 10.8 References.- 11.Optimum Design of Laminated Composite Materials.- 11.1 Mechanical Response of a Laminate.- Orthotropic Lamina.- Classical Laminated Plate Theory.- Bending, Extension, and Shear Coupling.- 11.2 Laminate Design.- Design of Laminates for In-plane Response.- Design of Laminates for Flexural Response.- 11.3 Stacking Sequence Design.- Graphical Stacking Sequence Design.- Penalty Function Formulation.- Integer Linear Programming Formulation.- Probabilistic Search Methods.- 11.4 Design Applications.- Stiffened Plate Design.- Aeroelastic Tailoring.- 11.5 Design Uncertainties.- 11.6 Exercises.- 11.7 References.- Name Index.

1,944 citations

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TL;DR: In this paper, the COC algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members, and numerical results obtained are shown to be in close agreement with analytical results.

Abstract: After outlining analytical methods for layout optimization and illustrating them with examples, the COC algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members. The numerical results obtained are shown to be in close agreement (up to twelve significant digits) with analytical results. Finally, the problem of generalized shape optimization (finding the best boundary topology and shape) is discussed.

1,412 citations