Shape Optimization Problems

Preface
Contents
Notation
	Use of Letters
	Sets
	Vectors and Matrices
	Domains and Functions
	Banach Spaces
	Function Spaces
1 Basics of Optimal Design
	1.1 Optimal Design Problem for a Stepped One-Dimensional Linear Elastic Body
		1.1.1 State Determination Problem
		1.1.2 An Optimal Design Problem
		1.1.3 Cross-Sectional Derivatives
		1.1.4 The Substitution Method
		1.1.5 The Direct Differentiation Method
		1.1.6 The Adjoint Variable Method
		1.1.7 Optimality Conditions
		1.1.8 Numerical Example
	1.2 Comparison of the Direct Differentiation Method and the Adjoint Variable Method
		1.2.1 The Direct Differentiation Method
		1.2.2 The Adjoint Variable Method
	1.3 An Optimal Design Problem of a Branched One-Dimensional Stokes Flow Field
		1.3.1 State Determination Problem
		1.3.2 An Optimal Design Problem
		1.3.3 Cross-Sectional Derivatives
		1.3.4 Optimality Conditions
		1.3.5 Numerical Example
	1.4 Summary
	1.5 Practice Problems
2 Basics of Optimization Theory
	2.1 Definition of Optimization Problems
	2.2 Classification of Optimization Problems
	2.3 Existence of a Minimum Point
	2.4 Differentiation and Convex Functions
		2.4.1 Taylor\'s Theorem
		2.4.2 Convex Functions
		2.4.3 Exercises in Differentiation and Convex Functions
	2.5 Unconstrained Optimization Problems
		2.5.1 A Necessary Condition for Local Minimizers
		2.5.2 Sufficient Conditions for Local Minimizers
		2.5.3 Sufficient Conditions for Global Minimizers
		2.5.4 Example of Unconstrained Optimization Problem
		2.5.5 Considerations Relating to the Solutions of Unconstrained Optimization Problems
	2.6 Optimization Problems with Equality Constraints
		2.6.1 A Necessary Condition for Local Minimizers
		2.6.2 The Lagrange Multiplier Method
		2.6.3 Sufficient Conditions for Local Minimizers
		2.6.4 An Optimization Problem with an Equality Constraint
		2.6.5 Direct Differentiation and Adjoint Variable Methods
		2.6.6 Considerations Relating to the Solution of Optimization Problems with Equality Constraints
	2.7 Optimization Problems Under Inequality Constraints
		2.7.1 Necessary Conditions at Local Minimizers
		2.7.2 Necessary and Sufficient Conditions for Global Minimizers
		2.7.3 KKT Conditions
		2.7.4 Sufficient Conditions for Local Minimizers
		2.7.5 Sufficient Conditions for Global Minimizers Using the KKT Conditions
		2.7.6 Example of an Optimization Problem Under an Inequality Constraint
		2.7.7 Considerations Relating to the Solutions of Optimization Problems Under Inequality Constraints
	2.8 Optimization Problems Under Equality and Inequality Constraints
		2.8.1 The Lagrange Multiplier Method for Optimization Problems Under Equality and Inequality Constraints
		2.8.2 Considerations Regarding Optimization Problems Under Equality and Inequality Constraints
	2.9 Duality Theorem
		2.9.1 Examples of the Duality Theorem
	2.10 Summary
	2.11 Practice Problems
3 Basics of Mathematical Programming
	3.1 Problem Setting
	3.2 Iterative Method
	3.3 Gradient Method
	3.4 Step Size Criterion
	3.5 Newton Method
	3.6 Augmented Function Methods
	3.7 Gradient Method for Constrained Problems
		3.7.1 Simple Algorithm
		3.7.2 Complicated Algorithm
	3.8 Newton Method for Constrained Problems
		3.8.1 Simple Algorithm
		3.8.2 Complicated Algorithm
	3.9 Summary
	3.10 Practice Problems
4 Basics of Variational Principles and Functional Analysis
	4.1 Variational Principles
		4.1.1 Hamilton\'s Principle
		4.1.2 Minimum Principle of Potential Energy
		4.1.3 Pontryagin\'s Minimum Principle
	4.2 Abstract Spaces
		4.2.1 Linear Space
			Set of All Continuous Functions
		4.2.2 Linear Subspaces
		4.2.3 Metric Space
			Denseness
			Separability
			Completeness
			Compactness
		4.2.4 Normed Space
			Banach Space
		4.2.5 Inner Product Space
			Hilbert Space
	4.3 Function Spaces
		4.3.1 Hölder Space
		4.3.2 Lebesgue Space
		4.3.3 Sobolev Space
			Schwartz Distribution
			Sobolev Space
		4.3.4 Sobolev Embedding Theorem
	4.4 Operators
		4.4.1 Bounded Linear Operator
		4.4.2 Trace Theorem
		4.4.3 Calderón Extension Theorem
		4.4.4 Bounded Bilinear Operators
		4.4.5 Bounded Linear Functional
		4.4.6 Dual Space
			Weak Complete and Dual Weak Complete
			Dual Space of Sobolev Space
		4.4.7 Rellich–Kondrachov Compact Embedding Theorem
		4.4.8 Riesz Representation Theorem
	4.5 Generalized Derivatives
		4.5.1 Gâteaux Derivative
		4.5.2 Fréchet Derivative
	4.6 Function Spaces in Variational Principles
		4.6.1 Hamilton\'s Principle
		4.6.2 Minimum Principle of Potential Energy
		4.6.3 Pontryagin\'s Minimum Principle
	4.7 Summary
	4.8 Practice Problems
5 Boundary Value Problems of Partial Differential Equations
	5.1 Poisson Problem
		5.1.1 Extended Poisson Problem
	5.2 Abstract Variational Problem
		5.2.1 Lax–Milgram Theorem
		5.2.2 Abstract Minimization Problem
	5.3 Regularity of Solutions
		5.3.1 Regularity of Given Functions
		5.3.2 Regularity of Boundary
	5.4 Linear Elastic Problem
		5.4.1 Linear Strain
		5.4.2 Cauchy Tensor
		5.4.3 Constitutive Equation
		5.4.4 Equilibrium Equations of Force
		5.4.5 Weak Form
		5.4.6 Existence of Solution
	5.5 Stokes Problem
	5.6 Abstract Saddle Point Variational Problem
		5.6.1 Existence Theorem of Solution
		5.6.2 Abstract Saddle Point Problem
	5.7 Summary
	5.8 Practice Problems
6 Fundamentals of Numerical Analysis
	6.1 Galerkin Method
		6.1.1 One-Dimensional Poisson Problem
		6.1.2 d-Dimensional Poisson Problem
		6.1.3 Ritz Method
		6.1.4 Basic Error Estimation
	6.2 One-Dimensional Finite Element Method
		6.2.1 Approximate Functions in Galerkin Method
		6.2.2 Approximate Functions in Finite Element Method
		6.2.3 Discretized Equations
		6.2.4 Exercise Problem
	6.3 Two-Dimensional Finite Element Method
		6.3.1 Approximate Functions in Galerkin Method
		6.3.2 Approximate Functions in Finite Element Method
		6.3.3 Discretized Equations
		6.3.4 Exercise Problem
	6.4 Various Finite Elements
		6.4.1 One-Dimensional Higher-Order Finite Elements
		6.4.2 Triangular Higher-Order Finite Elements
		6.4.3 Rectangular Finite Elements
		6.4.4 Tetrahedral Finite Elements
		6.4.5 Hexahedral Finite Elements
	6.5 Isoparametric Finite Elements
		6.5.1 Two-Dimensional Four-Node Isoparametric Finite Elements
		6.5.2 Gaussian Quadrature
	6.6 Error Estimation
		6.6.1 Finite Element Division Sequence
		6.6.2 Affine-Equivalent Finite Element Division Sequence
		6.6.3 Interpolation Error Estimation
		6.6.4 Error Estimation of Finite Element Solution
	6.7 Summary
	6.8 Practice Problems
7 Abstract Optimum Design Problem
	7.1 Linear Spaces of Design Variables
	7.2 State Determination Problem
	7.3 Abstract Optimum Design Problem
	7.4 Existence of an Optimum Solution
	7.5 Derivatives of Cost Functions
		7.5.1 Adjoint Variable Method
		7.5.2 Lagrange Multiplier Method
		7.5.3 Second-Order Fréchet Derivatives of Cost Functions
		7.5.4 Second-Order Fréchet Derivative of Cost Function Using Lagrange Multiplier Method
	7.6 Descent Directions of Cost Functions
		7.6.1 Abstract Gradient Method
		7.6.2 Abstract Newton Method
	7.7 Solution of Abstract Optimum Design Problem
		7.7.1 Gradient Method for Constrained Problems
		7.7.2 Newton Method for Constrained Problems
	7.8 Summary
8 Topology Optimization Problems of Density Variation Type
	8.1 Set of Design Variables
	8.2 State Determination Problem
	8.3 Topology Optimization Problem of θ-Type
	8.4 Existence of an Optimum Solution
	8.5 Derivatives of Cost Functions
		8.5.1 θ-Derivatives of Cost Functions
		8.5.2 Second-Order θ-Derivative of Cost Functions
		8.5.3 Second Order θ-Derivative of Cost Function Using Lagrange Multiplier Method
	8.6 Descent Directions of Cost Functions
		8.6.1 H1 Gradient Method
			Method Using the Inner Product of H1 Space
			Method Using Boundary Conditions
			Regularity of H1 Gradient Method
		8.6.2 H1 Newton Method
	8.7 Solution of Topology Optimization Problem of θ-Type
		8.7.1 Gradient Method for Constrained Problems
		8.7.2 Newton Method for Constrained Problems
	8.8 Error Estimation
	8.9 Topology Optimization Problem of Linear Elastic Body
		8.9.1 State Determination Problem
		8.9.2 Mean Compliance Minimization Problem
		8.9.3 θ-Derivatives of Cost Functions
		8.9.4 Second-Order θ-Derivatives of Cost Functions
		8.9.5 Second-Order θ-Derivative of Cost Function Using Lagrange Multiplier Method
		8.9.6 Numerical Example
	8.10 Topology Optimization Problem of Stokes Flow Field
		8.10.1 State Determination Problem
		8.10.2 Mean Flow Resistant Minimization Problem
		8.10.3 θ-Derivatives of Cost Functions
		8.10.4 Second-Order θ-Derivatives of Cost Functions
		8.10.5 Second-Order θ-Derivative of Cost Function Using Lagrange Multiplier Method
		8.10.6 Numerical Example
	8.11 Summary
	8.12 Practice Problems
9 Shape Optimization Problems of Domain Variation Type
	9.1 Set of Domain Variations and Definition of Shape Derivatives
		9.1.1 Initial Domain
		9.1.2 Sets of Domain Variations
		9.1.3 Definitions of Shape Derivatives
	9.2 Shape Derivatives of Jacobi Determinants
		9.2.1 Shape Derivatives of Domain Jacobi Determinant and Domain Jacobi Inverse Matrix
		9.2.2 Shape Derivatives of Boundary Jacobi Determinant and the Normal
	9.3 Shape Derivatives of Functionals
		9.3.1 Formulae Using Shape Derivative of a Function
		9.3.2 Formulae Using Partial Shape Derivative of a Function
	9.4 Variation Rules of Functions
	9.5 State Determination Problem
	9.6 Shape Optimization Problem of Domain Variation Type
	9.7 Existence of an Optimum Solution
	9.8 Derivatives of Cost Functions
		9.8.1 Shape Derivative of fi Using Formulae Based on Shape Derivative of a Function
		9.8.2 Second-Order Shape Derivative of fi Using Formulae Based on Shape Derivative of a Function
		9.8.3 Second-Order Shape Derivative of Cost Function Using Lagrange Multiplier Method
		9.8.4 Shape Derivative of fi Using Formulae Based on Partial Shape Derivative of a Function
	9.9 Descent Directions of Cost Functions
		9.9.1 H1 Gradient Method
			Method Using the Inner Product in H1 Space
			Method Using Boundary Condition
			Regularity of the H1 Gradient Method
		9.9.2 H1 Newton Method
	9.10 Solution to Shape Optimization Problem of Domain Variation Type
		9.10.1 Gradient Method for Constrained Problems
		9.10.2 Newton Method for Constrained Problems
	9.11 Error Estimation
	9.12 Shape Optimization Problem of Linear Elastic Body
		9.12.1 State Determination Problem
		9.12.2 Mean Compliance Minimization Problem
		9.12.3 Shape Derivatives of Cost Functions
			Shape Derivatives of f0 and f1 Using Formulae Based on Shape Derivative of a Function
			Second-Order Shape Derivatives of f0 and f1 Using Formulae Based on Shape Derivative of a Function
			Second-Order Shape Derivative of Cost Function Using Lagrange Multiplier Method
			Shape Derivatives of f0 and f1 Using Formulae Based on Partial Shape Derivative of a Function
		9.12.4 Relation with Optimal Design Problem of Stepped One-Dimensional Linear Elastic Body
		9.12.5 Numerical Example
	9.13 Shape Optimization Problem of Stokes Flow Field
		9.13.1 State Determination Problem
		9.13.2 Mean Flow Resistance Minimization Problem
		9.13.3 Shape Derivatives of Cost Functions
			Shape Derivative of f0 Using Formulae Based on Shape Derivative of a Function
			Second-Order Shape Derivative of f0 Using Formulae Based on Shape Derivative of a Function
			Second-Order Shape Derivative of Cost Function Using Lagrange Multiplier Method
			Shape Derivative of f0 Using Formulae Based on Partial Shape Derivative of a Function
		9.13.4 Relationship with Optimal Design Problem of One-Dimensional Branched Stokes Flow Field
		9.13.5 Numerical Example
	9.14 Summary
	9.15 Practice Problems
Appendices
	A.1 Basic Terminology
		A.1.1 Open Sets, Closed Sets and Bounded Sets
		A.1.2 Continuity of Functions
	A.2 Positive Definiteness of Real Symmetric Matrix
	A.3 Null Space, Image space and Farkas\'s Lemma
	A.4 Implicit Function Theorem
	A.5 Lipschitz Domain
	A.6 Heat Conduction Problem
		A.6.1 One-Dimensional Problem
		A.6.2 d-Dimensional Problem
	A.7 Classification of Second-Order Partial Differential Equations
	A.8 Divergence Theorems
	A.9 Inequalities
	A.10 Ascoli–Arzelà Theorem
Answers to Practice Problems
	Chapter 1
	Chapter 2
	Chapter 3
	Chapter 4
	Chapter 5
	Chapter 6
	Chapter 8
	Chapter 9
Afterword
References
Index