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Shape Derivative for Penalty-Constrained Nonsmooth–Nonconvex Optimization: Cohesive Crack Problem
A class of non-smooth and non-convex optimization problems with penalty constraints linked to variational inequalities is studied with respect to its...
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Shape Sensitivity of Eigenvalue Functionals for Scalar Problems: Computing the Semi-derivative of a Minimum
This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators...
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Basic Derivative Rules
The definition of the derivative function given in M-Box 10.2 is time-consuming to apply to every function...
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Derivative Analysis
After adding a new specific analytical protocol, this chapter applies the theory and methodological tools described in the first and second parts of...
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Shape optimization for interface identification in nonlocal models
Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider...
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The Derivative Graphically
In the previous chapter we estimated the slope of the tangent line at one specific point. We now take a qualitative look at the slopes of tangent...
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PDE-Constrained Shape Optimization: Toward Product Shape Spaces and Stochastic Models
Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which a...
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Shape Sensitivity Analysis of a 2D Fluid–Structure Interaction Problem
We study the shape differentiability of a general functional depending on the solution of a bidimensional stationary Stokes–elasticity system with...
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The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation
In this paper, we introduce the topological state derivative for general topological dilatations and explore its relation to standard optimal control...
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Shape Gradient Methods for Shape Optimization of an Unsteady Multiscale Fluid–Structure Interaction Model
We consider numerical shape optimization of a fluid–structure interaction model. The constrained system involves multiscale coupling of a...
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The Derivative
In studying SIR and other phenomena in Chap. 1, we contented ourselves with an intuitive, or heuristic, understanding of what a rate of change...
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On an effective approach in shape optimization problem for Stokes equation
In this paper, we deal with the numerical study of a shape optimization problem governed by Stokes system. More precisely, we propose an effective...
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Stochastic Shape Analysis
The chapter describes stochastic models of shapes from a Hamiltonian viewpoint, including Langevin models, Riemannian Brownian motions and stochastic...
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Network Design and Control: Shape and Topology Optimization for the Turnpike Property for the Wave Equation
The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint...
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Contact Shape Optimization
Contact shape optimization problems in 3D have a structure that can be effectively exploited by the TFETI-based methods introduced in Part III. The...
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How Do We Know the Shape of a Function?
Consider the graph of \(f(x)=x^3-3003x^2+3006000x\) in...
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Numerical Solution to a Free Boundary Problem for the Stokes Equation Using the Coupled Complex Boundary Method in Shape Optimization Setting
A new reformulation of a free boundary problem for the Stokes equations, which govern a viscous flow with an overdetermined condition on the free...
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Analysis of General Shape Optimization Problems in Nonlinear Acoustics
In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures....
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Geometric Aspects of Shape Optimization
We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical...
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PDE-Constrained Shape Optimization: Towards Product Shape Spaces and Stochastic Models
Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which a...