49Q10 Optimization of shapes other than minimal surfaces [See also 90C90]
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Language
- English (5)
Keywords
- Lagrange method, shape derivative, non-linear PDE, material derivative (1)
- Shape optimization (1)
- constrained minimization (1)
- distortion, phase transition, shape optimisation, speed method, transmission problem (1)
- distortion, phase transition, shape optimization, speed method, transmission problem (1)
- electrical impedance tomography (1)
- level set method (1)
- shape and topological derivative (1)
- structure optimization (1)
- velocity method (1)
Application Area
- C (5)
We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece
changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface.
A mechanical equilibrium problem for a material consisting of two components with dierent densities is considered. Due to the heterogeneous material densities, the
outer shape of the underlying workpiece can be changed by shifting the interface between
the subdomains. In this paper, the problem is modeled as a shape design problem for optimally compensating unwanted workpiece changes. The associated control variable is
the interface. Regularity results for transmission problems are employed for a rigorous
derivation of suitable first-order optimality conditions based on the speed method. The paper concludes with several numerical results based on a spline approximation of the
interface.
In this paper a new theorem is formulated
which allows a rigorous proof of the shape differentiability without the usage of the material derivative; the domain expression is automatically obtained and the boundary expression is easy to derive.
Furthermore, the theorem is applied to a cost function which depends on a quasi-linear transmission
problem. Using a Gagliardo penalization the existence of optimal shapes is established.
The well-known structure theorem of Hadamard-Zol\'esio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. However a volume representation (distributed shape derivative) is more general than the boundary form and allows to work with shapes having a lower regularity.
It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithm. In this paper we describe the numerous advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We give several examples of numerical applications such as the inverse conductivity problem and the level set method.
The ability of velocity methods to describe changes
of topology by creating defects like holes is investigated. For the
shape optimization energy-type objective functions are considered,
which depend on the geometry by means of state variables. The
state system is represented by abstract, quadratic, constrained
minimization problems stated over domains with defects. The velocity
method provides the shape derivative of the objective function
due to finite variations of a defect. Sufficient conditions are
derived which allow us to pass the shape derivative to the limit
with respect to diminishing defect, thus, to obtain the topological
derivative" of the objective function due to a topology change.
An illustrative example is presented for a circular hole bored at
the tip of a crack