On the Topological Derivative in Shape Optimization
SIAM Journal on Control and Optimization
Shapes and geometries: analysis, differential calculus, and optimization
Shapes and geometries: analysis, differential calculus, and optimization
The Topological Asymptotic for PDE Systems: The Elasticity Case
SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Level set method with topological derivatives in shape optimization
International Journal of Computer Mathematics - INNOVATIVE ALGORITHMS IN SCIENCE AND ENGINEERING
A Level Set Method in Shape and Topology Optimization for Variational Inequalities
International Journal of Applied Mathematics and Computer Science - Scientific Computation for Fluid Mechanics and Hyperbolic Systems
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Asymptotic formulae for the mechanical and electric fields in a piezoelectric body with a small void are derived and justified. Such results are new and useful for applications in the field of design of smart materials. In this way the topological derivatives of shape functionals are obtained for piezoelectricity. The asymptotic formulae are given in terms of the so-called polarization tensors (matrices), which are determined by the integral characteristics of voids. The distinguishing feature of the piezoelectricity boundary value problems under consideration is the absence of positive definiteness of a differential operator which is non-self-adjoint. Two specific Gibbs functionals of the problem are defined by the energy and the electric enthalpy. The topological derivatives are defined in different manners for each of the governing functionals. Actually, the topological derivative of the enthalpy functional is local, i.e., defined by the pointwise values of the mechanical and electric fields, which is contrary to the energy functional and some other suitable shape functionals which admit nonlocal topological derivatives, i.e., depending on the whole problem data. An example with weak interaction between mechanical and electric fields provides the analytic asymptotic expansions and can be used in numerical procedures of optimal design for smart materials.