# Higher order topological derivatives in elasticity

Research output: Contribution to journalArticle

### Abstract

The topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced into the domain. In this work, we compute higher order topological derivatives for elasticity problems, so that we are able to obtain better estimates of the response when holes of finite sizes are introduced in the domain. A critical element of our algorithm involves the asymptotic approximation for the stress on the hole boundary when the hole size approaches zero; it consists of a composite expansion that is based on the responses of elasticity problems on the domain without the hole and on a domain consisting of a hole in an infinite space. We present a simple example in which the higher order topological derivatives of the total potential energy are obtained analytically and by using the proposed asymptotic expansion. We also use the finite element method to verify the topological asymptotic expansion when the analytical solution is unknown.

Original language English (US) 3053-3066 14 International Journal of Solids and Structures 47 22-23 https://doi.org/10.1016/j.ijsolstr.2010.07.004 Published - Nov 1 2010

### Fingerprint

Topological Derivative
Higher order derivative
Elasticity
elastic properties
Derivatives
Elasticity Problem
Potential energy
expansion
Asymptotic Expansion
Finite element method
Functional Response
Composite materials
Asymptotic Approximation
Analytical Solution
finite element method
Finite Element Method
potential energy
Composite
Verify
Unknown

### Keywords

• Asymptotic analysis
• Higher order topological derivative
• Hole nucleation

### ASJC Scopus subject areas

• Modeling and Simulation
• Materials Science(all)
• Condensed Matter Physics
• Mechanics of Materials
• Mechanical Engineering
• Applied Mathematics

### Cite this

In: International Journal of Solids and Structures, Vol. 47, No. 22-23, 01.11.2010, p. 3053-3066.

Research output: Contribution to journalArticle

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