The topological derivative and its applications to fracture-based analysis and design

Alidoost, Kazem

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https://hdl.handle.net/2142/109561 Copy

Description

Title

The topological derivative and its applications to fracture-based analysis and design

Author(s)

Alidoost, Kazem

Issue Date

2020-09-15

Director of Research (if dissertation) or Advisor (if thesis)

• Tortorelli, Daniel A
• Geubelle, Philippe H

Doctoral Committee Chair(s)

• Tortorelli, Daniel A
• Geubelle, Philippe H

Committee Member(s)

• James, Kai
• Masud, Arif

Department of Study

Mechanical Sci & Engineering

Discipline

Theoretical & Applied Mechans

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Asymptotic Analysis
• Computational Mechanics
• Edge Cracks
• Energy Release Rate
• Surface Cracks
• Shape Optimization
• Topological Derivative

Abstract

This thesis discusses the topological derivative and its application to fracture-based analysis and design. The topological derivative describes the variation of a response functional with respect to infinitesimal changes in topology, such as the introduction of an infinitesimal crack or hole. In a previous work, Silva et al. [1] developed a first-order approximation of the energy release rate field in a two-dimensional domain associated with a small edge crack at any boundary location and any orientation. In this thesis, we extend this work. We first develop higher-precision approximations of the energy release rate field using higher-order topological derivatives, which allow the analyst to accurately treat longer cracks and determine the crack lengths for which the first-order approximation is accurate. These higher-order topological derivatives are calculated using the so-called topological-shape sensitivity method [2]. We next propose an approximation of the energy release rate field in a three-dimensional domain associated with a small surface crack of any boundary location, direction, and orientation combination using the topological derivative. This approximation is computationally attractive because it only requires a single analysis. By contrast, current boundary element and finite element based methods require an analysis for each crack length-location-direction combination. Furthermore, this approximation is evaluated on the non-cracked domain, obviating the need for refined meshes in the crack tip region. We conclude by leveraging the efficiency and simplicity of the proposed approximation to develop a fracture- and gradient-based shape optimization scheme for the design of fracture-resistant linearly elastic structures. A key characteristic of the shape optimization scheme presented in this thesis is that the domain and its boundary are defined implicitly using level-set functions constructed with the aid of R-functions, which allow for the use of differentiable Boolean operations to combine the level-set functions of predefined simple geometries. This adoption of R-functions has the dual impact of (i) allowing shapes to merge and/or separate and (ii) simplifying the computation of the shape velocity fields.

Graduation Semester

2020-12

Type of Resource

Thesis

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http://hdl.handle.net/2142/109561

Copyright and License Information

Copyright 2020 Kazem Alidoost

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