{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/109561"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/109561","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The topological derivative and its applications to fracture-based analysis and design","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Alidoost, Kazem"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Theoretical & Applied Mechans","degree_department":null,"school":null,"contributors":["Tortorelli, Daniel A","Geubelle, Philippe H","James, Kai","Masud, Arif"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-03-05T21:45:09Z","date_published":"2021-03-05T21:45:09Z","updated_at":"2026-07-22T22:24:50Z","subjects":["Asymptotic Analysis","Computational Mechanics","Edge Cracks","Energy Release Rate","Surface Cracks","Shape Optimization","Topological Derivative"],"languages":["en"],"rights":["Copyright 2020 Kazem Alidoost"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/109561","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Tortorelli, Daniel A","Geubelle, Philippe H","James, Kai","Masud, Arif"]},{"key":"dc:creator","label":"Author","values":["Alidoost, Kazem"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2021-03-05T21:45:09Z","2023-03-05T21:47:41Z","2020-09-15","2020-12"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Theoretical & Applied Mechans"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Asymptotic Analysis","Computational Mechanics","Edge Cracks","Energy Release Rate","Surface Cracks","Shape Optimization","Topological Derivative"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Kazem Alidoost"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/109561"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2022-12-01","The student, Kazem Alidoost, accepted the attached license on 2020-09-08 at 18:53.","The student, Kazem Alidoost, submitted this Dissertation for approval on 2020-09-08 at 19:07.","This Dissertation was approved for publication on 2020-09-15 at 14:16.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15804 on 2021-03-04 at 16:30:14","Made available in DSpace on 2021-03-05T21:45:09Z (GMT). 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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.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2022-12-01","The student, Kazem Alidoost, accepted the attached license on 2020-09-08 at 18:53.","The student, Kazem Alidoost, submitted this Dissertation for approval on 2020-09-08 at 19:07.","This Dissertation was approved for publication on 2020-09-15 at 14:16.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15804 on 2021-03-04 at 16:30:14","Made available in DSpace on 2021-03-05T21:45:09Z (GMT). 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