{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/92764"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/92764","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A WENO finite difference scheme for a new class of Hamilton-Jacobi equations in electroelastostatics","abstract":"\"Hamilton-Jacobi equations have repeatedly emerged in many fields of physics, most notably, optimal control, differential games, geometric optics, and image processing. This thesis presents a new numerical method to solve a new class of Hamilton-Jacobi equation that has recently appeared in the context of nonlinear electroelastostatics. In a pioneering contribution, Crandall and Lions (1983) proved that a certain type of first-order finite difference method converges to the viscosity solution of a special class of Hamilton-Jacobi equations. From then on several successful methods of high-order approximation have been proposed in the literature, including the so-called WENO finite difference schemes. These schemes, however, were developed and tested for special classes of Hamilton-Jacobi equations, which do not include the general type of Hamilton-Jacobi equation of interest in this work. The objective of this thesis is to extend high-order WENO finite difference schemes to the most general type of Hamilton-Jacobi equations involving non-periodic boundary conditions in the \"\"space\"\" variables. Following its derivation, the proposed WENO scheme is tested for several cases involving one and two \"\"space\"\" variables for which there are analytical solutions available for arbitrarily large values of the \"\"time\"\" variable. These numerical experiments provide insight into the stability and rate of convergence of the method as \"\"time\"\" increases. They also provide insight into how errors propagate into the domain of computation due to non-periodic boundary conditions. This thesis concludes with the application of the method to compute the effective stored-energy function of an elastomer containing an isotropic distribution of vacuous pores under arbitrary 3D deformations.\"","abstract_html":""Hamilton-Jacobi equations have repeatedly emerged in many fields of physics, most notably, optimal control, differential games, geometric optics, and image processing. This thesis presents a new numerical method to solve a new class of Hamilton-Jacobi equation that has recently appeared in the context of nonlinear electroelastostatics. In a pioneering contribution, Crandall and Lions (1983) proved that a certain type of first-order finite difference method converges to the viscosity solution of a special class of Hamilton-Jacobi equations. From then on several successful methods of high-order approximation have been proposed in the literature, including the so-called WENO finite difference schemes. These schemes, however, were developed and tested for special classes of Hamilton-Jacobi equations, which do not include the general type of Hamilton-Jacobi equation of interest in this work. The objective of this thesis is to extend high-order WENO finite difference schemes to the most general type of Hamilton-Jacobi equations involving non-periodic boundary conditions in the ""space"" variables. Following its derivation, the proposed WENO scheme is tested for several cases involving one and two ""space"" variables for which there are analytical solutions available for arbitrarily large values of the ""time"" variable. These numerical experiments provide insight into the stability and rate of convergence of the method as ""time"" increases. They also provide insight into how errors propagate into the domain of computation due to non-periodic boundary conditions. This thesis concludes with the application of the method to compute the effective stored-energy function of an elastomer containing an isotropic distribution of vacuous pores under arbitrary 3D deformations."","abstract_has_math":false,"creators":["Garnica, Alvaro David"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Civil Engineering","degree_department":null,"school":null,"contributors":["Lopez-Pamies, Oscar"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-11-10T17:50:10Z","date_published":"2016-11-10T17:50:10Z","updated_at":"2026-07-22T22:26:35Z","subjects":["Electroelastotstatics","Hamilton-Jacobi equations","Homogenization","WENO scheme","Elastic energy"],"languages":["en"],"rights":["Copyright 2016 Alvaro Garnica"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/92764","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Lopez-Pamies, Oscar"]},{"key":"dc:creator","label":"Author","values":["Garnica, Alvaro David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2016-11-10T17:50:10Z","2016-07-06","2016-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Civil Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"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":["Electroelastotstatics","Hamilton-Jacobi equations","Homogenization","WENO scheme","Elastic energy"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2016 Alvaro Garnica"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/92764"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"Hamilton-Jacobi equations have repeatedly emerged in many fields of physics, most notably, optimal control, differential games, geometric optics, and image processing. This thesis presents a new numerical method to solve a new class of Hamilton-Jacobi equation that has recently appeared in the context of nonlinear electroelastostatics. In a pioneering contribution, Crandall and Lions (1983) proved that a certain type of first-order finite difference method converges to the viscosity solution of a special class of Hamilton-Jacobi equations. From then on several successful methods of high-order approximation have been proposed in the literature, including the so-called WENO finite difference schemes. These schemes, however, were developed and tested for special classes of Hamilton-Jacobi equations, which do not include the general type of Hamilton-Jacobi equation of interest in this work. The objective of this thesis is to extend high-order WENO finite difference schemes to the most general type of Hamilton-Jacobi equations involving non-periodic boundary conditions in the \"\"space\"\" variables. Following its derivation, the proposed WENO scheme is tested for several cases involving one and two \"\"space\"\" variables for which there are analytical solutions available for arbitrarily large values of the \"\"time\"\" variable. These numerical experiments provide insight into the stability and rate of convergence of the method as \"\"time\"\" increases. They also provide insight into how errors propagate into the domain of computation due to non-periodic boundary conditions. This thesis concludes with the application of the method to compute the effective stored-energy function of an elastomer containing an isotropic distribution of vacuous pores under arbitrary 3D deformations.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2016-11-09 without embargo terms","The student, Alvaro Garnica, accepted the attached license on 2016-07-06 at 14:06.","The student, Alvaro Garnica, submitted this Thesis for approval on 2016-07-06 at 14:13.","This Thesis was approved for publication on 2016-07-06 at 15:25.","DSpace SAF Submission Ingestion Package generated from Vireo submission #9784 on 2016-11-09 at 10:22:55","Made available in DSpace on 2016-11-10T17:50:10Z (GMT). No. of bitstreams: 2 GARNICA-THESIS-2016.pdf: 2902677 bytes, checksum: bb510cdb5876f44c6e9ac971d23564a0 (MD5) LICENSE.txt: 4211 bytes, checksum: 233cdabe22aa8a64487a665e5e54631d (MD5) Previous issue date: 2016-07-06"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A WENO finite difference scheme for a new class of Hamilton-Jacobi equations in electroelastostatics"]}]}],"canonical_facts":{"dc:contributor":["Lopez-Pamies, Oscar"],"dc:creator":["Garnica, Alvaro David"],"dc:date":["2016-11-10T17:50:10Z","2016-07-06","2016-08"],"dc:description":["\"Hamilton-Jacobi equations have repeatedly emerged in many fields of physics, most notably, optimal control, differential games, geometric optics, and image processing. This thesis presents a new numerical method to solve a new class of Hamilton-Jacobi equation that has recently appeared in the context of nonlinear electroelastostatics. In a pioneering contribution, Crandall and Lions (1983) proved that a certain type of first-order finite difference method converges to the viscosity solution of a special class of Hamilton-Jacobi equations. From then on several successful methods of high-order approximation have been proposed in the literature, including the so-called WENO finite difference schemes. These schemes, however, were developed and tested for special classes of Hamilton-Jacobi equations, which do not include the general type of Hamilton-Jacobi equation of interest in this work. The objective of this thesis is to extend high-order WENO finite difference schemes to the most general type of Hamilton-Jacobi equations involving non-periodic boundary conditions in the \"\"space\"\" variables. Following its derivation, the proposed WENO scheme is tested for several cases involving one and two \"\"space\"\" variables for which there are analytical solutions available for arbitrarily large values of the \"\"time\"\" variable. These numerical experiments provide insight into the stability and rate of convergence of the method as \"\"time\"\" increases. They also provide insight into how errors propagate into the domain of computation due to non-periodic boundary conditions. This thesis concludes with the application of the method to compute the effective stored-energy function of an elastomer containing an isotropic distribution of vacuous pores under arbitrary 3D deformations.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2016-11-09 without embargo terms","The student, Alvaro Garnica, accepted the attached license on 2016-07-06 at 14:06.","The student, Alvaro Garnica, submitted this Thesis for approval on 2016-07-06 at 14:13.","This Thesis was approved for publication on 2016-07-06 at 15:25.","DSpace SAF Submission Ingestion Package generated from Vireo submission #9784 on 2016-11-09 at 10:22:55","Made available in DSpace on 2016-11-10T17:50:10Z (GMT). No. of bitstreams: 2 GARNICA-THESIS-2016.pdf: 2902677 bytes, checksum: bb510cdb5876f44c6e9ac971d23564a0 (MD5) LICENSE.txt: 4211 bytes, checksum: 233cdabe22aa8a64487a665e5e54631d (MD5) Previous issue date: 2016-07-06"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/92764"],"dc:language":["en"],"dc:rights":["Copyright 2016 Alvaro Garnica"],"dc:subject":["Electroelastotstatics","Hamilton-Jacobi equations","Homogenization","WENO scheme","Elastic energy"],"dc:title":["A WENO finite difference scheme for a new class of Hamilton-Jacobi equations in electroelastostatics"],"dc:type":["text"],"thesis:degree_discipline":["Civil Engineering"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:35Z"}