{"id":{"repo_id":"strathclyde","oai_identifier":"oai:strathclyde:jh343s35t"},"canonical_url":"https://search.dev.ndltd.org/etd/strathclyde/oai:strathclyde:jh343s35t","repository":{"repo_id":"strathclyde","name":"University of Strathclyde","base_url":"https://stax.strath.ac.uk/catalog/oai"},"display":{"title":"Patterns in an elastic bar","abstract":"We consider Yip's formulation of the Ericksen model for an elastic bar on an elastic foundation [63] which leads to the Euler-Lagrange equation for the functional ε(u) = ∫ between 0 and 1 (γu²xx +W(ux) + ɑu²)dx, where x is an element of the set (0, 1). with double Dirichlet boundary conditions. Here the potential W(p) = ((|p| - 1)²), is not differentiable at p = 0.;We define and prove existence and uniqueness of periodic solutions with any number n ≥ 0 of internal zeroes for all ɑ, γ > 0 and discuss the existence of non-periodic solutions.;The Euler-Lagrange equation contains conditions that make it diffcult to track, and then dropping one of them we obtain a weak formulation for this reduced problem,which we then prove it has a unique solution. Next, we use a combination of two numerical methods, namely the Finite Elements Method (FEM) to approximate the model and the Derivative Free Optimization (DFO) to find the location of the jump.","abstract_html":"We consider Yip's formulation of the Ericksen model for an elastic bar on an elastic foundation [63] which leads to the Euler-Lagrange equation for the functional ε(u) = ∫ between 0 and 1 (γu²xx +W(ux) + ɑu²)dx, where x is an element of the set (0, 1). with double Dirichlet boundary conditions. Here the potential W(p) = ((|p| - 1)²), is not differentiable at p = 0.;We define and prove existence and uniqueness of periodic solutions with any number n ≥ 0 of internal zeroes for all ɑ, γ > 0 and discuss the existence of non-periodic solutions.;The Euler-Lagrange equation contains conditions that make it diffcult to track, and then dropping one of them we obtain a weak formulation for this reduced problem,which we then prove it has a unique solution. Next, we use a combination of two numerical methods, namely the Finite Elements Method (FEM) to approximate the model and the Derivative Free Optimization (DFO) to find the location of the jump.","abstract_has_math":false,"creators":["Alruwaili, Abdulmohsen"],"institution":"University of Strathclyde","degree_name":"phd","degree_level":"doctoral-pg","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Grinfeld, Michael (Mathematician)","Barrenechea, Gabriel"],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020","date_published":"2020","updated_at":"2026-07-24T04:45:26Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.48730/g1ma-hz45"],"render_values":[{"text":"10.48730/g1ma-hz45","href":"https://doi.org/10.48730/g1ma-hz45","code":true}]},{"key":"dc:identifier","label":"Identifier","values":["T15539"],"render_values":[{"text":"T15539","href":null,"code":true}]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["201684026"],"render_values":[{"text":"201684026","href":null,"code":true}]}]},"links":{"outbound_url":"https://stax.strath.ac.uk/concern/theses/jh343s35t","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Grinfeld, Michael (Mathematician)","Barrenechea, Gabriel"]},{"key":"dc:creator","label":"Author","values":["Alruwaili, Abdulmohsen"]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["201684026"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020"]},{"key":"dc:date.issued","label":"Date","values":["2020"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Statistics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Strathclyde"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral-pg"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["phd"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["T15539"]},{"key":"dc:identifier.doi","label":"DOI","values":["10.48730/g1ma-hz45"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://stax.strath.ac.uk/concern/theses/jh343s35t"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We consider Yip's formulation of the Ericksen model for an elastic bar on an elastic foundation [63] which leads to the Euler-Lagrange equation for the functional ε(u) = ∫ between 0 and 1 (γu²xx +W(ux) + ɑu²)dx, where x is an element of the set (0, 1). with double Dirichlet boundary conditions. Here the potential W(p) = ((|p| - 1)²), is not differentiable at p = 0.;We define and prove existence and uniqueness of periodic solutions with any number n ≥ 0 of internal zeroes for all ɑ, γ > 0 and discuss the existence of non-periodic solutions.;The Euler-Lagrange equation contains conditions that make it diffcult to track, and then dropping one of them we obtain a weak formulation for this reduced problem,which we then prove it has a unique solution. Next, we use a combination of two numerical methods, namely the Finite Elements Method (FEM) to approximate the model and the Derivative Free Optimization (DFO) to find the location of the jump."]},{"key":"dc:description.abstract","label":"Abstract","values":["We consider Yip's formulation of the Ericksen model for an elastic bar on an elastic foundation [63] which leads to the Euler-Lagrange equation for the functional ε(u) = ∫ between 0 and 1 (γu²xx +W(ux) + ɑu²)dx, where x is an element of the set (0, 1). with double Dirichlet boundary conditions. Here the potential W(p) = ((|p| - 1)²), is not differentiable at p = 0.;We define and prove existence and uniqueness of periodic solutions with any number n ≥ 0 of internal zeroes for all ɑ, γ > 0 and discuss the existence of non-periodic solutions.;The Euler-Lagrange equation contains conditions that make it diffcult to track, and then dropping one of them we obtain a weak formulation for this reduced problem,which we then prove it has a unique solution. Next, we use a combination of two numerical methods, namely the Finite Elements Method (FEM) to approximate the model and the Derivative Free Optimization (DFO) to find the location of the jump."]},{"key":"dc:title","label":"Title","values":["Patterns in an elastic bar"]}]}],"canonical_facts":{"dc:contributor.advisor":["Grinfeld, Michael (Mathematician)","Barrenechea, Gabriel"],"dc:creator":["Alruwaili, Abdulmohsen"],"dc:creator.authoridentifier":["201684026"],"dc:date":["2020"],"dc:date.issued":["2020"],"dc:description":["We consider Yip's formulation of the Ericksen model for an elastic bar on an elastic foundation [63] which leads to the Euler-Lagrange equation for the functional ε(u) = ∫ between 0 and 1 (γu²xx +W(ux) + ɑu²)dx, where x is an element of the set (0, 1). with double Dirichlet boundary conditions. Here the potential W(p) = ((|p| - 1)²), is not differentiable at p = 0.;We define and prove existence and uniqueness of periodic solutions with any number n ≥ 0 of internal zeroes for all ɑ, γ > 0 and discuss the existence of non-periodic solutions.;The Euler-Lagrange equation contains conditions that make it diffcult to track, and then dropping one of them we obtain a weak formulation for this reduced problem,which we then prove it has a unique solution. Next, we use a combination of two numerical methods, namely the Finite Elements Method (FEM) to approximate the model and the Derivative Free Optimization (DFO) to find the location of the jump."],"dc:description.abstract":["We consider Yip's formulation of the Ericksen model for an elastic bar on an elastic foundation [63] which leads to the Euler-Lagrange equation for the functional ε(u) = ∫ between 0 and 1 (γu²xx +W(ux) + ɑu²)dx, where x is an element of the set (0, 1). with double Dirichlet boundary conditions. Here the potential W(p) = ((|p| - 1)²), is not differentiable at p = 0.;We define and prove existence and uniqueness of periodic solutions with any number n ≥ 0 of internal zeroes for all ɑ, γ > 0 and discuss the existence of non-periodic solutions.;The Euler-Lagrange equation contains conditions that make it diffcult to track, and then dropping one of them we obtain a weak formulation for this reduced problem,which we then prove it has a unique solution. Next, we use a combination of two numerical methods, namely the Finite Elements Method (FEM) to approximate the model and the Derivative Free Optimization (DFO) to find the location of the jump."],"dc:identifier":["T15539"],"dc:identifier.doi":["10.48730/g1ma-hz45"],"dc:identifier.uri":["https://stax.strath.ac.uk/concern/theses/jh343s35t"],"dc:publisher.department":["Department of Mathematics and Statistics"],"dc:publisher.institution":["University of Strathclyde"],"dc:title":["Patterns in an elastic bar"],"dc:type.qualificationlevel":["doctoral-pg"],"dc:type.qualificationname":["phd"]},"updated_at":"2026-07-24T04:45:26Z"}