{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/101764"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/101764","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Continuum Analytical Shape Sensitivity Analysis of 1-D Elastic Bar","abstract":"In this thesis, a continuum sensitivity analysis method is presented for calculation of shape sensitivities of an elastic bar. The governing differential equations and boundary conditions for the elastic bar are differentiated with respect to the shape design parameter to derive the continuum sensitivity equations. The continuum sensitivity equations are linear ordinary differential equations in terms of local or material shape design derivatives, otherwise known as shape sensitivities. One of the novelties of this work is the derivation of three variational formulations for obtaining shape sensitivities, one in terms of the local sensitivity and two in terms of the material sensitivity. These derivations involve evaluating (a) the variational form of the continuum sensitivity equations, or (b) the sensitivity of the variational form of the analysis equations. We demonstrate their implementation for various combinations of design velocity and global basis functions. These variational formulations are further solved using finite element analysis. The order of convergence of each variational formulation is determined by comparing the sensitivity solutions with the exact solutions for analytical test cases. This research focusses on 1-D structural equations. In future work, the three variational formulations can be derived for 2-D and 3-D structural and fluid domains.","abstract_html":"In this thesis, a continuum sensitivity analysis method is presented for calculation of shape sensitivities of an elastic bar. The governing differential equations and boundary conditions for the elastic bar are differentiated with respect to the shape design parameter to derive the continuum sensitivity equations. The continuum sensitivity equations are linear ordinary differential equations in terms of local or material shape design derivatives, otherwise known as shape sensitivities. One of the novelties of this work is the derivation of three variational formulations for obtaining shape sensitivities, one in terms of the local sensitivity and two in terms of the material sensitivity. These derivations involve evaluating (a) the variational form of the continuum sensitivity equations, or (b) the sensitivity of the variational form of the analysis equations. We demonstrate their implementation for various combinations of design velocity and global basis functions. These variational formulations are further solved using finite element analysis. The order of convergence of each variational formulation is determined by comparing the sensitivity solutions with the exact solutions for analytical test cases. This research focusses on 1-D structural equations. In future work, the three variational formulations can be derived for 2-D and 3-D structural and fluid domains.","abstract_has_math":false,"creators":["Nayak, Soumya Sambit"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mechanical Engineering","degree_department":"Mechanical Engineering","school":null,"contributors":[],"advisors":[],"committee_chairs":["Patil, Mayuresh J.","Acar, Pinar"],"committee_members":["Tafti, Danesh K."],"year":2021,"date_issued":"2021-01-06","date_published":"2021-01-06","updated_at":"2026-07-22T22:20:32Z","subjects":["Continuum Sensitivity Analysis","CASSA","material sensitivity","local sensitivity","design velocity","variational form","convergence"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:28843"],"render_values":[{"text":"vt_gsexam:28843","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/101764","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Patil, Mayuresh J.","Acar, Pinar"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Tafti, Danesh K."]},{"key":"dc:contributor.department","label":"Department","values":["Mechanical Engineering"]},{"key":"dc:creator","label":"Author","values":["Nayak, Soumya Sambit"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2021-01-07T09:00:19Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2021-01-07T09:00:19Z"]},{"key":"dc:date.issued","label":"Date","values":["2021-01-06"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mechanical Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Continuum Sensitivity Analysis","CASSA","material sensitivity","local sensitivity","design velocity","variational form","convergence"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:28843"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/101764"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, a continuum sensitivity analysis method is presented for calculation of shape sensitivities of an elastic bar. The governing differential equations and boundary conditions for the elastic bar are differentiated with respect to the shape design parameter to derive the continuum sensitivity equations. The continuum sensitivity equations are linear ordinary differential equations in terms of local or material shape design derivatives, otherwise known as shape sensitivities. One of the novelties of this work is the derivation of three variational formulations for obtaining shape sensitivities, one in terms of the local sensitivity and two in terms of the material sensitivity. These derivations involve evaluating (a) the variational form of the continuum sensitivity equations, or (b) the sensitivity of the variational form of the analysis equations. We demonstrate their implementation for various combinations of design velocity and global basis functions. These variational formulations are further solved using finite element analysis. The order of convergence of each variational formulation is determined by comparing the sensitivity solutions with the exact solutions for analytical test cases. This research focusses on 1-D structural equations. In future work, the three variational formulations can be derived for 2-D and 3-D structural and fluid domains."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["When solving an optimization problem, the extreme value of the performance metric of interest is calculated by tuning the values of the design variables. Some optimization problems involve shape change as one of the design variables. Change in shape leads to change in the boundary locations. This leads to a change in the domain definition and the boundary conditions. We consider a 1-D structural element, an elastic bar, for this study. Subsequently, we demonstrate a method for calculating the sensitivity of solution (e.g. displacement at a point) to change in the shape (length for 1-D case) of the elastic bar. These sensitivities, known as shape sensitivities, are critical for design optimization problems. We make use of continuum analytical shape sensitivity analysis to derive three variational formulations to compute these shape sensitivities. The accuracy and convergence of solutions is verified using a finite element analysis code. In future, the approach can be extended to multi-dimensional structural and fluid domain problems."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Continuum Analytical Shape Sensitivity Analysis of 1-D Elastic Bar"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Patil, Mayuresh J.","Acar, Pinar"],"dc:contributor.committeemember":["Tafti, Danesh K."],"dc:contributor.department":["Mechanical Engineering"],"dc:creator":["Nayak, Soumya Sambit"],"dc:date.accessioned":["2021-01-07T09:00:19Z"],"dc:date.available":["2021-01-07T09:00:19Z"],"dc:date.issued":["2021-01-06"],"dc:description.abstract":["In this thesis, a continuum sensitivity analysis method is presented for calculation of shape sensitivities of an elastic bar. The governing differential equations and boundary conditions for the elastic bar are differentiated with respect to the shape design parameter to derive the continuum sensitivity equations. The continuum sensitivity equations are linear ordinary differential equations in terms of local or material shape design derivatives, otherwise known as shape sensitivities. One of the novelties of this work is the derivation of three variational formulations for obtaining shape sensitivities, one in terms of the local sensitivity and two in terms of the material sensitivity. These derivations involve evaluating (a) the variational form of the continuum sensitivity equations, or (b) the sensitivity of the variational form of the analysis equations. We demonstrate their implementation for various combinations of design velocity and global basis functions. These variational formulations are further solved using finite element analysis. The order of convergence of each variational formulation is determined by comparing the sensitivity solutions with the exact solutions for analytical test cases. This research focusses on 1-D structural equations. In future work, the three variational formulations can be derived for 2-D and 3-D structural and fluid domains."],"dc:description.abstractgeneral":["When solving an optimization problem, the extreme value of the performance metric of interest is calculated by tuning the values of the design variables. Some optimization problems involve shape change as one of the design variables. Change in shape leads to change in the boundary locations. This leads to a change in the domain definition and the boundary conditions. We consider a 1-D structural element, an elastic bar, for this study. Subsequently, we demonstrate a method for calculating the sensitivity of solution (e.g. displacement at a point) to change in the shape (length for 1-D case) of the elastic bar. These sensitivities, known as shape sensitivities, are critical for design optimization problems. We make use of continuum analytical shape sensitivity analysis to derive three variational formulations to compute these shape sensitivities. The accuracy and convergence of solutions is verified using a finite element analysis code. In future, the approach can be extended to multi-dimensional structural and fluid domain problems."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:28843"],"dc:identifier.uri":["http://hdl.handle.net/10919/101764"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Continuum Sensitivity Analysis","CASSA","material sensitivity","local sensitivity","design velocity","variational form","convergence"],"dc:title":["Continuum Analytical Shape Sensitivity Analysis of 1-D Elastic Bar"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mechanical Engineering"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:32Z"}