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A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method

R. Ortigosa, D. Ruiz, Antonio Gil Orcid Logo, A. Donoso, J.C. Bellido

Computer Methods in Applied Mechanics and Engineering, Volume: 364, Start page: 112924

Swansea University Author: Antonio Gil Orcid Logo

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Abstract

This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solv...

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Published in: Computer Methods in Applied Mechanics and Engineering
ISSN: 0045-7825
Published: Elsevier BV 2020
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fullrecord <?xml version="1.0"?><rfc1807><datestamp>2020-02-13T11:58:11.4129895</datestamp><bib-version>v2</bib-version><id>53511</id><entry>2020-02-13</entry><title>A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method</title><swanseaauthors><author><sid>1f5666865d1c6de9469f8b7d0d6d30e2</sid><ORCID>0000-0001-7753-1414</ORCID><firstname>Antonio</firstname><surname>Gil</surname><name>Antonio Gil</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2020-02-13</date><deptcode>CIVL</deptcode><abstract>This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton&#x2013;Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton&#x2013;Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10 -12 .</abstract><type>Journal Article</type><journal>Computer Methods in Applied Mechanics and Engineering</journal><volume>364</volume><paginationStart>112924</paginationStart><publisher>Elsevier BV</publisher><issnPrint>0045-7825</issnPrint><keywords>Topology optimisation, SIMP method, Nonlinear elasticity, Polyconvexity</keywords><publishedDay>1</publishedDay><publishedMonth>6</publishedMonth><publishedYear>2020</publishedYear><publishedDate>2020-06-01</publishedDate><doi>10.1016/j.cma.2020.112924</doi><url/><notes/><college>COLLEGE NANME</college><department>Civil Engineering</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>CIVL</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2020-02-13T11:58:11.4129895</lastEdited><Created>2020-02-13T11:58:11.4129895</Created><authors><author><firstname>R.</firstname><surname>Ortigosa</surname><order>1</order></author><author><firstname>D.</firstname><surname>Ruiz</surname><order>2</order></author><author><firstname>Antonio</firstname><surname>Gil</surname><orcid>0000-0001-7753-1414</orcid><order>3</order></author><author><firstname>A.</firstname><surname>Donoso</surname><order>4</order></author><author><firstname>J.C.</firstname><surname>Bellido</surname><order>5</order></author></authors><documents><document><filename>53511__16629__5de85609706e4db6b3af7e324cec74ed.pdf</filename><originalFilename>ortigosa2020.pdf</originalFilename><uploaded>2020-02-19T15:10:53.8784672</uploaded><type>Output</type><contentLength>5106405</contentLength><contentType>application/pdf</contentType><version>Accepted Manuscript</version><cronfaStatus>true</cronfaStatus><embargoDate>2021-03-07T00:00:00.0000000</embargoDate><documentNotes>Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND).</documentNotes><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807>
spelling 2020-02-13T11:58:11.4129895 v2 53511 2020-02-13 A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method 1f5666865d1c6de9469f8b7d0d6d30e2 0000-0001-7753-1414 Antonio Gil Antonio Gil true false 2020-02-13 CIVL This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10 -12 . Journal Article Computer Methods in Applied Mechanics and Engineering 364 112924 Elsevier BV 0045-7825 Topology optimisation, SIMP method, Nonlinear elasticity, Polyconvexity 1 6 2020 2020-06-01 10.1016/j.cma.2020.112924 COLLEGE NANME Civil Engineering COLLEGE CODE CIVL Swansea University 2020-02-13T11:58:11.4129895 2020-02-13T11:58:11.4129895 R. Ortigosa 1 D. Ruiz 2 Antonio Gil 0000-0001-7753-1414 3 A. Donoso 4 J.C. Bellido 5 53511__16629__5de85609706e4db6b3af7e324cec74ed.pdf ortigosa2020.pdf 2020-02-19T15:10:53.8784672 Output 5106405 application/pdf Accepted Manuscript true 2021-03-07T00:00:00.0000000 Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND). true eng
title A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method
spellingShingle A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method
Antonio Gil
title_short A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method
title_full A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method
title_fullStr A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method
title_full_unstemmed A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method
title_sort A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method
author_id_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2
author_id_fullname_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2_***_Antonio Gil
author Antonio Gil
author2 R. Ortigosa
D. Ruiz
Antonio Gil
A. Donoso
J.C. Bellido
format Journal article
container_title Computer Methods in Applied Mechanics and Engineering
container_volume 364
container_start_page 112924
publishDate 2020
institution Swansea University
issn 0045-7825
doi_str_mv 10.1016/j.cma.2020.112924
publisher Elsevier BV
document_store_str 1
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description This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10 -12 .
published_date 2020-06-01T04:07:46Z
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