On the design of multimaterial structural topologies using Integer Programming
This paper presents a framework for the discrete design of optimal multimaterial structural topologies using integer design variables and mathematical programming. The structural optimization problems: compliance minimization subject to mass constraint, and mass minimization subject to compliance co...
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| Vydáno v: | Computer methods in applied mechanics and engineering Ročník 384; s. 114000 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Amsterdam
Elsevier B.V
01.10.2021
Elsevier BV |
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| ISSN: | 0045-7825, 1879-2138 |
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| Abstract | This paper presents a framework for the discrete design of optimal multimaterial structural topologies using integer design variables and mathematical programming. The structural optimization problems: compliance minimization subject to mass constraint, and mass minimization subject to compliance constraint are used to design the multimaterial topologies in this work. The extended SIMP interpolation is used to interpolate the different materials available for structural design, and the material phases in each element are represented using binary design variables, one variable per available material. The Topology Optimization of Binary Structure (TOBS) method (Sivapuram and Picelli, 2018) is employed, wherein the nonlinear objective/constraint functions of optimization are sequentially approximated (herein, linearized) to obtain a sequence of Integer Linear Programs (ILPs). A novel truncation error-regulating constraint in terms of the Young’s moduli of the elements is introduced to maintain the sequential approximations valid, by restricting large changes in successive structural topologies. A commercial branch-and-bound solver is used to solve the integer subproblems yielding perfectly binary solutions which guarantee discrete structural topologies with clear material interfaces at each iteration. Adjoint sensitivities are computed to generate the integer subproblems, and the sensitivities are filtered using a conventional mesh-independent sensitivity filter. Few examples show the design of multimaterial structures in the presence of design-dependent loads: hydrostatic pressure loads and self-weight loads. This work also demonstrates through few examples, convergence of optimal multimaterial topologies at inactive constraint values when different type of loadings simultaneously act on the structure.
•Integer programming is used in multimaterial structural optimization.•The optimized structures have clear boundaries and clear material interfaces.•A novel truncation error-regulating constraint is proposed for multimaterial problems.•Examples include models with design-dependent pressure and self-weight loads.•Convergence at inactive mass constraint values is demonstrated through few examples. |
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| AbstractList | This paper presents a framework for the discrete design of optimal multimaterial structural topologies using integer design variables and mathematical programming. The structural optimization problems: compliance minimization subject to mass constraint, and mass minimization subject to compliance constraint are used to design the multimaterial topologies in this work. The extended SIMP interpolation is used to interpolate the different materials available for structural design, and the material phases in each element are represented using binary design variables, one variable per available material. The Topology Optimization of Binary Structure (TOBS) method (Sivapuram and Picelli, 2018) is employed, wherein the nonlinear objective/constraint functions of optimization are sequentially approximated (herein, linearized) to obtain a sequence of Integer Linear Programs (ILPs). A novel truncation error-regulating constraint in terms of the Young's moduli of the elements is introduced to maintain the sequential approximations valid, by restricting large changes in successive structural topologies. A commercial branch-and-bound solver is used to solve the integer subproblems yielding perfectly binary solutions which guarantee discrete structural topologies with clear material interfaces at each iteration. Adjoint sensitivities are computed to generate the integer subproblems, and the sensitivities are filtered using a conventional mesh-independent sensitivity filter. Few examples show the design of multimaterial structures in the presence of design-dependent loads: hydrostatic pressure loads and self-weight loads. This work also demonstrates through few examples, convergence of optimal multimaterial topologies at inactive constraint values when different type of loadings simultaneously act on the structure. This paper presents a framework for the discrete design of optimal multimaterial structural topologies using integer design variables and mathematical programming. The structural optimization problems: compliance minimization subject to mass constraint, and mass minimization subject to compliance constraint are used to design the multimaterial topologies in this work. The extended SIMP interpolation is used to interpolate the different materials available for structural design, and the material phases in each element are represented using binary design variables, one variable per available material. The Topology Optimization of Binary Structure (TOBS) method (Sivapuram and Picelli, 2018) is employed, wherein the nonlinear objective/constraint functions of optimization are sequentially approximated (herein, linearized) to obtain a sequence of Integer Linear Programs (ILPs). A novel truncation error-regulating constraint in terms of the Young’s moduli of the elements is introduced to maintain the sequential approximations valid, by restricting large changes in successive structural topologies. A commercial branch-and-bound solver is used to solve the integer subproblems yielding perfectly binary solutions which guarantee discrete structural topologies with clear material interfaces at each iteration. Adjoint sensitivities are computed to generate the integer subproblems, and the sensitivities are filtered using a conventional mesh-independent sensitivity filter. Few examples show the design of multimaterial structures in the presence of design-dependent loads: hydrostatic pressure loads and self-weight loads. This work also demonstrates through few examples, convergence of optimal multimaterial topologies at inactive constraint values when different type of loadings simultaneously act on the structure. •Integer programming is used in multimaterial structural optimization.•The optimized structures have clear boundaries and clear material interfaces.•A novel truncation error-regulating constraint is proposed for multimaterial problems.•Examples include models with design-dependent pressure and self-weight loads.•Convergence at inactive mass constraint values is demonstrated through few examples. |
| ArticleNumber | 114000 |
| Author | Sivapuram, Raghavendra Yi, Bing Picelli, Renato Yoon, Gil Ho |
| Author_xml | – sequence: 1 givenname: Raghavendra orcidid: 0000-0002-1639-9349 surname: Sivapuram fullname: Sivapuram, Raghavendra email: rsivapur@eng.ucsd.edu organization: Structural Engineering, University of California San Diego, La Jolla, CA 92093, USA – sequence: 2 givenname: Renato surname: Picelli fullname: Picelli, Renato organization: Department of Mining and Petroleum Engineering, University of São Paulo, Praca Narçiso de Andrade s/n, Vila Mathias, Santos - SP, 11013-560, Brazil – sequence: 3 givenname: Gil Ho surname: Yoon fullname: Yoon, Gil Ho organization: School of Mechanical Engineering, Hanyang University, Seoul, South Korea – sequence: 4 givenname: Bing orcidid: 0000-0001-7102-4796 surname: Yi fullname: Yi, Bing organization: School of Traffic and Transportation Engineering, Central South University, Changsha, China |
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| Keywords | Multimaterial Extended SIMP Pressure loads Binary variables Inactive constraint Truncation error |
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| SubjectTerms | Binary variables Extended SIMP Hydrostatic pressure Inactive constraint Integer programming Interpolation Iterative methods Loads (forces) Mathematical programming Modulus of elasticity Multimaterial Optimization Pressure dependence Pressure loads Sensitivity Structural design Topology optimization Truncation error Truncation errors |
| Title | On the design of multimaterial structural topologies using Integer Programming |
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