International Journal of Nonlinear Analysis and Applications
Deriving a Topology Function to Be Used in the Reaction Diffusion Equation for Optimization of Structures with Thermal and Mechanical Boundary Conditions
Document Type : Research Paper
Authors
Mechanical engineering, Semnan university, Semnan, Iran
Abstract
A topological derivative of the Lagrangian is required for optimization of structures with thermal and mechanical boundary conditions by the level-set method using the reaction diffusion equation. In this study, drawing on the relationship between the shape derivative and the topological derivative, the topological derivative of the Lagrangian was obtained by Reynolds' transport theorem. Given that introducing holes to the topology creates boundaries, the derivative was found by incorporating the boundary integral into the Reynolds' transport theorem and analyzing the stress over the hole boundaries. The temperature was assumed to be dependent on topology in the present study under thermal and mechanical boundary conditions. Placing a hole in the structure affects the temperature of the remaining elements. Penalty factor is enforced on thermal conductivity for removed elements, and the result is taken into consideration in the Laplace's equation expressing the steady-state conductive heat transfer.
Keywords
[1] G. Allaire, F. De Gournay, F. Jouve, AM. Toader, Structural optimization using topological and shape sensitivity
via a level set method, Control. Cybern. 34(1) (2005) 59-80.
[2] G. Allaire, F. Jouve, A.M. Toader, Structural Optimization Using Sensitivity Analysis and a Level Set Method,
Journal of Computational Physics.194 (2004) 363-393.
[3] S. Amstutz, A. Novotny, Topological optimization of structures subject to Von Mises stress constraints, Struct.
Multidiscip. Optim. 41 (2010)407-420.
[4] M.P. Bendsoe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method,
Comput. Methods Appl. Mech. Engrg. 71 (1988) 197-224.
[5] M.P. Bendsoe, Optimal shape design as a material distribution problem, Struct. Optim. 1 (1989) 193-202.
[6] MP. Bendsoe, O. Sigmund, Material interpolation schemes in topology optimization, Arch. Appl. Mech. 69(9-10)
(1999) 635-654.
[7] MP. Bendsoe, O. Sigmund, Topology optimization: theory, methods and applications. Springer. Berlin. (2003).
[8] M. Burger, B. Hackl, W. Ring, Incorporating topological derivatives into level set methods, J. Comput. Phys.
194(1) (2004) 344-362.
[9] J. Cea, S. Garreau, P. Guillaume, M. Masmoudi, The shape and topological optimizations Connection, Comput.
Methods. Appl. Mech. Eng. 188(4) (2000) 713-726.[10] K.K. Choi, N.H. Kim, Structural Sensitivity Analysis and Optimization, Springer, 2005.
[11] H. Chung, O. Amir, H. A. Kim, Level-set topology optimization considering nonlinear thermoelasticity, Comput.
Methods Appl. Mech. Engrg. (2019).
[12] J. Deaton, R. V. Grandhi, Topology Optimization of Thermal Structures with Stress Constraints, presented at the
54th AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, Boston, MA, (2013).
[13] S. Deng, K. Suresh, J. Joo, Stress-Constrained Thermo-Elastic Topology Optimization: A Topological Sensitivity
Approach, presented at the Proceedings of the ASME IDETC/CIE Conference, Buffalo, NY, USA, (2014).
[14] H.A. Eschenauer, H.A. Kobelev, A. Schumacher, Bubble Method for Topology and Shape Optimization of Structures, Struct. Multidisc. Optim. 8 (1994) 42-51.
[15] R. Feijoo, A. Novotny, E. Taroco, C. Padra, The topological-shape sensitivity method in two- dimensional linear
elasticity topology design, in: Applications of Computational Mechanics in Structures and Fluids, 2005.
[16] R.A. Feijoo, A.A. Novotny, E. Taroco, C. Padra, The topological shape sensitivity method in two-dimensional
linear elasticity topology design, J. Comput. Methods Sci. Eng. (2005).
[17] T. Gao and W. Zhang, Topology optimization involving thermo-elastic stress loads, Struct. Multidiscip.Optim.
42 (2010) 725-738.
[18] S. Garreau, P. Guillaume, M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM.
J. Control. Optim. 39 (2001) 1756.
[19] L. Li, H. A. Kim, Multiscale Topology Optimization of Thermoelastic Structures using Level Set Method, American Institute of Aeronautics and Astronautics, (2020).
[20] D. Li, X. Zhang, Topology Optimization of Thermo-Mechanical Continuum Structure, presented at the 2010
IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Canada, (2010).
[21] X. Liu, C. Wang, Y. Zhou, Topology optimization of thermoelastic structures using the guide- weight method,
Sci. China Technol. Sci. 57(5) (2014) 968-979.
[22] D. J. Neiferd, R. V. Grandhi, Level-set Topology Optimization of Thermoelastic Structures - A Comparison of
Compliance, Strain Energy, and Stress Objectives, American Institute of Aeronautics and Astronautics, (2018).
[23] J. Nocedal, S.J. Wright, Numerical optimization, Springer, (1999).
[24] A.A. Novotny, R.A. Feij oo, E. Taroco, C. Padra, Topological sensitivity analysis, Comput. Methods Appl. Mech.
Engrg. 192 (2003) 803-829.
[25] A. Novotny, R. Feij oo, E. Taroco, C. Padra, Topological sensitivity analysis for three- dimensional linear elasticity
problem, Comput. Methods Appl. Mech. Engrg. 196 (2007) 4354- 4364.
[26] J. S. Osher, F. Santosa, Level Set Methods for Optimization Problems Involving Geometry and Constraints: I.
Frequencies of a Two-Density Inhomogeneous Drum, J. Comput. Phys. 171(1) (2001) 272-288.
[27] M. Otomori, T. Yamada, K. Izui, S. Nishiwaki, Matlab code for a level set-based topology optimization method
using a reaction diffusion equation, Struct. Multidisc. Optim. 2014.
[28] D. Reynolds, J.P. McConnachie, W. Bettess, C. Christie, J.W. Bull, Reverse Adaptivity - a New Evolutionary
Tool for Structural Optimization, Int. J. Numer. Methods Eng. 45 (1999) 529-552.
[29] H. Rodrigues, P. Fernandes, A material based model for topology optimization of thermo- elastic Structures,
International Journal for Numerical Methods in Engineering, 38 (1995) 1951- 1965.
[30] A. Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von Lopchpositionierungkrieterien, PhD thesis. Universit at-Gesamthochschule Siegen. Siegen. Germany. 1995.
[31] J.A. Sethian, A. Wiegmann, Structural Boundary Design via Level Set and Immersed Interface Methods, J.
Comput. Phys. 163(2) (2000) 489-528.
[32] J. Sokolowski, A. Zochowski, On the Topological Derivative in Shape Optimization, SIAM Journal of Control
Optimization. 37 (1999) 1251-1272.
[33] M. Stolpe, K. Svanberg, An alternative interpolation scheme for minimum compliance topology optimization,
Struct. Multidiscip. Optim 22 (2001) 116-124.
[34] N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire, Y. Br´echet, Material interface effects on the topology
optimization of multi-phase structures using a level set method, Struct. Multidiscip. Optim. (2014) 1-22.
[35] X. Wang , MY. Wang, D. Guo, Structural shape and topology optimization in a level-set-based framework of
region representation, Struct. Multidisc. Optim. 27(1) (2004) 1-19.
[36] MY. Wang, S. Chen, X. Wang, Y. Mei, Design of multimaterial compliant mechanisms using level-set methods,
J. Mech. Des. 127 (2005) 941-956.
[37] M.Y. Wang, X.M. Wang, D.M. Guo, A Level Set Method for Structural Topology Optimization, Comput. Methods
Appl. Mech. Eng. 192 (2003) 227-246.
[38] M.Y. Wang, X.M. Wang, ’Color’ Level Sets: a Multiphase Method for Structural Topology Optimization with
Multiple Materials, Comput. Meth. Appl. Mech. Eng. 193 (2004) 469-496.[39] Q. Xia, M.Y. Wang, Topology optimization of thermo-elastic structures using level set method, Comput. Mech.
42 (2008) 837-857.
[40] Y.M. Xie, G.P. Steven, A Simple Evolutionary Procedure for Structural Optimization, Comput. Struct. 5(1993)
885-896.
[41] Y.M. Xie, G.P. Steven, Evolutionary Structural Optimization, Springer-Verlag London Limited. UK. 1997.
[42] Xing and Xianghua, A Finite Element Based Level Set Method for Structural Topology Optimization, PHD
thesis. 2009.
[43] T. Yamada, K. Izui, S. Nishiwaki, A. Takezawa, A topology optimization method based on the level set method
incorporating a fictitious interface energy, Comput. Methods Appl. Mech. Eng. 199 (2010) 2876-2891.
[44] X. Yang, Y. Li, Topology optimization to minimize the dynamic compliance of a bi-material plate in a thermal
environment, Struct. Multidiscip. Optim. 47 (2012) 399-408.
[45] X. Yang,Y. Li, Structural topology optimization on dynamic compliance at resonance frequency in thermal
environments, Struct. Multidiscip. Optim.. 49 (2013) 81-91.
[46] M. Yulin, WA. Xiaoming, level set method for structural topology optimization and its Applications, Adv. Eng.
Softw. 35(7) (2004a) 415- 441.
[47] W. Zhang, J. Yang, Y. Xu, T. Gao, Topology optimization of thermoelastic structures: mean compliance minimization or elastic strain energy minimization, Struct. Multidiscip. Optim. 49(3) (2013) 417-429.
via a level set method, Control. Cybern. 34(1) (2005) 59-80.
[2] G. Allaire, F. Jouve, A.M. Toader, Structural Optimization Using Sensitivity Analysis and a Level Set Method,
Journal of Computational Physics.194 (2004) 363-393.
[3] S. Amstutz, A. Novotny, Topological optimization of structures subject to Von Mises stress constraints, Struct.
Multidiscip. Optim. 41 (2010)407-420.
[4] M.P. Bendsoe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method,
Comput. Methods Appl. Mech. Engrg. 71 (1988) 197-224.
[5] M.P. Bendsoe, Optimal shape design as a material distribution problem, Struct. Optim. 1 (1989) 193-202.
[6] MP. Bendsoe, O. Sigmund, Material interpolation schemes in topology optimization, Arch. Appl. Mech. 69(9-10)
(1999) 635-654.
[7] MP. Bendsoe, O. Sigmund, Topology optimization: theory, methods and applications. Springer. Berlin. (2003).
[8] M. Burger, B. Hackl, W. Ring, Incorporating topological derivatives into level set methods, J. Comput. Phys.
194(1) (2004) 344-362.
[9] J. Cea, S. Garreau, P. Guillaume, M. Masmoudi, The shape and topological optimizations Connection, Comput.
Methods. Appl. Mech. Eng. 188(4) (2000) 713-726.[10] K.K. Choi, N.H. Kim, Structural Sensitivity Analysis and Optimization, Springer, 2005.
[11] H. Chung, O. Amir, H. A. Kim, Level-set topology optimization considering nonlinear thermoelasticity, Comput.
Methods Appl. Mech. Engrg. (2019).
[12] J. Deaton, R. V. Grandhi, Topology Optimization of Thermal Structures with Stress Constraints, presented at the
54th AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, Boston, MA, (2013).
[13] S. Deng, K. Suresh, J. Joo, Stress-Constrained Thermo-Elastic Topology Optimization: A Topological Sensitivity
Approach, presented at the Proceedings of the ASME IDETC/CIE Conference, Buffalo, NY, USA, (2014).
[14] H.A. Eschenauer, H.A. Kobelev, A. Schumacher, Bubble Method for Topology and Shape Optimization of Structures, Struct. Multidisc. Optim. 8 (1994) 42-51.
[15] R. Feijoo, A. Novotny, E. Taroco, C. Padra, The topological-shape sensitivity method in two- dimensional linear
elasticity topology design, in: Applications of Computational Mechanics in Structures and Fluids, 2005.
[16] R.A. Feijoo, A.A. Novotny, E. Taroco, C. Padra, The topological shape sensitivity method in two-dimensional
linear elasticity topology design, J. Comput. Methods Sci. Eng. (2005).
[17] T. Gao and W. Zhang, Topology optimization involving thermo-elastic stress loads, Struct. Multidiscip.Optim.
42 (2010) 725-738.
[18] S. Garreau, P. Guillaume, M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM.
J. Control. Optim. 39 (2001) 1756.
[19] L. Li, H. A. Kim, Multiscale Topology Optimization of Thermoelastic Structures using Level Set Method, American Institute of Aeronautics and Astronautics, (2020).
[20] D. Li, X. Zhang, Topology Optimization of Thermo-Mechanical Continuum Structure, presented at the 2010
IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Canada, (2010).
[21] X. Liu, C. Wang, Y. Zhou, Topology optimization of thermoelastic structures using the guide- weight method,
Sci. China Technol. Sci. 57(5) (2014) 968-979.
[22] D. J. Neiferd, R. V. Grandhi, Level-set Topology Optimization of Thermoelastic Structures - A Comparison of
Compliance, Strain Energy, and Stress Objectives, American Institute of Aeronautics and Astronautics, (2018).
[23] J. Nocedal, S.J. Wright, Numerical optimization, Springer, (1999).
[24] A.A. Novotny, R.A. Feij oo, E. Taroco, C. Padra, Topological sensitivity analysis, Comput. Methods Appl. Mech.
Engrg. 192 (2003) 803-829.
[25] A. Novotny, R. Feij oo, E. Taroco, C. Padra, Topological sensitivity analysis for three- dimensional linear elasticity
problem, Comput. Methods Appl. Mech. Engrg. 196 (2007) 4354- 4364.
[26] J. S. Osher, F. Santosa, Level Set Methods for Optimization Problems Involving Geometry and Constraints: I.
Frequencies of a Two-Density Inhomogeneous Drum, J. Comput. Phys. 171(1) (2001) 272-288.
[27] M. Otomori, T. Yamada, K. Izui, S. Nishiwaki, Matlab code for a level set-based topology optimization method
using a reaction diffusion equation, Struct. Multidisc. Optim. 2014.
[28] D. Reynolds, J.P. McConnachie, W. Bettess, C. Christie, J.W. Bull, Reverse Adaptivity - a New Evolutionary
Tool for Structural Optimization, Int. J. Numer. Methods Eng. 45 (1999) 529-552.
[29] H. Rodrigues, P. Fernandes, A material based model for topology optimization of thermo- elastic Structures,
International Journal for Numerical Methods in Engineering, 38 (1995) 1951- 1965.
[30] A. Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von Lopchpositionierungkrieterien, PhD thesis. Universit at-Gesamthochschule Siegen. Siegen. Germany. 1995.
[31] J.A. Sethian, A. Wiegmann, Structural Boundary Design via Level Set and Immersed Interface Methods, J.
Comput. Phys. 163(2) (2000) 489-528.
[32] J. Sokolowski, A. Zochowski, On the Topological Derivative in Shape Optimization, SIAM Journal of Control
Optimization. 37 (1999) 1251-1272.
[33] M. Stolpe, K. Svanberg, An alternative interpolation scheme for minimum compliance topology optimization,
Struct. Multidiscip. Optim 22 (2001) 116-124.
[34] N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire, Y. Br´echet, Material interface effects on the topology
optimization of multi-phase structures using a level set method, Struct. Multidiscip. Optim. (2014) 1-22.
[35] X. Wang , MY. Wang, D. Guo, Structural shape and topology optimization in a level-set-based framework of
region representation, Struct. Multidisc. Optim. 27(1) (2004) 1-19.
[36] MY. Wang, S. Chen, X. Wang, Y. Mei, Design of multimaterial compliant mechanisms using level-set methods,
J. Mech. Des. 127 (2005) 941-956.
[37] M.Y. Wang, X.M. Wang, D.M. Guo, A Level Set Method for Structural Topology Optimization, Comput. Methods
Appl. Mech. Eng. 192 (2003) 227-246.
[38] M.Y. Wang, X.M. Wang, ’Color’ Level Sets: a Multiphase Method for Structural Topology Optimization with
Multiple Materials, Comput. Meth. Appl. Mech. Eng. 193 (2004) 469-496.[39] Q. Xia, M.Y. Wang, Topology optimization of thermo-elastic structures using level set method, Comput. Mech.
42 (2008) 837-857.
[40] Y.M. Xie, G.P. Steven, A Simple Evolutionary Procedure for Structural Optimization, Comput. Struct. 5(1993)
885-896.
[41] Y.M. Xie, G.P. Steven, Evolutionary Structural Optimization, Springer-Verlag London Limited. UK. 1997.
[42] Xing and Xianghua, A Finite Element Based Level Set Method for Structural Topology Optimization, PHD
thesis. 2009.
[43] T. Yamada, K. Izui, S. Nishiwaki, A. Takezawa, A topology optimization method based on the level set method
incorporating a fictitious interface energy, Comput. Methods Appl. Mech. Eng. 199 (2010) 2876-2891.
[44] X. Yang, Y. Li, Topology optimization to minimize the dynamic compliance of a bi-material plate in a thermal
environment, Struct. Multidiscip. Optim. 47 (2012) 399-408.
[45] X. Yang,Y. Li, Structural topology optimization on dynamic compliance at resonance frequency in thermal
environments, Struct. Multidiscip. Optim.. 49 (2013) 81-91.
[46] M. Yulin, WA. Xiaoming, level set method for structural topology optimization and its Applications, Adv. Eng.
Softw. 35(7) (2004a) 415- 441.
[47] W. Zhang, J. Yang, Y. Xu, T. Gao, Topology optimization of thermoelastic structures: mean compliance minimization or elastic strain energy minimization, Struct. Multidiscip. Optim. 49(3) (2013) 417-429.
)
Volume 12, Special Issue
December 2021Pages 873-891
- Receive Date: 09 March 2021
- Revise Date: 15 June 2021
- Accept Date: 28 June 2021