Математика и математическое моделирование. 2019; : 19-34
Топологическая оптимизация конструкций, состоящих из нескольких материалов с использованием модифицированного метода SIMP
https://doi.org/10.24108/mathm.0619.0000211Аннотация
В данной работе предлагается использование модифицированного метода SIMP для решения задачи топологической оптимизации конструкций из мультифазных материалов, то есть констукций, содержащих компоненты из более, чем двух различных материалов, которые необходимо распределить наилучшим образом так, чтобы получить наилучшие структурные характеристики, используя только одну переменную проектирования.
Многие из существующих задач топологической оптимизации рассматривают только один материал и пустоту, хотя для некоторых случаев интересно взглянуть на топологию конструкции с несколькими фазами материала, то есть на задачи топологической оптимизации из мультифазных материалов. Тем не менее, большинство известных сегодня подходов для решения задач топологической оптимизации конструкций из мультифазных материалов требуют введения дополнительных переменных проектирования, тем самым увеличивая вычислительные затраты.
В статье при решении задачи топологической оптимизации конструкций из мультифазных материалов для аппроксимации свойств материалов используется модифицированный метод SIMP. В данном подходе плотность материала рассматривается как независимая расчетная переменная и выбирается из непрерывного диапазона, после чего разделяется дискретными значениями плотностей каждой из фаз материалов. Другие свойства рассматриваются как непрерывные функции от плотности. Предлагаемый метод не требует введения дополнительных переменных аппроксимации материалов и обеспечивает стабильный переход из одной фазы материала в другую, также расчетные затраты не зависят от количества рассматриваемых материалов.
Для демонстрации простоты и эффективности предлагаемого решения в работе приведены примеры решения задач топологической оптимизации для различных конструкций из мультифазных материалов. Благодаря своей концептуальной простоте, предлагаемый метод может быть легко применен для любых существующих задач топологической оптимизации. Рассмотренные в статье примеры показывают, что с помощью данного решения могут быть получены надежные конструкции с улучшенными механическими характеристиками, которые могут быть использованы для решения реальных задач проектирования и изготовления сложных конструкций.
Список литературы
1. Bendsoe M.P., Kikuchi N. Generating optimal topologies in structural design using a homogenization method // Computer Methods in Applied Mechanics and Engineering. 1988. Vol. 71. No. 2. Pp. 197–224. DOI: 10.1016/0045-7825(88)90086-2
2. Pedersen P., Pedersen N.L. Interpolation/penalization applied for strength design of 3D thermoelastic structures // Structural and Multidisciplinary Optimization. 2012. Vol. 45. No. 6. Pp. 773–786. DOI: 10.1007/s00158-011-0755-3
3. Pereira A., Talischi C., Paulino G.H., Menezes I.F.M., Carvalho M.S. Fluid flow topology optimization in PolyTop : stability and computational implementation // Structural and Multidisciplinary Optimization. 2016. Vol. 54. No. 5. Pp.1345–1364. DOI: 10.1007/s00158-014-1182-z
4. Yongbo Deng, Yihui Wu, Zhenyu Liu. Topology optimization theory for laminar flow. Singapore: Springer, [2018]. 249 p.
5. Ning Chen, Dejie Yu, Baizhan Xia, Jian Liu, Zhengdong Ma. Microstructural topology optimization of structural-acoustic coupled systems for minimizing sound pressure level // Structural and Multidisciplinary Optimization. 2017. Vol. 56. No. 6. Pp. 1259–1270. DOI: 10.1007/s00158-017-1718-0
6. Rozvany G.I.N., Lewinski T. Topology optimization in structural and continuum mechanics. [Wien]: Springer, 2013. 471 p.
7. Stanford B., Beran P. Conceptual design of compliant mechanisms for flapping wings with topology optimization // AIAA J. 2011. Vol. 49. No. 4. Pp. 855–867. DOI: 10.2514/1.J050940
8. Yuhang Chen, Shiwei Zhou, Qing Li. Multiobjective topology optimization for finite periodic structures // Computers & Structures. 2010. Vol. 88. No. 11-12. Pp. 806–811. DOI: 10.1016/j.compstruc.2009.10.003
9. Yuhang Chen, Shiwei Zhou, Qing Li. Computational design for multifunctional microstructural composites // Intern. J. of Modern Physics B. 2009. Vol. 23. No. 7. Pp. 1345–1351. DOI: 10.1142/S0217979209060920
10. Takezawa A., Gil Ho Yoon, Seung Hyun Jeong, Makoto Kobashi, Mitsuru Kitamura. Structural topology optimization with strength and heat conduction constraints // Computer Methods in Applied Mechanics and Engineering. 2014. Vol. 276. Pp. 341–361. DOI: 10.1016/j.cma.2014.04.003
11. Gil Ho Yoon. Topological layout design of electro-fluid-thermal-compliant actuator // Computer Methods in Applied Mechanics and Engineering. 2012. Vol. 209-212. Pp. 28–44. DOI: 10.1016/j.cma.2011.11.005
12. Xiaoping Qian, Sigmund O. Topological design of electromechanical actuators with robustness toward over- and under-etching // Computer Methods in Applied Mechanics and Engineering. 2013. Vol. 253. Pp. 237–251. DOI: 10.1016/j.cma.2012.08.020
13. Vatanabe S.L., Paulino G.H., Silva E.C.N. Design of functionally graded piezocomposites using topology optimization and homogenization - Toward effective energy harvesting materials // Computer Methods in Applied Mechanics and Engineering. 2013. Vol. 266. Pp. 205–218. DOI: 10.1016/j.cma.2013.07.003
14. Sutradhar A., Paulino G.H., Miller M.J., Nguyen T.H. Topological optimization for designing patient-specific large craniofacial segmental bone replacements // Proc. of the Nat. Acad. of Sciences of the USA. 2010. Vol. 107. No. 30. Pp. 13222–13227. DOI: 10.1073/pnas.1001208107
15. Heesuk Kang, Long J.P., Urbiel Goldner G.D., Goldstein S.A., Hollister S.J. A paradigm for the development and evaluation of novel implant topologies for bone fixation: Implant design and fabrication // J. of Biomechanics. 2012. Vol. 45. No. 13. Pp. 2241–2247. DOI: 10.1016/j.jbiomech.2012.06.011
16. Bendsoe M.P., Sigmund O. Topology optimization: theory, methods and applications. 2nd ed. B.: Springer, 2004. 370 p.
17. Yin L., Ananthasuresh G.K. Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme // Structural and Multidisciplinary Optimization. 2001. Vol. 23. No. 1. Pp. 49–62. DOI: 10.1007/s00158-001-0165-z
18. Tavakoli R., Mohseni S.M. Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation // Structural and Multidisciplinary Optimization. 2014. Vol. 49. No. 4. Pp. 621–642. DOI: 10.1007/s00158-013-0999-1
Mathematics and Mathematical Modeling. 2019; : 19-34
Modified SIMP Method-Based Topological Optimization of Structures Consisting of Several Materials
https://doi.org/10.24108/mathm.0619.0000211Abstract
In this paper, we propose to use a modified SIMP method to solve the problem of topological optimization of structures from multiphase materials, i.e., structures containing components from more than two different materials, which must be distributed in the best way to obtain the best structural characteristics using only one design variable.
Most problems of topological optimization are concerned with only one material and void, although in some cases, it is of interest to have a look at the topology of structure with several phases of materials that is at the problems of topological optimization of the structure from multiphase materials. Nevertheless, most of presently known approaches to solve the problems of topological optimization of structures based on the multiphase materials involve introduction of additional design variables, thereby increasing computational costs.
The paper, when solving the problem of topological optimization of structures from multiphase materials, uses a modified SIMP method to approximate the properties of materials. In this approach a material density is considered as an independent calculated variable and selected from a continuous range, then separated by discrete values of densities of each material phase. Other properties are considered as continuous functions of density. The proposed method does not involve introduction of additional variables for materials approximation and ensures a stable transition from one phase of material to another and non-dependence of estimated costs on the amount of materials under consideration.
To demonstrate the simplicity and efficiency of the proposed solution, the paper gives the examples of solving topological optimization problems for various designs from multiphase materials. Due to its conceptual simplicity, the proposed method can be easily applied to any existing topological optimization problems. The examples described in the paper show that using this solution allows us to obtain reliable structures with improved mechanical characteristics, which can be used to solve real problems in designing and manufacturing of complex structures.
References
1. Bendsoe M.P., Kikuchi N. Generating optimal topologies in structural design using a homogenization method // Computer Methods in Applied Mechanics and Engineering. 1988. Vol. 71. No. 2. Pp. 197–224. DOI: 10.1016/0045-7825(88)90086-2
2. Pedersen P., Pedersen N.L. Interpolation/penalization applied for strength design of 3D thermoelastic structures // Structural and Multidisciplinary Optimization. 2012. Vol. 45. No. 6. Pp. 773–786. DOI: 10.1007/s00158-011-0755-3
3. Pereira A., Talischi C., Paulino G.H., Menezes I.F.M., Carvalho M.S. Fluid flow topology optimization in PolyTop : stability and computational implementation // Structural and Multidisciplinary Optimization. 2016. Vol. 54. No. 5. Pp.1345–1364. DOI: 10.1007/s00158-014-1182-z
4. Yongbo Deng, Yihui Wu, Zhenyu Liu. Topology optimization theory for laminar flow. Singapore: Springer, [2018]. 249 p.
5. Ning Chen, Dejie Yu, Baizhan Xia, Jian Liu, Zhengdong Ma. Microstructural topology optimization of structural-acoustic coupled systems for minimizing sound pressure level // Structural and Multidisciplinary Optimization. 2017. Vol. 56. No. 6. Pp. 1259–1270. DOI: 10.1007/s00158-017-1718-0
6. Rozvany G.I.N., Lewinski T. Topology optimization in structural and continuum mechanics. [Wien]: Springer, 2013. 471 p.
7. Stanford B., Beran P. Conceptual design of compliant mechanisms for flapping wings with topology optimization // AIAA J. 2011. Vol. 49. No. 4. Pp. 855–867. DOI: 10.2514/1.J050940
8. Yuhang Chen, Shiwei Zhou, Qing Li. Multiobjective topology optimization for finite periodic structures // Computers & Structures. 2010. Vol. 88. No. 11-12. Pp. 806–811. DOI: 10.1016/j.compstruc.2009.10.003
9. Yuhang Chen, Shiwei Zhou, Qing Li. Computational design for multifunctional microstructural composites // Intern. J. of Modern Physics B. 2009. Vol. 23. No. 7. Pp. 1345–1351. DOI: 10.1142/S0217979209060920
10. Takezawa A., Gil Ho Yoon, Seung Hyun Jeong, Makoto Kobashi, Mitsuru Kitamura. Structural topology optimization with strength and heat conduction constraints // Computer Methods in Applied Mechanics and Engineering. 2014. Vol. 276. Pp. 341–361. DOI: 10.1016/j.cma.2014.04.003
11. Gil Ho Yoon. Topological layout design of electro-fluid-thermal-compliant actuator // Computer Methods in Applied Mechanics and Engineering. 2012. Vol. 209-212. Pp. 28–44. DOI: 10.1016/j.cma.2011.11.005
12. Xiaoping Qian, Sigmund O. Topological design of electromechanical actuators with robustness toward over- and under-etching // Computer Methods in Applied Mechanics and Engineering. 2013. Vol. 253. Pp. 237–251. DOI: 10.1016/j.cma.2012.08.020
13. Vatanabe S.L., Paulino G.H., Silva E.C.N. Design of functionally graded piezocomposites using topology optimization and homogenization - Toward effective energy harvesting materials // Computer Methods in Applied Mechanics and Engineering. 2013. Vol. 266. Pp. 205–218. DOI: 10.1016/j.cma.2013.07.003
14. Sutradhar A., Paulino G.H., Miller M.J., Nguyen T.H. Topological optimization for designing patient-specific large craniofacial segmental bone replacements // Proc. of the Nat. Acad. of Sciences of the USA. 2010. Vol. 107. No. 30. Pp. 13222–13227. DOI: 10.1073/pnas.1001208107
15. Heesuk Kang, Long J.P., Urbiel Goldner G.D., Goldstein S.A., Hollister S.J. A paradigm for the development and evaluation of novel implant topologies for bone fixation: Implant design and fabrication // J. of Biomechanics. 2012. Vol. 45. No. 13. Pp. 2241–2247. DOI: 10.1016/j.jbiomech.2012.06.011
16. Bendsoe M.P., Sigmund O. Topology optimization: theory, methods and applications. 2nd ed. B.: Springer, 2004. 370 p.
17. Yin L., Ananthasuresh G.K. Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme // Structural and Multidisciplinary Optimization. 2001. Vol. 23. No. 1. Pp. 49–62. DOI: 10.1007/s00158-001-0165-z
18. Tavakoli R., Mohseni S.M. Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation // Structural and Multidisciplinary Optimization. 2014. Vol. 49. No. 4. Pp. 621–642. DOI: 10.1007/s00158-013-0999-1
