Обмеження на інтегральні міри напруженого стану в задачах топологічної оптимізації

Заголовок (англійською): 
Constraints on integral measures of stress state in topology optimization problems
Автор(и): 
Кришталь В.Ф.
Янчевський І.В.
Автор(и) (англ): 
Kryshtal V.F.
Yanchevskiy I.V.
Ключові слова (укр): 
топологічна оптимізація, механічне напруження, інтегральна міра, агрегатні функції, скінченні елементи
Ключові слова (англ): 
topology optimization, mechanical stress, integral measure, aggregate functions, finite elements
Анотація (укр): 
Топологічною оптимізацією (ТО) називають обчислювальний метод визначення розподілу матеріалу у заданій області проєктування для створення оптимальної форми деталі при заданих граничних умовах. В класичній постановці ТО у якості критерію пошуку зазначеного розподілу обрана мінімізація піддатливості деталі при обмеженнях на об’єм (масу) результату оптимізації. Більш наближеною до прикладного застосування є постановка задачі ТО, що передбачає мінімізацію об’єму деталі з урахуванням умови її міцності. Залучення агрегатних функцій для обчислення інтегральних мір напруженого стану має низку переваг над традиційною перевіркою максимального значення механічного напруження. У даній роботі представлені агрегатні функції, які отримали найбільше застосування в сучасних дослідженнях з питань ТО з урахуванням міцності оптимізованої деталі.
Анотація (англ): 
Topology optimization (TO) is a computational method of determining material distribution in a given design area to create the optimal shape of a part under given boundary conditions. The increased interest in the development of effective methods of designing parts of the optimal topology testifies to the relevance of these theoretical studies and the important applied value of the obtained results. In the classic formulation of maintenance, the minimization of the flexibility of the part under restrictions on the volume (mass) of the optimization result is chosen as a criterion for finding the specified distribution. Closer to practical application is the formulation of the maintenance problem, which involves minimizing the volume of the part, taking into account the condition of its strength. The inclusion of aggregate functions for the calculation of integral measures of the stress state has a number of advantages over the traditional check of the maximum value of mechanical stress: significant saving of time for solving the maintenance problem, reduction of computational costs and ensuring the stability of the computational process. This work presents and analyzes the specialization of the applied application of aggregate functions, which have been most widely used in modern research on maintenance issues, taking into account the strength of the optimized part. In particular, the P-norm and P-mean functions, the Kreiselmeier-Steinhauser functions, the smoothed Heaviside function, the measure of exceeded stresses, and the measure of uneven distribution of the stress state are described. The large number of options available in the literature for the mathematical formulation of limitations for integral measures of the stress state of designed parts indicates that the issue of developing a universal and effective method of designing parts, taking into account the criterion of its strength, remains open.
Публікатор: 
Київський національний університет будівництва і архітектури
Назва журналу, номер, рік випуску (укр): 
Опір матеріалів і теорія споруд, 2023, номер 110
Назва журналу, номер, рік випуску (англ): 
Strength of Materials and Theory of Structures, 2023, number 110
Мова статті: 
Українська
Формат документа: 
application/pdf
Документ: 
15-110_krishtal_v.f._yanchevskiy_i.v.pdf
Дата публікації: 
15 September 2023
Номер збірника: 
110
Університет автора: 
Національний технічний університет України «Київський політехнічний інститут імені Ігоря Сікорського», 03056, Україна, м. Київ-56, пр. Перемоги, 37
Литература: 
  1. Bendsøe M. P. Optimization of Structural Topology, Shape, And Material. – Berlin: Springer-Verlag, 1995. – 273 p. DOI: 10.1007/978-3-662-03115-5
  2. Sigmund O. A 99 line topology optimization code written in Matlab // Structural and Multidisciplinary Optimization. – 2001. – Vol. 21, Iss. 2. – P. 120-127. DOI: 10.1007/s0015800501
  3. Bendsoe M. P., Sigmund O. Topology optimization: theory, methods and applications. – Berlin: Springer-Verlag, 2004. – 370 p. DOI: 10.1007/978-3-662-05086-6_2.
  4. Andreassen E., Clausen A., Schevenels M., Lazarov B., Sigmund O. Efficient topology optimization in MATLAB using 88 lines of code // Struct. Multidisc. Optim.  – 2011. – Iss. 43. – Р. 1-16. DOI: 10.1007/S00158-010-0594-7.
  5. Liu K., Tovar A. An efficient 3D topology optimization code written in MatLab // Structural and Multidisciplinary Optimization. – 2014. – Vol. 50, Iss. 6. – P. 1175-1196. DOI: 10.1007/s00158-014-1107-x.
  6. Ferrari F., Sigmund O. A new generation 99 line Matlab code for compliance topology optimization and its extension to 3D // Struct. Multidisc. Optim.  – 2020. – Vol. 62, Iss. 4. – P. 2211-2228. DOI: 10.1007/s00158-020-02629-w.
  7. Huang X., Xie Y.M. Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. – John Wiley & Sons, Ltd, 2010. – 223 p. DOI:10.1002/9780470689486
  8. Duysinx P., Bendsøe M.P. Topology optimization of continuum structures with local stress constraints // Int. J. for Numerical Methods in Engineering. – 1998. –Vol. 43, Iss. 8. – P. 1453-1478. DOI: 10.1002/(SICI)1097-0207(19981230)43:8<1453::AID-NME480>3.0.CO;2-2.
  9. Duysinx P., Sigmund O. New developments in handling stress constraints in optimal material distribution. Conference Paper. 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. 1998.  DOI: 10.2514/6.1998-4906.
  10. De Leon D.M., Alexandersen J., O. Fonseca J.S., Sigmund O. Stress-constrained topology optimization for compliant mechanism design // Structural and Multidisciplinary Optimization. – 2015. – Vol. 52. – P. 929-943. DOI: 10.1007/S00158-015-1279-Z.
  11. Kim N., Dong T., Weinberg D.H., Dalidd J. Generalized optimality criteria method for topology optimization // Appl. Sci. – 2021. – Iss. 11. – P. 3175. DOI: 10.3390/app11073175.
  12. Cheng G., Guo X. ε-relaxed approach in structural topology optimization // Structural optimization. – 1997. – Vol. 13. – P. 258-266. DOI: 10.1007/BF01197454.
  13. Poon N.M.K., Martins J.R.R.A. An adaptive approach to constraint aggregation using adjoint sensitivity analysis // Struct. Multidisc. Optim. – 2007. – Vol. 34. – P. 61-73. DOI: 10.1007/s00158-006-0061-7.
  14. Duysinx P., Miegroet L.V., Lemaire E., Bruls O., Bruyneel M. Topology and generalized shape optimisation: why stress constraints are so important // Int. J. for Simulation and Multidisciplinary Design Optimization. – 2008. – Vol. 2. DOI: 10.1051/IJSMDO/2008034.
  15. Bruggi M. On an alternative approach to stress constraints relaxation in topology optimization // Structural and Multidisciplinary Optimization. – 2008. – Vol. 36. – P. 125-141. DOI: 10.1007/S00158-007-0203-6.
  16. París J., Navarrina F., Colominas I., Casteleiro M. Topology optimization of continuum structures with local and global stress constraints // Structural and Multidisciplinary Optimization. – 2009. – Vol. 39. – P. 419-437. DOI: 10.1007/S00158-008-0336-2.
  17. París J., Navarrina F., Colominas I., Casteleiro M. Block aggregation of stress constraints in topology optimization of structures // Adv. Eng. Softw. – 2010. – Vol. 41, Iss. 3. – P. 433-441. DOI: 10.1016/j.advengsoft.2009.03.006.
  18. Le C.H., Norato J.A., Bruns T.E., Ha C., Tortorelli D. Stress-based topology optimization for continua // Structural and Multidisciplinary Optimization. – 2010. – Vol. 41. – Iss. 4. – P. 605-620. DOI: 10.1007/S00158-009-0440-Y.
  19. Lee E., James K.A., Martins J.R. Stress-constrained topology optimization with design-dependent loading // Structural and Multidisciplinary Optimization. – 2012. – Vol. 46. – P. 647-661. DOI: 10.1007/S00158-012-0780-X.
  20. Luo Y., Wang M.Y., Kang Z. An enhanced aggregation method for topology optimization with local stress constraints // Computer Methods in Appl. Mech. and Eng. – 2013. – Vol. 254. – P. 31-41 DOI: 10.1016/j.cma.2012.10.019.
  21. Holmberg E., Torstenfelt B., Klarbring A. Stress constrained topology optimization // Struct. Multidisc. Optim. – 2013. – Vol. 48, Iss. 1. – P. 33-47. DOI: 10.1007/s00158-012-0880-7.
  22. Jeong S.H., Park S., Choi D., Yoon G.H. Toward a stress-based topology optimization procedure with indirect calculation of internal finite element information // Comput. Math. Appl. – 2013. – Vol. 66. – P. 1065-1081. DOI: 10.1016/j.camwa.2013.07.008.
  23. Biyikli E., To A.C. Proportional topology optimization: A new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB // PLoS ONE. – 2015. – Vol. 10, Iss. 12. – P. e0145041. DOI: 10.1371/journal.pone.0145041.
  24. Lee K.Y., Ahn K., Yoo J. A novel P-norm correction method for lightweight topology optimization under maximum stress constraints // Computers & Structures. – 2016. – Vol. 171. – P. 18-30. DOI: 10.1016/J.COMPSTRUC.2016.04.005.
  25. Verbart A., Langelaar M., Keulen F.V. A unified aggregation and relaxation approach for stress-constrained topology optimization // Structural and Multidisciplinary Optimization. – 2017. – Vol. 55. – P. 663-679. DOI: 10.1007/S00158-016-1524-0.
  26. Wang C., Qian X. Heaviside projection-based aggregation in stress-constrained topology optimization // Int. J. for Numerical Methods in Engineering. – 2018. – Vol. 115. – P. 849-871. DOI: 10.1002/nme.5828.
  27. Yang D., Liu H., Zhang W., Li S.C. Stress-constrained topology optimization based on maximum stress measures // Computers & Structures. – 2018. – Vol. 198. – P. 23-39. DOI: 10.1016/J.COMPSTRUC.2018.01.008.
  28. Da Silva G.A., Aage N., Beck A.T., Sigmund O. Local versus global stress constraint strategies in topology optimization: A comparative study // Int. J. for Numerical Methods in Engineering. – 2021. – Vol. 122. – P. 6003-6036. DOI: 10.1002/nme.6781.
  29. Yanchevskyi I.V., Kryshtal V.F. Integral criterion of the non-uniformity of stress distribution for the topology optimization of 2D-models // J. of Mech. Eng. – 2021. – Vol. 24, Iss. 1. – P. 65-74. DOI: 10.15407/pmach2021.01.065.
  30. Yang R., Chen C.J. Stress-based topology optimization // Structural optimization. – 1996. – Vol. 12. – P. 98-105. DOI: 10.1007/BF01196941.
  31. Kiyono C.Y., Vatanabe S.L., Silva E., Reddy J. A new multi-p-norm formulation approach for stress-based topology optimization design // Composite Structures. – 2016. – Vol. 156. – P. 10-19. DOI: 10.1016/J.COMPSTRUCT.2016.05.058.
  32. Lian H., Christiansen A.N., Tortorelli D.A., Sigmund O., Aage N. Combined shape and topology optimization for minimization of maximal von Mises stress // Struct. Multidisc. Optim. – 2017. – Vol. 55, Iss. 5. – P. 1541-1557. DOI: 10.1007/s00158-017-1656-x.
  33. Deng H., Vulimiri P.S., To A.C. An efficient 146-line 3D sensitivity analysis code of stress-based topology optimization written in MATLAB // Optimization and Engineering. – 2021. – Vol. 23. – P. 1733-1757. DOI: 10.1007/s11081-021-09675-3.
  34. Kranz M., Lüdeker J.K., Kriegesmann B. An empirical study on stress-based fail-safe topology optimization and multiple load path design // Struct. Multidisc. Optim. – 2021. – Vol. 64. – P. 2113-2134. DOI: 10.1007/s00158-021-02969-1.
  35. Fernández E., Collet M., Alarcón P., Bauduin S., Duysinx P. An aggregation strategy of maximum size constraints in density-based topology optimization // Structural and Multidisciplinary Optimization. – 2019. – Vol. 60, Iss. 5. – P. 2113-2130. DOI: 10.1007/s00158-019-02313-8.
  36. Kreisselmeier G., Steinhauser R. Systematic control design by optimizing a vector performance index // Int. Federation of Active Controls Syposium on Computer-Aided Design of Control Systems, Zurich, Switzerland. – 1979. – P. 113-117. DOI: 10.1016/B978-0-08-024488-4.50022-X.
  
References: 
  1. Bendsøe M. P. Optimization of Structural Topology, Shape, And Material. – Berlin: Springer-Verlag, 1995. – 273 p. DOI: 10.1007/978-3-662-03115-5
  2. Sigmund O. A 99 line topology optimization code written in Matlab // Structural and Multidisciplinary Optimization. – 2001. – Vol. 21, Iss. 2. – P. 120-127. DOI: 10.1007/s0015800501
  3. Bendsoe M. P., Sigmund O. Topology optimization: theory, methods and applications. – Berlin: Springer-Verlag, 2004. – 370 p. DOI: 10.1007/978-3-662-05086-6_2.
  4. Andreassen E., Clausen A., Schevenels M., Lazarov B., Sigmund O. Efficient topology optimization in MATLAB using 88 lines of code // Struct. Multidisc. Optim.  – 2011. – Iss. 43. – Р. 1-16. DOI: 10.1007/S00158-010-0594-7.
  5. Liu K., Tovar A. An efficient 3D topology optimization code written in MatLab // Structural and Multidisciplinary Optimization. – 2014. – Vol. 50, Iss. 6. – P. 1175-1196. DOI: 10.1007/s00158-014-1107-x.
  6. Ferrari F., Sigmund O. A new generation 99 line Matlab code for compliance topology optimization and its extension to 3D // Struct. Multidisc. Optim.  – 2020. – Vol. 62, Iss. 4. – P. 2211-2228. DOI: 10.1007/s00158-020-02629-w.
  7. Huang X., Xie Y.M. Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. – John Wiley & Sons, Ltd, 2010. – 223 p. DOI:10.1002/9780470689486
  8. Duysinx P., Bendsøe M.P. Topology optimization of continuum structures with local stress constraints // Int. J. for Numerical Methods in Engineering. – 1998. –Vol. 43, Iss. 8. – P. 1453-1478. DOI: 10.1002/(SICI)1097-0207(19981230)43:8<1453::AID-NME480>3.0.CO;2-2.
  9. Duysinx P., Sigmund O. New developments in handling stress constraints in optimal material distribution. Conference Paper. 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. 1998.  DOI: 10.2514/6.1998-4906.
  10. De Leon D.M., Alexandersen J., O. Fonseca J.S., Sigmund O. Stress-constrained topology optimization for compliant mechanism design // Structural and Multidisciplinary Optimization. – 2015. – Vol. 52. – P. 929-943. DOI: 10.1007/S00158-015-1279-Z.
  11. Kim N., Dong T., Weinberg D.H., Dalidd J. Generalized optimality criteria method for topology optimization // Appl. Sci. – 2021. – Iss. 11. – P. 3175. DOI: 10.3390/app11073175.
  12. Cheng G., Guo X. ε-relaxed approach in structural topology optimization // Structural optimization. – 1997. – Vol. 13. – P. 258-266. DOI: 10.1007/BF01197454.
  13. Poon N.M.K., Martins J.R.R.A. An adaptive approach to constraint aggregation using adjoint sensitivity analysis // Struct. Multidisc. Optim. – 2007. – Vol. 34. – P. 61-73. DOI: 10.1007/s00158-006-0061-7.
  14. Duysinx P., Miegroet L.V., Lemaire E., Bruls O., Bruyneel M. Topology and generalized shape optimisation: why stress constraints are so important // Int. J. for Simulation and Multidisciplinary Design Optimization. – 2008. – Vol. 2. DOI: 10.1051/IJSMDO/2008034.
  15. Bruggi M. On an alternative approach to stress constraints relaxation in topology optimization // Structural and Multidisciplinary Optimization. – 2008. – Vol. 36. – P. 125-141. DOI: 10.1007/S00158-007-0203-6.
  16. París J., Navarrina F., Colominas I., Casteleiro M. Topology optimization of continuum structures with local and global stress constraints // Structural and Multidisciplinary Optimization. – 2009. – Vol. 39. – P. 419-437. DOI: 10.1007/S00158-008-0336-2.
  17. París J., Navarrina F., Colominas I., Casteleiro M. Block aggregation of stress constraints in topology optimization of structures // Adv. Eng. Softw. – 2010. – Vol. 41, Iss. 3. – P. 433-441. DOI: 10.1016/j.advengsoft.2009.03.006.
  18. Le C.H., Norato J.A., Bruns T.E., Ha C., Tortorelli D. Stress-based topology optimization for continua // Structural and Multidisciplinary Optimization. – 2010. – Vol. 41. – Iss. 4. – P. 605-620. DOI: 10.1007/S00158-009-0440-Y.
  19. Lee E., James K.A., Martins J.R. Stress-constrained topology optimization with design-dependent loading // Structural and Multidisciplinary Optimization. – 2012. – Vol. 46. – P. 647-661. DOI: 10.1007/S00158-012-0780-X.
  20. Luo Y., Wang M.Y., Kang Z. An enhanced aggregation method for topology optimization with local stress constraints // Computer Methods in Appl. Mech. and Eng. – 2013. – Vol. 254. – P. 31-41 DOI: 10.1016/j.cma.2012.10.019.
  21. Holmberg E., Torstenfelt B., Klarbring A. Stress constrained topology optimization // Struct. Multidisc. Optim. – 2013. – Vol. 48, Iss. 1. – P. 33-47. DOI: 10.1007/s00158-012-0880-7.
  22. Jeong S.H., Park S., Choi D., Yoon G.H. Toward a stress-based topology optimization procedure with indirect calculation of internal finite element information // Comput. Math. Appl. – 2013. – Vol. 66. – P. 1065-1081. DOI: 10.1016/j.camwa.2013.07.008.
  23. Biyikli E., To A.C. Proportional topology optimization: A new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB // PLoS ONE. – 2015. – Vol. 10, Iss. 12. – P. e0145041. DOI: 10.1371/journal.pone.0145041.
  24. Lee K.Y., Ahn K., Yoo J. A novel P-norm correction method for lightweight topology optimization under maximum stress constraints // Computers & Structures. – 2016. – Vol. 171. – P. 18-30. DOI: 10.1016/J.COMPSTRUC.2016.04.005.
  25. Verbart A., Langelaar M., Keulen F.V. A unified aggregation and relaxation approach for stress-constrained topology optimization // Structural and Multidisciplinary Optimization. – 2017. – Vol. 55. – P. 663-679. DOI: 10.1007/S00158-016-1524-0.
  26. Wang C., Qian X. Heaviside projection-based aggregation in stress-constrained topology optimization // Int. J. for Numerical Methods in Engineering. – 2018. – Vol. 115. – P. 849-871. DOI: 10.1002/nme.5828.
  27. Yang D., Liu H., Zhang W., Li S.C. Stress-constrained topology optimization based on maximum stress measures // Computers & Structures. – 2018. – Vol. 198. – P. 23-39. DOI: 10.1016/J.COMPSTRUC.2018.01.008.
  28. Da Silva G.A., Aage N., Beck A.T., Sigmund O. Local versus global stress constraint strategies in topology optimization: A comparative study // Int. J. for Numerical Methods in Engineering. – 2021. – Vol. 122. – P. 6003-6036. DOI: 10.1002/nme.6781.
  29. Yanchevskyi I.V., Kryshtal V.F. Integral criterion of the non-uniformity of stress distribution for the topology optimization of 2D-models // J. of Mech. Eng. – 2021. – Vol. 24, Iss. 1. – P. 65-74. DOI: 10.15407/pmach2021.01.065.
  30. Yang R., Chen C.J. Stress-based topology optimization // Structural optimization. – 1996. – Vol. 12. – P. 98-105. DOI: 10.1007/BF01196941.
  31. Kiyono C.Y., Vatanabe S.L., Silva E., Reddy J. A new multi-p-norm formulation approach for stress-based topology optimization design // Composite Structures. – 2016. – Vol. 156. – P. 10-19. DOI: 10.1016/J.COMPSTRUCT.2016.05.058.
  32. Lian H., Christiansen A.N., Tortorelli D.A., Sigmund O., Aage N. Combined shape and topology optimization for minimization of maximal von Mises stress // Struct. Multidisc. Optim. – 2017. – Vol. 55, Iss. 5. – P. 1541-1557. DOI: 10.1007/s00158-017-1656-x.
  33. Deng H., Vulimiri P.S., To A.C. An efficient 146-line 3D sensitivity analysis code of stress-based topology optimization written in MATLAB // Optimization and Engineering. – 2021. – Vol. 23. – P. 1733-1757. DOI: 10.1007/s11081-021-09675-3.
  34. Kranz M., Lüdeker J.K., Kriegesmann B. An empirical study on stress-based fail-safe topology optimization and multiple load path design // Struct. Multidisc. Optim. – 2021. – Vol. 64. – P. 2113-2134. DOI: 10.1007/s00158-021-02969-1.
  35. Fernández E., Collet M., Alarcón P., Bauduin S., Duysinx P. An aggregation strategy of maximum size constraints in density-based topology optimization // Structural and Multidisciplinary Optimization. – 2019. – Vol. 60, Iss. 5. – P. 2113-2130. DOI: 10.1007/s00158-019-02313-8.
  36. Kreisselmeier G., Steinhauser R. Systematic control design by optimizing a vector performance index // Int. Federation of Active Controls Syposium on Computer-Aided Design of Control Systems, Zurich, Switzerland. – 1979. – P. 113-117. DOI: 10.1016/B978-0-08-024488-4.50022-X.