Structural topology optimization in linear and nonlinear elasticity using a discontinuous Galerkin finite element method

• Autorzy: Tan Yixin , Yan Lingkang , Zhu Shengfeng

Identyfikatory

DOI

10.2478/candc-2024-0012

• Języki publikacji: EN

Abstrakty

EN

A discontinuous Galerkin finite element method is developed for structural topology optimization with the level set method. The discontinuous Galerkin finite element method is employed to discretize and solve both the elasticity system, possible adjoint, and the transport equation of the level set function. Structural compliance and compliant mechanisms are considered for linear and nonlinear elastic structures. Numerical examples are provided to verify the effectiveness of the algorithm presented.

Słowa kluczowe

EN

topology optimization; level set method; discontinuous Galerkin finite element method; compliance; compliant mechanism; nonlinear elasticity

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2024
• Tom: Vol. 53, No. 1
• Strony: 283--315
• Opis fizyczny: Bibliogr. 29 poz., rys.

Twórcy

autor

Tan Yixin

• aSchool of Mathematical Sciences, East China Normal University, Shanghai 200241, China
• Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

autor

Yan Lingkang

• aSchool of Mathematical Sciences, East China Normal University, Shanghai 200241, China

autor

Zhu Shengfeng

• aSchool of Mathematical Sciences, East China Normal University, Shanghai 200241, China
• Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

Bibliografia

• Adams, T., Giani, S. and Coombs, W. (2016) Topology optimisation using level set methods and the discontinuous Galerkin method. In: Proc. Of the 24th UK Conf. of the Association for Computational Mechanics in Engineering (ACME-UK), Cardiff University, Cardiff: 6–9.
• Allaire, G., de Gournay, F., Jouve, F. and Toader, A. M. (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybernet. 34, 59–80.
• Allaire, G., Jouve, F. and Toader, A. M. (2004) Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393.
• Allaire, G., Dapogny, C. and Jouve, F. (2021) Shape and topology optimization. Handb. Numer. Anal. 22, 1–132.
• Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D. (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779.
• Bendsoe, M. P. and Sigmund, O. (2003) Topology Optimization: Theory, Methods, and Applications. Springer, Berlin.
• Buhl, T., Pedersen, C. and Sigmund, O. (2000) Stiffness design of geometrically nonlinear structures using topology optimization. Struct. Multidiscip. Optim. 19 (2) 93–104.
• Ciarlet, P. G. (1988) Mathematical Elasticity, Three-Dimensional Elasticity, vol. I. North-Holland, Amsterdam.
• Dapogny, C. and Frey, P. (2010) Computation of the signed distance function to a discrete contour on adapted triangulation. Calcolo, 49 1-27.
• Ern, A. and Guermond, J. L. (2006) Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM J. Numer. Anal. 44, 753–778.
• Evgrafov, A. (2018) Discontinuous Petrov-Galerkin methods for topology optimization. In: H. Rodrigues et al., eds., Proceedings of the 6th International Conference on Engineering Optimization. EngOpt 2018. Springer, Cham.
• Fulmanski, P., Laurain, A., Scheid, J.-F. and Sokołowski, J. (2008) Level set method with topological derivatives in shape optimization. Int. J. Comput. Math. 85, 1491–1514.
• Ganghoffer, J. F., Plotnikov, P. I. and Sokolowski, J. (2018) Nonconvex model of material growth: mathematical theory. Arch. Rational Mech. Anal. 230, 839–910.
• Hansbo, P. and Larson, M. G. (2022) Augmented Lagrangian approach to deriving discontinuous Galerkin methods for nonlinear elasticity problems. Int. J. Numer. Methods Eng. 123, 4407–4421.
• Hecht, F. (2012) New development in FreeFem++. J. Numer. Math. 20, 251–265.
• Howell, L. L. (2001) Compliant Mechanisms. John Wiley & Sons.
• Kim, N.-H. (2015) Introduction to Nonlinear Finite Element Analysis. Springer, US.
• Kwak, J. and Cho, S. (2005) Topological shape optimization of geometrically nonlinear structures using level set method. Comput. Struct. 83, 2257–2268.
• Laurain, A. (2018) A level set-based structural optimization code using FEniCS. Struct. Multidisc. Optim. 58, 1311–1334.
• Novotny, A. A. and Sokołowski, J. (2013) Topological Derivatives in Shape Optimization. Springer, Berlin.
• Novotny, A. A., Sokołowski, J. and Żochowski, A. (2019) Applications of the Topological Derivative Method. Springer, Cham.
• Osher, S. and Sethian, J. A. (1988) Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49.
• Qian, M. and Zhu, S. (2022) A level set method for Laplacian eigenvalue optimization subject to geometric constraints. Comput. Optim. Appl. 82, 499–524.
• Rivière, B., Shaw, S., Wheeler, M. F. and Whiteman, J. R. (2003) Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numer. Math. 95, 347–376.
• Sokołowski, J. and Zolésio, J. P. (1992) Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Heidelberg.
• Tan, Y. and Zhu, S. (2023) A discontinuous Galerkin level set method using distributed shape gradient and topological derivatives for multi-material structural topology optimization. Struct. Multidiscip. Optim. 66; Article number: 170.
• Tan, Y. and Zhu, S. (2024) Numerical shape reconstruction for a semi-linear elliptic interface inverse problem. East Asian J. Appl. Math. 14, 147–178.
• Zheng, J., Zhu, S. and Soleymani, F. (2024) A new efficient parametric level set method based on radial basis function-finite difference for structural topology optimization. Comput. Struct. 291, 107364.
• Zhu, S., Hu, X. and Wu, Q. (2018) A level set method for shape optimization in semilinear elliptic problems. J. Comput. Phys. 355, 104–120.