Modares Mechanical Engineering

Modares Mechanical Engineering

Topology Optimization of Structure in the Fluid-Structure Interaction Problem using the Level Set Method

Document Type : Original Research

Authors
Mohammad Ali Jahangiri 1
Reza Attarnejad 1
Nima Noei 2
1 School of Civil Engineering, University of Tehran
2 Leibniz Universität, Hannover, Germany
10.48311/mme.24.8.485
Abstract
This research focuses on topology optimization of fluid-structure interaction (FSI) problems using the level set method. To couple the fluid and structure equations, the Arbitrary Lagrangian-Eulerian (ALE) description is employed within a monolithic formulation. The use of ALE in FSI problems, while eliminating numerical instabilities caused by the convective term, enhances the speed and accuracy of finite element solutions in fluid-structure interaction. Additionally, considering the fluid in the unsteady state allows for the interpretation of optimal topology at any given moment of the analysis. The objective function of the optimal topology design problem is to minimize the structural compliance in the dry state, subject to a fixed volume of the design domain. To determine the normal velocity in the reaction-diffusion equation (RDE), adjoint sensitivity analysis based on pointwise gradients is used. The results obtained from this approach, compared to other topology optimization methods in the literature, demonstrate higher accuracy and clearer definition of structural boundaries.
Keywords
Topology optimization
Fluid-Structure Interaction
Level set method
Adjoint Sensitivity Analisys
Reaction-Diffusion Equation
finite element method

Subjects

[1] L. Shang, C. Hoareau, and A. Zilian, "Modeling and simulation of thin-walled piezoelectric energy harvesters immersed in flow using monolithic fluid–structure interaction," Finite Elements in Analysis and Design, vol. 206, p. 103761, 2022.
[2] Y. Wang, Z. Wang, Z. Xia, and L. H. Poh, "Structural Design Optimization Using Isogeometric Analysis: A Comprehensive Review," CMES-Computer Modeling in Engineering & Sciences, vol. 117, no. 3, 2018.
[3] M. P. Bendsøe and N. Kikuchi, "Generating optimal topologies in structural design using a homogenization method," Computer methods in applied mechanics and engineering, vol. 71, no. 2, pp. 197-224, 1988.
[4] G. I. Rozvany, M. Zhou, and T. Birker, "Generalized shape optimization without homogenization," Structural optimization, vol. 4, pp. 250-252, 1992.
[5] H. Mlejnek, "Some aspects of the genesis of structures," Structural optimization, vol. 5, pp. 64-69, 1992.
[6] M. P. Bendsøe, Optimization of structural topology, shape, and material. Springer, 1995.
[7] R. P. Sanches, "Evolutionary Topology Optimization of Fluid-structure Interaction Problems," Ph. D. thesis, University of Campinas, 2015.
[8] R. Sivapuram and R. Picelli, "Topology optimization of binary structures using integer linear programming," Finite Elements in Analysis and Design, vol. 139, pp. 49-61, 2018.
[9] J. A. Sethian, Level set methods and fast marching methods (no. 2). Cambridge Cambridge UP, 1999.
[10] S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations," Journal of computational physics, vol. 79, no. 1, pp. 12-49, 1988.
[11] T. Belytschko, S. Xiao, and C. Parimi, "Topology optimization with implicit functions and regularization," International Journal for Numerical Methods in Engineering, vol. 57, no. 8, pp. 1177-1196, 2003.
[12] N. Jenkins and K. Maute, "An immersed boundary approach for shape and topology optimization of stationary fluid-structure interaction problems," Structural and Multidisciplinary Optimization, vol. 54, pp. 1191-1208, 2016.
[13] R. Picelli, W. M. Vicente, and R. Pavanello, "Evolutionary topology optimization for structural compliance minimization considering design-dependent FSI loads," Finite Elements in Analysis and Design, vol. 135, pp. 44-55, 2017.
[14] G. H. Yoon, "Topology optimization for stationary fluid–structure interaction problems using a new monolithic formulation," International journal for numerical methods in engineering, vol. 82, no. 5, pp. 591-616, 2010.
[15] G. H. Yoon, "Stress-based topology optimization method for steady-state fluid–structure interaction problems," Computer Methods in Applied Mechanics and Engineering, vol. 278, pp. 499-523, 2014.
[16] C. S. Andreasen and O. Sigmund, "Topology optimization of fluid–structure-interaction problems in poroelasticity," Computer Methods in Applied Mechanics and Engineering, vol. 258, pp. 55-62, 2013.
[17] R. Picelli, W. Vicente, and R. Pavanello, "Bi-directional evolutionary structural optimization for design-dependent fluid pressure loading problems," Engineering Optimization, vol. 47, no. 10, pp. 1324-1342, 2015.
[18] R. Picelli, S. Ranjbarzadeh, R. Sivapuram, R. d. S. Gioria, and E. C. N. Silva, "Topology optimization of binary structures under design-dependent fluid-structure interaction loads," Structural and Multidisciplinary Optimization, vol. 62, pp. 2101-2116, 2020.
[19] F. Feppon, G. Allaire, F. Bordeu, J. Cortial, and C. Dapogny, "Shape optimization of a coupled thermal fluid–structure problem in a level set mesh evolution framework," SeMA Journal, vol. 76, pp. 413-458, 2019.
[20] T. Vu-Huu, P. Phung-Van, H. Nguyen-Xuan, and M. A. Wahab, "A polytree-based adaptive polygonal finite element method for topology optimization of fluid-submerged breakwater interaction," Computers & Mathematics with Applications, vol. 76, no. 5, pp. 1198-1218, 2018.
[21] H. Li et al., "Three-dimensional topology optimization of a fluid–structure system using body-fitted mesh adaption based on the level-set method," Applied Mathematical Modelling, vol. 101, pp. 276-308, 2022.
[22] G. H. Yoon, "A new monolithic design approach for topology optimization for transient fluid–structure interaction system," Computer Methods in Applied Mechanics and Engineering, vol. 403, p. 115729, 2023.
[23] S. Ranjbarzadeh, R. Picelli, R. Gioria, and E. C. N. Silva, "Topology optimization of structures subject to non-Newtonian fluid–structure interaction loads using integer linear programming," Finite Elements in Analysis and Design, vol. 202, p. 103690, 2022.
[24] G. A. Holzapfel, "Nonlinear solid mechanics: a continuum approach for engineering science," ed: Kluwer Academic Publishers Dordrecht, 2002.
[25] P. Sun, C.-S. Zhang, R. Lan, and L. Li, "An advanced ALE-mixed finite element method for a cardiovascular fluid–structure interaction problem with multiple moving interfaces," Journal of Computational Science, vol. 50, p. 101300, 2021.
[26] J. Hron and S. Turek, "A monolithic FEM/multigrid solver for an ALE formulation of fluid-structure interaction with applications in biomechanics," in Fluid-Structure Interaction: Modelling, Simulation, Optimisation: Springer, 2006, pp. 146-170.
[27] J. San Martín, L. Smaranda, and T. Takahashi, "Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time," Journal of computational and applied mathematics, vol. 230, no. 2, pp. 521-545, 2009.
[28] A. M. Winslow, "''Equipotential''zoning of two-dimensional meshes," California Univ., Livermore (USA). Lawrence Livermore Lab., 1963.
[29] M. Alnæs et al., "The FEniCS project version 1.5," Archive of numerical software, vol. 3, no. 100, 2015.
[30] C. Taylor and P. Hood, "A numerical solution of the Navier-Stokes equations using the finite element technique," Computers & Fluids, vol. 1, no. 1, pp. 73-100, 1973.
[31] Z. Ge, M. Feng, and Y. He, "Stabilized multiscale finite element method for the stationary Navier–Stokes equations," Journal of mathematical analysis and applications, vol. 354, no. 2, pp. 708-717, 2009.
[32] T. Yamada, K. Izui, S. Nishiwaki, and A. Takezawa, "A topology optimization method based on the level set method incorporating a fictitious interface energy," Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 45-48, pp. 2876-2891, 2010.
[33] H. A. Jahangiry, M. Gholhaki, H. Naderpour, and S. M. Tavakkoli, "Isogeometric level set-based topology optimization for geometrically nonlinear plane stress problems," Computer-Aided Design, vol. 151, p. 103358, 2022.
Modares Mechanical Engineering
Volume 24, Issue 8
July 2024
Pages 485-498