Skip to main content Accessibility help

Topology Optimization of Capillary, Two-Phase Flow Problems

  • Yongbo Deng (a1), Zhenyu Liu (a2) and Yihui Wu (a1)


This paper presents topology optimization of capillary, the typical two-phase flow with immiscible fluids, where the level set method and diffuse-interface model are combined to implement the proposed method. The two-phase flow is described by the diffuse-interface model with essential no slip condition imposed on the wall, where the singularity at the contact line is regularized by the molecular diffusion at the interface between two immiscible fluids. The level set method is utilized to express the fluid and solid phases in the flows and the wall energy at the implicit fluid-solid interface. Based on the variational procedure for the total free energy of two-phase flow, the Cahn-Hilliard equations for the diffuse-interface model are modified for the two-phase flow with implicit boundary expressed by the level set method. Then the topology optimization problem for the two-phase flow is constructed for the cost functional with general formulation. The sensitivity analysis is implemented by using the continuous adjoint method. The level set function is evolved by solving the Hamilton-Jacobian equation, and numerical test is carried out for capillary to demonstrate the robustness of the proposed topology optimization method. It is straightforward to extend this proposed method into the other two-phase flows with two immiscible fluids.


Corresponding author

*Corresponding author. Email addresses: Deng), Liu), Wu)


Hide All
[1] Steven, G. P., Li, Q. and Xie, Y. M., Evolutionary topology and shape design for physical field problems, Comput. Mech., 26 (2000), 129139.
[2] Borrvall, T. and Petersson, J., Topology optimization of fluid in Stokes flow, Int. J. Numer. Meth. Fluids, 41 (2003), 77107.
[3] Bendsoe, M. P. and Kikuchi, N., Generating optimal topologies in optimal design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71 (1988), 197224.
[4] Sigmund, O., A 99-line topology optimization code written in Matlab, Struct. Multidisc. Optim., 21 (2001), 120127.
[5] Sigmund, O., On the design of compliant mechanisms using topology optimization, Mech. Struct. Mach., 25 (1997), 495526.
[6] Saxena, A., Topology design of large displacement compliantmechanisms withmultiple materials and multiple output ports, Struct. Multidisc. Optim., 30 (2005), 477490.
[7] Bendsoe, M. and Sigmund, O., Topology Optimization-Theory Methods and Applications, Springer, 2003.
[8] Gersborg-Hansen, A., Bendsoe, M. P. and Sigmund, O., Topology optimization of heat conduction problems using the finite volume method, Struct. Multidisc. Optim., 31 (2006), 251259.
[9] Nomura, T., Sato, K., Taguchi, K., Kashiwa, T. and Nishiwaki, S., Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique, Int. J. Numer. Methods Eng., 71 (2007), 12611296.
[10] Sigmund, O. and Hougaard, K. G., Geometric properties of optimal photonic crystals, Phys. Rev. Lett., 100 (2008), 153904.
[11] Duhring, M. B., Jensen, J. S. and Sigmund, O., Acoustic design by topology optimization, J. Sound Vibr., 317 (2008), 557575.
[12] Akl, W., El-Sabbagh, A., Al-Mitani, K. and Baz, A., Topology optimization of a plate coupled with acoustic cavity, Int. J. Solids Struct., 46 (2008), 20602074.
[13] Xie, Y. M. and Steven, G. P., Evolutionary structural optimization, Springer, 1997.
[14] Tanskanen, P., The evolutionary structural optimization method: theoretical aspects, Comput. Methods Appl. Mech. Engrg., 191 (2002), 4748.
[15] Allaire, G., Shape Optimization by the Homogenization Method, Springer-Verlag, New York, 2002.
[16] Rozvany, G. I. N., Aims scope methods history and unified terminology of computer-aided optimization in structural mechanics, Struct. Multidisc. Optim., 21 (2001), 90108.
[17] Bendsoe, M. P. and Sigmund, O., Material interpolations in topology optimization, Arch. Appl. Mech., 69 (1999), 635654.
[18] Guest, J. K. and Prevost, J. H., Topology optimization of creeping fluid flows using a Darcy-Stokes finite element, Int. J. Numer. Methods Eng., 66 (2006), 461484.
[19] Gersborg-Hansen, A., Sigmund, O. and Haber, R. B., Topology optimization of channel flow problems, Struct. Multidisc. Optim., 30 (2005), 181192.
[20] Wang, M. Y., Wang, X. and Guo, D., A level set method for structural optimization, Comput. Methods Appl. Mech. Engrg., 192 (2003), 227246.
[21] Allaire, G., Jouve, F. and Toader, A., Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194 (2004), 363393.
[22] Zhou, S. and Li, Q., A variational level set method for the topology optimization of steadystate Navier-Stokes flow, J. Comput. Phys., 227 (2008), 1017810195.
[23] Duan, X., Ma, Y. and Zhang, R., Shape-topology optimization for Navier-Stokes problemusing variational level set method, J. Comput. Appl. Math., 222 (2008), 487499.
[24] Liu, Z. and Korvink, J. G., Adaptive moving mesh level set method for structure optimization, Engrg. Optim., 40 (2008), 529558.
[25] Xing, X., Wei, P. and Wang, M. Y., A finite element-based level set method for structural optimization, Int. J. Numer. Methods Engrg., 82 (2010), 805842.
[26] Kreissl, S., Pingen, G. and Maute, K., An explicit level-set approach for generalized shape optimization of fluids with the lattice Boltzmann method, Int. J. Numer. Meth. Fluids, 65 (2011), 496519.
[27] Bourdin, B. and Chambolle, A., Optimisation topologique de structures soumises à des forces de pression, Actes du 32ème Congrèes National d’Analyse Numérique, SMAI (ed.), 2000.
[28] Bourdin, B. and Chambolle, A., Design-dependent loads in topology optimization, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 1948.
[29] Burger, M. and Stainko, R., Phase-field relaxation of topology optimization with local stress constraints, SIAM J. Control Optim., 45 (2006), 14471466.
[30] Blank, L., Garcke, H., Sarbu, L., Srisupattarawanit, T., Styles, V. and Voig, A., Phase-field approaches to structural topology optimization, International Series of Numerical Mathematics, 160 (2012), 245256.
[31] Gain, A. L. and Paulino, G. H., Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation, Struct. Multidisc. Optim., 46 (2012), 327342.
[32] Zhou, S. and Wang, M. Y., Multimaterial structural topology optimization with a generalized CahnCHilliard model of multiphase transition, Struct. Multidisc. Optim., 33 (2007), 89111.
[33] Liu, J., Dedè, L., Evans, J. A., Borden, M. J. and Hughes, T. J. R., Isogeometric analysis of the advective Cahn-Hilliard equation: Spinodal decomposition under shear flow, J. Comput. Phys., 242 (2013), 321350.
[34] Takezawa, A., Nishiwaki, S. and Kitamura, M., Shape and topology optimization based on the phase field method and sensitivity analysis, J. Comput. Phys., 229 (2010), 26972718.
[35] Deng, Y., Liu, Z. and Wu, Y., Optimization of unsteady incompressible Navier-Stokes flows using variational level set method, Int. J. Numer. Meth. Fluids, 71 (2013), 14751493.
[36] Deng, Y., Liu, Z. and Wu, Y., Topology optimization of steady and unsteady incompressible Navier-Stokes flows driven by body forces, Struct. Multidisc. Optim., 47 (2013), 555570.
[37] Olesen, L. H., Okkels, F. and Bruus, H., A high-level programming-language implementation of topology optimization applied to steady-state Navier-Stokes flow, Int. J. Numer. Methods Eng., 65 (2006), 9751001.
[38] Deng, Y., Liu, Z., Zhang, P., Liu, Y. and Wu, Y., Topology optimization of unsteady incompressible Navier-Stokes flow, J. Comput. Phys., 230 (2011), 66886708.
[39] Makhija, D., Pingen, G., Yang, R. and Maute, K., Topology optimization of multi-component flows using a multi-relaxation time lattice Boltzmann method, Comput. Fluids, 67 (2012), 104114.
[40] Alexandersen, J., Aage, N., Andreasen, C. S. and Sigmund, O., Topology optimisation for natural convection problems, Int. J. Numer. Meth. Fluids, 76 (2014), 699721.
[41] Victor, M. S., Manuel, G. V. and Clayton, J. R., Wetting and spreading dynamics, CRC Press, 2007.
[42] Gueyffier, D., Li, J., Nadim, A., Scardovelli, R. and Zaleski, S., Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows, J. Comput. Phys., 152 (1999), 423456.
[43] Glimm, J., Grove, J.W., Li, X. L., Shyue, K. M., Zhang, Q. and Zeng, Y., Three-dimensional front tracking, SIAM J. Sci. Comput., 19 (1998), 703727.
[44] Peskin, C. S. and McQueen, D. M., Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. Comput. Phys., 37 (1980), 113132.
[45] Peskin, C. S., The immersed boundary method, Acta Num., 11 (2002), 139.
[46] Chang, Y. C., Hou, T. Y., Merriman, B. and Osher, S., A level set formulation of Eulerian interface capturingmethods for incompressible fluid flows, J. Comput. Phys., 124 (1996), 449464.
[47] Arienti, M. and Sussman, M., An embedded level set method for sharp-interface multiphase simulations of Diesel injectors, Int. J. Multiphas. Flow, 59 (2014), 114.
[48] Engberga, R. F. and Kenig, E. Y., An investigation of the influence of initial deformation on fluid dynamics of toluene droplets in water, Int. J. Multiphas. Flow, 76 (2015), 144157.
[49] Osher, S. and Fedkiw, R. P., Level set methods and dynamic implicit surfaces, Springer-Verlag, New York, 2002.
[50] Sethian, J. A. and Smereka, P., Level set methods for fluid interfaces, Annu. Rev. Fluid Mech., 35 (2003), 341372.
[51] Amiri, H. A. A. and Hamouda, A. A., Evaluation of level set and phase field methods in modeling two phase flow with viscosity contrast through dual-permeability porous medium, Int. J. Multiphas. Flow, 52 (2013), 2234.
[52] Anderson, D. M., McFadden, G. B. and Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139165.
[53] Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96127.
[54] Kim, J. S., A continuous surface tension force formulation for diffuse-interface models, J. Comput. Phys., 204 (2005), 784804.
[55] Qian, T., Wang, X. P. and Sheng, P., Molecular hydrodynamics of the moving contact line in two-phase immiscible flows, Comm. Comput. Phys., 1 (2006), 152.
[56] Jacqmin, D., Contact-line dynamics of a diffuse fluid interface, J. Fluid Mech., 402 (2000), 5788.
[57] Cahn, J. W. and Hilliard, J., Free energy of a nonuniform system: I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258267.
[58] Sethian, J. A., Level set methods and fast marching methods evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge University Press, 1999.
[60] Cahn, J.W., On spinodal decomposition, Acta Metall., 9 (1961), 795801.
[61] Cahn, J.W., Critical-point wetting, J. Chem. Phys., 66 (1977), 36673672.
[62] Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. and Hu, H. H., Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing, J. Comput. Phys., 219 (2006), 4767.
[63] Yue, P., Feng, J. J., Liu, C. and Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293317.
[64] Nocedal, J. and Wright, S., Numerical optimization, 2nd edition, Springer, 2000.
[65] Elman, H. C., Silvester, D. J. and Wathen, A. J., Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, 2006.
[66] Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 2003.
[67] Giles, M. B., Pierce, N. A., An introduction to the adjoint approach to design, Flow Turbul. Combust., 65 (2000), 393415.
[68] Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S., Optimization with PDE Constraints, Springer: Berlin, 2009.
[69] Mohammadi, B. and Pironneau, O., Applied Shape Optimization for Fluids, Oxford University Press, USA: Oxford, 2010.
[70] Defay, R. and Prigogine, I., Surface Tension and Adsorption, Longmans, Green & Co Ltd, London, 1966.
[71] Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, Pergamon Press, Oxford, 1987.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed