Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T12:09:51.909Z Has data issue: false hasContentIssue false
  • English
  • Français

Topology Optimization of Capillary, Two-Phase Flow Problems

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Incompressible viscous fluids Two-phase and multiphase flows

Published online by Cambridge University Press:  31 October 2017

Yongbo Deng ,
Zhenyu Liu  and
Yihui Wu
Yongbo Deng*
Affiliation:
State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP), Chinese Academy of Sciences, Changchun 130033, China
Zhenyu Liu*
Affiliation:
Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP), Chinese Academy of Sciences, Changchun 130033, China
Yihui Wu*
Affiliation:
State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP), Chinese Academy of Sciences, Changchun 130033, China
*
*Corresponding author. Email addresses:[email protected](Y. Deng), [email protected](Z. Liu), [email protected](Y. Wu)
*Corresponding author. Email addresses:[email protected](Y. Deng), [email protected](Z. Liu), [email protected](Y. Wu)
*Corresponding author. Email addresses:[email protected](Y. Deng), [email protected](Z. Liu), [email protected](Y. Wu)
Get access

Abstract

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.

Keywords

Capillarytwo-phase flowtopology optimizationdiffuse-interface modellevel set method

MSC classification

Secondary: 76D55: Flow control and optimization 76D45: Capillarity (surface tension) 76T10: Liquid-gas two-phase flows, bubbly flows 65K10: Optimization and variational techniques 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 5 , November 2017 , pp. 1413 - 1438
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Steven, G. P., Li, Q. and Xie, Y. M., Evolutionary topology and shape design for physical field problems, Comput. Mech., 26 (2000), 129–139.CrossRefGoogle Scholar
[2] Borrvall, T. and Petersson, J., Topology optimization of fluid in Stokes flow, Int. J. Numer. Meth. Fluids, 41 (2003), 77–107.CrossRefGoogle Scholar
[3] Bendsoe, M. P. and Kikuchi, N., Generating optimal topologies in optimal design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71 (1988), 197–224.CrossRefGoogle Scholar
[4] Sigmund, O., A 99-line topology optimization code written in Matlab, Struct. Multidisc. Optim., 21 (2001), 120–127.CrossRefGoogle Scholar
[5] Sigmund, O., On the design of compliant mechanisms using topology optimization, Mech. Struct. Mach., 25 (1997), 495–526.CrossRefGoogle Scholar
[6] Saxena, A., Topology design of large displacement compliantmechanisms withmultiple materials and multiple output ports, Struct. Multidisc. Optim., 30 (2005), 477–490.CrossRefGoogle Scholar
[7] Bendsoe, M. and Sigmund, O., Topology Optimization-Theory Methods and Applications, Springer, 2003.Google Scholar
[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), 251–259.CrossRefGoogle Scholar
[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), 1261–1296.CrossRefGoogle Scholar
[10] Sigmund, O. and Hougaard, K. G., Geometric properties of optimal photonic crystals, Phys. Rev. Lett., 100 (2008), 153904.CrossRefGoogle ScholarPubMed
[11] Duhring, M. B., Jensen, J. S. and Sigmund, O., Acoustic design by topology optimization, J. Sound Vibr., 317 (2008), 557–575.CrossRefGoogle Scholar
[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), 2060–2074.CrossRefGoogle Scholar
[13] Xie, Y. M. and Steven, G. P., Evolutionary structural optimization, Springer, 1997.CrossRefGoogle Scholar
[14] Tanskanen, P., The evolutionary structural optimization method: theoretical aspects, Comput. Methods Appl. Mech. Engrg., 191 (2002), 47–48.CrossRefGoogle Scholar
[15] Allaire, G., Shape Optimization by the Homogenization Method, Springer-Verlag, New York, 2002.CrossRefGoogle Scholar
[16] Rozvany, G. I. N., Aims scope methods history and unified terminology of computer-aided optimization in structural mechanics, Struct. Multidisc. Optim., 21 (2001), 90–108.CrossRefGoogle Scholar
[17] Bendsoe, M. P. and Sigmund, O., Material interpolations in topology optimization, Arch. Appl. Mech., 69 (1999), 635–654.Google Scholar
[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), 461–484.CrossRefGoogle Scholar
[19] Gersborg-Hansen, A., Sigmund, O. and Haber, R. B., Topology optimization of channel flow problems, Struct. Multidisc. Optim., 30 (2005), 181–192.CrossRefGoogle Scholar
[20] Wang, M. Y., Wang, X. and Guo, D., A level set method for structural optimization, Comput. Methods Appl. Mech. Engrg., 192 (2003), 227–246.CrossRefGoogle Scholar
[21] Allaire, G., Jouve, F. and Toader, A., Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194 (2004), 363–393.CrossRefGoogle Scholar
[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), 10178–10195.CrossRefGoogle Scholar
[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), 487–499.CrossRefGoogle Scholar
[24] Liu, Z. and Korvink, J. G., Adaptive moving mesh level set method for structure optimization, Engrg. Optim., 40 (2008), 529–558.CrossRefGoogle Scholar
[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), 805–842.CrossRefGoogle Scholar
[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), 496–519.CrossRefGoogle Scholar
[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.Google Scholar
[28] Bourdin, B. and Chambolle, A., Design-dependent loads in topology optimization, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 19–48.Google Scholar
[29] Burger, M. and Stainko, R., Phase-field relaxation of topology optimization with local stress constraints, SIAM J. Control Optim., 45 (2006), 1447–1466.CrossRefGoogle Scholar
[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), 245–256.CrossRefGoogle Scholar
[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), 327–342.CrossRefGoogle Scholar
[32] Zhou, S. and Wang, M. Y., Multimaterial structural topology optimization with a generalized CahnCHilliard model of multiphase transition, Struct. Multidisc. Optim., 33 (2007), 89–111.CrossRefGoogle Scholar
[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), 321–350.CrossRefGoogle Scholar
[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), 2697–2718.CrossRefGoogle Scholar
[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), 1475–1493.CrossRefGoogle Scholar
[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), 555–570.CrossRefGoogle Scholar
[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), 975–1001.CrossRefGoogle Scholar
[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), 6688–6708.CrossRefGoogle Scholar
[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), 104–114.CrossRefGoogle Scholar
[40] Alexandersen, J., Aage, N., Andreasen, C. S. and Sigmund, O., Topology optimisation for natural convection problems, Int. J. Numer. Meth. Fluids, 76 (2014), 699–721.CrossRefGoogle Scholar
[41] Victor, M. S., Manuel, G. V. and Clayton, J. R., Wetting and spreading dynamics, CRC Press, 2007.Google Scholar
[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), 423–456.CrossRefGoogle Scholar
[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), 703–727.CrossRefGoogle Scholar
[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), 113–132.CrossRefGoogle Scholar
[45] Peskin, C. S., The immersed boundary method, Acta Num., 11 (2002), 1–39.CrossRefGoogle Scholar
[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), 449–464.CrossRefGoogle Scholar
[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), 1–14.CrossRefGoogle Scholar
[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), 144–157.CrossRefGoogle Scholar
[49] Osher, S. and Fedkiw, R. P., Level set methods and dynamic implicit surfaces, Springer-Verlag, New York, 2002.Google Scholar
[50] Sethian, J. A. and Smereka, P., Level set methods for fluid interfaces, Annu. Rev. Fluid Mech., 35 (2003), 341–372.CrossRefGoogle Scholar
[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), 22–34.CrossRefGoogle Scholar
[52] Anderson, D. M., McFadden, G. B. and Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139–165.CrossRefGoogle Scholar
[53] Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96–127.CrossRefGoogle Scholar
[54] Kim, J. S., A continuous surface tension force formulation for diffuse-interface models, J. Comput. Phys., 204 (2005), 784–804.CrossRefGoogle Scholar
[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), 1–52.Google Scholar
[56] Jacqmin, D., Contact-line dynamics of a diffuse fluid interface, J. Fluid Mech., 402 (2000), 57–88.CrossRefGoogle Scholar
[57] Cahn, J. W. and Hilliard, J., Free energy of a nonuniform system: I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.CrossRefGoogle Scholar
[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.Google Scholar
[59] http://www.comsol.com Google Scholar
[60] Cahn, J.W., On spinodal decomposition, Acta Metall., 9 (1961), 795–801.CrossRefGoogle Scholar
[61] Cahn, J.W., Critical-point wetting, J. Chem. Phys., 66 (1977), 3667–3672.CrossRefGoogle Scholar
[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), 47–67.CrossRefGoogle Scholar
[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), 293–317.CrossRefGoogle Scholar
[64] Nocedal, J. and Wright, S., Numerical optimization, 2nd edition, Springer, 2000.Google Scholar
[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.Google Scholar
[66] Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 2003.CrossRefGoogle Scholar
[67] Giles, M. B., Pierce, N. A., An introduction to the adjoint approach to design, Flow Turbul. Combust., 65 (2000), 393–415.CrossRefGoogle Scholar
[68] Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S., Optimization with PDE Constraints, Springer: Berlin, 2009.Google Scholar
[69] Mohammadi, B. and Pironneau, O., Applied Shape Optimization for Fluids, Oxford University Press, USA: Oxford, 2010.Google Scholar
[70] Defay, R. and Prigogine, I., Surface Tension and Adsorption, Longmans, Green & Co Ltd, London, 1966.Google Scholar
[71] Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, Pergamon Press, Oxford, 1987.Google Scholar