×

zbMATH — the first resource for mathematics

Optimal control of sliding droplets using the contact angle distribution. (English) Zbl 1464.35171
MSC:
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
65K10 Numerical optimization and variational techniques
76D05 Navier-Stokes equations for incompressible viscous fluids
76T99 Multiphase and multicomponent flows
76D55 Flow control and optimization for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Abels and D. Breit, Weak solutions for a non-Newtonian diffuse interface model with different densities, Nonlinearity, 29 (2016), pp. 3426-3453, https://doi.org/10.1088/0951-7715/29/11/3426. · Zbl 1354.35084
[2] H. Abels, D. Depner, and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), pp. 453-480, https://doi.org/10.1007/s00021-012-0118-x. · Zbl 1273.76421
[3] H. Abels, D. Depner, and H. Garcke, On an incompressible Navier-Stokes / Cahn-Hilliard system with degenerate mobility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), pp. 1175-1190, http://www.sciencedirect.com/science/article/pii/S0294144913000243. · Zbl 1347.76052
[4] H. Abels, H. Garcke, and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, https://doi.org/10.1142/S0218202511500138. · Zbl 1242.76342
[5] R. A. Adams and J. H. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. 140, Elsevier, Amsterdam, 2003, https://www.elsevier.com/books/sobolev-spaces/adams/978-0-12-044143-3. · Zbl 1098.46001
[6] A. Al-Sharafi, B. S. Yilbas, H. Ali, and N. Alaqeeli, A water droplet pinning and heat transfer characteristics on an inclined hydrophobic surface, Scientific Reports, 8 (2018), 3061, https://doi.org/10.1038/s41598-018-21511-w.
[7] S. Aland, Time integration for diffuse interface models for two-phase flow, J. Comput. Phys., 262 (2014), pp. 58-71, https://doi.org/10.1016/j.jcp.2013.12.055. · Zbl 1349.82065
[8] S. Aland and F. Chen, An efficient and energy stable scheme for a phase-field model for the moving contact line problem, Internat. J. Numer. Methods Fluids, 81 (2016), pp. 657-671, https://doi.org/10.1002/fld.4200.
[9] M. Aln\aes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. Rognes, and G. Wells, The FEniCS project, version 1.5, Arch. Numer. Software, 3 (2015), https://doi.org/10.11588/ans.2015.100.20553.
[10] H. Alt, Linear Functional Analysis, Springer, New York, 2016, https://doi.org/10.1007/978-1-4471-7280-2. · Zbl 1358.46002
[11] P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 15-41, https://doi.org/10.1137/S0895479899358194. · Zbl 0992.65018
[12] P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, Hybrid scheduling for the parallel solution of linear systems, Parallel Comput., 32 (2006), pp. 136-156, https://doi.org/10.1016/j.parco.2005.07.004.
[13] H. Antil, M. Hintermüller, R. Nochetto, T. Surowiec, and D. Wegner, Finite horizon model predictive control of electrowetting on dielectric with pinning, Interfaces Free Bound., 19 (2017), pp. 1-30, https://doi.org/10.4171/IFB/375. · Zbl 1368.49004
[14] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Users Manual, Tech. report ANL-95/11, Revision 3.9, Argonne National Laboratory, 2018, http://www.mcs.anl.gov/petsc.
[15] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Web Page, http://www.mcs.anl.gov/petsc, 2018.
[16] S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser, Basel, 1997, pp. 163-202. · Zbl 0882.65154
[17] J. Barrett, H. Garcke, and R. Nürnberg, Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid, Math. Comp., 75 (2005), pp. 7-41, https://doi.org/10.1090/S0025-5718-05-01802-8. · Zbl 1078.74050
[18] H. Bonart, C. Kahle, and J.-U. Repke, Comparison of energy stable simulation of moving contact line problems using a thermodynamically consistent Cahn-Hilliard Navier-Stokes model, J. Comput. Phys., 399 (2019), 108959, https://doi.org/10.1016/j.jcp.2019.108959. · Zbl 1453.76062
[19] H. Bonart, C. Kahle, and J.-U. Repke, Optimal control of droplets on a solid surface using distributed contact angles, Langmuir, 36 (2020), pp. 8894-8903, https://doi.org/10.1021/acs.langmuir.0c01242.
[20] M. K. Chaudhury and G. M. Whitesides, How to make water run uphill, Science, 256 (1992), pp. 1539-1541, https://doi.org/10.1126/science.256.5063.1539.
[21] X. Cheng, K. Promislow, and B. Wetton, Asymptotic Behaviour of Time Stepping Methods for Phase Field Models, arXiv:1905.02299v1, 2019.
[22] P. Colli, G. Gilardi, and J. Sprekels, On a Cahn-Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl., 197 (2018), pp. 1445-1475, https://doi.org/10.1007/s10231-018-0732-1. · Zbl 1406.35163
[23] P. Colli and A. Signori, Boundary Control Problem and Optimality Conditions for the Cahn-Hilliard Equation with Dynamic Boundary Conditions, arXiv:1905.00203v1, 2019.
[24] F. Demengel and G. Demengel, Korn’s Inequality in \(L^p\), in Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, New York, 2012, pp. 371-434, https://doi.org/10.1007/978-1-4471-2807-6_7. · Zbl 1239.46001
[25] V. Dhamo, Optimal Boundary Control of Quasilinear Elliptic Partial Differential Equations: Theory and Numerical Analysis, PhD thesis, TU Berlin, 2012, http://dx.doi.org/10.14279/depositonce-3204. · Zbl 1260.49049
[26] A. K. Epstein, A. I. Hochbaum, P. Kim, and J. Aizenberg, Control of bacterial biofilm growth on surfaces by nanostructural mechanics and geometry, Nanotechnology, 22 (2011), 494007, https://doi.org/10.1088/0957-4484/22/49/494007.
[27] S. Frigeri, M. Grasselli, and J. Sprekels, Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential, Appl. Math. Optim., 81 (2020), pp. 899-931, https://doi.org/10.1007/s00245-018-9524-7. · Zbl 1440.35235
[28] I. Fumagalli, N. Parolini, and M. Verani, Optimal control in ink-jet printing via instantaneous control, Comput. Fluids, 172 (2017), pp. 264-273, https://doi.org/10.1016/j.compfluid.2018.05.021. · Zbl 1410.76061
[29] C. G. Gal, M. Grasselli, and A. Miranville, Cahn-Hilliard-Navier-Stokes systems with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), 50, https://doi.org/10.1007/s00526-016-0992-9. · Zbl 1372.35140
[30] M. Gao and X.-P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), pp. 1372-1386, https://doi.org/10.1016/j.jcp.2011.10.015. · Zbl 1408.76138
[31] H. Garcke, M. Hinze, and C. Kahle, A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow, Appl. Numer. Math., 99 (2016), pp. 151-171, http://www.sciencedirect.com/science/article/pii/S0168927415001324. · Zbl 1329.76168
[32] H. Garcke, M. Hinze, and C. Kahle, Optimal control of time-discrete two-phase flow driven by a diffuse-interface model, ESAIM Control Optim. Calc. Var., 25 (2019), 13, https://doi.org/10.1051/cocv/2018006. · Zbl 1437.35575
[33] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer Ser. Comput. Math. 5, Springer, New York, 1986. · Zbl 0585.65077
[34] C. Gräßle, M. Hintermüller, M. Hinze, and T. Keil, Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System, https://arxiv.org/abs/1907.04285, 2019.
[35] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Appl. Math. 69, SIAM, Philadelphia, 2011. · Zbl 1231.35002
[36] G. Grün, On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities, SIAM J. Numer. Anal., 51 (2013), pp. 3036-3061, http://epubs.siam.org/doi/abs/10.1137/130908208. · Zbl 1331.35277
[37] G. Grün, F. Guillén-Gonzáles, and S. Metzger, On fully decoupled convergent schemes for diffuse interface models for two-phase flow with general mass densities, Commun. Comput. Phys., 19 (2016), pp. 1473-1502, https://doi.org/10.4208/cicp.scpde14.39s. · Zbl 1373.76089
[38] G. Grün and F. Klingbeil, Two-phase flow with mass density contrast: Stable schemes for a thermodynamic consistent and frame indifferent diffuse interface model, J. Comput. Phys., 257 (2014), pp. 708-725, http://www.sciencedirect.com/science/article/pii/S0021999113007043. · Zbl 1349.76210
[39] F. Guillén-Gonzáles and G. Tierra, Splitting schemes for a Navier-Stokes-Cahn-Hilliard model for two fluids with different densities, J. Comput. Math., 32 (2014), pp. 643-664, https://doi.org/10.4208/jcm.1405-m4410. · Zbl 1324.76032
[40] F. Guillén-González and G. Tierra, On linear schemes for a Cahn-Hilliard diffuse interface model, J. Comput. Phys., 234 (2013), pp. 140-171, https://doi.org/10.1016/j.jcp.2012.09.020. · Zbl 1284.35025
[41] D. Han and X. Wang, A second order in time, decoupled, unconditionally stable numerical scheme for the Cahn-Hilliard-Darcy system, J. Sci. Comput., 77 (2018), pp. 1210-1233, https://doi.org/10.1007/s10915-018-0748-0. · Zbl 1407.65158
[42] Q. He, R. Glowinski, and X.-P. Wang, A least-squares/finite element method for the numerical solution of the Navier-Stokes-Cahn-Hilliard system modeling the motion of the contact line, J. Comput. Phys., 230 (2011), pp. 4991-5009, https://doi.org/10.1016/j.jcp.2011.03.022. · Zbl 1416.76111
[43] M. Hintermüller, M. Hinze, and C. Kahle, An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system, J. Comput. Phys., 235 (2013), pp. 810-827, https://doi.org/10.1016/j.jcp.2012.10.010. · Zbl 1291.65300
[44] M. Hintermüller, M. Hinze, C. Kahle, and T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn-Hilliard-Navier-Stokes system, Optim. Eng., 19 (2018), pp. 629-662, https://doi.org/10.1007/s11081-018-9393-6.
[45] M. Hintermüller, M. Hinze, and M. H. Tber, An adaptive finite element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem, Optim. Methods Softw., 25 (2011), pp. 777-811, https://doi.org/10.1080/10556788.2010.549230. · Zbl 1366.74070
[46] M. Hintermüller and T. Keil, Optimal Control of Geometric Partial Differential Equations, WIAS preprint-2612, https://doi.org/10.20347/WIAS.PREPRINT.2612, 2019.
[47] M. Hintermüller, T. Keil, and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with nonmatched fluid densities, SIAM J. Control Optim., 55 (2017), pp. 1954-1989, https://doi.org/10.1137/15M1025128. · Zbl 1368.49022
[48] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, Math. Model. Theory Appl. 23, Springer, New York, 2009, https://www.springer.com/mathematics/book/978-1-4020-8838-4. · Zbl 1167.49001
[49] L. Hou, N. R. Smith, and J. Heikenfeld, Electrowetting manipulation of any optical film, Appl. Phys. Lett., 90 (2007), 251114, https://doi.org/10.1063/1.2750544.
[50] C. Huh and L. Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci., 35 (1971), pp. 85-101, https://doi.org/10.1016/0021-9797(71)90188-3.
[51] S. Hysing, S. Turek, D. Kuzmin, N. Parolini, E. Burman, S. Ganesan, and L. Tobiska, Quantitative benchmark computations of two-dimensional bubble dynamics, Internat. J. Numer. Methods Fluids, 60 (2009), pp. 1259-1288, https://doi.org/10.1002/fld.1934. · Zbl 1273.76276
[52] P. Knopf and M. Ebenbeck, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, ESAIM Control Optim. Calc. Var., 26 (2020), https://doi.org/10.1051/cocv/2019059. · Zbl 1451.35233
[53] A. Laurain and S. W. Walker, Droplet footprint control, SIAM J. Control Optim., 53 (2015), pp. 771-799, https://doi.org/10.1137/140979721. · Zbl 1319.35310
[54] A. Logg, K.-A. Mardal, and G. Wells, eds., Automated Solution of Differential Equations by the Finite Element Method-The FEniCS Book, Lect. Notes Comput. Sci. Eng. 84, Springer, New York, 2012, https://doi.org/10.1007/978-3-642-23099-8. · Zbl 1247.65105
[55] F. Mugele and J. Heikenfeld, Electrowetting, Wiley, Weinheim, 2018, https://doi.org/10.1002/9783527412396.
[56] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer, New York, 2012. · Zbl 1246.35005
[57] M. G. Pollack, A. D. Shenderov, and R. B. Fair, Electrowetting-based actuation of droplets for integrated microfluidics, Lab Chip, 2 (2002), https://doi.org/10.1039/b110474h.
[58] T. Qian, X.-P. Wang, and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), pp. 333-360. · Zbl 1178.76296
[59] J. Shen, J. Xu, and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), pp. 407-416, https://doi.org/10.1016/j.jcp.2017.10.021. · Zbl 1380.65181
[60] J. Shen, X. Yang, and H. Yu, Efficient energy stable numerical schemes for a phase field moving contact line model, J. Comput. Phys., 284 (2015), pp. 617-630, https://doi.org/10.1016/j.jcp.2014.12.046. · Zbl 1351.76184
[61] D. ’t Mannetje, S. Ghosh, R. Lagraauw, S. Otten, A. Pit, C. Berendsen, J. Zeegers, D. van den Ende, and F. Mugele, Trapping of drops by wetting defects, Nature Commun., 5 (2014), 3559, https://doi.org/10.1038/ncomms4559.
[62] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Grad. Stud. Math. 112, AMS, Providence, RI, 2010. · Zbl 1195.49001
[63] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), pp. 25-57, https://doi.org/10.1007/s10107-004-0559-y. · Zbl 1134.90542
[64] A. Wächter and C. Laird, IPOPT Webpage, https://github.com/coin-or/Ipopt, 2019.
[65] S. Walker and B. Shapiro, A control method for steering individual particles inside liquid droplets actuated by electrowetting, Lab Chip, 5 (2005), pp. 1404-1407, https://doi.org/10.1039/B513373B.
[66] S. W. Walker, A mixed formulation of a sharp interface model of Stokes flow with moving contact lines, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 969-1009, https://doi.org/10.1051/m2an/2013130. · Zbl 1299.76064
[67] S. W. Walker, B. Shapiro, and R. H. Nochetto, Electrowetting with contact line pinning: Computational modeling and comparisons with experiments, Phys. Fluids, 21 (2009), 102103, https://doi.org/10.1063/1.3254022. · Zbl 1183.76554
[68] X. Wu, G. van Zwieten, and K. van der Zee, Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models , Int. J. Numer. Methods Biomed. Eng., 30 (2014), pp. 180-203, https://doi.org/10.1002/cnm.2597.
[69] J. Xu, Y. Li, S. Wu, and A. Bousquet, On the stability and accuracy of partially and fully implicit schemes for phase field modeling, Comput. Methods Appl. Mech. Engrg., 345 (2019), pp. 826-853, https://doi.org/10.1016/j.cma.2018.09.017. · Zbl 1440.80003
[70] X. Xu, Y. Di, and H. Yu, Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines, J. Fluid Mech., 849 (2018), pp. 805-833, https://doi.org/10.1017/jfm.2018.428. · Zbl 1415.76666
[71] X. Yang and L. Ju, Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model, Comput. Methods Appl. Mech. Engrg., 318 (2017), pp. 1005-1029, https://doi.org/10.1016/j.cma.2017.02.011. · Zbl 1439.76029
[72] H. Yu and X. Yang, Numerical approximations for a phase-field moving contact line model with variable densities and viscosities, J. Comput. Phys., 334 (2017), pp. 665-686, https://doi.org/10.1016/j.jcp.2017.01.026. · Zbl 1375.76201
[73] E. Zeidler, Applied Functional Analysis, Appl. Math. Sci. 109, Springer, New York, 1995, https://doi.org/10.1007/978-1-4612-0821-1. · Zbl 0834.46003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.