×

A new approach for solving nonlinear algebraic systems with complementarity conditions. Application to compositional multiphase equilibrium problems. (English) Zbl 07431569

Summary: We present a new method to solve general systems of equations containing complementarity conditions, with a special focus on those arising in the thermodynamics of multicomponent multiphase mixtures at equilibrium. Indeed, the unified formulation introduced by A. Lauser et al. [“A new approach for phase transitions in miscible multi-phase flow in porous media”, Adv. Water Resour. 34, No. 8, 957–966 (2011; doi:10.1016/j.advwatres.2011.04.021)] has recently emerged as a promising way to automatically handle the appearance and disappearance of phases in porous media compositional multiphase flows. From a mathematical viewpoint and after discretization in space and time, this leads to a system consisting of algebraic equations and nonlinear complementarity equations. Due to the nonsmoothness of the latter, semismooth and smoothing methods commonly used for solving such a system are often slow or may not converge at all. This observation led us to design a new strategy called NPIPM (NonParametric Interior-Point Method). Inspired from interior-point methods in optimization, the technique we propose has the advantage of avoiding any parameter management while enjoying theoretical global convergence. This is validated by extensive numerical tests, in which we compare NPIPM to the Newton-min method, the standard reference for almost all reservoir engineers and thermodynamicists.

MSC:

65-XX Numerical analysis
76-XX Fluid mechanics

Software:

PLCP; SQPlab
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Acary, V.; Brogliato, B., (Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics, vol. 35 (2008), Springer: Springer Berlin) · Zbl 1173.74001
[2] Aganagić, M., Newton’s method for linear complementarity problems, Math. Program., 28, 3, 349-362 (1984) · Zbl 0533.90088
[3] Arenas, F. E.; Martinez, H. J.; Pérez, R., A local Jacobian smoothing method for solving nonlinear complementarity problems, Univ. Sci., 25, 1, 149-174 (2020)
[4] Auslender, A.; Cominetti, R.; Haddou, M., Asymptotic analysis for penalty and barrier methods in convex and linear programming, Math. Oper. Res., 22, 1, 43-62 (1997) · Zbl 0872.90067
[5] Beaude, L.; Brenner, K.; Lopez, S.; Masson, R.; Smai, F., Non-isothermal compositional liquid gas Darcy flow: Formulation, soil-atmosphere boundary condition and application to high-energy geothermal simulations, Comput. Geosci., 23, 3, 443-470 (2019) · Zbl 1419.76604
[6] Ben Gharbia, I., Résolution de problèmes de complémentarité.: Application à un écoulement diphasique dans un milieu poreux (2012), Université Paris Dauphine: Université Paris Dauphine Paris IX, URL: http://tel.archives-ouvertes.fr/tel-00776617
[7] Ben Gharbia, I.; Flauraud, É., Study of compositional multiphase flow formulation using complementarity conditions, Oil Gas Sci. Technol., 74, 43 (2019)
[8] Ben Gharbia, I.; Flauraud, É.; Michel, A., Study of compositional multi-phase flow formulations with cubic EOS, (SPE Reservoir Simulation Symposium, 23-25 February, Houston, Texas, USA, vol. 2 (2015)), 1015-1025
[9] Ben Gharbia, I.; Gilbert, J. C., Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix, Math. Program., 134, 2, 349-364 (2012) · Zbl 1254.90252
[10] Ben Gharbia, I.; Gilbert, J. C., An algorithmic characterization of P-matricity, SIAM J. Matrix Anal. Appl., 34, 904-916 (2013) · Zbl 1281.65088
[11] Ben Gharbia, I.; Gilbert, J. C., An algorithmic characterization of P-matricity II: Adjustments, refinements, and validation, SIAM J. Matrix Anal. Appl., 40, 800-813 (2019) · Zbl 07099842
[12] Ben Gharbia, I.; Haddou, M.; Tran, Q. H.; Vu, D. T.S., An analysis of the unified formulation for the equilibrium problem of compositional multiphase mixtures (2021), URL: https://hal-ifp.archives-ouvertes.fr/hal-03059788, under revision
[13] Ben Gharbia, I.; Jaffré, J., Gas phase appearance and disappearance as a problem with complementarity constraints, Math. Comput. Simulation, 99, 28-36 (2014) · Zbl 07312578
[14] Bonnans, F., (Optimisation continue: cours et problèmes corrigés. Optimisation continue: cours et problèmes corrigés, Mathématiques appliquées pour le Master (2006), Dunod)
[15] Bonnans, J. F.; Gilbert, J. C.; Lemaréchal, C.; Sagastizábal, C. A., (Numerical Optimization: Theoretical and Practical Aspects. Numerical Optimization: Theoretical and Practical Aspects, Universitext (2006), Springer Verlag: Springer Verlag Berlin) · Zbl 1108.65060
[16] Boyd, S.; Vandenberghe, L., (Convex Optimization. Convex Optimization, Berichte über verteilte messysteme (2004), Cambridge University Press: Cambridge University Press Cambridge, UK) · Zbl 1058.90049
[17] Bui, Q. M.; Elman, H. C., Semi-smooth Newton methods for nonlinear complementarity formulation of compositional two-phase flow in porous media, J. Comput. Phys., 407, Article 109163 pp. (2020) · Zbl 07504691
[18] Chen, S.; Pang, L.-P.; Li, D., An inexact semismooth Newton method for variational inequality with symmetric cone constraints, J. Ind. Manag. Optim., 11, 3, 733-746 (2015) · Zbl 1401.90237
[19] Coats, K. H., An equation of state compositional model, SPE J., 20, 5, 363-376 (1980)
[20] Cobzaş, Ş.; Miculescu, R.; Nicolae, A., (Lipschitz Functions. Lipschitz Functions, Lecture Notes in Mathematics, vol. 2241 (2019), Springer: Springer Cham, Switzerland) · Zbl 1431.26002
[21] Deiters, U. K.; Kraska, T., (High-Pressure Fluid Phase Equilibria: Phenomenology and Computation. High-Pressure Fluid Phase Equilibria: Phenomenology and Computation, Supercritical Fluid Science and Technology, vol. 2 (2012), Elsevier: Elsevier Amsterdam), 370
[22] Dreves, A.; Facchinei, F.; Fischer, A.; Herrich, M., A new error bound result for generalized Nash equilibrium problems and its algorithmic application, Comput. Optim. Appl., 59, 63-84 (2014) · Zbl 1307.91117
[23] Dreves, A.; Facchinei, F.; Kanzow, C.; Sagratella, S., On the solution of the KKT conditions of generalized Nash equilibrium problems, SIAM J. Optim., 21, 3, 1082-1108 (2011) · Zbl 1230.90176
[24] Dussault, J.-P.; Frappier, M.; Gilbert, J. C., Polyhedral Newton-Min Algorithms for Complementarity ProblemsResearch Report (2019), Inria Paris; Université de Sherbrooke: Inria Paris; Université de Sherbrooke Québec, Canada, URL: https://hal.archives-ouvertes.fr/hal-02306526
[25] Dussault, J.-P.; Frappier, M.; Gilbert, J. C., A lower bound on the iterative complexity of the Harker and Pang globalization technique of the Newton-min algorithm for solving the linear complementarity problem, EURO J. Comput. Optim., 7, 4, 359-380 (2019) · Zbl 07194686
[26] Facchinei, F.; Pang, J. S., (Finite-Dimensional Variational Inequalities and Complementarity Problems, I. Finite-Dimensional Variational Inequalities and Complementarity Problems, I, Springer Series in Operations Research (2003), Springer: Springer New York) · Zbl 1062.90001
[27] Facchinei, F.; Pang, J. S., (Finite-Dimensional Variational Inequalities and Complementarity Problems, II. Finite-Dimensional Variational Inequalities and Complementarity Problems, II, Springer Series in Operations Research (2003), Springer: Springer New York) · Zbl 1062.90002
[28] Fischer, A., A special Newton-type optimization method, Optimization, 24, 3-4, 269-284 (1992) · Zbl 0814.65063
[29] Gfrerer, H.; Outrata, J., On a semismooth Newton method for solving generalized equations, SIAM J. Optim., 31, 1, 489-517 (2021) · Zbl 1462.90141
[30] Haddou, M., A new class of smoothing methods for mathematical programs with equilibrium constraints, Pac. J. Optim., 5, 1, 86-96 (2009) · Zbl 1161.90488
[31] Haddou, M.; Maheux, P., Smoothing methods for nonlinear complementarity problems, J. Optim. Theory Appl., 160, 3, 711-729 (2014) · Zbl 1396.90085
[32] Huang, S.; Wan, Z., A new nonmonotone spectral residual method for nonsmooth nonlinear equations, J. Comput. Appl. Math., 313, 82-101 (2017) · Zbl 1356.90139
[33] Izmailov, A. F.; Solodov, M. V., (Newton-Type Methods for Optimization and Variational Problems. Newton-Type Methods for Optimization and Variational Problems, Springer Series in Operations Research and Financial Engineering (2014), Springer: Springer Cham, Switzerland) · Zbl 1304.49001
[34] Kräutle, S., The semismooth Newton method for multicomponent reactive transport with minerals, Adv. Water Resour., 34, 1, 137-151 (2011)
[35] Kröner, A.; Kunisch, K.; Vexler, B., Semismooth Newton methods for optimal control of the wave equation with control constraints, SIAM J. Control Optim., 49, 2, 830-858 (2011) · Zbl 1218.49035
[36] Lauser, A.; Hager, C.; Helmig, R.; Wohlmuth, B., A new approach for phase transitions in miscible multi-phase flow in porous media, Adv. Water Resour., 34, 8, 957-966 (2011)
[37] Lusetti, I., Numerical Methods for Compositional Multiphase Flow Models with Cubic EOSTechnical Report (2016), IFPEN
[38] Mannel, F.; Rund, A., A hybrid semismooth quasi-Newton method for nonsmooth optimal control with PDEs, Optim. Eng. (2020)
[39] Masson, R.; Trenty, L.; Zhang, Y., Formulations of two phase liquid gas compositional Darcy flows with phase transitions, Int. J. Finite Vol., 11, 1-34 (2014), URL: http://ijfv.math.cnrs.fr/IMG/pdf/gazliqcomp-ijfv-1.pdf · Zbl 1490.76155
[40] Masson, R.; Trenty, L.; Zhang, Y., Coupling compositional liquid gas Darcy and free gas flows at porous and free-flow domains interface, J. Comput. Phys., 321, 708-728 (2016) · Zbl 1349.65376
[41] Mehrotra, S., On the implementation of a primal-dual interior point method, SIAM J. Optim., 2, 4, 575-601 (1992) · Zbl 0773.90047
[42] Mifflin, R., Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15, 6, 959-972 (1977) · Zbl 0376.90081
[43] Milzarek, A.; Ulbrich, M., A semismooth Newton method with multidimensional filter globalization for \(l \text{\_} 1\)-optimization, SIAM J. Optim., 24, 1, 298-333 (2014) · Zbl 1295.49022
[44] Orbey, H.; Sandler, S. I., (Modeling Vapor-Liquid Equilibria: Cubic Equations of State and their Mixing Rules. Modeling Vapor-Liquid Equilibria: Cubic Equations of State and their Mixing Rules, Cambridge Series in Chemical Engineering (1998), Cambridge University Press)
[45] Peton, N., Comparaison de plusieurs formulations pour les écoulements multiphasiques et compositionnels en milieu poreuxTechnical Report (2015), IFPEN
[46] Qi, L.; Sun, J., A nonsmooth version of Newton’s method, Math. Program., 58, 1-3, 353-367 (1993) · Zbl 0780.90090
[47] Qi, L.; Sun, D.; Ulbrich, M., (Semismooth and Smoothing Newton Methods. Semismooth and Smoothing Newton Methods, Springer Series in Operations Research and Financial Engineering (2022), Springer: Springer New York)
[48] Ulbrich, M., (Semismooth newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. Semismooth newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, MOS-SIAM Series on Optimization (2011), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia) · Zbl 1235.49001
[49] Vu, D. T.S., Numerical Resolution of Algebraic Systems with Complementarity Conditions. Application to the Thermodynamics of Compositional Multiphase Mixtures (2020), Université Paris-Saclay, URL: https://tel.archives-ouvertes.fr/tel-02987892
[50] Wright, S. J., Primal-Dual Interior-Point Methods (1997), SIAM: SIAM Philadelphia · Zbl 0863.65031
[51] Wright, M. H., The interior-point revolution in optimization: History, recent developments, and lasting consequences, Bull. Amer. Math. Soc., 42, 1, 39-56 (2005) · Zbl 1114.90153
[52] Zhu, J.; Hao, B., A new smoothing method for solving nonlinear complementarity problems, Open Math., 17, 1, 104-119 (2019) · Zbl 1430.90533
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.