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 Lauser et al. (2011) 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.

我们提出了一种新方法来求解包含互补条件的一般方程组，特别关注平衡时多组分多相混合物的热力学中出现的那些方程。事实上，Lauser 等人引入的统一公式。(2011) 最近成为自动处理多孔介质组合多相流中相的出现和消失的有前途的方法。从数学的角度来看，经过空间和时间的离散化，这导致了一个由代数方程和非线性互补方程组成的系统。由于后者的非平滑性，通常用于求解此类系统的半平滑和平滑方法通常很慢或可能根本不收敛。这一观察使我们设计了一种称为 NPIPM 的新策略（非参数内点法）。受优化中的内点方法的启发，我们提出的技术具有在享受理论全局收敛的同时避免任何参数管理的优点。这通过广泛的数值测试得到了验证，我们将 NPIPM 与 Newton-min 方法进行了比较，后者是几乎所有油藏工程师和热力学家的标准参考。