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A new approach for solving nonlinear algebraic systems with complementarity conditions. Application to compositional multiphase equilibrium problems

机译:互补条件求解非线性代数系统的新方法。 应用于组成多相均衡问题

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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.
机译:我们提出了一种解决包含互补条件的方程的一般系统的新方法,特别关注多组分多相混合物在平衡时产生的那些产生的方法。实际上,Lauser等人介绍了统一的制定。 (2011年)最近出现为自动处理多孔介质组成多相流动中相位的外观和消失的有希望的方法。从数学观点和空间和时间的离散化之后,这导致由代数方程和非线性互补方程组成的系统。由于后者的非空间,常用于解决这种系统的半球和平滑方法通常是慢的或可能不会收敛。此观察指导我们设计了一种名为NPIPM(非参数内部点方法)的新策略。灵感来自优化中的内部点方法,我们提出的技术具有避免任何参数管理的优势,同时享受理论全球收敛。这是通过广泛的数值测试验证,其中我们将NPIPM与Newton-Min方法进行比较,几乎所有水库工程师和热力学家的标准参考。

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