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PhD in Applied Mathematics at IFPEN

Employer
Global Academy Jobs
Location
France
Closing date
Jun 30, 2017

Job Details

Subject: Numerical resolution of algebraic systems with complementarity conditions. Application to the thermodynamics of compositional multiphase mixtures
Description: In reservoir simulations, a delicate issue arises from the laws of thermodynamics relative to hydrocarbon mixtures. For compositional multiphase flow, the difficulty lies in handling the appearance and disappearance of phases for different components. The traditional dynamic approach, called "variable switching", considers only the unknowns of the present phases and the equations for them. It is heavy to implement and costly in CPU time, because the "switching" occurs constantly, even from a Newton iteration to another.

An alternative approach, called "unified formulation", enables one to keep a fixed set of unknowns and equations regardless of the context. From the theoretical viewpoint, this is an important step and a promising lead that the IFPEN teams have been following for the last few years. From the practical viewpoint, as the new formalism involves several unsmooth "complementarity" equations, it is necessary after discretization to resort to "semi-smooth" Newton methods, which exhibit a pathological oscillatory behavior during phase transitions. In short, all we have done is to pass the buck to the nonlinear solver. The commonly used remedy in this case is to restart the Newton process with a smaller time-step, which is of course a waste of CPU time and which eventually makes the unified formulation less interesting.

Nevertheless, on the grounds of some recent works on Newton variants (not specifically dedicated to complementary equations), we can legitimately hope that this numerical challenge can be overcome. The objective of this thesis is to design new nonlinear solvers that are better suited to our systems and that guarantee convergence without incurring an exaggerated price to be paid in execution time.
Keywords: Complementarity conditions, Linear and nonlinear solver, Newton method, Homotopy continuation, implicit discretization.

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