Mika Juntunen

A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H(div) velocityand continuous capillary pressure

We consider the slightly compressible two-phase flow problem in a porous medium with capillary pressure. The problem is solved using the implicit pressure, explicit saturation (IMPES) method, and the convergence is accelerated with iterative coupling of the equations. We use discontinuous Galerkin to discretize both the pressure and saturation equations. We apply two improvements, which are projecting the flux to the mass conservative H(div)-space and penalizing the jump in capillary pressure in the saturation equation. We also discuss the need and use of slope limiters and the choice of primary variables in discretization. The methods are verified with two- and three-dimensional numerical examples. The results show that the modifications stabilize the method and improve the solution.

A global jacobian method for mortar discretizations of nonlinear porous media flows

We describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method. There are two main ideas: (1) linearize the global system in both subdomain and interface variables simultaneously to yield a single Newton iteration; and (2) algebraically eliminate subdomain velocities (and optionally, subdomain pressures) to solve linear systems for the 1st (or the 2nd) Schur complements. Solving the 1st Schur complement system gives the multiscale solution without the need to solve an interface iteration. Solving the 2nd Schur complement system gives a linear interface problem for a nonlinear model. The methods are less complex than a previously developed nonlinear mortar algorithm, which requires two nested Newton iterations and a forward difference approximation. Furthermore, efficient linear preconditioners can be applied to speed up the iteration. The methods are implemented in parallel, and a numerical study is performed ...

On the connection between the stabilized Lagrange multiplier and Nitsche's methods

We derive Nitsche’s method for the domain decomposition through the stabilized Lagrange multiplier method. Taking material parameters carefully into account this derivation naturally introduces parameter weighted average flux and stabilizing terms to Nitsche’s method. We show stability and a priori analyses in the mesh dependent norms for both the stabilized method and Nitsche’s method, and discuss connections between the proposed methods.

Quantifying uncertainty in material damage from vibrational data

The response of a vibrating beam to a force depends on many physical parameters including those determined by material properties. Damage caused by fatigue or cracks results in local reductions in stiffness parameters and may drastically alter the response of the beam. Data obtained from the vibrating beam are often subject to uncertainties and/or errors typically modeled using probability densities. The goal of this paper is to estimate and quantify the uncertainty in damage modeled as a local reduction in stiffness using uncertain data. We present various frameworks and methods for solving this parameter determination problem. We also describe a mathematical analysis to determine and compute useful output data for each method. We apply the various methods in a specified sequence that allows us to interface the various inputs and outputs of these methods in order to enhance the inferences drawn from the numerical results obtained from each method. Numerical results are presented using both simulated and experimentally obtained data from physically damaged beams.

Two-phase flow in complicated geometries

We study modeling two-phase flow in complicated geometries. We use modern mesh generation techniques to improve the quality of the mesh and at the same time both reduce the number of elements and capture the geometry accurately. The generated meshes consist of orthogonally optimized general hexahedras. To model the flow in general hexahedras, we use the multipoint flux mixed finite element method. As a test problem we use the Frio experiment data.

Reparameterization for statistical state estimation applied to differential equations

The ensemble Kalman filter is a widely applied data assimilation technique useful for improving the forecast of computational models. The main computational cost of the ensemble Kalman filter comes from the numerical integration of each ensemble member forward in time. When the computational model involves a partial differential equation, the degrees of freedom of the solution in the discretization of the spatial domain are oftentimes used for the representation of the state of the system, and the filter is applied to this state vector. We propose a method of approximating the state of a partial differential equation in a representation space developed separately from the numerical method. This representation space represents a reparameterization of the state vector and can be chosen to retain desirable physical features of the solutions. We apply the ensemble Kalman filter to this representation of the state, and numerically demonstrate that acceptable results are obtained with substantially smaller ensemble sizes.

On the local mesh size of Nitsche’s method for discontinuous material parameters

We propose Nitsche’s method for discontinuous parameters that takes the local mesh sizes of the non-matching meshes carefully into account. The method automatically adapts to the changing material parameters and mesh sizes. With continuous parameters, the method compares to the classical Nitsche’s method. With large discontinuity, the method approaches assigning Dirichlet boundary conditions with Nitsche’s method.