Some multiscale methods for partial differential equations by J. Xu and A. Zhou Terascale implicit methods for partial differential equations by D. E. Keyes Boundary conditions for fully implicit two-phase flow models by M. Peszynska, E. W. Jenkins, and M. F. Wheeler Phase-field models of solidification by G. B. McFadden Singularity formation in free-surface Stokes flows by Q. Nie, S. Tanveer, T. F. Dupont, and X. Li A note on cache memory methods for multigrid in three dimensions by C. C. Douglas and D. T. Thorne.

Recent advances in numerical methods for partial differential equations and applications : proceedings of the 2001 John H. Barrett memorial lectures, trends in computational mathematics, May 10-12, 2001, the University of Tennessee, Knoxville, TN

Motivated by applications in subsurface CO2 sequestration, we investigate constrained optimal control problems with partially miscible two-phase flow in porous media. The objective is, e.g., to maximize the amount of trapped CO2 in an underground reservoir after a fixed period of CO2 injection, where the time-dependent injection rates in multiple wells are used as control parameters. We describe the governing two-phase two-component Darcy flow PDE system and formulate the optimal control problem. For the discretization we use a variant of the BOX method, a locally conservative control-volume FE method. The timestep-wise Lagrangian of the control problem is implemented as a functional in the PDE toolbox Sundance, which is part of the HPC software Trilinos. The resulting MPI parallelized Sundance state and adjoint solvers are linked to the interior point optimization package IPOPT. Finally, we present some numerical results in a heterogeneous model reservoir.

Optimal Control of Partially Miscible Two-Phase Flow with Applications to Subsurface CO2 Sequestration

The goal of the present paper is to apply control techniques to determine the Blake cavitation threshold for adiabatic bubbles. Considering state space representation of Rayleigh-Plesset (RP) equation, a stability condition is derived to predict the threshold for cavitation which was determined beforehand by mechanical bias. Moreover, sliding mode control has been utilized to regulate the radius of a single spherical bubble to the desired value to prevent collapse occurrence which has a great importance in some industrial applications. Simulation results are presented to show the effectiveness of the proposed method.

Stability analysis and sliding mode control of a single spherical bubble dynamics

This paper transforms the Keller–Miksis (KM) model, which is the main equation describing the bubble behavior, into a state space representation and presents a stability condition for the system using the linearized form of the KM equation. The dynamic of the cavitation bubble is analyzed for different gases, and its burst time is measured. Then, a sliding mode controller (SMC) is designed for the nonlinear system to regulate the radius of a single spherical bubble and prevent collapse occurrence, which has a great importance in some industrial applications. The system's robustness in the presence of uncertainties in bubble parameters is one of its most significant control objectives obtained in the controller designed in this paper. A comparison is also made between this controller and the controller previously designed for the simpler bubble model (Rayleigh–Plesset (RP) model). Also, the chattering created by SMC is eliminated by two methods. Simulation results are presented to show the effectiveness of the proposed method.

Stability analysis and sliding mode control of a single spherical bubble described by Keller–Miksis equation

Rate optimization of fractional flow reservoir model based on the continuous adjoint method

With the target of optimizing CO2 sequestration in underground reservoirs, we investigate constrained optimal control problems with partially miscible two-phase flow in porous media Our objective is to maximize the amount of trapped CO2 in an underground reservoir after a fixed period of CO2 injection, while time-dependent injection rates in multiple wells are used as control parameters We describe the governing two-phase two-component Darcy flow PDE system, formulate the optimal control problem and derive the continuous adjoint equations For the discretization we apply a variant of the so-called BOX method, a locally conservative control-volume FE method that we further stabilize by a periodic averaging feature to reduce oscillations The timestep-wise Lagrange function of the control problem is implemented as a variational form in Sundance, a toolbox for rapid development of parallel FE simulations, which is part of the HPC software Trilinos We discuss the BOX method and our implementation in Sundance The MPI parallelized Sundance state and adjoint solvers are linked to the interior point optimization package IPOPT, using limited-memory BFGS updates for approximating second derivatives Finally, we present and discuss different types of optimal control results

Adjoint based optimal control of partially miscible two-phase flow in porous media with applications to CO2 sequestration in underground reservoirs

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Optimal Control of Partially Miscible Two-Phase Flow with Applications to Subsurface CO2 Sequestration

Adjoint based optimal control of partially miscible two-phase flow in porous media with applications to CO2 sequestration in underground reservoirs