Fully implicit black-oil simulations result in huge, often very-illconditioned, linear systems of equations for different unknowns (e.g., pressure and saturations). It is well-known that the underlying Jacobian matrices contain both hyperbolic and nearly elliptic subsystems (corresponding to saturations and pressure, respectively). Because a reservoir simulation is typically driven by the behavior of the pressure, constrained-pressure-residual (CPR)type two-stage preconditioning methods to solve the coupled linear systems are a natural choice and still belong to the most popular approaches. After a suitable extraction and decoupling, the computationally most costly step in such two-stage methods consists in solving the elliptic subsystems accurately enough. Algebraic multigrid (AMG) provides a technique to solve elliptic linear equations very efficiently. Hence, in recent years, corresponding CPR-AMG approaches have been extensively used in practice.Unfortunately, if applied in a straightforward manner, CPR-AMG does not always work as expected. In this paper, we discuss the reasons for the lack of robustness observed in practice, and present remedies. More precisely, we will propose a preconditioning strategy (based on a suitable combination of left and right preconditioning of the Jacobian matrix) that aims at a compromise between the solvability of the pressure subproblem by AMG and the needs of the outer CPR process. The robustness of this new preconditioning strategy will be demonstrated for several industrial test cases, some of which are very ill-conditioned. Furthermore, we will demonstrate that CPR-AMG can be interpreted in a natural way as a special AMG process applied directly to the coupled Jacobian systems.

Fully implicit petroleum reservoir simulations result in huge, often very ill-conditioned linear systems of equations to solve for different unknowns, for example, pressure and saturations. It is well known that the full system matrix contains both hyperbolic as well as nearly elliptic sub-systems. Since the solution of the coupled system is mainly determined by the solution of their elliptic (typically pressure) components, (CPR-type) two-stage preconditioning methods still belong to the most popular approaches to tackle such coupled systems. After a suitable extraction and decoupling, the numerically most costly step in such two-stage methods consists in solving these elliptic sub-systems. It is known that algebraic multigrid (AMG) provides a technique to solve elliptic linear equations very efficiently. The main advantage of AMG-based solvers -their numerical scalability -makes them particularly efficient for solving huge linear systems.Depending on the application, the system's properties range from simple to highly indefinite. Unfortunately decoupling pressure and saturation related parts may introduce further difficulties. Consequently, in complex industrial simulations, the application of AMG to elliptic sub-systems might not be straightforward. In fact, an important goal in defining an efficient two-stage preconditioning strategy consists in extracting elliptic sub-systems that are suitable for an efficient AMG solution and, at the same time, ensure a fast overall convergence of the two-stage approach.The importance of this will be demonstrated for several industrial cases. In particular, some of these cases are very hard to solve by AMG if applied in a standard way.Preliminary results for a CPR-type coupling of SAMG to CMG's PARASOL, a variable degree variable ordering ILU preconditioner using FGMRES, are compared to using PARASOL by itself. Alternative preconditioning operators will be presented giving elliptic sub-systems which are not only more suitable for applying AMG efficiently but also help accelerate the CPR-type process. Comparisons with one-level iterative methods will show the acceleration by AMG is highly superior. Finally, a strategy is presented that combines all linear solver parts in one single AMG-iteration. In this sense CPR can be seen as a special case of AMG for systems. This, in turn, yields a -formally -very simple but simultaneously very flexible solution approach.

Summary System-algebraic multigrid (AMG) provides a flexible framework for linear systems in simulation applications that involve various types of physical unknowns. Reservoir-simulation applications, with their driving elliptic pressure unknown, are principally well-suited to exploit System-AMG as a robust and efficient solver method. However, the coarse grid correction must be physically meaningful to speed up the overall convergence. It has been common practice in constrained-pressure-residual (CPR) -type applications to use an approximate pressure/saturation decoupling to fulfill this requirement. Unfortunately, this can have significant effects on the AMG applicability and, thus, is not performed by the dynamic row-sum (DRS) method. This work shows that the pressure/saturation decoupling is not necessary for ensuring an efficient interplay between the coarse grid correction process and the fine-level problem, demonstrating that a comparable influence of the pressure on the different involved partial-differential equations (PDEs) is much more crucial. As an extreme case with respect to the outlined requirement, linear systems from compositional simulations under the volume-balance formulation will be discussed. In these systems, the pressure typically is associated with a volume balance rather than a diffusion process. It will be shown how System-AMG can still be used in such cases.

System-AMG provides a flexible framework for linear systems in simulation applications that involve various different types of physical unknowns. Reservoir simulation applications, with their driving elliptic pressure unknown, are principally well-suited to exploit System-AMG as a robust and efficient solver method. However, in order to be efficient and robust, the coarse grid correction process of AMG on the one hand needs to be possible, i.e., the matrix needs to fulfill certain requirements. It has been demonstrated earlier how to ensure this by the dynamic rowsumming (DRS) method. On the other hand, the coarse grid correction must be physically meaningful in order to speed up the overall convergence. It has been common practice in CPR-type applications to use an approximate pressure-saturation decoupling to fulfill this requirement. This, however, can have drastic impacts on AMG’s applicability and, thus, is not performed by the DRS-method. In this work, we are going to see that the pressure-saturation decoupling indeed is not necessary for ensuring an efficient interplay between the coarse grid correction process and the fine-level problem. We will find that a comparable influence of the pressure on the different involved PDEs is much more crucial. As an extreme case w.r.t. the outlined requirement, we will discuss linear systems from compositional simulations under the volume balance formulation. In these systems, the pressure typically is associated with a volume-balance, rather than a diffusion process. The corresponding coarse grid correction does, hence, not provide any benefit regarding the overall convergence: the other PDEs involve pressure-based diffusive parts that have a drastically different structure than the volume-balance has. We will see how System-AMG can still be used in such cases.

This work deals with the parametric optimization of the position and form of a conformal cooling used in the injection molding industry. Based on a literature survey, an optimization routine concerning the parameter optimization of cooling system designs was developed and implemented with the help of the software package Moldflow. The main objective of the optimization is to reduce the cooling time; the second is to obtain an optimized homogeneous temperature distribution over the complete tool surface. To enable a comparison of the new close-contour solution with a classical manufacturing process, an optimized cooling system simulation, based on a conventional manufacturing solution, was established. It can be shown that the optimized close-contour cooling design offers significant advantages that cannot be exploited using classical manufacturing. Finally, the additive manufacturing of a prototype in the framework of powder bed fusion is documented as a proof of concept.