**Organizer & chair: Jaap van der Vegt (Universiteit Twente)**

Waaier 1, 10:15 – 12:15

## 10:15-10:45

### The numerical prediction of extreme hydrodynamic wave loads

*Arthur E.P. Veldman, Rijksuniversiteit Groningen*

Non-linear free-surface phenomena, such as extreme waves and violent sloshing, and their impact on the endangered offshore and coastal constructions have long been subjects that could only be studied with experimental methods. Nowadays, computational luid dynamics tools can make a significant contribution to these flow problems. The presentation will sketch the computational modeling aspects of the simulation method ComFLOW, which is developed in close connection with MARIN, Deltares and the offshore industry. The method solves the one- and two-phase Navier-Stokes equations with a volume of fluid based free-surface treatment. Special attention will be paid to recent algorithmic refinements. Its current capabilities will be validated against a diverse series of experiments.

## 10:45-11:15

### Adaptive grids for non-monotone waves in an extended Richards model

*Paul Zegeling, Universiteit Utrecht*

In this talk we will emphasize the importance of both analysis and computation in relation to a bifurcation problem in a non-equilibrium Richards model from hydrology. The extension of the Richards PDE for the water saturation to take into account dynamic memory effects was suggested by Hassanizadeh and Gray in the 90's. This gives rise to an extra third-order mixed space-time-derivative term in the PDE. Theory from applied analysis can, however, only used to predict the general solution behavior. We show that numerical adaptive grid PDE solutions can be nicely compared with the experimental observations from the laboratory, both in one dimension (non-monotone waves) and in two space dimensions (instabilities).

## 11:15-11:45

### Goal-Adaptive Methods for Fluid-Structure Interaction

*Harald van Brummelen, Technische Universiteit Eindhoven*

The numerical solution of fluid-structure-interaction problems poses a paradox, in that most of the computational resources are consumed by the subsystem that is of least practical interest, viz., the fluid. Goal-adaptive discretization methods provide a paradigm to bypass this paradox. Based on the solution of a dual problem, the contribution of local errors to the error in a specific goal functional is estimated, and only the regions that yield a dominant contribution are refined.

In the present work, we address two fundamental complications in goal adaptivity for fluid-structure-interaction problems. The first pertains to the free-boundary character of fluid-structure-interaction problems, which induces so-called shape-derivatives in the linearized adjoint problem. The second complication concerns the treatment of the interface conditions, which has nontrivial consequences for the dual problem. Numerical results are presented to illustrate these different aspects.

## 11:45-12:15

### Preconditioners for the discretized incompressible Navier-Stokes equations

*C. Vuik, M. ur Rehman and A. Segal, Technische Universiteit Delft*

In this talk we consider simulation with the incompressible Navier Stokes equations. After discretization by the Finite Element Method and linearization (using Picard or Newton-Raphson) one obtains a large sparse linear system. Due to the incompressibility constraint a large zero block appears on the main diagonal of the system matrix. This type of problems is also known as a saddle point problem. Straightforward application of direct or iterative solvers leads to breakdown or divergence of the methods. For large 3 dimensional problems only iterative methods are feasible due to large time and memory requirements of direct solvers.

For the iterative methods several Krylov subspace solvers can be used as there are: Bi-CGSTAB, GMRES, GCR and PCG. We also use a recently described Krylov solver: IDR(s). The most important part is always to find a suitable preconditioner. As a preconditioner we consider an adaptation of the SIMPLE(R) method, which is mostly used for Finite Volume Methods, but can also be used for problems originating from Finite Elements Methods. It appears that the Modified SIMPLER (MSIMPLER) preconditioner leads to the best results comparing with other known preconditioners. The resulting method depends only weakly on the grid size. Finally, we consider Stokes problems with variable viscosity. Applications are from geomechanics, for instance modeling the flow in the inner parts of the earth. A combination of the Schur complement method, the pressure mass matrix as preconditioner and the Multi-Level (ML) method for the subdomain problems leads to an iterative solution method, which is independent of the grid size, the variation in the viscosity and has a scalable parallel behavior.