Bärbel Holm

Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

Binary-fluid–solid interaction based on the Navier–Stokes–Cahn–Hilliard Equations

We consider amodel for binary-fluid-solid interaction based on a diffuse-interface model for the binary fluid and a hyperelastic-materialmodel for the solid. The diffuse-interface binary-fluidmodel is described by the quasi-incompressible Navier-Stokes-Cahn-Hilliard equations with preferential-wetting boundary conditions at the fluid-solid interface. The fluid traction on the interface includes a capillary-stress contribution in addition to the regular viscous-stress and pressure contributions. The dynamic interface condition comprises the traction exerted by the nonuniform solid-fluid surface tension in accordance with the Young-Laplace law for the solid-fluid interface. The solid is modeled as a hyperelastic material. We present a weak formulation of the aggregated binary-fluid-solid interaction problem, based on an arbitrary Lagrangian-Eulerian formulation of theNavier-Stokes-Cahn-Hilliard equations and a properweak evaluation of the binary-fluid traction and of the solid-fluid surface tension. We also present an analysis of the essential properties of the binary-fluid-solid interaction problem, including a dissipation relation for the complete fluid-solid interaction problem. To validate the presented binary-fluid-solid interactionmodel, we consider numerical simulations for the elasto-capillary interaction of a droplet with a soft solid substrate and present a comparison to corresponding experimental data.

4. Numerical methods for unsteady thermal fluid structure interaction

We discuss thermal fluid-structure interaction processes, where a simulation of the time-dependent temperature field is of interest. Thereby, we consider partitioned coupling schemes with a Dirichlet-Neumann method. We present an analysis of the method on a model problem of discretized coupled linear heat equations. This shows that for large quotients in the heat conductivities, the convergence rate will be very small. The time dependencymakes the use of time-adaptive implicitmethods imperative. This gives rise to the question as to how accurately the appearing nonlinear systems should be solved, which is discussed in detail for both the nonlinear and linear case. The efficiency of the resulting method is demonstrated using realistic test cases. (Less)

Fluid-Structure Interaction: Modeling, Adaptive Discretisations and Solvers

The objective of this work is to examine the potential of isogeometric methods in the context of multidisciplinary shape optimization. We introduce a shape optimization problem based on a coupled fluid-structure system, whose geometry is defined by NURBS (Non-Uniform Rational B-Spline) curves. This shape optimization problem is then solved by using either an isogeometric approach, or a classical grid-based approach. In spite of the fact that optimization results do not show any major differences, conceptional advantages of the new isogeometric method become apparent. In particular, control points of the spline can be directly handled as design variables without the need of a spline-fit and consequently geometry errors can be excluded at every stages of the optimization loop.

5. Recent development of robust monolithic fluid-structure interaction solvers

In the last few years, from the modeling point of view, the monolithic approach for fluid-structure interaction problems in many different application fields has been adopted by more and more researchers. Meanwhile, the development of monolithic solvers in the solution procedure for solving such coupled fluid-structure interaction problems all at once is in general a very hard task and has received a lot of attention. Due to the coupling conditions on the interface, it is challenging to design efficient preconditioners for the linearized coupled system of equations, that are robust with respect to the mesh size, time step size and material parameters. Further, it is nontrivial to realize scalable parallel implementations for solving such large scale coupled systems, which requires special care for handling the interface conditions. In this survey, we present an overview of some recent results on robust monolithic fluidstructure interaction solvers, that are mainly based on the block factorization, geometric and algebraic multigrid, and domain decomposition methods.