Philipp Birken

A time-adaptive fluid-structure interaction method for thermal coupling

The thermal coupling of a fluid and a structure is of great significance for many industrial processes. As a model for cooling processes in heat treatment of steel we consider the surface coupling of the compressible Navier-Stokes equations bordering at one part of the surface with the heat equation in a solid region. The semi-discrete coupled system is solved using stiffly stable SDIRK methods of higher order, where on each stage a fluid-structure-coupling problem is solved. For the resulting method it is shown by numerical experiments that a second order convergence rate is obtained. This property is further used to implement a simple time-step control, which saves considerable computational time and, at the same time, guarantees a specified maximum error of time integration.

Termination criteria for inexact fixed-point schemes

We analyze inexact fixed-point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed-point iteration. Important applications are the Picard iteration and partitioned fluid-structure interaction. For the analysis, the iteration is modeled as a perturbed fixed-point iteration, and existing analysis is extended to the nested case x=F(S(x)). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property. Numerical results demonstrate the effectiveness of the approach with the nonstandard termination criterion. (Less)

A comparison of Rosenbrock and ESDIRK methods combined with iterative solvers for unsteady compressible flows

In this article, we endeavour to find a fast solver for finite volume discretizations for compressible unsteady viscous flows. Thereby, we concentrate on comparing the efficiency of important classes of time integration schemes, namely time adaptive Rosenbrock, singly diagonally implicit (SDIRK) and explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) methods. To make the comparison fair, efficient equation system solvers need to be chosen and a smart choice of tolerances is needed. This is determined from the tolerance TOL that steers time adaptivity. For implicit Runge-Kutta methods, the solver is given by preconditioned inexact Jacobian-free Newton-Krylov (JFNK) and for Rosenbrock, it is preconditioned Jacobian-free GMRES. To specify the tolerances in there, we suggest a simple strategy of using TOL/100 that is a good compromise between stability and computational effort. Numerical experiments for different test cases show that the fourth order Rosenbrock method RODASP and the fourth order ESDIRK method ESDIRK4 are best for fine tolerances, with RODASP being the most robust scheme.

Solving Nonlinear Systems Inside Implicit Time Integration Schemes for Unsteady Viscous Flows

Historically, the computation of steady flows has been at the forefront of the development of computational fluid dynamics (CFD). This began with the design of rockets and the computation of the bow shock at supersonic speeds and continued with the aerodynamic design of airplanes at transonic cruising speed [14]. Only in the last decade, increasing focus has been put on unsteady flows, which are more difficult to compute. This has several reasons. First of all, computing power has increased dramatically and for 5,000 Euro it is now possible to obtain a machine that is able to compute about a minute of realtime simulation of a nontrivial unsteady three dimensional flow in a day. As a consequence, ever more nonmilitary companies are able to employ numerical simulations as a standard tool for product development, opening up a large number of additional applications. Examples are the computation of tunnel fires [4], flow around wind turbines [29], fluid-structure-interaction like flutter [10], flows inside nuclear reactors [25], wildfires [24], hurricanes and unsteady weather phenomenas [23], gas quenching [20] and many others. More computing capacities will open up further possibilities in the next decade, which suggests that the improvement of numerical methods for unsteady flows should start in earnest now. Finally, the existing methods for the computation of steady states, while certainly not at the end of their development, have matured, making the consideration of unsteady flows interesting for a larger group of scientists. In this article, we will focus on the computation of laminar viscous flows, as modelled by the Navier-Stokes equations.

A study of multigrid smoothers used in compressible CFD based on the convection diffusion equation

A study of multigrid smoothers used in compressible CFD based on the convection diffusion equation

Convergence analysis of coupling iterations for the unsteady transmission problem with mixed discretizations

We analyze the convergence rate of the Dirichlet-Neumann iteration for the fully discretized one dimensional unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping intervals. The Laplacian is discretized using finite differences on one interval and finite elements on the other and the implicit Euler method is used for the time discretization. Following previous analysis where finite elements where used on both subdomains, we provide an exact formula for the spectral radius of the iteration matrix for this specific mixed discretizations. We then show that these tend to the ratio of heat conductivities in the semidiscrete spatial limit, but to a factor of the ratio of the products of density and specific heat capacity in the semidiscrete temporal one. In the previous finite element analysis, the same result was obtained in the semidiscrete spatial limit but the factor in the temporal limit was lower. This explains the fast convergence previously observed for cases with strong jumps in the material coefficients. Numerical results confirm the analysis. (Less)

Accuracy in a Finite Volume Godunov Type Method

The standard Godunov type method used in computational fluid dynamics shows accuracy problems for low Mach number flows and for the kinetic energy at the highest wave numbers resolvable on a given grid. Both drawbacks become visible when simulating the decay of isotropic turbulence at the low Mach numbers typical for the respective experimental investigations. A modification to cure both problems is proposed by Thornber et al. [10] with a mathematical motivation in case of a special fifth order reconstruction. The theoretical results are repeated here. Numerical results are achieved for schemes not investigated in that literature, namely AUSMDV and AUSM + -up which includes already modifications for low Mach number flows. First experiences with Thornber’s modification confirm the positive influence in combination with AUSMDV even if the reconstruction is only of second order. In combination with AUSM + -up Thornber’s modification provides too little damping when used without subgrid scale modelling.

Preconditioned Smoothers for the Full Approximation Scheme for the RANS Equations

We consider multigrid methods for finite volume discretizations of the Reynolds averaged Navier–Stokes equations for both steady and unsteady flows. We analyze the effect of different smoothers based on pseudo time iterations, such as explicit and additive Runge–Kutta (AERK) methods. Furthermore, by deriving them from Rosenbrock smoothers, we identify some existing schemes as a class of additive W (AW) methods. This gives rise to two classes of preconditioned smoothers, preconditioned AERK and AW, which are implemented the exact same way, but have different parameters and properties. This derivation allows to choose some of these based on results for time integration methods. As preconditioners, we consider SGS preconditioners based on flux vector splitting discretizations with a cutoff function for small eigenvalues. We compare these methods based on a discrete Fourier analysis. Numerical results on pitching and plunging airfoils identify AW3 as the best smoother regarding overall efficiency. Specifically, for the NACA 64A010 airfoil steady-state convergence rates of as low as 0.85 were achieved, or a reduction of 6 orders of magnitude in approximately 25 pseudo-time iterations. Unsteady convergence rates of as low as 0.77 were achieved, or a reduction of 11 orders of magnitude in approximately 70 pseudo-time iterations.

On Goal Oriented Time Adaptivity using Local Error Estimates

We consider adaptive time discretization methods for ordinary differential equations where one aims to control the error in a quantity of interest of the form J(u) = ∫ j(u(t))dt with j : Rd -> R. In this setting we propose a new timestep controller based on local error estimates of the quantity of interest. The new method converges when the tolerance goes to zero.We experimentally compare the new scheme with the classic norm-based time-adaptivity based on local error estimates as well as the dual-weighted residual (DWR) method. The results show significantly lower efficiency for the DWR method. The local error based schemes are similarly efficient, with the new scheme showing significant improvement in some cases.

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)