1,370 research outputs found
On the convergence rate of the Dirichlet-Neumann iteration for unsteady thermal fluid structure interaction
We consider the Dirichlet-Neumann iteration for partitioned simulation of
thermal fluid-structure interaction, also called conjugate heat transfer. We
analyze its convergence rate for two coupled fully discretized 1D linear heat
equations with jumps in the material coefficients across these. These are
discretized using implicit Euler in time, a finite element method on one
domain, a finite volume method on the other one and variable aspect ratio. We
provide an exact formula for the spectral radius of the iteration matrix. This
shows that for large time steps, the convergence rate is the aspect ratio times
the quotient of heat conductivities and that decreasing the time step will
improve the convergence rate. Numerical results confirm the analysis and show
that the 1D formula is a good estimator in 2D and even for nonlinear thermal
FSI applications.Comment: 29 pages, 20 figure
Fast Solvers for Unsteady Thermal Fluid Structure Interaction
We consider time dependent thermal fluid structure interaction. The
respective models are the compressible Navier-Stokes equations and the
nonlinear heat equation. A partitioned coupling approach via a
Dirichlet-Neumann method and a fixed point iteration is employed. As a refence
solver a previously developed efficient time adaptive higher order time
integration scheme is used.
To improve upon this, we work on reducing the number of fixed point coupling
iterations. Thus, first widely used vector extrapolation methods for
convergence acceleration of the fixed point iteration are tested. In
particular, Aitken relaxation, minimal polynomial extrapolation (MPE) and
reduced rank extrapolation (RRE) are considered. Second, we explore the idea of
extrapolation based on data given from the time integration and derive such
methods for SDIRK2. While the vector extrapolation methods have no beneficial
effects, the extrapolation methods allow to reduce the number of fixed point
iterations further by up to a factor of two with linear extrapolation
performing better than quadratic.Comment: 17 page
Waveform Relaxation with asynchronous time-integration
We consider Waveform Relaxation (WR) methods for partitioned time-integration
of surface-coupled multiphysics problems. WR allows independent
time-discretizations on independent and adaptive time-grids, while maintaining
high time-integration orders. Classical WR methods such as Jacobi or
Gauss-Seidel WR are typically either parallel or converge quickly.
We present a novel parallel WR method utilizing asynchronous communication
techniques to get both properties. Classical WR methods exchange discrete
functions after time-integration of a subproblem. We instead asynchronously
exchange time-point solutions during time-integration and directly incorporate
all new information in the interpolants. We show both continuous and
time-discrete convergence in a framework that generalizes existing linear WR
convergence theory. An algorithm for choosing optimal relaxation in our new WR
method is presented.
Convergence is demonstrated in two conjugate heat transfer examples. Our new
method shows an improved performance over classical WR methods. In one example
we show a partitioned coupling of the compressible Euler equations with a
nonlinear heat equation, with subproblems implemented using the open source
libraries DUNE and FEniCS
A multirate Neumann-Neumann waveform relaxation method for heterogeneous coupled heat equations
An important challenge when coupling two different time dependent problems is
to increase parallelization in time. We suggest a multirate Neumann-Neumann
waveform relaxation algorithm to solve two heterogeneous coupled heat
equations. In order to fix the mismatch produced by the multirate feature at
the space-time interface a linear interpolation is constructed. The heat
equations are discretized using a finite element method in space, whereas two
alternative time integration methods are used: implicit Euler and SDIRK2. We
perform a one-dimensional convergence analysis for the nonmultirate fully
discretized heat equations using implicit Euler to find the optimal relaxation
parameter in terms of the material coefficients, the stepsize and the mesh
resolution. This gives a very efficient method which needs only two iterations.
Numerical results confirm the analysis and show that the 1D nonmultirate
optimal relaxation parameter is a very good estimator for the multirate 1D case
and even for multirate and nonmultirate 2D examples using both implicit Euler
and SDIRK2.Comment: 32 pages, 12 figure
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