Extrapolation methods for accelerating unsteady partitioned fluid-structure interaction simulations

The advantages of an unsteady implicit partitioned Fluid-Structure Interaction Scheme are, especially for large time step sizes, accompanied by a high number of inner iterations needed. This work investigates the use of extrapolation techniques in between time steps to obtain a smaller number of inner iterations. In contrast to previous works not only the structural displacement but also the fluid field variables are considered for extrapolation. The different extrapolation approaches are compared within a numerical benchmark.


INTRODUCTION
In solving partitioned Fluid-Structure Interaction (FSI) problems, the implicit approach has shown great advantages in terms of time step restrictions and stability.Especially when exploiting its ability to use large time steps, a main challenge of this approach is to reduce the number of FSI iterations within every time step.This challenge has been investigated by numerous people.
Mainly the methods can be divided into two types.Methods working on acceleration within individual time steps and methods integrating several time steps.The former include, among others, adaptive under-relaxation of Aitken type as in [13,6,5,14,15,9], reduced order modeling [13], steepest descent methods [5], or vector extrapolation [6].
Methods using multiple time steps are typically based on extrapolating the converged FSI solutions.Extrapolation in between time steps produce a more sophisticated first guess and thus reduce the number of FSI iterations needed in the current time step to reach convergence.This has been implemented into partitioned implicit [2] and partitioned explicit FSI solvers [3], but attention has been singly on the structural parameters as in [8,9] or the displacement of the fluid field as in [14].In [11] extrapolation has been implemented for the structural displacement and the fluid pressure.
This work introduces a method of force extrapolation.The extrapolation routines for the structural parameters are used for the fluid force acting on the structure at the FSI interface.Compared to the classical displacement extrapolation, the force extrapolation results in a greater acceleration.Furthermore, there are virtually no additional computational costs, as the same functions and memory blocks, as being used for the displacement, can be employed.Singly, one additional structure solution per time step is necessary.
In the following chapter, the underlying conservation equations are given in the Arbitrary Lagrangian-Eulerian formulation (ALE), then the native FSI algorithm is introduced.Thereafter the extrapolation functions are presented as well as the implementation for displacement and force extrapolation.After specifying the underlying solution software and the FSI test case, numerical results of all implementations are shown and compared for 3 different time step sizes.

FLUID-STRUCTURE INTERACTION SYSTEM
In the following a subscript f refers to the fluid and a subscript s to the structure.Let Ω be our computational FSI domain described in the ALE framework, which has a disjoint partition into the structure domain Ω s and the fluid domain Let the velocity and pressure denoted by (v, p) be well defined with all boundary conditions on the respective remaining boundaries ∂Ω f \Γ.Then, the fluid part of the coupled problem can be stated as the Navier-Stokes equations for incompressible Newtonian fluids in the ALE framework (1) (2) with v g the ALE grid velocity, ρ f the fluid density, n the outer normal vector, f f outer forces acting on the fluid, (3) the Cauchy stress tensor, p the pressure, I the identity, and µ f the dynamic viscosity.
As for the structure part, the elasticity equation with the St. Venant Kirchhoff material model is used.Let X be the reference configuration and x = χ(X, t) the position in the current configuration.The structure part of the coupled problem can be denoted as (4) with f s the outer forces acting on the structure, ρ s the structure density, (5) the second Piola-Kirchhoff stress tensor, λ s , and µ s the Lam constants, the Green-Lagrange strain, (7) the deformation gradient, and the structure velocity.The FSI boundary conditions for fluid and structure on Γ then follow as (9)

EXTRAPOLATION IN FSI
Let (φ) n i be the discrete counterpart to a time and space dependent variable φ at FSI iteration i, at time step n and ε FSI the FSI convergence criterion.

NATIVE FSI ALGORITHM
One time step of the native partitioned implicit FSI implementation is depicted in figure 1.The ith iteration of the nth time step can be described as follows.First the Navier-Stokes equation is solved for velocity and pressure (v) n i , (p) n i in the fluid domain.Using this solution the force on the FSI boundary (σ f ) n i (n f ) n i is derived.Implementing the force as boundary condition, the elasticity equation is solved for the displacement (χ) n i on the structure domain.Then the procedure checks for convergence.If i ≠ 1 and (10) FSI convergence is achieved and the next time step is calculated.Otherwise, the structure solution is under-relaxed by a constant under-relaxation factor α: Then, the fluid grid is displaced according to (χ -) n i and a new grid velocity (v g ) n i+1 in the fluid domain as well as a new fluid boundary condition on Γ is computed.

EXTRAPOLATION FUNCTIONS
Extrapolation is a well known technique to accelerate many kinds of unsteady processes, as it can provide a sophisticated initial guess to the underlying solver.In implicit partitioned FSI computations it is mainly used in two areas.Either it is applied within a time step, i.e. as vector extrapolation (see U. Küttler and W. Wall [6]), or to generate a more promising initial structural displacement in the current time step using polynomial extrapolation as in J. Vierendeels et al. [13] or M. Schäfer et al. [9].
As polynomial extrapolation functions have been proven of value in this field, see [2,11,3,14], they are also used in this work.To demonstrate the effect of different orders of extrapolation, four different orders are implemented, namely the first four Lagrange polynomial interpolation functions.
Let (φ) n * be the state of variable (φ) at FSI convergence in time step n.Then the Lagrangian interpolation functions state: : These functions are used for both displacement and force extrapolation.Higher order extrapolation suggest a higher order of accuracy, thus less iterations to convergence, however, their implementation requires more memory and computing time.

EXTRAPOLATED DISPLACEMENT
In the case of displacement extrapolation the structural displacement χ on Γ is altered in the first FSI iteration of every time step.Singly the variables on the FSI boundary, as opposed to the entire fluid region Ω f , are extrapolated in order to minimize memory consumption.The course of this algorithm in time step n is depicted in figure 2.
In contrast to the native algorithm, the fluid solution, derivation of force on the boundary, and solution of the structure problem are skipped.Instead, the first structure solution (χ) n  1 is extrapolated by one of the equations ( 12) to (15).Additionally, no under-relaxation is applied to the displacements, as this would damp the advantage of extrapolation.The following iterations are carried out according to the non-extrapolated case.

EXTRAPOLATED FORCE
In the case of fluid force extrapolation, the boundary forces σ f n f on Γ acting on the structure are extrapolated.This affects the FSI procedure in the same way as extrapolating the fluid variables v and p on the entire domain Ω f , but saves memory, as singly values on the interface have to be stored.A sketch of this algorithm in time step n is given in figure 3.
Here, the first FSI iteration omits the fluid solution and the derivation of force on the FSI boundary.In this case, the force (σ f ) n 1 (n f ) n 1 is extrapolated by one of the equations ( 12) to (15).Once again the under-relaxation of the structure solution is omitted in the first iteration, as for the displacement case.Again, the following iterations are identical to the non-extrapolated case.Compared to the displacement extrapolation this introduces an additional structure solve per time step.However, the gain in accuracy outweighs this additional solve in terms of CPU time needed for convergence.

SOLVERS
The fluid structure coupling is realized by an implicit partitioned approach.under-relaxation is applied to the structural displacements by a constant Under-relaxation factor.This ensures that the acceleration of the FSI procedure originates singly from the initial guess in every time step, thus the applied extrapolation.
The fluid is solved by the in house finite volume code FASTEST [4].It utilizes the SIMPLE algorithm by S.V. Patankar and D. B. Spalding [7], the linearized fix point problem is solved using an incomplete ILU factorization and a geometric multi grid is implemented to accelerate the procedure.The code operates on block structured hexahedral grids.As for time stepping, a second order backwards difference scheme is used.
The non-FSI grid boundaries are moved by interpolating splines and a transfinite interpolation scheme is used to smooth the inner-block grids.
The structure problem is solved by the finite element code FEAP by R. L. Taylor [10].It uses Newton iteration and the resulting linear system is solved by a conjugate gradient solver.The second order transient Newmark scheme is used for temporal integration.
The coupling and exchange of forces and displacements on the non-matching grids is realized by the code coupling interface MpCCI [1].

BENCHMARK TEST CASE
A commonly used FSI test case is chosen to compare the different approaches described above.This test case consists of a channel flow around a rigid cylinder with an elastic bar attached to it.All parameters are set according to the FSI Benchmark by S. Turek et al. [12].
The channel has the dimensions 2.5 × 0.41 m 2 , with a slightly off symmetric positioned cylinder at (0.2 m, 0.2 m), radius 0.05 m and a bar attached to the cylinder of size 0.35 × 0.02 m 2 .Figure 4 shows the configuration, with its fluid and structure domain.Boundary conditions for the fluid are: A parabolic inflow profile (16) with v _ = 2 m/s, zero-gradient outlet condition, and no-slip on all other surfaces.The structure is clamped at the cylinder.The fluid field is initialized with constant velocity v f = (v _ , 0) T , the structure is initially at rest without any loading.Figures 8, 9, and 10 show the average CPU times per time step.The native FSI algorithm coincides with the 0th order displacement extrapolation, as the new time step starts its calculation on the grid displaced in the last time step.
As expected, all extrapolation algorithms show faster convergence than the native FSI algorithm.The force case outperforms the displacement case for small orders of extrapolation independent of the time step size.
For small time step sizes the CPU time needed for the displacement case approaches the time needed for the force case as the order of extrapolation increases.This behavior reduces with increasing time step size.The advantage of force over displacement extrapolation is clearly seen at the largest time step ∆t = 0.005 s.There, the maximum improvement of the displacement extrapolation is reached at 3rd order.This is more than 2 times slower than the 3rd order force extrapolation.
For ∆t = 0.005 s the 3rd order force extrapolation is more than 2.5 times faster than the native algorithm.
The maximum improvement for time step ∆t = 0.002 s shows the 2nd order force extrapolation, which is more than 3 times faster than the native algorithm.The fastest displacement case is at 3rd order extrapolation.For the time step ∆t = 0.001 s the maximum improvement can be seen with the 1st order force extrapolation, which is more than 3.7 times faster than the native algorithm.The best convergence for the displacement case can be found with 3rd order extrapolation.

CONCLUSIONS
In this article the extrapolation of forces acting on the structure at the FSI boundary in order to accelerate unsteady implicit partitioned FSI computations is described and examined.This is compared to the well known acceleration technique of extrapolating the displacement on the FSI boundary.Numerical tests based on an FSI Benchmark show the superior performance of the force extrapolation for different time step sizes.This advancement can be achieved at minimal computational cost as the same memory blocks and extrapolation procedures as for the displacement extrapolation can be employed.
In addition, different extrapolation schemes based on the Lagrange interpolation polynomials are compared.It is shown that the impact of a higher degree of extrapolation is increasing with the time step size.This leads to more efficient implicit solvers, as greater time step sizes can be used and less FSI iterations per time step are needed.
As for future prospects, the combination of force extrapolation with other extrapolation techniques can be studied.Furthermore, this approach can be combined with other acceleration methods acting within single time steps.As these two kinds of methods are completely independent a superposition of speedup is expected.

Figure 5 Figure 6 Figure 7
Figure 5 Section of the fluid and structure grid.

Figure 8 Figure 9 Figure 10
Figure 8 Average CPU time for one time step ∆t = 0.001.
algorithm with displacement extrapolation.

Table 1
Average bar tip displacement

Table 2
Average force on acting on Γ