Fluid-structure interaction (FSI) problems depend on parameters such as the solid shear modulus, the fluid density and the fluid viscosity. Parameter-dependent FSI discretizations allow to observe the reaction of an FSI model to a change of such parameters. A parameter-dependent discretization of a linear FSI problem yields many linear systems to be approximated. These equations can be translated to one single matrix equation. The solution, a matrix, can be approximated by a low-rank method as discussed in . But as soon as nonlinear FSI problems are considered, such a translation is not possible anymore.
The proposed method extends the low-rank framework of  to nonlinear problems. It splits the parameter set into disjoint subsets. On each of these subsets, the Newton approximation for the problem related to the upper median parameter is computed to approximate the Jacobian matrix for all problems related to the subset. This allows to formulate a Newton step as a matrix equation. The Newton update, a matrix, can be approximated by a low-rank method and the global approximation to the parameter-dependent nonlinear FSI problem is achieved by stacking the approximations on the disjoint subsets column-wise.
2 The Nonlinear Problem
Let be open subsets of with , . We use the stationary Navier-Stokes equations [4, Chapter 184.108.40.206] to model the fluid part in and the stationary Navier-Lamé equations [4, Problem 2.23] for the solid part in . The interface is , the boundary part where Neumann outflow conditions hold and the boundary part where Dirichlet conditions hold . The weak formulation of the coupled nonlinear FSI problem with a vanishing right hand side reads
With , an extension of the Dirichlet data on , the trial function is the velocity, the deformation and the pressure. The test functions are (divergence equation), (momentum equation) and (deformation equation). The scalar product on and is denoted by and , respectively. The parameters involved are the kinematic fluid viscosity , the fluid density , the solid shear modulus and the first Lamé parameter .
3 Discretization and Linearization
Assume we are interested in discretizing the nonlinear FSI problem described in (1) parameter-dependently with respect to shear moduli given by the set
Consider a finite element discretization on , a matching mesh of the domain , with a total number of degrees of freedom. Let be a discretization matrix of all linear operators involved in (1) with fixed parameters , , and . Let be the discretization matrix of the operator
The nonlinear part in (1), the convection term, requires a linearization technique.
3.1 Linearization with Newton Iteration
For a linearization by means of the Newton iteration, we need the Jacobian matrix of the operator . In our finite element space, every unknown consists of a pressure , a velocity and a deformation . The discrete test space also has dimension and every unknown there can be written as . The Jacobian matrix of in our finite element space, evaluated at , is
be the right hand side vector that depends on the desired Dirichlet boundary conditions of the nonlinear FSI problem. Consider the FSI problem related to a fixed shear modulusfor some first.
If we start with an initial guess , for instance , at Newton step , the equation
is to be solved for . The approximation at linearization step then is
evaluates all operators in (1) at the pressure, velocity and deformation of the approximation of the previous linearization step and the shear modulus .
4 Newton Iteration and Low-rank Methods
In order to approximate a set of problems at one time, (3) has to be translated to a matrix equation. For this, first of all, we split the parameter set into disjoint subsets.
If we perform a Newton step for a set of problems at one time, the same Jacobian matrix is used for the whole set. Therefore, the solutions to these different problems should not differ too much from each other. The method suggested in this paper splits the given parameter set into disjoint subsets, each of them containing adjacent parameters.
By , we denote the index of the upper median parameter of the set . After the parameter set is split into the subsets , we compute the Newton approximation of the problem related to the upper median parameter for all up to some given accuracy . is then used as initial guess for one Newton step.
4.1 The Matrix Equation
With and , the matrix equation that is to be solved for on every subset is
denotes the identity matrix of size. In (4), the initial guess for the Newton step is
The approximation at the next linearization step is
The global approximation for the whole parameter-dependent problem then is
The right hand side: is not a matrix of low rank and would have to be evaluated for all columns of separately in a second Newton step. Thus, the right hand side matrix would not have low-rank structure either.
The Jacobian matrix: Since all columns of the initial guess coincide, the Jacobian matrix in (4) is correct for all equations related to the parameter set . But the columns of differ from each other. A second Newton step would then become what is, in the literature, often called an inexact Newton step [4, Remark 5.7].
4.2 Low-rank Methods
Consider only the column related to the parameter index in (4):
Assume that is fixed and is invertible for all . and depend linearly on . and are analytic matrix- and vector-valued functions, respectively. Due to [2, Theorem 2.4]
, the singular value decay of the matrixin (4) is exponential. Algorithm 1 exploits this fact and approximates in (4) by a low-rank matrix.
5 Numerical Results
A jetty in a channel with the geometric configuration
is considered. The left Dirichlet inflow is given by the velocity profile
At , the do nothing boundary condition holds. At , deformation and velocity in normal direction is prohibited. Everywhere else on , the velocity and the deformation vanish. For the Navier-Stokes equations, stabilized Stokes elements [4, Lemma 4.47] are used.
The nonlinear FSI problem (1) is discretized with bilinear finite elements with respect to
The fixed first Lamé parameter is . With these parameters, solid configurations with Poisson ratios between and are covered. The fluid density is and the kinematic fluid viscosity is .
5.2 Comparison ChebyshevT with Standard Newton
A server operating CentOS 7 with 2 AMD EPYC 7501 and 512GB RAM, MATLAB® 2017b in combination with the htucker MATLAB toolbox  and the finite element toolkit GASCOIGNE  was used to compare Algorithm 1 with Newton iterations applied consecutively. The parameter set was split into subsets.
5.2.1 Preconditioner and Eigenvalue Estimation
After the Newton approximations are available for all , the LU decomposition of the mean-based preconditioner [5, Chapter 3.2] of is computed separately on every subset
. To estimate the parametersand for the ChebyshevT method [5, Algorithm 3]
, the eigenvalues of the matrices
are taken into consideration. For all subsets, they were estimated to and for a small number of degrees of freedom within seconds (computation time for the Newton approximations included). Every approximation is related to a certain shear modulus. Therefore, all problems differ by the Poisson ratio of the solid. The y-axis in Figure 1, on the other hand, corresponds to the relative residual norm
of the approximation for . Algorithm 1 was applied with and to a problem with . Therefore, the global approximation rank is . In comparison to this, standard Newton iterations were applied to the separate problems consecutively where for every Newton iteration, the last approximation served as initial guess for the next Newton iteration.
The approximations obtained by the Standard Newton iterations within hours ( Newton steps) provided, as visualized in Figure 1, relative residuals with norms smaller than each. Algorithm 1 took minutes ( Newton steps) to compute the low-rank approximation. In addition to the seconds for the eigenvalue estimation, the Newton steps to compute for all took, in total, minutes and the Newton steps for the matrix equations (4) took, in total, minutes.
Low-rank methods can be used to compute approximations to parameter-dependent nonlinear FSI discretizations, in particular, if each of the subsets, the parameter set is split into, does contain problems that do not differ too much from each other. The Newton step on the subset uses the same Jacobian matrix and the same initial guess for the whole subset. It has to provide acceptable convergence within one single step not only for the upper median problem.
Whether the results can be improved by choosing the subsets or the approximation ranks on these subsets adaptively, is still open. Moreover, how these low-rank methods can be applied to fully nonlinear FSI problems that use, in addition to the Navier-Stokes equations on the fluid, for instance, the St. Venant Kirchhoff model equations [4, Definition 2.18] on the solid is an open problem. Then, the right hand side in (4) would have to be approximated.
This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 314838170, GRK 2297 MathCoRe.
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