A Newton Method for Fluid-Structure Interaction Using Full Jacobians Based on Automatic Form Differentiation
Paper in proceedings, 2012
The study of fluid-structure interaction (FSI) problems is becoming increasingly important both as part of design/engineering and in the modeling of biomedical processes. Examples include the design of new fighter aircraft, the study of the dynamics of heart valves, and the design of prosthetic heart valves. FSI problems are highly coupled and highly nonlinear problems which are challenging to solve. Furthermore, the solution of the discretized system of nonlinear equations is particularly challenging in cases where the solid and the fluid have densities of similar size; this is typically the case for the simulation of biomedical processes involving the deformation of tissue. In such cases, a simple fixed point iteration, in which the solution from a fluid solver is used to impose Neumann boundary conditions for a structure (elasticity), followed by an update of the fluid domain based on the structure solution (via the solution of an auxiliary problem for the update of the fluid mesh), may fail to converge. Instead, a more coupled approach such as a Newton or quasi-Newton method must be employed. In this note, we study the use of Newton's method to solve the fully coupled FSI problem. Typically, a Lagrangian formulation is used to describe the solid; that is, the solid equations are solved on a fixed reference domain (the initial configuration), while an ALE (Arbitrary Lagrangian-Eulerian) formulation is used to describe the fluid. This means that the fluid domain is changing throughout the simulation of a time-dependent problem. The differentiation of the FSI problem, which is required to formulate Newton's method, therefore involves a differentiation with respect to the changing domain of the fluid problem. Such shape differentiation can indeed be used to derive the full Jacobian of the FSI problem; see Fernández and Moubachir . We here study an alternative approach based on mapping the fluid problem back to the initial configuration of the fluid domain. This alternative is advantageous since it allows the use of straightforward differentiation on a fixed domain. This also allows the use of existing tools for automatic differentiation of finite element variational forms such as those developed as part of the FEniCS Project [5-7]. The FEniCS form language UFL  is a domain-specific language for finite element variational forms which allows the FSI problem to be expressed in a language close to the mathematical notation. Forms may be differentiated automatically, and automatically assembled into matrices and vectors. The methodology is here applied to the fully nonlinear time-dependent FSI problem modeled by the incompressible Navier-Stokes equations and the St. Venant-Kirchoff nonlinear hyperelastic model.