# Domain Decomposition Methodology with Robin Interface Matching Conditions for Solving Strongly Coupled Problems

## Abstract

In the case of strongly coupled problems like fluid-structure models in aero-elasticity or aero-thermo-mechanics, a standard solution methodology is based on so called Dirichlet-Neumann iterations. This means that, for instance, the velocity at the interface between the two media is imposed in the fluid, the solution of the fluid problem gives a pressure that is imposed at the boundary of the structure, and then the solution of the problem in the structure gives a new velocity to be imposed to the fluid. This method is not always stable, depending on the relative properties of the media, unless a suitable relaxation parameter is introduced. In order to enforce both velocity and pressure continuity at the interface, the matching conditions can be formulated, like in domain decomposition methods, in a mixed form. This means that the boundary conditions derived in one physical domain from the other one is of Robin type. With Robin boundary condition, an interface stiffness, in the case of velocity-pressure conditions, is introduced. The optimal choice for this stiffness can be proved to be, in the case of linear problems, the so called ”Dirichlet-Neumann” operator of the opposite domain, this means for the discrete equations, the static condensation on the interface of the domain stiffness matrix. Of course, the static condensation cannot be performed in practice, since it is extremely expensive and that the resulting matrix is dense. But it can be approximated in several ways. The underlying general idea behind that methodology is the following: with Robin boundary conditions on the interface, a constitutive law is imposed on the boundary of each media that should optimally exactly represent the interaction with the other media.

## Keywords

Domain decomposition strongly coupled fluid-structure coupling## References

- 1.Funaro, D., Quarteroni, A., Zanolli, P.: An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods. SIAM J. Numer. Anal. 25(6), 1213–1236 (1998)CrossRefMathSciNetGoogle Scholar
- 2.Gander, M.J., Halpern, L., Nataf, F.: Optimal Schwarz Waveform Relaxation for the One Dimensional Wave Equation. SIAM J. Numer. Anal. 41, 1643–1681 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
- 3.Magoules, F., Roux, F.-X., Series, L.: Algebraic Approximation of Dirichlet-to-Neumann Maps for the Equations of Linear Elasticity. Computer Methods in Applied Mechanics and Engineering 195(29-32), 3742–3759 (2006)zbMATHCrossRefMathSciNetGoogle Scholar