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

Abstract

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.

On the convergence rate of the Dirichlet–Neumann iteration for unsteady thermal fluid–structure interaction

“…One computes S (1) and S (2) by inserting the corresponding matrices specified above in (32) and (33) …”

“…A convergence analysis of the Dirichlet-Neumann iteration for an FE-FE discretization of a steady heat transfer problem can be found in [32]. Asymptotically, it is found to be the quotient of heat conductivities.…”

Global-Local non intrusive analysis with robin parameters: application to plastic hardening behavior and crack propagation in 2D and 3D structures

“…Figure 3 shows a flowchart of the iterative method implementation of the global/local with Robin parameters. The Robin parameter is chosen as the stiffness of an element strip [33], as presented in Figure 5. Robin operators k L and k G are determined independently: k L is the linear elastic stiffness of the strip on the side of the complementary model from the interface and k G is the linear elastic stiffness of the strip on the side of the local model from the interface.…”

A coupling numerical methodology for weakly transient conjugate heat transfer problems

“…On this basis, a transition of the amplification factor was identified recently and as a result, optimal coefficients have been derived in steady CHT procedures [15]. Other models are available such as the energy method [16] or the matrix analysis [17]. This demonstrates that a great deal of effort has been dedicated to determine robust and efficient fluid-solid interface conditions.…”