A symmetric positive definite formulation for monolithic fluid structure interaction

Numerically stable fluid–structure interactions between compressible flow and solid structures

“…Recent developments in [16] discuss a modification of the implicit coupling methodology for incompressible flow by separating out the coupling forces as implicit variables λ (similar to immersed boundary methods), decomposing the symmetric damping force into D = C T C and solving forV s = CV n+1 s . The symmetric positive-definite system they obtain can be modified for compressible flow in a manner similar to Section 4.1 to obtain , as compared to a high-resolution simulation, at time 4s.…”

“…whereĜ and −Ĝ T are the volume weighted gradient and divergence operators respectively, β is the diagonal matrix of fluid dual cell masses, and K T is the matrix of 1s and 0s mapping λ to the appropriate fluid velocity scalars (see [16] for more details). Note that in order to avoid confusion in notation we renamed a few operators.…”

“…Note that in order to avoid confusion in notation we renamed a few operators. In particular W and J in [16] correspond to the K and W we use here, respectively. This system is both symmetric and positive-definite.…”

“…Unlike previous methods, however, we apply these coupled interactions implicitly by adding the constraint to the pressure system and combining it with any implicit solid forces in order to obtain a strongly coupled, symmetric indefinite system (similar to [17], which only handles incompressible flow). We also show that, under a few reasonable assumptions, this system can be made symmetric positive-definite by following the methodology of [16]. Because our method handles the fluidstructure interactions implicitly, we avoid introducing any new time step restrictions and obtain stable results even for high density-to-mass ratios, where explicit methods struggle or fail.…”

A positive-definite cut-cell method for strong two-way coupling between fluids and deformable bodies

“…However, these boundary conditions are once again applied at regular grid faces as opposed to the true contact interface, and the resultant linear system remains symmetric inde nite. A positive-de nite formulation of the same problem was nally presented by Robinson-Mosher et al [2011] through linear algebraic manipulations that rely on construction of a symmetric factorization of the solid's damping matrix; our method is more general in avoiding this requirement. In addition to our use of a cut-cell approach that better accounts for solid geometry, the present work is also distinguished from the preceding three approaches by our use of corotational linear elasticity which allows us to incorporate the full implicit solid dynamics into the coupled pressure projection; Robinson-Mosher et al [2011] include only the solid damping.…”