An Immersed Boundary Proper Generalized Decomposition (IB-PGD) for Fluid–Structure Interaction Problems

Proper generalized decomposition (PGD), a method looking for solutions in separated forms, was proposed recently for solving highly multidimensional problems. In the PGD, the unknown fields are constructed using separated representations, so that the computational complexity scales linearly with the dimension of the model space instead of exponential scaling as in standard grid-based methods. The PGD was proven to be effective, reliable and robust for some simple benchmark fluid–structure interaction (FSI) problems. However, it is very hard or even impossible for the PGD to find the solution of problems having complex boundary shapes (i.e., problems of fluid flow with arbitrary complex geometry obstacles). The paper hence further extends the PGD to solve FSI problems with arbitrary boundaries by combining the PGD with the immersed boundary method (IBM) to give a so-called immersed boundary proper generalized decomposition (IB-PGD). In the IB-PGD, a forcing term constructed by the IBM is introduced to Navier–Stokes equations to handle the influence of the boundaries and the fluid flow. The IB-PGD is then applied to solve Poisson’s equation to find the fluid pressure distribution for each time step. The numerical results for three problems are presented and compared to those of previous publications to illustrate the robustness and effectiveness of the IB-PGD in solving complex FSI problems.