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Variational Implementation of Immersed Finite Element Methods

TitleVariational Implementation of Immersed Finite Element Methods
Publication TypePreprint
AuthorsHeltai, L, Costanzo, F
Document NumberarXiv:1110.2063v1;

Dirac-delta distributions are often crucial components of the solid-fluid\\r\\ncoupling operators in immersed solution methods for fluid-structure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-delta distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational\\r\\nstructure of the problem is exploited to avoid Dirac-delta distributions at\\r\\nboth the continuous and the discrete level. In this paper, we generalize the\\r\\nFEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic\\r\\nof differential type or purely elastic, and (iii) the solid to be and either\\r\\ncompressible or incompressible. At the continuous level, our variational\\r\\nformulation combines the natural stability estimates of the fluid and\\r\\nelasticity problems. In immersed methods, such stability estimates do not\\r\\ntransfer to the discrete level automatically due to the non- matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.

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