%0 Journal Article
%J Computers and Mathematics with Applications 69 (2015) 1167–1188
%D 2015
%T Benchmarking the Immersed Finite Element Method for Fluid-Structure Interaction Problems
%A Roy Saswati
%A Luca Heltai
%A Francesco Costanzo
%X We present an implementation of a fully variational formulation of an immersed methods for fluid-structure interaction problems based on the finite element method. While typical implementation of immersed methods are characterized by the use of approximate Dirac delta distributions, fully variational formulations of the method do not require the use of said distributions. In our implementation the immersed solid is general in the sense that it is not required to have the same mass density and the same viscous response as the surrounding fluid. We assume that the immersed solid can be either viscoelastic of differential type or hyperelastic. Here we focus on the validation of the method via various benchmarks for fluid-structure interaction numerical schemes. This is the first time that the interaction of purely elastic compressible solids and an incompressible fluid is approached via an immersed method allowing a direct comparison with established benchmarks.
%B Computers and Mathematics with Applications 69 (2015) 1167–1188
%G en
%1 34633
%2 Mathematics
%4 1
%# MAT/08
%$ Submitted by Luca Heltai (heltai@sissa.it) on 2015-04-20T15:26:58Z
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%R 10.1016/j.camwa.2015.03.012
%0 Report
%D 2012
%T A Fully Coupled Immersed Finite Element Method for Fluid Structure Interaction via the Deal.II Library
%A Luca Heltai
%A Saswati Roy
%A Francesco Costanzo
%K Finite Element Method
%K Immersed Boundary Method
%K Immersed Finite Element Method
%X We present the implementation of a solution scheme for fluid-structure\\r\\ninteraction problems via the finite element software library deal.II. The\\r\\nsolution scheme is an immersed finite element method in which two independent discretizations are used for the fluid and immersed deformable body. In this type of formulation the support of the equations of motion of the fluid is extended to cover the union of the solid and fluid domains. The equations of motion over the extended solution domain govern the flow of a fluid under the action of a body force field. This body force field informs the fluid of the presence of the immersed solid. The velocity field of the immersed solid is the restriction over the immersed domain of the velocity field in the extended equations of motion. The focus of this paper is to show how the determination of the motion of the immersed domain is carried out in practice. We show that our implementation is general, that is, it is not dependent on a specific choice of the finite element spaces over the immersed solid and the extended fluid domains. We present some preliminary results concerning the accuracy of the proposed method.
%I SISSA
%G en
%U http://hdl.handle.net/1963/6255
%1 6172
%2 Mathematics
%3 Functional Analysis and Applications
%4 1
%# MAT/08 ANALISI NUMERICA
%$ Submitted by Luca Heltai (heltai@sissa.it) on 2012-09-28T11:06:13Z\\nNo. of bitstreams: 1\\n1209.2811v1.pdf: 4766431 bytes, checksum: a46803f7f8daf3195359f65c9161b944 (MD5)
%0 Journal Article
%J Computer Methods in Applied Mechanics and Engineering. Volume 229-232, 1 July 2012, Pages 110-127
%D 2012
%T Variational implementation of immersed finite element methods
%A Luca Heltai
%A Francesco Costanzo
%K Turbulent flow
%X Dirac-delta distributions are often crucial components of the solid-fluid coupling 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 structure of the problem is exploited to avoid Dirac-delta distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM 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 of differential type or purely elastic, and (iii) the solid to be and either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer 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.

%B Computer Methods in Applied Mechanics and Engineering. Volume 229-232, 1 July 2012, Pages 110-127
%I Elsevier
%G en
%U http://hdl.handle.net/1963/6462
%1 6389
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Lucio Lubiana (lubiana@sissa.it) on 2013-02-04T16:21:56Z\\nNo. of bitstreams: 0
%R 10.1016/j.cma.2012.04.001