Variational implementation of immersed finite element methods

Luca Heltai, Francesco Costanzo

Research output: Contribution to journalArticle

22 Citations (Scopus)

Abstract

Dirac-. δ 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-. δ distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-. δ 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-. δ 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 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.

Original languageEnglish (US)
Pages (from-to)110-127
Number of pages18
JournalComputer Methods in Applied Mechanics and Engineering
Volume229-232
DOIs
StatePublished - Jul 1 2012

Fingerprint

finite element method
Finite element method
fluids
Fluids
Fluid structure interaction
boundary element method
Boundary element method
estimates
formulations
operators
Newtonian fluids
interpolation
Mathematical operators
Elasticity
Interpolation
elastic properties
interactions

All Science Journal Classification (ASJC) codes

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications

Cite this

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abstract = "Dirac-. δ 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-. δ distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-. δ 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-. δ 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 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.",
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Variational implementation of immersed finite element methods. / Heltai, Luca; Costanzo, Francesco.

In: Computer Methods in Applied Mechanics and Engineering, Vol. 229-232, 01.07.2012, p. 110-127.

Research output: Contribution to journalArticle

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