A Fictitious domain approach with spread interface for elliptic problems with general boundary conditions

By Isabelle Ramière, Philippe Angot, Michel Belliard. 2007

161 citations recorded by the source corpus.

The aim of this article is to solve second-order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach . The main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain. As the spatial discretization is being performed in the fictitious domain, this method allows the use of structured meshes. The discretization is not boundary-fitted to the original physical domain. This paper describes several ways to impose Dirichlet, Robin or Neumann boundary conditions on a spread immersed interface, without locally modifying the numerical scheme and without using Lagrange multipliers. The numerical applications focus on difiusion and convection problems in the unit disk, with Dirichlet or Robin boundary conditions. For such problems, analytical solutions can be determined for a correctly chosen source term. The numerical resolution is performed using a Q1 Finite Element scheme. The spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vinicity of the immersed boundary. The L2 norm of the errors is computed in order to evaluate the capability of the method. Immersed boundaries are found in many industrial applications like two-phase flow simulations, fluid/ structure interaction, etc. This article represents a first step towards the simulation of these kinds of applications.

Subfields: Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/87360f592aad08d0ce3e6dd2c31ae0f106fa8480

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In Computational Mechanics

Citations in the corpus

An extended stochastic finite element method for solving stochastic partial differential equations on random domains (2008, background): “Let us mention that fictitious domain methods [22–25], recently extended to the stochastic framework [26], start with similar reformulation of the problem and definition of approximation spaces.”
Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method (2012, methodology): “The fictitious domain approach has been widely used in conjunction with penalty methods [12, 13], the mortar approach [14], Lagrange multipliers [11, 15–18], Nitsche’s method [19], extended finite elements [20–23], discontinuous Galerkin [24] and spectral methods [25–27] to address problems of struc”
The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics (2012, background): “Instead of choosing a very small value, we increase parameter ̨ of Equation (3) in fict by some orders of magnitude, so that Dirichlet conditions applied to the boundaries of the finite cell mesh are directly carried on to the boundary of phys [9, 25, 26].”
The p-Version of the Finite Element and Finite Cell Methods (2017, methodology): “On the other hand, these operations lend themselves to parallel computation; see, for example, Rank et al. (2001). Furthermore, it has been shown that a substantial reduction in time can be achieved if special integration techniques are used (Nübel et al.”