Immersed boundaries without boundary locking: A DG-based approach

Gustavo C. BuscagliaICMC, Univ. São Paulo at São Carlos, Brazil

Adrian J. Lew and Ramsharan RangarajanMechanics and Computation, Stanford Univ.

Colloquium on Immersed Boundary MethodsAmsterdam, 15 June 2009

Overview

➢ Boundary-fitting and non-boundary-fitting methods. Boundary conditions at immersed boundaries.

➢A crucial underlying difficulty of Dirichlet conditions at immersed boundaries: Locking.

➢ The proposed method: Inevitable discontinuities and the usefulness of DG.

➢ Some tests: 2nd-order without pain. Reaction-diffusion without stabilization. Elasticity.

➢ Further work. Conclusions.

Boundary-fitting and non-boundary-fitting methods

Boundary-fitting mesh

Boundary-fitting and non-boundary-fitting methods

Non-boundary-fitting mesh

Non-boundary-fitting methods

➢ Immersed-boundary methods: Simplest approach

Solve on internal mesh (gray).

Crude (staircase) approximation of boundaries >> Avoid or Refine

Non-boundary-fitting mesh

Non-boundary-fitting methods

➢ Immersed-boundary methods: Virtual boundary method

Imposition of Dirichlet conditions by feedback control.Goldstein et al (1993), Beyer & LeVeque (1992) ... Peskin (1972)

BC: approximate

Stability issues

regularize

Non-boundary-fitting methods

Immersed-boundary methods: Direct imposition of Dirichlet conditions

Two equations for point Q!!

Incompatibility

A crucial underlying difficulty

In FEM framework:

Who is Vh?

Simplest approach(first order)

Improvement??No! At most first order!

Vh has no obvious basis. But most importantly its interpolation error in H1 is O(h1/2). The approximation locks as one tends towards this Vh.

The subspace of the FEM space of continuous functions defined on the global mesh that vanishes on the boundary is not good for interpolation.

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The proposed method: Discontinuous-Galerkin-based Immersed Boundary FEM

Enforce the Dirichlet conditions exactly while not requiring theapproximation space Vh to consist of continuous functions.

where consists of interior elements, and of boundary elements

The proposed space:

The interpolatory degrees of freedom at the boundary are recovered by switching to a discontinuous space (that nevertheless satisfies the BC)

Discontinuous-Galerkin formulation

Convergence test

Poisson problem

Second order convergence is attainedNotice also the locking of uhc

Stabilization?

Poisson problem as before

Notice minimum near beta = 10, but beta=0 looks like a safe, preferrable choice.

Apparently unnecessary (and harmful)

Stabilizing term (classical)

Reaction-dominated diffusion: The pathology on a conforming mesh.

Overshooting when the mesh cannot resolve the boundary layer (of order sqrt(gamma)).

Reaction-dominated diffusion: The behavior of the proposed method

Overshooting occurs just within the boundary elements. The interior elements exhibit practically no overshoot (<0.004).

Extension to elasticity

Linear elasticity: Torsion of a spherical shell

Extension to elasticity

Linear elasticity: Torsion of a spherical shell

Extension to elasticity

Nonlinear elasticity:

Compression of a femur

Extension to elasticity

Nonlinear elasticity:

Compression of a femur

Image-based simulation

Image-based geometries pipeline

Image-based simulation

Further workImmersed-boundary methods in fluid mechanics.

Moving-boundary problems.

Higher-order interpolation.

Mathematical analysis.

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Thank you!