Immersed boundaries without boundary locking: A DG-based approach Gustavo C. Buscaglia ICMC, Univ. São Paulo at São Carlos, Brazil Adrian J. Lew and Ramsharan Rangarajan Mechanics and Computation, Stanford Univ. Colloquium on Immersed Boundary Methods Amsterdam, 15 June 2009
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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.
.....
Thank you!
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