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LOCAL DIVERGENCE-FREE IMMERSED FINITE ELEMENT-DIFFERENCE METHOD USING COMPOSITE B-SPLINES.

Lianxia Li1, Cole Gruninger1, Jae H Lee1

  • 1Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA.

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Summary
This summary is machine-generated.

Composite B-spline (CBS) kernels improve fluid-structure interaction simulations by maintaining volume conservation, enhancing accuracy, and reducing computational costs. These kernels offer stable and efficient simulations for deforming elastic solids without extra stabilization.

Keywords:
Immersed boundaryPrimary: 58F15, 58F17Secondary: 53C35composite B-spline kernels (CBS)fluid-structure interactionimmersed finite element-finite difference methodisotropic kernelvolume conservationvolumetric stabilization

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Area of Science:

  • Computational Fluid Dynamics (CFD)
  • Fluid-Structure Interaction (FSI)
  • Numerical Methods

Background:

  • Immersed boundary (IB) methods rely on regularized delta functions for information transfer between fluid and solid domains.
  • Traditional isotropic kernels in IB methods often fail to preserve the divergence-free condition, leading to incompressibility errors in solid mechanics.
  • Existing volumetric stabilization techniques add complexity to simulations of large deformations in incompressible elastic structures.

Purpose of the Study:

  • To evaluate the performance of Composite B-spline (CBS) kernels in immersed boundary simulations.
  • To compare the volume conservation and accuracy of CBS kernels against traditional isotropic (IB and B-spline) kernels.
  • To assess the necessity of additional volumetric stabilization when using CBS kernels for large deformations.

Main Methods:

  • Implemented and tested Composite B-spline (CBS) kernels within the immersed boundary framework.
  • Conducted benchmark tests using various flow scenarios: elastic band, pressurized membrane, compressed block, Cook's membrane, slanted channel flow, and Turek-Hron problem.
  • Validated the methodology with a bioprosthetic heart valve fluid-structure interaction model in a pulse duplicator.

Main Results:

  • CBS kernels demonstrated superior volume conservation compared to conventional isotropic kernels, negating the need for explicit volumetric stabilization.
  • The accuracy of CBS kernels on coarser grids was comparable to that of IB and B-spline kernels on finer grids.
  • CBS kernels showed improved performance with smaller mesh ratio factors and were less sensitive to relative grid spacing variations than isotropic kernels.

Conclusions:

  • Composite B-spline kernels offer a stable, accurate, and efficient alternative for immersed boundary simulations of fluid-structure interaction.
  • CBS kernels inherently maintain discrete divergence-free properties, simplifying simulations involving large deformations of elastic solids.
  • The study advocates for the use of CBS kernels to avoid specialized volumetric stabilization treatments in complex FSI problems.