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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.
Advances in Computational Science and Engineering
|August 22, 2025
Summary
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.
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.
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