Local divergence-free immersed finite element-difference method using composite B-splines

In the class of immersed boundary (IB) methods, the choice of the regularized delta function plays a crucial role in transferring information between fluid and solid domains through interpolation and spreading operators. Most prior work using the IB method has used isotropic kernels that do not pres...

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Vydáno v:Advances in Computational Science and Engineering Ročník 4; s. 16 - 56
Hlavní autoři: Li, Lianxia, Gruninger, Cole, Lee, Jae H., Griffith, Boyce E.
Médium: Journal Article
Jazyk:angličtina
Vydáno: United States 01.06.2025
Témata:
isotropic kernel
fluid-structure interaction
immersed finite element-finite difference method
volume conservation
Immersed boundary
composite B-spline kernels (CBS)
volumetric stabilization
Secondary: 53C35
Primary: 58F15, 58F17
ISSN:2837-1739, 2837-1739
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Abstract In the class of immersed boundary (IB) methods, the choice of the regularized delta function plays a crucial role in transferring information between fluid and solid domains through interpolation and spreading operators. Most prior work using the IB method has used isotropic kernels that do not preserve the divergence-free condition of the velocity field, leading to loss of incompressibility of the solid when interpolating the Eulerian velocity to Lagrangian markers. One approach to addressing this issue in IB simulations involving large deformations of immersed incompressible elastic structures is to use a volumetric stabilization approach, such as adding a volumetric energy term and using modified invariants in the structure's constitutive model. Composite B-spline (CBS) kernels offer an alternative approach by inherently maintaining the discrete divergence-free property. This work evaluates the performance of CBS kernels in terms of their volume conservation and accuracy, comparing them with several traditional isotropic kernel functions using a construction introduced by Peskin (referred to as IB kernels) and B-spline (BS) kernels. Benchmark tests include pressure-loaded and shear-dominated flows, such as an elastic band under differential pressure loads, a pressurized membrane, a compressed block, Cook's membrane, a slanted channel flow, and a modified Turek-Hron problem. Additionally, we validate our methodology using a complex fluid-structure interaction model of bioprosthetic heart valve dynamics in a pulse duplicator. Results demonstrate that CBS kernels achieve superior volume conservation compared to conventional isotropic kernels, eliminating the need for additional volumetric stabilization techniques typically required to address instabilities arising from volume conservation errors. Further, it is common that the accuracy provided by CBS kernels on coarser grids is comparable to that provided by IB and BS kernels on finer grids. Unlike IB and BS kernels, which perform better with larger mesh ratio factors between solid and fluid grids, CBS kernels show improved results with smaller mesh ratio factors. Additionally, the study reveals that although wider kernels provide more accurate results across all methods, CBS kernels are less sensitive to variations in relative grid spacings than isotropic kernels. This study highlights the advantages of CBS kernels in achieving stable, accurate, and efficient FSI simulations without requiring specialized volumetric stabilization treatments when simulating large deformations of elastic solids immersed in fluid.
AbstractList In the class of immersed boundary (IB) methods, the choice of the regularized delta function plays a crucial role in transferring information between fluid and solid domains through interpolation and spreading operators. Most prior work using the IB method has used isotropic kernels that do not preserve the divergence-free condition of the velocity field, leading to loss of incompressibility of the solid when interpolating the Eulerian velocity to Lagrangian markers. One approach to addressing this issue in IB simulations involving large deformations of immersed incompressible elastic structures is to use a volumetric stabilization approach, such as adding a volumetric energy term and using modified invariants in the structure's constitutive model. Composite B-spline (CBS) kernels offer an alternative approach by inherently maintaining the discrete divergence-free property. This work evaluates the performance of CBS kernels in terms of their volume conservation and accuracy, comparing them with several traditional isotropic kernel functions using a construction introduced by Peskin (referred to as IB kernels) and B-spline (BS) kernels. Benchmark tests include pressure-loaded and shear-dominated flows, such as an elastic band under differential pressure loads, a pressurized membrane, a compressed block, Cook's membrane, a slanted channel flow, and a modified Turek-Hron problem. Additionally, we validate our methodology using a complex fluid-structure interaction model of bioprosthetic heart valve dynamics in a pulse duplicator. Results demonstrate that CBS kernels achieve superior volume conservation compared to conventional isotropic kernels, eliminating the need for additional volumetric stabilization techniques typically required to address instabilities arising from volume conservation errors. Further, it is common that the accuracy provided by CBS kernels on coarser grids is comparable to that provided by IB and BS kernels on finer grids. Unlike IB and BS kernels, which perform better with larger mesh ratio factors between solid and fluid grids, CBS kernels show improved results with smaller mesh ratio factors. Additionally, the study reveals that although wider kernels provide more accurate results across all methods, CBS kernels are less sensitive to variations in relative grid spacings than isotropic kernels. This study highlights the advantages of CBS kernels in achieving stable, accurate, and efficient FSI simulations without requiring specialized volumetric stabilization treatments when simulating large deformations of elastic solids immersed in fluid.
In the class of immersed boundary (IB) methods, the choice of the regularized delta function plays a crucial role in transferring information between fluid and solid domains through interpolation and spreading operators. Most prior work using the IB method has used isotropic kernels that do not preserve the divergence-free condition of the velocity field, leading to loss of incompressibility of the solid when interpolating the Eulerian velocity to Lagrangian markers. One approach to addressing this issue in IB simulations involving large deformations of immersed incompressible elastic structures is to use a volumetric stabilization approach, such as adding a volumetric energy term and using modified invariants in the structure's constitutive model. Composite B-spline (CBS) kernels offer an alternative approach by inherently maintaining the discrete divergence-free property. This work evaluates the performance of CBS kernels in terms of their volume conservation and accuracy, comparing them with several traditional isotropic kernel functions using a construction introduced by Peskin (referred to as IB kernels) and B-spline (BS) kernels. Benchmark tests include pressure-loaded and shear-dominated flows, such as an elastic band under differential pressure loads, a pressurized membrane, a compressed block, Cook's membrane, a slanted channel flow, and a modified Turek-Hron problem. Additionally, we validate our methodology using a complex fluid-structure interaction model of bioprosthetic heart valve dynamics in a pulse duplicator. Results demonstrate that CBS kernels achieve superior volume conservation compared to conventional isotropic kernels, eliminating the need for additional volumetric stabilization techniques typically required to address instabilities arising from volume conservation errors. Further, it is common that the accuracy provided by CBS kernels on coarser grids is comparable to that provided by IB and BS kernels on finer grids. Unlike IB and BS kernels, which perform better with larger mesh ratio factors between solid and fluid grids, CBS kernels show improved results with smaller mesh ratio factors. Additionally, the study reveals that although wider kernels provide more accurate results across all methods, CBS kernels are less sensitive to variations in relative grid spacings than isotropic kernels. This study highlights the advantages of CBS kernels in achieving stable, accurate, and efficient FSI simulations without requiring specialized volumetric stabilization treatments when simulating large deformations of elastic solids immersed in fluid.In the class of immersed boundary (IB) methods, the choice of the regularized delta function plays a crucial role in transferring information between fluid and solid domains through interpolation and spreading operators. Most prior work using the IB method has used isotropic kernels that do not preserve the divergence-free condition of the velocity field, leading to loss of incompressibility of the solid when interpolating the Eulerian velocity to Lagrangian markers. One approach to addressing this issue in IB simulations involving large deformations of immersed incompressible elastic structures is to use a volumetric stabilization approach, such as adding a volumetric energy term and using modified invariants in the structure's constitutive model. Composite B-spline (CBS) kernels offer an alternative approach by inherently maintaining the discrete divergence-free property. This work evaluates the performance of CBS kernels in terms of their volume conservation and accuracy, comparing them with several traditional isotropic kernel functions using a construction introduced by Peskin (referred to as IB kernels) and B-spline (BS) kernels. Benchmark tests include pressure-loaded and shear-dominated flows, such as an elastic band under differential pressure loads, a pressurized membrane, a compressed block, Cook's membrane, a slanted channel flow, and a modified Turek-Hron problem. Additionally, we validate our methodology using a complex fluid-structure interaction model of bioprosthetic heart valve dynamics in a pulse duplicator. Results demonstrate that CBS kernels achieve superior volume conservation compared to conventional isotropic kernels, eliminating the need for additional volumetric stabilization techniques typically required to address instabilities arising from volume conservation errors. Further, it is common that the accuracy provided by CBS kernels on coarser grids is comparable to that provided by IB and BS kernels on finer grids. Unlike IB and BS kernels, which perform better with larger mesh ratio factors between solid and fluid grids, CBS kernels show improved results with smaller mesh ratio factors. Additionally, the study reveals that although wider kernels provide more accurate results across all methods, CBS kernels are less sensitive to variations in relative grid spacings than isotropic kernels. This study highlights the advantages of CBS kernels in achieving stable, accurate, and efficient FSI simulations without requiring specialized volumetric stabilization treatments when simulating large deformations of elastic solids immersed in fluid.
Author Li, Lianxia
Gruninger, Cole
Lee, Jae H.
Griffith, Boyce E.
AuthorAffiliation 4 Computational Medicine Program, University of North Carolina School of Medicine, Chapel Hill, NC, USA
2 Department of Biomedical Engineering, University of North Carolina, Chapel Hill, NC, USA
5 McAllister Heart Institute, University of North Carolina School of Medicine, Chapel Hill, NC, USA
1 Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
3 Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina, Chapel Hill, NC, USA
AuthorAffiliation_xml – name: 3 Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina, Chapel Hill, NC, USA
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  givenname: Cole
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  surname: Griffith
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Keywords isotropic kernel
fluid-structure interaction
immersed finite element-finite difference method
volume conservation
Immersed boundary
composite B-spline kernels (CBS)
volumetric stabilization
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Snippet In the class of immersed boundary (IB) methods, the choice of the regularized delta function plays a crucial role in transferring information between fluid and...
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Title Local divergence-free immersed finite element-difference method using composite B-splines
URI http://www.aimsciences.org/article/id/67fcd90c55a5ae148e358355
https://www.ncbi.nlm.nih.gov/pubmed/40842861
https://www.proquest.com/docview/3246305375
https://pubmed.ncbi.nlm.nih.gov/PMC12365689
Volume 4
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