Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917008157704192 |
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| author | Gruninger, Cole Griffith, Boyce E. |
| author_facet | Gruninger, Cole Griffith, Boyce E. |
| contents | This paper presents an approach to enhance volume conservation in the immersed boundary (IB) method by using regularized delta functions derived from composite B-splines. The conventional IB method, while effective for fluid-structure interaction applications, has long been challenged by poor volume conservation, particularly evident in simulations of pressurized, closed membranes. We demonstrate that composite B-spline regularized delta functions significantly enhance volume conservation through two complementary properties: they provide continuously divergence-free velocity interpolants and maintain the gradient character of forces corresponding to mean pressure jumps across interfaces. By correctly representing these forces as discrete gradients, they eliminate a key source of spurious flows that typically plague immersed boundary computations. Our approach maintains the local nature of the classical IB method, avoiding the computational overhead associated with the non-local Divergence-Free Immersed Boundary (DFIB) method's construction of an explicit velocity potential which requires additional Poisson solves for interpolation and force spreading operations. We show that sufficiently regular composite B-spline kernels maintain initial volumes to within machine precision. We provide a detailed analysis of the relationship between kernel regularity and the accuracy of force spreading and velocity interpolation operations. Our findings indicate that composite B-splines of at least $C^1$ regularity produce results comparable to the DFIB method in dynamic simulations, with errors in volume conservation dominated by truncation error of the time-stepping scheme. The proposed approach requires minimal modifications to an existing IB code, making it an accessible improvement for a wide range of applications in computational fluid dynamics and fluid-structure interaction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading Gruninger, Cole Griffith, Boyce E. Numerical Analysis Computational Physics Fluid Dynamics This paper presents an approach to enhance volume conservation in the immersed boundary (IB) method by using regularized delta functions derived from composite B-splines. The conventional IB method, while effective for fluid-structure interaction applications, has long been challenged by poor volume conservation, particularly evident in simulations of pressurized, closed membranes. We demonstrate that composite B-spline regularized delta functions significantly enhance volume conservation through two complementary properties: they provide continuously divergence-free velocity interpolants and maintain the gradient character of forces corresponding to mean pressure jumps across interfaces. By correctly representing these forces as discrete gradients, they eliminate a key source of spurious flows that typically plague immersed boundary computations. Our approach maintains the local nature of the classical IB method, avoiding the computational overhead associated with the non-local Divergence-Free Immersed Boundary (DFIB) method's construction of an explicit velocity potential which requires additional Poisson solves for interpolation and force spreading operations. We show that sufficiently regular composite B-spline kernels maintain initial volumes to within machine precision. We provide a detailed analysis of the relationship between kernel regularity and the accuracy of force spreading and velocity interpolation operations. Our findings indicate that composite B-splines of at least $C^1$ regularity produce results comparable to the DFIB method in dynamic simulations, with errors in volume conservation dominated by truncation error of the time-stepping scheme. The proposed approach requires minimal modifications to an existing IB code, making it an accessible improvement for a wide range of applications in computational fluid dynamics and fluid-structure interaction. |
| title | Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading |
| topic | Numerical Analysis Computational Physics Fluid Dynamics |
| url | https://arxiv.org/abs/2408.08280 |