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Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading

Cole Gruninger, Boyce E. Griffith

TL;DR

The paper tackles artificial volume loss in the immersed boundary method by introducing composite B-spline regularized delta functions that simultaneously deliver continuously divergence-free velocity interpolation and gradient-preserving force spreading. This local, kernel-based approach preserves pressure-driven force gradients on the MAC grid, enabling near machine-precision volume conservation for pressurized membranes without the Poisson-solve overhead of the DFIB method. Across several tests—including advection in Taylor vortices, a quasi-static pressurized circle, parametrically excited membranes, and membranes in lid-driven cavity flow—the high-regularity composite B-spline kernels achieve performance comparable to or better than DFIB, with area errors largely governed by time stepping rather than spatial discretization. Practically, the method requires only a minimal modification to existing IB codes (changing the delta function) and remains computationally efficient, broadening applicability to three-dimensional FSI problems with complex boundary conditions.

Abstract

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.

Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading

TL;DR

The paper tackles artificial volume loss in the immersed boundary method by introducing composite B-spline regularized delta functions that simultaneously deliver continuously divergence-free velocity interpolation and gradient-preserving force spreading. This local, kernel-based approach preserves pressure-driven force gradients on the MAC grid, enabling near machine-precision volume conservation for pressurized membranes without the Poisson-solve overhead of the DFIB method. Across several tests—including advection in Taylor vortices, a quasi-static pressurized circle, parametrically excited membranes, and membranes in lid-driven cavity flow—the high-regularity composite B-spline kernels achieve performance comparable to or better than DFIB, with area errors largely governed by time stepping rather than spatial discretization. Practically, the method requires only a minimal modification to existing IB codes (changing the delta function) and remains computationally efficient, broadening applicability to three-dimensional FSI problems with complex boundary conditions.

Abstract

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 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.
Paper Structure (22 sections, 2 theorems, 66 equations, 27 figures, 1 table)

This paper contains 22 sections, 2 theorems, 66 equations, 27 figures, 1 table.

Table of Contents

  1. Introduction
  2. Continuous Equations of Motion
  3. Discrete Equations of Motion
  4. Eulerian Spatial Discretization
  5. Mimetic Properties of the Staggered MAC grid
  6. Lagrangian Spatial Discretization
  7. Regularized Delta Functions
  8. Velocity Interpolation and Force Spreading Operations
  9. Non-local DFIB Velocity Interpolation and Force Spreading
  10. Local IB Velocity Interpolation and Force Spreading
  11. Temporal Discretization
  12. Area Conservation Measurements
  13. Numerical Tests and Discussion
  14. Advecting an Immersed Boundary Using Taylor Vortices
  15. A Pressurized Circular Membrane at Equilibrium
  16. ...and 7 more sections

Key Result

Theorem 1

For any zero-mean, discretely divergence-free velocity field $\textbf{u}_{i,j}$ defined on the faces $\mathbb{F}$ of the MAC grid ($\nabla_h\cdot \textbf{u}_{i,j} = 0$), there exists a scalar potential $a_{i,j}$ defined on the grid nodes $\mathbb{N}$ such that Moreover, this potential is unique up to a constant factor.

Figures (27)

  • Figure 1: Illustration of the edge-centered and cell center locations defined about the Cartesian cell $\mathbf{x}_{i,j}$.
  • Figure 2: Illustration of the composite B-spline kernel $\mathrm{BS}_3\mathrm{BS}_2$ used for velocity interpolation and force spreading operations. Left: the composite $\mathrm{BS}_3\mathrm{BS}_2$ kernel used to interpolate $x$-components of velocity and spread $x$-components of Lagrangian force density. Right: the composite $\mathrm{BS}_3\mathrm{BS}_2$ kernel used to interpolate $y$-components of velocity and spread $y$-components of Lagrangian force density.
  • Figure 3: Mean relative area errors $\overline{\Delta A}\left(\mathbf{X}_{\text{tracer}}\right)$ for regularized delta functions in the IB method and for the DFIB method.
  • Figure 4: Semi-log plot of time-dependent relative area errors.
  • Figure 5: Semi-log plot of the error $||\textbf{F}-\textbf{F}_{\text{exact}}||_{L^2}$ for the $\text{BS}_4$$\text{BS}_3$, $\text{BS}_6$$\text{BS}_5$, $\text{IB}_4$ regularized delta functions and the DFIB method for a uniform Cartesian grid discretization with $h = 1/128$ and Lagrangian mesh discretizations corresponding to $M_{\text{fac}} = 2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}$.
  • ...and 22 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 1