Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading (2408.08280v3)

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.

• The paper introduces composite B-splines to ensure divergence-free velocity interpolation that markedly improves volume conservation in immersed boundary simulations.
• It demonstrates that high-regularity kernels achieve second-order time convergence and nearly perfect volume conservation, as validated by numerical experiments.
• The approach simplifies integration into existing systems, offering a computationally efficient alternative to more intensive methods like the divergence-free immersed boundary method.

Local Divergence-Free Velocity Interpolation for the Immersed Boundary Method Using Composite B-Splines

The paper "Local divergence-free velocity interpolation for the immersed boundary method using composite B-splines" by Cole Gruninger and Boyce E. Griffith addresses a longstanding issue in the immersed boundary (IB) method related to volume conservation. Specifically, it proposes the use of composite B-splines to maintain a divergence-free property in the velocity field, which is crucial for accurately simulating fluid-structure interactions (FSI).

Overview

Problem Statement

The immersed boundary method is widely utilized for simulating systems involving FSI. It employs a Lagrangian representation for the structure and an Eulerian representation for the fluid. The interaction between these representations is mediated by convolutions with delta function kernels. A known limitation of the conventional IB method is its inability to conserve volume accurately, especially in simulations involving pressurized, closed membranes.

Contributions

This research focuses on enhancing volume conservation through local, divergence-free velocity interpolation using composite B-spline delta functions. These functions preserve more geometric and physical properties of the flow field compared to traditional methods. The key contributions of the paper are:

• Introduction of tensor product composite B-splines for determining the regularized delta functions.
• Detailed analysis of the relationship between kernel regularity and the accuracy of force spreading and velocity interpolation operations.
• Demonstration that composite B-spline regularized delta functions can significantly enhance volume conservation, rivaling the non-local Divergence-Free Immersed Boundary method introduced by Bao et al.

Key Results and Numerical Experiments

The authors carried out a series of numerical experiments to validate their approach. They showed that using sufficiently regular composite B-spline kernels can maintain initial volumes to within machine precision. Notable findings include:

• Pure Advection Test: The paper demonstrates that for purely advective cases, composite B-splines of $C^0$ regularity and above exhibit second-order convergence in time. In contrast, the conventional IB kernel fails to maintain this accuracy under smaller time steps due to discrepancies in velocity interpolation.
• Quasi-Static Pressurized Membrane: Composite B-splines of higher regularity matched the performance of the DFIB method by conserving the initial volume within machine precision. For less regular kernels, volume conservation was improved compared to traditional IB methods but still fell short of the highest regularity kernels.
• Dynamic Simulations of Parametrically Excited Membranes: The paper found that composite B-splines of $C^1$ regularity and higher performed comparably to the DFIB method. Lower regularity kernels, however, struggled with maintaining higher time-stepping accuracy.

Theoretical Implications

By adopting composite B-splines, this work lays the foundation for combining the benefits of divergence-free interpolation with local computational schemes. The paper illuminated that the regularity of the kernel function critically impacts both the error in volume conservation and the quality of interpolated force densities. Kernels of higher regularity reduce the magnitude of spurious vorticity and errors in the Lagrangian force densities, which results in better physical fidelity and more accurate simulations.

Practical Implications and Future Developments

One of the practical benefits of this research is the ease of adoption. Implementing composite B-spline regularized delta functions in existing IB code requires minimal changes. This makes the method accessible for a wide range of IB method applications without the significant computational overhead associated with methods like DFIB, which require additional Poisson solves.

Moving forward, future research can expand upon these findings by exploring quadrature rules less sensitive to kernel regularity, potentially improving computational efficiency. Further work might also explore adaptive timestepping strategies or alternative interpolation methods to enhance accuracy when dealing with discontinuities in the interpolated velocity fields.

Conclusion

The introduction of composite B-splines for velocity interpolation in the immersed boundary method represents a significant advancement in the quest for accurate volume conservation in FSI simulations. By ensuring that the interpolated velocities are divergence-free, this approach not only matches but in some cases surpasses the volume conservation capabilities of more computationally intensive methods like DFIB. The minimal modifications required to integrate this approach into existing IB frameworks make it a compelling option for both current and future applications in computational fluid dynamics and fluid-structure interaction.