arXiv subjects

Cole Gruninger

Publications and source records attributed to Cole Gruninger.

6 recordsLinked to original sources

arXiv2026

Coupling-Aware Vanka Smoothing for Multigrid Preconditioning of the Implicit Immersed Boundary Equations

The immersed boundary (IB) method models fluid--structure interaction using the natural Lagrangian and Eulerian formulations of structural mechanics and fluid dynamics, respectively, but explicit time discretization of the IB force imposes a stiffness-dependent upper bound on the time-step size. Treating these forces implicitly removes this restriction, but the resulting coupled linear systems become increasingly difficult to solve as the structural stiffness increases. Algebraically eliminating the Lagrangian degrees of freedom yields a reduced Eulerian velocity--pressure IB system. Here we introduce a coupling-aware Vanka (CAV) smoothing strategy to enable effective multigrid preconditioning of this system. CAV patches are built as unions of standard pressure-centered Vanka patches, with the graph of the Eulerian elasticity matrix determining which patches are combined. Under grid refinement, CAV patch sizes remain bounded, and the computational cost of each multigrid cycle thereby grows linearly with the number of Eulerian degrees of freedom. Tests using target-point, membrane, and beam force laws show that CAV-preconditioned FGMRES reduces the relative residual by ten orders of magnitude in $9--15$ iterations, with little growth under grid refinement. In a nonlinear benchmark modeling flow past a flexible fiber, the average number of FGMRES iterations per Newton solve increases only from $8.6$ to $9.5$ as the Eulerian grid is refined from $32\times32$ to $256\times256$ cells. To our knowledge, CAV provides the first robust multigrid strategy for time-dependent implicit IB formulations.

math.NA↗

arXiv2026

Composite B-Spline Current Deposition and Interpolation Operators for Thin-Wire Finite-Difference Time-Domain Simulations

Holland-Simpson thin-wire finite-difference time-domain (FDTD) simulations of obliquely oriented closed-loop antennas exhibit persistent low-frequency parasitic currents because the current-deposition operator fails to conserve charge. This deposition operator, together with an interpolation operator that samples the tangential electric field along the wire, can be realized as regularizations of distributions: the wire current is deposited as a source term by integrating it against a regularized delta function along the wire, and the electric field is sampled back to the wire by integrating it against the same regularized delta function. We show that charge conservation requires the deposited current to be discretely divergence-free when the wire carries a constant current, and we introduce a family of composite B-spline regularizations that satisfy this condition to machine precision. Exact evaluation of the coupling line integrals is possible because the B-spline kernels are piecewise polynomial with a priori-known breakpoints, allowing composite Gauss-Legendre quadrature with subinterval breakpoints at every grid-plane crossing. Taking the interpolation operator as the discrete adjoint of the deposition operator preserves skew-symmetry and ensures that a discretely irrotational electric field drives no net electromotive force around a closed loop. Numerical experiments on a center-fed dipole and on circular and square loop antennas show that the proposed regularizations yield orientation-independent impedance values consistent with known characteristics, whereas a simple trilinear regularization produces unphysical parasitic low-frequency currents in closed loops.

math.NA↗

arXiv2024

Local Divergence-Free Immersed Finite Element-Difference Method Using Composite B-Splines

In the class of immersed boundary (IB) methods, the choice of the delta function plays a crucial role in transferring information between fluid and solid domains. Most prior work 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 velocity to Lagrangian markers. To address this issue, in simulations involving large deformations of incompressible hyperelastic structures immersed in fluid, researchers often use stabilization approaches such as adding a volumetric energy term. Composite B-spline (CBS) kernels offer an alternative by maintaining the discrete divergence-free property. This work evaluates CBS kernels in terms of volume conservation and accuracy, comparing them with isotropic kernel functions using a construction introduced by Peskin (IB kernels) and B-spline (BS) kernels. Benchmark tests include pressure-loaded and shear-dominated flows, such as an elastic band under pressure loads, a pressurized membrane, a compressed block, Cook's membrane, and a slanted channel flow. Additionally, we validate our methodology using a complex fluid-structure interaction model of bioprosthetic heart valve dynamics. Results demonstrate that CBS kernels achieve superior volume conservation compared to isotropic kernels, eliminating the need for stabilization techniques. Further, CBS kernels converge on coarser fluid grids, while IB and BS kernels need finer grids for comparable accuracy. Unlike IB and BS kernels, which perform better with larger mesh ratios, CBS kernels improve with smaller mesh ratios. Wider kernels provide more accurate results across all methods, but CBS kernels are less sensitive to grid spacing variations than isotropic kernels.

math.NA↗

arXiv2024

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

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.

math.NA↗

arXiv2023

Flagellum Pumping Efficacy in Shear-Thinning Viscoelastic Fluids

Microorganism motility often takes place within complex, viscoelastic fluid environments, e.g., sperm in cervicovaginal mucus and bacteria in biofilms. In such complex fluids, strains and stresses generated by the microorganism are stored and relax across a spectrum of length and time scales and the complex fluid can be driven out of its linear response regime. Phenomena not possible in viscous media thereby arise from feedback between the "swimmer" and the complex fluid, making swimming efficiency co-dependent on the propulsion mechanism and fluid properties. Here we parameterize a flagellar motor and filament properties together with elastic relaxation and nonlinear shear-thinning properties of the fluid in a computational immersed boundary model. We then explore swimming efficiency over this parameter space. One exemplary insight is that motor efficiency (measured by the volumetric flow rate) can be boosted vs.\ degraded by moderate vs.\ strong shear-thinning of the viscoelastic environment.

physics.flu-dyn↗

arXiv2023

Benchmarking the Immersed Boundary Method for Viscoelastic Flows

We present and analyze a series of benchmark tests regarding the application of the immersed boundary (IB) method to viscoelastic flows through and around non-trivial, stationary geometries. The IB method is widely used for the simulation of biological fluid dynamics and other modeling scenarios where a structure is immersed in a fluid. Although the IB method has been most commonly used to model systems with viscous incompressible fluids, it also can be applied to visoelastic fluids, and has enabled the study of a wide variety of dynamical problems including the settling of vesicles and the swimming of elastic filaments in fluids modeled by the Oldroyd-B constuitive equation. However, to date, relatively little work has explored the accuracy or convergence properties of the numerical scheme. Herein, we present benchmarking results for an IB solver applied to viscoelastic flows in and around non-trivial geometries using the idealized Oldroyd-B and more realistic, polymer-entanglement-based Rolie-Poly constitutive equations. We use two-dimensional numerical test cases along with results from rheology experiments to benchmark the IB method and compare it to more complex finite element and finite volume viscoelastic flow solvers. Additionally, we analyze different choices of regularized delta function and relative Lagrangian grid spacings which allow us to identify and recommend the key choices of these numerical parameters depending on the present flow regime.

physics.flu-dyn↗