Boyce E. Griffith

F. B. Balboa, J. B. BelL, R. Delgado-Buscalioni et al.

We develop numerical schemes for solving the isothermal compressible and incompressible equations of fluctuating hydrodynamics on a grid with staggered momenta. We develop a second-order accurate spatial discretization of the diffusive, advective and stochastic fluxes that satisfies a discrete fluctuation-dissipation balance, and construct temporal discretizations that are at least second-order accurate in time deterministically and in a weak sense. Specifically, the methods reproduce the correct equilibrium covariances of the fluctuating fields to third (compressible) and second (incompressible) order in the time step, as we verify numerically. We apply our techniques to model recent experimental measurements of giant fluctuations in diffusively mixing fluids in a micro-gravity environment [A. Vailati et. al., Nature Communications 2:290, 2011]. Numerical results for the static spectrum of non-equilibrium concentration fluctuations are in excellent agreement between the compressible and incompressible simulations, and in good agreement with experimental results for all measured wavenumbers.

Boyce E. Griffith, Xiaoyu Luo

The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach employs a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach.

Amneet Pal Singh Bhalla, Matthew G. Knepley, Mark F. Adams et al.

The immersed boundary (IB) method is a widely used approach to simulating fluid-structure interaction (FSI). Although explicit versions of the IB method can suffer from severe time step size restrictions, these methods remain popular because of their simplicity and generality. In prior work (Guy et al., Adv Comput Math, 2015), some of us developed a geometric multigrid preconditioner for a stable semi-implicit IB method under Stokes flow conditions; however, this solver methodology used a Vanka-type smoother that presented limited opportunities for parallelization. This work extends this Stokes-IB solver methodology by developing smoothing techniques that are suitable for parallel implementation. Specifically, we demonstrate that an additive version of the Vanka smoother can yield an effective multigrid preconditioner for the Stokes-IB equations, and we introduce an efficient Schur complement-based smoother that is also shown to be effective for the Stokes-IB equations. We investigate the performance of these solvers for a broad range of material stiffnesses, both for Stokes flows and flows at nonzero Reynolds numbers, and for thick and thin structural models. We show here that linear solver performance degrades with increasing Reynolds number and material stiffness, especially for thin interface cases. Nonetheless, the proposed approaches promise to yield effective solution algorithms, especially at lower Reynolds numbers and at modest-to-high elastic stiffnesses.

Cole Gruninger, Boyce E. Griffith

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. Together with an interpolation operator that samples the tangential electric field along the wire, this deposition operator can be realized as a regularization of distributions against a regularized delta function supported on the wire. 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 achievable because the B-spline kernels are piecewise polynomial with breakpoints known a priori, 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 naive trilinear regularization produces unphysical parasitic low-frequency currents in closed loops.

Keon Ho Kim, Boyce E. Griffith

The immersed peridynamics (IPD) method is a fluid-structure interaction (FSI) model to simulate fluid-driven material damage and failure of an immersed structure, in which a peridynamic (PD) constitutive correspondence model is employed within a classical immersed boundary (IB)-type framework to describe stresses, forces, and structural deformations of a structural body, instead of classical continuum mechanics. This paper introduces an extension of the IPD method to simulate fluid-driven structural deformation, damage, and failure of anisotropic materials with complex geometries. We use quadrature rules attached to finite element (FE) meshes to generate both the PD points and their associated weights, which are used to approximate the PD integrals. We demonstrate that non-uniform discretizations improve both accuracy and volume conservation of hyperelastic materials along with accurately represented boundaries. To capture realistic biomaterial behaviors, we incorporate hyperelastic constitutive models including both isotropy and anisotropy into the proposed IPD method. In addition, a ductile failure model is adopted to simulate realistic failure processes of anisotropic materials. For non-failure cases, our numerical simulations demonstrate that the extended IPD method yields comparable accuracy with similar numbers of structural degrees of freedom for different choices of peridynamic horizon sizes. For failure tests, we investigate the effect of a fiber orientation on deformations and failure processes using realistic biomaterial models with varying fiber directions. We further demonstrate that the developed method generates grid-converged simulations of damage growth, crack formation and propagation, and rupture under large deformations, including purely fluid-driven failure processes.

Ali Hasan, Ebrahim M. Kolahdouz, Andinet Enquobahrie et al.

Each year, approximately 300,000 heart valve repair or replacement procedures are performed worldwide, including approximately 70,000 aortic valve replacement surgeries in the United States alone. This paper describes progress in constructing anatomically and physiologically realistic immersed boundary (IB) models of the dynamics of the aortic root and ascending aorta. This work builds on earlier IB models of fluid-structure interaction (FSI) in the aortic root, which previously achieved realistic hemodynamics over multiple cardiac cycles, but which also were limited to simplified aortic geometries and idealized descriptions of the biomechanics of the aortic valve cusps. By contrast, the model described herein uses an anatomical geometry reconstructed from patient-specific computed tomography angiography (CTA) data, and employs a description of the elasticity of the aortic valve leaflets based on a fiber-reinforced constitutive model fit to experimental tensile test data. Numerical tests show that the model is able to resolve the leaflet biomechanics in diastole and early systole at practical grid spacings. The model is also used to examine differences in the mechanics and fluid dynamics yielded by fresh valve leaflets and glutaraldehyde-fixed leaflets similar to those used in bioprosthetic heart valves. Although there are large differences in the leaflet deformations during diastole, the differences in the open configurations of the valve models are relatively small, and nearly identical hemodynamics are obtained in all cases considered.