arXiv subjects

Boyce E. Griffith

Publications and source records attributed to Boyce E. Griffith.

At least 37 records · Page 2Linked to original sources

arXiv2023

A Model of Fluid-Structure and Biochemical Interactions for Applications to Subclinical Leaflet Thrombosis

Subclinical leaflet thrombosis (SLT) is a potentially serious complication of aortic valve replacement with a bioprosthetic valve in which blood clots form on the replacement valve. SLT is associated with increased risk of transient ischemic attacks and strokes and can progress to clinical leaflet thrombosis. SLT following aortic valve replacement also may be related to subsequent structural valve deterioration, which can impair the durability of the valve replacement. Because of the difficulty in clinical imaging of SLT, models are needed to determine the mechanisms of SLT and could eventually predict which patients will develop SLT. To this end, we develop methods to simulate leaflet thrombosis that combine fluid-structure interaction and a simplified thrombosis model that allows for deposition along the moving leaflets. Additionally, this model can be adapted to model deposition or absorption along other moving boundaries. We present convergence results and quantify the model's ability to realize changes in valve opening and pressures. These new approaches are an important advancement in our tools for modeling thrombosis in which they incorporate both adhesion to the surface of the moving leaflets and feedback to the fluid-structure interaction.

math.NA↗

arXiv2023

A Nodal Immersed Finite Element-Finite Difference Method

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. This method uses a finite element (FE) method to approximate the stresses and forces on a structural mesh and a finite difference (FD) method to approximate the momentum of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. Force spreading and velocity interpolation both require projecting data onto the finite element space. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to accommodate mass lumping with higher-order shape functions. Our theoretical results are confirmed by numerical benchmarks, including standard solid mechanics tests and examination of a dynamic model of a bioprosthetic heart valve.

math.NA↗

arXiv2023

A sharp interface Lagrangian-Eulerian method for flexible-body fluid-structure interaction

This paper introduces a sharp-interface approach to simulating fluid-structure interaction involving flexible bodies described by general nonlinear material models and across a broad range of mass density ratios. This new flexible-body immersed Lagrangian-Eulerian (ILE) approach incorporates the geometrical and domain solution flexibility of the immersed boundary (IB) method with an accuracy comparable to body-fitted approaches that sharply resolve flows and stresses up to the fluid-structure interface. Unlike many IB methods, our ILE formulation uses distinct momentum equations for the fluid and solid subregions with a Dirichlet-Neumann coupling strategy that connects fluid and solid subproblems through simple interface conditions. We use a penalty method involving two representations of the fluid-structure interface. These two representations are connected by approximate Lagrange multiplier forces that impose kinematic interface conditions. This approach also enables the use of multi-rate time stepping, which allows us to take different time step sizes for the fluid and structure subproblems. Our fluid solver relies on an immersed interface method for discrete surfaces to impose stress jump conditions along complex interfaces. The dynamics of the volumetric structural mesh are determined using a standard finite element approach to large-deformation nonlinear elasticity via a nearly incompressible solid mechanics formulation. This formulation also readily accommodates compressible structures for cases in which part of the solid boundary does not contact the incompressible fluid. Comparisons are made with computational and experimental benchmarks. We also demonstrate the capabilities of this methodology by applying it to model the transport and capture of a cylindrical blood clot in an inferior vena cava filter.

math.NA↗

arXiv2022

How swimming style affects schooling of two fish-like wavy hydrofoils

Fish swimming style and schooling behaviour are two important aspects of underwater locomotion. How swimming style affects fish schooling is investigated by a representative problem setup consisting of two tethered NACA0012 hydrofoils undulating in a free stream flow at various wavelengths $ λ^* = 0.8 - 8 $, front-back distance $ D = 0,\ 0.25,\ 0.5,\ 0.75 $, phase difference $ ϕ/π= 0,\ 0.5,\ 1,\ 1.5 $, and lateral gap distance $ G = 0.25,\ 0.3,\ 0.35 $ with fixed Reynolds number $ Re = 5000 $, Strouhal number $ St = 0.4 $, and maximum amplitude $ A_{max} = 0.1 $. In total, 336 cases were simulated by open-source software IBAMR based on immersed boundary method. The increase in wavelength generally leads to higher thrust and lateral force, consistent with a previous study regarding a single swimmer. The highest Froude efficiency is obtained at intermediate wavelength $ λ^* = 2 $ for either individual or two foils as a group. At side-by-side arrangement $ D = 0 $, the thrust force upon the two foils can be equivalent, indicating a stable formation. At staggered arrangement, $ D > 0 $, the follower can take significant advantage of the leader in locomotion performance by tuning phase difference. The follower's benefit decreases with relative distance. Various combinations of wavelength and relative distance can lead to distinct flow structures, indicating a tunable stealth capacity of the schooling swimmers.

physics.flu-dyn↗

arXiv2022

An Immersed Interface Method for Discrete Surfaces

Fluid-structure systems occur in a range of scientific and engineering applications. The immersed boundary(IB) method is a widely recognized and effective modeling paradigm for simulating fluid-structure interaction(FSI) in such systems, but a difficulty of the IB formulation is that the pressure and viscous stress are generally discontinuous at the interface. The conventional IB method regularizes these discontinuities, which typically yields low-order accuracy at these interfaces. The immersed interface method(IIM) is an IB-like approach to FSI that sharply imposes stress jump conditions, enabling higher-order accuracy, but prior applications of the IIM have been largely restricted to methods that rely on smooth representations of the interface geometry. This paper introduces an IIM that uses only a C0 representation of the interface,such as those provided by standard nodal Lagrangian FE methods. Verification examples for models with prescribed motion demonstrate that the method sharply resolves stress discontinuities along the IB while avoiding the need for analytic information of the interface geometry. We demonstrate that only the lowest-order jump conditions for the pressure and velocity gradient are required to realize global 2nd-order accuracy. Specifically,we show 2nd-order global convergence rate along with nearly 2nd-order local convergence in the Eulerian velocity, and between 1st-and 2nd-order global convergence rates along with 1st-order local convergence for the Eulerian pressure. We also show 2nd-order local convergence in the interfacial displacement and velocity along with 1st-order local convergence in the fluid traction. As a demonstration of the method's ability to tackle complex geometries,this approach is also used to simulate flow in an anatomical model of the inferior vena cava.

math.NA↗

arXiv2022

On the Lagrangian-Eulerian Coupling in the Immersed Finite Element/Difference Method

The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior numerical work investigating the impact of the choice of kernel function on the accuracy of the methodology has been limited. This work systematically studies the effect of the choice of regularized delta function in several FSI benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses a finite element structural discretizations combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. The IFED formulation evaluates the regularized delta function on a collection of interaction points that can be chosen to be denser than the nodes of the Lagrangian mesh, and this study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations. Our results indicate that kernels satisfying a commonly imposed even-odd condition require higher resolution to achieve similar accuracy as kernels that do not satisfy. We also find that narrower kernels are more robust and that structural meshes that are substantially coarser than the Cartesian grid can yield high accuracy for shear-dominated cases but not for cases with large normal forces. We verify our results in a large-scale FSI model of a bovine pericardial bioprosthetic heart valve in a pulse duplicator.

math.NA↗

arXiv2021

The Projection Extension Method: A Spectrally Accurate Technique for Complex Domains

An essential ingredient of a spectral method is the choice of suitable bases for test and trial spaces. On complex domains, these bases are harder to devise, necessitating the use of domain partitioning techniques such as the spectral element method. In this study, we introduce the Projection Extension (PE) method, an approach that yields spectrally accurate solutions to various problems on complex geometries without requiring domain decomposition. This technique builds on the insights used by extension methodologies such as the immersed boundary smooth extension and Smooth Forcing Extension (SFE) methods that are designed to improve the order of accuracy of the immersed boundary method. In particular, it couples an accurate extension procedure, that functions on arbitrary domains regardless of connectedness or regularity, with a least squares minimization of the boundary conditions. The resulting procedure is stable under iterative application and straightforward to generalize to higher dimensions. Moreover, it rapidly and robustly yields exponentially convergent solutions to a number of challenging test problems, including elliptic, parabolic, Newtonian fluid flow, and viscoelastic problems.

math.NA↗

arXiv2021

A Hybrid Semi-Lagrangian Cut Cell Method for Advection-Diffusion Problems with Robin Boundary Conditions in Moving Domains

We present a new discretization for advection-diffusion problems with Robin boundary conditions on complex time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme that uses a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the $L^1$, $L^2$, and $L^\infty$ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to handle conversion of one concentration field to another across a moving boundary.

math.NA↗

arXiv2021

The Smooth Forcing Extension Method: A High-Order Technique for Solving Elliptic Equations on Complex Domains

High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral methods. These tools make it difficult to apply these techniques to higher dimensions. In contrast, fixed Cartesian grid methods, such as the immersed boundary (IB) method, are easy to apply and generalize, but typically are low-order accurate. In this study, we introduce the Smooth Forcing Extension (SFE) method, a fixed Cartesian grid technique that builds on the insights of the IB method, and allows one to obtain arbitrary orders of accuracy. Our approach relies on a novel Fourier continuation method to compute extensions of the inhomogeneous terms to any desired regularity. This is combined with the highly accurate Non-Uniform Fast Fourier Transform for interpolation operations to yield a fast and robust method. Numerical tests confirm that the technique performs precisely as expected on one-dimensional test problems. In higher dimensions, the performance is even better, in some cases yielding sub-geometric convergence. We also demonstrate how this technique can be applied to solving parabolic problems and for computing the eigenvalues of elliptic operators on general domains, in the process illustrating its stability and amenability to generalization.

math.NA↗

arXiv2021

A one-sided direct forcing immersed boundary method using moving least squares

This paper presents a one-sided immersed boundary (IB) method using kernel functions constructed via a moving least squares (MLS) method. The resulting kernels effectively couple structural degrees of freedom to fluid variables on only one side of the fluid-structure interface. This reduces spurious feedback forcing and internal flows that are typically observed in IB models that use isotropic kernel functions to couple the structure to fluid degrees of freedom on both sides of the interface. The method developed here extends the original MLS methodology introduced by Vanella and Balaras (J Comput Phys, 2009). Prior IB/MLS methods have used isotropic kernel functions that coupled fluid variables on both sides of the boundary to the interfacial degrees of freedom. The original IB/MLS approach converts the cubic spline weights typically employed in MLS reconstruction into an IB kernel function that satisfies particular discrete moment conditions. This paper shows that the same approach can be used to construct one-sided kernel functions (kernel functions are referred to as generating functions in the MLS literature). We also examine the performance of the new approach for a family of kernel functions introduced by Peskin. It is demonstrated that the one-sided MLS construction tends to generate non-monotone interpolation kernels with large over- and undershoots. We present two simple weight shifting strategies to construct generating functions that are positive and monotone, which enhances the stability of the resulting IB methodology. Benchmark cases are used to test the order of accuracy and verify the one-sided IB/MLS simulations in both two and three spatial dimensions. This new IB/MLS method is also used to simulate flow over the Ahmed car model, which highlights the applicability of this methodology for modeling complex engineering flows.

math.NA↗

arXiv2021

A sharp interface Lagrangian-Eulerian method for rigid-body fluid-structure interaction

This paper introduces a sharp interface method to simulate fluid-structure interaction (FSI) involving rigid bodies immersed in viscous incompressible fluids. The capabilities of this methodology are demonstrated for a range of benchmark test cases along with large-scale models of biomedical FSI. The numerical approach developed herein, which we refer to as an immersed Lagrangian-Eulerian method, integrates aspects of partitioned and immersed FSI formulations by solving separate momentum equations for the fluid and solid subdomains, as in a partitioned formulation, while also using non-conforming discretizations of the dynamic fluid and structure regions, as in an immersed formulation. A Dirichlet-Neumann coupling scheme is used, in which the motion of the immersed solid is driven by fluid traction forces evaluated along the fluid-structure interface, and the motion of the fluid along that interface is constrained to match the solid velocity and thereby satisfy the no-slip condition. To develop a practical numerical method, we adopt a penalty approach that approximately imposes the no-slip condition along the fluid-structure interface. Our fluid-structure interaction scheme relies on an immersed interface method for discrete geometries, which enables the accurate determination of both velocities and stresses along complex fluid-structure interfaces. Unlike commonly used partitioned FSI methods, which can suffer from so-called added mass effect instabilities, our methodology retains stability for test cases involving extremely small, nearly equal, equal, and large solid-fluid density ratios without requiring subiterations or complex handling of the pressure. Biomedical FSI demonstration cases are also presented including the dynamics of a bileaflet mechanical heart valve in a pulse duplicator, and transport of blood clots in a patient-averaged anatomical model of the inferior vena cava.

math.NA↗

arXiv2020

Models for plasma kinetics during simultaneous therapeutic plasma exchange and extracorporeal membrane oxygenation

This paper focuses on the derivation and simulation of mathematical models describing new plasma fraction in blood for patients undergoing simultaneous extracorporeal membrane oxygenation and therapeutic plasma exchange. Models for plasma exchange with either veno-arterial or veno-venous extracorporeal membrane oxygenation are considered. Two classes of models are derived for each case, one in the form of an algebraic delay equation and another in the form of a system of delay differential equations. In special cases, our models reduce to single compartment ones for plasma exchange that have been validated with experimental data. We also show that the algebraic delay equations are forward Euler discretizations of the delay differential equations, with timesteps equal to transit times through model compartments. Numerical simulations are performed to compare different model types, to investigate the impact of plasma device port switching on the efficiency of the exchange process, and to study the sensitivity of the models to their parameters.

math.DS↗

arXiv2020

An immersed interface-lattice Boltzmann method for fluid-structure interaction

An immersed interface-lattice Boltzmann method (II-LBM) is developed for modelling fluid-structure systems. The key element of this approach is the determination of the jump conditions that are satisfied by the distribution functions within the framework of the lattice Boltzmann method when forces are imposed along a surface immersed in an incompressible fluid. In this initial II-LBM, the discontinuity related to the normal portion of the interfacial force is sharply resolved by imposing the relevant jump conditions using an approach that is analogous to imposing the corresponding pressure jump condition in the incompressible Navier-Stokes equations. We show that the jump conditions for the distribution functions are the same in both single-relaxation-time and multi-relaxation-time LBM formulations. Tangential forces are treated using the immersed boundary-lattice Boltzmann method (IB-LBM). The performance of the II-LBM method is compared to both the direct forcing IB-LBM for rigid-body fluid-structure interaction, and the classical IB-LBM for elastic interfaces. Higher order accuracy is observed with the II-LBM as compared to the IB-LBM for selected benchmark problems. Because the jump conditions of the distribution function also satisfy the continuity of the velocity field across the interface, the error in the velocity field is much smaller for the II-LBM than the IB-LBM. The II-LBM is also demonstrated to provide superior volume conservation when simulating flexible boundaries.

physics.comp-ph↗

arXiv2019

A sharp interface method for an immersed viscoelastic solid

The immersed boundary-finite element method (IBFE) is an approach to describing the dynamics of an elastic structure immersed in an incompressible viscous fluid. In this formulation, there are discontinuities in the pressure and viscous stress at fluid-structure interfaces. The standard immersed boundary approach, which connects the Lagrangian and Eulerian variables via integral transforms with regularized Dirac delta function kernels, smooths out these discontinuities, which generally leads to low order accuracy. This paper describes an approach to accurately resolve pressure discontinuities for these types of formulations, in which the solid may undergo large deformations. Our strategy is to decompose the physical pressure field into a sum of two pressure-like fields, one defined on the entire computational domain, which includes both the fluid and solid subregions, and one defined only on the solid subregion. Each of these fields is continuous on its domain of definition, which enables high accuracy via standard discretization methods without sacrificing sharp resolution of the pressure discontinuity. Numerical tests demonstrate that this method improves rates of convergence for displacements, velocities, stresses, and pressures, as compared to the conventional IBFE method. Further, it produces much smaller errors at reasonable numbers of degrees of freedom. The performance of this method is tested on several cases with analytic solutions, a nontrivial benchmark problem of incompressible solid mechanics, and an example involving a thick, actively contracting torus.

math.NA↗

arXiv2019

A robust incompressible Navier-Stokes solver for high density ratio multiphase flows

This paper presents a robust, adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined Cartesian grids that adapt to the evolving interfaces and track regions of high vorticity. The algorithm combines the interface capturing level set method with a variable-coefficient incompressible Navier-Stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude. The discretization approach ensures second-order pointwise accuracy for both velocity and pressure with several physical boundary treatments, including velocity and traction boundary conditions. The paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver. To ensure the stability of the numerical scheme in the presence of high density and viscosity ratios, we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations. This consistency is achieved by solving an additional mass balance equation, which we approximate via a strong stability preserving Runga-Kutta time integrator and by employing the same mass flux (obtained from the mass equation) in the discrete momentum equation. The scheme uses higher-order total variation diminishing (TVD) and convection-boundedness criterion (CBC) satisfying limiter to avoid numerical fluctuations in the transported density field. The high-order bounded convective transport is done on a dimension-by-dimension basis, which makes the scheme simple to implement. We also demonstrate through several test cases that the lack of consistent mass and momentum transport in non-conservative formulations, which are commonly used in practice, or the use of non-CBC satisfying limiters can yield very large numerical error and very poor accuracy for convection-dominant high density ratio flows.

physics.comp-ph↗

arXiv2018

Transition in swimming direction in a model self-propelled inertial swimmer

We propose a reciprocal, self-propelled model swimmer at intermediate Reynolds numbers ($Re$). Our swimmer consists of two unequal spheres that oscillate in antiphase generating nonlinear steady streaming (SS) flows. We show computationally that the SS flows enable the swimmer to propel itself, and also switch direction as $Re$ increases. We quantify the transition in the swimming direction by collapsing our data on a critical $Re$ and show that the transition in swimming directions corresponds to the reversal of the SS flows. Based on our findings, we propose that SS can be an important physical mechanism for motility at intermediate $Re$.

physics.flu-dyn↗

arXiv2017

Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology

In standard models of cardiac electrophysiology, including the bidomain and monodomain models, local perturbations can propagate at infinite speed. We address this unrealistic property by developing a hyperbolic bidomain model that is based on a generalization of Ohm's law with a Cattaneo-type model for the fluxes. Further, we obtain a hyperbolic monodomain model in the case that the intracellular and extracellular conductivity tensors have the same anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is equivalent to a cable model that includes axial inductances, and the relaxation times of the Cattaneo fluxes are strictly related to these inductances. A purely linear analysis shows that the inductances are negligible, but models of cardiac electrophysiology are highly nonlinear, and linear predictions may not capture the fully nonlinear dynamics. In fact, contrary to the linear analysis, we show that for simple nonlinear ionic models, an increase in conduction velocity is obtained for small and moderate values of the relaxation time. A similar behavior is also demonstrated with biophysically detailed ionic models. Using the Fenton-Karma model along with a low-order finite element spatial discretization, we numerically analyze differences between the standard monodomain model and the hyperbolic monodomain model. In a simple benchmark test, we show that the propagation of the action potential is strongly influenced by the alignment of the fibers with respect to the mesh in both the parabolic and hyperbolic models when using relatively coarse spatial discretizations. Accurate predictions of the conduction velocity require computational mesh spacings on the order of a single cardiac cell. We also compare the two formulations in the case of spiral break up and atrial fibrillation in an anatomically detailed model of the left atrium, and [...].

math.NA↗

arXiv2017

An Immersed Boundary Method with Divergence-Free Velocity Interpolation and Force Spreading

The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid-structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an Eulerian representation of the fluid and a Lagrangian representation of the structure. The Lagrangian and Eulerian frames are coupled by integral transforms with delta function kernels. The discretized IB equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure, and to spread structural forces to the fluid. It is well-known that the conventional IB method can suffer from poor volume conservation since the interpolated Lagrangian velocity field is not generally divergence-free, and so this can cause spurious volume changes. In practice, the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries. The aim of this paper is to greatly reduce the volume error of the IB method by introducing velocity-interpolation and force-spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least C1 and satisfies a continuous divergence-free condition, and that the force-spreading operator is the adjoint of the velocity-interpolation operator. We confirm through numerical experiments in two and three spatial dimensions that this new IB method is able to achieve substantial improvement in volume conservation compared to other existing IB methods, at the expense of a modest increase in the computational cost. Further, the new method provides smoother Lagrangian forces (tractions) than traditional IB methods. The method presented here is restricted to periodic computational domains. Its generalization to non-periodic domains is important future work.

math.NA↗