A Gaussian-like immersed-boundary kernel with three continuous derivatives and improved translational invariance

Bao, Yuanxun; Kaye, Jason; Peskin, Charles S.

doi:10.1016/j.jcp.2016.04.024

Cited by 55 publications

(59 citation statements)

References 7 publications

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“…Figure shows the lift and drag coefficients as functions of time for N =32, 64, and 128 and for M fac =1, 2, and 4 using the four‐point regularized kernel function . Similar results are shown in Figure for the similar three‐point kernel, whereas Figure shows results for a recently developed six‐point kernel with higher continuity order . By construction, the structural meshes obtained for a fixed value of N are nested versions of each other (ie, they can be viewed as obtained via Lagrangian mesh refinement).…”

Section: Numerical Resultsmentioning

confidence: 57%

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Hybrid finite difference/finite element immersed boundary method

Griffith

Luo

2017

Numer Methods Biomed Eng

131

184

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SUMMARY 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.

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Section: Numerical Resultsmentioning

confidence: 57%

“…Similar to Figures and , but here, using a six‐point kernel function with higher continuity order . As with the four‐point kernel, the best accuracy is obtained for larger values of M fac .…”

Section: Numerical Resultsmentioning

confidence: 59%

Hybrid finite difference/finite element immersed boundary method

Griffith

Luo

2017

Numer Methods Biomed Eng

131

184

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“…In the latter case, each end of the fiber makes contact with a chordal tree. 3. The leaflets are supported by a system of chordae tendineae, which anchor into two papillary muscles.…”

Section: Assumptionsmentioning

confidence: 99%

“…Equations (18) and (19) are interaction equations; they couple the Eulerian and Lagrangian descriptions to each other through convolutions with the Dirac delta function. The regularized delta function for all simulations is the 5-point delta function derived in [3].…”

Section: Periodic In Flow Directionmentioning

confidence: 99%

Modeling the mitral valve

Kaiser

McQueen

Peskin

2019

Numer Methods Biomed Eng

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This work is concerned with modeling and simulation of the mitral valve, one of the four valves in the human heart. The valve is composed of leaflets, the free edges of which are supported by a system of chordae, which themselves are anchored to the papillary muscles inside the left ventricle. First, we examine valve anatomy and present the results of original dissections. These display the gross anatomy and information on fiber structure of the mitral valve. Next, we build a model valve following a designbased methodology, meaning that we derive the model geometry and the forces that are needed to support a given load, and construct the model accordingly. We incorporate information from the dissections to specify the fiber topology of this model. We assume the valve achieves mechanical equilibrium while supporting a static pressure load. The solution to the resulting differential equations determines the pressurized configuration of the valve model. To complete the model we then specify a constitutive law based on a stress-strain relation consistent with experimental data that achieves the necessary forces computed in previous steps. Finally, using the immersed boundary method, we simulate the model valve in fluid in a computer test chamber. The model opens easily and closes without leak when driven by physiological pressures over multiple beats. Further, its closure is robust to driving pressures that lack atrial systole or are much lower or higher than normal.

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“…In order to facilitate comparison with the BE model, we estimate the radius of the fiber simulated using the IB method to be r fiber = r hydro h, where r hydro is the hydrodynamic radius [30] of the regularized delta function and h is the grid spacing for the Cartesian grid on which the fluid equations are solved. For these simulations, we use the 6-point delta function defined in [2], whose hydrodynamic radius is r hydro ≈ 1.47h. Note that because the radius of the fiber is set by the grid-spacing h, it is not possible to perform a refinement study at a fixed fiber radius.…”

Section: Immersed Boundary Simulationsmentioning

confidence: 99%

Coarse graining the dynamics of immersed and driven fiber assemblies

Stein

Shelley

2019

Phys. Rev. Fluids

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An important class of fluid-structure problems involve the dynamics of ordered arrays of immersed, flexible fibers. While specialized numerical methods have been developed to study fluid-fiber systems, they become infeasible when there are many, rather than a few, fibers present, nor do these methods lend themselves to analytical calculation. Here, we introduce a coarse-grained continuum model, based on local-slender body theory, for elastic fibers immersed in a viscous Newtonian fluid. It takes the form of an anisotropic Brinkman equation whose skeletal drag is coupled to elastic forces. This model has two significant benefits: (1) the density effects of the fibers in a suspension become analytically manifest, and (2) it allows for the rapid simulation of dense suspensions of fibers in regimes inaccessible to standard methods. As a first validation, without fitting parameters, we achieve very reasonable agreement with 3D Immersed Boundary simulations of a bed of anchored fibers bent by a shear flow. Secondly, we characterize the effect of density on the relaxation time of fiber beds under oscillatory shear, and find close agreement to results from full numerical simulations. We then study buckling instabilities in beds of fibers, using our model both numerically and analytically to understand the role of fiber density and the structure of buckling transitions. We next apply our model to study the flow-induced bending of inclined fibers in a channel, as has been recently studied as a flow rectifier, examining the nature of the internal flows within the bed, and the emergence of inhomogeneous permeability. Finally, we extend the method to study a simple model of metachronal waves on beds of actuated fibers, as a model for ciliary beds. Our simulations reproduce qualitatively the pumping action of coordinated waves of compression through the bed. I. INTRODUCTIONMany fundamental hydrodynamic phenomena, particularly in biology, involve the interaction of structured arrays of immersed fibers, often anchored to a substrate [21]. For example, in eukaryotic cells, arrays of aligned microtubules in the spindle orchestrate the segregation of chromosomes, while those around centrosomes help position the spindle prior to cell division [14]. During midoogenesis of Drosophila, kinesin motors interacting with the microtubule cytoskeleton drive largescale coherent flows known as cytoplasmic streaming [10]. Another example is the beds of driven cilia that pump fluid, move mucus, or propel microorganisms [4]. In microfluidic engineering, fabricated arrays of flexible fibers are the elements of proposed soft flow rectifiers [1]. A central aspect of all of these examples is that the relevant dynamics is collective and not well-described by the dynamics of a single fiber.Given the importance of fluid-fiber systems, specialized computational methods have been developed to treat them, most especially in the zero Reynolds limit where flows are governed by the Stokes equation. These approaches include the use of nonlocal slender body the...

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A Gaussian-like immersed-boundary kernel with three continuous derivatives and improved translational invariance

Cited by 55 publications

References 7 publications

Hybrid finite difference/finite element immersed boundary method

Hybrid finite difference/finite element immersed boundary method

Modeling the mitral valve

Coarse graining the dynamics of immersed and driven fiber assemblies

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