A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method)

Fedkiw, Ronald; Aslam, Tariq D.; Merriman, Barry; Osher, Stanley

doi:10.1006/jcph.1999.6236

Cited by 1,687 publications

(1,389 citation statements)

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“…Standard finite differences cannot be applied across the interface due to the jump boundary conditions on Σ. The ghost cell method was developed to deal with this issue when solving elliptical problems by creating "ghost" computational points and using those ghost points in standard finite difference discretizations [15,[20][21][22]32]. In [34] and [36], we extended the ghost cell method to attain second-order accuracy on interior problems (i.e., p is constant in Ω c ) with boundary conditions that depend upon the geometry (e.g., curvature) and without a jump condition on the normal derivative.…”

Section: The Ghost Cell Methodsmentioning

confidence: 99%

“…Suppose we desire the normal vector at a computational node point (x i , y j ). If the level set is sufficiently smooth at the four points of {(x i−1 , y j ), (x i , y j−1 ), (x i+1 , y j ), (x i , y j+1 )} (i.e., Q < η at those points), then we use the standard normal vector discretization (15) where ε is a small positive number used to avoid division by zero; we use ε ∼ 10 −16 in our work.…”

Section: Calculating Geometric Quantitiesmentioning

confidence: 99%

“…However, by considering the normal derivative jump condition, we shall have two equations for p ℓ and p r , allowing us to completely eliminate them from the extrapolation. The proper discretization of the normal derivative jump [D∇p · n] across the interface has been an open problem since the introduction of the ghost cell method for the Poisson problem [15,21,32]. Suppose that we wish to discretize [D∇p · n] at the point x Σ = (x Σ , y j ) = (x i + θΔx, y j ) from the preceding discussion.…”

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

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A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasisteady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.

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Section: The Ghost Cell Methodsmentioning

confidence: 99%

Section: Calculating Geometric Quantitiesmentioning

confidence: 99%

Section: Determining P ℓ From the Jump Boundarymentioning

confidence: 99%

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A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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“…This is accomplished by first re-constructing the Cartesian velocity components at the volume centers-(i, j, k) nodes-by interpolating the contravariant velocity components and using Eq. (8). With the Cartesian velocity components available at the volume centers the convective and viscous terms (C (u m ) and D(u m ) for m = 1, 2, 3) can be readily discretized using the discretization method of choice (see below for details) in the same manner as in a non-staggered mesh.…”

Section: Hybrid Staggered/non-staggered Approach In Curvilinear Coordmentioning

confidence: 99%

“…The former methods are known as immersed boundary formulations and tend to smear a solid boundary across few grid nodes due to the discrete delta function formulation they employ to introduce the effect of the boundary on the equations of motion [2]. The latter class of methods, on the other hand, treats solid boundaries as sharp interfaces utilizing either Cartesian, cut-cell formulations [3,4] or hybrid Cartesian/Immersed Boundary (HCIB) approaches (see [5,6,1,7] among others)-the reader is referred to [8,9] for more detailed discussion of this class of methods. Regardless on whether a diffused or a sharp interface formulation is employed, however, all available non-boundary conforming methods solve the Navier-Stokes equations in a background coordinate-conforming mesh, such as a Cartesian (e.g.…”

Section: Introductionmentioning

confidence: 99%

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

Sotiropoulos

2007

Journal of Computational Physics

346

348

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A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

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Numerical Implementation for the Filling and Packing Simulation

Zhou

Zhang

et al. 2013

Computer Modeling for Injection Molding

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A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method)

Cited by 1,687 publications

References 30 publications

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

Numerical Implementation for the Filling and Packing Simulation

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