Discretization of Dirac delta functions in level set methods

Engquist, Björn; Tornberg, Anna‐Karin; Tsai, Richard

doi:10.1016/j.jcp.2004.09.018

Cited by 190 publications

(236 citation statements)

References 15 publications

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“…4.1 using our discrete Heaviside function. For comparison, we also considered the Heaviside function from Sussman and Fatemi [51] (96) with δ = Δx, and the Heaviside function in recent work by Engquist et al [14]:…”

Section: Discussionmentioning

confidence: 99%

“…Traditional stencil New (1st order) stencil New (higher order) stencil Table 5 Convergence of a ghost cell problem using a Heaviside function, computed with the continuous approximation by Sussman and Fatemi in [51] (left), the continuous approxmation by Engquist et al in [14] (middle), and our discrete Heaviside approximation (right) …”

Section: Discussionmentioning

confidence: 99%

“…To calculate the curvature κ i,j at a computational node point (x i , y j ), we evaluated Q at each of the nine grid points in {(x i+k , y j+ℓ ) : −1 ≤ k, ℓ ≤ 1}. If Q < η at each of these points, then the level set function was deemed sufficiently smooth to calculate the curvature κ i,j using the standard 9-point curvature stencil (14) which is second order accurate where the level set function is smooth [34,36]. If we could calculate the curvature κ i,j at enough nearby points to compute a bicubic or bilinear interpolation at x Σ , then we used that interpolated curvature value.…”

Section: Calculating Geometric Quantitiesmentioning

confidence: 99%

See 2 more Smart Citations

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: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Calculating Geometric Quantitiesmentioning

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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“…To account for the fact that we do not have access to the distance to the interface Σ but rather to an approximation of β tanh( dist(Σ,x) β ), see Section 2.2.2, we rescale δ (x) as suggested in Engquist et al [15], Tornberg [33] and consider instead:…”

Section: Data For the Navier-stokes Solvermentioning

confidence: 99%

Numerical simulations of bouncing jets

Bonito

Guermond

Lee

2015

Numerical Methods in Fluids

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Bouncing jets are fascinating phenomenons occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non-Newtonian fluids when the jets falls in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier-Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is done with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing.

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“…The accuracy of the numerical integration associated with different regularized delta functions has been analysed by several groups [17][18][19]. In particular, Tornberg and Engquist [18] formulated several requirements that the regularized delta functions must satisfy in order to ensure that the error of integration is bounded by O(h q ), where h is the mesh size and q a positive integer.…”

Section: Introductionmentioning

confidence: 99%

On surface tension modelling using the level set method

Shepel¹,

Smith

2008

Numerical Methods in Fluids

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SUMMARYThe paper describes and compares the performance of two options for numerically representing the surface tension force in combination with the level set interface-tracking method. In both models, the surface tension is represented as a body force, concentrated near the interface, but the technical implementation is different: the first model is based on a traditional level set approach in which the force is distributed in a band around the interface using a regularized delta function, whereas in the second, the force is partly distributed in a band around the interface and partly localized to the actual computational cells containing the interface. A comparative study, involving analysis of several two-phase flows with moving interfaces, shows that in general the two surface tension models produce results of similar accuracy. However, in the particular case of merging and pinching-off of interfaces, the traditional level set model of surface tension produces an error that results in non-converging solutions for film-like interfaces (i.e. ones involving large contact areas). In contrast, the second model, based on the localized representation of the surface tension force, displays consistent first-order convergence.

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Discretization of Dirac delta functions in level set methods

Cited by 190 publications

References 15 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

Numerical simulations of bouncing jets

On surface tension modelling using the level set method

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