A gradient augmented level set method for unstructured gridsfrey/papers/levelsets/Bockmann... ·...

Journal of Computational Physics 258 (2014) 47–72

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Journal of Computational Physics

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A gradient augmented level set method for unstructured grids

Arne Bøckmann ∗, Magnus Vartdal

University of Oslo (UiO), Department of Mathematics, PO Box 1053, Blindern, NO-0316 Oslo, Norway

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 October 2012Received in revised form 26 June 2013Accepted 14 October 2013Available online 21 October 2013

Keywords:Level setUnstructured gridCompact stencilGradient augmentedFast marching method

We present and test a gradient augmented level set method for unstructured grids, whichhas been designed to allow for easy integration in parallel flow solvers. The means to thisend is a Hermite formulation, where the gradient is stored and advected as an independentvariable, along with the level set function. The method uses narrow-band storage inconjunction with a Fast Marching Method, which is also designed to take advantage of thegradient augmentation. We demonstrate through standard passive advection test cases thatthe proposed method is able to compute interface motion with performance comparableto popular high-performing methods on structured grids.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

Modern fluid simulation of practical problems usually involve complex geometries, variable spatial resolution and parallelcomputing. The introduction of phase interfaces adds to the complexity. Whereas there exists an abundance of methodsand algorithms to track fluid interfaces, much fewer possess the generality that allow them to be implemented with anunstructured, parallel solver.

Volume Of Fluid (VOF) methods and Level Set methods are two of the most popular interface capturing approaches usedon stationary grids. In the original VOF method [1], the volume fraction of fluid in the computational cell is used to identifyand track the surface, ideally conserving mass. However, VOF encounters a common problem: a characteristic of a fluid sur-face or interface is that it represents a discontinuity, which does not readily lend itself to polynomial interpolation schemes.While several variants of shock capturing schemes exist, these introduce significant numerical diffusion when applied todiscontinuities. Thus, in [1], the surface is reconstructed in a geometric fashion, where chunks of fluid are transfered fromone cell to the other. To maintain a sharp interface, additional ad-hoc procedures are introduced after advection. This is arelatively crude process, and it is particularly challenging on unstructured grids. Thus, while conserving mass, VOF has inmany cases difficulties with accurately representing the motion of an interface; see [2] for a modern example.

Level set methods, on the other hand, convert the scalar function, in which the interface is embedded, to a form moresuitable for polynomial interpolation schemes. This is done through the solution of a reinitialization equation – most oftenan eikonal equation – which converts the level set function into a signed distance function. Since the distance to theinterface is not a conserved quantity, level set methods using this approach do not conserve mass exactly. While the signeddistance function represents the interface in a relatively smooth manner, numerical diffusion is still present, and this oftencauses unacceptably high local mass loss or gain, especially in regions of high curvature.

Several strategies exist to counteract this. The conceptually simplest is to increase the resolution of the grid near theinterface, either through adaptive mesh refinement [3], or a finer, separate background grid [4]. Another approach is to

* Corresponding author.E-mail address: [email protected] (A. Bøckmann).

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48 A. Bøckmann, M. Vartdal / Journal of Computational Physics 258 (2014) 47–72

increase the numerical accuracy. The WENO-type high-order, shock-capturing schemes [5] are popular for structured grids.Implementation on unstructured grids is however more complicated [6], and a general drawback of WENO schemes is thatthe numerical stencils are rather large, thus complicating data transfer in parallel settings.

A simpler approach is the use of Lagrangian particles to track the interface. The Particle Level Set method [7] is onesuch method, where particles of finite radius are placed tangent to the interface to correct numerical error in the Euleriansolution. By using Lagrangian interface tracking in conjunction with the level set method, topology changes may be simpli-fied as compared to a purely Lagrangian method. While the original Particle Level Set method used a WENO scheme for thegrid-based advection, it was discovered in [8] that since the accuracy mainly depends on the particle correction, a first ordersemi-Lagrangian scheme for the grid-based quantities is sufficient. Lagrangian augmentation of the interface advection hasmany attractive features. But like all of the discussed methods, it introduces an additional layer of complexity. Moreover,as the interface is stretched or compressed, so are the particles. This necessitates particle seeding or deletion. When seedingnew particles, some kind of interpolant must be used to approximate the interface position from the grid data, and theaccuracy of the interface reconstruction is limited by this interpolant. Due to the relatively limited information containedin each particle (position, radius and sign), and the fact that the corrections are made on a per-particle basis, a relativelydense particle distribution is required. In [7], a default of 64 particles per cell per phase was used in 3D. Particles wereseeded in a band that spanned around three cells from the interface, yielding a large number of particles. In [9], gradientand curvature information were also embedded in Lagrangian particles and used in conjunction with a level set method.The information transfer from particle to grid was however done by a first order Taylor expansion only, and it remainsunresolved how reseeding and topology change should be treated in a robust manner.

If local mass conservation is a pressing issue, the signed distance function may be replaced by a smeared-out approx-imation of the volume fraction [10,11], however, this may decrease the accuracy of the numerical schemes. Global massconservation can always be assured by iterating a scheme that moves the interface in the normal direction [12]. This maybe an acceptable approach if the local mass loss/gain is low, but over longer periods of time leads to unphysical results.

As mentioned above, high-order schemes can reduce the mass loss, but these often require large stencils. Cubic Hermiteinterpolation is one of the exceptions – all data required to construct the interpolant is contained within a single cell.The gradients may be obtained by a transport equation, which is derived by taking the gradient of the scalar advectionequation, or its discrete analogue. While maintaining compact stencils, this also introduces additional effective resolution,at the expense of increased data storage. Transporting the gradient as an independent quantity has been used in CubicInterpolated Pseudo-particle/Constrained Interpolation Profile (CIP) [13,14] methods for some time, however, only recentlyhas this approach spread to level set [15] and VOF methods [16]. In [15], a semi-Lagrangian scheme based on tensor productcubic Hermite splines was used. This required a structured grid, and the issue of reinitialization was not treated. Specificallydesigned for use with the advection scheme proposed in [15], a reinitialization procedure using a Fast Marching Method(FMM) was introduced in [17]. Combined, these two should constitute a good choice for structured grids.

In [18], flow simulations using semi-Lagrangian gradient transport and the tetrahedral cubic Hermite interpolant de-scribed in [19] were presented, though interface advection was not treated. Other interesting approaches to high-ordersemi-Lagrangian advection schemes are found in [20,21]. In [20], a method for structured grids which uses a narrow-bandformulation with arbitrary order Gauss–Lobatto stencils is presented. In [21], a conservative, 2D unstructured scheme isformulated using a combination of point values updated by a semi-Lagrangian scheme, and area average updated by a FiniteVolume scheme. These are only a few examples of the versatility of semi-Lagrangian schemes. An additional advantage isthat they are not limited by the Courant–Friedrichs–Lewy (CFL) condition, as the characteristics may be traced back acrossseveral cells during one time step [22].

In the present work, we extend the work of [15] to general 3D unstructured grids, and include reinitialization of thesigned distance property, which is integrated into the gradient-augmented setting. This is done by a Fast Marching Methodwhich computes both the level set function and its gradient. For the FMM, we apply Huygens’ principle to obtain a localapproximation of the distance function. This approach requires local solutions of a small optimization problem, and differsfrom the Finite Difference (FD) approach used in [17]. Huygens’ principle leads to the same type of update rule as presentedin e.g. [23,24], which are based on a control-theoretic approach. We use a narrow-band method with storage based on atetrahedral subgrid, meaning that general, unstructured primal grids are decomposed into tetrahedra.

To avoid overshoots and generation of artifacts in high-curvature areas, we develop a gradient limiting algorithm whichis demonstrated to be important for the overall accuracy. The proposed method can readily be incorporated into a parallelsetting, as we adhere strictly to compact stencils. While this has been done, the detailed parallel implementation andperformance are not the focus of the present work. It should be noted that in the Finite Element framework, anotherapproach with many of the same attractive features we seek already exists. The Discontinuous Galerkin Method has beenapplied with the level set approach on unstructured grids [25,26] with excellent results.

The rest of the article is organized as follows. A brief description of the level set method, narrow-band and grid structureis given in Section 2. Cubic Hermite interpolants for 1-, 2- and 3-simplices are described in Section 3, followed by anexplanation of the semi-Lagrangian advection scheme for function value and gradient in Section 4 and the reinitializationprocedure for the level set function in Section 5. We test the method on standard passive advection cases in Section 6, anddiscuss the overall method and results in Section 7. To promote readability, much of the discretization details, examples andtheoretical support are left to Appendices.

A. Bøckmann, M. Vartdal / Journal of Computational Physics 258 (2014) 47–72 49

Fig. 1. Grid structure.

2. Level set method and grid structure

The level set method, introduced in [27], treats interface advection in n dimensions by embedding the (n − 1)-dimensional interface in an n-dimensional function Φ(x, t), here referred to as the level set function. The interface is de-scribed by an isosurface of Φ . This dictates that the material derivative of Φ equals zero, or in other terms, that Φ isadvected passively:

∂Φ

∂t+ u · ∇Φ = 0, (1)

where u is the advecting velocity field. Using the zero-isosurface, the set of points Γ describing the interface are given by

Γ (t) = {x ∈ Ω: Φ(x, t) = 0

}, (2)

where Ω is a domain contained in Rn . Since the zero-isosurface of Φ is the main interest, values of Φ away from the zero-

isosurface play a less important role. In practical, discretized applications, Φ can be chosen freely away from the interfaceas long as it has a diminutive or no effect on the zero-isosurface found by interpolation, and consequently as a minimalrequirement has the correct sign. As values of Φ outside the extent of the interpolation schemes used to describe the zero-isosurface do not affect the zero-isosurface, it suffices to advance Eq. (1) in a narrow band across the zero-isosurface [28].This can greatly reduce the computational cost and storage requirements. To prevent numerical diffusion from propagatinglarge irregularities from new members of the band, the linear Fast Marching Method described in Section 5 is applied toassign monotonically increasing values of |Φ| to the new band members, based on values in the existing band members.Instability near the edge of the band was also noted as a potential problem in [29], which used smoothing to prevent it.A cut-plane of prismatic triangular cells containing an interface in a narrow-band structure is shown in Fig. 1(b). The bandis updated after every advection step; new members are added, and existing members are removed to maintain a constantthickness in terms of cells. The storage points in the band are the grid entities edge, face and cell [30,31], along with thenodes from the primal grid. These partition general polyhedra into tetrahedra, where every tetrahedron is made up of anode, edge, face and cell. Edge is the midpoint between two nodes, face is the average position of the nodes describing apolyhedral face, and cell is the average position of all nodes of the polyhedron, see Fig. 1(a). A 2D variant of this structureincluding the narrow-band is seen in Fig. 1(b). We will refer to this tetrahedral grid as the ‘subgrid’.

3. Interpolation

The present work requires cubic Hermite interpolants for 1-, 2- and 3-simplices, i.e., line segments, triangles and tetra-hedra. While a derivation of these are given in Appendix B, we introduce the concepts and methods required to evaluatethe interpolants here. It is advantageous to work with barycentric coordinates, which for tetrahedra are defined by

p = T λ,

p =[

x1

], T =

[x1 x2 x3 x41 1 1 1

], λ =

⎡⎢⎣

λ1λ2λ3λ4

⎤⎥⎦ , x =

[ x1x2x3

], (3)

where xi is the position of vertex i, x is an arbitrary point, and λ are the corresponding barycentric coordinates. Conversely,

λ = T −1 p. (4)


Similar expressions hold mutatis mutandis for the barycentric coordinates of triangles and line segments. Barycentric coor-dinates are often referred to as volume (tetrahedron) or area (triangle) coordinates. For a point x, the physical interpretationof the barycentric coordinate λi pertaining to vertex i is the ratio of the volume of the tetrahedron formed by x and allvertices except i to the volume of the whole tetrahedron, assuming that x is located inside the tetrahedron. Thus, the fourbarycentric coordinates describing any point x sum to one, which is also seen from Eq. (3). While we will derive a cubicinterpolant, the barycentric coordinates are in fact exactly the vertex weights for linear interpolation. A function f (λ) givenin barycentric coordinates can be differentiated in Cartesian coordinates using the chain rule. Using Einstein’s summationconvention,

∂ f (λ)

∂xi= ∂ f (λ)

∂λm

∂λm

∂xi= ∂ f (λ)

∂λmT −1

mi ,

∂ f (λ)

∂xi∂x j= ∂ f (λ)

∂λm∂λnT −1

mi T −1nj . (5)

In the following, the summation convention is only invoked when stated explicitly. For more information on the topics ofthis section, the reader should consult e.g. [32].

3.1. Cubic Hermite interpolation on simplices

We express the cubic Hermite interpolants for simplices in Bernstein–Bézier form (B-form). This simplifies several pro-cedures associated with the interpolant, such as differentiation and geometric interpretation. Moreover, all of the Bernsteinbasis polynomials are linearly independent. The Bernstein basis polynomials and corresponding coefficients employ a multi-index notation, where each index is associated with a vertex of the simplex. The index-combinations that are not givenexplicitly below are found by permutation of the multi-index, which merely corresponds to rearranging the vertex num-bering. For example, for the tetrahedral interpolant, c2001 may be found by swapping the second and fourth index on bothsides of the equation for c2100. For notational convenience, we have also assigned a multi-index to the following vertexdata: position x, function value f and gradient g . These should be interpreted as barycentric coordinates – e.g. x010 is theposition of vertex 2 in a triangle. The 1- 2- and 3-simplex interpolants are given below.

3.1.1. 1-simplex interpolant (line segment)

P (λ) =∑

i+ j=3

ci j B3i j,

B3i j = 3!

i! j!λi1λ

j2,

c30 = f10,

c21 = f10 + 1

3(x01 − x10) · g10. (6)

3.1.2. 2-simplex interpolant (triangle)

P (λ) =∑

i+ j+k=3

ci jk B3i jk,

B3i jk = 3!

i! j!k!λi1λ

j2λ

k3,

c300 = f100,

c210 = f100 + 1

3(x010 − x100) · g100,

c111 = 1

4(c210 + c201 + c120 + c102 + c021 + c012) − 1

6(c300 + c030 + c003). (7)

3.1.3. 3-simplex interpolant (tetrahedron)

P (λ) =∑

i+ j+k+l=3

ci jkl B3i jkl,

B3i jkl = 3!

i! j!k!l!λi1λ

j2λ

k3λ

l4,

c3000 = f1000,


c2100 = f1000 + 1

3(x0100 − x1000) · g1000,

c1110 = 1

4(c2100 + c2010 + c1200 + c1020 + c0210 + c0120) − 1

6(c3000 + c0300 + c0030). (8)

3.2. Gradient limiting

To suppress overshoots, a gradient limiter is imposed on the interpolants given by Eqs. (6)–(8). The importance of gradi-ent limiting is further discussed and demonstrated in Section 6. It changes the gradients locally on a ‘per simplex’ basis, andthus breaks the nodal C1 continuity. The gradient limiter is applied edge-wise to all edges in the simplex. We will use themulti-index notation of the 1-simplex, noting that the edges of 2- and 3-simplices may be identified and reduced to twoindices by the following procedure: For an m-simplex, remove m − 1 of the entries in the multi-index, e.g. j and l in themulti-index i jkl. Then, exactly four of the Bézier-coefficients will have multi-indices 30, 21, 12 and 03. These four determinethe interpolant along an edge of the simplex; if j and l are removed as in the above example, the edge is between vertex 1and 3. The other edges are obtained by all permutations of the two removed indices. For notational convenience, we definethe following differences:

D1 = 3(c21 − c30),

D2 = 3(c03 − c12),

D3 = 3

2(c12 − c30),

D4 = 3

2(c03 − c21),

D5 = c03 − c30. (9)

The gradient limiting procedure is given in Algorithm 1. The Bézier coefficients c21 and c12 affect the end-point gradients,but not the function value. Altering these two associated with one edge does not affect the gradient along any of theother edges, where the associated basis polynomials equal zero. Algorithm 1 may thus be applied sequentially. The Béziercoefficients with three ones in the multi-index should be computed after limiting the gradient, as they are given in termsof potentially gradient-limited Bézier coefficients.

Algorithm 1 Gradient limiting.if (D1 − D3)(D1 − D5) < 0 then

c12 ← 2c21 − c30

else if (D2 − D4)(D2 − D5) < 0 thenc21 ← 2c12 − c03

else if (D1 − D5)(D2 − D5) � 0 thenc21 ← 2

3 c30 + 13 c03

c12 ← 23 c03 + 1

3 c30

end if

While the form of the gradient limiter is partially heuristic, it is based on considerations that arise from the studyof one-dimensional distance functions (see Sections 5 and B.4). When these are discretely sampled and approximated bycubic Hermite interpolation, overshoots are produced if the function is sampled near a gradient discontinuity. In level setmethods, these occur close to the interface for poorly resolved geometrical features, and in order not to produce additionalmass, overshoots should be suppressed. An illustrating example is given in Section 6.2.

4. Semi-Lagrangian transport

A semi-Lagrangian formulation is used to advect the level set scalar and gradient. While a fully Lagrangian methodstores the data in arbitrary points following the velocity field, a semi-Lagrangian method stores the data in stationarycomputational nodes. The advection is carried out by backtracing pathlines. We transport both a scalar function and itsgradient. The tetrahedral cubic Hermite interpolant from the previous section is used to approximate the scalar and gradientfield between nodes. The passive scalar advection equation reads

DΦ

Dt= 0,

D

Dt= ∂

∂t+ u · ∇. (10)

In the above equation, DDt denotes the material derivative, i.e., the time derivative following a material element, and u

denotes the advective velocity field. Taking the gradient of Eq. (10), one obtains the advection equation for the gradient:


DΨ

Dt= −∇u · Ψ ,

Ψ = ∇Φ. (11)

Eqs. (10) and (11) describe time derivatives along the pathlines given by

Dx

Dt= u

(x(t), t

). (12)

At this point, it is useful to anchor the material point x some place in space and time. Following [15], we thus replace x(t)by X(x, t, τ ), where x denotes the point on the pathline X corresponding to time t , and τ is a free time variable providingall points on the pathline. Thus,

dX

dτ= u

(X(x, t, τ ), τ

),

X(x, t, t) = x. (13)

From Eq. (13), one can find the point of departure. That is, the material point xd whose position, when advected by u,coincides with the arbitrarily chosen point x after a time step:

xd = X(x, t + �t, t). (14)

Advancing Eq. (10) one time step �t amounts to evaluating

Φ(x, t + �t) = Φ(xd, t). (15)

The time-update for Φ in a grid node located at xg is thus found by choosing X(xg, t + �t, t + �t) = xg and integratingEq. (13) backwards numerically, from τ = t + �t to τ = t , followed by an evaluation of Eq. (15).

In Eq. (11), there is a source term that changes along the pathline. While Eq. (11) can be solved directly to transport thegradient, one can also take the gradient of the discretized scalar transport equation (15) to obtain a simpler update rule forthe gradient. Noting that the point of departure is a function of x, one can use the chain rule to express the gradient of thescalar function at the next time step in terms of the gradient at the point of departure:

Ψ (x, t + �t) = ∇xd · Ψ (xd, t). (16)

This is referred to in [15] as a superconsistent scheme, providing better coherence between the advection of Ψ and Φ , and ithas the additional advantage that Ψ only needs to be evaluated once per time step, regardless of the chosen time integrationscheme. The term ∇xd however, depends on the time integration scheme.

Like [15], we use a third order Total Variation Diminishing Runge–Kutta scheme (TVD RK3) [33] for the solution ofEq. (13):

x1 = x − �t u(x, t + �t),

x2 = 3

4x + 1

4x1 − 1

4�t u(x1, t),

xd ≈ 1

3x + 2

3x2 − 2

3�t u

(x2, t + 1

2�t

). (17)

Using the TVD RK3 scheme, ∇xd in Eq. (16) is found as

∇x1 = I − �t ∇u(x, t + �t),

∇x2 = 3

4I + 1

4∇x1 − 1

4�t ∇x1 · ∇u(x1, t),

∇xd ≈ 1

3I + 2

3∇x2 − 2

3�t ∇x2 · ∇u

(x2, t + 1

2�t

). (18)

∇xd can be computed in parallel with xd in Eq. (17). Unless the velocity field is given analytically, the velocity and velocitygradient must be interpolated in space and time from the nodal values. The procedure for advancing Φ and Ψ one timestep in a grid node is summarized in Algorithm 2.


Algorithm 2 Advection – updating grid node g from time t to t + �t .x ← xg .Solve Eqs. (17)–(18).Find the tetrahedron T in which xd is located, and number its vertices from 1 to 4.Find the barycentric coordinates λ of xd with respect to T from Eqs. (3)–(4).Evaluate Eqs. (15)–(16). Approximate Φ(xd, t) by the interpolant P (λ) from Eq. (8), subject to Algorithm 1. Approximate ∇Φ(xd, t) by Eq. (5) appliedto P (λ).

5. Reinitialization of the level set function

When advecting an initially smooth level set function, it becomes distorted, and the regions containing the interface maybe smeared out or strongly compressed. To maintain a numerically well-defined interface, the initial smoothness thus needsto be restored from time to time. The process of restoring the initial smoothness is usually referred to as reinitializationin the level set literature. It is of great importance that the reinitialization minimally disturbs the zero-level set, lest theinterface position is altered. The most common approach is to let the level set function away from the interface approximatea signed distance function, i.e., a function whose absolute value is the distance to the closest point on the interface (zero-levelset), and whose sign changes as one crosses the interface. For this purpose, there exist several methods. The most commonis the solution of an eikonal equation, which describes the travel time of an advancing front emanating from the interface:

‖∇Φ‖ = 1,

Φ(xΓ ) = 0, ∀xΓ ∈ Γ, (19)

where Γ is the set of points belonging to the interface. To obtain a single-valued solution of Eq. (19) corresponding tothe first arrival time, a small diffusive term ε∇2Φ can be added to the right hand side [34], which yields the ‘viscositysolution’ [35] as ε approaches zero. To account for the sign change across the interface,

sgn(Φ(x)

) = sgn(Φ0(x)

), (20)

where sgn is the signum function, and Φ0 is the initial level set function from which Γ is found. Assuming a uniformpropagation of unit velocity, the travel time of the advancing front to reach any point in the domain gives the distance tothe closest point on the interface. While the eikonal equation may be treated as a time dependent PDE and solved iterativelyfrom an initial state [36], it may also be solved in a single pass due to the upwind-nature of the front propagation problem.This requires a strict adherence to the upwind causality, which can be enforced through a priority queue of the grid nodesright outside the front. Such a method is called a Fast Marching Method (FMM), as it ‘marches’ the solution outwards fromthe interface. It is similar in nature to Dijkstra’s algorithm [37], but whereas the latter is restricted to a path following thenodal connectivity, an FMM approximates the Euclidean distance.

The remaining part of this section is structured as follows. The methodology of the FMM is given in Section 5.1, the localtravel time approximation is described in Section 5.2, the initialization of the FMM is described in Section 5.3, a discussionof the expected accuracy is given in Section 5.4, and a brief explanation of how the algorithm can be parallelized is givenin Section 5.5. More details on the method and its relation to the eikonal equation are given in Appendix A.

5.1. The Fast Marching Method

As the Fast Marching Method sequentially ‘marches’ the solution outwards from the interface, a book-keeping systemis needed to keep track of which nodes are already processed and which nodes are prospective candidates for update, i.e.,which nodes lie close to the current front position. Grid nodes are classified into three categories:

accepted : Nodes that lie inside the front, and have received their finaltravel time/distance value.

close : Nodes that lie outside, but adjacent to the front.far : Nodes that lie outside the front, and are not in the close set.

In the current setting, ‘adjacent’ means immediate neighbors on the grid. Note that the names of the three sets vary inthe literature. To ensure the data transfer is a strict upwinding process, the close set is sorted according to tentative traveltimes computed from the accepted set. The sorting places the nodes in a priority queue, and at every step of the algorithm,the close node with the smallest tentative value (i.e., the node that is first encountered by the moving front), is removedfrom the close set and added to the accepted set. All of its neighbors that were in the far set are then transfered to the closeset. Tentative values are computed for these, and if some of the neighbors were in the original close set, their values arerecomputed to take into account the newly available information. The process is repeated until all of the nodes are in theaccepted set.

The sorting process associated with the close nodes is a potential bottleneck. But since few changes are made to theorder in a single step of the algorithm, an effective binary tree structure known as a ‘heap’ may be used. For detaileddescriptions of the heap structure and associated operations, the reader is referred to [38]. For a comprehensive review ofFast Marching Methods, see [34].


Fig. 2. Accepted and close nodes.

5.2. Local travel time approximation

In the following discussion, it is important to keep in mind the equivalence between travel time and distance due to theunit propagation velocity, as the two will be used interchangeably.

The level set function is in our method always discretized on a tetrahedral grid. A node in the close set may have either 1,2 or 3 local neighbors in an adjacent tetrahedron that belong to the accepted set. By adjacent, we mean a tetrahedron inwhich this node is a vertex, and we will refer to it as the ‘close node’. The accepted nodes in an adjacent tetrahedron willform either a point, a line segment or a triangle (0-, 1- or 2-simplex respectively), and in the cases of 2 or 3 neighbors,we assume that the accepted solution may be interpolated on the simplex formed by the accepted neighbors. We will referto such a simplex as an ‘accepted simplex’, and its convex hull is denoted by T . An example is given in Fig. 2. Here, the closenode C1 has two accepted 2-simplices; (A1, A2, A3) and (A2, A3, A4). The close node C2 has only one accepted 1-simplex(A3, A4).

We use Huygens’ principle [39] as formulated in [40] to determine the value of the distance function at the close node:

‘. . . the wave front is the envelope of secondary waves whose centers are themselves on a previous wave front.’

With a constant propagation velocity, the secondary wave fronts are spherical. Huygens’ principle has difficulties explain-ing the implied presence of backward-traveling waves, and they are thus neglected. However, our sole interest in the firstarrival time makes the use of Huygens’ principle unproblematic, and the idea of its use with level sets is not new.

The arrival time f at the close node for the spherical front that is emitted from a point x in the accepted simplex, is thetime at which the spherical front is emitted plus the distance from the emitter, as the front propagates with unit velocity.Using an interpolant P to approximate the distance function in T , the arrival time can be estimated as

f (x) ≈ P (x) + D(x), (21)

where D(x) is the Euclidean distance from x to the close node’s position xc .The envelope of such fronts describes a section of the main front emanating from the accepted simplex. To obtain the first

arrival time at the close node of the fronts emanating from T , one must find the point x in T that minimizes f . To considerthis point valid in updating the close node, it must satisfy two criteria:

1. x must lie in T , i.e., all of its barycentric coordinates with respect to the accepted simplex must be nonnegative.2. f (x) must be greater than the distance function in all of the accepted nodes from which it is computed.

The first criterion is to avoid extrapolation into neighboring tetrahedra, where a local interpolant should be used instead.The second criterion ensures that the close node is updated from upwind values only. To obtain the first arrival time at xc

of the fronts emanating from all adjacent accepted simplices, we solve

F = minA,x∈T A

f A(x), (22)

where T A is the convex hull of an adjacent accepted simplex A, and f A includes the appropriate expressions for P and Don A.

Assuming that only one point in any T A admits the minimum value F , we define xe to be this point, i.e., the point thatemits the spherical wavelet that first reaches the close node:

xe = arg minA,x∈T A

f A(x). (23)

In practice, if by chance several points provide F , the implemented code picks one based on the node indexing. Eq. (22)represents a constrained minimization problem. Using an unconstrained minimization algorithm, one must search for localminima in A, and all of its faces. A face is in this respect any simplex of lower dimension that can be formed from a subsetof the vertices in A. If A is a triangle, the search for local minima must thus include the edges of the triangle and the


Fig. 3. Variables and local coordinate system (see Appendix A) used to determine the travel time in a tetrahedron. A spherical wavelet emanates from x,and reaches the close node in D time.

vertices, and if A is a line segment, the search must also include the vertices (end-points). It is important to note that sincef is not necessarily a convex function, the minimization algorithm may fail to converge to the true minimum. A good initialguess is thus important, see Appendix A.

To facilitate the use of cubic Hermite interpolation, we also store the distance function’s gradient. When the point ofemission xe providing F is found, tentative values of Φc and the gradient Ψ c of the level set function in the close node areupdated according to

Φc = F ,

Ψ c = xc − xe

‖xc − xe‖ , (24)

where xc is the position vector of the close node. These are not terminal values unless Φc is the smallest value of Φ in theclose set when all close nodes have been updated in this manner. In that case, the close node is transfered to the acceptedset. The update rule for Ψ in Eq. (24) is a consequence of the properties of the distance function; its gradient has unitmagnitude, and the gradient’s direction is normal to the propagating front. Our approach is illustrated in Fig. 3.

Similar update rules are found in [23,24]. In [41], Fermat’s principle was applied to arrive at the same type of updaterule, and it was shown that with a linear interpolant on Cartesian grids, it reduces to the update rule obtained by FDdiscretization of the eikonal equation. The work of [41] was extended to general unstructured tetrahedral grids in [42]. Thepresent work may be seen as an extension of [42] to cubic Hermite interpolation stencils. Note the equivalence betweenFermat’s principle and Huygens’ principle [43], and that the FMM itself is a consequence of Huygens’ principle [44].

In the present work, we advance both the distance function and its gradient for use in the cubic Hermite interpolant.While improving the accuracy, this necessitates the use of a robust optimization algorithm; f may in fact have severallocal minima in high-curvature regions and near colliding fronts. As both the gradient and Hessian of P and D are easilycomputed, we use Newton’s method with a trust-region approach. We use the algorithm library known as TOMS611 [45]1

as a ‘black-box’ solver, employing the HUMSL subroutine described in [45]. The iterative method needs an initial guess forx which is readily provided by using linear rather than cubic Hermite interpolation on the accepted simplex. Linear interpo-lation provides a unique solution unless the linear gradient has a magnitude larger than unity. Otherwise, the projection ofthe close node down on the accepted simplex is used as an initial guess. See Appendix A for details.

Where two advancing fronts meet, the gradient of the distance function is discontinuous. In high-curvature regionsand thin films, this happens near the interface. Clearly, a polynomial interpolant is not well suited here, and the order ofaccuracy is reduced, as noted in [15]. As demonstrated in Section 6.2, the FMM is especially prone to overshoots in theseareas, and the interpolants given in Eqs. (6)–(7) are thus subject to the gradient limiting of Algorithm 1.

Finally, we note that the requirement of compact stencils has one shortcoming in the FMM context – there might notexist a sufficient number of upwind nodes for a proper 3D reconstruction of the advancing front. The problem is illustratedin 2D in [46], where it is shown that with a linear front, a strictly acute triangulation is required to ensure a sufficientnumber of upwind nodes with compact stencils. Alternatively, a remeshing method is proposed in [46], to ensure acuteangles. In [42], the upwind requirement is instead relaxed, noting that this might be an acceptable approach if the front issufficiently smooth. In the present work, we reconstruct the front from one or two nodes if a proper reconstruction cannotbe done from three accepted nodes. We consider this an acceptable approach, even though O (h) and O (1) errors may belocally introduced in Φ and Ψ respectively, where h is the grid spacing. As the test cases in Section 6 demonstrate, accuratesolutions and good convergence in the L1 error norm are still obtained.

1 Source code available on Netlib: http://www.netlib.org/toms/611.

http://www.netlib.org/toms/611


5.3. Initialization

An initial accepted set is required to start the FMM. Rather than labeling a set of nodes as accepted – as in e.g. [47] –we use edge intersections of the zero-isosurface. The edge intersections replace neighbor nodes in the connectivity graph.As an example, if an edge intersection is found between node 1 and 2, node 2 is replaced by the edge intersection in node1’s neighbor list, and vice versa. The edge intersections constitute the initial accepted set, and carry information on position,gradient (interface normal) and level set value; the latter is obviously always zero. This requires keeping a list of interfaceintersections and corresponding interface normals. In the FMM routine, every time the code attempts to access neighborinformation on a node that is located on the other side of the interface, it should be redirected to the edge intersectionlist. The edge intersections partition the domain into two parts; one positive and one negative, between which informationcannot be exchanged. Hence, to avoid the clutter of locally applying the correct sign in the numerical procedures, thepositive and negative sides are flagged initially, and the initial interface normals on the negative side are reversed. Thepositive and negative sides are then treated in two separate passes using the same ‘positive’ discretization, before the signsof function value and derivatives on the negative side are reverted.

To obtain the edge intersections, the cubic Hermite interpolant of Eq. (6) is used. This results in a cubic expression whosereal roots describe the positions of the edge intersections. An initial gradient is also required. The gradient component alongthe edge is found by differentiating the interpolant, while the edge-orthogonal components are found by linear interpolationto the point of intersection. The two are then added, and the resulting gradient vector is normalized to account for thedistance function property. The gradient limiting of Algorithm 1 is applied to the cubic Hermite interpolant before solutionof the cubic equation. Initialization of an edge between nodes i and j is algorithmically summarized in Algorithm 3.

Algorithm 3 Initialization algorithm.For an edge between nodes i and j, label node i as vertex 1 and node j as vertex 2.Obtain the Bézier coefficients from Eq. (6).Subject the Bézier coefficients to Algorithm 1.a ← 3c21 − c30 − 3c12 + c03

b ← 3c30 + 3c12 − 6c21

c ← −3c30 + 3c21

d ← c30

T ← {t ∈R: 0 � t � 1 ∧ P (t) = at3 + bt2 + ct + d = 0}if T =∅ then

t1 ← inf(T )

t2 ← sup(T )

for k = 1,2 doxint

k ← tkx j + (1 − tk)xi

v1 ← x j−xi

‖x j−xi‖2 P ′(tk)

v2 ← tkΨ j + (1 − tk)Ψ i

v3 ← v1 + v2 − ((x j−xi )·v2)(x j−xi )

‖x j−xi‖2

Ψ intk ← v3‖v3‖

end forReplace the position and gradient in node j as a neighbor of i by xint

1 and Ψ int1 .

Replace the position and gradient in node i as a neighbor of j by xint2 and Ψ int

2 .else

returnend ifreturn

In Algorithm 3, t is λ2, obtained by substituting λ1 = 1 − λ2. t = 0 thus gives the position of node 1, and t = 1 gives theposition of node 2. As can be seen, if several zero-contour crossings are found along an edge, the ones closest to each nodeon either side are used to replace the neighbor connectivity.

5.4. Expected accuracy

Assuming a smooth distance function, a sufficient number of upwind neighbors and the use of the cubic Hermite in-terpolant of Eq. (7), we expect the local approximation of Φ according to Eq. (22) to introduce O (h3) errors due to thetruncation error in Eq. (B.4). This means that the estimated point of emission xe will be offset from the true point of emis-sion by an O (h3) distance, where h is the grid spacing. From this, we can determine the expected accuracy of Ψ , found byEq. (24). As h decreases towards zero, the computed gradient will be rotated an angle of O (h2) as compared to the truegradient. This leaves the gradient with an O (h2) error. The FMM can overall not be more accurate than the initial acceptedset, as errors in this set will be propagated to the entire domain. As the univariate cubic Hermite interpolant is used todetermine the initial zero-level set, the errors introduced in Φ are O (h4). The linear interpolation used to find the edge-orthogonal gradient components introduces an O (h2) error in Ψ . We thus expect, under optimal conditions, convergencerates of third order in Φ , and second order in Ψ in a band of constant thickness with respect to the number of cells around


the interface. This may however be reduced by non-smooth geometry and grids with highly obtuse angles, as noted inSection 5.2. If the FMM is advanced in the whole domain, a reduction of one in the order of convergence can be expected.This is because the number of cells away from the interface increases by h−1, which is multiplied by the local discretizationerror to take error accumulation into account.

5.5. Parallelization

While the FMM is in essence a serial algorithm, it is clear that different parts of the front may be advanced in parallel,as two parts of the front located some distance apart are not mutually dependent until they eventually merge. Spatiallypartitioning the advancement of the FMM into several processes, the processes may run in parallel, while the FMM isexecuted serially on each process. Upwind conflicts may thus arise. When process A receives a value of Φ from anotherprocess B at a shared node, this value must be compared to the largest accepted value existing on process A. If the receivedvalue is smaller, some of the accepted values on process A may not have been computed based on the correct causality,and the solution must be stepwise reverted on process A until the value received from B is larger than the largest acceptedvalue on A. A parallel method based on the approach of [48] is implemented with the present discretization. While we willnot go into further detail on parallel implementation and performance, we note that the only amendment required to themethod of [48] is the transfer of gradients at shared nodes.

6. Numerical tests

In this section we test the method’s performance. As error indicators in test cases where the primary interest is inter-face motion, we use volume and shape preservation. In cases where the aim is to assess the accuracy everywhere in thecomputational domain we use the L1 and L∞ error norms of Φ and Ψ .

The total positive volume is approximated by applying the Heaviside function to a piecewise linear reconstruction of Φ

in each tetrahedron. The Hermite interpolant is first used to assign values of Φ to a grid that is roughly four times finer,applying the partitioning method described in Section 2 recursively. The volume of the positive phase is then found by

V+ =∫Ω

H(Φ(x)

)dx, (25)

where the integral is approximated by a linear, geometric reconstruction of the interface. As a measure of the average errorin interface position, we use the error measure suggested in [49]:

EΓ = 1

A

∫Ω

∣∣H(Φex(x)

) − H(Φcomp(x)

)∣∣dx, (26)

where Φex denotes the exact solution and Φcomp denotes the computed solution. The integral in Eq. (26) over one tetra-hedron is approximated by |V ex+ − V comp

+ |, where V+ is computed as in Eq. (25), but the integration is constrained to thetetrahedron. This simplification under-predicts the error when the computed and exact interface intersect. Sensitivity studieshave shown that this effect is insignificant.

We define the error in Φ and Ψ in node i as

eiΦ = ∣∣Φex(xi) − Φcomp(xi)

∣∣,eiΨ = ∥∥Ψ ex(xi) − Ψ comp(xi)

∥∥. (27)

The L1 and L∞ error norms are evaluated according to

L1 = 1

N

∑i

ei,

L∞ = maxi

ei, (28)

where N is the number of nodes under consideration.In the following tests, we use the exact analytic expressions for the velocity and velocity derivatives to solve the transport

equations for the level set scalar and gradient.

6.1. Deformation field test – convergence

To assess the global convergence rate of the advection scheme, we use the velocity field suggested in [50,51] with thesmooth scalar field used in [15]. As the scalar field is smooth, there should be no need for gradient limiting. However, as thesmooth function is distorted by the velocity field, the discretized field might not be perceived as smooth everywhere, and


Table 1Results from the deformation field test, with gradient limiting.

h L1 error in Φ p L1 error in Ψ p L∞ error in Φ p L∞ error in Ψ p

1/16 2.09 × 10−4 – 6.05 × 10−3 – 1.36 × 10−3 – 1.06 × 10−1 –1/32 2.87 × 10−5 2.9 1.06 × 10−3 2.5 2.05 × 10−4 2.7 2.79 × 10−2 1.91/64 3.84 × 10−6 2.9 1.57 × 10−4 2.8 3.50 × 10−5 2.6 1.28 × 10−2 1.11/128 4.92 × 10−7 3.0 2.35 × 10−5 2.7 5.81 × 10−6 2.6 9.05 × 10−4 3.8

Table 2Results from the deformation field test, without gradient limiting.


1/16 1.76 × 10−4 – 4.01 × 10−3 – 9.49 × 10−4 – 1.62 × 10−2 –1/32 2.64 × 10−5 2.7 7.38 × 10−4 2.4 1.61 × 10−4 2.6 4.54 × 10−3 1.81/64 3.62 × 10−6 2.9 1.13 × 10−4 2.7 2.31 × 10−5 2.8 9.85 × 10−4 2.21/128 4.67 × 10−7 3.0 1.62 × 10−5 2.8 3.06 × 10−6 2.9 2.97 × 10−4 1.7

Fig. 4. Equally spaced isocontours of Φ at t = 1.

gradient limiting might be triggered by Algorithm 1. We include cases both with and without application of Algorithm 1,to check its effect on the accuracy and convergence rate. The velocity field is 2D, and given by the stream function

ψ(x, y) = 1

πsin2(πx) sin2(π y) cos

(πt

T

), (29)

with T = 2. The resulting velocity field is constrained to a [0,1] × [0,1] box, with vanishing velocity on the boundaries. Theadvected scalar should return to its initial condition at t = 2, which is where we measure the error. The initial scalar fieldis given by

Φ(x, y) = exp(−(

(x − x0)2 + (y − y0)

2)) − exp(−r2

0

),

x0 = 0.5, y0 = 0.75, r0 = 0.15. (30)

Fig. 4 shows the scalar field in its most distorted state. We use a 3D discretization for a 2D problem, and the domainis thus given unit length over a single cell in the z-direction, with periodic boundary conditions. A constant time step of�t = 0.002 is used with Cartesian primal grids of spacings h = 1/16, h = 1/32, h = 1/64 and h = 1/128. The resultingtetrahedral subgrid is twice as fine. The results with and without gradient limiting are summarized in Tables 1 and 2respectively, where p denotes the estimated order of convergence. Without gradient limiting, the order of convergence isaround three for Φ and two for Ψ in the L∞ error norm. This is as expected – the triangular and tetrahedral cubic Hermiteinterpolants’ accuracy are limited by the approximation of the ‘bubble functions’ by Eqs. (B.4)–(B.7). While Φ shows thesame convergence in the L1 error norm, Ψ converges slightly faster than expected. While we have not found an obviousreason, this might be caused by local error cancellation effects as the velocity field is reversed. With the gradient limitingalgorithm applied, the L1 error norms converge at approximately the same rates as without gradient limiting. But the L∞error norm of Ψ shows a less predictable behavior, as expected.


Fig. 5. Top row: Interface at t = 0 (black) and t = 8 (red). Bottom row: Interface at t = 4. Left to right: Case 1, 2 and 3. (For interpretation of the referencesto color in this figure, the reader is referred to the web version of this article.)

6.2. Deformation field test – behavior with under-resolved interface

To test how the proposed method behaves when the interface geometry is insufficiently resolved, the deformation fieldtest case was is run with T = 8. This stretches the interface much more, and using a 64 × 64 subgrid, the results are directlycomparable to [15]. Three tests are run: one without reinitialization or gradient limiting (Case 1), one with reinitializationevery time step and gradient limiting (Case 2), and one with reinitialization every time step, but without gradient limiting(Case 3). The results are visualized2 in Fig. 5. With a 64 × 64 subgrid, it is clear that the thinnest parts of the spiral are toothin to be properly resolved on the grid. If the FMM cannot find an interface crossing along an edge, the nodes of this edgewill have the level set function overwritten by characteristics coming other parts of the interface. Thus, subgrid featuresare filtered out with the FMM, while they are retained (although they may be too small to produce interface intersections)without the FMM. When the velocity field is reversed, subgrid features which do not contribute to the interface in terms ofgrid intersections may grow back to a size where they contribute to the interface. With an FMM, they are permanently lost.

As can be seen from Fig. 5, the case without reinitialization gives the best results. Compared to [15], the results aresimilar. When reinitialization is applied, the results are more similar to those produced by the fifth order WENO schemeimplemented in [15], which includes reinitialization. Here, the mass loss is substantial, which is a characteristic of non-conservative level set methods when one phase is stretched into features that are too small to be resolved on the grid.A physical example of this is atomization, for which conservative schemes should be used [11]. Finally, reinitializationwithout gradient limiting produces an excessive amount of mass, giving practically useless results. The effect is especiallypronounced in thin regions or near high interface curvature (see also Section 6.5). The deformation field test case thusprovides a proper stress test of the gradient limiting algorithm.

6.3. Computing the signed distance function of a spherical interface

To assess the accuracy of the proposed FMM, we use it to compute the signed distance and its gradient field from aspherical interface at different grid resolutions. This problem has a very simple analytical solution:

Φ = R − ‖x0 − x‖,Ψ = x0 − x

‖x0 − x‖ , (31)

where R is the radius of the sphere and x0 is its center point. We choose R = 0.15 and x0 = (0.5,0.5,0.5) in a [0,1] ×[0,1] × [0,1] domain with a Cartesian primary grid. Three sets of convergence tests are run:

2 The apparent jaggedness is an artifact from the linear interpolation used in the visualization software.


Table 3Convergence results from set 1.


1/8 8.70 × 10−5 – 2.49 × 10−3 – 3.01 × 10−4 – 9.21 × 10−3 –1/16 1.01 × 10−5 3.1 4.47 × 10−4 2.5 6.87 × 10−5 2.1 8.29 × 10−3 0.21/32 1.37 × 10−6 2.9 1.04 × 10−4 2.1 3.29 × 10−5 1.0 1.03 × 10−2 −0.31/64 4.22 × 10−7 1.7 2.80 × 10−5 1.9 1.62 × 10−5 1.0 1.11 × 10−2 −0.11/128 1.26 × 10−7 1.7 8.00 × 10−6 1.8 7.99 × 10−6 1.0 1.07 × 10−2 0.1



1/8 9.79 × 10−3 – 4.48 × 10−2 – 1.72 × 10−2 – 4.63 × 10−1 –1/16 5.02 × 10−3 1.0 2.58 × 10−2 0.8 9.22 × 10−3 0.9 4.72 × 10−1 0.01/32 2.50 × 10−3 1.0 1.37 × 10−2 0.9 5.62 × 10−3 0.7 4.76 × 10−1 0.01/64 1.26 × 10−3 1.0 7.12 × 10−3 0.9 3.48 × 10−3 0.7 4.79 × 10−1 0.01/128 6.30 × 10−4 1.0 3.65 × 10−3 1.0 2.11 × 10−3 0.7 4.82 × 10−1 0.0



1/8 1.13 × 10−5 – 6.95 × 10−4 – 5.11 × 10−4 – 3.33 × 10−2 –1/16 8.61 × 10−7 3.7 1.46 × 10−4 2.3 4.57 × 10−5 3.5 8.60 × 10−3 2.01/32 9.90 × 10−8 3.1 3.96 × 10−5 1.9 2.28 × 10−5 1.0 8.60 × 10−3 0.01/64 1.23 × 10−8 3.0 1.07 × 10−5 1.9 1.14 × 10−5 1.0 8.60 × 10−3 0.01/128 1.56 × 10−9 3.0 2.81 × 10−6 1.9 5.71 × 10−6 1.0 8.60 × 10−3 0.0

Table 6Convergence results from the FMM initialization procedure.


1/8 4.87 × 10−5 – 6.82 × 10−3 – 1.80 × 10−4 – 2.48 × 10−2 –1/16 2.34 × 10−6 4.4 2.17 × 10−3 1.7 1.24 × 10−5 3.9 6.20 × 10−3 2.01/32 1.67 × 10−7 3.8 5.31 × 10−4 2.0 9.67 × 10−7 3.7 2.19 × 10−3 1.51/64 9.43 × 10−9 4.1 1.22 × 10−4 2.1 6.23 × 10−8 4.0 5.62 × 10−4 2.01/128 5.72 × 10−10 4.0 2.95 × 10−5 2.0 3.97 × 10−9 4.0 1.42 × 10−4 2.0

Set 1To initialize the FMM, the analytical solution is prescribed at every grid point. This only serves to obtain the initial

accepted set, which is found by Algorithm 3. The update procedure proceeds as described in Section 5. Cubic Hermiteinterpolation is used to approximate the solution on accepted simplices.

Set 2As set 1, but with linear interpolation approximating the solution on accepted simplices.

Set 3The whole FMM procedure is discarded to isolate the effect of the local travel time approximation. Cubic Hermite inter-

polation is used to approximate the solution on accepted simplices, which have vertex values set to the analytical solution.Thus, no initialization is required.

While we expect third order convergence in Φ in the L1 error norm in the third set of tests, we expect slower con-vergence in the first set of tests. This is because error accumulates as the front propagates outwards. To reach a point adistance L from the initial interface, the front must propagate across approximately L/h cells, where h is the grid spacing.As the grid is refined, more steps must be taken to reach the same point. Results from the three sets of tests are given inTables 3–5, and the errors produced by the initialization algorithm are given in Table 6. p denotes the estimated order ofconvergence. In the error norms for Ψ , the center point, where Ψ is undefined, is not included.

The slow convergence in the L∞ norm of Φ may be explained by the analytical distance function. For the grid nodesclosest to the center of the sphere, the third derivatives of the distance function increase as h−2, where h is the gridspacing. In the Taylor expansion of Eq. (B.4), the leading order error term is multiplied by the third derivative, yieldingfirst order convergence of Φ in the nodes closest to the center of the sphere. The L∞ norm of Ψ does not converge, asexpected since Ψ converges one order slower than Φ . The major difference between the approximately constant L∞ norms


Fig. 6. Isosurfaces of Φ with the 8 × 8 × 8 primal grid, �Φ = 0.1. The zero-isosurface is shaded darker. Results with linear interpolation (left), and cubicHermite interpolation (right).

Fig. 7. Volume and interface error development for repeated reinitialization.

in Ψ between the linear and cubic Hermite interpolants should be noted. For the case with neighbors set to the analyticalsolution, the L1 norm of Φ shows third order convergence in set 3, as expected from the accuracy of the triangular cubicHermite interpolant. The effect of error accumulation is clearly visible in Table 4. While the linear interpolant introducesO (h2) errors in Φ locally, the L1 error norm shows first order convergence, due to the accumulation of errors. This effect isstrong with the linear interpolant, as the distance from the interface is consistently under-predicted on the concave side, andover-predicted on the convex side. This is not the case with the triangular cubic Hermite interpolant, and error accumulationdoes not strongly influence the convergence rate in the L1 norm on the coarser grids. A visual comparison between theresults produced by linear and cubic Hermite interpolation is given in Fig. 6, where the above-mentioned differences maybe seen. It should also be noted that contrary to locally planar fronts, where an appropriate number of upwind neighborscan always be found given the absence of obtuse grid angles, this is not necessarily the case with a locally curved front.O (h) errors in Φ may thus be introduced locally, and propagated into other parts of the domain. Due to these two effects,it is difficult to predict the convergence in Φ , and we rely on empirical investigation. The convergence of Ψ appears to bemuch less prone to error accumulation than Φ . The initialization of the FMM converges at the expected rate.

In the tests above, the narrow-band was extended to the whole domain. This has been done because the proposed FMMis applicable as a standalone solver, and may be applied to other problems than interface advection where the distance fieldaway from the interface may be of interest. In the context of interface advection, the main interest is to reinitialize thenodes closest to the interface, and slower convergence due to error accumulation is thus not an issue.

6.4. Repeated reinitialization

As reinitialization involves a resampling of the interface location, it is clear that it may introduce additional numericaldiffusion. To better see the effect of frequent reinitialization, we apply the Fast Marching Method repeatedly to a stationaryinterface, and measure the volume and interface error as a function of the number of applications. The interface has theshape of an octant of a sphere of radius 0.3; a length which is discretized by approximately 15 subgrid spacings. The primalgrid is unstructured tetrahedral. We sequentially apply 100 reinitializations and extract the interface error and ratio ofcurrent to initial volume. The resulting development is shown in Fig. 7. The initial rapid growth in error is caused by edgesmoothing. After 100 iterations, around 1% of the analytical volume is lost, and the interface position is off by an average ofroughly 0.2% of the radius. The initial and final shapes are shown in Fig. 8.


Fig. 8. The initial shape (left) after 100 reinitializations (right).

Table 7Parameters in the Zalesak’s disk test.

Domain [0,100] × [0,100] (×[0,1])Disk center (50,75)

Disk radius 15Notch with 5Top of notch y-coordinate 85u(x, y) −Ω(y − 50)

v(x, y) Ω(x − 50)

Ω π600

Table 8Results from Zalesak’s disk after one revolution.

Gradient limiter on/off Reinit. on/off Resolution Area loss [%] EΓ EΓ order

on off 25 × 25 −11.30 7.38 × 10−1 –50 × 50 −0.32 8.86 × 10−2 3.1

100× 100 0.13 1.58 × 10−2 2.5

on 25× 25 0.41 3.10 × 10−1 –50 × 50 0.07 6.84 × 10−2 2.2

100× 100 0.10 2.27 × 10−2 1.6

off off 100× 100 0.23 2.63 × 10−2 –on 100× 100 0.04 3.47 × 10−2 –

6.5. Zalesak’s disk

Zalesak’s disk [52] is a widely used 2D advection test, testing the ability of an advection scheme to preserve the shapeand area of a notched disk in a velocity field describing solid body rotation. The shape contains both smooth curves andconvex/concave corners, as well as two parallel straight lines of relatively narrow spacing on coarse grids. As our proposedmethod is 3D in general, we use one cell of unit length in the z-direction. The geometry of the notched disk is as givenin [52], and summarized in Table 7. We use a unit time step �t for all spatial resolutions, and the angular frequency Ω isset such that one full revolution takes place every 1200 steps.

We run tests with three different resolutions and reinitialization every time step on/off. Additionally, we test the effectof the gradient limiter by turning it off, and running the highest resolution with and without reinitialization. To avoidcontamination from the fringes of the interface band, the band spans around nine primal grid cells on both sides of theinterface in the cases without reinitialization. With reinitialization every time step, the band spans around two cells oneither side of the interface-containing cells. The primal grid structure is unstructured prismatic triangular, and the subgridon which the level set solution is computed is, as always, tetrahedral and roughly twice as fine. The given grid sizes referto the total number of primal grid spacings along x- and y-aligned boundaries, and the resolution is similar in the interiorof the domain. It should be noted that as opposed to a Cartesian primal grid, the prismatic triangular grid may producetetrahedra with obtuse angles. After one revolution, we extract EΓ , which is a measure of the average error in interfaceposition, and the change in area (volume). Both are found as described in the beginning of this section. The results aregiven in Table 8. Convergence without and with reinitialization is shown in Figs. 9(a) and 9(b), where the shapes withdifferent resolutions, after one revolution, are superimposed. Fig. 9(c) shows the effect of turning off the gradient limiter,and Fig. 9(d) shows the exact solution plotted over the primal grid.

As noted in [7,53], numerical diffusion in the sharply convex and concave parts approximately cancel out in terms ofarea preservation, so conserved area is not necessarily a good indicator of the quality of the solution.

As a ‘fair’ comparison with other works, we compare our resolution on the primal grid to twice the resolution in otherworks, noting that the tetrahedral subgrid on which the level set method is discretized is roughly twice as fine as the primalgrid. Note that additional effective resolution is introduced by the storage of gradient vectors. Compared to the results of [7],


Fig. 9. Results from Zalesak’s disk. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

the fifth order WENO scheme is outperformed, most notably on the coarsest grid, where the whole disk has vanished afterone revolution. Our results are more similar to those of the Particle Level Set method where, when comparing the caseswith reinitialization, we have smaller or equal interface errors, and better area preservation. The Zalesak’s disk test case wasalso used in [54], on a Cartesian grid with a third order WENO method for advection. Here, reinitialization was done by anFMM, which was initialized by a constrained minimization method for nodes near the interface. While EΓ was not used asan error estimate, visualization of the disk’s shape indicates accuracy similar to that of the current method. In [54], arealosses are stated to be 0.652% and 0.222% for 100 × 100 and 200 × 200 grids respectively – higher than the losses obtainedwith the current method on the 50 × 50 and 100 × 100 grids.

While Table 8 lists the accuracy, it provides little insight into the effects of the reinitialization and gradient limiter.A visual representation of the solution is therefore an important complement. Figs. 9(a) and 9(b) reveal that reinitializationserves to counteract numerical diffusion near the notch at the coarsest resolution, while it slightly increases numericaldiffusion in corners at the finest resolution. It also appears to provide more symmetry about the notch for the lowerresolutions. It can be seen that smoothing near corners appear to constitute most of the interface error. Here, the gradientis discontinuous and the accuracy typically drops to first order (see Section 6.3), which explains the slower convergenceon finer grids. Fig. 9(c) shows the importance of gradient limiting, especially with reinitialization, where the absence ofthe gradient limiter results in spurious wriggles near corners. In terms of interface accuracy, both cases without gradientlimiting are less accurate than their gradient limited counterparts.

6.6. Enright test

With the analytical velocity field proposed in [51], the ‘Enright test’ [7], stretches a sphere into a thin film and thenback again. This has become a very popular test with 3D level set methods. Depending on the spatial resolution, thestretching may or may not produce topology change, and areas of high curvature arise. We use tetrahedral primal grids,


Table 9Parameters in the Enright test.

Domain [0,1] × [0,1] × [0,1]Sphere center (0.35,0.35,0.35)

Sphere radius 0.15u(x, y, z, t) 2 sin2(πx) sin(2π y) sin(2π z) cos( πt

3 )

v(x, y, z, t) − sin(2πx) sin2(π y) sin(2π z) cos( πt3 )

w(x, y, z, t) − sin(2πx) sin(2π y) sin2(π z) cos( πt3 )

Table 10Results from the Enright test.

Resolution Volume loss [%] EΓ EΓ order

25 × 25 × 25 24.68 1.47 × 10−2 –50 × 50 × 50 0.73 2.29 × 10−3 2.7

Fig. 10. The Enright test on an approximately 50 × 50 × 50 primal grid. From left to right: t = 0.0, t = 0.6, t = 1.5, t = 2.4, t = 3.0.

with approximate resolutions 25 × 25 × 25 and 50 × 50 × 50. The geometrical setup and velocity field are given in Table 9.The simulation is stopped at t = 3, at which the initial sphere should ideally be recovered. We use constant time steps�t = 6 × 10−4, so that the total simulation is resolved by 5000 steps. Reinitialization is applied every 200 steps, which ismore realistic for practical use than every time step, as in the Zalesak’s disk test case. The bandwidth is approximately 4cells on either side of the interface. Results are given in Table 10.

At the end of the simulation, the sphere has almost reattained its original shape, except a spurious ridge on one side,and a dent on the other. Snapshots from the time evolution is shown in Fig. 10. A very similar end configuration whichincludes the spurious ridge is shown in [55], where the result has been obtained by a particle based method. The interfaceexperiences the most stretching at the midpoint of the simulation. With both grid resolutions, the thin interface is poorlyresolved. Parts of it are thus filtered out, resulting in volume loss. At 0.73% on the 50 × 50 × 50 primal grid, the volume lossis less than that obtained both with the fifth order WENO scheme and the Particle Level Set method on a 100 × 100 × 100grid [7].

7. Conclusive summary and future work

In the present work, we have described a method for gradient augmented level set advection and reinitialization on un-structured grids. As demonstrated through the test cases in Section 6, the proposed method provides high performance onunstructured grids, both in terms of interface accuracy and volume preservation. Applied to the advection of smooth func-tions such as in Section 6.1, the proposed gradient limiter has little effect on the accuracy and convergence. When appliedto geometrical shapes with sharp or under-resolved features, as in Sections 6.2 and 6.5, it suppresses the appearance ofspurious interface artifacts. The proposed Fast Marching Method has been demonstrated to work as intended; it introducesvery little numerical diffusion, and in some cases, it counteracts numerical diffusion. While the method should not be usedwith low-quality grids, modern meshing software can generate high-quality unstructured grids that are adequate for usewith the proposed method. Note that the method is likely to perform better on a ‘native’ tetrahedral grid than the presentsubgrid construction, as the connectivity graph is more regular. This can be seen in Fig. 1(b), where the majority of nodesin the (2D) primal grid have six neighbors, whereas nodes with only four neighbors are evenly distributed in the subgrid.The number of neighbor nodes are important in the FMM, as discussed in Section 5.2. From the reported results, we judgethe method suitable for practical flow simulations that involve a high degree of interface deformation. In extreme caseshowever, for instance in flows that are completely dominated by small bubbles or thin films, volume loss might become anissue. While the obvious solution is to increase the grid resolution, this is not always computationally feasible. A possiblesolution that has the potential to almost eliminate numerical diffusion and mass loss is the introduction of Lagrangian parti-cles. These may track both interface location and gradient vector [9], and upon inclusion in the numerical stencils, one suchparticle can determine the four Bézier coefficients which are otherwise approximated by Eq. (B.4). Preliminary result havebeen promising, and suggest that such an approach relies much less on a dense particle distribution than e.g. the ParticleLevel Set method [7]. An ongoing study of the present method applied to flow simulations has shown highly encouragingresults.


Acknowledgements

The current work is based on a tentative implementation of a parallel semi-Lagrangian narrow-band level set methodin the CFD code CDP [30]. The implementation was based on linear interpolation, both in the advection and Fast MarchingMethod used for reinitialization. The authors would like to express their gratitude to Associate Professor Mikael Mortensenat the University of Oslo for a critical review of the article manuscript.

The current work is sponsored by The Norwegian Research Council grant NFR 1912 04/V30 ‘Wave-current-body interac-tion’ and the DNV Education Fund.

Appendix A. Discretization by Huygens’ principle on a tetrahedron

A.1. Relation to the eikonal equation

Eq. (22) satisfies the eikonal equation commonly used to reinitialize the level set function:

∇Φ · ∇Φ = 1. (A.1)

To see this, we note that ∇Φ does not change along its own direction unless colliding fronts are present. This can beseen by taking the gradient of Eq. (A.1), which yields

∇∇Φ · ∇Φ = 0. (A.2)

As in Section 5.2, we define T to be the convex hull of the accepted simplex:

T ={

x ∈R3: x =

n+1∑k=1

akxk ∧n+1∑k=1

ak = 1 ∧ ak � 0

}, (A.3)

where xk denotes a vertex coordinate, and n ∈ {0,1,2}. In the following, we assume n = 2, i.e., that the accepted simplex isa triangle. Assuming that Eq. (A.1) is satisfied in the whole tetrahedron, and that the gradient is continuous, the gradientthrough xc can be backtraced to find a point of intersection x in T that has the same gradient. For this purpose, we introducea local Cartesian coordinate system with origin and basis vectors e1 and e2 in the same plane as T . Approximating thein-plane components of the gradient in T by the gradient of some interpolant P , the component along e3 must be scaledsuch that the gradient has unit magnitude to satisfy Eq. (A.1). Since the close node is further from the interface thanthe accepted nodes, the gradient must point in the close node’s general direction, i.e., it must have a positive componentalong e3. The gradient g(x) may thus be approximated by

g(x) ≈ ∂ P

∂x1e1 + ∂ P

∂x2e2 +

√1 −

(∂ P

∂x1

)2

−(

∂ P

∂x2

)2

e3, (A.4)

assuming that the magnitude of the in-plane gradient of P is equal to or less than one. A possible numerical solution of theeikonal equation at the close node may be found by looking for a point x in T that has a gradient vector given by Eq. (A.4),whose extension from x passes through the close node xc . At such a point, an exact Taylor expansion of the distance functionfrom this point to the close node amounts to adding the distance D = ‖xc − x‖, as the gradient is of unit magnitude, andunchanging along its own direction. If f (xmin) = P (xmin) + D(xmin) is a local minimum of f (see Eq. (21)) in the interiorof T , and has a gradient g(xmin) given by Eq. (A.4) which is parallel to xc − xmin, the solution to Eq. (22) complies withthe eikonal equation. This is the case, and it can be seen by taking the dot product of the gradient g(xmin) and the vectorxc − xmin. At xmin, we have

∂ P

∂x j= ∂ D

∂x j= xi

j − xminj

D, j = 1,2, (A.5)

where xi is the orthogonal projection of xc onto the plane of T . The distance vector between xmin and the close node is

d = xc − xmin = (xi

1 − xmin1

)e1 + (

xi2 − xmin

2

)e2 + D

√1 −

(xi

1 − xmin1

D

)2

−(

xi2 − xmin

2

D

)2

e3. (A.6)

Substituting Eq. (A.5) into Eq. (A.4) and taking the dot product with Eq. (A.6) yields

g(xmin) · d = D. (A.7)

Since ‖g‖ = 1 and ‖d‖ = D , g(xmin) and xc − xmin must be parallel. The same analysis applies on the edges of T . Here,the gradient must be assumed to lie in the plane formed by xc and two of the accepted nodes. We previously mentionedthat the gradient is unchanging along its own direction unless colliding fronts are present. If colliding fronts are present,


the minimization over T in Eq. (22) ensures that the front that emanates from xmin = xe is not blocked by other frontsemanating from T . This, of course, requires that the minimization procedure converges to the true minimum. The FMM andminimization over adjacent accepted simplices A ensure that the front emanating from xe is not blocked by fronts comingfrom outside of T , and thus, the gradient is continuous on the line segment between xc and xe , and the above assumptionsare valid.

A.2. Update procedure and discretization

We will describe the update procedure assuming three accepted neighbors; the procedure is analogous for two neighbors,and for one neighbor, there is no need to search for a minimum of f , as there is only one eligible point. To solve Eq. (22)on a tetrahedron, we use the local Cartesian coordinate system with origin and basis vectors e1 and e2 in the plane of T .In the local coordinate system, we define the following points: xc is the close node, xi is the orthogonal projection of xc

onto the plane of T , and x is some point in the plane of T , at which we hope to find a local minimum of f (x). We alsodefine D = ‖xc − x‖ and h = ‖xc − xi‖. P (x) is taken as the gradient limited version of Eq. (7). The setup described above isillustrated in Fig. 3. To apply Newton’s method to find the minimum of f (x), we need the first and second derivatives of Pand D . For P , these are found from Eq. (5). For D , they are given by

D =√(

xi1 − x1

)2 + (xi

2 − x2)2 + h2,

∂ D

∂x1= − xi

1 − x1

D,

∂ D

∂x2= − xi

2 − x2

D,

∂2 D

∂x21

= 1

D− (xi

1 − x1)2

D3,

∂2 D

∂x22

= 1

D− (xi

2 − x2)2

D3,

∂2 D

∂x1∂x2= − (xi

1 − x1)(xi2 − x2)

D3. (A.8)

Newton’s method requires an initial guess x0. For this, we discard the gradient information at the accepted nodes, anduse a linear interpolant on the nodal function values. The gradient of this interpolant is constant everywhere, and there willthus be exactly one point where the eikonal-complying gradient, whose component along e3 is scaled such that the gradienthas unit magnitude, extends through the update node, see Section A.1. Having obtained the eikonal-complying gradient, thegradient vector is backtraced from the update node, and the point of intersection with the accepted simplex is taken as thestarting point. Under certain circumstances, the linear in-plane gradient may have a magnitude greater than one, and doeshence not comply with the eikonal equation. If that happens, xi is instead chosen as an initial guess. The linear in-planegradient may be found by differentiating the linear interpolant P lin constructed by barycentric coordinates. Using Einstein’ssummation convention,

P lin(λ) = λ jΦ j,

∂ P lin

∂xi= T −1

ji Φ j, (A.9)

where the matrix T is found from the triangle variant of Eq. (3)

T =[

x1 x2 x31 1 1

], (A.10)

x1, x2 and x3 are the vertex positions in the 2D coordinate system formed by e1 and e2, and j is a summation index overthe vertices. The third gradient component is positive along e3, such that the linear gradient vector g lin has unit magnitude:

g lin = T −1j1 Φ je1 + T −1

j2 Φ je2 +√

1 − (T −1

j1 Φ j)2 − (

T −1j2 Φ j

)2e3, (A.11)

where Einstein’s summation convention again has been invoked. Backtracing glin from the close node to the point of inter-section with the accepted triangle yields

x0 = xi − h

glin

(glin

1 e1 + glin2 e2

). (A.12)

3


While providing an initial guess, the above equations may also be used as a standalone local, linear update rule, by updatingΦ in the close node according to

Φ(xc) ≈ P lin(x0) + ∥∥xc − x0

∥∥. (A.13)

As mentioned above, if the square root in Eq. (A.11) is complex, x0 is replaced by xi . When this happens, however, it isan indication that the FMM has locally failed to approximate the distance function satisfactorily. When using the linearapproximation above to assign values to new band members (Section 2), a complex square root is taken as a failure, andthe minimum search is continued on the edges and vertices.

It is interesting to note that the linear approach given by Eqs. (A.9)–(A.13) is equivalent to the first order Finite Differenceapproach given in [46]. In [46], a linear variation in Φ along the edges from the vertices of the accepted triangle to the closenode is assumed. These provide linearly independent directions, and thus determine a constant gradient in the wholetetrahedron. Combined with the requirement of a unit magnitude gradient, the problem has two solutions; one where thegradient has a positive component along e3 and one with a negative component, the former being the meaningful traveltime estimate. These are the solutions of the quadratic equation presented in [46]. The same assumptions are made inthe above equations, and empirical investigation has confirmed that the two approaches produce identical results down tomachine precision. As an alternative to Eq. (A.13), it is equally acceptable to find the gradient by Eq. (A.11) and approximateΦ(xc) by a Taylor expansion from one of the accepted nodes, which produces exactly the same result.

A.3. Optimization algorithm

Newton’s method for optimization is an iterative method based on a quadratic Taylor expansion around at the currentiterate point, and finding the minimum of the Taylor-expansion. This yields a linear system of equations:

Hiju j = −gi,

Hij = ∂2 f

∂xi∂x j

(xn),

gi = ∂ f

∂xi

(xn),

u = xn+1 − xn, (A.14)

where n is the current iterate, and the summation convention is invoked. Eq. (A.14) alone may of course diverge, or convergeto a local maximum or saddle point. The most common methods applied to ensure convergence to a local minimum arecalled line-search methods and trust region methods. TOMS611 applies the latter. Trust region methods define a ‘trustedregion’, within which the objective function may adequately be approximated by a quadratic function. The region is re-determined in every iteration, based on the ratios of the objective function to the Taylor expansion. Minimization is thenconstrained to the trusted region.

Appendix B. A Hermite interpolant for tetrahedral data

To construct a tetrahedral cubic Hermite interpolant, we follow a similar procedure as with a tensor product interpolant– i.e., we build bi- and trivariate interpolants from the univariate case.

B.1. The univariate Hermite interpolant

We express the well-known cubic Hermite interpolant in Bernstein–Bézier form (B-form), which is convenient to workwith when the number of variables increases. The univariate nth degree B-form in barycentric coordinates reads

P (λ) =∑

i+ j=n

ci j Bni j,

Bnij = n!

i! j!λi1λ

j2. (B.1)

There is only one independent variable here, as λ1 +λ2 = 1. If 0 � λ1, λ2, the interpolant is constrained to a line segment,or a 1-simplex. The cubic Hermite interpolant, which is also the basis in the tensor product approach, is obtained by

n = 3,

c30 = f10,

c21 = f10 + 1(x01 − x10) · g10, (B.2)

3


where f10, g10 and x10 are the function value, gradient and Cartesian position vector at λ = (λ1, λ2) = (1,0), i.e., the vertexdata and coordinates of vertex 1. The remaining two coefficients – c03 and c12 – are found by permuting the entries of themulti-index.

B.2. Extension to bivariate Hermite interpolation

The bivariate B-form in barycentric coordinates reads

P (λ) =∑

i+ j+k=n

ci jk Bni jk,

Bnijk = n!

i! j!k!λi1λ

j2λ

k3, (B.3)

which expresses a cubic function if n = 3. If 0 � λ1, λ2, λ3, the interpolant is constrained to a triangle, or a 2-simplex.For λm = 0, we require that the univariate polynomial along the edge that excludes vertex m be recovered. For example,if λ3 = 0, then Bn

ijk = Bnij if k = 0, otherwise, Bn

ijk = 0. This applies similarly if λ2 = 0, which should recover the univariatepolynomial along edge 1–3. We can use this directly to determine all of the coefficients but c111, simply by adding a thirdzero-index in Eq. (6), and use index permutation to obtain the other coefficients.

To find c111, we follow [56], where the function at the barycenter of the triangle is approximated by a Taylor expansion:

f i jk = 1

3( f i + f j + fk) − 1

6

(g i · (xi − xi jk) + g j · (x j − xi jk) + gk · (xk − xi jk)

) + O(h3),

xi jk = 1

3(xi + x j + xk), (B.4)

where h is a characteristic length of the triangle. The vertex data in the above equation may instead be expressed in termsof the Bézier coefficients:

f i jk ≈ 1

6(c210 + c201 + c120 + c102 + c021 + c012). (B.5)

Equating Eq. (B.3) evaluated in λ = ( 13 , 1

3 , 13 ), and with n = 3 to f i jk yields

1

27

∑i+ j+k=3

6

i! j!k! ci jk = f i jk. (B.6)

Approximating f i jk by Eq. (B.5) and solving for c111 yields

c111 = 1

4(c210 + c201 + c120 + c102 + c021 + c012) − 1

6(c300 + c030 + c003). (B.7)

It should be noted that not all of the coefficients in the tensor product approach presented in [15] are readily obtainedfrom stored vertex data, and finite differences are used in a similar way to approximate the missing vertex data. In thepresent work, the requirement of compact stencils makes the order of accuracy drop by one as compared to the workof [15].

B.3. Extension to trivariate Hermite interpolation

The trivariate B-form in barycentric coordinates reads

P (λ) =∑

i+ j+k+l=n

ci jkl Bni jkl,

Bnijkl = n!

i! j!k!l!λi1λ

j2λ

k3λ

l4, (B.8)

which expresses a cubic function if n = 3. If 0 � λ1, λ2, λ3, λ4, the interpolant is constrained to a tetrahedron, or a 3-simplex.For λm = 0, we require that the bivariate polynomial along the face that excludes vertex m be recovered. We can use thisdirectly to determine all of the coefficients simply by adding a fourth zero-index in Eq. (7), and use index permutation toobtain the other coefficients. The resulting interpolant is equivalent to those given in [19,56].


B.4. Gradient limiting

A distance function will have discontinuous first derivatives at points that do not have a unique closest point on theinterface. Representing the distance function in terms of propagating fronts (as in the FMM) these points are where two ormore fronts collide, and are manifested by ‘kinks’ in the distance function. In areas of high curvature and thin films, thesekinks appear near the interface. The distance function across a kink is poorly described by cubic Hermite interpolation,and the resulting overshoots tend to increase the size of small-scale features as reconstructed by the interpolant. Whilewe can accept the degraded accuracy near kinks, we cannot accept the overshoots. Implementation of the method with aflow solver has revealed that overshoots may cause small-scale features to grow over time, which leads to results that areobviously unphysical. Instead, we prefer small scale features to be preserved, or filtered out. To accomplish this, we limitthe gradients of the distance function locally. The procedure is inspired by a common formulation of cubic Hermite splinesin Computer-Aided Geometric Design (CAGD). It represents a concavity/convexity constraint.

First, we translate the univariate cubic Hermite interpolant to a graph in the xy plane. While this may be done byplotting y(λ1, λ2) = P (λ1, λ2) against x(λ1, λ2) = λ1x1 + λ2x2, it may also be done in a more geometric fashion by replacingthe coefficients ci j by points ai j in the xy plane. This is done by setting

ai j =(

i

nx1 + j

nx2, ci j

),

x(λ1, λ2) =∑

i+ j=n

ai jn!

i! j!λi1λ

j2. (B.9)

By the Binomial theorem, the x-components of the first approach are recovered. The Binomial theorem can also be used toshow that∑

i+ j=n

n!i! j!λ

i1λ

j2 = 1. (B.10)

Since λ1 and λ2 are always positive between x1 and x2, any point on the graph between x1 and x2 lies within the convexhull of the points ai j . The x-components of ai j are evenly spaced along the line segment between x1 and x2, with spacing|x2−x1|

3 . The y-components ci j have a very useful geometric interpretation. For the cubic Hermite polynomial, we have

c30 = f (x1),

c03 = f (x2),

c21 − c3013 (x2 − x1)

= df

dx(x1),

c03 − c1213 (x2 − x1)

= df

dx(x2). (B.11)

The piecewise linear graph through a30, a21, a12 and a03 is the control polygon of the cubic Hermite interpolant expressedas a Bézier curve – a polygon describing the general shape of the curve. The points are referred to as control points. FromEq. (B.11), it can be seen that this polygon has end-point function values and first derivatives coinciding with the curve.A graphical explanation of why the cubic Hermite interpolant often over-predicts the size of small-scale features can beseen in Fig. 11(a). Here, the function and gradient of a 1D distance function are sampled near a small-scale feature – say,a 1D bubble. Due to the small size, the distance function exhibits a kink near the interface. To reproduce the sampledgradient, a21 must lie outside the actual distance function’s graph. The convex hull of the points ai j thus encompasses anarea outside the distance function’s graph – as does the interpolant. As can be seen, this happens if the distance functionand gradient are sampled closer than 1

3 h from a kink, where h is the grid spacing. In Fig. 11(a), this gives the interpolant azero-contour outside the original distance function, and the size of the feature is thus overestimated.

One possible way to remedy the overshoot is to decrease the y-coordinate of a21 so that it lies on the extension of theline segment between a12 and a03: c21 = 2c12 − c03. Then, a21 lies on the distance function’s graph, see Fig. 11(b). Thisrepresents one end of the bounds for c21 to preserve convexity/concavity; the other is c21 = 1

2 (c30 + c12), which we donot use. To identify problematic end-point gradients that require limiting, we look for intersections between the extensionof one end-segment of the control polygon with the other. Such an intersection is identified by looking at differences inslopes between the control points, represented by D1 to D5 in Eq. (9). We will now demonstrate the mechanisms of thegradient limiter by a simple experiment, which is illustrated in Fig. 12. Starting with a convex control polygon, Fig. 12(a), andgradually increasing the left end-point gradient, the gradient limiter will start limiting the right end-point gradient whenthe control polygon is no longer convex to retain convexity, Fig. 12(b). Continuing increasing the left end-point gradient,both end-point gradients of the gradient limited control polygon will eventually equal the linear gradient between theend-points. Further increasing the left end-point gradient will result in an S-shaped control polygon. At this point, an edge


Fig. 11. Elimination of overshoot by gradient limiting.

Fig. 12. Mechanisms of the gradient limiter.


may potentially have three interface intersections, which is a level of subgrid detail beyond what may physically be resolvedin a satisfactory manner. We thus limit both end-point gradients to the linear gradient, Fig. 12(c) which is taken care ofin the last two statements of Algorithm 1. Algorithm 1 is designed for the univariate cubic Hermite interpolant. But asthe multivariate versions reduce to the univariate interpolant along edges, Algorithm 1 may be used edge-wise on themultivariate interpolants. This will ensure convexity/concavity along all edges.

References

[1] C.W. Hirt, B.D. Nichols, Volume of fluid (vof) method for the dynamics of free boundaries, J. Comput. Phys. 39 (1) (1981) 201–225.[2] M. Huang, B. Chen, L. Wu, A slic–vof method based on unstructured grid, Microgravity Sci. Technol. 22 (3) (2010) 305–314.[3] X. Zheng, J. Lowengrub, A. Anderson, V. Cristini, Adaptive unstructured volume remeshing – ii: Application to two- and three-dimensional level-set

simulations of multiphase flow, J. Comput. Phys. 208 (2) (2005) 626–650.[4] M. Herrmann, Refined level set grid method for tracking interfaces, in: Annual Research Briefs, Center for Turbulence Research, NASA Ames/Stanford

University, 2005, pp. 3–18.[5] X. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1) (1994) 200–212.[6] M. Dumbser, M. Käser, Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput.

Phys. 221 (2) (2007) 693–723.[7] D. Enright, R. Fedkiw, J. Ferziger, I. Mitchell, A hybrid particle level set method for improved interface capturing, J. Comput. Phys. 183 (1) (2002)

83–116.[8] D. Enright, F. Losasso, R. Fedkiw, A fast and accurate semi-Lagrangian particle level set method, Comput. Struct. 83 (6) (2005) 479–490.[9] S. Ianniello, A. Di Mascio, A self-adaptive oriented particles Level-Set method for tracking interfaces, J. Comput. Phys. 229 (4) (2010) 1353–1380.

[10] E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J. Comput. Phys. 210 (1) (2005) 225–246.[11] O. Desjardins, V. Moureau, H. Pitsch, An accurate conservative level set/ghost fluid method for simulating turbulent atomization, J. Comput. Phys.

227 (18) (2008) 8395–8416.[12] Y. Chang, T. Hou, B. Merriman, S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys.

124 (2) (1996) 449–464.[13] H. Takewaki, A. Nishiguchi, T. Yabe, Cubic interpolated pseudo-particle method (cip) for solving hyperbolic-type equations, J. Comput. Phys. 61 (2)

(1985) 261–268.[14] T. Yabe, F. Xiao, T. Utsumi, The constrained interpolation profile method for multiphase analysis, J. Comput. Phys. 169 (2) (2001) 556–593.[15] J. Nave, R. Rosales, B. Seibold, A gradient-augmented level set method with an optimally local, coherent advection scheme, J. Comput. Phys. 229 (10)

(2010) 3802–3827.[16] M. Raessi, J. Mostaghimi, M. Bussmann, Advecting normal vectors: A new method for calculating interface normals and curvatures when modeling

two-phase flows, J. Comput. Phys. 226 (1) (2007) 774–797.[17] D. Salac, The augmented fast marching method for level set reinitialization, Bulletin of the American Physical Society 56.[18] N. Tanaka, T. Yamasaki, T. Taguchi, Accurate and robust fluid analysis using cubic interpolation with volume/area coordinates (civa) method on unstruc-

tured grids, JSME Int. J. Ser. B 47 (4) (2004) 672–680.[19] N. Tanaka, Development of a highly accurate interpolation method for mesh-free flow simulations i. Integration of gridless, particle and cip methods,

Int. J. Numer. Methods Fluids 30 (8) (1999) 957–976.[20] O. Desjardins, H. Pitsch, A spectrally refined interface approach for simulating multiphase flows, J. Comput. Phys. 228 (5) (2009) 1658–1677.[21] S. Ii, M. Shimuta, F. Xiao, A 4th-order and single-cell-based advection scheme on unstructured grids using multi-moments, Comput. Phys. Commun.

173 (1–2) (2005) 17–33.[22] M. Lentine, J.T. Grétarsson, R. Fedkiw, An unconditionally stable fully conservative semi-Lagrangian method, J. Comput. Phys. 230 (8) (2011) 2857–2879.[23] J. Tsitsiklis, Efficient algorithms for globally optimal trajectories, IEEE Trans. Autom. Control 40 (9) (1995) 1528–1538.[24] J. Sethian, A. Vladimirsky, Ordered upwind methods for static Hamilton–Jacobi equations: Theory and algorithms, SIAM J. Numer. Anal. 41 (1) (2003)

325–363.[25] J. Grooss, J. Hesthaven, A level set discontinuous Galerkin method for free surface flows, Comput. Methods Appl. Mech. Eng. 195 (25) (2006) 3406–3429.[26] E. Marchandise, J. Remacle, N. Chevaugeon, A quadrature-free discontinuous Galerkin method for the level set equation, J. Comput. Phys. 212 (1) (2006)

338–357.[27] S. Osher, J.A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations, J. Comput. Phys. 79

(1988) 12–49.[28] D. Adalsteinsson, J. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys. 118 (2) (1995) 269–277.[29] P. Gomez, J. Hernandez, J. Lopez, On the reinitialization procedure in a narrow-band locally refined level set method for interfacial flows, Int. J. Numer.

Methods Eng. 63 (10) (2005) 1478–1512.[30] F. Ham, K. Mattsson, G. Iaccarino, Accurate and stable finite volume operators for unstructured flow solvers, in: Annual Research Briefs, Center for

Turbulence Research, NASA Ames/Stanford University, 2006, pp. 243–261.[31] F. Ham, K. Mattsson, G. Iaccarino, P. Moin, Towards time-stable and accurate les on unstructured grids, in: Complex Effects in Large Eddy Simulations,

2007, pp. 235–249.[32] M. Lai, L. Schumaker, Spline Functions on Triangulations, vol. 110, Cambridge University Press, 2007.[33] C. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77 (2) (1988) 439–471.[34] J. Sethian, Fast marching methods, SIAM Rev. (1999) 199–235.[35] M. Crandall, P. Lions, Viscosity solutions of Hamilton–Jacobi equations, Trans. Am. Math. Soc. (1983) 1–42.[36] M. Sussman, P. Smereka, S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys. 114 (1) (1994)

146–159.[37] E. Dijkstra, A note on two problems in connexion with graphs, Numer. Math. 1 (1) (1959) 269–271.[38] R. Segdewick, K. Wayne, Algorithms, 4th edition, Addison-Wesley Professional, 2011.[39] C. Huygens, Traité de la lumière [1690], Project Gutenberg, 2005.[40] H. Piaggio, The mathematical theory of Huygens’ principle, Nature 145 (1940) 531–532.[41] S. Fomel, A variational formulation of the fast marching eikonal solver, in: SEP-95: Stanford Exploration Project, 1997, pp. 127–147.[42] P.G. Lelièvre, C.G. Farquharson, C.A. Hurich, Computing first-arrival seismic traveltimes on unstructured 3-d tetrahedral grids using the fast marching

method, Geophys. J. Int. 184 (2) (2011) 885–896.[43] A. de Witte, Equivalence of Huygens’ principle and Fermat’s principle in ray geometry, Am. J. Phys. 27 (1959) 293–301.[44] J. Sethian, Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. Comput. Phys. 169 (2) (2001)

503–555.

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib4869727431393831323031s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6875616E6732303130736C6963s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7A68656E67323030356164617074697665s1


http://refhub.elsevier.com/S0021-9991(13)00700-6/bib686572726D616E6E32303035726566696E6564s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib686572726D616E6E32303035726566696E6564s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6C6975313939347765696768746564s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib64756D6273657232303037617262697472617279s1


http://refhub.elsevier.com/S0021-9991(13)00700-6/bib656E726967687432303032687962726964s1



http://refhub.elsevier.com/S0021-9991(13)00700-6/bib69616E6E69656C6C6F3230313073656C66s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib4F6C73736F6E32303035323235s1



http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6368616E67313939366C6576656Cs1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6368616E67313939366C6576656Cs1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib74616B6577616B69313938356375626963s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib74616B6577616B69313938356375626963s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7961626532303031636F6E73747261696E6564s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6E617665323031306772616469656E74s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6E617665323031306772616469656E74s1



http://refhub.elsevier.com/S0021-9991(13)00700-6/bib74616E616B61323030346163637572617465s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib74616E616B61323030346163637572617465s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7431393939646576656C6F706D656E74s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7431393939646576656C6F706D656E74s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6465736A617264696E7332303039737065637472616C6C79s1



http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6C656E74696E6532303131756E636F6E646974696F6E616C6C79s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7473697473696B6C697331393935656666696369656E74s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7365746869616E323030336F726465726564s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7365746869616E323030336F726465726564s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib67726F6F7373323030366C6576656Cs1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6D61726368616E646973653230303671756164726174757265s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6D61726368616E646973653230303671756164726174757265s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib4F736865723139383866726F6E7473s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib4F736865723139383866726F6E7473s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6164616C737465696E73736F6E3139393566617374s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib676F6D657A323030357265696E697469616C697A6174696F6Es1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib676F6D657A323030357265696E697469616C697A6174696F6Es1



http://refhub.elsevier.com/S0021-9991(13)00700-6/bib68616D32303037746F7761726473s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib68616D32303037746F7761726473s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6C61693230303773706C696E65s1



http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6372616E64616C6C31393833766973636F73697479s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib737573736D616E313939346C6576656Cs1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib737573736D616E313939346C6576656Cs1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib64696A6B73747261313935396E6F7465s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib73656764657769636B32303131616C676F726974686D73s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7069616767696F313934306D617468656D61746963616Cs1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib666F6D656C31393937766172696174696F6E616Cs1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6C656C696576726532303131636F6D707574696E67s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6C656C696576726532303131636F6D707574696E67s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib64657769747465313935396571756976616C656E6365s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7365746869616E3230303165766F6C7574696F6Es1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7365746869616E3230303165766F6C7574696F6Es1


[45] D.M. Gay, Algorithm 611: Subroutines for unconstrained minimization using a model/trust-region approach, ACM Trans. Math. Softw. 9 (4) (1983)503–524.

[46] J. Sethian, A. Vladimirsky, Fast methods for the eikonal and related Hamilton–Jacobi equations on unstructured meshes, Proc. Natl. Acad. Sci. 97 (11)(2000) 5699.

[47] D. Chopp, Some improvements of the fast marching method, SIAM J. Sci. Comput. 23 (1) (2001) 230–244.[48] M. Herrmann, A domain decomposition parallelization of the fast marching method, in: Annual Research Briefs, Center for Turbulence Research, NASA

Ames/Stanford University, 2003, pp. 213–225.[49] M. Sussman, E. Fatemi, An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow,

SIAM J. Sci. Comput. 20 (4) (1999) 1165–1191.[50] J. Bell, P. Colella, H. Glaz, A second-order projection method for the incompressible Navier–Stokes equations, J. Comput. Phys. 85 (2) (1989) 257–283.[51] R. LeVeque, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal. (1996) 627–665.[52] S. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys. 31 (3) (1979) 335–362.[53] M. Herrmann, A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids, J. Comput. Phys. 227 (4) (2008)

2674–2706.[54] P.A. Gremaud, C.M. Kuster, Z. Li, A study of numerical methods for the level set approach, Appl. Numer. Math. 57 (5–7) (2007) 837–846.[55] S. Hieber, P. Petros Koumoutsakos, A Lagrangian particle level set method, J. Comput. Phys. 210 (1) (2005) 342–367.[56] P. Ciarlet, The Finite Element Method for Elliptic Problems, vol. 4, North Holland, 1978.

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib67617931393833616C676F726974686Ds1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib67617931393833616C676F726974686Ds1



http://refhub.elsevier.com/S0021-9991(13)00700-6/bib63686F707032303031736F6D65s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib686572726D616E6E32303033646F6D61696Es1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib686572726D616E6E32303033646F6D61696Es1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib737573736D616E31393939656666696369656E74s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib737573736D616E31393939656666696369656E74s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib62656C6C313938397365636F6E64s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib6C6576657175653139393668696768s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib7A616C6573616B3139373966756C6C79s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib686572726D616E6E3230303862616C616E636564s1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib686572726D616E6E3230303862616C616E636564s1


http://refhub.elsevier.com/S0021-9991(13)00700-6/bib486965626572323030354C616772616E6769616Es1

http://refhub.elsevier.com/S0021-9991(13)00700-6/bib636961726C65743139373866696E697465s1

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