Redistancing the Augmented Level Set Method

Transcription

1 ILASS-Americas 23rd Annual Conference on Liquid Atomization and Spray Systems, Ventura, CA, May 2011 Redistancing the Augmented Level Set Method L. Anumolu and M.F. Trujillo Department of Mechanical Engineering University of Wisconsin Madison, Madison, WI Abstract Re-initialization of the level set function, through the solution of a Hamilton-Jacobi equation, has been found to lead to erroneous displacements of the zero level set iso-surface or interface. In the present work, we adopt an augmented-gradient methodology for solving this Hamilton-Jacobi equation, which updates both the level set function and its gradient. Even with this relatively accurate method, artificial displacements of the interface occur, leading to mass errors. Objects having coarse grid resolution are particularly vulnerable to this type of error. To address this issue, a hybrid scheme is proposed, where the underlying Hermite interpolation polynomials in nodes neighboring the interface are used to anchor the interface; thereby preventing erroneous movements. Away from this region the gradient-augmented solution for the re-distancing is employed. The results for 1D and 2D test cases show significant improvement with this hybrid treatment. Moreover, the introduction of a re-distancing scheme also improves the interface capturing behavior of the augmented level set method presented by Nave et al. [JCP 229:2010]. Corresponding Author: trujillo@engr.wisc.edu

2 Introduction The level set approach [1] has gained considerable popularity as a means of implicitly capturing the kinematics of a gas-liquid interface undergoing topological changes [2, 3]. Under this method, the level set function, ϕ, is generally defined as a signed distance function and the following transport equation is solved ϕ + u ϕ = 0, (1) t where, u, is the underlying flow field. We will interpret the ϕ = 0 iso-surface as denoting the material interface. A common error with the level set approach is the loss of mass that occurs due to erroneous displacements of the zero level set iso-surface, or the production of structures that are well below numerical resolution. One avenue that has been taken to avoid this issue is to combine the Volume-of- Fluid [4] and level set methods in the work of Sussman and Fatemi [5]. Other options include the use of Lagrangian particles within the level set method as reported by Enright et al. [6] and Wang et al. [7], or the use of a similar hybrid front tracking and level set method suggested by Nourgaliev et al. [8]. Improvements to the numerical accuracy for solving the ϕ transport equation, as in the gradient-augmented level set presented by Nave et al. [9], also contribute to the reduction of mass errors. Regardless of the method utilized, at some point the level set function will show significant distortion from its initial signed distance function characteristic, i.e ϕ will deviate substantially from one. In such cases small deviations in the level set function may cause large changes in the interface location, making the interface identification particularly susceptible to errors. It is therefore useful to recover the signed distance condition by replacing the deformed level set function with a new function without disturbing the zero level set [10]. This process is called reinitialization. This can be done by solving the Eikonal equation, ϕ = 1 using a fast marching method [11] or by recovering the property ϕ = 1 through the solution of the following Hamilton-Jacobi equation [2]. ϕ τ = sign(ϕo )(1 ϕ ), (2) where τ is the pseudo time for reinitialization and ϕ o is the initial deformed level set function. It can be shown that the analytical solution of this equation does not perturb the zero level iso-surface (i.e. ϕ o (x) = 0) as the re-initialization proceeds. In contrast, its numerical solution does resulting in artificial mass loss or gain. Since for practical problems, the only solution method is numerical, a need exists for preventing these artificial displacements. With this objective in mind, Sussman and Fatemi [12] proposed a scheme that anchors the interface by imposing local mass constraint. Russo and Smereka [13] suggested a simple modification by making use of truly upwind information for solving the reinitialization PDE (Eq. 2). Min [14] extended Russo s truly upwind method to higher order numerical accuracy by improving the calculation of gradients. In the present work, we present a hybrid treatment for re-initializing the level set function, which is based on a direct solution of the distance function for nodes neighboring the interface and an implementation of a gradient-augmented Hamilton-Jacobi method for nodes outside this region. This gradientaugmented approach was first introduced by van Leer [15] and has appeared in the solution of level set problems in the work of Nave et al. [9]. Here we apply this approach to the solution of the Hamilton- Jacobi equation. In the following section we describe our reinitialization scheme in this gradientaugmented framework and illustrate the artificial displacement of the interface that results when Eq. (2) is solved in the entire domain. This is followed by the presentation of our hybrid treatment, which is able to successfully address these artificial displacements. Various examples are used to illustrate the behavior. Gradient Augmented Level set Reinitialization In the work of Nave et al. [9], the characteristic form of the Courant-Isaacson-Rees (CIR) [16] method was used to solve for the level set function and its gradient, ϕ = ψ in a coupled fashion, without involving a re-initialization phase. This combined solution resulted in noticeable improvement in accuracy when compared to methods such as the 5th order WENO scheme [17]. Furthermore, the method is local and numerically stable. In our solution of the redistancing PDE, Eq. (2), we will employ a similar gradient-augmented approach. First, we take the gradient of Eq. (2) to yield an evolution equation for ϕ = ψ, namely ϕ τ +S(ϕo ) ϕ ϕ ψ = S(ϕo ) S(ϕ o ) ψ, (3) and we rewrite the reinitialization PDE (Eq. 2) as ϕ τ + S(ϕo ) ϕ ϕ ϕ = S(ϕo ). (4) 2

3 Putting both of these expressions in characteristic form, we have dϕ dτ = S(ϕo ), (5a) and dψ dτ = S(ϕo ) S(ϕ o ) ψ, (5b) along dx c dτ = S(ϕo ) ψ ψ, (5c) where S(ϕ o ) = sign(ϕ o ). We now proceed to solve these equations using the CIR methodology [16]. As an illustration of the procedure, consider a two-dimensional domain, where each node is denoted by indices (i, j). The field is described by ϕ n i,j and ψn i,j corresponding to time level τ n. Beginning with the initial field, we solve the characteristic equation Eq. (5c) backwards in time, such that at time level τ n + τ, each node (i, j) has been intercepted by its corresponding characteristic, i.e. x i,j = x c (τ n + τ). This is done using the following 3rd order Runge-Kutta method (RK) j 1 x (j) c = α jk x (k) c k=0 β jk τs(ϕ o ) ψ(k) ψ (k) ; j = 1, 2, 3, (6) where x c (τ n + τ) = x (0) c = x i,j and x c (τ n ) = x (3). The RK coefficients are represented by α jk and β jk. Here the origin of the characteristic is denoted by x c (τ n ) (= x f ). This location generally lies somewhere inside a given computational cell. In the following step, we integrate Eqs. (5a) and (5b) for ϕ and ψ, respectively from τ n to τ n + τ using the RK method. This is accomplish by solving the following equations This is obtained through fourth order Hermite interpolation, namely P (x c ) = 1 1 α=0 β=0 ϕi+α,j+β f α+1 (η x )f β+1 (η y ) β=2 ϕi+α,j+β 2 f α+1 (η x )f β+1 (η y ) y + 3 α=0 y 1 α=2 β=0 ϕi+α 2,j+β x + 3 α=2 f α+1 (η x )f β+1 (η y ) x 3 β=2 ϕi+α 2,j+β 2 xy f α+1 (η x )f β+1 (η y ) x y, (9) where P (x c ) represents either ϕ or ψ. When P (x c ) represents ψ then basis polynomials in Eq. (9) have to be differentiated with respect to x and y, and η x = x c x i x, η y = y c y j. y In the case of P (x c (τ n )) = ϕ(x c (τ n )), ϕ x, ϕ y and ϕ xy are the partial derivatives of ϕ, x and y are the respective cell spacing in the x and y directions, and basis polynomials are given by f β (η) = 1 3η 2 + 2η 3 if β = 1 3η 2 2η 3 if β = 2 η 2η 2 + η 3 if β = 3 η 2 + η 3 if β = 4 (10) The node values for ϕ in Eq. (9) are indicated by a superscript and their respective positions in a computation cell are illustrated in Fig. 1. j 1 ϕ (j) = α jk ϕ (k) + β jk τs (ϕ o ) ; j = 1, 2, 3. (7) k=0 and j 1 ψ (j) = α jk ψ (k) + β jk τ( S (ϕ o ) k=0 S (ϕ o ) ψ (k) ); j = 1, 2, 3. (8) resulting in the automatic update of the nodal values ϕ n+1 i,j and ψ n+1 i,j. This step requires that we determine the corresponding values for ϕ and ψ at x c (τ n ). Figure 1 Solution of Eqs. (6), (7) and (8) represents a relatively high accurate approach for re-initializing the level set function and its gradient. The number of pseudo-time steps employed is directly related to the extent of the re-initialization domain with respect to the interface. This is by virtue of the characteristics which propagate sign distance function behavior at velocities given by S(ϕ o ) ψ from the interface. We ψ 3

4 illustrate its performance with selected 1D and 2D test cases. The first test case is a 1D example taken from Czajkowski and Desjardins [18]. The initial field is given by function ϕ o = 0.05 sinh( 18x 5 ) and we computed re-initialization with 23 grid points spanning the entire domain. The results after re-initialization are shown globally in Fig. 2a and locally in Fig. 2b. These are contrasted to the exact solution. It is clear that re-initialization has effectively produced ϕ = 1 behavior everywhere. Nevertheless, as a result of this process the interface has erroneously shifted. This error is not surprising. As explained in the introduction, it has previously been discussed in the work of Sussman and Fatemi [12], Russo and Smereka [13] and Min [14]. atomization calculations, where the resulting small droplets are generally not sufficiently resolved due to computational restrictions. As suggested in the results of Fig. 3, it is this small droplet population that is more prone to drastic mass errors stemming from re-initialization using only the Hamilton-Jacobi equation. (a) (b) (a) Grid: 8x8. (b) Grid: 50x50. Figure 3: Reinitialization of a circular disk employing (a) 4 nodes and (b) 25 nodes across the disk diameter (Initial-Red, After reinitialization-using 200 time steps τ = x 2 Blue). The results clearly indicated that structures with low level of grid resolution are particularly susceptible to the interface drift error study in the present work. Figure 2: (a) Farfield and (b) nearfield view of the signed distance function generated using the gradient augmented scheme. A second case test consist of the re-initialization of the level set function beginning with the following 2D field ϕ o = (x x o ) 2 + (y y o ) 2 r. (11) where (x o, y o ) is the center and r is the radius of the disc. This level set field already satisfies the ϕ = 1 condition. This test case was implemented in the work of Sussman and Fatemi. [12], where their objective was to develop a reinitialization scheme to control the erroneous drift of the interface. In our calculations shown in Fig. 3, we employ two levels of grid resolution for the re-initialization process. The coarse and fine staggered mesh resolution corresponds to 4 (8x8 domain) and 25 (50x50 domain) grid cells spanning the diameter of the disk, respectively. In both cases artificial displacement of the interface occurs. However, this is much more noticeable for the coarser case; hence, objects with insufficient level of resolution are more susceptible to this type of error. This is particularly relevant in spray Hybrid Treatment From the test cases illustrated in the previous section, it is clear that the gradient augmented redistancing scheme Eqs. (5a), (5b) and (5c) fails to prevent erroneous interface displacement. In fact, any numerical solution of the Hamilton-Jacobi equation tends to incur this type of error as elaborated in [13, 12]. The underlying cause will be elucidated by analysis in a future publication. For the moment, we restrict the current work towards an immediate solution to this problem. Our approach is to directly solve for the shortest distance between all nodes neighboring the interface and the interface itself using the already existing Hermite interpolation polynomial basis. Once this has been done, the calculated distance is assigned to the corresponding level set function for each interface-neighboring nodes. This essentially anchors the level set field in a band around the interface. Subsequently, the Hamilton-Jacobi equation is solved for a prescribed time for the remaining nodes, where this prescribed time can be used as a way of controlling the extent of the re-initialized domain. While in theory the direct method applied to the interface-neighboring nodes can be equally applied to a larger number of nodes, in practice it is computationally more efficient 4

5 to employ the Hamilton-Jacobi method. Therefore, only the neighboring nodes, which are responsibe for the erroneous shift in the interface, are re-initialized with the direct method. The two methods combined provide a solution to the artificial interface motion, while minimizing computational expense. The first step in anchoring the neighboring nodes is their identification. This is done by solving the characteristic equation (5c) backwards in time to determine the origin (x f ) of each characteristic, which after τ intercepts its corresponding node. As an illustration, consider the cell depicted in Fig. 1, where the characteristic is shown originating from some point x f in the direction S (ϕ o ) ψ Since ψ. ϕ (x i,j+1 ) ϕ (x f ) < 0, node (i, j + 1) is a neighboring node. We now employ the Hermite interpolation basis along this characteristic line to solve for the shortest distance to the interface. Note that since this characteristic propagates normal to the interface, it already provides the direction of this shortest path. For the 2D geometry shown in Fig. 1, the Hermite basis polynomials are given by where a α (η x ) = and b α (η y ) = 1 3ηx 2 + 2ηx 3 if α = 1 3ηx 2 2ηx 3 if α = 2 η x 2ηx 2 + ηx 3 if α = 3 ηx 2 + ηx 3 if α = 4 1 3ηy 2 + 2ηy 3 if α = 1 3ηy 2 2ηy 3 if α = 2 η y 2ηy 2 + ηy 3 if α = 3 ηy 2 + ηy 3 if α = 4 η x = x x i x, η y = y y j y., (12), (13) Let us denote the coefficients of the above basis polynomials a α (η x ) and b α (η y ), respectively by A β α and Bα, β where β corresponds to the power of the corresponding term. For instance, for a 1 (η x ) = 1 3ηx 2 +2ηx 3 (α = 1), the coefficients A β α are A 3 1 = 2, A 2 1 = 3, A 1 1 = 0 and A 0 1 = 1. Along the characteristic direction, η y is a function of η x (η y = mη x + C), which can be substituted into b α (η y ). With this substitution, a α (η x ) and b α (η y ) can be introduced into the general expression for P (x) (Eq. 9) resulting in a sixth order polynomial with respect to η x. The coefficients of this polynomial can be expressed as C ζ = c ζ 11 ϕi,j + c ζ 12 ϕi,j+1 + c ζ 13 ϕi,j y +c ζ 21 ϕi+1,j + c ζ 22 ϕi+1,j+1 + c ζ 23 ϕi+1,j y y y + c ζ 14 ϕi,j+1 y y +c ζ 24 ϕi+1,j+1 y y + c ζ 31 ϕi,j x x + c ζ 32 ϕi,j+1 x x +c ζ 33 ϕi,j xy x y + c ζ 34 ϕi,j+1 xy x y + c ζ 41 ϕi+1,j x x +c ζ 42 ϕi+1,j+1 x x + c ζ 43 ϕi+1,j xy x y + c ζ 44 ϕi+1,j+1 xy x y, (14) where ζ [1, 7] N denotes (ζ 1) order for the corresponding term. Here, the values for c ζ γδ are given by c 1 γδ = A1 γb 1 δ, c 2 γδ = A1 γb 2 δ + A2 γb 1 δ, c 3 γδ = A1 γb 3 δ + A3 γb 1 δ + A2 γb 2 δ, c 4 γδ = A1 γb 4 δ + A2 γb 3 δ + A3 γb 2 δ + A4 γb 1 δ,, c 5 γδ = A2 γb 4 δ + A3 γb 3 δ + A4 γb 2 δ, c 6 γδ = A3 γb 4 δ + A4 γb 3 δ, and c 7 γδ = A4 γb 4 δ, with γ = (1, 2, 3, 4) and δ = (1, 2, 3, 4). Roots of the 6 th order polynomial are readily obtained from intrinsic functions in MATLAB. We discard all complex or imaginary roots, or roots outside the computational cell in question. Of the remaining valid roots, we choose the minimum one and assigned this to ϕ i,j+1. This process is repeated for all nodes neighboring the interface. Nodes that are farther removed from the interface are updated with the gradient augmented reinitialization scheme presented previously. We return to our former 1D case (i.e. ϕ o = 0.05 sinh(18x/5)), and apply the hybrid reinitialization. The results are shown in Fig. 4 which clearly mark the substantial improvement incurred by anchoring the interface. A more quantitative comparison is afforded through the use of the L 2 error norm, defined as L 2 = N (ϕ (x i ) ϕ exact (x i )) 2 (15) i=1 for N grid cells. In Table 1, the comparison between hybrid and Hamilton-Jacobi (conventional) re-initialization are compared for N=7, 20 and 49, which confirm the previous graphical findings. Furthermore the values clearly indicate that grid resolution plays a significant role in exacerbating reinitialization errors. 5

6 Grid cells Hybrid Conventional E E E E E E-2 Table 1: L 2 error comparison between the hybrid and conventional re-initialization scheme. (a) Conventional reinitialization scheme. (b) Hybrid reinitialization scheme. Figure 5: Mass errors are significantly reduced with the implementation of the hybrid re-initialization procedure. (a) Figure 4: (a) Farfield view of the comparison between hybrid and non-hybrid schemes and (b) near field view. Similarly, we return to the 2D case, corresponding to the ϕ o field given in Eq. (11). As discovered previously, the coarse grid re-initialization yielded unacceptable results. The hybrid treatment is now contrasted to these previous findings using the conventional re-initialization. The results are shown in Fig. 5, which again show significant improvement. Quantitative findings are determined with the L 1 error defined as Ω (b) 1 L H ( ϕ numerical) H ( ϕ expect) dxdy (16) where a numerical integration is performed following the procedure outlined by Sussman and Fatemi [12]. Their re-initialization procedure is compared to the one proposed here, and the results are shown in Table 2 after 4 and 8 steps of the re-initialization process. We report an improvement with the current method. x Steps L 1 Error L 1 Error [12] E E E E-04 Table 2: L 1 Error calculated from Eq. (16) and compared to result from [12], ( τ = x). Advection with re-initialization: In the original presentation of the augmentedlevel-set method, Nave et al. [9] argued that the method was sufficiently accurate for the omission of a re-initialization phase. While this may be true for some cases, the standard advection of the Zalesak disk [19] show noticeable improvement when exercising the hybrid re-initialization scheme. This is shown in Fig. 6. During hybrid re-initialization, the field was reinitialized for 6 pseudo time steps ( τ = x) every 314 advection steps. The case was executed in a domain of size 100x100, where the disk was centered at (50, 75) with a radius equal to 15. The width and height of the notch was 5 and 25, and the underlying flow field was given by u = π (100 y) 314 and v = π 314 (x 100). (17) As displayed in Fig. 6, the notch was preserved after 5 rotations, where it was lost in the pure advection case. To display the degree by which the level set function departs from ϕ = 1 condition, maps of this quantity are shown in Fig. 7 for both respective cases. Without re-initialization errors incurred during the advection of the interface lead to severe distortion of the level set function. Furthermore, errors originating from the boundary also shown signs of disturbing the ϕ = 1 condition. With proper re-initialization, these erroneous effects are greatly diminished. As a last test, we study the advection of a har- 6

7 (a) Advected for 1 rotation. (a) Advected with hybrid (b) Advected without reinireinitialization scheme. tialization. Figure 8: A = 4; H = 2 (Initial-Red, Advection with reinitialization-blue, Advection with out reinitialization-green). Figure 6: Zalesak disk advected for 5 rotations (Initial-Red, After 1 rotation-blue, After 4 rotations-black, After 5 rotations-green). (a) ψ for Fig (6a). Conclusions and Future work A semi-lagrangian method was presented for level set re-distancing. where the level set function ϕ and its gradient ϕ are updated simultaneously. Unlike the truly upwind schemes proposed in [13, 14] our method benefits from a local stencil and enjoys 4th order accuracy by virtue of the Hermite interpolation polynomials. The scheme presented addresses effectively the erroneous drift in the interface that plagues the conventional re-initialization procedure. This is accomplished by implementing a hybrid scheme, where the level set function is obtained directly from the Hermite polynomials for nodes neighboring the interface. This essentially anchors the interface and prevents artificial motion. For nodes away from this region, the re-distancing procedure employs a gradient-augmented methodology inspired by the advection scheme presented by Nave et al. [9]. In cases, where grid resolution is relatively coarse, re-initialization errors have the potential to dominate advection errors. It is in these cases, where the re-initialization scheme presented is truly beneficial. (b) ψ for Fig (6b). Figure 7: ψ at the near interface for 5th rotation of the Zalesak problem. monic disc, described by 2 2 (x xo ) + (y yo ) (r + A cos (4Hθ)) (18) where A = 4 is the harmonic amplitude and H = 2 are the number of harmonics per quarter circle. For this problem the domain size is 100x100, and the disk is centered at (50, 75) with a radius equal to 15. The flow field is the same as Zalesak s problem given in Eq. (17). Fig. 8 displays the results after 1 and 2 rotations corresponding to pure advection and hybrid re-initialization. In the case of re-initialization, the procedure was applied every quarter rotation resulting in a noticeable improvement. However, the improvement is not as drastic as the previous 1D case or circular disk reinitialization under coarse grid resolution. We find that for certain cases the error due to advection can easily overwhelm errors due to artificial interface displacement during re-initialization. Hence, even though these re-initialization errors are drastically minimized most of the error, originating from advection, dominates the solution. ϕo = (b) Advected for 2 rotations. Acknowledments Authors would like to acknowledge the support of the Office of Naval Research through Mark Spector, code 331, and the Caterpillar corporation. References [1] S. Osher and J. Sethian. J. Comput. Phys., 79:12 49, [2] M. Sussman, P. Smereka, and S. Osher. J. Comput. Phys., 114: , [3] M. Sussman, A.S. Almgren, J.B. Bell, P. Colella, and L.H. Howell. J. Comput. Phys., 148:81 124,

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