17中英文雙語(yǔ)外文文獻(xiàn)翻譯成品高階時(shí)間行進(jìn)式局部水平集函數(shù)的重新初始化

1、<p>　　外文標(biāo)題：High-order time-marching reinitialization for regional level-set functions</p><p>　　外文作者：Shucheng Pan, Xiuxiu Lyu, Xiangyu Y. Hu ?, Nikolaus A. Adams</p><p>　　文獻(xiàn)出處：《Journal of Co

2、mputational Physics》 , 2017 , 354</p><p>　　英文4879單詞，23887字符，中文7482漢字。</p><p>　　此文檔是外文翻譯成品，無(wú)需調(diào)整復(fù)雜的格式哦！下載之后直接可用，方便快捷！只需二十多元。</p><p>　　High-order time-marching reinitialization for reg

3、ional level-set functions</p><p>　　Shucheng Pan, Xiuxiu Lyu, Xiangyu Y. Hu ?, Nikolaus A. Adams</p><p><b>　　Abstract</b></p><p>　　In this work, the time-marching reiniti

4、alization method is extended to compute the unsigned distance function in multi-region system s involving arbitrary number of regions. High order and interface preservation are achieved by applying a simple mapping that

5、transform s the regional level-set function to the level-set function and a high-order two- step reinitialization method which is a combination of the closest point ?nding procedure and the HJ-WENO scheme. The convergenc

6、e failure of the clos</p><p>　　Keywords: Reinitialization , Regional level-set function, High-order accuracy, Interface preserving</p><p>　　Introduction</p><p>　　The level-set metho

7、d [15] is a well-established interface-capturing method and is being widely used for multiphase ?ow computation, image processing and computer vision [14]. A reinitialization process is employed to replace the distorted

8、level-set function φ0 : Rd → R during advection by the signed distance function φ : Rd → R which is the solution of the Eikonal equation,</p><p>　　Successful numerical methods for directly solving this stati

9、onary boundary value problem include the fast m arching method [21] and the fast sweeping method [27]. One can also transform Eq. (1) to a time-m arching problem [25,24],which is a Hamilton–Jacobi (HJ) equation with a di

10、scontinuous coefficient across the interface. </p><p>　　Numerical approximations of Eq. (2) may exhibit oscillations or interface shifting [18], as the discretization of the derivatives across the interface

11、may employ erroneous level-set information from the other side of the interface. High-order schemes specially developed for HJ equations [11] m ay suffer from order degeneration and large truncation errors [18,5]. Modi?c

12、ations [24,18,13,5,9,3] have been proposed to cope with spurious displacement of the interface and successfully reduce the m ass </p><p>　　The level-set method has been extended to simulate the interface evo

13、lution of a multi-region system involving arbitrary number of regions by using multiple level-set functions [26,23] or a single regional level-set function [28,20,17], i.e.,a combination of the unsigned distance function

14、 ?(x) ≥ 0 and the integer region indicator χ (x), where x ∈ is a point in the computational domain . For multiple level-set functions, reinitialization is applied to each level-set function [23].</p><p>　　Re

15、initialization for the regional level-set function can be accomplished by the fast m arching method [20]. Although the time-m arching method is considered to be m ore costly, it is m ore ?exible and easier to parallelize

16、 [3]. When applied to the regional level-set method, high-order accuracy, to our knowledge, has not been demonstrated in the literature. For example, the time-m arching reinitialization method used in Ref. [17] to regula

17、rize the regional level-set functions is limited to 1st or</p><p>　　Rein itialization of a regional level-set function</p><p>　　Let X = {r ∈N|1 ≤ r ≤ N } be the index set for all regions. Region

18、al level-set reinitialization corresponds to ?nding the solution of on each region domain χ which is an open subset of , such that Eq. (1) is decomposed into N sub-problems. </p><p>　　A good reinitialization

19、 method for regional level-set functions should preserve the interface and achieve high-order accuracy with a computational effort weakly depending on the number of the regions N . The displacement of the interface m ay

20、lead to a sign change of ?, corresponding to a region-indicator change, which is di?cult to handle for N > 2. Thus, we require that the developed method does not change the sign of ? so that the indicator χ remains in

21、variant during the reinitialization. Alt</p><p>　　Direct implementation of the time-m arching reinitialization method on a regional level-set function gives</p><p>　　which is incorrect near the

22、interface for reinitialization as ? gives the wrong characteristic direction across the interface. To address this issue, we em ploy a mapping Cr : R × N → R de?ned as</p><p>　　It transform s the region

23、al level-set function to a level-set function for each region χ r . Eq. (4) is reformulated as which is a standard HJ equation.</p><p>　　Now, reinitialization can be perform ed region by region [17], or by s

24、olving N scalar evolution equations [3]. Both have computational cost scaling linearly with N due to global mapping from ? and χ to φ. The same cost can be achieved with time-m arching reinitialization methods [25,13,5,9

25、].</p><p>　　To reduce the computational cost to be approximately the same as for the original level-set reinitialization method, we can locally apply the mapping on every stencil of the spatial discretizatio

26、n schemes. Considering a 2D m ulti-region system de?ned by ?i, j and χ i, j on a uniform Cartesian grid, with i and j being the indices of the coordinate directions, we apply a mapping for each grid point on the stencil

27、of the discretization scheme, where ? t k t , ? t l t , and 2t + 1 is the required st</p><p>　　Note that Eq. (7) is a local operation and the local level-set functions, φr+ k, j and φr, j+ l, are temporary v

28、ariables, unlike the regional level-set (?, χ ) which is de?ned globally. Thus (?, χ ) at other grid points remains invariant. Now the information is propagated from the region boundary ? χ i, j to the interior of the re

29、gion domain χ i, j . The mapping Cr serves the same purpose as the signum term in Eq. (2). The semi-discrete form of Eq. (6) is where HG is the Godunov num erical Ham i</p><p>　　If N = 2, this is identical t

30、o traditional time-m arching level-set formulation which can be solved by existing high-order schemes [16,11]. However directly applying these high-order schemes requires additional operations when N > 2. When the sol

31、ution of Eq. (8) changes its sign after one time-step, updating of χ is required, which is not encountered with the original level-set problem.We emphasize that unlike traditional level-set reinitialization where interfa

32、ce-preservation serves to achieve </p><p>　　High-order two-step reinitialization method</p><p>　　To ensure interface-preservation and high-order accuracy, we perform 2 operations. First, as with

33、 the initialization step of the fast m arching method [21], we tag the grid point (i, j) adjacent to the interface as Alive, others as Far. A cell [i ? 1, i] × [ j ? 1, j] is a cut-cell if it contains an Alive grid

34、point. Inside every cut-cell, the level-set function is approximated by a bicubic polynomial [4] whose coefficients are determined by interpolation using all 16 grid-point values around th</p><p>　　A modi?ed

35、 Newton method [4] is used to ?nd the closest point x that satis?es pr (x ) = 0 and pr (x ) · (x ? x) = 0 simultaneously, where x is the location of an Alive grid point. The distance of an Alive grid point (i, j) to

36、 its region boundary is ?i, j =m in p∈P r xi, j ? xp with χ r = χ i, j , where P r is the set of all polynomials in cut-cells that share the grid point (i, j).</p><p>　　The second step is to update iterative

37、ly all Far grid-point values by solving Eq. (8). To ensure that global accuracy is at least the sam e as that of the 4th-order accurate approximation of Alive grid values, we em ploy a 5th-order HJ-WENO schem e [11] for

38、approximation of the derivatives in HG,</p><p>　　where+ x φr, j = φr+ 1, j ? φr, j . The weighting factors, w 0 , w 1 and w 2 , are de?ned in Ref. [11]. Thus a fully 4th-order reinitialization method with a

39、low-order two-step regional level-set reinitialization method which uses piecewise linear functions to approximate Alive values and an upwind ?nite difference scheme for Far values. The high-order subcell ?x method in Re

40、f. [13] is applied to the solution of Eq. (8) for reducing the interface oscillation. Note that we replace the one-sided E</p><p>　　Note that bicubic interpolation degenerates to 1st-order accuracy near trip

41、le points. In this case, one can use piecewise linear interpolation which gives 2nd-order accuracy. A sim ple possibility is to em ploy a com m on triangulation algorithm , such as m arching cubes [12] or m arching tetra

42、hedra [1], after mapping Cr . Such simple methods, however, may generate invalid interface segments in a cut-cell with m ore than two regions. As shown in Fig. 1, the interface deviates from the expected s</p><

43、;p>　　Extension to three dimensions (3D) encounters an issue with the closest point ?nding algorithm of Ref. [4] which m ay not converge within a prescribed number of iterations, as reported in Ref. [10] for two region

44、 cases. This issue becomes m ore serious near multiple junctions due to the existence of discontinuities in the multi-region cases. Without additional modi?cation, the multi-region 3D test cases with an initially highly

45、distorted level-set function such as that in Fig. 4(f) fail to conve</p><p>　　Numerical examples</p><p>　　In this section, we assess the capability and accuracy of our time-m arching regional le

46、vel-set reinitialization method by a range of numerical examples. Both 2D and 3D cases are considered. For all test cases, Eq. (8) is solved by a 3rd-order strongly stable Runge–Kutta scheme [22] with a CFL number of 0.5

47、.</p><p>　　Two-region system</p><p>　　Two cases with 2 regions are considered to test the accuracy and interface-preserving property of our reinitialization method for regional level-set functio

48、ns. First, we test a simple case with an initial unsigned regional level-set function ?0,χ</p><p>　　given by on a rectangular domain [0, 10] × [0, 10]. </p><p>　　For comparison of accuracy,

49、 we compute the error measures, L1 (?), L∞ (?), L1 (κ) and L∞ (κ), where κ = ·| φr | is the mean curvature. The errors are measured on the full domain, with the kink of (5, 5) excluded. As shown in Fig. 2(a), the hi

50、gh-order two-step method achieves 4th-order accuracy for error norm s of ? and the corresponding 2nd-order accuracy for error norm s of κ, which are higher than those for the subcell ?x method (3rd-order ? and 1st-order

51、κ) and the low-order method (1st-order</p><p>　　Multi-region system</p><p>　　The ?rst case is a 2D circle of radius r = 0.3 which is divided into two equal parts on the computational domain [0,

52、1] × [0, 1] with initial regional level-set function being which means the ?0 function is not consistent with the χ function, as shown in Fig. 3(a).</p><p>　　After reinitialization, the unsigned level-s

53、et function matches the χ function very well. Error measures, L1 (?) and L∞ (?), are computed within a narrow band,{(i, j)|?i, j < 3 x ∧ xi, j ? xs < 0.05}, where xi, j and xs are the locations of grid point and tr

54、iple points, respectively. As expected, in Fig. 3(c), we achieve the same order of accuracy as with that for the 2-region cases. The errors of the 3D case with indicate similar accuracy, and the sub?gure shows that the b

55、icubic interpolation e</p><p>　　We consider cases with random ly generated Voronoi diagram s to test the capability and performance of our method for a multi-region system with arbitrary N . The diagram s ar

56、e either generated inside a circle or on a square with a periodic con?guration. Both cases have a region number N ranging from 5 to 100 and the initial ? is assigned to be ?0 = ed ? 1, where d is the distance to the Voro

57、noi edges. The reinitialization is perform ed on a domain [0, 10] × [0, 10] with 512 × 512 grid points. A</p><p>　　Level-set motion with reinitialization</p><p>　　Different types of ?o

58、ws are tested to demonstrate the ability of the present regional level-set reinitialization method. To make sure the numerical error generated during advection is not dominant, the advection of the interface is solved by

59、 a high-order regional level-set method [17], where the 5th-order HJ-WENO scheme [11] and 3rd-order SSP Runge–Kutta scheme [22] are used to discretize the level-set advection equation. Reinitialization is perform ed afte

60、r every time step.</p><p>　　The ?rst case is the normal driven ?ow with 2 regions, where the velocity u is determ ined by the normal direction n . A circle with a radius of r = 0.2 at t = 0 expands with a un

61、iform speed to r = 0.4 at t = 0.2. As expected, a 4th-order method for level-set functions produces 3rd-order errors, as u is the ?rst derivative of the level-set function, see Fig. 5(a)</p><p>　　The second

62、example is the triple-point advection under constant velocity u = (1, 0). A 2D circle of radius r = 0.2 is divided into two equal parts and transported by the ?ow ?eld until t = 0.4. As shown in Fig. 5(b), the L1 and L∞

63、norm s indicate that the reinitialization errors dominate. To assess the robustness of the reinitialization method in m ore complex ?ows, we</p><p>　　em ploy the vortex ?ow u = ( ?? y , ? ??x ), where = sin2

64、 (π x) cos2 (π y) cos(π t / 3)/ π [2]. The circle of the previous case is located at (0.5, 0.75) with a radius of 0.15. The initial two triple points move due to the background ?ow ?elds, and the interface deform s into

65、a ?lament wrapping around the center of the domain. At t = 3 the interface reaches its maximum deformation, and each region becomes a thin ?lament with sharp corners, as shown in Fig. 5(d). Then the interface wraps back

66、int</p><p>　　Finally we consider 3D cases with velocity u = (1, 0, 0). First we extend the 2D triple-point advection to 3D, where a sphere of radius r = 0.2 is divided into two equal parts. This system is tr

67、ansported until t = 0.4. Second we advect the 3D cases with random ly generated Voronoi diagram s, Fig. 4(f), and N = 5 until t = 0.2. As shown in Fig. 6, the L1 and L∞ norm s indicate that the reinitialization errors do

68、minate. The extracted interface network (? = 0) and interior iso-surfaces (? = 0.02 ? </p><p>　　Concluding remark s</p><p>　　In this short note we demonstrate how to achieve high-order accuracy

69、for computing distance functions in m ulti-region system s involving arbitrary number of regions by solving the time-m arching reinitialization equation. We em ploy a simple map of the regional level-set function onto th

70、e level-set function inside the stencil of a ?nite difference scheme, fol- lowed by solving the discretized HJ equation by a high-order two-step reinitialization method, which is a combination of a closest point</p>

71、;<p>　　Ackn owledgemen ts</p><p>　　This work is supported by China Scholarship Council under No. 201306290030 and No. 201406120010. XYH acknowledges funding from National Natural Science Foundation of

72、 China (No. 11628206) and Deutsche Forschungsgemeinschaft (HU 1527/6-1). NAA acknowledges funding from the European Research Council (ERC) under the Horizon 2020 grant agreement 667483. The authors thank Dr. Matt Elsey f

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98、 Hu , Nikolaus A. Adams</p><p><b>　　摘要</b></p><p>　　在本文中，時(shí)間推進(jìn)式的重新初始化方法被延伸到應(yīng)用于計(jì)算包括區(qū)域任意數(shù)的多區(qū)域系統(tǒng)中的無(wú)符號(hào)距離函數(shù)。通過(guò)應(yīng)用簡(jiǎn)單的映射來(lái)完成高階和界面保存，該映射將局部水平集函數(shù)轉(zhuǎn)換為水平集函數(shù)并用高階兩步重新初始化方法，該方法融合了最近點(diǎn)尋找步驟與HJ -WENO算法。通過(guò)

99、采用提出的多結(jié)處理和方向優(yōu)化算法來(lái)解決三維最近點(diǎn)尋找步驟收斂失敗的問(wèn)題。通過(guò)簡(jiǎn)單的幾例測(cè)試表明我們的方法在重新初始化區(qū)域水平集函數(shù)以及嚴(yán)格界面保存屬性上展現(xiàn)了4階精度。在更復(fù)雜的情形下，使用隨機(jī)生成圖像的重新初始化結(jié)果顯示了我們的方法對(duì)區(qū)域N的任意數(shù)的計(jì)算能力，其計(jì)算與區(qū)域N無(wú)關(guān)。所提出的方法已被應(yīng)用于具有不同類型流體的動(dòng)態(tài)界面計(jì)算，結(jié)果證明了它的高精度性和魯棒性。</p><p>　　關(guān)鍵詞：重新初始化，局部水