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附錄 英文資料翻譯
一個(gè)接口安裝有限元的水平集方法算法，實(shí)施，分析和應(yīng)用
1.1移動(dòng)界面及其數(shù)學(xué)說(shuō)明
移動(dòng)界面是不同媒介之間分隔的界限，它們一般發(fā)生在科學(xué)工程領(lǐng)域和日常生活中，例如冰和水的邊界溫度變化 、水包油泡沫動(dòng)作的邊界移動(dòng)，更多的例子包括晶體生長(zhǎng)表面、相固固相變的界限、網(wǎng)站接口區(qū)域界限、以及區(qū)分不同部位的材料界線等。晶界在多晶體、鐵磁材料的磁疇壁中、兩相流體流量、以及在液態(tài)運(yùn)動(dòng)時(shí)大生物分子表面生成。
在這里，在廣義上可以理解為：一個(gè)接口可以是一個(gè)幾何面有沒(méi)有厚度或者尖接口，它也可以指一個(gè)分散接口，可以有一定的厚度，如幾個(gè)原子的直徑。相比與那些固定接觸點(diǎn)，移動(dòng)接處點(diǎn)的屬性越來(lái)越重要，作為尺度越來(lái)越小。隨著現(xiàn)代技術(shù)的需求，大量的設(shè)備在規(guī)模要小得多，在移動(dòng)界面的研究變得越來(lái)越重要。
一個(gè)移動(dòng)的接處點(diǎn)可以通過(guò)一個(gè)精確的數(shù)學(xué)數(shù)據(jù)描述接口，也就是說(shuō)，一個(gè)沒(méi)有任何厚度的表面，或通過(guò)場(chǎng)函數(shù)的描述，不斷在不同區(qū)域的值與突出的區(qū)域在一個(gè)區(qū)間連續(xù)過(guò)渡。后者代表一個(gè)接口。后者通常被稱(chēng)為擴(kuò)散描述接口描述。在這篇論文中，我們會(huì)研究大幅接口描述移動(dòng)接口。
研究一個(gè)移動(dòng)的接口，有如下方程：
V(t) =dx(t)dt
這樣一個(gè)點(diǎn)的速度（矢量）定義為該接觸點(diǎn)的公式，然后由正常的速度（正常的COM分量速度矢量在每個(gè)接口點(diǎn)）方程式去表述，正常的速度通常被稱(chēng)為移動(dòng)界面運(yùn)動(dòng)規(guī)律。因此，就可以在數(shù)學(xué)上確定移動(dòng)界面某些運(yùn)動(dòng)規(guī)律管接口的正常速度。
在一般情況下，運(yùn)動(dòng)規(guī)律，從基礎(chǔ)物理、化學(xué)等中得到。如果我們用??=（t）來(lái)定義，就可以得到運(yùn)動(dòng)規(guī)律 Vn = ?H,其中Vn= Vn的（的x，t）表示正常速度（即正常分量速度矢量）的點(diǎn)x在時(shí)間t∈??中的平均曲率。在許多應(yīng)用中，還涉及到更通常速度的物理量。因此，偏微分方程（PDE）通常必須滿(mǎn)足其中一項(xiàng)某些數(shù)量的接觸點(diǎn)的描述作用，如溫度場(chǎng)。在這種情況下，通過(guò)接觸點(diǎn)介紹運(yùn)動(dòng)關(guān)系公式里的接口一側(cè)偏微分方程的解。作為接口不斷發(fā)展了偏微分方程必須在不斷發(fā)展的領(lǐng)域解決。該接觸點(diǎn)可能是一個(gè)復(fù)雜的形狀，比如能隨時(shí)間變化合并或分裂的拓?fù)浣Y(jié)構(gòu)。計(jì)算科學(xué)家面臨的挑戰(zhàn)就是準(zhǔn)確地解決這些問(wèn)題。
圖1.1：與邊界?典型域。界面顯示分離??地區(qū) -和+。
1.2.1標(biāo)記方法
該標(biāo)記方法是一種離散拉格朗日方法，在一個(gè)接觸面上設(shè)點(diǎn)的集合。在二維空間的點(diǎn)連鎖與線段在一起，形成一條曲線。在移動(dòng)界面上，這些標(biāo)記每個(gè)點(diǎn)的速度是在每一個(gè)時(shí)刻決定的，然后該點(diǎn)在時(shí)間上向前移動(dòng)少量（見(jiàn)圖1.2），這種方法的缺點(diǎn)是難以拓?fù)渥兓幚?，另一個(gè)缺點(diǎn)是接觸面的定位精度取決于密切標(biāo)記粒子沿界面的地方。隨著時(shí)間的推移，某些指標(biāo)可能會(huì)移動(dòng)生成到了更遠(yuǎn)的分界面位置附近的更多的錯(cuò)誤（見(jiàn)圖1.3）。因此，任何算法需要定期檢查安置粒子以確保他們足夠接近以達(dá)到預(yù)期準(zhǔn)確性。此外，如果有太多并攏沿界面，某些標(biāo)記顆粒可能需要被刪除或重新分配。另一個(gè)有缺點(diǎn)的標(biāo)記的方法是，它可以成為基于曲率運(yùn)動(dòng)不穩(wěn)定的接口，界面的曲率計(jì)算的小錯(cuò)誤將增長(zhǎng)時(shí)間。圖1.2：標(biāo)記方法的說(shuō)明。 （一）標(biāo)記（點(diǎn)）代表界面位置。 （二）在每一個(gè)時(shí)刻每個(gè)標(biāo)記在其指定的移動(dòng)速度到新的位置。 （三）在其新的位置接口。
1.2.2固體單元的流體方法
與此相反的標(biāo)記方法，固體單元的流體的方法跟蹤接口有內(nèi)在的聯(lián)系。它使用計(jì)算域的單元格網(wǎng)格，每個(gè)單元格中包含一個(gè)介于0和1的值。
圖1.3：一個(gè)標(biāo)記方法的缺點(diǎn)。 （一）兩個(gè)接口接近對(duì)方。 （二）標(biāo)記需要?jiǎng)h除后接口重疊。
該接口單元經(jīng)過(guò)包含一個(gè)值介于0和1的接口在某處經(jīng)過(guò)該單元格。該接口的位置可以被近似為基礎(chǔ)在單元格中的值，但是許多較小的單元可能需要得到想要的精度（見(jiàn)圖1.4）。如正常量的方向和曲率該接口是很難用這種方法計(jì)算準(zhǔn)確。盡管有這樣的缺點(diǎn)，但是這種方法已經(jīng)成功的被用來(lái)模擬移動(dòng)界面，特別是傳播火焰的界面描述。
圖1.4：固體體積的流體方法的說(shuō)明。
（一）接口圍陰影區(qū)。
（二）網(wǎng)格單元格的值由陰影填充單元比例計(jì)算地區(qū)。
1.2.3水平集方法
水平集方法，首先在[OS88]介紹的是一個(gè)通過(guò)歐拉方法含蓄地描述了接口。這種方法的出發(fā)點(diǎn)是為了說(shuō)明一個(gè)移動(dòng)的表面在t時(shí)刻，作為一個(gè)輔助集的零水平含蓄函數(shù)φ=φ（的x，t）。例如，如果是三維表面在任何時(shí)間t（如球體），那么該函數(shù)φ（x，t）是一個(gè)函數(shù)的三維空間，即x的定義有三個(gè)坐標(biāo)。
為方便起見(jiàn)，我們可以假設(shè)函數(shù)正面的有效數(shù)字在一側(cè)的接口以及對(duì)方負(fù)值。我們稱(chēng)φ為一級(jí)別設(shè)置功能。很明顯，這些功能有很大的不確定。水平集功能的進(jìn)化，其零水平集始終代表了這樣一種方式時(shí)間接口的位置。對(duì)已知的水平集方法的優(yōu)勢(shì)之一是使它進(jìn)行移動(dòng)接口拓?fù)渥兓ㄒ?jiàn)圖1.5）。
水平集函數(shù)的演化是由底層運(yùn)動(dòng)規(guī)律的運(yùn)動(dòng)界面正常速度生成，如果水平設(shè)置功能是表示一個(gè)點(diǎn)上的X接口和時(shí)間t為φ=φ，則方程確定水平集函數(shù)為?tφ+ Vn的|?φ|=0。其中?t表示時(shí)間導(dǎo)數(shù)和梯度?表示空間。這方程被稱(chēng)為水平集方程。一個(gè)人需要延長(zhǎng)正常速度提示（通常只在接口上確定）遠(yuǎn)離接口，這樣的水平集方程可以解決一個(gè)適當(dāng)?shù)目臻g域。
推導(dǎo)水平集方程和水平集方法的詳細(xì)資料載于第2章。
圖1.5：一個(gè)例子集函數(shù)的水平隨著時(shí)間的推移（左欄）擬訂與零的水平代表著該接口（右相應(yīng)地塊列）。
1.3本論文的主要貢獻(xiàn)
在這篇論文中的項(xiàng)目，我們開(kāi)發(fā)了一個(gè)新的水平集方法。這是一接口安裝有限元的水平集方法。對(duì)這種情況的主要特點(diǎn)方法是：網(wǎng)格：我們使用格點(diǎn)與頂點(diǎn)在一個(gè)統(tǒng)一的基礎(chǔ)網(wǎng)格。在每個(gè)時(shí)間步長(zhǎng)的網(wǎng)格細(xì)化到產(chǎn)品的接口貼網(wǎng)而前plicitly定位運(yùn)動(dòng)界面。雖然它需要額外的時(shí)間來(lái)完善在每個(gè)時(shí)間步長(zhǎng)網(wǎng)，某些計(jì)算變得更加容易。重新初始化的新方法：我們重新初始化水平集函數(shù)求解泊松方程類(lèi)型的問(wèn)題。我們保持平穩(wěn)的水平集函數(shù)通過(guò)實(shí)施跨梯度跳轉(zhuǎn)條件界面，然后有最小二乘法解決一超定系統(tǒng)。這種方法產(chǎn)生一個(gè)新的水平集函數(shù)、平穩(wěn)區(qū)域和整個(gè)接口，平穩(wěn)過(guò)渡非常重要，因?yàn)樗黾恿藴?zhǔn)確性曲率計(jì)算，詮釋了一些例子來(lái)演示如何重新初始化工作。
新方法的速度擴(kuò)展：水平集方法有賴(lài)于速度函數(shù)附近的接口定義，不只是在接口處，因此，如果接口的速度只在接口上已知時(shí)，我們需要擴(kuò)大這一速度遠(yuǎn)離流暢的界面。我們解決拉普拉斯的內(nèi)部和外部的接口上使用方程速度接口作為邊界條件，這是一個(gè)簡(jiǎn)單的方式來(lái)擴(kuò)展速度。如果需要，我們可以平滑施加梯度跳到對(duì)面的條件接口和解決最小二乘。 （這類(lèi)似于我們的重新初始化方法。）
曲率近似數(shù)值??分析：我們也提出了一種新方法計(jì)算出的水平集函數(shù)的曲率。這相當(dāng)于美國(guó)荷蘭國(guó)際標(biāo)準(zhǔn)中有限先和二階導(dǎo)數(shù)差逼近，但這種方法可能在某些應(yīng)用中比較簡(jiǎn)單，因?yàn)檫^(guò)程中的其他步驟該算法在計(jì)算中使用的數(shù)量可能被重復(fù)使用。一個(gè)證明和這一計(jì)算的準(zhǔn)確性例子可以給出。凝固問(wèn)題上的應(yīng)用：在這里，我們證明了算法用它來(lái)解決Stefan問(wèn)題。樹(shù)突狀凝固模型的例子，一個(gè)凍結(jié)物塊放入冷液體中。包括的影響有各向同性、各向異性，不過(guò)沒(méi)有表面張力為藍(lán)本，其他的例子顯示了不同的表面張力各向異性的相位角的影響。
這些例子表明，如果沒(méi)有那些接口配件時(shí)，用我們的方法產(chǎn)生了此前由一個(gè)嚴(yán)格的有限差分格式制作的類(lèi)似的效果。應(yīng)用分子溶劑：在這里，我們證明了算法用它來(lái)尋找最佳的溶劑系統(tǒng)溶質(zhì)溶劑接觸面。我們使用平衡溶劑化系統(tǒng)變分隱式溶劑模型非極性分子，每個(gè)分子由一組特定的原子在計(jì)算上組成。初始界面是封閉的原子移動(dòng)的方向，減少了自由能。包括空閑的能源對(duì)溶質(zhì)和溶質(zhì)溶劑型接口的范德華能源相互作用。我們的例子演示了我們的方法如何收集最佳溶質(zhì)溶劑溶劑化系統(tǒng)的各種接觸面。
附錄 英文資料翻譯
1.1 MovAn interface-fitted finite element based level set method Algorithm, implementation, analysis and applicationsing Descriptions
Moving interfaces are boundaries that separate different media that are deforming or flowing. They occur commonly in science, engineering and daily life. For instance, an ice-water boundary moves during the change of temperature and an oil bubble moves in water. More examples include growing crystal surfaces, phase boundaries in solid-solid phase transformations such as precipitate and martensite interfaces,domain boundaries that separate different parts of material such as grain boundaries in polycrystals, domain walls in ferromagnetic materials, two phase fluid flows, and surfaces of large biomolecules moving in water. Here interfaces are understood in a broad sense: an interface can be a geometrical surface that has no thickness—a sharp interface; it can also mean a diffuse interface that can have certain thickness, e.g., of a few atomic diameters.
Compared with those of bulk phases, the properties of moving interfaces can be more and more important as the length scales become smaller and smaller. As modern technologies demand heavily that devices be much smaller in scale, the study of moving interfaces has become more and more important. A moving interface can be mathematically described by a sharp interface, a surface without any thickness, or by a field function that takes constant values in different regions with a sharp but continuous transition from one region to another, representing an interface. The latter description is often called diffused interface description. In this thesis, we will consider a sharp interface description of moving interfaces.
Consider a moving interface which is represented by the set of points ?? =(t) which depends on time t. Let x(t) be an arbitrary point that remains on ??. The velocity (vector) of such a point is defined by
V(t) =dx(t)dt. (1.1.1)
The interface motion is then determined by the normal velocity (the normal com-ponent of the velocity vector) at each point of the interface. Equations that determine the normal velocities are often called motion laws for moving interfaces.
Thus, moving interfaces are mathematically determined by certain motion laws governing the normal velocity of the interface. In general, motion laws are given from the underlying physics, chemistry, etc. Motion by mean curvature is a com-mon and important example of such motion laws. If we denote by ?? = (t) the geometrical surface at time t, then the motion law is
Vn = ?H,
where Vn = Vn(x, t) denotes the normal velocity (the normal component of the velocity vector) of the point x ∈ ?? at time t, and H is the mean curvature.
In many applications, normal velocity also involves more physical quantities.Therefore partial differential equations (PDEs) typically must be satisfied on either side of the interface for certain quantities such as the temperature field. In such a case, the movement of the interface is then described in terms of the relationship between the solutions of the PDEs on either side of the interface. As the interface continually evolves the PDEs must be solved on continually evolving domains. The interface may be a complex shape and can change topologically over time by merging or breaking apart. The challenge for computational scientists is to accurately solve such problems.
1.2 Different Types of Numerical Methods for
Interface Motion
Let us imagine two regions separated by an interface that is moving in time according to a certain motion law. In two dimensions the interface would be a curve separating two areas. In three dimensions the interface would be a surface separating two volumes. We would like to track this moving interface numerically. We can designate an interior region and exterior region and call them ? and +. The interface separating ? and + is designated by ??, and the boundary of the computational domain is designated by ? (see Figure 1.1).
Figure 1.1: A typical domain with boundary ?. Interface ?? shown separating regions ? and +.
1.2.1 Marker Method
The marker method is a Lagrangian method that discretizes the interface with a set of points located on the interface. In two dimensions the points are chained together with line segments to form a curve. To move the interface, at each of these marker points the velocity is determined at each time step. The points are then moved forward in time a small amount (see Figure 1.2).
One drawback of this method is that topological changes are difficult to handle. A sophisticated method must therefore be employed to check if two regions of ? join together or if one region splits into two. Another drawback is that the accuracy of the location of the interface depends on how closely the marker particles are place along the interface. As time progresses, some markers may move farther apart creating more error in the location of the interface near those points (see Figure 1.3).
So any algorithm needs to periodically check the placement of the particles to insure that they are sufficiently close together to have the desired accuracy. Also, some marker particles may need to be removed or redistributed if there are too many close together along the interface. Yet another drawback of the marker method is that it can become unstable with curvature based motion of the interface. Small errors in the calculated curvature of the interface will grow in time.
(c)
Figure 1.2: Illustration of the marker method. (a) Markers (dots) represent the interface location. (b) At each time step each marker is moved at its specified velocity to its new location. (c) Interface in its new location.
1.2.2 Volume-of-Fluid Method
In contrast to the marker method, the volume-of-fluid method tracks the interface implicitly. It uses a grid of cells on the computational domain, and each cell contains a value between 0 and 1. Cells completely contained in ? have a value
Figure 1.3: One drawback of the marker method. (a) Two interfaces approaching each other. (b) Markers need to be removed after interfaces overlap.
of 1 and those completely contained in + have value 0. Cells that the interface passes through contain a value between 0 and 1 based on where the interface passes through that cell. The interface location can then be approximated based on the values in the cells, but many smaller cells may be required to get a desired accuracy (see Figure 1.4). Quantities such as the normal direction and curvature of the interface are difficult to calculate accurately with this method. In spite of the drawbacks, this method has been used to model moving interfaces successfully,in particular, flame front propagation [Cho80, Set84].
Figure 1.4: Illustration of the volume-of-fluid method. (a) Interfaces enclosing shaded areas. (b) Grid cell values based on proportion of cell filled by shaded areas.
1.2.3 Level Set Method
The level set method, first introduced in [OS88] is an Eulerian method that describes the interface implicitly. The starting point of this method is to describe a moving surface ?? = (t) at time t implicitly as the zero level set of an auxiliary function φ = φ(x, t), i.e., (t) = {x : φ(x, t) = 0}. For instance, if ?? is a three-dimensional surface (such as a sphere) then the function φ(x, t) at any time t is a function defined on the three-dimensional space, i.e., x has three coordinates.
For convenience, we can assume that the function takes on positive values on one side of the interface and negative values on the other side. We shall call φ a level set function of ??. Clearly such functions are vastly nonunique. The level set function evolves over time in such a way that its zero level set always represents the location of the interface. One of the known advantages of the level set method is that it captures naturally, and easily, topological changes of moving interfaces(see Figure 1.5).
The evolution of the level set function is determined by the underlying motion law governing the normal velocity Vn of the moving interface. If the level set function is denoted by φ = φ(x, t) for a point x on the interface and time t, then the equation determining the level set function is given by ?tφ + Vn|?φ| = 0.
where ?t denotes the time derivative and ? denotes the spatial gradient. This equation is called the level set equation. One needs to extend the normal velocity Vn (usually determined only on the interface) away from the interface so that the level set equation can be solved in an appropriate space domain. The derivation of the level set equation and more details of the level set method are given in Chapter 2.
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