Simulation Details for 2D

Transcription

1 Appendix B Simulation Details for 2D In this appendix we add some details two-dimensional simulation method. The details provided here describe the method used to obtain results reported in Chapters 3 and 5. Brakke method The Surface Evolver program [48] developed and maintained by Ken Brakke is a powerful collection of tools to study minimal surfaces in arbitrary dimension. As such, it is often used to model grain growth in two [155, 50, 49] and three dimensions [111, 156]. In Chapter 3 we compared the numerical error involved in using Brakke s method with that involved in using the method proposed in this thesis. We briefly describe the method used by Brakke in Surface Evolver. The two options effective area and area normalization are used in order to most accurately simulate motion by curvature flow. We describe the motion of boundary nodes for this setting. This will allow us to calculate the way in which the radius of a regular polygon changes in one time step using this method, a result critical in Section 3.5 for calculating the relative and absolute errors involved in this method. We consider the motion of boundary node σ in this setting. We begin by considering two unit vectors e 1 and e 2 pointing away from σ in the direction of the two adjacent edges. Without any normalization, node σ will move with a velocity e 1 + e 2. However, in this setting there exists a resistance to this motion from the adjacent edges. We therefore scale this vector by dividing it by half of some measurement of the adjacent edges. Brakke points out that the resisting motion comes from the component of the edges that are normal to the direction of the motion. Therefore, if p = p n (edges) is the length of the projection of the adjacent edges onto the normal direction 166

2 of e 1 + e 2, then we scale the e 1 + e 2 by 1/p. Figure B.1: Regular polygonal grain with m sides and of radius r. In the case of a regular polygon with m sides and radius r, as illustrated in Figure B.1, e 1 + e 2 always points in the direction of the center of the polygon. Since each of e 1 and e 2 are unit vectors, we have e 1 + e 2 =2sin(π/m). The length of the projection of two adjacent edges onto direction normal to e 1 + e 2 is then 2 cos(π/m)sin(π/m). Therefore, we are left with a vector pointing in the direction of the center of the polygon and with a magnitude 1/(r cos(π/m)). If we take into account the mobility of the grain boundary M and the surface tension γ, then after one time step of size t, the radius changes: r = Mγ t r cos( π m ) (B.1) for a regular polygon with n sides and of radius r. This equation is used in Section 3.5. Further information about this program and the algorithms used can be found in [48]. Proposed method Data structure We use two basic data structures, one representing a node and one representing a grain. Edges are represented implicitly through these two other data structures. We provide here a pseudo-code representation of a data structure representing a node. The id number, as well as the coordinates 167

3 class CNode { int id; double x, y; double dx, dy; int valence; // classifies all nodes by valence, either 2 or 3 CGrain* neighbors[3]; // pointers to all neighboring grains } x, y and velocities dx, dy are straightforward. The valence variable stores how many neighbors (grains or nodes) a particular node has; this will provide information about whether a node is a vertex node or a boundary node. The CGrain vector stores references to all three neighboring grains. The data structure for each grain is somewhat similar. class CGrain { int id; vector <CNode*> bnodes; } For each grain with a given id, we store only an ordered list of (references to) CNodes. This information is enough to calculate all properties of interest of a given grain. As a simple example, we can calculate how many triple junctions lay on the boundary of a grain by counting the number of its associated CNodes that have valence 3. Time step, minimum and maximum edge length On each time step, we set the size of the longest l max and shortest l min allowable edges in the system. If N is the number of grains in the system, then: l max = 1 5 N (B.2) and l min = N. (B.3) We set the step size t based on this: t = l 2 min/20. (B.4) On every step, if a grain has an area smaller than 2πl 2 min then we erase it. If an edge grows to be longer than l max, then we refine it on the following step, placing an extra boundary node at its 168

4 midpoint. If an edge shrinks to be shorter than l min then we remove one of its nodes, and move the other node to the midpoint of that edge segment. Node motions We begin by explaining how we move individual boundary nodes. Figure B.2 illustrates a typical boundary node σ, adjacent to two edges e 1 and e 2. Here we consider the edges as edges and also as Figure B.2: A typical boundary node σ, adjacent to two edges e 1 and e 2. vectors, pointing away from the boundary node σ. The area of the shaded parallelogram is e 1 e 2. For reasons of numerical stability, we choose to move boundary nodes in the direction of e 1 + e 2.If we moved node σ by exactly e 1 + e 2, then we would change the areas of the two adjacent grains by ± e 1 e 2. However, we only want to change the area of the adjacent grains by ±α t, as described in Chapter 3. Therefore, in one time step of size t, wemoveσ with a displacement vector: σ = α t e 1 e 2 (e 1 + e 2 ). (B.5) Determining the motion of a triple point is very similar. Figure B.3 shows a triple point τ and three neighboring edges and grains. We need to move the triple node τ in such a way that changes Figure B.3: A triple-node τ where Grains 1, 2, and 3 meet, where e i are the vectors from the triple-node to the nearest nodes on each of the three boundaries, and α i are the turning angles, or exterior angles with respect to each of the three grains. 169

5 the areas of each grain by an amount α i π/3, where α i is the turning angle of that grain at that triple point. We can make similar calculations to those we made for boundary nodes. We are then left with a system of three equations in two unknown spatial variables: e 1 e 2 e 2 e 3 dv =2Mγ t α 1 π 3 α 2 π 3 (B.6) e 3 e 1 α 3 π 3 This system is not overdetermined because only two equations are independent. This leaves us then: 1 τ =2Mγ t e 1 e 2 α 1 π 3 (B.7) e 3 e 1 α 3 π 3 Removing small faces When the area of a face is less than 2πlmin 2, then we delete it. Most of the time, this face has only two or three sides. If it has more than three sides, then we remove sides one at a time until only three are left. When there are three sides left to a face, then we remove it by collapsing its three corners to one triple node. This topological change is illustrated in Figure B.4. Figure B.4: The collapse of a three-sided face to a triple point. First we remove all edge nodes; next, a new point is placed at the geometric average of the three triple-points. Other neighboring boundary nodes are left in place. Removing a two-sided face is done in a very similar manner. This topological change is illustrated in Figure B.5. Figure B.5: The collapse of a two-sided face to an edge. We begin by removing the edge nodes and then remove two triple nodes, leaving the other neighboring nodes in place. 170