... including image processing and mesh generation algorithms. ... requires that the computational domain be discretized into a collection of vertices and edges.
submitted to MSMSE
Finite Element 3D Mesh Generation From Experimental Microstructures M. C. Demirel1,2,3, D. C. George2, and A. D. Rollett3, and B. R. Schlei2
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Materials Science and Technology, MST-8, Los Alamos National Laboratory, NM, 87545 2
Theoretical Division, T-1, Los Alamos National Laboratory, NM, 87545
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Carnegie Mellon University, Department of Materials Science & Engineering, PA, 15213
Simulation and predictions of material processes at multi-length scales require a detailed understanding of microstructures. While a meaningful study requires large data sets, the size of data sets has been limited due to the lack of automated image processing and mesh generation algorithms. We obtained as images a relatively large population of data from 19 samples of Aluminum foils by Electron Backscattered Diffraction (EBSD) technique. The experimentally obtained data were analyzed in detail and compared with existing theoretical distribution models. We describe a novel set of algorithms that extract boundary information from these images and provide a complete boundary description to the mesh generation step. The mesh generation step in turn produces a 3D mesh as input for the finite element code and maintains mesh quality as the simulation proceeds. We show that proper microstructural characterization of materials and prediction of evolution requires a careful implementation of several key steps including image processing and mesh generation algorithms.
_______________________________________________ This paper will be submitted to Journal of Applied Physics
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INTRODUCTION Accurate representations of microstructure are important across a wide range of disciplines for the visualization of structures, the prediction of properties and the simulation of changes in structure. Numerical representations are particularly useful because of the power of modern computers to depict complex shapes in addition to computing the response of materials to external and internal forces. No single standard approach has yet been found that can accommodate all problem types. Most materials are polycrystalline and anisotropic and their microstructures generally comprise irregular polygons. Their properties depend on microstructure in various ways. Some properties as exemplified by elastic moduli are dominated by the lattice orientations and thus the crystallographic texture. Still other properties depend on the transmission of fracture, diffusion or fluxes across the body. In cases where the boundaries between grains and/or phases offer the path of least resistance, the transmission becomes a percolation problem. Similarly, some evolution problems can be reduced to computation of the behavior of volume elements without regard to the neighborhood of that element (e.g. crystal plasticity). For solidification, grain growth or recrystallization, however, the behavior of the interfaces is the dominant step. In this latter problem then, accurate representation of the interfacial network is the key to accurate and successful problem solving.
Numerical solution and simulation of a PDE-based application typically requires that the computational domain be discretized into a collection of vertices and edges. This discretization can take a number of different forms ranging from logically rectangular and multi-block structured grids to unstructured meshes consisting of simple geometric entities such as triangles or
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tetrahedra. In this paper, we developed a new tool, which produces unstructured computational meshes using the topological information derived from experimental data.
Several steps are required in order to generate an input for the simulation from experimental result. As an example, we will demonstrate the current technique on several images that are obtained from EBSD experiments. These microstructures were obtained from 19 Aluminum film samples. First, the surface of each sample is scanned and the image is recorded by backscattered contrast and corrected for instrumental distortions. Second, the image analysis algorithm then locates the grain boundaries and the triple junctions. This information is essential for the mesh generation and characterization of the topology of the grain boundary network. Finally, the meshing of the image is accomplished by the Los Alamos Grid Toolbox, LaGriT. In the following sections, a brief summary of region enclosing contour extraction is presented. This is followed by a description of mesh generation by LaGriT.
REGION ENCLOSING CONTOUR EXTRACTION FROM BOUNDARY PIXELS In the fields of digital image processing, computer vision and pattern recognition, the application of edge detection algorithms is important for the extraction of multiple touching regions in images. However, the enclosure of the regions with discrete contours is not straightforward. In this paper, we apply a novel region-enclosing contour method 1 , which is based on the use of unstructured grids. The elements of the unstructured grids are triangles, which carry inherent topological information 1 .
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We desire to process the given grain edge pixels such that we obtain as a final result a set of contours, where (i) each contour encloses a grain (region) counter-clockwise with a minimum number of supporting points according to the initially given edge pixels, and where (ii) the whole area of the image has been taken under consideration (Figure 1D). In the following, we shall restrict our discussion to a small bi-level image as shown in figure 1A without loss of generality.
In a first step, the initially given boundary pixels in the bi-level image are enclosed with dilated contours 2 , 3 (Figure 1A). The algorithm mentioned here is very fast and completely reliable while extracting (relatively smooth) dilated contours from image blobs. In addition, the produced contours are perfect in that they are always non-intersecting and non-degenerate. The dilated contours and their supporting point set form the input to a Constrained Delaunay Tesselation (CDT) 4 (“constrained”, because a subset of the triangular mesh edges, which is the initial set of dilated contour edges, may not be altered). The CDT of the contours is the key step that allows for the shapes’ skeleton extraction (Figure 1B) and see ref [1] for more detail).
The shape skeleton shown in figure 1B encloses only three of the nine visible grains fully. Because it is our intent to construct region-enclosing contours for all nine visible grains, a frame is added around the original image. The added outer frame consists of a set of single vectors that are arranged counter-clockwise around the original image1 . The point set that supports the frame’s vector set is sampled at the rate of pixels available along a side of the image and it always includes the four corners of the image. In general, the shape skeleton is disconnected from the image
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(vector set) frame, i.e., there are gaps present between limb-like arcs of the skeleton and the image frame.
In order to close the gaps a CDT is applied to the combined skeleton and image frame. This newly formed Delaunay triangular mesh provides a set of edges that bridge the gap between a terminal point and a frame point (see ref [1] for more detail). On this set one imposes a criterion. For instance, one could either select the shortest edge, or the edge, which best preserves the orientation of the terminating skeleton segment in a skeletons’ limb. For this current application we choose the latter method, i.e., we try to preserve the orientation of the last line segment for a limb-like arc as much as possible when connecting it to the frame (Figure 1C).
After insertion of these gap edges, the grain regions are completely enclosed with contours. Finally, the supporting points of the region enclosing the contours are down-sampled1 and an unstructured input grid for grain-growth simulations is generated (Figure 1D). The unstructured grid of Figure 1D illustrates the down-sampling technique, but is not used in the grid generation step. The ordered sets of down sampled nodes that define the grain boundaries are input to the grid generation step. In ref [1], a grain boundary image with 25,518 grain edge pixels was processed according to this method. The shape processing resulted in only 1,368 region-enclosing contour support points for 177 grain regions (accurate coverage of all grains required only 2,678 Delaunay triangles), where the maximum processing time was below 5 seconds on a 800 MHz Pentium III CPU, with a LINUX operating system.
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GRID GENERATION TOOL (LaGriT) From the results of the contour extraction stage, we construct a three-dimensional, unstructured finite element grid using Los Alamos Grid Toolbox, LaGriT 5 . LaGriT is a three-dimensional grid generation and optimization software package especially appropriate for applications requiring moving boundaries and/or that are geometrically complex. The first step in grid generation for our application is to separately triangulate each grain from the grain boundaries that are given by the contour extraction algorithms as sets of ordered pairs of coordinates (x,y). The result of the first step is a set of two-dimensional planar triangular meshes, one for each grain. In the second step, we extrude each grain’s two-dimensional mesh in a direction perpendicular to the plane, resulting in a three-dimensional prism mesh for each grain; then each prism is divided into three tetrahedra. Note that the extrusion step doubles the number of nodes, but that converting the prisms to tetrahedra adds no additional nodes to the mesh. Third, the exterior boundary is extracted from each grain’s three-dimensional volume mesh; each boundary mesh is a triangular mesh that is topologically two-dimensional but geometrically three-dimensional. The fourth step results in a single conformal three-dimensional mesh. The individual boundary meshes created in step three are used as surfaces and the area inside each boundary mesh is identified as a separate geometric region. When meshing, LaGriT creates conforming interfaces where regions meet, and this capability in this context results in each grain being meshed into a separate region. All nodes from the boundary meshes are copied into the combined mesh. At this point additional nodes may be added to the mesh. In the example given in Figure 3A, a node is added in the interior of each grain. Tetrahedral elements are formed in the combined mesh using a Watson point insertion algorithm 6 . The boundaries of the individual grains are preserved. The grid generation process is
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flexible and it is possible to omit the first step and begin with the downsized triangulated grid as shown in Figure 1D.
The quality of the initial three-dimensional mesh is quite poor, but LaGriT optimization tools are used to improve mesh quality. The optimization process, which we call graph massage 7 , accepts as input a refine length, a merge length and a damage length. Edges longer than the refine length are bisected; edges shorter than the merge length will be merged by eliminating one of the end nodes of the edge; however, no merge operation will occur if the result is damage to a grain boundary greater than damage length. Damage is measured as the length of the projection from the node to be merged away to the new edge created by removing the node. Graph massage also includes a node smoothing option. This option moves locations of nodes to reduce the number of needle and sliver elements by increasing element inscribed radii; it obeys the damage length constraint for nodes on grain boundaries. The last tool in the graph massage toolkit is reconnection, which can perform 2 to 3, 3 to 2, and 4 to 4 face interchanges of elements using the criterion of increasing the inscribed radii of the elements involved in the reconnection. 4 to 4 reconnections may occur across an interface, and, in these cases, the reconnection must obey the damage length constraint. For reconnections, the damage is the distance between the two planes that sandwich the edges to be interchanged. In addition to improving the initial mesh quality; LaGriT optimization tools are also used during simulation to handle topological events such as grains disappearing and to maintain mesh quality as the grain boundaries move. Figures 2 and 3 show the steps involved in generating a three-dimensional mesh from the output of the shape processing step.
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The number of finite elements used in the simulation depends on the number of grains. As an example, the first microstructure consists of 26 grains, and, after massage, the three-dimensional finite element mesh contained 3570 tetrahedral elements. The mesh lines conform to the grain boundaries both before and after the massage step.
RESULTS Schematic description of generating an input for the simulation from experiment result is shown in Figure 4. As an example, we chose an Aluminum sample with 339 grains (please see ref [8] for experimental details). The microstructure of the sample shown in Figure 4A is columnar, therefore experimental data are collected from the surface of the sample, and then the microstructure information is extruded in the third dimension (Figure 4D) in order to obtain complete microstructure. However, an extension of our technique that combines algorithms for extracting bounding surfaces from 3-dimensional voxel data with the LaGriT grid generation tools may be applied to general 3-dimensional microstructures, i.e. non-columnar grain microstructures provided the experimental data are available in three dimensions.
Using the tools described above, the EBSD experimental data were digitized and then several averages were calculated. Several images of microstructures were recorded from EBSD experiments. These microstructures were obtained from 19 Aluminum film samples. The average cross sectional grain area distribution was calculated from the experimental microstructure. The equivalent grain size distribution is plotted in figure 5A. Grain size distribution in aluminum was
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also analyzed by Carpenter et al. 9 at a much smaller length scale (nm) and their figure 2 is reproduced in figure 5B. The distribution is similar to that observed in Aluminum thin films with nanometer grain size by Carpenter et al. 9 . We also compared the current data with the existing theoretical log-normal distribution model, f(x),
(1)
where x is the equivalent grain diameter, s is the natural logarithm of the standard deviation and m is natural logarithm of the mean of the log-normal distribution. Two parameters (s, and m) are calculated using the experimental data. The standard deviation, s and mean, m, of the distribution are 0.45 and 4.95 respectively. There are other distribution models in the literature proposed for the grain growth distribution 1 0, 1 1, 1 2, 1 3. According to Carpenter et al. 9 , the best fit to this type of experimental data is the log-normal distribution.
The digitalized output for experimental microstructure information can be used to simulate microstructure behavior. We are developing a three dimensional finite element code for simulating the microstructure behavior at grain scale 1 4, 1 5. Currently, our simulation model uses curvature driven motion implemented by a three dimensional gradient-weighted moving finite elements (GWMFE) code 1 5, 1 6. With the tools developed in this paper, we obtained several inputs for the microstructure simulation code. The comparison between computed results and the experiments
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provided important details related to microstructure behavior. Our recent results showed a strong similarity between growth experiments and anisotropic three-dimensional simulations 8 .
CONCLUSION Region enclosing contour extraction, and mesh generation and maintenance are essential to transform information from experimental data into computer simulation input. In this paper, we have demonstrated a technique to accomplish this transformation. We have described the tools developed to implement the technique, and have applied it to a simulation of microstructure evolution. This technique can be easily applied to other experimental information in order to produce fast and reliable computer input.
ACKNOWLEDGEMENT This research is support by Los Alamos National Laboratory (DOE, W-7405-ENG-36), the MRSEC program of NSF (DMR-0079996), and the Computational Materials Science Network (US-DOE).
REFERENCES 1
B. R. Schlei, in Region enclosing contours from edge pixels, Seattle, 2002.
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B. R. Schlei and L. Prasad, “A Parallel Algorithm for Dilated Contour Extraction from Bilevel Images,” Report No. Los Alamos Preprint LA-UR-00-309 (2000).
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B. R. Schlei, DICONEX - Dilated Contour Extraction Code, Version 1.0 (Los Alamos Natl. Lab., Los Alamos, 2002).
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P. L. George and H. Borouchaki, Hermes (1998).
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D. George, LaGriT user’s manual, http://www.t12.lanl.gov/~lagrit (1995).
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D. F. Watson, Comput. J. 24, 167-172 (1981).
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A. P. Kuprat and D. C. George, in Maintaining Tetrahedral Mesh Quality in Response to Time-dependent Topological and Geometrical Deformation, 1998 (International Society of Grid Generation (ISBN 0-9651627)), p. 589-598.
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M. C. Demirel, A. P. Kuprat, D. C. George, and A. D. Rollett, Physical Review Letters, submitted (2002).
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D. T. Carpenter, J. R. Codner, K. Barmak, and J. M. Rickman, Materials Letters 41, 296302 (1999).
10 D. Weaire, J. P. Kermode, and J. Wejchert, Phil. Mag. B 53, L101-L105 (1986). 11 H. V. Atkinson, Acta. Metall. 35, 2671 (1987). 12 H. J. Frost, Mat. Sci. Forum 94, 903 (1992). 13 C. S. Pande, Acta. Metall. 38, 945 (1990). 14 N. N. Carlson and K. Miller, Siam Journal of Scientific Computing 19, 766-798 (1998). 15 A. Kuprat, Siam Journal on Scientific Computing 22, 535 - 560 (2000). 16 M. C. Demirel, A. P. Kuprat, D. C. George, G. B. Straub, and A. D. Rollett, Interface Science 10, 137 (2002).
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Figure 1 a.
Figure 1 b.
Figure 1 c.
Figure 1 d.
Figure 1. Shape Processing Steps: A) Original image (gray boundary pixels) and dilated contours, B) CDT of the shape enclosed by the dilated contours and its shape skeleton, C) CDT for the shape skeleton and the image frame. The gaps are closed with the line segments, which are drawn with larger line width, D) Original image superimposed with down-sampled region-enclosing contours and a CDT grid. Note, that the region-enclosing contours stay within the limits defined by the gray pixels.
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Figure 2. Three steps in mesh generation. The left figure shows the ordered set of nodes defining a typical grain boundary. The center figure shows the triangulated single planar grain defined by the nodes in the preceding step. The right figure shows the surface mesh of a single grain generated by extruding the mesh created in the preceding step and then extracting the surface.
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Figure 3. The left figure shows the three-dimensional mesh of a 26 grain example before graph massage and the right shows the mesh after graph massage. An interior node has been added to each grain.
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Figure 4. Schematic description of mesh generation from experimental data: A) EBSD Inverse Pole Figure, B) Region Enclosing Contour Extraction, C) Triangulation and Mesh Generation (LaGriT), D) Extrusion of the mesh in the 3rd dimension
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Figure 5. Grain size distribution for the initial experimental microstructure of the Aluminum foil and the log-normal distribution fit (m=4.95 and s=0.45, please see the text for the definitions). 5170 grains were used to calculate the experimental distribution.
