**4. From surfaces to FE models**

The complexity of the reconstructed surfaces, together with a rather large number of grains, essentially prevents the direct use of the finite element meshing capabilities of either Amira or even professional mesh generators such as for example ABAQUS/CAE (Simulia, 2010). A framework for the automatic meshing has been therefore developed (Simonovski & Cizelj, 2011a) using the Python scripting language and ABAQUS/CAE meshing tools, which are fully accessible through Python.

The framework can be applied to both analytical models of structures (e.g. 3D Voronoi tessellations) and to data obtained from experimental techniques. In both cases the surface structures are defined by the triangle-based surfaces, bounding the volume of individual grains. In the case of voxel-based data these surfaces are reconstructed with Amira. For

the surface triangles that they contain. Next, individual ACIS SAT file is imported into ABAQUS/CAE pre-processor, assigned seeds and its surfaces are meshed using triangular FE elements of a selected size. Mesh density is user controlled. The coarsest possible mesh is defined by the triangles of the reconstructed surface. Finer meshes can be obtained by using several FE elements per each triangle of the reconstructed surface. Meshing individual surfaces is independent upon each other which provides for an efficient parallelization of the process. A surface between the adjoining grains needs to be meshed only once since the shared

Grain-Scale Modeling Approaches for Polycrystalline Aggregates 57

Fig. 8. Creating a conformal volume mesh between grains by meshing the grain boundary surfaces and imprinting the obtained surface meshes on the corresponding grain boundaries.

grains 1 and 9 Grain 8 mesh

Cohesive elements between

grains 1 and 8

Grain 9 mesh

Fig. 9. Detail of the FE mesh showing cohesive elements between the grains.

To obtain a conformal mesh between adjoining grains the meshed surfaces are imprinted onto the corresponding grains, see Fig. 8. This is done by replacing the reconstructed surface of a given grain with an appropriate FE meshed surface and saving the grains' new geometry

Cohesive elements between

surface is identical for both grains.

Grain 6 mesh

Cohesive elements between

grains 1 and 6

Grain 1 mesh

Fig. 7. Geometry reconstructed from experimental data: grain 1 from the wire data set.

analytical models such as 3D Voronoi tessellations, the spatial structure is generated by the underlying analytical model (e.g. Qhull algorithm (*Qhull code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection about a Point*, n.d.) implemented in (Petriˇc, 2010)) and surface reconstruction is not needed.

Before starting the FE meshing procedure the surface triangles aspect ratios are checked. Surface triangle aspect ratio is defined in this work as the ratio of the circumscribed circle and the inscribed circle of a triangle. Triangles with aspect ratio of more than 1000 are removed by collapsing triangle's shortest edge, removing the triangle from the structure and updating the vertices and triangles. The procedure is performed iteratively until the worst aspect ratio is above 1000. This approach improves the FE mesh quality. The triangle-based surfaces are also checked for possible errors like intersections and corrected, if necessary, by slight displacement of appropriate triangle's corner points.

The ability of exporting the reconstructed surfaces into a standard CAD format that can be read by FE pre-processors is lacking in many visualization tools. Instead, the user is encouraged to use the tool's built in meshers, which often do not match the capabilities of dedicated FE mesh engines. Furthermore, FE pre-processors do not support export formats of the visualization tools. So there is a basic difficulty of importing the reconstructed geometry into FE pre-processors. This issue is circumvented here by developing a function for exporting the reconstructed surfaces into ACIS SAT file that can be imported into practically any FE pre-processor. The surfaces of each grain are therefore saved into ACIS SAT file with all 8 Polycrystalline Materials

Grain 1: 8 198 triangles Wire: 299 102 triangles Case: 300K

Fig. 7. Geometry reconstructed from experimental data: grain 1 from the wire data set.

analytical models such as 3D Voronoi tessellations, the spatial structure is generated by the underlying analytical model (e.g. Qhull algorithm (*Qhull code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection about a Point*, n.d.) implemented in

Before starting the FE meshing procedure the surface triangles aspect ratios are checked. Surface triangle aspect ratio is defined in this work as the ratio of the circumscribed circle and the inscribed circle of a triangle. Triangles with aspect ratio of more than 1000 are removed by collapsing triangle's shortest edge, removing the triangle from the structure and updating the vertices and triangles. The procedure is performed iteratively until the worst aspect ratio is above 1000. This approach improves the FE mesh quality. The triangle-based surfaces are also checked for possible errors like intersections and corrected, if necessary, by slight

The ability of exporting the reconstructed surfaces into a standard CAD format that can be read by FE pre-processors is lacking in many visualization tools. Instead, the user is encouraged to use the tool's built in meshers, which often do not match the capabilities of dedicated FE mesh engines. Furthermore, FE pre-processors do not support export formats of the visualization tools. So there is a basic difficulty of importing the reconstructed geometry into FE pre-processors. This issue is circumvented here by developing a function for exporting the reconstructed surfaces into ACIS SAT file that can be imported into practically any FE pre-processor. The surfaces of each grain are therefore saved into ACIS SAT file with all

Grain 1: 4 210 triangles Wire: 149 417 triangles Case: 150K

Grain 1: 802 triangles Wire: 29 866 triangles

Case: 30K

(Petriˇc, 2010)) and surface reconstruction is not needed.

displacement of appropriate triangle's corner points.

the surface triangles that they contain. Next, individual ACIS SAT file is imported into ABAQUS/CAE pre-processor, assigned seeds and its surfaces are meshed using triangular FE elements of a selected size. Mesh density is user controlled. The coarsest possible mesh is defined by the triangles of the reconstructed surface. Finer meshes can be obtained by using several FE elements per each triangle of the reconstructed surface. Meshing individual surfaces is independent upon each other which provides for an efficient parallelization of the process. A surface between the adjoining grains needs to be meshed only once since the shared surface is identical for both grains.

Fig. 8. Creating a conformal volume mesh between grains by meshing the grain boundary surfaces and imprinting the obtained surface meshes on the corresponding grain boundaries.

Fig. 9. Detail of the FE mesh showing cohesive elements between the grains.

To obtain a conformal mesh between adjoining grains the meshed surfaces are imprinted onto the corresponding grains, see Fig. 8. This is done by replacing the reconstructed surface of a given grain with an appropriate FE meshed surface and saving the grains' new geometry

100 grains 500 grains

Grain-Scale Modeling Approaches for Polycrystalline Aggregates 59

1000 grains 5000 grains

Isotropic elasticity, anisotropic elasticity and anisotropic elasticity with crystal plasticity constitutive laws are commonly used for bulk grains. Since isotropic elasticity can not account for the effects due to different crystallographic orientations of the grains this is not covered

Fig. 10. FE mesh examples of 3D Voronoi tessellations.

here. Overview of the other two constitutive laws is given below.

**5. Constitutive models**

**5.1 Bulk grains**

with all the imprinted meshes into new ACIS SAT files. The process is repeated for all the grains. Updated ACIS SAT geometry files of individual grains with imprinted surface meshes are imported into a ABAQUS/CAE, assigned appropriate surface definitions, material properties, loads and boundary conditions. Exactly one FE seed per each edge is assigned to preserve the FE meshed surfaces, obtained in the previous step. The number of FE per edge is not allowed to increase/decrease thus automatically creating conformal meshes between adjoining constituents. FE volume-meshing is performed next using ABAQUS/CAE built-in mesher. All the information that has been generated in the previous steps (topology, common surfaces between the constituents, material properties,...) has been saved using the Python pickle module and is now used to hierarchically define all the properties, including node, element and surface sets. Generating FE models of individual grains is independent upon each other which again provides for an efficient parallelization.

In the last step, zero thickness layers of cohesive elements are inserted between the adjacent grains. Layers of zero thickness triangular cohesive elements are inserted between the nodes occupying the same position on the adjacent surfaces. The triangular cohesive elements are oriented to conform with the tetrahedral elements on both surfaces. The nodes, elements, set, surfaces,... and all other definitions are also updated to reflect the new configuration. Fig. 9 illustrates mesh of adjacent grains with inserted zero thickness cohesive layers. Further details on the procedures employed in automatic and parallel generation of the finite element meshes are available in (Simonovski & Cizelj, 2011a).

Fig. 10 shows the obtained FE models for the 3D Voronoi tessellations given in Fig. 3. The mesh quality factors are given in Table 1.

Fig. 11 illustrates the constructed FE model of the 400 *μ*m diameter stainless steel wire. The top figure corresponds to the first experimental series, containing 362 grains. The model contains 903 199 finite elements: 796 105 linear solid tetrahedra elements and 107 094 cohesive elements. The bottom figure corresponds to the second experimental series, containing 1334 grains. The model contains 3 395 769 finite elements: 2 976 828 linear solid tetrahedra elements and 418 941 cohesive elements. The mesh quality factors are given in Table 2.


Table 1. 3D Voronoi FE models: mesh quality factors comparison.

10 Polycrystalline Materials

with all the imprinted meshes into new ACIS SAT files. The process is repeated for all the grains. Updated ACIS SAT geometry files of individual grains with imprinted surface meshes are imported into a ABAQUS/CAE, assigned appropriate surface definitions, material properties, loads and boundary conditions. Exactly one FE seed per each edge is assigned to preserve the FE meshed surfaces, obtained in the previous step. The number of FE per edge is not allowed to increase/decrease thus automatically creating conformal meshes between adjoining constituents. FE volume-meshing is performed next using ABAQUS/CAE built-in mesher. All the information that has been generated in the previous steps (topology, common surfaces between the constituents, material properties,...) has been saved using the Python pickle module and is now used to hierarchically define all the properties, including node, element and surface sets. Generating FE models of individual grains is independent upon

In the last step, zero thickness layers of cohesive elements are inserted between the adjacent grains. Layers of zero thickness triangular cohesive elements are inserted between the nodes occupying the same position on the adjacent surfaces. The triangular cohesive elements are oriented to conform with the tetrahedral elements on both surfaces. The nodes, elements, set, surfaces,... and all other definitions are also updated to reflect the new configuration. Fig. 9 illustrates mesh of adjacent grains with inserted zero thickness cohesive layers. Further details on the procedures employed in automatic and parallel generation of the finite element meshes

Fig. 10 shows the obtained FE models for the 3D Voronoi tessellations given in Fig. 3. The

Fig. 11 illustrates the constructed FE model of the 400 *μ*m diameter stainless steel wire. The top figure corresponds to the first experimental series, containing 362 grains. The model contains 903 199 finite elements: 796 105 linear solid tetrahedra elements and 107 094 cohesive elements. The bottom figure corresponds to the second experimental series, containing 1334 grains. The model contains 3 395 769 finite elements: 2 976 828 linear solid tetrahedra elements

Solid elements 91 140 143 588 528 860 4 517 884 Cohesive elements 13 363 34 136 105 468 787 976 All elements 104 503 177 724 634 328 5 305 860

Min angle < 5 [◦] 38 (0.0417 %) 1021 (0.7111 %) 1571 (0.2970 %) 7847 (0.1737 %) Max angle > 170 [◦] 0 (0 %) 11 (0.0077 %) 4 (0.0007 %) 27 (0.0006 %) Aspect ratio > 10 43 (0.0472 %) 1103 (0.7682 %) 1742 (0.3294 %) 8651 (0.1915 %)

Worst min angle [◦] 2.52 0.02 0.1 0.05 Worst max angle [◦] 162.09 179.9 172.5 174.27 Worst aspect ratio 27.48 1156 512.4 1028

100 grains 500 grains 1000 grains 5000 grains

and 418 941 cohesive elements. The mesh quality factors are given in Table 2.

Table 1. 3D Voronoi FE models: mesh quality factors comparison.

each other which again provides for an efficient parallelization.

are available in (Simonovski & Cizelj, 2011a).

mesh quality factors are given in Table 1.

*Number of elements*

*Number of elements with*

*Values of*
