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Default splitting algorithm with all-quadrilateral transition elements and no degenerate elements.

Regions of non-localization are shown. Either the REMESH,SPLIT command or the REMESH,SPLIT,,,TRAN,QUAD command can apply here.

Splitting with single-layer transition and degenerate elements.

In this case, the REMESH,SPLIT,,,TRAN,DEGE command applies.

4.7.1.3.5. Mesh-Transition Options for 3-D Mesh Splitting

Unlike in 2-D splitting, command-driven transition element control is not supported in 3-D splitting. The program generates the transitions automatically.

For tetrahedral meshes, transitions are generated for horizontal rezoning and partial remeshing.

For any tetrahedral element, if all four or any three nodes are selected by the program as candidates for splitting, the entire element is split into eight child tetrahedra (as shown in Figure 4.8: Splitting Tetrahedral Linear Elements (SOLID285)) (p. 108)). However, different transition element schemes are developed for cases when only one or two nodes in a tetrahedron are selected for splitting.

The transition process is a two-phase operation: the first phase usually creates one or more tetrahedral elements and pyramids and /or prisms (depending on the splitting template), and the second phase further decomposes the pyramids and/or prisms into child tetrahedra.

The following figure shows the first phase of the transitions for parent tetrahedra when one and two nodes are selected for splitting:

Figure 4.10: Phase 1 Transition Creation: Tetrahedra with One and Two Nodes Selected for Splitting

In the first case, the selected node is 1. From the parent tetrahedron (1-2-3-4), one child tetrahedron (1-5-6-7) and one child prism (5-6-7-2-3-4) are created. In the second case, nodes 1 and 4 are selected for splitting, creating two child tetrahedra (1-5-6-7 and 4-8-9-4), one child pyramid (5-8-9-6-7), and one child prism (2-5-8-3-9-6).

In the second phase of transition creation, the prisms and pyramids are further split into tetrahedra, as shown in the following two figures:

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Figure 4.11: Phase 2(a) Transition Creation: Prism (Wedge) Element Is Split into Three Tetrahedra

The prism (1-2-6-4-3-5) is split into three tetrahedra (1-2-6-4, 2-4-3-6, and 3-4-5-6). No new nodes are created. The diagonals are created to maintain compatibility with the neighboring elements and are selected on the basis of best element shape metrics of the child elements.

Figure 4.12: Phase 2(b) Transition Creation: Pyramid Element Is Split into Two Tetrahedra

The pyramid (1-2-3-4-5) is split into tetrahedra (1-2-3-5 and 1-3-4-5). No new nodes are created. The diagonal is created to maintain compatibility with the neighboring elements and are selected on the basis of best element shape metrics of the child elements.

4.7.1.3.6.Improving the Local Topology of Tetrahedral Meshes via Edge and Face Swapping

Splitting a mesh ensures that child elements inherit all properties from parent elements. For 3-D splitting, a deformed parent tetrahedral element produces eight distorted child tetrahedra. If high deformation exists at the solution substep at which splitting occurs, the mapping operation (MAPSOLVE) may not converge. It is also possible that the solution may not proceed much further after the analysis restart,

as the child elements themselves are distorted.

To improve tetrahedral element quality, the program modifies the local topography of the elements automatically when the REMESH,SPLIT command executes. The modifications use edgeand face-swap

operators to change the connectivity of a group of contiguous tetrahedral elements to improve the shape metrics of the elements.

The swap operations are usually referred to as i-j swapping, where i is the number of tetrahedrons before swapping and j is the number of tetrahedrons after swapping. Support is available for 2-2, 2-3, 3-2, and 4-4 swaps.

The following figure shows an example of 2-3 and 2-2 swapping:

Figure 4.13: Edge/Face Swapping for Tetrahedral Elements

4.7.1.3.7. Improving Tetrahedral Element Quality via Mesh Morphing

In addition to swapping, the program also performs mesh morphing to improve tetrahedral element quality after splitting occurs. As with swapping, morphing occurs automatically on the split elements when the REMESH,SPLIT command executes.

The mesh-morphing operation maintains the mesh connectivity but moves the nodes to improve element shape metrics. The operation uses the following cotangent-weighted Laplacian operator to perform an iterative node-wise morphing:

where:

L(v) = Node movement vector

N = Number of elements adjacent to patch node i

V = Position vector of node

The weight function is based on the cotangent of the face angles of the tetrahedral elements sharing the node i, shown in the following figure: