InitiateEdgeConnectivity

The InitiateEdgeConnectivity method initializes edge connectivity information in an AbstractMesh object. This method identifies all unique edges in the mesh, assigns them global edge numbers, and establishes the mapping between local element edges and these global edges.

Purpose

This method creates a critical data structure that identifies and numbers all edges in the mesh uniquely. Edge connectivity is essential for:

• Supporting higher-order finite element methods
• Implementing edge-based numerical schemes
• Constructing hierarchical mesh refinement
• Computing edge-based quantities (jumps, averages)
• Supporting p-refinement techniques
• Facilitating Nédélec (edge) finite elements
• Handling mesh adaptation algorithms

Method Signature

INTERFACE
  MODULE SUBROUTINE obj_InitiateEdgeConnectivity(obj)
    CLASS(AbstractMesh_), INTENT(INOUT) :: obj
  END SUBROUTINE obj_InitiateEdgeConnectivity
END INTERFACE

• obj (INTENT(INOUT)) - The AbstractMesh_ object for which to create the edge connectivity data

Algorithm

1. If edge connectivity is already initialized (obj%isEdgeConnectivityInitiated is true), the method returns immediately
2. Verifies the element data is properly allocated
3. Creates a binary tree to efficiently identify unique edges
4. For each active element in the mesh:
   • Retrieves the reference element's edge connectivity pattern
   • Allocates global edge and edge orientation arrays in the element data structure
   • For each edge in the element:
     • Identifies the two nodes defining the edge
     • Creates a sorted version of the edge (for consistent identification)
     • Attempts to insert the edge into the binary tree
     • If the edge is new, assigns it a new global edge ID
     • If the edge already exists, retrieves its existing global ID
     • Records edge orientation based on node ordering
5. Updates the total number of edges in the mesh (obj%tEdges)
6. Deallocates the temporary binary tree

Implementation Details

• Uses a binary tree data structure (EdgeDataBinaryTree_) to efficiently identify unique edges
• Edge orientation is stored as +1 or -1 to indicate whether the local edge direction matches the global edge direction
• A sorted version of each edge (nodes in ascending order) is used for consistent edge identification
• Global edge IDs are sequential starting from 1
• The method skips inactive elements

Data Structures Modified

• obj%elementData(iel)%ptr%globalEdges - For each element, stores the global edge numbers
• obj%elementData(iel)%ptr%edgeOrient - For each element, stores the orientation of each edge
• obj%tEdges - Updates the total number of unique edges in the mesh
• obj%isEdgeConnectivityInitiated - Flag indicating edge connectivity has been initialized

Edge Orientation

Edge orientation is stored as:

• +1 if the local edge direction (from first to second node) matches the global edge direction
• -1 if the local edge direction is opposite to the global edge direction

This orientation is crucial for ensuring consistent integration and assembly along edges.

Performance Considerations

• Time complexity is O(n × e × log(E)) where n is the number of elements, e is the average number of edges per element, and E is the total number of unique edges
• Memory usage scales with the total number of edges
• The binary tree structure optimizes the identification of unique edges
• For large meshes with many edges, this operation can be computationally intensive

Usage Example

TYPE(Mesh_) :: mesh
! ... Initialize mesh ...

! Build edge connectivity
CALL mesh%InitiateEdgeConnectivity()

! Now we can access global edge numbers and orientations
! For example, to get global edge numbers for an element:
globalEdges = mesh%elementData(elemNum)%ptr%globalEdges

Dependencies

This method depends on:

• ReferenceElement_Method for retrieving geometric parameters
• EdgeData_Class for edge data structure
• EdgeDataBinaryTree_Class for efficiently identifying unique edges
• SortUtility for consistent edge ordering

Notes

• Edge connectivity is a prerequisite for many advanced finite element methods
• Edge orientation information is essential for maintaining consistency in numerical integration
• This method creates a unique global numbering for all edges in the mesh
• The binary tree approach ensures efficient handling of large meshes