InitiateExtraNodetoNodes

Inheritence

This method is inherited from the AbstractMesh class.

InitiateExtraNodetoNodes

The InitiateExtraNodetoNodes method builds an extended node-to-node connectivity mapping in an AbstractMesh object. Unlike the standard node-to-node connectivity, this extended mapping includes "second-degree" connections that are not directly connected through a shared element but are neighbors of neighbors.

This routine generate the node to nodes mapping

• This mapping is stored inside obj%nodeData%extraGlobalNodeNum

• For a local node number i, obj%nodeData(i)%ExtraGlobalNodeNum denotes the global node data surrounding the local node number used for edge-based stabilization. This list does not include self node.

• It needs information about nodeToNodes, nodeToElements, and elementToElements. Therefore,

• If nodeToNodes is not initiated, then this method initiates it.

• If nodeToElements is not initiated, then this method initiates it.

• If elementToElements is not initiated, then this method initiates it.

Purpose

This method creates an enhanced connectivity graph that captures wider nodal neighborhoods, which is especially valuable for:

• Implementing jump-based stabilization techniques in finite element methods
• Supporting extended stencil operations in numerical schemes
• Enhancing solution smoothing algorithms
• Implementing advanced discontinuous Galerkin methods
• Improving shock-capturing schemes
• Creating patch-based reconstruction techniques
• Supporting multi-scale methods that require information beyond immediate neighbors

Interface

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

• obj (INTENT(INOUT)) - The AbstractMesh_ object for which to create the extended node-to-node connectivity map

Algorithm

1. Verify the method hasn't already been executed and that prerequisites are initialized
2. For each node in the mesh:
   • Get its immediate connected nodes (standard node-to-node connections)
   • Get all elements that contain this node
   • For each of these elements:
     • Get all elements that share a face with this element (element-to-element connectivity)
     • Filter out elements that are already connected to the current node
     • For each of these "second-degree" elements:
       • Get all nodes of this element
       • Filter out nodes that are already directly connected to the current node
       • Add these filtered nodes to the extra connectivity list
   • Remove any duplicate nodes from the extended connectivity list
   • Store the extended node list in the node data structure

Implementation Details

• Requires three prerequisite connectivity maps:
  • Node-to-node connectivity
  • Node-to-element connectivity
  • Element-to-element connectivity
• Uses boolean masks to efficiently filter nodes and elements
• Employs the APPEND utility for combining node lists
• Uses RemoveDuplicates to ensure each extra node appears only once

Data Structures Modified

• obj%nodeData(iLocalNode)%ptr%extraGlobalNodes - For each node, stores the list of "second-degree" connected global node numbers
• obj%isExtraNodeToNodesInitiated - Flag indicating the extended node-to-node mapping has been built

Performance Considerations

• Time complexity is higher than standard connectivity maps due to the multi-level traversal required
• Memory usage can be significantly higher than standard node-to-node connectivity as it captures a wider neighborhood
• For complex meshes with many elements per node, this operation can be computationally intensive
• The method employs efficient filtering through boolean masks to minimize unnecessary operations

Usage Example

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

! Build prerequisite connectivity (will be automatically called if not already initialized)
CALL mesh%InitiateNodeToNodes()
CALL mesh%InitiateNodeToElements()
CALL mesh%InitiateElementToElements()

! Build extended node-to-node connectivity
CALL mesh%InitiateExtraNodeToNodes()

! Now extended node connectivity is available for advanced numerical schemes

Dependencies

This method depends on:

• Standard node-to-node connectivity (obj%InitiateNodeToNodes())
• Node-to-element connectivity (obj%InitiateNodeToElements())
• Element-to-element connectivity (obj%InitiateElementToElements())
• NodeData_SetExtraGlobalNodes for storing the extended connectivity

Notes

• The method specifically seeks nodes that are not immediate neighbors but are connected through element-to-element relationships
• This extended connectivity is particularly valuable for numerical methods that require wider stencils or jump terms
• Unlike standard connectivity which is symmetric (if A connects to B, then B connects to A), the extra connectivity may not always be symmetric