Lecture Notes in Computational Science and Engineering - Bioserver

12 Gudula Rünger
Data and communication management for adaptive finite element methods
The efficient parallel implementation of a hexahedral adaptive finite element
method has been presented in [16]. In this approach, the communication is
encapsulated in order to provide a general mechanism for repartitioning in
graph based algorithms.
The main characteristics inducing irregularity to adaptive hexahedral FEM
are adaptively refined data structures and hanging nodes. A hanging node is
a vertex of an element that can be a mid node of a face or an edge. Such
nodes can occur when hexahedral elements are refined irregularly, i.e. when
neighboring volumes have different levels of refinement. For a correct numerical
computation, hanging nodes require projections onto nodes of other refinement
levels during the solution process. The adaptive refinement process
with computations on different refinement levels creates hierarchies of data
structures and requires the explicit storage of these structures including their
relations. These characteristics lead to irregular communication behavior and
load imbalances in a parallel program.
The task pool team approach from Sect. 2 provides a concept which allows
any kind of data references or communication patterns, including strong irregular
communication. For adaptive FEM, however, the locality of references
and communication is slightly different than described for the algorithms presented
in the last section.
• The adaptive FEM has a data oriented program structure with dynamically
growing (or shrinking) data structures. Based on the input data a
suitable initial distribution of data can be chosen at program start.
• The computational effort varies dynamically according to the dynamic
behavior of the data. The process is guided by the FEM algorithm with
appropriate error estimation methods.
• The data dependencies are of local character, i.e. there are dependencies to
neighboring cells in the mesh or graph data structures. This leads to local
communication patterns in a parallel program with only a few neighboring
processors.
• For efficient parallel execution, load redistribution is required during runtime.
But load increases resulting from adaptive refinement are usually
concentrated on a few processors. Appropriate load redistribution may affect
all processors but are of local nature between neighboring processors.
• Communication occurs for remote data access when data are needed for
calculation. Also shared data structures have to be maintained consistently
during program run. In most cases the exact time of communication is
not known in advance. Message sizes or communication partners vary and
depend on the input. However, the communication partners work synchronously.
A parallel FEM implementation can exploit the locality of references and
local communication pattern. A suitable data management and communication
layer for adaptive three-dimensional hexahedral FEM is presented in [18].