Keywords: isogeometric analysis, NURBS, parallelization, domain decomposition, weighted graph of isogeometric mesh, memory distributed computing platforms.

computationally very intensive task, demands of which can be alleviated by performing it in a parallel computing environment. Typical parallel application decreases the demands on memory and other resources by spreading the task over several mutually interconnected computers and speeds up the response of the application by distributing the computation to individual processors.

The similarity between the IGA and FEA allows the adoption of the same parallelization concepts and paradigms. However, by comparing computational costs of the IGA and FEA, one can identify some differences which have an impact on the overall parallel performance. On one hand, the costs of the IGA are shifted from the global computation (solution of the overall system of equations) toward the local computation (on the element (knot span) level). This is the consequence of the fact that to obtain a solution of the same quality the IGA needs usually much fewer unknowns, which is compensated by the much more demanding integration (per element) due to the typically higher order of the basis functions. On the other hand, as a result of the larger support of isogeometric basis functions, the contributions to characteristic components (such as the stiffness matrix, load vector, etc.) related to a single unknown may generally come from much more elements depending again on the degree of the polynomial approximation. This has two effects. Firstly, the bandwidth of the resulting system of equations is larger, which implies that the use of iterative parallel solvers might be more appropriate. Secondly, the number of shared nodes (control points), assembling contributions from different subdomains, or shared elements, contributing to nodes on different subdomains, is greater, which indicates that the communication costs related to the global assembly of the characteristic components will also increase. However, if the number of unknowns in the IGA is significantly smaller compared to the FEA (while still delivering the solution of the same quality), the impact of these two effects on the performance should be rather limited without influencing significantly the overall scalability (for large enough problems).

Similarly as for the FEA, the construction of properly balanced partitioning of the computational isogeometric mesh is the key ingredient for the efficient parallel processing. However, compared to the FEA, the decomposition of the isogeometric mesh is rather non-trivial. The difficulty is that the construction of the graph properly representing the isogeometric mesh is not straightforward, which is the consequence of the relation between the computational knot spans and control points in the isogeometric mesh.

This paper describes the parallelization strategy of the IGA using the node-cut based domain decomposition. In this context, the paper introduces a novel concept how to construct a weighted dual graph of the NURBS-based isogeometric mesh that can be decomposed by standard graph-based partitioning methods. The proposed methodology is illustrated on a simple academic example.

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