Keywords: domain decomposition methods, hierarchical substructuring, parallel computing, mesh deformation, FETI.

Solving highly heterogeneous structural mechanics problems with a large number of degrees of freedom (HPC simulations) is a real issue in engineering work, because of the required time and memory. Non-overlapping domain decomposition methods such as the FETI (Finite Elements Tearing and Interconnecting) or BDD (Balanced Domain Decomposition) methods have been developed in order to allocate the problems on distributed memory clusters with a large number of processors and to make mechanical calculations parallel.

Two difficulties are encountered when applying domain decomposition methods. First, the mesh generation is most often a sequential process applied to the full domain. Second, the linear system resulting from the partitioning of the mesh may be poorly conditioned, leading to slow convergence. Recently developed techniques such as adapted coarse spaces (e.g. FETIGenEO) or multipreconditioning (e.g. AMPFETI) enable to restore good convergence rate, at the cost of extra computations.

In this study, we try to mitigate these two difficulties by proposing a new hierarchical substructuring method which aims at making the mesh preprocessing step parallel and at improving the condition number of the linear system to be solved by generating regular interfaces adapted to the heterogeneity.

The method is schematically based on a reversal of meshing and substructuring steps associated with a possibly locally structured discretization. From a discrete geometry (i.e. CAD or discrete CAD), a first coarse mesh is generated. This coarse mesh does not represent the underlying geometry but allows to define a well-shaped partitioning. Then, the subdomains, defined as a union of coarse elements, are distributed on the compute cores together with the underlying subdomain CAD information. The fine mesh of the structure is then obtained by parallel refining while ensuring the interface compatibility between subdomains. Finally, mesh deformation techniques — based on RBF (Radial Basis Functions) interpolation — are applied to the fine mesh in order to be as close as possible to the underlying geometry.

We will discuss the performance of the new decomposition compared to automatic graph partitioners, in terms of parallel efficiency and convergence of the FETI method applied to the interface problem.

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