Oliver Rheinbach

Stephan Köhler, Oliver Rheinbach

For complex geometries, the coarse problem of geometric multigrid can be too large to be solved by a direct solver. Here, we report on the use of domain decomposition applied to the multigrid coarse problem. Additive overlapping Schwarz methods are domain decomposition methods for the iterative solution of partial differential equations whose numerical and parallel scalability can be improved by the addition of coarse levels. A successful coarse space for such methods, inspired by iterative substructuring, is the generalized Dryja--Smith--Widlund (GDSW) space. A monolithic two-level overlapping Schwarz preconditioner based on a GDSW coarse space has been introduced for the solution of saddle-point problems arising from incompressible fluid problems, and has subsequently been extended to a three-level method. In the present work, for the first time, we consider a monolithic four-level overlapping Schwarz preconditioner, obtained by applying the two-level monolithic GDSW construction recursively three times, so that the second- and third-level coarse problems are themselves treated by overlapping Schwarz and only the smallest fourth-level coarse problem is solved by a sparse direct method. Numerical results are presented for a three-dimensional incompressible stationary Stokes problem with a Carreau-type non-Newtonian viscosity model posed on the complex geometry of an extrusion die, on up to $4\,000$\,MPI ranks, comparing the four-level preconditioner with its two-level and three-level counterparts in both roles. This work is part of the StroemungsRaum project, funded by the German Bundesministerium für Forschung, Technologie und Raumfahrt (BMFTR, formerly BMBF) as part of the SCALEXA program on new methods and technologies for exascale computing.

Matthias Brändel, Oliver Rheinbach

Data-driven surrogate models are an alternative to numerical homogenization of heterogeneous materials. In this contribution, a supervised learning approach is presented for predicting effective Lamé parameters of hyperelastic composites from low-dimensional microstructural descriptors. The data set is based on previously published numerical homogenization results for ensembles of two-phase stochastic microstructures generated by planar Boolean models, covering variations of inclusion shape, phase contrast, and area fraction; see Brändel, Brands, Maike, Rheinbach, Schröder, Schwarz and Stoyan (2022). A neural network is trained on combinations of scalar and curve-valued statistical descriptors, including the area fraction, a derived scalar shape descriptor $τ$, the two-point correlation function $S_2(r)$, and the lineal-path function $\ell(z)$. Additional data representing limiting cases of the parameter space are incorporated to stabilize training and improve extrapolation behavior. The surrogate is evaluated by leave-one-grain-type-out cross-validation in order to assess generalization to unseen grain geometries. Numerical results demonstrate that additional descriptors can reduce relative errors. A predictor trained with $τ$ and $S_2(r)$ provides a compact representation with good quantitative accuracy and regular dense response behavior. Adding the lineal-path function $\ell(z)$ further reduces the error at the available data points, indicating that it is a promising additional descriptor; however, dense post-training response evaluations show that improved pointwise accuracy does not automatically guarantee physically admissible behavior between sampled parameter values. This motivates future work on physically constrained surrogate models, loss formulations, bounded output parametrizations, and a more systematic representation of curve-valued geometric descriptors.

Stephan Köhler, Oliver Rheinbach

Additive overlapping Schwarz Methods are iterative methods of the domain decomposition type for the solution of partial differential equations. Numerical and parallel scalability of these methods can be achieved by adding coarse levels. A successful coarse space, inspired by iterative substructuring, is the generalized Dryja-Smith-Widlund (GDSW) space. In https://doi.org/10.1137/18M1184047, based on the GDSW approach, two-level monolithic overlapping Schwarz preconditioners for saddle point problems were introduced. We present parallel results up to 32768 MPI ranks for the solution of incompressible fluid problems for a Poiseuille flow example on the unit cube and a complex extrusion die geometry using a two- and a three-level monolithic overlapping Schwarz preconditioner. These results are achieved through the combination of the additive overlapping Schwarz solvers implemented in the Fast and Robust Overlapping Schwarz (FROSch) library https://doi.org/10.1007/978-3-030-56750-7_19, which is part of the Trilinos package ShyLU https://doi.org/10.1109/IPDPS.2012.64, and the FEATFLOW library http://www.featflow.de using a scalable interface for the efficient coupling of the two libraries. This work is part of the project StroemungsRaum - Novel Exascale-Architectures with Heterogeneous Hardware Components for Computational Fluid Dynamics Simulations, funded by the German Bundesministerium fur Forschung, Technologie und Raumfahrt BMFTR (formerly BMBF) as part of the program on New Methods and Technologies for Exascale Computing (SCALEXA).