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Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Within the PUM global-local enrichment scheme [1, 2] different physical models can exist to capture multiscale behavior. For instance, we consider classical linear elasticity globally and local zones where fractures occur. The elastic fields of the undamaged media provide appropriate boundary data for local PD simulations on a subdomain containing the crack tip to grow the crack path. Once the updated crack path is found, the elastic field in the body and surrounding the crack is updated using PUM basis with appropriate enrichment near the crack. The subdomain for the PD simulation is chosen to include the current crack tip as well as nearby features that will influence crack growth. This paper is part II of this series and validates the combined PD/PUM simulator against the experimental results presented in [3]. The presented results show that we can attain good agreement between experimental and simulation data with a local PD subdomain that is moving with the crack tip and adaptively chosen size.

On the way to Exascale, programmers face the increasing challenge of having to support multiple hardware architectures from the same code base. At the same time, portability of code and performance are increasingly difficult to achieve as hardware architectures are becoming more and more diverse. Today's heterogeneous systems often include two or more completely distinct and incompatible hardware execution models, such as GPGPU's, SIMD vector units, and general purpose cores which conventionally have to be programmed using separate tool chains representing non-overlapping programming models. The recent revival of interest in the industry and the wider community for the C++ language has spurred a remarkable amount of standardization proposals and technical specifications in the arena of concurrency and parallelism. This recently includes an increasing amount of discussion around the need for a uniform, higher-level abstraction and programming model for parallelism in the C++ standard targeting heterogeneous and distributed computing. Such an abstraction should perfectly blend with existing, already standardized language and library features, but should also be generic enough to support future hardware developments. In this paper, we present the results from developing such a higher-level programming abstraction for parallelism in C++ which aims at enabling code and performance portability over a wide range of architectures and for various types of parallelism. We present and compare performance data obtained from running the well-known STREAM benchmark ported to our higher level C++ abstraction with the corresponding results from running it natively. We show that our abstractions enable performance at least as good as the comparable base-line benchmarks while providing a uniform programming API on all compared target architectures.

In this paper we present an algebraic dimension-oblivious two-level domain decomposition solver for discretizations of elliptic partial differential equations. The proposed parallel solver is based on a space-filling curve partitioning approach that is applicable to any discretization, i.e. it directly operates on the assembled matrix equations. Moreover, it allows for the effective use of arbitrary processor numbers independent of the dimension of the underlying partial differential equation while maintaining optimal convergence behavior. This is the core property required to attain a sparse grid based combination method with extreme scalability which can utilize exascale parallel systems efficiently. Moreover, this approach provides a basis for the development of a fault-tolerant solver for the numerical treatment of high-dimensional problems. To achieve the required data redundancy we are therefore concerned with large overlaps of our domain decomposition which we construct via space-filling curves. In this paper, we propose our space-filling curve based domain decomposition solver and present its convergence properties and scaling behavior. The results of numerical experiments clearly show that our approach provides optimal convergence and scaling behavior in arbitrary dimension utilizing arbitrary processor numbers., Comment: 28 pages, 11 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2103.03315

Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a PUM scheme for handling problems with zones of linear elasticity and zones where fractures occur. Here, the peridynamic (PD) model is used in regions of fracture and smooth PUM is used in the surrounding linear elastic media. The method is a so-called global-local enrichment strategy. The elastic fields of the undamaged media provide appropriate boundary data for the localized PD simulations. The first steps for a combined PD/PUM simulator are presented. In part I of this series, we show that the local PD approximation can be utilized to enrich the global PUM approximation to capture the true material response with high accuracy efficiently. Test problems are provided demonstrating the validity and potential of this numerical approach.

We propose a simple domain decomposition method for $d$-dimensional elliptic PDEs which involves an overlapping decomposition into local subdomain problems and a global coarse problem. It relies on a space-filling curve to create equally sized subproblems and to determine a certain overlap based on the one-dimensional ordering of the space-filling curve. Furthermore we employ agglomeration and a purely algebraic Galerkin discretization in the construction of the coarse problem. This way, the use of $d$-dimensional geometric information is avoided. The subproblems are dealt with in an additive, parallel way, which gives rise to a subspace correction type linear iteration and a preconditioner for the conjugate gradient method. To make the algorithm fault-tolerant we store on each processor, besides the data of the associated subproblem, a copy of the coarse problem and also the data of a fixed amount of neighboring subproblems with respect to the one-dimensional ordering of the subproblems induced by the space-filling curve. This redundancy then allows to restore the necessary data if processors fail during the computation. Theory supports that the convergence rate of such a linear iteration method stays the same in expectation, and only its order constant deteriorates slightly due to the faults. We observe this in numerical experiments for the preconditioned conjugate gradient method in slightly weaker form as well. Altogether, we obtain a fault-tolerant, parallel and efficient domain decomposition method based on space-filling curves which is especially suited for higher-dimensional elliptic problems., Comment: 61 pages, 26 figures

Quantum mechanical calculations require the repeated solution of a Schr\"odinger equation for the wavefunctions of the system. Recent work has shown that enriched finite element methods significantly reduce the degrees of freedom required to obtain accurate solutions. However, time to solution has been adversely affected by the need to solve a generalized eigenvalue problem and the ill-conditioning of associated systems matrices. In this work, we address both issues by proposing a stable and efficient orbital-enriched partition-of-unity method to solve the Schr\"odinger boundary-value problem in a parallelepiped unit cell subject to Bloch-periodic boundary conditions. In our proposed PUM, the three-dimensional domain is covered by overlapping patches, with a compactly-supported, non-negative weight function, that is identically equal to unity over some finite subset of its support associated with each patch. This so-called flat-top property provides a pathway to devise a stable approximation over the whole domain. On each patch, we use $p$-th degree orthogonal polynomials that ensure $p$-th order completeness, and in addition include eigenfunctions of the radial solution of the Schr\"odinger equation. Furthermore, we adopt a variational lumping approach to construct a block-diagonal overlap matrix that yields a standard eigenvalue problem and demonstrate accuracy, stability and efficiency of the method., Comment: 24 pages, 12 figures