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Applications of a hyper-graph grammar system in adaptive finite-element computations

Gurgul P., Jopek K., Pingali K., Paszyńska A.

International Journal of Applied Mathematics and Computer Science

2018

Vol. 28, no. 3

569--582

This paper describes application of a hyper-graph grammar system for modeling a three-dimensional adaptive finite element method. The hyper-graph grammar approach allows obtaining a linear computational cost of adaptive mesh transformations and computations performed over refined meshes. The computations are done by a hyper-graph grammar driven algorithm applicable to three-dimensional problems. For the case of typical refinements performed towards a point or an edge, the algorithm yields linear computational cost with respect to the mesh nodes for its sequential execution and logarithmic cost for its parallel execution. Such hyper-graph grammar productions are the mathematical formalism used to describe the computational algorithm implementing the finite element method. Each production indicates the smallest atomic task that can be executed concurrently. The mesh transformations and computations by using the hyper-graph grammar-based approach have been tested in the GALOIS environment. We conclude the paper with some numerical results performed on a shared-memory Linux cluster node, for the case of three-dimensional computational meshes refined towards a point, an edge and a face.

Using the Hypergraph Grammar for generation of quasi optimal element partition trees in two dimensions

Ryzner Jakub, Paszyński Maciej, Paszyńska Anna

Computer Methods in Materials Science

2018

Vol. 18, No. 1

29--40

The paper presents the graph grammar model of Finite Element Method allowing for speeding up performed numerical simulations. In the presented approach, the finite element mesh operations are performed together with operations generating so-called element partition tree. The element partition tree sets the ordering of matrix operations performed by solver in order to solve the computational problem. The quality of element partition tree influences the computational time of the solver. Our method allows for generation good quality element partition trees for h-adaptive Finite Element Method. The paper is concluded with numerical results confirming the quality of generated element partition trees.

Hypergrammar-based parallel multi-frontal solver for grids with point singularities

Gurgul P., Paszyński M., Paszyńska A.

Computer Science

2015

Vol. 16 (1)

75--102

This paper describes the application of hypergraph grammars to drive a linear computational cost solver for grids with point singularities. Such graph grammar productions are the first mathematical formalisms used to describe solver algorithms, and each indicates the smallest atomic task that can be executed in parallel, which is very useful in the case of parallel execution. In particular,the partial order of execution of graph grammar productions can be found, and the sets of independent graph grammar productions can be localized. They can be scheduled set by set into a shared memory parallel machine. The graph-grammar-based solver has been implemented with NVIDIA CUDA for GPU. Graph grammar productions are accompanied by numerical results for a 2D case. We show that our graph-grammar-based solver with a GPU accelerator is, by order of magnitude, faster than the state-of-the-art MUMPS solver.