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2 2015

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Graph Transformation Systems for Modeling Three Dimensional Finite Element Method. Part I

Ryszka I., Paszyńska A., Grabska E., Sieniek M., Paszyński M.

Fundamenta Informaticae

2015

Vol. 140, nr 2

129--172

In this paper we present several graph transformation systems modeling three dimensional h-adaptive Finite Element Method (3D h-FEM) algorithms with tetrahedral finite elements. In our approach a computational mesh is represented by a composite graph and mesh operations are expressed by the graph transformation rules. Each graph transformation system is responsible for different kind of operations. In particular, there is a graph transformation system expressing generation of an initial mesh, generating element matrices and elimination trees for interfacing with direct solver algorithm, a graph transformation system deciding which elements have to be further refined, as well as a graph transformation system responsible for execution of mesh refinements. These graph transformation systems are tested using a graph transformation tool (called GRAGRA), which provides a graphical environment for defining graphs, graph transformation rules and graph transformation systems. In this paper we illustrate the concepts by using an exemplary derivation for a three dimensional projection problem, based on a set of graph transformation rules.

Graph Transformation Systems for Modeling Three Dimensional Finite Element Method. Part 2

Ryszka I., Paszyńska A., Grabska E., Sieniek M., Paszyński M.

Fundamenta Informaticae

2015

Vol. 140, nr 2

173--203

In this paper we introduce formal definitions for several graph transformation systems modeling three dimensional h-adaptive Finite Element Method (3D h-FEM) algorithms with tetrahedral finite elements. We introduce a composite graph representation of the computational mesh and graph transformation rules expressing the mesh operations. In particular, there are graph transformation rules expressing the generation of the initial mesh consisting with tetrahedral finite elements, graph transformation rules expressing the construction of an elimination tree for interfacing with multi-frontal direct solver algorithm, graph transformation rules selecting sub-graph representing finite elements for further refinements, graph transformation rules responsible for execution of mesh refinements. We also discuss several benefits of using graph transformation system instead of classical FEM approach, including the benefits from the viewpoint of multi-frontal direct solvers.