Graph Grammar Based Model for Three Dimensional Multi-physics Simulations

Abstract

This chapter presents a graph grammar model for the generation and refinements of three dimensional computational meshes as well as for the solution of multi-physics problems over them. It is assumed that the computational meshes consists in tetrahedral and prism elements, and the tetrahedral elements can be further broken into four new tetrahedral and two new pyramid elements, and the prism elements can be further broken into four new prism elements. The graph grammar expresses the generation and adaptation algorithms as a sequence of graph grammar productions. It is assumed that the computational mesh is represented as a graph, and the mesh generation is expressed as a sequence of execution of graph grammar productions, starting from an initial graph with a single node. The graph grammar based algorithm is then utilized to generate the three dimensional mesh representing the simplified model of the human head with internal ear.We concentrate on the solution of the linear elasticity coupled with acoustics (the problem of propagation of acoustic waves over the simplified model of the human head). In the following part of the chapter we introduce the graph grammar productions transforming the computational mesh into a sequence of element frontal matrices, being the input for the multi-frontal solver algorithm. Finally, the overview on the out-of-core solver algorithm is presented, and the numerical results with comparison to the state-of-the art MUMPS solver are described.