Anna Paszyńska

Parallel multi-frontal solver for p adaptive finite element modeling of multi-physics computational problems

The paper presents a parallel direct solver for multi-physics problems. The solver is dedicated for solving problems resulting from adaptive finite element method computations. The concept of finite element is actually replaced by the concept of the node. The computational mesh consists of several nodes, related to element vertices, edges, faces and interiors. The ordering of unknowns in the solver is performed on the level of nodes. The concept of the node can be efficiently utilized in order to recognize unknowns that can be eliminated at a given node of the elimination tree. The solver is tested on the exemplary three-dimensional multi-physics problem involving the computations of the linear acoustics coupled with linear elasticity. The three-dimensional tetrahedral mesh generation and the solver algorithm are modeled by using graph grammar formalism. The execution time and the memory usage of the solver are compared with the MUMPS solver.

A Graph Grammar Model of the hp Adaptive Three Dimensional Finite Element Method. Part I

The first part of our paper presents a composite programmable graph grammar model for the self-adaptive two dimensional hp Finite Element Method algorithms (2D hp-FEM) with mixed triangular and rectangular finite elements. The two dimensional model is a starting point for the three dimensional model of self-adaptive hp-FEM presented in the second part of this paper. A computational mesh is represented by a composite graph. The operations performed over the mesh are expressed by the graph grammar rules. The three dimensional model is based on the extension of the two dimensional model with rectangular finite elements. In the second part of this paper, we conclude the presentation with numerical examples concerning the generation of the optimal mesh for simulation of the Step-and-Flash Imprint Lithography (SFIL).

Graph Transformations for Modeling hp-Adaptive Finite Element Method with Triangular Elements

The paper presents composition graph (CP-graph) grammar, which consists of a set of CP-graph transformations, suitable for modeling triangular finite element mesh transformations utilized by the self-adaptive hpFinite Element Method (FEM). The hpadaptive FEM allows to utilize distributed computational meshes, with finite elements of various size (thus hstands for element diameter) and polynomial orders of approximation varying locally, on finite elements edges and interiors (thus pstands for polynomial order of approximation). The computational triangular mesh is represented by attributed CP-graph. The proposed graph transformations model the initial mesh generation, procedure of hrefinement (breaking selected finite elements into son elements), and prefinement (adjusting polynomial orders of approximation on selected element edges and interiors). The graph grammar has been defined and verified by implemented graph grammar transformation software tool.

Graph transformations for modeling parallel hp-adaptive finite element method

The paper presents composition graph (CP-graph) grammar, which consists of a set of CP-graph transformations, suitable for modeling all aspects of parallel hp adaptive Finite Element Method (FEM) computations. The parallel hp adaptive FEM allows to utilize distributed computational meshes, with finite elements of various size (thus h stands for element diameter) and polynomial orders of approximation varying locally, on finite elements edges and interiors (thus p stands for polynomial order of approximation). The computational mesh is represented by attributed CP-graph. The proposed graph transformations model the initial mesh generation, procedure of h refinement (breaking selected finite elements into son elements), and p refinement (adjusting polynomial orders of approximation on selected element edges and interiors), as well as partitioning of computational mesh into sub-domains and enforcement of mesh regularity rules over the distributed data structure.

Graph Transformations for Modeling hp-Adaptive Finite Element Method with Mixed Triangular and Rectangular Elements

The paper presents composition graph (CP-graph) grammar, which consists of a set of CP-graph transformations, suitable for modeling transformations of two dimensional meshes with rectangular elements mixed with triangular elements. The mixed meshes are utilized by the self-adaptive hp Finite Element Method (FEM) extended to support triangular and rectangular elements. The hp -FEM generates a sequence of mixed triangular and rectangular element meshes providing exponential convergence of the numerical error with respect to the mesh size. This is done be executing several h or p refinements over an initial mesh. The mixed finite element mesh is represented by attributed CP-graph. The proposed graph transformations model the initial mesh generation as well as mesh refinements. The proposed extended graph grammar has been defined and verified by using implemented software.

Preventing deadlock during anisotropic 2D mesh adaptation in hp-adaptive FEM

Abstract The paper presents a grammar for anisotropic two-dimensional mesh adaptation in hp -adaptive Finite Element Method with rectangular elements. Expressing mesh transformations as grammar productions is useful for concurrency analysis thanks to exhibiting the partial causality order (Lamport relationship) between atomic operations. It occurs that a straightforward approach to modeling this process via grammar productions leads to potential deadlock in h -adaptation of the mesh. This fact is shown on a Petri net model of an exemplary adaptation. Therefore auxiliary productions are added to the grammar in order to ensure that any sequence of productions allowed by the grammar does not lead to a deadlock state. The fact that the enhanced grammar is deadlock-free is proven via a corresponding Petri net model. The proof has been performed by means of reachability graph construction and analysis. The paper is concluded with numerical simulations of magnetolluric measurements where the deadlock problem occurred.

Out-of-core multi-frontal solver for multi-physics hp adaptive problems

Abstract In this paper we present the out-of-core solver algorithm for three dimensional (3D) multi-physics problems solved by the Finite Element Method (FEM). The solver is able to solve problems where 3D meshes contain finite elements of different kind (tetrahedral, prism and pyramid elements) with the number of equations and polynomial orders of approximation varying locally on finite element edges, faces, and interiors. The solver works at the level of nodes, representing blocks of the global matrix associated with different vertices, edges, faces, and interiors of different elements. The solver minimizes the memory usage by dumping out all local systems from all nodes of the entire elimination tree during the elimination phase. The systems are going to be reutilized later during the backward substitution stage. The solver is tested on a challenging computational problem: acoustics of the human head. The memory usage of the solver is compared against that of the MUMPS solver.

Hypergraph Grammars in hp-adaptive Finite Element Method☆

Abstract The paper presents the hypergraph grammar for modelling the hp-adaptive finite element method algorithm with rectangular elements. The finite element mesh is represented by a hypergraph. All mesh transformations are modelled by means of hypergraph grammar rules. These rules allow to generate the initial mesh, to assign values of polynomial order to the element nodes, to generate the matrix for each element, to solve the problem and to perform the hp-adaptation.

Quasi-Optimal elimination trees for 2D grids with singularities

We construct quasi-optimal elimination trees for 2D finite element meshes with singularities. These trees minimize the complexity of the solution of the discrete system. The computational cost estimates of the elimination process model the execution of the multifrontal algorithms in serial and in parallel shared-memory executions. Since the meshes considered are a subspace of all possible mesh partitions, we call these minimizers quasi-optimal. We minimize the cost functionals using dynamic programming. Finding these minimizers is more computationally expensive than solving the original algebraic system. Nevertheless, from the insights provided by the analysis of the dynamic programming minima, we propose a heuristic construction of the elimination trees that has cost O(Ne log(Ne)), where Ne is the number of elements in the mesh. We show that this heuristic ordering has similar computational cost to the quasi-optimal elimination trees found with dynamic programming and outperforms state-of-the-art alternatives in our numerical experiments.

An extension of vose's markov chain model for genetic algorithms

The paper presents an extension of Voses Markov chain model for genetic algorithm (GA). The model contains not only standard genetic operators such as mutation and crossover but also two new operators - translation to the left/right and permutation of bits. The presented model can be used for finding the transition matrices and for the investigation of asymptotic properties by using Markov transition functions. The ergodity of the Markov chain describing the GA with new operators, translation to the left/right and permutation, is shown. The model is specialized for a case of Bentleys GA. For this GA the ergodity of the Markov chains and the asymptotic correctness in the probabilistic sense are shown. To model other aspects of the Bentleys GA (effective fitness, total transmission probability) the microscopic Exact Poli GP Schema Theory for Subtree-Swapping Crossover is used.