Maciej Woźniak

Dynamics with Matrices Possessing Kronecker Product Structure.

Abstract In this paper we present an application of Alternating Direction Implicit (ADI) algorithm for solution of non-stationary PDE-s using isogeometric finite element method. We show that ADI algorithm has a linear computational cost at every time step. We illustrate this approach by solving two example non-stationary three-dimensional problems using explicit Euler and Newmark time-stepping scheme: heat equation and linear elasticity equations for a cube. The stability of the simulation is controlled by monitoring the energy of the solution.

Fast GPU integration algorithm for isogeometric finite element method solvers using task dependency graphs

This article analyzes the integration for isogeometric finite element method solvers. In particular, it shows that isogeometric solvers with higher order B-splines spend significant amount of time for generation of the element frontal matrices when executed sequentially on CPU. The integration algorithm is represented as a sequence of basic undividable computational tasks and the dependency relation between them is identified. The basic tasks are defined for particular steps of the integration algorithm, for given integration points. In this article we show how to prepare independent sets of tasks that can be automatically scheduled and executed concurrently in a GPU card. This is done with the help of the graph expressing the dependency between tasks, constructed for the integration algorithm. The algorithm is implemented on GPU and tested on a sequence of numerical examples concerning the two dimensional isogeometric L2xa0−xa0projection problems for the MRI scans of the human head. The execution time of the concurrent GPU integration is compared with the sequential integration executed on CPU.

An Algorithm for Tensor Product Approximation of Three-Dimensional Material Data for Implicit Dynamics Simulations

In the paper, a heuristic algorithm for tensor product approximation with B-spline basis functions of three-dimensional material data is presented. The algorithm has an application as a preconditioner for implicit dynamics simulations of a non-linear flow in heterogeneous media using alternating directions method. As the simulation use-case, a non-stationary problem of liquid fossil fuels exploration with hydraulic fracturing is considered. Presented algorithm allows to approximate the permeability coefficient function as a tensor product what in turn allows for implicit simulations of the Laplacian term in the partial differential equation. In the consequence the number of time steps of the non-stationary problem can be reduced, while the numerical accuracy is preserved.

Coupled isogeometric Finite Element Method and Hierarchical Genetic Strategy with balanced accuracy for solving optimization inverse problem

Abstract The liquid fossil fuel reservoir exploitation problem (LFFEP) has not only economical signification but also strong natural environment impact. When the hydraulic fracturing technique is considered from the mathematical point of view it can be formulated as an optimization inverse problem, where we try to find optimal locations of pumps and sinks to maximize the amount of the oil extracted and to minimize the contamination of the groundwater. In the paper, we present combined solver consisting of the Hierarchical Genetic Strategy (HGS) with variable accuracy for solving optimization problem and isogeometric finite element method (IGA-FEM) with different mesh size for modeling a non-stationary flow of the non-linear fluid in heterogeneous media. The algorithm was tested and compared with the strategy using Simple Genetic Algorithm (SGA) as optimization algorithm and the same IGA-FEM solver for solving a direct problem. Additionally, a parallel algorithm for non-stationary simulations with isogeometric L2 projections is discussed and preliminarily assessed for reducing the computational cost of the solvers consisting of genetic algorithm and IGA-FEM algorithm. The theoretical asymptotic analysis which shows the correctness of algorithm and allows to estimate computational costs of the strategy is also presented.

Comparison of the structure of equation systems and the GPU multifrontal solver for finite difference, collocation and finite element method

Abstract The article is an in-depth comparison of numerical solvers and corresponding solution pro- cesses of the systems of algebraic equations resulting from finite difference, collocation, and finite element approximations. The paper considers recently developed isogeometric versions of the collocation and finite element methods, employing B-splines for the computations and ensuring Cp − xa01 continuity on the borders of elements for the B-splines of the order p . For solving the systems, we use our GPU implementation of the state-of-the-art parallel multifrontal solver, which leverages modern GPU architectures and allows to reduce the complexity. We analyze the structures of linear equation systems resulting from each of the methods and how different matrix structures lead to different multifrontal solver elimination trees. The paper also considers the flows of multifrontal solver depending on the originally employed method.

Hypergraph Grammar based Adaptive Linear Computational Cost Projection Solvers for Two and Three Dimensional Modeling of Brain

Abstract In this paper we present a hypergraph grammar model for transformation of two and three dimensional grids. The hypergraph grammar concerns the proces of generation of uniform grids with two or three dimensional rectangular or hexahedral elements, followed by the proces of h refinements, namely breaking selected elements into four or eight son elements, in two or three dimensions, respectively. The hypergraph grammar presented in this paper expresses also the two solver algorithms. The first one is the projection based interpolation solver algorithm used for computing H1 or L2 projections of MRI scan of human head, in two and three dimensions. The second one is the multi-frontal direct solver utilized in the loop of the Euler scheme for solving the non-stationary problem modeling the three dimensional heat transport in the human head generated by the cellphone usage.