Application of projection-based interpolation algorithm for non-stationary problem

In this paper we present a solver for non-stationary problems using L2 projection and h-adaptations. The solver utilizes the Euler time integration scheme for time evolution mixed with the projection based interpolation techniques for solving the L2 projections problem at every time step. The solver is tested on the model problem of the heat transfer in L-shape domain. We show that our solver delivers linear computational cost at every time step

Application of a Hierarchical Chromosome Based Genetic Algorithm to the Problem of Finding Optimal Initial Meshes for the Self-Adaptive hp-FEM

The paper presents an algorithm for finding the optimal initial mesh for the self-adaptive hp Finite Element Method (hp-FEM) calculations. We propose the application of the hierarchical chromosome based genetic algorithm for optimal selection of the initial mesh. The selection of the optimal initial mesh will optimize the convergence rate of the numerical error of the solution over the sequence of meshes generated by the self-adaptive hp-FEM. This is especially true in the case when material data are selected as a result of some stochastic algorithm and it is not possible to design optimal initial mesh by hand. The algorithm has been tested on the non-stationary mass transport problem modeling phase transition phenomenon

Petrov-Galerkin formulation equivallent to the residual minimization method for finding an optimal test function

Numerical solutions of Partial Differential Equations with Finite Element Method have multiple applications in science and engineering. Several challenging problems require special stabilization methods to deliver accurate results of the numerical simulations. The advection-dominated diffusion problem is an example of such problems. They are employed to model pollution propagation in the atmosphere. Unstable numerical methods generate unphysical oscillations, and they make no physical sense. Obtaining accurate and stable numerical simulations is difficult, and the method of stabilization depends on the parameters of the partial differential equations. They require a deep knowledge of an expert in the field of numerical analysis. We propose a method to construct and train an artificial expert in stabilizing numerical simulations based on partial differential equations. We create a neural network-driven artificial intelligence that makes decisions about the method of stabilizing computer simulations. It will automatically stabilize difficult numerical simulations in a linear computational cost by generating the optimal test functions. These test functions can be utilized for building an unconditionally stable system of linear equations. The optimal test functions proposed by artificial intelligence will not depend on the right-hand side, and thus they may be utilized in a large class of PDE-based simulations with different forcing and boundary conditions. We test our method on the model one-dimensional advection-dominated diffusion problem

CONVERGENCE OF ITERATIVE SOLVERS FOR NON-LINEAR STEP-AND-FLASH IMPRINT LITHOGRAPHY SIMULATIONS

The paper presents the analysis of the iterative solvers utilized to solve the non-linear problemof Step-and-Flash Imprint Lithography (SFIL) a modern patterning process. The simulationsconsists in solving molecular statics problem for the polymer network, with quadratic potentials.The model distinguishes the strong interparticle interactions between particles forminga polymer network, and weak interactions between remaining particles. It also allows for largedeformations, which all together implies the non-linear model. To illustrate the convergenceof the iterative solvers, we present snapshots of the deformation of the sample being subjectto the iterative solution. We claim that the animation is an interesting way of illustratingthe convergence of the iterative solvers

One-dimensional fully automatic h-adaptive isogeometric finite element method package

This paper deals with an adaptive finite element method originally developedby Prof. Leszek Demkowicz for hierarchical basis functions. In this paper, weinvestigate the extension of the adaptive algorithm for isogeometric analysisperformed with B-spline basis functions. We restrict ourselves to h-adaptivity,since the polynomial order of approximation must be fixed in the isogeometriccase. The classical variant of the adaptive FEM algorithm, as delivered by thegroup of Prof. Demkowicz, is based on a two-grid paradigm, with coarse andfine grids (the latter utilized as a reference solution). The problem is solved independentlyover a coarse mesh and a fine mesh. The fine-mesh solution is thenutilized as a reference to estimate the relative error of the coarse-mesh solutionand to decide which elements to refine. Prof. Demkowicz uses hierarchicalbasis functions, which (though locally providing C p−1 continuity) ensure onlyC 0 on the interfaces between elements. The CUDA C library described in thispaper switches the basis to B-spline functions and proposes a one-dimensionalisogeometric version of the h-adaptive FEM algorithm to achieve global C p−1continuity of the solution

Applications of Alternating Direction Solver for simulations of time-dependent problems

This paper deals with application of Alternating Direction solver (ADS) to nonstationarylinear elasticity problem solved with isogeometric FEM. Employingtensor product B-spline basis in isogeometric analysis under some restrictionsleads to system of linear equations with matrix possessing tensor product structure.Alternating Direction Implicit algorithm is a direct method that exploitsthis structure to solve the system in O (N ), where N is a number of degreesof freedom (basis functions). This is asymptotically faster than state-of-theartgeneral purpose multi-frontal direct solvers. In this paper we also presentthe complexity analysis of ADS incorporating dependence on order of B-splinebasis

h-RELATION PERSONALIZED COMMUNICATION STRATEGY

This paper considers the communication patterns arising from the partition of geometricaldomain into sub-domains, when data is exchanged between processors assigned to adjacentsub-domains. It presents the algorithm constructing bipartite graphs covering the graphrepresentation of the partitioned domain, as well as the scheduling algorithm utilizing thecoloring of the bipartite graphs. Specifically, when the communication pattern arises from thepartition of a 2D geometric area, the planar graph representation of the domain is partitionedinto not more than two bipartite graphs and a third graph with maximum vertex valency 2,by means of the presented algorithm. In the general case, the algorithm finds h−1 or fewerbipartite graphs, where h is the maximum number of neighbors. Finally, the task of messagescheduling is reduced to a set of independent scheduling problems over the bipartite graphs.The algorithms are supported by a theoretical discussion on their correctness and efficiency

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

This paper describes the application of hypergraph grammars to drive linear computationalcost solver for grids with point singularities. Such graph grammar productions are the rstmathematical formalism used to describe solver algorithm and each of them indicates thesmallest atomic task that can be executed in parallel, which is very useful in case of parallelexecution. In particular the partial order of execution of graph grammar productions can befound, and the sets of independent graph grammar productions can be localized. They canbe scheduled set by set into shared memory parallel machine. The graph grammar basedsolver has been implemented with NIVIDIA CUDA for GPU. Graph grammar productionsare accompanied by numerical results for 2D case. We show that our graph grammar basedsolver with GPU accelerator is order of magnitude faster than state of the art MUMPSsolver

Parallel Fast Isogeometric Solvers for Explicit Dynamics

This paper presents a parallel implementation of the fast isogeometric solvers for explicit dynamics for solving non-stationary time-dependent problems. The algorithm is described in pseudo-code. We present theoretical estimates of the computational and communication complexities for a single time step of the parallel algorithm. The computational complexity is O(p^6 N/c t_comp) and communication complexity is O(N/(c^(2/3)t_comm) where p denotes the polynomial order of B-spline basis with Cp-1 global continuity, N denotes the number of elements and c is number of processors forming a cube, t_comp refers to the execution time of a single operation, and t_comm refers to the time of sending a single datum. We compare theoretical estimates with numerical experiments performed on the LONESTAR Linux cluster from Texas Advanced Computing Center, using 1 000 processors. We apply the method to solve nonlinear flows in highly heterogeneous porous media