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

Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines

This paper derives theoretical estimates of the computational cost for isogeometric multi-frontal direct solver executed on parallel distributed memory machines. We show theoretically that for the Cp-1 global continuity of the isogeometric solution, both the computational cost and the communication cost of a direct solver are of order O(log(N)p2) for the one dimensional (1D) case, O(Np2) for the two dimensional (2D) case, and O(N4/3p2) for the three dimensional (3D) case, where N is the number of degrees of freedom and p is the polynomial order of the B-spline basis functions. The theoretical estimates are verified by numerical experiments performed with three parallel multi-frontal direct solvers: MUMPS, PaStiX and SuperLU, available through PETIGA toolkit built on top of PETSc. Numerical results confirm these theoretical estimates both in terms of p and N. For a given problem size, the strong efficiency rapidly decreases as the number of processors increases, becoming about 20% for 256 processors for a 3D example with 1283 unknowns and linear B-splines with C0 global continuity, and 15% for a 3D example with 643 unknowns and quartic B-splines with C3 global continuity. At the same time, one cannot arbitrarily increase the problem size, since the memory required by higher order continuity spaces is large, quickly consuming all the available memory resources even in the parallel distributed memory version. Numerical results also suggest that the use of distributed parallel machines is highly beneficial when solving higher order continuity spaces, although the number of processors that one can efficiently employ is somehow limited

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

Comparison of the Structure of Equation Systems and the GPU Multifrontal Solver for Finite Difference, Collocation and Finite Element Method

AbstractThe 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−1 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

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

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

Weisheit als Gabe des Heiligen Geistes nach Thomas von Aquin

Mądrość jako dar Ducha Świętego według św. Tomasza z Akwin