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

Heuristic algorithm to predict the location of C^{0} separators for efficient isogeometric analysis simulations with direct solvers

We focus on two and three-dimensional isogeometric finite element method computations with tensor product Ck B-spline basis functions. We consider the computational cost of the multi-frontal direct solver algorithm executed over such tensor product grids. We present an algorithm for estimation of the number of floating-point operations per mesh node resulting from the execution of the multi-frontal solver algorithm with the ordering obtained from the element partition trees. Next, we propose an algorithm that introduces C0 separators between patches of elements of a given size based on the stimated number of flops per node. We show that the computational cost of the multi-frontal solver algorithm executed over the computational grids with C0 separators introduced is around one or two orders of magnitude lower, while the approximability of the functional space is improved. We show O(NlogN) computational complexity of the heuristic algorithm proposing the introduction of the C0 separators between the patches of elements, reducing the computational cost of the multi-frontal solver algorithm

Deep learning driven self-adaptive hp finite element method

The fi nite element method (FEM) is a popular tool for solving engineering problems governed by Partial Di fferential Equations (PDEs). The accuracy of the numerical solution depends on the quality of the computational mesh. We consider the self-adaptive hp-FEM, which generates optimal mesh refi nements and delivers exponential convergence of the numerical error with respect to the mesh size. Thus, it enables solving di ficult engineering problems with the highest possible numerical accuracy. We replace the computationally expensive kernel of the refi nement algorithm with a deep neural network in this work. The network learns how to optimally re fine the elements and modify the orders of the polynomials. In this way, the deterministic algorithm is replaced by a neural network that selects similar quality refi nements in a fraction of the time needed by the original algorithm

Fast parallel IGA-ADS solver for time-dependent Maxwell's equations

We propose a simulator for time-dependent Maxwell's equations with linear computational cost. We employ B-spline basis functions as considered in the isogeometric analysis (IGA). We focus on non-stationary Maxwell's equations defined on a regular patch of elements. We employ the idea of alternating-directions splitting (ADS) and employ a second-order accurate time-integration scheme for the time-dependent Maxwell's equations in a weak form. After discretization, the resulting stiffness matrix exhibits a Kronecker product structure. Thus, it enables linear computational cost LU factorization. Additionally, we derive a formulation for absorbing boundary conditions (ABCs) suitable for direction splitting. We perform numerical simulations of the scattering problem (traveling pulse wave) to verify the ABC. We simulate the radiation of electromagnetic (EM) waves from the dipole antenna. We verify the order of the time integration scheme using a manufactured solution problem. We then simulate magnetotelluric measurements. Our simulator is implemented in a shared memory parallel machine, with the GALOIS library supporting the parallelization. We illustrate the parallel efficiency with strong and weak scalability tests corresponding to non-stationary Maxwell simulations.EXPERTIA (KK-2021/00048) SIGZE (KK-2021/00095) PDC2021-121093-I0

The value of continuity: Refined isogeometric analysis and fast direct solvers

We propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce . C0-separators to reduce the interconnection between degrees of freedom in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method "refined Isogeometric Analysis (rIGA)". To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing the Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between . p2 and . p3, with . p being the polynomial order of the discretization. Numerical results indicate that our proposed refined isogeometric analysis delivers a speed-up factor proportional to . p2. In a . 2D mesh with four million elements and . p=5, the linear system resulting from rIGA is solved 22 times faster than the one from highly continuous IGA. In a . 3D mesh with one million elements and . p=3, the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis

Zbieżność solwerów iteracyjnych dla nieliniowych symulacji procesu nanolitografii przez naświetlanie i wyciskanie

The paper presents the analysis of the iterative solvers utilized to solve the non-linear problem of Step-and-Flash Imprint Lithography (SFIL) a modern patterning process. The simulations consists in solving molecular statics problem for the polymer network, with quadratic potentials. The model distinguishes the strong interparticle interactions between particles forming a polymer network, and weak interactions between remaining particles. It also allows for large deformations, which all together implies the non-linear model. To illustrate the convergence of the iterative solvers, we present snapshots of the deformation of the sample being subject to the iterative solution. We claim that the animation is an interesting way of illustrating the convergence of the iterative solvers.Artykuł analizuje zbieżność solwerów iteracyjnych dla nieliniowych symulacji procesu nanolitografii przez naświetlanie i wyciskanie. Symulacje polegają na rozwiązaniu zadania statyki cząsteczkowej dla sieci polimerów, w którym przyjęto kwadratowe potencjały międzycząsteczkowe, rozróżniono silniejsze oddziaływania pomiędzy cząstkami tworzącymi łańcuchy polimerów oraz słabsze oddziaływania pomiędzy pozostałymi cząstkami, a także dopuszczono występowanie dużych odkształceń, co implikuje model nieliniowy. W celu ilustracji zbieżności solwerów przedstawiono wizualizacje odksztalceń sieci polimerów w kolejnych iteracjach. Taka animacja jest interesującą metodą ilustracji zbieżności solwerów iteracyjnych

Applications of alternating direction solver for simulations of time-dependent problems

This paper deals with the application of an Alternating Direction Solver (ADS) to a non-stationary linear elasticity problem solved with the isogeometric finite element method (IGA-FEM). Employing a tensor product B-spline basis in isogeometric analysis under some restrictions leads to a system of linear equations with a matrix possessing a tensor product structure. The ADI algorithm is a direct method that exploits this Kronecker product structure to solve the system in O (N), where N is the number of degrees of freedom (basis functions). This is asymptotically faster than state-of-the-art, general-purpose, multi-frontal direct solvers when applied to explicit dynamics. In this paper, we also present a complexity analysis of the ADS incorporating dependence on the B-spline basis of order p

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

In this paper, we present a solver for non-stationary problems using L 2 projec- tion and h -adaptations. The solver utilizes the Euler time integration scheme for time evolution mixed with projection-based interpolation techniques for solving the L 2 projection problem at every time step. The solver is tested on the model problem of a heat transfer in an L-shape domain. We show that our solver delivers linear computational cost at every time step