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

Strategia komunikacji oparta na relacji h

This paper considers the communication patterns arising from the partition of geometrical domain into sub-domains, when data is exchanged between processors assigned to adjacent sub-domains. It presents the algorithm constructing bipartite graphs covering the graph representation of the partitioned domain, as well as the scheduling algorithm utilizing the coloring of the bipartite graphs. Specifically, when the communication pattern arises from the partition of a 2D geometric area, the planar graph representation of the domain is partitioned into 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 fewer bipartite graphs, where h is the maximum number of neighbors. Finally, the task of message scheduling 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.W artykule omówiono problem szeregowania komunikacji pomiędzy procesorami przypisanymi do poddziedzin otrzymanych w wyniku podziału obszaru na podobszary, przy założeniu, że dane wymieniane są pomiędzy sąsiadującymi podobszarami. W artykule przedstawiony został algorytm tworzenia grafów dwudzielnych w oparciu o grafową reprezentację obszaru podzielonego na podobszary. Przedstawiono również algorytm szeregowania bazujący na kolorowaniu skonstruowanych grafów dwudzielnych. W szczególności, kiedy rozważamy komunikację w obrębie obszarów dwuwymiarowych, graf reprezentujący podzielony obszar dwuwymiarowy jest grafem planarnym, i rozważany algorytm zdekomponuje go na dwa grafy dwudzielne oraz trzeci graf o maksymalnej walencji wierzchołka równej 2. W ogólnym przypadku (np. gdy rozważamy obszary trójwymiarowe) przedstawiony algorytm znajdzie h — 1 lub mniej grafów dwudzielnych, gdzie h oznacza maksymalną liczbę sąsiadujących podobszarów. Zadanie szeregowania komunikatów zostało zredukowane do niezależnych zadań szeregowania na grafach dwudzielnych. Artykuł podsumowuje analiza teoretyczna poprawności i efektywności omówionych algorytmów

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