The cost of continuity: performance of iterative solvers on isogeometric finite elements

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using $C^0$ B-splines, which span traditional finite element spaces, and $C^{p-1}$ B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size $h$ and polynomial order of approximation $p$. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most \slfrac{33p^2}{8} times more expensive for the more continuous space, although for moderately low $p$, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high $p$. Preconditioning options can be up to $p^3$ times more expensive to setup, although this difference significantly decreases for some popular preconditioners such as Incomplete LU factorization

Modelo numérico eficiente para flujo electrocinético en sistemas microfluídicos con geometrias complejas

Microfluidic devices like those used in chemical and biomedical applications basically consist of different networks of microchannels that interconnect chambers and reservoirs. The transport of fluids throughout the network is driven by pressure gradients, electric fields, or a combination of the two, which yields to the so-called electrokinetic flow. Analytical and numerical models have been used to aid in the design and simulation before fabrication with MEMS technology. Efficient numerical models are required since typical microchannel dimensions are in the range of several micrometers in width and depth and some centimeters in length. The numerical solution is carried out by using PETSC-FEM, for which we have developed a python interface for pre- and postprocessing using third-parties programs (Tetgen, Mayavi). A parallelizable preconditioner for Domain Decomposition Methods (DDM) by means of Finite Element discretization of Navier-Stokes equations is used to improve the convergence of problems with different scales like in microfluidic problems.Fil: Kler, Pablo Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Guarnieri, Fabio Ariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Dalcin, Lisandro Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentin

Modelo numerico eficiente para flujo electrocinetico en sistemas microfluidicos con geometrias complejas

Microfluidic devices like those used in chemical and biomedical applications basically consist of different networks of microchannels that interconnect chambers and reservoirs. The transport of fluids throughout the network is driven by pressure gradients, electric fields, or a combination of the two, which yields to the so-called electrokinetic flow. Analytical and numerical models have been used to aid in the design and simulation before fabrication with MEMS technology. Efficient numerical models are required since typical microchannel dimensions are in the range of several micrometers in width and depth and some centimeters in length. The numerical solution is carried out by using PETSC-FEM, for which we have developed a python interface for pre- and postprocessing using third-parties programs (Tetgen, Mayavi). A parallelizable preconditioner for Domain Decomposition Methods (DDM) by means of Finite Element discretization of Navier-Stokes equations is used to improve the convergence of problems with different scales like in microfluidic problems.Fil: Kler, Pablo Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Guarnieri, Fabio Ariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentina. Universidad Nacional de Entre Ríos; ArgentinaFil: Dalcin, Lisandro Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentin

A preconditioner for the Schur complement matrix

A preconditioner for iterative solution of the interface problem in Schur Complement Domain Decomposition Methods is presented. This preconditioner is based on solving a global problem in a narrow strip around the interface. It requires much less memory and computing time than classical Neumann–Neumann preconditioner and its variants, and handles correctly the flux splitting among subdomains that share the interface. The aim of this work is to present a theoretical basis (regarding the behavior of Schur complement matrix spectra) and some simple numerical experiments conducted in a sequential environment as a motivation for adopting the proposed preconditioner. Efficiency, scalability, and implementation details on a production parallel finite element code [Sonzogni V, Yommi A, Nigro N, Storti M. A parallel finite element program on a Beowulf cluster. Adv Eng Software 2002;33(7–10):427–43; Storti M, Nigro N, Paz R, Dalcín L. PETSc-FEM: a general purpose, parallel, multi-physics FEM program, 1999–2006] can be found in works [Paz R, Storti M. An interface strip preconditioner for domain decomposition methods: application to hydrology. Int J Numer Methods Eng 2005;62(13):1873–94; Paz R, Nigro N, Storti M. On the efficiency and quality of numerical solutions in cfd problems using the interface strip preconditioner for domain decomposition methods. Int J Numer Methods Fluids, in press].Fil: Storti, Mario Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Dalcin, Lisandro Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Paz, Rodrigo Rafael. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Yommi, Alejandra Karina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Sonzogni, Victorio Enrique. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Nigro, Norberto Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentin

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

On error-based step size control for discontinuous Galerkin methods for compressible fluid dynamics

We study temporal step size control of explicit Runge-Kutta methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc