Javier Principe

Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling

Purpose – The purpose of this paper is to describe a variational multiscale finite element approximation for the incompressible Navier‐Stokes equations using the Boussinesq approximation to model thermal coupling.Design/methodology/approach – The main feature of the formulation, in contrast to other stabilized methods, is that the subscales are considered as transient and orthogonal to the finite element space. These subscales are solution of a differential equation in time that needs to be integrated. Likewise, the effect of the subscales is kept, both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation.Findings – This strategy allows the approaching of the problem of dealing with thermal turbulence from a strictly numerical point of view and discussion important issues, such as the relationship between the turbulent mechanical dissipation and the turbulent thermal dissipation.Originality/value – The treatment...

A variational subgrid scale model for transient incompressible flows

We introduce in this paper a variational subgrid scale model for the solution of the incompressible Navier–Stokes equations. With respect to classical multiscale-based stabilisation techniques, we retain the subgrid scale effects in the convective term and integrate the subgrid scale equation in time. The method is applied to the Navier–Stokes equations in an accelerating frame of reference and with Dirichlet (essential), Neumann (natural) and mixed boundary conditions. The concrete objective of the paper is to test a numerical algorithm for solving the non-linear subgrid scale equation and the introduction of the subgrid scale into the grid scale equation. The performance of the technique is demonstrated through the solution of two numerical examples: one to test the tracking of the subgrid scale in the convection term and the other to investigate the effects of considering the subgrid scale transient.

A Highly Scalable Parallel Implementation of Balancing Domain Decomposition by Constraints

In this work we propose a novel parallelization approach of two-level balancing domain decomposition by constraints preconditioning based on overlapping of fine-grid and coarse-grid duties in time. The global set of MPI tasks is split into those that have fine-grid duties and those that have coarse-grid duties, and the different computations and communications in the algorithm are then rescheduled and mapped in such a way that the maximum degree of overlapping is achieved while preserving data dependencies among them. In many ranges of interest, the extra cost associated to the coarse-grid problem can be fully masked by fine-grid related computations (which are embarrassingly parallel). Apart from discussing code implementation details, the paper also presents a comprehensive set of numerical experiments that includes weak scalability analyses with structured and unstructured meshes for the three-dimensional Poisson and linear elasticity problems on a pair of state-of-the-art multicore-based distributed-m...

Multilevel Balancing Domain Decomposition at Extreme Scales

In this paper we present a fully distributed, communicator-aware, recursive, and interlevel-overlapped message-passing implementation of the multilevel balancing domain decomposition by constraints (MLBDDC) preconditioner. The implementation highly relies on subcommunicators in order to achieve the desired effect of coarse-grain overlapping of computation and communication, and communication and communication among levels in the hierarchy (namely, interlevel overlapping). Essentially, the main communicator is split into as many nonoverlapping subsets of message-passing interface (MPI) tasks (i.e., MPI subcommunicators) as levels in the hierarchy. Provided that specialized resources (cores and memory) are devoted to each level, a careful rescheduling and mapping of all the computations and communications in the algorithm lets a high degree of overlapping be exploited among levels. All subroutines and associated data structures are expressed recursively, and therefore MLBDDC preconditioners with an arbitrar...

A stabilized finite element approximation of low speed thermally coupled flows

Purpose – The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number models with particular emphasis on the numerical implementation of the algorithm developed.Design/methodology/approach – The formulation, that allows us to consider convection dominated problems using equal order interpolation for all the valuables of the problem, is based on the subgrid scale concept. The full Newton linearization strategy gives rise to monolithic treatment of the coupling of variables whereas some fixed point schemes permit the segregated treatment of velocity‐pressure and temperature. A relaxation scheme based on the Armijo rule has also been developed.Findings – A full Newtown linearization turns out to be very efficient for steady‐state problems and very robust when it is combined with a line search strategy. A segregated treatment of velocity‐pressure and temperature happens to be more appropriate for transient probl...

A finite element dynamical nonlinear subscale approximation for the low Mach number flow equations

In this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows. The physical model is described by the low Mach number equations, which are obtained as a limit of the compressible Navier-Stokes equations in the small Mach number regime. In contrast to the commonly used Boussinesq approximation, this model permits to take volumetric deformation into account. Although the former is more general than the latter, both systems have similar mathematical structure and their numerical approximation can suffer from the same type of instabilities. We propose a stabilized finite element approximation based on the variational multiscale method, in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed. Modeling the subscale and taking its effect on the coarse scale problem into account results in a stable formulation. The quality of the final approximation (accuracy, efficiency) depends on the particular model. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms. The first ingredient permits to obtain an improved time discretization scheme (higher accuracy, better stability, no restrictions on the time step size). The second ingredient permits to prove global conservation properties. It also allows us to approach the problem of dealing with thermal turbulence from a strictly numerical point of view. Numerical tests show that nonlinear and dynamic subscales give more accurate solutions than classical stabilized methods.

FEMPAR: An Object-Oriented Parallel Finite Element Framework

FEMPAR is an open source object oriented Fortran200X scientific software library for the high-performance scalable simulation of complex multiphysics problems governed by partial differential equations at large scales, by exploiting state-of-the-art supercomputing resources. It is a highly modularized, flexible, and extensible library, that provides a set of modules that can be combined to carry out the different steps of the simulation pipeline. FEMPAR includes a rich set of algorithms for the discretization step, namely (arbitrary-order) grad, div, and curl-conforming finite element methods, discontinuous Galerkin methods, B-splines, and unfitted finite element techniques on cut cells, combined with h-adaptivity. The linear solver module relies on state-of-the-art bulk-asynchronous implementations of multilevel domain decomposition solvers for the different discretization alternatives and block-preconditioning techniques for multiphysics problems. FEMPAR is a framework that provides users with out-of-the-box state-of-the-art discretization techniques and highly scalable solvers for the simulation of complex applications, hiding the dramatic complexity of the underlying algorithms. But it is also a framework for researchers that want to experience with new algorithms and solvers, by providing a highly extensible framework. In this work, the first one in a series of articles about FEMPAR, we provide a detailed introduction to the software abstractions used in the discretization module and the related geometrical module. We also provide some ingredients about the assembly of linear systems arising from finite element discretizations, but the software design of complex scalable multilevel solvers is postponed to a subsequent work.

Dissipative Structure and Long Term Behavior of a Finite Element Approximation of Incompressible Flows with Numerical Subgrid Scale Modeling

In this chapter we summarize a finite element formulation for incompressible flows based on a two-scale decomposition of the velocity field, where the sub-grid scales are modeled numerically. The main features of the formulation are the choice for the space of the sub-grid scales, their time dependency and the fact that they are accounted for in all the terms where they appear. We present the main results obtained for this formulation, with emphasis on its dissipative structure and stability behavior in the long term, which give arguments to support the claim that it is able to model turbulent flows without any additional turbulence model.

On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections

In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain.

Finite element dynamical subgrid-scale model for low Mach number flows with radiative heat transfer

Purpose – The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative heat transfer. For high-temperature flows this coupling can have strong effects on the temperature and velocity fields. Design/methodology/approach – The basic numerical formulation has been proposed in previous works. It is based on the variational multiscale (VMS) concept in which the unknowns of the problem are divided into resolved and subgrid parts which are modeled to consider their effect into the former. The aim of the present paper is to extend this modeling to the case in which the low Mach number equations are coupled with radiation, also introducing the concept of subgrid scales for the radiation equations. Findings – As in the non-radiative case, an important improvement in the accuracy of the numerical scheme is observed when the nonlinear effects of the subgrid scales are taken into account. Besides it is possible to show...