A full discretisation of the time-dependent Boussinesq (buoyancy) model with nonlinear viscosity

Aldbaissy, Rim; Hecht, Frédéric; Mansour, Gihane; Sayah, Toni

doi:10.1007/s10092-018-0285-0

Cited by 18 publications

(15 citation statements)

References 14 publications

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“…Since in the present toolbox we use only iterative solvers (GMRES), this modification is not essential. It is in exchange important when LU-type direct solvers without pivoting are used, essentially for 2D calculations (UMFPACK solver in Rakotondrandisa et al (2020); Aldbaissy et al (2018), SuperLU in Woodfield et al (2019)). Finally, the penalty constant γ has no role in stabilizing the method.…”

Section: Finite-element Formulationmentioning

confidence: 99%

Parallel finite-element codes for the simulation of two-dimensional and three-dimensional solid–liquid phase-change systems with natural convection

Sadaka

Rakotondrandisa

Tournier

et al. 2020

Computer Physics Communications

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We present and distribute a FreeFem++ Toolbox for the parallel computing of twoor three-dimensional liquid-solid phase-change systems involving natural convection. FreeFem++ (www.freefem.org) is a free finite-element software available for all existing operating systems. We use the recent library ffddm that makes available in FreeFem++ state-of-the-art scalable Schwarz domain decomposition methods (DDM). The single domain approach used in our previous contribution [A. Rakotondrandisa, G. Sadaka, I. Danaila, A finite-element Toolbox for the simulation of solid-liquid phase-change systems with natural convection, Computer Physics Communications, Vol. 253, p. 107188, 2020] is adapted for the use of the DDM method. As a result, the computational time is considerably reduced for 2D configurations and furthermore 3D problems become affordable. The numerical method is based on an enthalpy-porosity model. The same set of equations is solved in both liquid and solid phases: the incompressible Navier-Stokes equations with Boussinesq approximation for thermal effects. A Carman-Kozeny-type penalty term is added to the momentum equations to bring progressively the velocity to zero into the solid. Model equations are discretized using Galerkin triangular or tetrahedral finite elements. The coupled system of equations is integrated in time using a second-order Gear implicit scheme. The resulting discrete equations are solved using a Newton algorithm. The DDM approach is based on an overlapping Schwarz method. The mesh is first split in subdomains using Scotch or Metis libraries. The final linear system is then solved in parallel using a GMRES Krylov method, with a Restricted Additive Schwarz (RAS) preconditioner. The mesh is adapted during the computation using metrics control. The 3D-mesh adaptivity uses the mmg (www.mmgtools.org) open source library. Parallel 2D and 3D computations of benchmark cases of increasing difficulty are presented: natural convection of air, natural convection of water, melting or solidification of a phase-change material, and, finally, a water freezing case. For each case, careful validations are provided and the performance of the code is assessed. The robustness of the Toolbox in 3D is also demonstrated by adapting the number of processors to the number of tetrahedra, which can considerably vary after the mesh adaptation.

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Section: Finite-element Formulationmentioning

confidence: 99%

Parallel finite-element codes for the simulation of two-dimensional and three-dimensional solid–liquid phase-change systems with natural convection

Sadaka

Rakotondrandisa

Tournier

et al. 2020

Computer Physics Communications

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“…A spectral discretization of the Navier-Stokes equation coupled with the heat equation has been proposed in [3], in the stationary case, and in [4] in the unsteady one. Finite Element approximation of the time dependent Boussinesq model with nonlinear viscosity depending on the temperature has been studied in [6]. Recently, the finite element approximation of the heat equation coupled with Stokes equations with nonlinear slip boundary conditions has been analyzed in [35].…”

Section: Introductionmentioning

confidence: 99%

Virtual Element Method for the Navier--Stokes Equation coupled with the Heat Equation

Antonietti¹,

Vacca²,

Verani³

2022

Preprint

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“…The chosen time discretisation is the backward differentiation formula of degree 2 (BDF2), which for k = 2 gives a method of order 2 in space and time. Existence of discrete solutions is established by the Brouwer fixed-point theory similarly as in [18], and the error analysis in the semi-discrete and fully-discrete settings is adapted from the theory of [5] for the Boussinesq equations.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

Divergence-conforming methods for transient doubly-diffusive flows: A priori and a posteriori error analysis

Bürger,

Khan,

Méndez

et al. 2021

Preprint

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The analysis of the double-diffusion model and H(div)-conforming method introduced in [Bürger, Méndez, Ruiz-Baier, SINUM (2019), 57:1318-1343 is extended to the time-dependent case. In addition, the efficiency and reliability analysis of residual-based a posteriori error estimators for the steady, semi-discrete, and fully discrete problems is established. The resulting methods are applied to simulate the sedimentation of small particles in salinity-driven flows. The method consists of Brezzi-Douglas-Marini approximations for velocity and compatible piecewise discontinuous pressures, whereas Lagrangian elements are used for concentration and salinity distribution. Numerical tests confirm the properties of the proposed family of schemes and of the adaptive strategy guided by the a posteriori error indicators.

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A full discretisation of the time-dependent Boussinesq (buoyancy) model with nonlinear viscosity

Cited by 18 publications

References 14 publications

Parallel finite-element codes for the simulation of two-dimensional and three-dimensional solid–liquid phase-change systems with natural convection

Parallel finite-element codes for the simulation of two-dimensional and three-dimensional solid–liquid phase-change systems with natural convection

Virtual Element Method for the Navier--Stokes Equation coupled with the Heat Equation

Divergence-conforming methods for transient doubly-diffusive flows: A priori and a posteriori error analysis

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