A two-dimensional adaptive remeshing method for solving melting and solidification problems with convection

Belhamadia, Youssef; Fortin, A.; Briffard, Thomas

doi:10.1080/10407782.2019.1627837

Cited by 22 publications

(6 citation statements)

References 33 publications

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“…4 the lower values of the errors obtained with this discretization). Similar discretizations using Taylor-Hood finite elements for the fluid flow and P 2 for the temperature were used in recent contributions by Woodfield et al (2019); Belhamadia et al (2019). Figure 4: Space accuracy of the numerical scheme tested using the Burggraf manufactured solution.…”

Section: Space Accuracy: Burggraf Stationary Flow With Thermal Effectsmentioning

confidence: 99%

“…Zimmerman and Kowalski (2018) implemented the variable viscosity model suggested by Danaila et al (2014) in a different finite-element framework using FEniCS and an AMR (Adaptive Mesh Refinement) technique based on a dual-weighted residual method. In a very recent contribution, Belhamadia et al (2019) derived a time-dependent adaptive remeshing method for phase-change problem with convection based on an error estimator applicable to second or higher order variables.…”

Section: Introductionmentioning

confidence: 99%

“…Temperature is discretized using P 2 or P 1 finite elements. The choice of using Taylor-Hood finite elements for the fluid flow and P 2 for the temperature was also made in Woodfield et al (2019); Belhamadia et al (2019). It offers second order accuracy and (inf-sup) stability of the space discretization of the fluid flow.…”

Section: Introductionmentioning

confidence: 99%

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A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

Rakotondrandisa

Sadaka

Danaila

2020

Computer Physics Communications

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We present and distribute a new numerical system using classical finite elements with mesh adaptivity for computing two-dimensional liquid-solid phase-change systems involving natural convection. The programs are written as a toolbox for FreeFem++ (www3.freefem.org), a free finite-element software available for all existing operating systems. The code implements a single domain approach. The same set of equations is solved in both liquid and solid phases: the incompressible Navier-Stokes equations with Boussinesq approximation for thermal effects. This model describes naturally the evolution of the liquid flow which is dominated by convection effects. To make it valid also in the solid phase, a Carman-Kozeny-type penalty term is added to the momentum equations. The penalty term brings progressively (through an artificial mushy region) the velocity to zero into the solid. The energy equation is also modified to be valid in both phases using an enthalpy (temperature-transform) model introducing a regularized latent-heat term. Model equations are discretized using Galerkin triangular finite elements. Piecewise quadratic (P2) finite-elements are used for the velocity and piecewise linear (P1) for the pressure. For the temperature both P2 or P1 discretizations are possible. The coupled system of equations is integrated in time using a second-order Gear scheme. Non-linearities are treated implicitly and the resulting discrete equations are solved using a Newton algorithm. An efficient mesh adaptivity algorithm using metrics control is used to adapt the mesh every time step. This allows us to accurately capture multiple solid-liquid interfaces present in the domain, the boundary-layer structure at the walls and the unsteady convection cells in the liquid. We present several validations of the toolbox, by simulating benchmark cases of increasing difficulty: natural convection of air, natural convection of water, melting of a phase-change material, a melting-solidification cycle, and, finally, a water freezing case. Other similar cases could be easily simulated with this toolbox, since the code structure is extremely versatile and the syntax very close to the mathematical formulation of the model.

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Section: Space Accuracy: Burggraf Stationary Flow With Thermal Effectsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

Rakotondrandisa

Sadaka

Danaila

2020

Computer Physics Communications

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“…Regarding the design and rigorous analysis of residual-based a posteriori error estimators for flow-transport couplings, the literature is predominantly focused on the stationary case (see, e.g., [3,6,7,9,13,22,37,42] and the references therein). Only a few results are available for the time-dependent regime, from which we mention the adaptive mixed method for Richards equation in porous media [15], the remeshing scheme based on goal-oriented adaptivity for solidification problems advanced in [14], the collection of adaptive schemes for reactive flow discussed in [16] and for heat transfer in [30]. However, none of these theoretical frameworks is directly applicable to (1.1) using divergence-conforming approximations.…”

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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“…Adaptive FE methods were suggested for classical two-phase Stefan problem in 2D and 3D (Belhamadia et al, 2004a,b). For phasechange systems with convection, adaptivity strategies were suggested and tested only for 2D problems (Belhamadia et al, 2012;Danaila et al, 2014;Belhamadia et al, 2019). An attempt to adapt the mesh in 3D simulations for melting phenomena was undertaken by Zimmerman and Kowalski (2018) using an AMR (Adaptive Mesh Refinement) technique based on a dual-weighted residual method.…”

Section: Introductionmentioning

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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A two-dimensional adaptive remeshing method for solving melting and solidification problems with convection

Cited by 22 publications

References 33 publications

A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection

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

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

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