We present and distribute a FreeFem++ Toolbox for the parallel computing of two-or 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] 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.

中文翻译：

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用于模拟具有自然对流的二维和三维固液相变系统的并行有限元代码

我们提出并分发了一个 FreeFem++ 工具箱，用于并行计算涉及自然对流的二维或三维液固相变系统。FreeFem++ (www.freefem.org) 是一个免费的有限元软件，适用于所有现有的操作系​​统。我们使用最新的库 ffddm，它在 FreeFem++ 最先进的可扩展 Schwarz 域分解方法 (DDM) 中可用。我们之前的贡献中使用的单域方法 [A. Rakotondrandisa, G. Sadaka, I. Danaila, 用于模拟固液相变系统的有限元工具箱，适用于 DDM 方法的使用。因此，2D 配置的计算时间显着减少，而且 3D 问题变得可以负担得起。数值方法基于焓-孔隙率模型。在液相和固相中求解同一组方程：不可压缩的 Navier-Stokes 方程与热效应的 Boussinesq 近似。Carman-Kozeny 类型的惩罚项被添加到动量方程中，以逐渐使固体的速度为零。模型方程使用伽辽金三角形或四面体有限元进行离散化。使用二阶齿轮隐式方案对耦合方程组进行时间积分。所得离散方程使用牛顿算法求解。DDM 方法基于重叠 Schwarz 方法。首先使用 Scotch 或 Metis 库将网格划分为子域。然后使用 GMRES Krylov 方法和受限加法施瓦茨 (RAS) 预处理器并行求解最终线性系统。网格在计算过程中使用度量控制进行调整。3D 网格自适应使用 mmg (www.mmgtools.org) 开源库。提出了难度越来越大的基准案例的并行 2D 和 3D 计算：空气的自然对流、水的自然对流、相变材料的熔化或凝固，以及最后的水冻结案例。对于每种情况，都会提供仔细的验证并评估代码的性能。工具箱在 3D 中的稳健性还通过将处理器数量调整为四面体数量来证明，四面体数量在网格调整后可能会发生很大变化。空气的自然对流、水的自然对流、相变材料的熔化或凝固，最后是水冻结的情况。对于每种情况，都会提供仔细的验证并评估代码的性能。工具箱在 3D 中的稳健性还通过将处理器数量调整为四面体数量来证明，四面体数量在网格调整后可能会发生很大变化。空气的自然对流、水的自然对流、相变材料的熔化或凝固，最后是水冻结的情况。对于每种情况，都会提供仔细的验证并评估代码的性能。工具箱在 3D 中的稳健性还通过将处理器数量调整为四面体数量来证明，四面体数量在网格调整后可能会发生很大变化。