This paper explores, for the first time, the application of the novel mesh-free local radial basis function collocation method (LRBFCM) to the solution of a multi-physics problem in three dimensions. A related benchmark problem is solved by considering the natural convection of an incompressible Newtonian fluid in a differentially heated cubic cavity with and without the application of a magnetic field. The research is limited to typical magnetic fields used in the magnetohydrodynamic processing of liquid metals. For this purpose the assumption of small magnetic Reynolds numbers Re_m ≪ 1 is made. Spatial discretization is performed by local non-uniform collocation with scaled multiquadrics radial basis functions (RBFs) with the shape parameter set to a constant value and the explicit Euler formula used to perform the time stepping. The involved temperature, velocity and pressure fields are represented on overlapping seven-nodded sub-domains. The pressure-velocity coupling is resolved by the fractional step method. The originality of the contribution represents LRBFCM solution of the classic three-dimensional steady natural convection benchmark for Rayleigh numbers from 10⁵ to 10⁷ and Prandtl number 0.71, and its extension to Prandtl number 0.1, and Hartman numbers 0, 10, 50 and 100. The accuracy of the LRBFCM is found to be comparable with the published benchmark results obtained using established numerical methods.

本文首次探讨了新颖的无网格局部径向基函数配置方法（LRBFCM）在三维多物理场问题求解中的应用。通过考虑不可加热的牛顿流体在有或没有施加磁场的情况下在差热立方腔中的自然对流来解决相关的基准问题。该研究仅限于液态金属的磁流体动力学处理中使用的典型磁场。为此，假定雷诺磁数Re _m小 is 1被制作。空间离散化是通过将局部非均匀搭配与可缩放的多二次径向基函数（RBF）配合使用的，其中形状参数设置为常数，并且使用显式欧拉公式执行时间步长。涉及的温度，速度和压力场在重叠的七点子域上表示。压力-速度耦合通过分步法解决。贡献的独创性代表了瑞利数从10 ⁵到10 ⁷的经典三维稳定自然对流基准的LRBFCM解决方案 Prandtl数为0.71，其扩展为Prandtl数为0.1，Hartman数为0、10、50和100。发现LRBFCM的精度与使用已建立的数值方法获得的已发布基准结果相当。