Development of the numerical scheme and finite element analysis for thermo-electrically conducting fluids

在热电流体磁场数值格式中，提出了原始变量法。在上述方案中需要强调的一点是有效地满足磁通守恒定律。在本研究中，我们比较了两种磁场的数值格式，即交叉螺旋度法和数值残差法，重点讨论了速度和磁场的泊松方程的收敛性。此外，还定量地阐明了低普朗特数流体的周期性与洛伦兹力的对流抑制效应之间的关系。利用最大熵法计算了流体的动能、熵值、回力值和磁交叉螺旋度的频率特性。在有限差分法领域发展起来的有限解析方法的基础上，提出了一种新的混合流线迎风有限元方法。为了得到最优的形状函数，我们在稳定平流扩散方程的微分算子上引入了伴随微分算子。满足这些微分算子的形函数是相互对偶的。其中一个函数精确地插值了对流主导流中的函数，另一个函数则是一个混合流-流-逆风权重函数。并且，我们定义了一个离散的del算子来减少计算机的存储空间。从而实现了有限元法公式的简化和计算速度的提高。

The primitive variable methods have been proposed in the numerical scheme of magnetic field of thermo-electrically conducting fluids. The point to be emphasized in the above-mentioned scheme is to satisfy the conservation law of magnetic flux efficiently. In the present research, we have compared the two numerical schemes of magnetic field, namely the cross helicity method and the numerical residual method, concentrating on convergence of Poisson's equation of the velocity and magnetic fields. Moreover, the relationship between the periodicity of low prandtl number fluids and the convective inhibitation effect by Lorentz force has been clarified quantitatively. The frequency characteristics of kinetic energy, enstrophy, palinstrophy and hydromagnetic cross helicity are calculated by the maximum entropy method. Furthermore, we have presented a new hybrid-streamline-upwind finite-element method, which is based on the finite analytic method developed in the field of the finite difference method. In order to obtain an optimal shape function, we have introduced an adjoint differential operator to the differential operator for a steady advection-diffusion equation. The shape functions which satisfy these differential operators are mutually dual. One of them interpolates accurately the functions appearing in convection-dominated flows, the other becomes a hybrid-streamlime-upwind weight function. And, we have defined a discrete del operator for the reduction of memory storage in the computer.As a result, we have achieved simplicity of formulation and high-speed calculation of the finite-element method.