Local Plus Global Adaptivity in Moving Node Finite Element Methods

9706353基思·米勒计划致力于移动节点有限元方法的局部和全局适应性研究。他计划首先(与普渡大学的尼尔·卡尔森合作)将2-D梯度加权移动有限元的局部适应性(通过节点的连续移动)与改进的全局适应性(通过增加、删除和重新连接节点，基于丰富的几何标准和局部误差估计)相结合。在这里，他们正在建立他们在开发GWMFE方法方面的长期和富有成效的合作。这种方法特别适合于发展急剧移动锋面的问题，特别是需要解决锋面精细结构的问题。GWMFE在本地移动节点以解决解决方案的尖锐特性方面做得非常好，但如果我们要拥有真正健壮和灵活的代码，显然需要全局自适应。在这里，他们将对米勒的学生库普拉特在他92年的论文中引入的部分全局适应性进行根本性的改变和补充，这大大增加了该方法的健壮性。其次，Miller计划发展一种具有流线扩散的修正GWMFE；这将纠正目前方法在某些稳态对流问题上存在的一些问题，其中GWMFE网格随流动向下游漂移，而不是进入稳定状态。这种修改后的GWMFE在一维计算中工作得非常好；成功地将其扩展到多维，将是对慢速瞬变和近稳态流体计算的重大进步。采用三角形网格的二维有限元方法通常计算偏微分方程组或偏微分方程组的解的分段线性近似，即近似解的图形是一个具有平面三角面的演化曲面。对于标准有限元方法，网格是指定和固定的；然而，对于GWMFE方法，允许网格变形，网格的节点自己决定如何移动。对于锋面急剧移动的问题，节点因此可以自动集中在前部并随其移动。通过这种方法，可以在解的关键区域获得高分辨率，同时使用比标准方法少得多的节点和大得多的时间步长。例如，硅芯片制造中掺杂砷离子的高度非线性扩散，半导体器件开关状态时空穴、电子和电压的纳秒演化的漂移-扩散方程，以及油层泛滥的“黑油”方程。GWMFE方法在这些问题和许多其他问题上都取得了惊人的成功，即使在逻辑上固定的网格的约束下也是如此；即网格自动变形，但其节点的数量和互联保持不变。然而，逻辑上固定的网格对于许多重要问题是不够的，显然，通过增加全局自适应(根据需要插入和删除节点)，该方法的效率和稳健性可以大大增强。这需要大量的代码开发，但GWMFE的局部和全局适应性的组合应该会产生一种准确和健壮的组合方法，使用更少的节点，从而使具有移动的尖锐特征的重要问题类能够有效地计算，而这些移动的尖锐特征目前是数值计算无法访问的。

9706353 Keith Miller Miller plans to work on local plus global adaptivity for moving node finite element methods. He plans first (in collaboration with Neil Carlson of Purdue) to combine the local adaptivity of 2-D gradient-weighted Moving Finite Elements (through the continuous movement of its nodes) with an improved global adaptivity (through addition, deletion and reconnection of nodes, based upon enriched geometric criteria and upon local error estimates). Here they are building upon their longstanding and fruitful collaboration in developing the GWMFE method. This method is especially suited to problems which develop sharp moving fronts, especially problems where one needs to resolve the fine-scale structure of the fronts. GWMFE does an extremely fine job of moving its nodes around locally to resolve the sharp features of the solution, but clearly global adaptivity is needed if we are to have truly robust and flexible codes. Here they would be making fundamental changes and additions to the partial global adaptivity introduced by Miller's student Kuprat in his thesis of '92, which added so greatly to the robustness of the method. Miller plans second to develop a revised GWMFE with streamline diffusion; this should correct some problems which the present method has on certain steady-state convection problems in which the GWMFE grid drifts downstream with the flow rather than coming to a steady-state. This revised GWMFE works extremely well in 1-D; success in extending it to multidimensions would be a significant advance for slow-transient and near steady-state fluid computations. Finite element (FE) methods with a triangular grid in 2-D typically compute a piecewise linear approximation to the solution of a partial differential equation (PDE) or system of PDEs; that is, the graph of the approximate solution is an evolving surface with planar triangular faces. For standard FE methods the grid is specified and fixed; however, for the GWMFE method the grid is allowed to deform and the nodes of the grid decide for themselves how to move. On problems with sharp moving fronts the nodes can thus automatically concentrate in the front and move with it. In this way one attains high resolution in the critical regions of the solution while using far fewer nodes and far larger time steps than with standard methods. Examples are the highly nonlinear diffusion of doped arsenic ions in the manufacture of silicon chips, the drift-diffusion equations for the nanosecond evolution of holes, electrons and voltages as a semiconductor device switches states, and the "black oil" equations for flooding of oil reservoirs. The GWMFE method has been strikingly successful on these and many other problems even within the constraints of a logically-fixed grid; that is, the grid deforms automatically, but the number and interconnections of its nodes remain fixed. However, a logically-fixed grid is inadequate for many important problems and it is apparent that the efficiency and robustness of the method can be greatly enhanced by adding global adaptivity (the insertion and deletion of nodes as needed). This requires a good deal of code development, but the combination of local plus global adaptivity in GWMFE should yield a combined method which is accurate and robust, which uses far fewer nodes, and which thereby renders efficiently computable important classes of problems with moving sharp features which are presently inaccessible to numerical computation.