Local Plus Global Adaptivity in Moving Node Finite Element Methods

移动节点有限元方法中的局部加全局自适应

• 批准号: 9706353
• 负责人: Keith Miller
• 金额: $ 7.07万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706353&HistoricalAwards=false
• 关键词: Local; Plus; Global; Adaptivity; Moving

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.

9706353基思米勒 米勒计划研究移动节点有限元方法的局部加全局自适应性。他计划首先（与普渡大学的Neil Carlson合作）将2-D梯度加权移动有限元的局部自适应性（通过其节点的连续移动）与改进的全局自适应性（通过节点的添加，删除和重新连接，基于丰富的几何标准和局部误差估计）结合联合收割机。在这里，他们正在建立在他们的长期和富有成效的合作，在开发GWMFE方法。这种方法特别适合于发展尖锐的运动锋的问题，特别是需要解决锋的精细尺度结构的问题。GWMFE在局部移动节点以解决方案的尖锐特征方面做得非常好，但如果我们要拥有真正强大和灵活的代码，显然需要全局自适应性。在这里，他们将对米勒的学生库普拉特在92年的论文中引入的部分全局自适应性进行根本性的修改和补充，这大大增加了该方法的鲁棒性。米勒计划第二步发展一种带有流线扩散的修正的GWMFE;这将纠正当前方法在某些稳态对流问题上存在的一些问题，在这些问题中，GWMFE网格随流动向下游漂移，而不是达到稳态。这种修订后的GWMFE工程非常好，在1-D的成功，将其扩展到多维将是一个显着的进步缓慢瞬态和近稳态流体计算。 二维三角形网格的有限元（FE）方法通常计算偏微分方程（PDE）或PDE系统的解的分段线性近似;也就是说，近似解的图形是具有平面三角形面的演化曲面。对于标准的有限元方法，网格是指定和固定的;然而，对于GWMFE方法，网格可以变形，网格的节点自己决定如何移动。在具有急剧移动前沿的问题中，节点可以 因此，自动集中在前面，并与它一起移动。通过这种方式，在解决方案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，半导体器件状态转换时空穴、电子和电压纳秒演化的漂移扩散方程，以及油藏注水的“黑油”方程。GWMFE方法在这些问题和许多其他问题上取得了惊人的成功，即使在逻辑固定网格的约束下也是如此;也就是说，网格自动变形，但其节点的数量和互连保持固定。然而，一个逻辑上固定的网格是不够的，许多重要的问题，它是显而易见的，该方法的效率和鲁棒性可以大大提高通过添加全局自适应性（插入和删除节点的需要）。这需要大量的代码开发，但是GWMFE中的局部和全局自适应性的组合应该产生一种精确和鲁棒的组合方法， 更少的节点，从而有效地呈现可计算的重要类别的问题与移动尖锐的功能，目前无法访问的数值计算。