Novel mixed and DG methods

新颖的混合和 DG 方法

• 批准号: 1211635
• 负责人: Jay Gopalakrishnan
• 金额: $ 16.12万
• 依托单位: Portland State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-11-18 至 2013-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211635&HistoricalAwards=false
• 关键词: Novel; mixed; DG; methods

Ever increasing demands on computational solution techniques necessitate development of new and improved methods. Two classes of methods, increasingly being used in simulation of physical and engineering systems, are finite element methods of the mixed type and the discontinuous Galerkin (DG) type. Building sound mathematical foundations for these methods increases their reliability, reveals avenues to improve them, and helps discover radically new methods. In this spirit, five lines of research are proposed on the following topics: (i) mixed methods (ii) discontinuous Petrov-Galerkin (DPG) schemes(iii) hybridizable discontinuous Galerkin (HDG) methods (iv) simulation of photonic membranes, and (v) complex axisymmetric simulations. The first deals with new stress elements and their implications in mixed methods for elasticity with weakly imposed stress symmetry. The second pursues a new DPG paradigm in the design of schemes where optimal test spaces are automatically computed. The third, concerns DG methods that mimic mixed methods, yet having the added advantage of flexible stabilization, and continues a line of research previously supported by the foundation. Both source problems and eigenproblems are considered. The remaining two lines of research, considers applications in need of new mathematical developments, e.g., (iv) needs good nonlinear eigensolvers and (v) needs sound treatment of singularities.Methods for computer simulation are an indispensable tool in modern scientific research. The proposed research brings fresh mathematical ingredients that spawn novel simulation methods. These mathematical techniques have the advantage of being broadly applicable. Accordingly, several disparate application areas can be targeted, including solid mechanics, transport phenomena, fluid flow, wave propagation, triggered lightning, and nanophotonic membranes. To detail a few examples, application of the new methods to fluid flow, through industrial and academic collaborations, can potentially benefit the aircraft industry. Reliable simulation methods can inexpensively guide experimentation of next generation nanophotonic devices. Finally, human resource development is integrated into the activities through training and participation of graduate students in the research.

对计算解决技术不断增长的需求需要开发新的和改进的方法。 越来越多地用于物理和工程系统模拟的两类方法是混合型和不连续伽辽金（DG）型有限元方法。 为这些方法建立良好的数学基础可以提高它们的可靠性，揭示改进它们的途径，并有助于发现全新的方法。 本着这种精神，提出了以下主题的五个研究方向：（i）混合方法（ii）不连续彼得罗夫-伽辽金（DPG）方案（iii）可杂交不连续伽辽金（HDG）方法（iv）光子膜模拟，以及(v) 复杂的轴对称模拟。第一个涉及新的应力元素及其在弱施加应力对称性弹性混合方法中的影响。第二个在方案设计中追求新的 DPG 范式，其中自动计算最佳测试空间。第三个涉及模仿混合方法的 DG 方法，但具有灵活稳定的额外优势，并继续了基金会之前支持的一系列研究。源问题和特征问题都被考虑。 其余两条研究方向考虑了需要新数学发展的应用，例如，（iv）需要良好的非线性本征解算器，（v）需要对奇点进行良好的处理。计算机模拟方法是现代科学研究中不可或缺的工具。 拟议的研究带来了新的数学成分，催生了新颖的模拟方法。这些数学技术具有广泛适用的优点。因此，可以针对几个不同的应用领域，包括固体力学、传输现象、流体流动、波传播、触发闪电和纳米光子膜。详细举几个例子，通过工业和学术合作将新方法应用于流体流动，可能会给飞机工业带来好处。 可靠的模拟方法可以廉价地指导下一代纳米光子器件的实验。 最后，通过研究生的培训和参与研究，将人力资源开发融入到活动中。