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.

对计算解决方案技术的需求不断提高，需要开发新的和改进的方法。 两类的方法，越来越多地用于模拟物理和工程系统，是混合类型和不连续的Galerkin（DG）类型的有限元方法。 为这些方法建立声音数学基础，提高了它们的可靠性，揭示了改善它们的途径，并有助于发现彻底的新方法。 以这种精神，提出了以下主题的五项研究：（i）混合方法（ii）不连续的彼得 - 盖奶酪（DPG）方案（iii）可混合的不连续的盖素（HDG）方法（IV）方法（IV）方法（IV）光子膜和（V）复杂轴对比的轴对比模拟。第一批涉及新的应力元素及其对弹性弱的应力对称性弹性的混合方法的影响。第二个在设计自动计算最佳测试空间的方案的设计中追求新的DPG范式。第三个是涉及模仿混合方法的DG方法，但具有柔性稳定的额外优势，并继续了以前由基金会支持的研究线。源问题和本本概述都被考虑。 其余的两条研究是考虑需要新数学发展的应用，例如（iv）需要良好的非线性特征素，（v）需要对奇异性的合理处理。用于计算机模拟的方法是现代科学研究中必不可少的工具。 提出的研究带来了新的数学成分，这些成分产生了新型的模拟方法。这些数学技术具有广泛适用的优势。因此，可以针对几个不同的应用区域，包括固体力学，运输现象，流体流动，波传播，触发闪电和纳米膜膜。为了详细说明一些例子，通过工业和学术合作将新方法应用于流体流程，可能会使飞机行业受益。 可靠的仿真方法可以廉价地指导下一代纳米光子设备的实验。 最后，通过研究生参与研究的培训和参与，将人力资源开发纳入活动。