Hemivariational Inequalities: Numerical Methods and Applications

半变分不等式：数值方法及应用

• 批准号: 1521684
• 负责人: Weimin Han
• 金额: $ 16.78万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1521684&HistoricalAwards=false
• 关键词: Hemivariational; Inequalities; Numerical; Methods; Applications

Certain systems in physical sciences and engineering cannot be described by mathematical equations but are instead described by more complicated relations known as inequalities (stating that one quantity is greater than or smaller than another). Solving a system of inequalities numerically can be very challenging. This research project aims to develop a rigorous and comprehensive mathematical theory, as well as reliable and efficient numerical methods, for the numerical solution of a class known as hemivariational inequalities. In addition to the resultant advances in computational mathematics, the research will have impact in the petroleum industry through application to analysis of borehole pumping systems. Numerical simulations are used by the petroleum industry to infer the downhole oil pump conditions based on measured surface conditions. For deviated (non-vertical) oil wells, methods for reliable simulation have yet to be developed, mainly due to the complicated friction and dynamics in deviated wells. Appropriate models can be cast in the form of hemivariational inequalities; this research project is expected to facilitate practically useful numerical simulations for diagnostics of oil pump conditions in deviated wells. Collaboration with researchers in the petroleum industry will ensure the transfer of the new mathematical results to applications. The results from the project are expected to help correctly control downhole oil pumps to save energy and avoid pump damage. Graduate students will actively participate in all aspects of the research and will thus be trained in high level mathematics and numerical methods on challenging and important problems from applications. Inequality problems in mechanics can be divided into two main classes: that of variational inequalities, which is concerned with convex energy functionals (potentials), and that of hemivariational inequalities, which is concerned with nonsmooth and nonconvex energy functionals (superpotentials). Through the formulation of hemivariational inequalities, problems involving nonmonotone, nonsmooth, and multivalued constitutive laws, forces, and boundary conditions can be treated successfully. During the last three decades, hemivariational inequalities were shown to be very useful across a wide variety of disciplines, ranging from nonsmooth mechanics, physics, and engineering, to economics. However, relatively little work on the numerical analysis of hemivariational inequalities has been done. In this research project, a comprehensive theory will be developed for the numerical solution of various hemivariational inequalities, including several families of elliptic, parabolic, and hyperbolic hemivariational inequalities. For each family of hemivariational inequalities, numerical schemes will be introduced based on the finite element method for spatial discretization and finite differences for temporal discretization. Convergence of the numerical solutions will be shown under the basic solution regularity, and error estimates will be derived under appropriate solution regularity assumptions. The error estimates will be of optimal order when linear elements are used. Numerical experiments will be performed to illustrate convergence orders predicted by the theory. Results from this project (such as a posteriori error analysis, adaptive algorithms, and discontinuous Galerkin methods) will form a solid foundation for further developing numerical methods to solve hemivariational inequalities.

物理科学和工程学中的某些系统不能用数学方程来描述，而是用更复杂的关系来描述，称为不等式（指出一个量大于或小于另一个量）。 以数值方式求解不等式系统可能非常具有挑战性。该研究项目旨在开发严格而全面的数学理论以及可靠且高效的数值方法，用于半变分不等式的数值求解。除了计算数学方面取得的进展外，该研究还将通过应用于井下泵送系统分析对石油工业产生影响。石油工业使用数值模拟根据测量的表面条件推断井下油泵的条件。对于斜井（非垂直），可靠的模拟方法尚未开发出来，主要是由于斜井中复杂的摩擦和动力学。适当的模型可以以半变分不等式的形式建立；该研究项目预计将有助于进行实用的数值模拟，以诊断斜井中的油泵状况。与石油行业研究人员的合作将确保新的数学结果转化为应用。该项目的结果预计将有助于正确控制井下油泵，以节省能源并避免泵损坏。 研究生将积极参与研究的各个方面，从而接受高水平数学和数值方法的培训，以解决应用中具有挑战性和重要的问题。力学中的不等式问题可分为两大类：变分不等式，涉及凸能量泛函（势）；半变分不等式，涉及非光滑和非凸能量泛函（超势）。通过半变分不等式的表述，可以成功地处理涉及非单调、非光滑和多值本构定律、力和边界条件的问题。在过去的三十年中，半变分不等式被证明在从非光滑力学、物理学、工程学到经济学等各种学科中都非常有用。然而，对半变分不等式的数值分析所做的工作相对较少。在这个研究项目中，将开发一种综合理论来解决各种半变分不等式的数值解，包括椭圆、抛物线和双曲半变分不等式的几个族。对于每族半变分不等式，将基于空间离散的有限元方法和时间离散的有限差分引入数值格式。在基本解规律下将显示数值解的收敛性，并在适当的解规律假设下导出误差估计。当使用线性元素时，误差估计将具有最佳阶数。将进行数值实验来说明理论预测的收敛阶数。该项目的结果（例如后验误差分析、自适应算法和不连续伽辽金方法）将为进一步开发解决半变分不等式的数值方法奠定坚实的基础。