Collaborative Research: Multiphysics modeling and analysis of thermo-visco-acoustic equations with applications to the design of trace gas sensors

微量气体传感器可用于检测和识别非常少量的气体，应用于大气化学、环境和工业排放监测、爆炸物检测、工业过程控制和非侵入性医疗诊断等不同领域。微量气体传感器的大规模应用要求传感器系统紧凑、便携、高效、灵敏、经济、可靠。石英增强型光声光谱（QEPAS）传感器有望实现这些目标。特别是，QEPAS传感器可以小到几立方毫米，而基于其他敏感光谱技术的传感器需要几十到几百立方厘米的大电池体积。QEPAS传感器使用石英音叉来探测激光光束与微量气体相互作用时产生的微弱声波。在广泛部署QEPAS传感器之前，需要克服的一个主要工程挑战是提高其灵敏度并降低其生产成本。该项目的总体目标是为QEPAS传感器开发一个计算模型，这是对现有模型的重大改进，然后使用该模型来确定提高QEPAS传感器灵敏度的成本效益设计。该项目的主要数学挑战是开发有效的计算方法来求解构成模型基础的多物理场方程。该项目将为两名数学研究生提供计算科学方面的广泛培训，这些研究生在物理和工程应用、数学建模、数值分析和并行计算方面具有互补的专业知识。QEPAS传感器采用共振振动的石英音叉来检测微弱的声压波和热干扰，这些干扰是由激光束的光辐射与微量气体相互作用产生的。该项目将涉及开发和分析计算方法，以解决描述热粘声流体与共振振动机械结构（石英音叉）之间相互作用的亥姆霍兹方程系统。该模型将用于数值优化QEPAS信号作为传感器几何参数的函数。粘性流体对音叉阻尼的累积效应将根据系统的几何参数和物理常数进行计算。因此，该模型将允许通过改变音叉几何形状的QEPAS传感器的现实优化。此外，在某些情况下，热扩散波可以支配音叉表面的声压波，这种现象被称为光-热-声共振探测（ROTADE）。目前这些传感器的数学描述不能同时捕获QEPAS和ROTADE现象，尽管实验数据表明，根据激光束沿音叉轴的位置，两种类型的微量气体传感都可能发生。新模型将允许同时模拟两种类型的传感器系统。初步的分析和计算结果表明，由于方程参数小，解的波数大，用标准有限元方法求解模型中的方程是无效的。由于方程的有限元离散化，小参数会产生病态线性系统，而高波数会在计算解中引起大的相位误差（污染误差）。这个项目将通过开发和分析多物理场亥姆霍兹系统的块预调节器来提高计算数学的知识。此外，将最初提出的用于标量亥姆霍兹方程的高阶有限元和内部惩罚稳定方法扩展到多物理场亥姆霍兹系统，从而开发减少污染误差的方法。所开发的技术将与更一般的耦合亥姆霍兹系统相关，例如在研究薄体附近的热现象，助听器换能器和微电子机械装置的设计中出现的系统。

Trace gas sensors can be used to detect and identify very small quantities of gases for applications in such diverse fields as atmospheric chemistry, environmental and industrial emissions monitoring, explosives detection, industrial process control, and non-invasive medical diagnostics. The large-scale adoption of trace gas sensors requires sensor systems that are compact, portable, efficient, sensitive, cost-effective and highly reliable. Quartz Enhanced Photoacoustic Spectroscopy (QEPAS) sensors hold promise as a technology that may achieve many of these goals. In particular, QEPAS sensors can be as small as several cubic millimeters, whereas sensors based on other sensitive spectroscopic techniques require large cell volumes of tens to hundreds of cubic centimeters. QEPAS sensors use a quartz tuning fork to detect weak sound waves that are generated when a beam of light from a laser interacts with a trace gas. A major engineering challenge to overcome before QEPAS sensors can be widely deployed is to increase their sensitivity and lower their production cost. The overall goal of this project is to develop a computational model for QEPAS sensors that is a significant enhancement over existing models, and to then use this model to determine cost-effective designs that increase the sensitivity of QEPAS sensors. The major mathematical challenge of the project is to develop efficient computational methods to solve the multiphysics equations that form the basis of the model. The project will provide broad training in computational science for two mathematics graduate students from faculty mentors with complementary expertise in the physics and engineering of the application, mathematical modeling, numerical analysis, and parallel computing.QEPAS sensors employ a resonantly vibrating quartz tuning fork to detect weak acoustic pressure waves and thermal disturbances which are generated when optical radiation from a laser beam interacts with a trace gas. The project will involve the development and analysis of computational methods to solve a system of Helmholtz equations that describes the interaction between a thermo-visco-acoustic fluid and a resonantly vibrating mechanical structure (a quartz tuning fork). The model will be used to numerically optimize the QEPAS signal as a function of the geometric parameters of the sensor. The cumulative effect of the damping of the tuning fork by the viscous fluid will be computed in terms of the geometric parameters of the system and physical constants. Consequently, the model will allow for realistic optimization of QEPAS sensors by varying the tuning fork geometry. Furthermore, in some situations, the thermal diffusion wave can dominate the acoustic pressure wave on the surface of the tuning fork, in a phenomenon known as Resonant Opto-Thermo-Acoustic DEtection (ROTADE). Current mathematical descriptions of these sensors cannot capture both QEPAS and ROTADE phenomena simultaneously, although experimental data indicates that depending on the position of the laser beam along the tuning fork axis, both types of trace gas sensing may occur. The new model will allow for simultaneous simulation of both types of sensor systems. Preliminary analytical and computational results show that standard finite element methods for solving the equations in the model are ineffective due to small parameters in the equations and the high wave number of the solution. The small parameters produce an ill-conditioned linear system resulting from the finite element discretizations of the equations, while the high wave numbers can cause large phase errors in the computed solution (pollution error). This project will advance knowledge in computational mathematics by developing and analyzing block preconditioners for the multiphysics Helmholtz system. In addition, methods for reducing the pollution error will be developed by extending higher-order finite element and interior penalty stabilization methods originally proposed for scalar Helmholtz equations to the multiphysics Helmholtz system. The techniques developed will be relevant for more general coupled Helmholtz systems such as those which arise in the study of thermal phenomena near thin bodies, the design of hearing aid transducers and micro-electrical-mechanical devices.