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

痕量气体传感器可用于检测和识别非常少量的气体，应用于各种领域，如大气化学、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。痕量气体传感器的大规模采用要求传感器系统紧凑、便携、高效、灵敏、成本效益高和高度可靠。石英增强型光声光谱(QEPAS)传感器有望实现这些目标中的许多目标。特别是，QEPAS传感器可以小到几个立方毫米，而基于其他敏感光谱技术的传感器需要几十到数百立方厘米的大单元体积。QEPAS传感器使用石英音叉来检测激光光束与痕量气体相互作用时产生的微弱声波。在QEPAS传感器能够广泛部署之前，需要克服的一个主要工程挑战是提高它们的灵敏度和降低它们的生产成本。该项目的总体目标是为QEPAS传感器开发一个计算模型，该模型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该模型基础的多物理方程。该项目将为两名数学研究生提供广泛的计算科学培训，他们来自教师导师，他们在应用程序的物理和工程、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传感器使用共振振动的石英音叉来检测激光光束的光辐射与痕量气体相互作用时产生的微弱声波压力波和热扰动。该项目将包括开发和分析计算方法，以求解描述热粘性声学流体和共振机械结构(石英音叉)之间相互作用的亥姆霍兹方程组。该模型将被用来作为传感器几何参数的函数对QEPAS信号进行数值优化。根据系统的几何参数和物理常数，计算粘性流体对音叉的累积减振效果。因此，该模型将允许通过改变音叉几何形状来现实地优化QEPAS传感器。此外，在某些情况下，热扩散波可能主导音叉表面的声波压力波，这一现象称为共振光热声学检测(ROTADE)。目前对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，根据激光沿音叉轴的位置，这两种类型的痕量气体传感都可能发生。新模型将允许同时模拟这两种类型的传感器系统。初步的分析和计算结果表明，由于方程中的参数较小，且解的波数较高，采用标准有限元方法求解模型中的方程是无效的。较小的参数会导致方程的有限元离散而产生病态的线性系统，而较高的波数会导致较大的相位误差(污染误差)。这个项目将通过开发和分析多物理Helmholtz系统的块预处理器来提高计算数学方面的知识。此外，还将通过将最初针对标量Helmholtz方程提出的高阶有限元和内部惩罚稳定化方法推广到多物理Helmholtz系统来开发减小污染误差的方法。所开发的技术将适用于更一般的耦合亥姆霍兹系统，例如在研究薄体附近的热现象、助听器换能器和微型机电设备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.