Mesoscale Numerical Methods for Certain Types of Implicit Partial Differential Equations

研究者开发了一种综合的计算工具，用于模拟复合结构在热机械载荷下微观结构演变的动力学。他使用一阶隐式偏微分方程和特定类型的狄利克雷边界条件，并使用一种新的子网格投影法近似微观结构动力学。他将这种计算技术应用于基于马氏体相变的振动主动控制和降噪。隐式偏微分方程代表了一类高阶非线性的常微分方程和偏微分方程和系统，如Eikonal或Hamilton-Jacobi方程。它们与在相变研究中起关键作用的偏微分包裹体密切相关。通常，隐式偏微分方程的解不是光滑的，它们不是唯一的，而且它们通常包含大量的竞争尺度。这三个不同的特点对设计适合的数值方法来求解隐式偏微分方程提出了明确的挑战。最近基于贝尔范畴论证的工作表明，存在用标准解析方法无法得到的此类方程的解。存在论本身并不是建设性的，它没有提供任何关于如何构建选择原则的暗示，也没有提供一个广义解的概念。在正常使用中，机械受到周期性的应力。这导致声波在材料中传播。由于这些波与整个机器的势能相比很小，因此将其转换为热能是一种有效的降噪方法。因此，可以部分或全部使用高阻尼材料来完成这项任务。形状记忆合金表现出如此显著的阻尼特性。这是一种特殊的合金，当冷却或施加压力时，它们的微观结构会从坚硬的、旋转对称的相转变为延展性的、不对称的相。这些理想的阻尼性能是在马氏体相的孪晶边界内运动的结果，以及非共格奥氏体-马氏体界面的运动，并且明显依赖于温度。研究者基于隐式偏微分方程进行计算建模，以理解和主动控制形状记忆合金中的相变。其应用包括非侵入性手术中尖端设备的振动控制、超声波探测器、各种航天器平台的稳定以及驾驶舱的降噪。

The investigator develops a comprehensive computational tool for modeling of the kinetics of microstructural evolution under thermo-mechanical loading in compound structures. He approximates the microstructural kinetics using first order implicit partial differential equations with a certain specific type of Dirichlet boundary conditions, and using a novel subgrid projection method. He applies this computational technique to provide an active control of vibrations and noise reduction based on the martensitic phase transformation. Implicit partial differential equations represent a large class of ordinary and partial differential equations and systems that are nonlinear in the highest derivatives, such as the Eikonal or Hamilton-Jacobi equations. They are closely related to partial differential inclusions that play a crucial role in the study of phase transitions. Typically, the solutions of imlicit partial differential equations are not smooth, they are not unique, and often they incorporate enormous amount of competing scales. These three distinctive features present a definitive challenge to the design of suitable numerical methods applicable to finding solutions of implicit partial differential equations. Recent work based on the Baire category argument shows that there exist solutions of such equations that cannot be obtained by standard analytical approaches. The existence theory itself is not constructive, does not yield any hint as to how to construct selection principles, and it does not provide a notion of a generalized solution.In regular use, machinery is subjected to periodic stresses. This results in acoustic waves travelling through the material. Since these waves are small in comparison to the potential energy of the overall machine, conversion to heat is an effective method of noise reduction. Thus highly damping materials may be used either in part or in full to accomplish this task. Shape memory alloys exhibit such significant damping properties. These are special alloys that change their microstructure from that of a stiff, rotationally symmetric phase to a ductile, less symmetric phase when cooled or put under stress. These desirable damping properties are a result of movement within the twin boundaries in the martensite phase, as well as the motion of the incoherent austenite-martensite interface, and are significantly temperature dependent. The investigator undertakes computational modeling to understand and actively control the phase transitions in shape memory alloys, based on implicit partial differential equations. The applications include such possibilities as controlled vibration of a cutting edge in non-invasive surgery, ultrasonic wave detectors, stabilization of platforms on various spacecraft as well as acoustic suppression in cockpits.