Mesoscale Numerical Methods for Certain Types of Implicit Partial Differential Equations

研究人员开发了一个全面的计算工具，用于模拟复合结构在热机械载荷下的微观结构演化动力学。他使用一阶隐式偏微分方程组和一种特定类型的Dirichlet边界条件，并使用一种新的亚网格投影法来近似微观结构动力学。他应用这种计算技术来提供基于马氏体相变的振动和降噪的主动控制。隐式偏微分方程组代表了一大类在最高导数上是非线性的常微分方程组和偏微分方程组，如Eikonal方程或Hamilton-Jacobi方程。它们与在相变研究中起关键作用的偏差型包裹体密切相关。通常，不合法的偏微分方程解是不光滑的，它们不是唯一的，而且它们往往包含大量相互竞争的尺度。这三个明显的特点对设计适用于求解隐式偏微分方程解的合适的数值方法提出了决定性的挑战。最近基于Baire范畴论证的工作表明，存在用标准分析方法无法获得的此类方程的解。存在论本身不是建设性的，没有给出任何关于如何构造选择原则的提示，它也没有提供一个广义解决方案的概念。在常规使用中，机器受到周期性的应力。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性材料来完成这一任务。形状记忆合金表现出如此显著的减振性能。这是一种特殊的合金，当冷却或施加压力时，它们的微观结构从刚性的旋转对称相转变为延展性较差的对称性相。这些理想的减振性能是马氏体相中孪晶界内运动的结果，也是非共格奥氏体-马氏体界面上运动的结果，并且与温度密切相关。为了理解和主动控制形状记忆合金的相变，研究人员基于隐式偏微分方程组进行了计算模拟。这些应用包括非侵入性手术中尖端受控振动、超声波探测器、各种航天器平台的稳定以及驾驶舱内的声学抑制等。

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.