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变动力学。 他近似的微结构动力学使用一阶隐式偏微分方程与某种特定类型的Dirichlet边界条件，并使用一种新的亚网格投影方法。他应用这种计算技术提供了一种基于马氏体相变的振动和噪音降低的主动控制。 隐式偏微分方程代表了一大类在最高阶导数中为非线性的常微分方程和偏微分方程以及系统，例如Eikonal或Hamilton-Jacobi方程。 它们与偏微分包含密切相关，而偏微分包含在相变研究中起着至关重要的作用。 通常，不合法偏微分方程的解是不光滑的，它们不是唯一的，并且它们经常包含大量的竞争尺度。 这三个独特的功能提出了一个明确的挑战，设计合适的数值方法适用于寻找隐式偏微分方程的解决方案。 最近的工作的基础上Baire类参数表明，存在的解决方案，这些方程不能得到标准的分析方法。 存在理论本身不是构造性的，它没有给出任何关于如何构造选择原理的提示，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。 这导致声波穿过材料。由于这些波与整个机器的势能相比很小，因此转换为热量是降低噪音的有效方法。 因此，高阻尼材料可以部分或全部用于完成该任务。 形状记忆合金表现出如此显著的阻尼特性。 这些是特殊的合金，当冷却或置于应力下时，其微观结构从刚性的旋转对称相变为韧性的不对称相。 这些理想的阻尼性能是马氏体相中的孪晶界内的运动以及不相干的马氏体-马氏体界面的运动的结果，并且显著地依赖于温度。 研究人员进行计算建模，以了解和主动控制形状记忆合金的相变，基于隐式偏微分方程。 这些应用包括非侵入性手术中切割边缘的受控振动、超声波探测器、各种航天器上平台的稳定以及驾驶舱中的声学抑制等可能性。

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.