An Analytical and Numerical Study of Dynamic Materials

该项目致力于动态材料的开发和研究，即在空间和时间上进行微尺度组装的材料复合材料。分析和计算手段将被用来分析这种材料在许多空间维度和时间上的有效性质。在解析上，通过对具有时空变化的高阶系数的相关双曲型系统应用齐次化，将在麦克斯韦理论的框架内确定各向同性介质二元混合物的有效参数的可达范围。在计算上，将使用原始方程的直接数值模拟来更好地理解波在具有复杂微结构的非均匀介质中的传播物理。这两种方法结合在一起，将导致对动态环境下的时空最优材料设计的正确表述和分析。通过允许材料属性的时空可变性，有可能创造出纯空间设计无法实现的效果。例如，通过适当控制弹性结构的设计因素，即其质量和刚度，可以有选择地屏蔽其主要部分，使其免受冲击或其他动力因素造成的破坏性动态扰动的侵袭。骚乱可以被有目的地引导到其影响不那么有害的地区。另一个例子与电磁或声波波导有关。为了正常工作，传统的波导必须具有相对于信号波长的特定最小直径。如果波导太窄或波的频率太低，那么传播就不可能了。然而，通过允许波导填充的材料特性在空间和时间上变化，将有可能传输所有频率的波。

This project focuses on the development and study of dynamic materials, i.e., material composites assembled on a microscale in space and time. Both analytic and computational means will be used to analyze the effective properties of such materials in many spatial dimensions and time. Analytically, by applying homogenization to the relevant hyperbolic systems with spatio-temporally varying senior coefficients, attainable bounds will be specified for the effective parameters of the binary mixtures of isotropic dielectrics in the framework of Maxwell's theory. Computationally, a direct numerical simulation of the original equations will be used to better understand the physics of wave propagation through heterogeneous media with complex microstructure. Together, both approaches will lead to a correct formulation and analysis of optimal material design in space time in response to a dynamic environment.By allowing spatio-temporal variability in the material properties, it is possible to create effects that are unachievable through purely spatial design. For example, by appropriately controlling the design factors of an elastic construction, i.e., its mass and stiffness, it is possible to selectively screen major parts of it from the invasion of destructive dynamic disturbances caused by impactsor other dynamic factors. Disturbances can be purposefully guided into regions where their effect is less pernicious. Another example is related to electromagnetic or acoustic waveguides. To function properly, a conventional waveguide must have a certain minimum diameter relative to the wavelength of the signal. If the waveguide is too narrow or the frequency of the waves is too low, then propagation will not be possible. However, by allowing the material properties of the filling of the waveguide to vary both in space and in time, it will become possible to transmit waves of all frequencies.