An Analytical and Numerical Study of Dynamic Materials

该项目侧重于动态材料的开发和研究，即在空间和时间的微观尺度上组装的材料复合材料。分析和计算两种方法将用于分析这些材料在许多空间维度和时间上的有效特性。解析地，通过对具有时空变化的高级系数的相关双曲系统应用均匀化，将在麦克斯韦理论的框架下为各向同性介电体二元混合物的有效参数指定可达到的界限。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观结构的非均质介质中的传播物理。总之，这两种方法将导致正确的制定和分析在时空中响应动态环境的最佳材料设计。通过允许材料属性的时空变化，可以创造出纯粹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，可以选择性地屏蔽其主要部分，使其免受冲击或其他动态因素引起的破坏性动态干扰的入侵。干扰可以有目的地引导到影响不那么有害的地区。另一个例子与电磁波导或声波导有关。为了正常工作，传统的波导必须有一个相对于信号波长的最小直径。如果波导太窄或波的频率太低，那么传播将不可能。然而，通过允许波导填充物的材料特性在空间和时间上变化，就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.