**General Interest**

We are interested in understanding dynamic systems through theoretical, computational, and experimental tools. One of our recent efforts is to explore the application of modern deep learning tools to problems in mechanics and physics. Our recent work (under review) on eigenvalue computations using CNNs is the very first application of such tools to any mechanics problem.

A major focus is on understanding the propagation of waves in complex microstructured solids and liquids and on creating materials which can be used to control their temporal and spatial characteristics. We are also interested in other dynamic systems related to solid structures such as aerodynamically induced flutter instabilities and their control.

**Deep Learning, Mechanics, and Physics**

We have recently shown how to apply convolutional neural networks (CNN) to problems in mechanics. The specific model problem we were concerned with was the prediction of eigenvalues of a phononic crystal. We were able to show that CNNs can vast overpower traditional NN architectures such as MLPs (deep or shallow) in prediction accuracy and generalization capability. Furthermore, they can be trained with a fraction of the data required by MLPs. This study is the first application of modern DL architectures to any problem in the broad area of mechanics.

**Computational Phononics**

We have developed a complete suite of algorithms for phononic bandstructure computations. These algorithms are based on fast variational solutions^{1,2} to the elastodynamic eigenvalue problem which are further accelerated through GPU computations^{3} over a cluster of distributed NVIDIA GPUs. We can calculate and sort thousands of phononic eigensurfaces for arbitrarily complex 3-D unit cells and extract not only bandgap locations but also properties which are essential to thermal calculations. These include density of states, phase and group velocities over very large frequency domains (calculations by Yan Lu at IIT).

**Optimization of Phononic Crystals**

In collaboration with Prof. James Guest from John’s Hopkins we have implemented topology optimization routines for the optimization of phononic crystals. Several other groups are studying the optimization of phononic crystals. Our research differs in it being directed towards 3-D phononic crystals for the control of bulk waves. This has significant computational requirements which we fulfill through GPU computations and highly efficient vectorization techniques. Below is an example result of topology optimization implemented for maximizing the bandgap in 3-D FCC phononic crystals made of Tungsten-Carbide and Epoxy phases (calculations by Yan Lu at IIT, Yang Yang at JHU).

**Waves in Microstructured Media**

Appropriately designed microstructures can be used to control waves in precise ways. These controls can be both spatial or temporal. Spatial control refers to controlling the trajectory of waves whereas temporal control refers to controlling their frequency content. We use COMSOL, MATLAB, Python, and FEniCS to simulate the propagation of waves (both transient and harmonic) in phononic and metamaterial structures. Some applications are in the cloaking of waves, frequency filtering and noise control, and imaging (calculations by Valentin Serey at IIT, Yan Lu at IIT).

**Homogenization and Theoretical Work**

We are interested in the area of homogenization as it applies to the dynamics of composites. Coherent dynamic homogenization principles are essential for metamaterials and we have contributed to the development of field averaging based elastodynamic homogenized constitutive parameters. These parameters are invariant under coordinate transformations and homogenization itself. They display the unusual characteristics required from metamaterials (negative density, stiffness etc.) and are important candidates for realizing transformation elastodynamic devices such as cloaks and concentrators.

Our group is also interested in investigating theoretical bounds on metamaterial properties and performance of transformation devices. These bounds emerge from fundamental principles of causality and also from results of such areas as scattering theory.

**Dynamics**

More generally we have interest in the analysis and control of dynamic systems, seeing wave mechanics as a specific area within the more general area of dynamics. Dynamics is further seen merely as the solution space of multi-parameter partial differential equations which connects it to deep mathematical fields. Some ongoing considerations include the analysis, experiment, and control of flow induced flutter instabilities in flying wing aircrafts.