Research > Multi-Scale Problems


We will target the solutions of scientifically important problems that have multi-scale properties, features, and complexity. We will model such systems at various scales. Starting with the atomic level, we will model large numbers of features at any level so that the correct physics for the next higher level will be revealed. If the next higher level is molecular scale, a large number of molecules will be simulated in order to obtain the correct physics in the next higher level. Depending on the system under consideration, the next higher level may be porous membrane of a biological system, for instance. Multi-physics governing equations at each scale will be solved. Continuity of the governing equations from one scale to another will be insured, instead of jumping from one set of equations to the next one, which usually involves approximations due to the large numbers of degrees of freedom (DOFs) that is typically avoided. Instead of avoiding such complexity and resorting to approximations, correct physical behavior will be modeled at every scale with inputs from the lower scale.

Discontinuity Across Scales: A system involving several levels of details and complexity is usually analyzed at different scales, usually for the sake of simplicity. An example is medical imaging. Different imaging modalities produce images with different resolutions. For instance, optical coherence tomography (OCT) produces images with a resolution of the order of 1-10 micrometers (10-6-10-5 m). With microwave imaging, it is possible to obtain resolutions in the order of sub-millimeters to centimeters (10-4-10-2 m). With acoustic/ultrasound imaging, resolution in the order of centimeters is possible (10-2 m). Such images are discontinuous as we go from one modality to another, from one scale to another. We aim to bridge this gap by solving such large problems that resolution in one level (e.g., 10-3 m) will be constructed with the resolution at a finer level (e.g., 10-6 m).