# Biomedical Modeling

**Multiscale and multiphysics modeling**

**Propagation of noise/fluctuations across scales and model components**

Construction and analysis of a tightly-coupled domain decomposition for highly nonlinear hydrogen diffusion through a three-layer membrane subject to a stochastic boundary concentration. Analysis of noise propagation in a hybrid atomistic-continuum solver for highly nonlinear systems, with the one-dimensional stochastic Ginzburg-Landau equation acting as a testbed.

**Granular Flows**

We extend continuum-scale models of granular materials under dynamic compaction to account for thermal and stress localization probabilistically. Various dynamic compaction models are explored. At the pore-scale, a stochastic pore collapse Carroll-Holt model is developed to characterize the relationship between stress and thermal fluctuations. A stochastically-associative flow rule for elastoplastic deformations is derived by adding a random term in the constitutive relation that depends on various microstructural features. At the macroscale, a multiscale Baer-Nunziato type model is used to study the effect of a random initial microstructure on thermal localization. Using the efficient MLMC algorithm, we find the joint probability distribution of pore surface temperature and porosity; deducing the probability of thermal initiation events.

**Flow in rough fractures**

We model flow through rough fractures as flow through a channel with random walls. The stochastic boundaries are transformed to deterministic boundaries rendering the deterministic Stokes flow equation to stochastic. The resulting stochastic partial differential equations are analyzed with traditional uncertainty quantification techniques. The analysis of flow through rough channels is extended to complex fluids as non-Newtonian fluid and particle-laden flows.

**Acoustofluidics**

We are performing stability analysis on acoustofluidics at the micro and nano scales using the energy perturbation method. Non-linear analysis is performed to understand the stability of specific solutions. Bifurcation parameters are identified and numerical techniques are used to complement the study and validate experimental results. Direct applications are acoustic waves in batteries to prohibit/delay the formation of dendrites on the anode/cathode implying a longer battery life.

**Inverse Modeling**

An accurate and efficient reconstruction of the release history of contaminants is essential for regulatory and remedial efforts. Most of these efforts rely on measurements of contaminant concentration to identify sources and release history of the contaminant. Usually the measurements are corrupted by measurement errors. To solve this inverse problem, our research focuses on sampling the source concentration in a Bayesian framework based on the measurements taken at a later time, the governing equation, and certain forms of the measurement errors.

*Exponential time differencing for stiff systems*

The scheme can be applied to fluid flow in reservoir simulation, which has proven to be stiff. I am investigating the possible application of exponential time differencing on the IMPES (Implicit Pressure Explicit Saturation) scheme in the reservoir simulation. Such a scheme is faster yet less stable than the fully implicit method, because the time step required for the convergence of pressure is prohibitively small. Using an exponential time differencing numerical scheme will replace solving pressure implicitly.