### Mathematical and Computer Sciences

### Computational analysis tools for multiscale engineering systems (Petzold)

### Sharp gradients and interface tracking (Ceniceros, Gibou, Liu)

### Stochastic partial differential equations (Birnir)

### Homogenization (Birnir)

### Cluster computing and computational grid computing environments (Wolski, Yang)

### Semi-automatic generation of graphical user interfaces for scientific computing (Petzold)

### Fast Solvers (Chandrasekaran)

### Computation in Complex Fluids

### High Resolution Simulation of Free Surface Flows (Gibou)

### High Resolution Simulation of Multiphase Flows with Phase Change (Gibou)

Over the last two decades, there has been on ongoing quest for new computational methods to solve multiphase flows with phase change. This thrust has been motivated in part by the energy industry as phase change processes allow fluids to store and release large amounts of heat energy. Other applications include the study of condensation in the context of manned space flight dehumidification systems, particularly difficult to study experimentally in microgravity environments and of considerable interest to NASA. The study of phase change with physical experiments remains a challenge, mainly because of the small time and length scales associated with these processes. Consequently, such studies are limited to empirical correlations of specific cases. Theoretical results, starting with the work of Rayleigh have offered some insight on the nature of simple solutions and have provided revealing stability analyses. However, they rely on considerable simplifications.

Numerical simulations offer a promising avenue and several approaches have been introduced in the last two decades. The main challenges for a direct numerical simulation come from the fact that the interface location must be calculated as part of the solution process and because discontinuities in materials properties across the interface must be preserved. Finally, the problem involves dissimilar length scales with smaller scales influencing larger ones so that nontrivial pattern formation dynamics can be expected to occur on all intermediate scales. This results in a highly nonlinear problem that is very sensitive to numerical errors and prone to numerical instabilities.

Prof. Gibou is developing efficient numerical methods for the simulation of multiphase flows with phase change. In particular, he has developed with his co-workers at UCSB (Banerjee and Chen) and at Lockheed Martin (Nguyen), the first numerical algorithm that treat properly interfacial phase change in the sharp limit. The goal is to pursue this work to consider three dimensional flows with applications to various physical studies of interest at NASA and DOE national laboratories (see http://www1.engr.ucsb.edu/~fgibou/).

### Multi-scale Computational Methods for Polymeric Fluids and Soft Materials (Ceniceros)

### Numerical Methods for Multi-phase Flows and Free Surface Phenomena (Ceniceros)

### Field-theoretic computer simulations (Fredrickson)

### Korteweg stresses in miscible fluid flows (Meiburg)

### Flow of macromolecular fluids (Leal)

The dynamics of macromolecular fluids in flow is critical in many materials processing applications, as well as biological and other naturally occurring systems. The goal of computational simulation is prediction not only of continuum flow variables (u, p), but also the corresponding microstructural state and stress distributions since these control both the flow and transport properties of the fluid, as well as the properties of any product that results from this flow. The unusual feature of macromolecular liquids (and, indeed, all “non-Newtonian” fluids) is that internal relaxation processes are slow, and thus the microstructural state can be modified greatly from the equilibrium configuration by interaction with a flow, with major changes in the macroscopic properties.

The computational problem is thus to solve the Cauchy equations of motion, together with material model equations that describe the coupling of the microstructural states of the material with flow. The transition from microstructure to macroflow occurs via the relationship between stress and the microstructural state of the material. The state of the material at each material point is described via a statistical distribution function and the latter is either calculated directly by solving a multidimensional advection-diffusion equation, or a corresponding stochastic “Langevin-type” equation. Alternatively, one can attempt to derive equations for the leading moments of the distribution function starting from the fundamental statistical mechanical models, but this involves closure approximations that may change the mathematical character of the problem. There are a large variety of challenging computational problems associated with each of the possible approaches; solving an advection-diffusion based model, solving the stochastic DE model, or introducing closures or other approximations. Both in the configuration space for micro-variables, and in physical space for flow, Lagrangian or particle-based techniques (such as “smooth particle hydrodynamics”) appear to be advantageous, and amenable to parallelization, but there are unresolved fundamental issues. The huge size of the problems is also a major issue. For a fully 3D, time-dependent flow, the multi-dimensional, time-dependent configuration-space problem must be solved at enough material points to provide adequate spatial resolution for the stress. The configuration problem for each material point is itself multidimensional in the configuration space independent variables and time. This is a large problem under any circumstances. However, in some key materials, the microstructural state can also develop very short length scales, even in a flow domain where one would expect smooth variations on longer length scales. For example, in liquid crystalline polymers, instabilities in the flow lead to disclinations, and the onset of a very short length scale “polydomain” structure. This second type of problem represents special challenges for simulations.

### Interface dynamics (Leal)

### Computation in Microscale Engineering

### Mixing in microchannels (Mezic)

### Bubbles and bubble migration in microdevices (Homsy)

### Computation in Materials

### Analysis and simulations of complex materials (Garcia-Cervera)

### High Resolution Simulation of the Stefan Problem (Gibou)

### Computation in Systems Biology

### Application of systems engineering tools to biological problems (Doyle)

### Multiscale simulation of complex biological systems (Petzold)

### Image Segmentation with Application to Radiotherapy (Gibou)

Segmentation is the art of automatically separating an image into different regions in a fashion that mimics the human visual system. It is therefore a broad term that is highly dependent on the application at hand, e.g. one might want to segment each object individually, groups of objects, parts of objects, etc. In order to segment a particular image, one must first identify the intended result before a set of rules can be chosen to target this goal. The human eye uses low-level information such as the presence of boundaries, regions of different intensity or colors, brightness and texture, etc., but also mid-level and high-level cognitive information, for example, to identify objects or to group individual objects together. As a direct consequence, there are a wide variety of approaches to the segmentation problem, and many successful algorithms have been proposed and developed to simulate a number of these different processes. My research on this topic has focused on a class of method known as deforma ble model based on energy minimization.

A natural field of application for such an algorithm is in medicine. Three-dimensional conformal radiotherapy (3DCRT) and intensity-modulated radiation therapy (IMRT) are being widely developed and implemented for clinical applications. These procedures depend upon intense use of patient imaging. The availability of spiral computerized tomography (CT) scanners has made practical the acquisition of large patient image sets consisting of around one hundred reconstructed planes. Most frequently these three-dimensional studies are fused with a treatment planning CT to transfer the target volume onto the treatment planning CT. Using this radiotherapy technology, the radiation oncologist can prescribe dose distributions that conform closely to tumor target volumes. With computerized treatment planning, it is also possible to reduce the dose that neighboring normal anatomical structures receive during the course of the radiotherapy procedure. However, the implementation of this technology is hampered by the effort required to segment tumor volumes and normal anatomical structures such that they are numerically represented in the computers. More often than not, these structures must be segmented on workstations by drawing closed contours around the cross-sections of the anatomy as perceived by the operator in axial CT reconstructions. The construction of a series of such closed polygons in consecutive CT reconstruction planes (or slices) constitutes the process of anatomical structure segmentation as it is most commonly implemented for radiotherapy treatment planning. Software tools that support this procedure are provided in most commercial treatment planning systems. These tools use the current state-of-the-art image display and graphic interaction techniques. Nevertheless, the segmentation process is still a subjective and time-consuming part of the treatment planning process.

Prof. Gibou is developing real-time segmentation algorithms that take into account prior knowledge of the organ to be segmented. The key idea is to incorporate the structure of the target organ into the segmentation, while processing three-dimensional data. The benefit of this approach is that the human time that is required during the manual segmentation process can be cut down drastically while still retaining the desired accuracy. This work is in close collaboration with researchers at Stanford University (see http://www1.engr.ucsb.edu/~fgibou/