Numerical methods for various moving boundary problems is the primary theme of my research. This involves both the development and improvement of existing algorithms, and also the use of these methods to study new applications. Since moving boundary problems exist in many different fields, this leads to research contributions to a diverse array of disciplines. A recent addition to my research area has been in computational neuroscience. In this work, we are exploring ways to increase the speed and capacity of detailed neuronal simulations.
The common thread through the moving boundary problems are the level set method and the fast marching method due to Osher and Sethian (1985). In the level set method, the boundary is represented implicitly as an isosurface of a function, a so-called level set function, of a higher dimension (see illustration to the right). The resulting evolution is then mapped into an evolution of the level set function. The advantages of this method are that the method handles changes of topology naturally without special rules for collision detection, can be easily adapted to any number of dimensions, and can treat the formation of corners and cusps in the boundary correctly through the use of methods borrowed from hyperbolic conservation laws.
The fast marching method also represents a moving interface implicitly, but in this method, the level set function is also the time of crossing map. This method encapsulates the entire evolution of the surface with a single time-independent function. While it is typically only used for monotonic speed functions, i.e. fronts which move only in one direction, recent work shows that this restriction can be overcome in some instances (see reference 3 below). Where applicable, the fast marching method is much faster than the level set method because it can compute the entire evolution in a single pass through the mesh. These methods are also utilized as subprocedures in support of the level set method.
References
Bacterial biofilms, which are the aggregation of bacteria on solid surfaces surrounded by gas or liquid, are the most ubiquitous form of life on the planet. Biofilms have an impact on our lives on a daily basis. Biofilms are responsible for billions of dollars in industrial damage annually, are a common cause of nosocomial (hoepital borne) infections, are a leading cause of death in people with cystic fibrosis, and are the cause of diseases such as Legionaire's disease. Biofilms are also used to our benefit through manufacturing of some cleaning products, improved nutrient uptake in agricultural crops, bioremediation, and treatment of industrial waste water. Understanding the growth and development of bacterial biofilms is a critical piece to decreasing their negative, and increasing their positive impacts on society.
Our focus in bacterial biofilms has been concentrated on two particular biofilm systems. In the first system (picture at the right), we are studying Pseudomonas aeruginosa biofilms. These bacteria are responsible for the majority of deaths related to cystic fibrosis, but are also commonly found to be destructive agents in almost any tissue that has a comprimised immune system. These biofilms are an example of a system which exhibits a behavior known as quorum sensing. Quorum sensing is a mechanism by which the bacteria can monitor their local population density so that they can perform a common task. In this case, a coordinated attack upon the host cell. On the right is a simulated biofilm showing the onset of quorum sensing.
The second system we are studying concerns the symbiotic system of autotrophic and heterotrophic bacteria used in activated sludge waste water treatment reactors. In these systems, autotrophic bacteria are used to efficiently remove nitrogen from industrial waste water, and heterotrophic bacteria live off the autotrophs by-products and provide protection from environmental stresses. Improving our understanding of this system will lead to improved throughput in activated sludge reactors.
Our work in this area has been a combination of mathematical modeling and simulation. The simulations have been carried out using a combination of the level set method coupled with the eXtended Finite Element Method (X-FEM). This combination of methods is a powerful tool for problems where there are sharp boundary layers and other local phenomena around interfaces that cannot be adequately captured on regular meshes, and leads to higher order accuracy at the boundary where the accuracy is needed.
References
We have been developing new algorithms for spatially adaptive neuronal simulations with the intent to improve the speed of existing algorithms. For some systems, we have shown up to 80% decrease in computational cost without loss of accuracy. Furthermore, the algorithms we have developed are easily adpated to a multi-processor parallel environment that will have much better load-balancing properties than can be achieved with earlier methods. The basic principle that we have adopted is that the new algorithm scales with the amount of activity in the neurons, and not with the number or size of the neurons. As the size and scale of neuronal systems increases, the cost of this algorithm will scale an order of magnitude better than other methods, making this ideal for large scale neuronal simulation environments. We are currently in the process of adapting these algorithms to large scale computer architectures.
In addition to algorithm development, we are applying this algorithm to study neurological disorders such as Parkinson's disease. In this work, we are studying the role of deep brain stimulation in the treatment of Parkinson's disease, and attempting to tune neurostimulators to produce maximum benefit with a minimal voltage by varying stimulator frequency, wave patterns, and strength in a virtual environment. This work is being done in collaboration with neurosurgeons in the Northwestern Feinberg School of Medicine.
References
A long standing collaboration with Natarajan Sukumar, currently at UC Davis, and others has been focused on applying the combination of level set or fast marching methods to problems in fracture mechanics. In this framework, a level set representation is used to evolve a crack surface, while the X-FEM is used to compute the strain field and the resulting crack tip velocity. We have recently produced a fully three-dimensional fatigue crack growth algorithm that can simulate non-planar crack growth in solid bodies. This is a substantial computational challenge that we have recently overcome. An illustration of such a growth is shown on the right.
References
Robert R. McCormick School of Engineering and Applied Science
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