Regularization methods are numerical techniques based on exact solutions of partial differential equations that are derived in the same way as fundamental solutions when the Dirac delta is replaced by a smooth approximation, such as a Gaussian. The vortex blob method is an example. The resulting solutions of the PDEs are bounded approximations to the fundamental solutions. This project advances the theory, the computational implementation and the application of these methods to problems of Stokes flows. Regularized high-order elements, such as doublets and quadrupoles, are systematically derived, the required properties of blobs are proposed and families of blobs with such properties are developed. The regularized fluid velocity expression forms the basis of new numerical methods, whose convergence and efficiency properties are analyzed. Performance-based guidelines for choosing the numerical parameters are provided. Finally, several applications of the methods include the dynamics of microorganism populations and the large-scale flow patterns they generate, the motion of cilia as they beat to generate an organized net flow, flows across permeable walls, and more.
The investigator develops mathematical tools necessary to design new reliable ways to generate computer simulations of biological phenomena involving microorganisms in a liquid environment. These include the motion of bacteria in the human body, the synchronized motion of cilia in the lungs, the transport and filtration of chemicals transported by blood, and other biologically important situations. The computer simulation of these phenomena complement experimental studies done in medical laboratories and are extremely useful, particularly for understanding the behavior of the human body under circumstances that cannot be tested in vivo. Often, computer simulations can detect responses that occur when parameters are outside the range typically used in the laboratory, and thus point to new experiments that should be pursued. The methods developed in this work increase our understanding of how E. coli may move around the body or how defects in the lung cilia can translate into decreased functionality and possibly disease.
