Biomolecular structure and interactions in aqueous environment are determined by a complicated interplay between physical and chemical forces including solvation, electrostatics, van der Waals forces, the hydrophobic effect and covalent bonding. Among them, electrostatics has been of particular interest due to its long-range nature and the tradeoff between desolvation and interaction effects [1]. In addition, electrostatic interactions play a significant role within a biomolecule as well as between biomolecules, making the balance between the two vital to the understanding of macromolecular systems. As a result, much effort has been devoted to accurate modeling and simulation of biomolecule electrostatics. One important application of this work is to compute the structure of electrostatic interactions for a biomolecule in an electrolyte solution, as well as the potential that the molecule generates in space. There are two valuable uses for these simulations. First, it provides a full picture of the electrostatic energetics of a biomolecular system, improving our understanding of how electrostatics contributes to stability, specificity, function, and molecular interaction [2]. Second, these simulations serve as a tool for molecular design, since electrostatic complementarity is an important feature of interacting molecules. Through examination of the electrostatics and potential field generated by a protein molecule, for example, it may be possible to suggest improvements to other proteins or drug molecules that interact with it, or perhaps even design new interacting molecules de novo [3]. There are two approaches in simulating a protein macromolecule in an aqueous solution with nonzero ionic strength. Discrete/atomistic approaches based on Monte-Carlo or molecular dynamics simulations treat the macromolecule and solvent explicitly at the atomic level. Therefore, an enormous number of solvent molecules are required to provide reasonable accuracy, especially when electric fields far away from macroscopic surface are of interest, leading to computational infeasibility. In this work, we adopt instead an approach based on a continuum description of the macromolecule and solvent. Although the continuum model of biomolecule electrostatics is widely used, the numerical techniques used to evaluate the model do not exploit fast solver approaches developed for analyzing integrated circuit interconnect. I will describe the formulation used for analyzing biomolecule electrostatics, and then derive an integral formulation of the problem that can be rapidly solved with precorrected-FFT method [4]. / Singapore-MIT Alliance (SMA)

In this paper a precorrected FFT-Fast Multipole Tree (pFFT-FMT) method for solving the potential flow around arbitrary three dimensional bodies is presented. The method takes advantage of the efficiency of the pFFT and FMT algorithms to facilitate more demanding computations such as automatic wake generation and hands-off steady and unsteady aerodynamic simulations. The velocity potential on the body surfaces and in the domain is determined using a pFFT Boundary Element Method (BEM) approach based on the Green’s Theorem Boundary Integral Equation. The vorticity trailing all lifting surfaces in the domain is represented using a Fast Multipole Tree, time advected, vortex participle method. Some simple steady state flow solutions are performed to demonstrate the basic capabilities of the solver. Although this paper focuses primarily on steady state solutions, it should be noted that this approach is designed to be a robust and efficient unsteady potential flow simulation tool, useful for rapid computational prototyping. / Singapore-MIT Alliance (SMA)

In this paper, the precorrected-FFT method is used to solve the electromagnetic scattering from two-dimensional cavities of arbitrary shape. The integral equation is discretized by the method of moments and the resultant matrix equation is solved iteratively by the generalized conjugate residual method. Instead of directly computing the matrix-vector multiplication, which requires N² operations, this approach reduces the computation complexity to O(N log N) as well as avoids the storage of large matrices. At the same time, a technique known as the complexifying k is applied to accelerate the convergence of the iterative method in solving this resonance problem. Some examples are considered and excellent agreements of radar cross sections between these computed using the present method and those from the direct solution are observed, demonstrating the feasibility and efficiency of the present method. / Singapore-MIT Alliance (SMA)

This paper presents an accurate and efficient method-of-moments solution of the electrical-field integral equation (EFIE) for large, three-dimensional, arbitrarily shaped objects. In this method, the generalized conjugate residual method (GCR) is used to solve the matrix equation iteratively and the precorrected-FFT technique is then employed to accelerate the matrix-vector multiplication in iterations. The precorrected-FFT method eliminates the need to generate and store the usual square impedance matrix, thus leading to a great reduction in memory requirement and execution time. It is at best an O(N log N) algorithm and can be modified to fit a wide variety of systems with different Green’s functions without excessive effort. Numerical results are presented to demonstrate the accuracy and computational efficiency of the technique. / Singapore-MIT Alliance (SMA)

Efficiency is one of the key issues in numerical simulation of large-scale problems with complex 3-D geometry. Traditional domain based methods, such as finite element methods, may not be suitable for these problems due to, for example, the complexity of mesh generation. The Boundary Element Method (BEM), based on boundary integral formulations (BIE), offers one possible solution to this issue by discretizing only the surface of the domain. However, to date, successful applications of the BEM are mostly limited to linear and continuum problems. The challenges in the extension of the BEM to nonlinear problems or problems with non-continuum boundary conditions (BC) include, but are not limited to, the lack of appropriate BIE and the difficulties in the treatment of the volume integrals that result from the nonlinear terms. In this thesis work, new approaches and techniques based on the BEM have been developed for 3-D nonlinear problems and Stokes problems with slip BC. For nonlinear problems, a major difficulty in applying the BEM is the treatment of the volume integrals in the BIE. An efficient approach, based on the precorrected-FFT technique, is developed to evaluate the volume integrals. In this approach, the 3-D uniform grid constructed initially to accelerate surface integration is used as the baseline mesh to evaluate volume integrals. The cubes enclosing part of the boundary are partitioned using surface panels. No volume discretization of the interior cubes is necessary. This grid is also used to accelerate volume integration. Based on this approach, accelerated BEM solvers for non-homogeneous and nonlinear problems are developed and tested. Good agreement is achieved between simulation results and analytical results. Qualitative comparison is made with current approaches. Stokes problems with slip BC are of particular importance in micro gas flows such as those encountered in MEMS devices. An efficient approach based on the BEM combined with the precorrected-FFT technique has been proposed and various techniques have been developed to solve these problems. As the applications of the developed method, drag forces on oscillating objects immersed in an unbounded slip flow are calculated and validated with either analytic solutions or experimental results.

We present the application of the boundary integral equation method for solving the motion of biological cell or particle under Stokes flow in the presence of electrostatic field. The huge dense matrix-vector product from the boundary integral method poses a computationally challenging problem for solving the large system of equations generated. In our work, we used the precorrected-FFT (pFFT) method to reduce the computational time and memory usage drastically, so that large scale simulations can be performed quickly on a personal computer. Results on the force field acting on the particle, as well as the behavior of the particle through cell trap are presented. / Singapore-MIT Alliance (SMA)