Alan E. Lindsay

Kelsey L. DiPietro, Ronald D. Haynes, Weizhang Huang et al.

Numerical and analytical methods are developed for the investigation of contact sets in electrostatic-elastic deflections modeling micro-electro mechanical systems. The model for the membrane deflection is a fourth-order semi-linear partial differential equation and the contact events occur in this system as finite time singularities. Primary research interest is in the dependence of the contact set on model parameters and the geometry of the domain. An adaptive numerical strategy is developed based on a moving mesh partial differential equation to dynamically relocate a fixed number of mesh points to increase density where the solution has fine scale detail, particularly in the vicinity of forming singularities. To complement this computational tool, a singular perturbation analysis is used to develop a geometric theory for predicting the possible contact sets. The validity of these two approaches are demonstrated with a variety of test cases.

Alan E. Lindsay, Bryan Quaife, Laura Wendelberger

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green's functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.

Jeremy G. Hoskins, Alan E. Lindsay, Manas Rachh

In this paper we present a new boundary integral equation formulation for the solution of the elastostatic traction boundary value problem in two and three dimensions. The approach relies on the introduction of new layer potentials, called string kernels, which are based on modifications of the Boussinesq-Cerruti family of half-space solutions. We prove that the resulting integral equations are second-kind integral equations, and show that they are well-behaved in the incompressible limit. We illustrate the performance of the method with several numerical examples.

Andrew J. Bernoff, Alan E. Lindsay

In 1977 Berg & Purcell published a landmark paper entitled "Physics of Chemoreception" which examined how a bacterium can sense a chemical attractant in the fluid surrounding it. At small scales the attractant molecules move by Brownian motion and diffusive processes dominate. This example is the archetype of diffusive signaling problems where an agent moves via a random walk until it either strikes or eludes a target. Berg & Purcell modeled the target as a sphere with a set of small circular targets (pores) that can capture a diffusing agent. They argued that, in the limit of small radii and wide spacing, each pore could be modeled independently as a circular pore on an infinite plane. Using a known exact solution, they showed the capture rate to be proportional to the combined perimeter of the pores. In this paper we study how to improve this approximation by including inter-pore competition effects and verify this result numerically for a finite collection of pores on a plane or sphere. Asymptotically we have found the corrections to the Berg-Purcell formula that account for enhancement of capture due to the curvature of the spherical target and inhibition of capture due to spatial interactions of the pores. Numerically we develop a spectral boundary element method for the exterior mixed Neumann-Dirichlet boundary value problem. Our formulation reduces the problem to a linear integral equation, specifically a Neumann to Dirichlet map, which is supported only on the individual pores. The difficulty is that both the kernel and the flux are singular, a notorious obstacle in such problems. A judicious choice of singular boundary elements allows us to resolve the flux singularity at the edge of the pore. In biological systems there can be thousands of receptors whose radii are 0.1% the cell radius. Our numerics resolve this realistic limit with an accuracy of roughly one part in 10^8.