Browsing by Author "Driscoll, Tobin A."
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Item Approximations in Canonical Electrostatic MEMS Models(Kluwer Academic Publishers (see: http://www.sherpa.ac.uk/romeo.php for publisher's conditions for archiving in an institutional repository), 2004-09-28) Pelesko, John A.; Driscoll, Tobin A.The mathematical modeling and analysis of electrostatically actuated micro- and nanoelectromechanical systems (MEMS and NEMS) has typically relied upon simplified electrostatic field approximations to facilitate the analysis. Usually, the small aspect ratio of typical MEMS and NEMS devices is used to simplify Laplace's equation. Terms small in this aspect ratio are ignored. Unfortunately, such an approximation is not uniformly valid in the spatial variables. Here, this approximation is revisited and a uniformly valid asymptotic theory for a general "drum shaped" electrostatically actuated device is presented. The structure of the solution set for the standard non-uniformly valid theory is reviewed and new numerical results for several domain shapes presented. The effect of retaining typically ignored terms on the solution set of the standard theory is explored.Item Computing Eigenmodes of Elliptical Operators Using Radial Basis Functions(Department of Mathematical Sciences, 2003-03-21) Platte, Rodrigo B.; Driscoll, Tobin A.Radial basis function (RBF) approximations have been successfully used to solve boundary-value problems numerically. We show that RBFs can also be used to compute eigenmodes of elliptic operators. Special attention is given to the Laplacian operator in two dimensions. We include techniques to avoid degradation of the solution near the boundaries and corner singularities. Numerical results compare favorably to basic finite element methods.Item Eigenvalue stability of radial basis function discretizations for time-dependent problems(Department of Mathematical Sciences, 2005) Platte, R.B.; Driscoll, Tobin A.Differentiation matrices obtained with infinitely smooth radial basis function (RBF) collo- cation methods have, under many conditions, eigenvalues with positive real part, preventing the use of such methods for time-dependent problems. We explore this difficulty at theoretical and practical levels. Theoretically, we prove that differentiation matrices for conditionally positive definite RBFs are stable for periodic domains. We also show that for Gaussian RBFs, special node distributions can achieve stability in 1-D and tensor-product nonperiodic domains. As a more practical approach for bounded domains, we consider differentiation matrices based on least-squares RBF approximations and show that such schemes can lead to stable methods on less regular nodes. By separating centers and nodes, least-squares techniques open the possibility of the separation of accuracy and stability characteristics.Item An improved Schwarz-Chrisoffel Toolbox for MATLAB(Mathematical Sciences Department, 2003) Driscoll, Tobin A.The Schwarz-Christoffel Toolbox (SC Toolbox) for MATLAB, rst released in 1994, made possible the interactive creation and visualization of conformal maps to regions bounded by polygons. The most recent release supports new features, including an object-oriented command-line interface model, new algorithms for multiply elongated and multiple-sheeted regions, and a module for solving Laplace's equation on a polygon with Dirichlet and homogeneous Neumann conditions. Brief examples are given to demonstrate the new capabilities.Item Isospectral Shapes with Neumann and Alternating Boundary Conditions(Department of Mathematical Sciences, 2003) Driscoll, Tobin A.; Gottlieb, H.P.W.The best-known negative answer to Mark Kac's question, "Can one hear the shape of a drum?" is a pair of octagons discovered by Gordon, Webb, and Wolpert. Their nonconstructive proof of Dirichlet isospectrality has since been supplemented by experiment and high-accuracy computation of many of the eigenvalues of these drums. In this paper we compute the Neumann modes of these regions, also known to be identical, to high precision. Additionally, we carry out the computations for two cases in which the boundary conditions alternate between Dirichlet and Neumann around the sides. There is overwhelming numerical evidence that the regions remain isospectral, though to our knowledge there has been no analytic demonstration of this fact.Item Polynomials and their Potential Theory for Gaussian Radial Basis Function Interpolation(Department of Mathematical Sciences, 2004) Driscoll, Tobin A.; Platte, Rodrigo B.We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these radial functions are susceptible to the Runge phenomenon, not only in the limit of increasingly flat functions, but also in the finite shape parameter case. We show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations. Using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices, without imposing restrictions on the shape parameter or number of points.