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The award supported Peter Bosler as a postdoc in the Department of Mathematics at the University of Michigan. The project developed a novel Lagrangian particle method (LPM) for geophysical fluid flow simulations on a rotating sphere. The method is potentially relevant to Naval operations that rely on accurate and efficient modeling of atmosphere and ocean dynamics. We developed new remeshing and refinement schemes to overcome the loss of accuracy due to distortion in the particle distribution. The results were disseminated through published articles and presentations at conferences and seminars.

This young investigator award from the Air Force Office of Scientific Research to Daniel Forger played a pivotal role in his career. Many discoveries were made and are described in over 10 published papers or manuscripts in preparation. Some research highlights include: 1) discovering that timekeeping in mammals is more complex than originally thought, 2) discovering on how to optimally stimulate neuronal (and in particular the circadian) systems and 3) mathematical methods for studying biological clocks. Forger remains very grateful to the AFOSR for the opportunity to perform this work., AFOSR Young Investigator Award.

This project draws together a team of researchers to develop new grid-free tools for plasma simulations. The objectives are: (1) to develop a grid-free field solver, fluid model, and kinetic model, (2) to evaluate these tools in comparison with traditional mesh-based methods, and (3) to demonstrate the capability of the grid-free approach in an application of USAF interest. The field solver will use boundary integral methods and a recently developed tree code algorithm to compute the electrostatic force induced by a set of charged particles. The kinetic model will use tree-based statistical procedures for inter-particle collisions. Various hybrid grid-free simulations will be possible using these tools. The resulting grid-free approach will have several advantages over traditional mesh-based methods: it will eliminate the need for elaborate volume-meshing routines required to handle complex geometry, and will avoid the spurious numerical artifacts in the particle distribution that can arise in mesh-based schemes.

This grant provided support for a postdoc at the University of Michigan to assist in developing a grid-free particle method for electrostatic plasma simulations. The aim of the work is to substantially improve the accuracy and efficiency of these simulations. The proposed method is an alternative to traditional mesh-based methods such as particle-in-cell (PlC). In the new approach, the standard Eulerian formulation of the Vlasov-Poisson equation is replaced by a Lagrangian formulation in which the charge flow map is the key unknown quantity. Discretizing the Lagrangian formulation leads to a grid-free particle method. The investigators made progress in formulating and testing this approach. Numerical results are presented for the cold one-stream and two-stream instabilities. The method is especially well suited for tracking charge transport and resolving fine structures in phase space.

An examination of the relationship between the scattering matrix (the Fourier transform of the scattering operator) and the integral equations used in the Singularity Expansion Method (SEM) established that only the complex poles off the axis are intrinsically associated with the scatterer. While it has been known in special cases that those on the axis do not contribute to the field, this appears to be the first time this relationship has been clearly exhibited. Since the scattering matrix can be shown to be analytic in a half-plane containing the axis, any integral equation should exhibit the same properties for this region. The relationship between the eigenvalues of the integral equations of SEM and the complex eigenvalues of the associated partial differential equations -- whether scalar or vector. In particular, the integral equations of SEM have at most two eigenvalues + or - and these are functions of the at-most-denumerable number of complex eigenvalues of the associated differential equations. The theory of non-self-adjoint operators in Hilbert space could have significant implications for that part of the electrical engineering formalism known as the Eigenmode Expansion Method (EEM) particularly in relevance to the use of this formalism in developing equivalent circuits. (jhd)

A wake is modelled by a vortex sheet carrying positive and negative circulation. The sheet's evolution is computed by the vortex-blob method. Initial conditions and circulation density for the vortex sheet are chosen to simulate some of the wake patterns observed in the soap-film experiments of Couder et.al.(2,3).

The report summarizes progress made in several, areas, including: estimates for finite element methods, numerical simulation of nonlinear dispersive waves, numerical models for plasticity, numerical methods for the transport equation, and computation of fluid-flow problems. (Author)

Eigenfunction expansions for nonselfadjoint operators are important for scalar and electromagnetic wave scattering. Two methods: the Singularity Expansion Method (SEM), and the Eigenmode Expansion Method (EEM) which had been developed separately were studied. Criteria for their validity were established; moreover, the poles of the Green's function of the SEM are in 1-1 correspondence with the zeros of the eigenvalues of the EEM. A constructive numerical process for determining the poles of the Green's function was developed. Among several other results was the establishment of variational principles for the spectrum of compact nonselfadjoint operators. Another research area was the singularity behavior of eigenfunction expansions of various Green's functions in electromagnetic theory. The principal result shows that in an eigenfunction of a typical Green's function the point singularity of a Green's function is represented by a layer of surface singularity. This characteristic is analogous to the Gibbs' phenomenon where the representation of a discontinuous function by an orthogonal expansion creates the spikes which are absent in the original function. (Author)

Several problems relevant in the application of mechanics to real-world problems were treated. The work included theoretical analysis of solutions of partial differential equations, detailed studies of numerical techniques for approximately solving differential equations arising in mechanics, as well as actual numerical computations using such numerical schemes to model experimental phenomena. Application areas impacted by this research include heat conduction and fluid mechanics. Also closely related are problems arising in the study of controlled fusion. In addition, the work has added significantly to theoretical understanding of numerical methods for solving various partial differential equations. (Author)

A new finite element method is proposed for the numerical solution of a class of initial-boundary value problems for first order hyperbolic systems in one space dimension. An application of our procedure to a system modeling gas flow in a pipe is discussed. Asymptotic error estimates are derived in the L sq norm in space. (Author)

This paper gives a theory of spectral approximation for closed operators in Banach spaces. The perturbation theory developed in this paper is applied to the study of a finite element procedure for approximating the spectral properties of a differential system modeling a fluid in a rotating basin. (Author)

During this period the nine scientists associated with the project produced 76 papers of which 35 have appeared in journals at this time. Many invited lectures, editorships and a chair in mathematics attest to the ability of the principal investigators. In the area of entrainment of frequency in the large involving oscillations of physical systems at resonance and across point of resonance 14 papers were produced. Pareto optimization theorems in optimal control theory account for three papers. Six papers on optical resonance of laser beams in crystals and 11 papers on evolution equations and related problems in shock wave theory, semiconductors, diffusion, combustion kinetics and ecological systems give an idea of the diversity of areas addressed by the functional analytic techniques of Professors Ceasari, Smoller Suryanarayana, Kannan and associates. (Author)

This report includes a list of papers completed and published, a description of the activities of each of the investigators, and a discussion of some of the most significant results obtained.

The various theoretical results outlined are concerned with such matters as the structure of graphs having a maximum and minimum number of 'cliques' as used in clustering and classification schemes, with the enumeration of acyclic configurations of generalizations of mathematical models of the biological growth cells and paving problems in statistical mechanics, with properties of 'anticritical' graphs such as point and line-connectivity, with duals of tree graphs, with enumeration of random walks in graphs in terms of the moments of the eigenvalues of their adjacency matrices, and with determining the minimum number of prints of a digraph which must be removed so that the result is acyclic, i.e., finding the minimum feedback vertex set of a digraph. The report also provides a listing of the publications in which the detailed accomplishments are presented., See also report dated 30 Aug 74, AD/A-003 983.

Research on both internal and free-surface flows related to experimental situations was continued. Part of this research was based on using a low-order code developed previously, jointly by Pritchard and Scott. Funds for the programmer allowed two substantial new programming projects to be initiated during this time. One concerned investigation of techniques for implementing a quartic finite element method for viscous, incompressible, two-dimensional flows. The other involved a streamline-diffusion finite element method for inviscid flows.