Advances and Mathematical Issues in Large Scale Simulation - IMS

ADVANCES AND MATHEMATICAL ISSUES IN LARGE SCALE SIMULATION
(December 2002 - March 2003)
and
(October - November 2003)

Abstracts

Heterogeneous Multiscale Methods
Bjorn Engquist, Princeton University

The heterogeneous multiscale method is framework for a class of computational techniques for solving problems with a large variety of scales. We will discuss the theory of convergence as well as applications, for example, to composite materials and the coupling of molecular dynamics and fluid equations.

Multiscale Modeling and Computation
Thomas Yizhao Hou, California Institute of Technology

Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the range of length scales in the underlying physical problems. Here, we introduce a dynamic multiscale method for computing nonlinear partial differential equations with multiscale solutions. The main idea is to construct semi-analytic multiscale solutions local in space and time, and use them to construct the coarse grid approximation to the global multiscale solution. Such approach overcomes the common difficulty associated with the memory effect and the non-unqiueness in deriving the global averaged equations for incompressible flows with multiscale solutions. It provides an effective multiscale numerical method for computing incompressible Euler and Navier-Stokes equations with multiscale solutions. In a related effort, we introduce a new class of numerical methods to solve the stochastically forced Navier-Stokes equations. We will demonstrate that our numerical method can be used to compute accurately high order statistical quantities more efficiently than the traditional Monte-Carlo method.

Design and Analysis of Millimeter-Wave Microwave and Integrated Antennas
Tomasz Grzegorczyk, Research Laboratory of Electronics, Center for Electromagnetic Research and Applications, Massachusetts Institute of Technology, USA

Metamaterials with negative refraction index (NRI) behavior have been essentially realized as a periodic arrangement of split-ring resonators and rods. This behavior has been verified both experimentally and numerically, based on a simple extension of Maxwell's equations to negative permittivity and permeability regimes. Yet, important limitations of these metamaterials nowadays are high losses and narrow bandwidth. These effects are directly related to the geometry of the periodic structure. In this talk, we shall review the basic principles governing these metamaterials (negative index of refraction, backward phase propagation, etc), and go deeper into the understanding of loss mechanisms. This will be done in essentially two ways: 1) Identifying the influence of various geometrical parameters on the losses mechanisms. 2) Considering the metamaterial as a bulk material with homogeneous permittivity and permeability, and retrieving these parameters as function of frequency. Eventually, some experimental data will be shown to corroborate the theoretical predictions.

Propagation and Losses in Negative Refraction Index Materials
Tomasz Grzegorczyk, Research Laboratory of Electronics, Center for Electromagnetic Research and Applications, Massachusetts Institute of Technology, USA

Telecommunications have experienced a very distinctive shift toward higher frequencies over the past years, moving the antenna technology from microwave to millimeter waves. The major impacts of this phenomenon has been: 1) a change in the technology itself, from standard circuit boards (e.g. microstrip antennas) to micro processes, 2) a change of the antennas configurations, from planar to three dimensional. A direct consequence of these impacts have been the necessity to update the theoretical models. In the case of planar antennas for example, the integral equation method combined with the Method of Moments have been proven to yield very accurate results. Yet, the generalization of this approach toward three dimensional geometries still draws a lots of attention, essentially because of computational efficiency issues. In this talk, we shall present a real case of an integrated antenna designed to operate at 75 GHz, and which geometry requires a generalization of the aforementioned technique to three-dimensional structures. Therefore, in parallel to the technological presentation of the antenna, we shall emphasize the theoretical challenges encountered, and the solution adopted to overcome them.

An Introduction to the Fast-MoM
Alireza Baghi-Wadji, Institute for Industrial Electronics and Materials Science Vienna University of Technology, Austria

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FEM analysis of waveguides loaded with ferrite magnetized in an arbitrary direction
Zhou Le Zhu, Department of Electronics, Peking University, China

In this presentation, a simple edge-element based FEM (finite element method) is introduced for directly calculating phase constants of waveguides loaded with ferrite magnetized in an arbitrary direction. In order to solve the quadratic eigenvalue equation, which appears in the case where the magnetized direction is not parallel to the propagation, a simple and effective iteration approach is proposed with no increase in the number of degrees of freedom. Calculated results for two example structures proved the accuracy and efficiency of the formulations and proposed approach. Presentation also shows some results of interests for sandwich coupled-slot lines with ferrite layers, which seem to present some new and potential applications

A Pseudo-spectral Analysis of Photonic Crystals
Alireza Baghi-Wadji, Institute for Industrial Electronics and Materials Science Vienna University of Technology, Austria

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Advanced Finite Element Method and Hybrid Fast Method for Computational Electromagnetics
Jian Ming Jin, Department of Electrical and Computer Engineering, University of Illinois at urbana-Champaign

During the past decade, significant advances have been made in the finite element method for computational electromagnetics. As a result, large and complex problems of scattering, antennas, and microwave circuits and devices can be analyzed accurately and efficiently using the finite element method. Problems involving millions of unknowns have been handled on workstations and devices consisting of inhomogeneous anisotropic material have been simulated successfully. The major advances include (1) the development of higher-order vector finite elements, which make it possible to obtain highly accurate and efficient solutions of vector wave equations; (2) the development of perfectly matched layers as an absorbing boundary condition; (3) the development of hybrid techniques that combine the finite element and asymptotic methods for the analysis of large and complex problems that were unsolvable in the past; (4) the development of fast integral solvers, such as the fast multipole method, and their integration with the finite element method; and finally (5) the development of the finite element method in time domain for transient analysis. The objective of this seminar is to summarize these advances and demonstrate the increased capabilities of the finite element methods to solve complex computational electromagnetics problems.

The Finite Element Method for Computational Electromagnetics: Recent Progress, Current Status, and Future Directions
Jian Ming Jin, Department of Electrical and Computer Engineering, University of Illinois at urbana-Champaign

This talk shall give the latest development of finite element method for electromagnetic computation and the fast algorithm applications in CEM as well as the trend and development in future.

FEM-FMA Hybrid Method and its Application to Analysis of EM Scattering from 3-D Complex Objects
Zhou Le Zhu, Department of Electronics, Peking University, China

A fast hybrid method, which combines edge-element based finite element method and fast multipole algorithm (FEM-FMA), is presented to analyze electromagnetic scattering from 3-D complex objects in this presentation. The method has all the advantages of FEM-BI, edge-elements and FMA, therefore it is very suitable for analysis of EM scattering from large and complex objects. In the presentation, firstly we outline the formula for the edge-element based FEM and MOM hybrid method for general 3-D complex objects. Then we replace MOM with FMA to form FEM-FMA fast hybrid method. Finally we use the hybrid method to calculate the RCS of 3-D conductive targets coated with anisotropic materials. The numerical results demonstrate a good effect of this method on accuracy, speed and memory requirement.

Application of B-Spline and Dual B-Spline Wavelets to EM Scattering problems
Alireza Baghi-Wadji, Institute for Industrial Electronics and Materials Science Vienna University of Technology, Austria

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Solving Antenna and Scattering Problems Using Fast Multipole Method
Le-Wei LI, Department of Electrical & Computer Engineering, National University of Singapore

Development of fast algorithms has been a very fast growing research area. A few different methodologies were developed, including the order-reduced finite element method and matrix-free integral equation method. Among those popularly used approaches, the matrix-free integral equation method is the best known and fast growing one. This integral equation method is implemented usually in different ways, each forming a different method to come up those popular ones of Fast Multipole Method (FMM), Adaptive Integral Method (AIM), and Pre-corrected Fast Fourier Transform (P-FFT) Method. This talk will focus on the FMM. It will cover the brief introduction to the fast multipole method, its implementation into codes, and applications to electromagnetic scattering by various objects of arbitrary shape and large-scale, and also to large-scaled antenna design. Potential of this method and its accuracy with restrictions are discussed as well.

Boundary Element Method in Computational Acoustics
Ih Jeong-Guon, Center for Noise and Vibration Control (NOVIC), Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology

In this seminar, several research topics in computing the acoustic field by using the BEM will be talked about. The topics being conveyed to the audiences are some of study results that have been dealt with for recent ten years in the Acoustics Lab. at KAIST. First, the non-singular boundary integral equation for acoustic problems has been tackled. A weakly singular boundary integral equation and a non-singular one are derived for acoustic problems. By using the one-dimensional wave propagation mode, the degree of the singularities appearing in the conventional boundary integral equation can be reduced. Since the strong singularity is removed in the weakly singular equation, the jump term of the field pressure expression in the very vicinity of the boundary disappears. It is shown that the weak singularity can be also removed for a smooth boundary in the non-singular representation, and thus special treatments for singular integral are no longer needed. Second, the effective use of the multi-domain BEM (MBEM) has been studied for analyzing the complicated and large scaled acoustic systems. If a part of the boundary of such system were only to be changed for analyzing the effect of boundary perturbations, the rest of the system would be largely unchanged in many cases. The fact that some parts of the cavity model never change can be utilized in accelerating the involved computations. The MBEM using the sub-domain modularization technique is proposed for the fast calculation of the field response in the specified sub-domain of the interior acoustic field, in which each sub-domain is regarded as a completely closed single-domain if the acoustic variables are determined on each boundary. Third, a simplified BEM for dealing with high-frequency sound has been studied on. The boundary integral equation is modified into a quadratic form to enable the prediction of sound levels in the 1/3-octave band analysis. Monopole and dipole source terms in the conventional BEM are transformed into the auto- and cross-spectra of two vibrating sources, in which the cross-spectra are eventually neglected by assuming that the correlation coefficients involved are negligible. The over-determination process for overcoming non-uniqueness in exterior radiation problems is unnecessary in the present method because phase information can be ignored.

Efficient and Accurate Boundary Methods for Computational Optics
Christian Hafner, Laboratory for Electromagnetics Fields and Microwave Electronics, Dept of Information Technology and Electrical Engineering, ETH (Swiss Federal Institute of Technology)

The Multiple Multipole Program (MMP) is the an advanced boundary method for computational electromagnetics and optics. Its vicinity to analytic methods allows one to obtain highly accurate and reliable results. After a short introduction of the fundamentals of MMP, additional techniques are introduced that allow one to drastically improve the performance for various applications. This includes a special consideration of ill-conditioned matrices, the so-called connection concept, an advanced eigenvalue solver with a special eigenvalue tracing procedure, and a parameter estimation technique. Special attention is paid to the error estimation and validation of the results. In a second section, the most recent MMP version of the MaX-1 software with its advanced modelling, visualization, and animation features is presented and it is demonstrated how this software is used for handling complicated projects. After this, some typical applications of computational optics with a focus on photonic bandgap computations and photonic crystal structures are presented.

Computational Acoustics for Vehicle Interior Noise
Ichiro Hagiwara, Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology

In this presentaion I will mainly present about "new Component mode Synthesis Method". But I will also introduce my research about "Reduction of Noise Inside a Cavity by Piezoelectric Actuators". This presentation abstract is as follows: A new analytical model is developed for the reduction of noise inside a cavity using distributed piezoelectric actuators. A modal coupling method is used to establish the governing equations of motion of the fully coupled acoustics-structure-piezoelecric patch system. Two performance functions relating "gloval" and "local" optimal control of sound pressure levels(SPL) respectively are applied to obtain the control laws. The discussions on associated control mechanism show that both the mechanisms of modal amplitude suppression and modal rearrangement may sometimes coexist in the imprementation of optoimal noise control".

Virtualization-Aware Application Framework for Multiscale Continuum/Atomistic/Quantum- Mechanical Simulations on a Grid
Aiichiro Nakano, Collaboratory for Advanced Computing and Simulations, Department of Computer Science, Department of Physics & Astronomy, Department of Materials Science & Engineering, University of Southern California

Grid of globally distributed commodity PC clusters is becoming a major platform for high-performance scientific computing. On a Grid, applications will likely be virtualized, i.e., the user will not know on which computers the application code is running. To support virtualization, applications need to be scalable from a single processor to thousands of processors, their performance needs to be portable from one architecture to another, and they need to be adaptive to dynamically changing computing resources.

We are developing a virtualization-aware application framework based on data-locality principles to perform multiscale materials simulations on a Grid. The multiscale simulation approach seamlessly integrates: Quantum mechanical (QM) calculation based on the density functional theory (DFT); classical atomistic simulation based on the molecular dynamics (MD) method; and continuum mechanics based on the finite element (FE) method. The framework combines: Linear-scaling algorithms based on space-time multiresolution techniques; topology-preserving computational-space decomposition with wavelet-based adaptive load balancing; and spacefilling-curve-based data compression for compressed software pipelines. Scalability of these applications has been tested up to 6.4 billion-atom MD and 0.44 million-electron DFT calculations.

Our multiscale simulation code has been Grid-enabled using: A modular, additive hybridization scheme; divide-and-conquer scheme based on multiple QM clustering; and computation/communication overlapping. A multidisciplinary, collaborative simulation has been performed on a Grid of globally distributed PC clusters in the US and Japan. The Gridified multiscale simulation code has been used to study environmental effects of water molecules on fracture in silicon.

I will also describe our large-scale atomistic simulations of nanostructured materials and nanoscale devices. These include sintering and fracture of nanophase ceramics and nanocomposites, nanoindentation, hypervelocity impact, oxidation of metallic nanoparticles, and colloidal and epitaxial quantum dot structures.

Work supported by NSF, DOE, ARL, DoD DURINT, and NASA.

Piezoelectric Hybrid Element And Applications: Crack-tip Field Analysis and Fracture Assessment
Wu Chang-Chun, School of Civil Engineering & Mechanics, Shanghai Jiao Tong University, China

In the analysis crack-tip singularity, the conventional isoparametric element (PZT-Q4) shows poor behavior. Instead, the hybrid element is developed for the electro-elastic analysis. The hybrid model shows a reliable behavior and can recover the singularity at the crack-tip. The suggested hybrid element is used in the piezoelectric crack-tip analysis, and a series of crack tip singularity are well numerically represented.

In many cases, instead of the stress criterion, one prefers use the path- independent integrals as the crack parameter. But the conventional compatible element can only provide the lower bound which cannot be used in the safety estimation for a crack system.

An assessment approach for the piezoelectric crack problem is suggested. The dual path-independent integrals and the relative bound theorems are established. For the numerical implementation, a penalty-equilibrium hybrid piezoelectric element is suggested, such that the upper bound solution is available

Incompatible Discrete Model and Application to Computational Plasticity & Strain Gradient Materials
Wu Chang-Chun, School of Civil Engineering & Mechanics, Shanghai Jiao Tong University, China

Some new concepts for incompatible finite element approach are presented. The suggested patch test condition (PTC) cannot only ensure the convergence of the incompatible numerical solutions, but also can be used to develop new incompatible model. In such a way, some incompatible models of high performance are formulated. In the concerned computational plasticity, the conventional isoparametric elements show very poor solutions, and the incompressible locking problem cannot be avoided. The suggested incompatible elements are able to simulate the incompressible deformation and provided a series of rational plastic solutions. As a further application, the suggested incompatible numerical approach is successfully developed to the multiscale simulation of the strain gradient materials

An introduction to molecular dynamics
Gerhard Zumbusch, Friedrich-Schiller-Universität Jena Institut für Angewandte Mathematik, Germany

A wide range of phenomena can be modeled with particles. From atomar scale to planetary movement particles represent the center of gravity of objects. They obey Newton´s laws of motion. Together with interaction governed by point to point forces, a system of particles can accurately simulate physics. In the talk the basic ingredients of molecular dynamics codes and a range of experiments are presented.

Multilevel and tree type algorithms
Gerhard Zumbusch, Friedrich-Schiller-Universität Jena Institut für Angewandte Mathematik, Germany

Large scale numerical simulations occur in different applications, be it the solution of equations systems stemming from partial differential equations or integral equations or form the force evaluation in molecular dynamics. Efficient algorithms for these tasks can be constructed by the concept of multiscale methods. A hierarchy of scales is introduced and at each scale the problem is considered. This approach includes multigrid solvers, multiscale analysis and algorithms like the fast-multipole method.

Parallel algorithms
Gerhard Zumbusch, Friedrich-Schiller-Universität Jena Institut für Angewandte Mathematik, Germany

Large scale numerical simulations require large computers which are usually (massive) parallel computers. However, parallel computers require parallel algorithms. This is especially true for distributed memory computers which also require a different programming model. A range of strategies for the parallelisation is presented from static domain decomposition in molecular dynamics to dynamic load balancing in tree type algorithms and multilevel methods.

Molecular forces and time integration
Gerhard Zumbusch, Friedrich-Schiller-Universität Jena Institut für Angewandte Mathematik, Germany

This talk complements the introductory talk on molecular dynamics. As an extension of the basic molecular dynamics code with straight forward Lennard-Jones Potential different models of inter-molecular force are discussed. This includes forces for atomic bouds in molecules, atomic interaction in metal and certain more refined carbon-hydrogen bonds. Further extensions of the basic code include an in depth treatment of time integration including a sound interpretation of long time simulation results.

Fracturing and structure change -classical molecular dynamics simulation
Tamio Ikeshoji, Research Institute for Computational Sciences, Japan

Fracture is the biggest damage of materials. But, its atomic scale mechanism is not well known, since large scale simulations are needed. Molecular dynamics can handle only small volume and short time, but it gives an atomic level information, which can be used to develop stronger structural materials. Fracturing by fatigue with classical molecular dynamicssimulations will be shown. Some other examples of structure formation processes analyzed by molecular dynamics will be also shown.

Catalytic reactions on solid surface-first principles molecular dynamics simulation
Tamio Ikeshoji, Research Institute for Computational Sciences, Japan

Catalysis is an important material not only in chemical industries but also in various fields. In fuel cells, for instance, electrode catalysis is very important to have a high performance. In order to simulate reactions, electronic state calculations (so-called ab-initio calculations) are necessary, since reactions are highly related to the electronic structures. Molecular dynamics is also necessary to be combined to the ab-initio calculations. Then, first principles molecular dynamics simulations must be performed to simulate reactions. Some examples of the simulations of fuel cell reactions and others will be shown as well as explanation of first principles molecular dynamics itself.

Computational Sciences in RICS, AIST
Tamio Ikeshoji, Research Institute for Computational Sciences, Japan

RICS (Research Institute for Computational Sciences) in AIST covers a wide range of computations; calculations for electronic state, electron transport, molecular dynamics, macroscopic dynamics, etc. They are based on quantum, classical, statistics, and continuum dynamics. Calculation targets are for nanotechnology, chemical reactions, biological systems, two phase flow, solid-liquid problems, etc. In this seminar, several topics of activities in RICS will be shown. PCP (Parallel Computing Platform) and TACPACK (web-base queueing system of classical and first principles molecular dynamics) are also explained.

How to calculate local pressure and energy flux by molecular dynamics
Tamio Ikeshoji, Research Institute for Computational Sciences, Japan

Local pressure are often used to analyze the simulation results by molecular dynamics. But, it was believed for a long time that there is no definite way in its calculation. We solved somehow about this problem. Local energy flux has also the similar problem. The recent work on this topic will be shown.

Atomistic Modeling - Lattice Statics and Molecular Dynamics
Vijay B. Shenoy, Indian Institute of Science, India

This lecture will deal with simulation methodogies for mateials starting from an atomistic perspective. Topics covered will include quantum mechanics based total energy calculations, tight binding method, atomistic models (without direct appeal to electronic structure).

Coarse Grained Approaches - The Quasicontinuum Method
Vijay B. Shenoy, Indian Institute of Science, India

Approaches to developed effective models that allow for efficient simulations will be the topic of this lecture. The recently developed quasicontinuum method will be described, including outstanding problems

Pattern Formation in Thin Films
Vijay B. Shenoy, Indian Institute of Science, India

Pattern formation in thin films is of interest both from sceintific and technical viewpoints. This talk will introduce the are of pattern formation. A particular example of patterning of soft films will be dealt with in detail. Outstanding problems will be discussed.

Size Dependent Properties of Nanostructures
Vijay B. Shenoy, Indian Institute of Science, India

A key challenge in the developement of nanoscale devices is the understanding of how properties of matter depend on the size of the materials. Some theoretical ideas that allow the prediction of size-dependence of properties will be covered, after the introduction of the experimental results.

BEM-based Holographic Reconstruction with Field Expansion Technique
In-Youl Jeon, Korea Advanced Institute of Science and Technology (KAIST)

The BEM-based near-field acoustic holography (NAH) can effectively reconstruct the vibro-acoustic field radiating from an arbitrarily shaped object in an inverse manner. In applying BEM-based NAH, the overdetermined transfer matrix is involved, thus the number of measurements increases greatly as the increase of BEM mesh numbers. In many practical cases, the total measurement time and associated effort depend on the complexity of the problem. The purpose of this study is the extension of a priori measured data to decrease the measurement effort and enlarge the size of transfer matrix to overcome the rank-deficient problem. The additional field information would be obtained by linear geometrical interpolation or field expansion using the orthogonal spherical basis function. For validating the present method, a three-dimensional box example is considered. Consequently, it is found that the reconstruction error can be considerably reduced and the calculation effort can be decreased with the use of additional field information.

A Simplified Hologrphic Reconstruction Using the Inverse BEM
In-Youl Jeon, Korea Advanced Institute of Science and Technology (KAIST)

It has been well known that the BEM-based near-field acoustic holography can reconstruct the vibro-acoustic field of a radiating source with the appropriate measurement points distributed in non-separable coordinate. However, the modeling error of noise source would greatly affect the accuracy of reconstruction field in case of the sound radiation from a complex vibrating structure in which both airborne and structural sources present altogether. In this study, a hypothetical simple surface, which encloses the noise source in a conformal way, is modeled instead of using the real sophisticated surface model, which results a reduction of the size of system matrix. Consequently, the singularity of system matrix would be reduced, thus the resolution of reconstruction field would be dramatically improved. A simple example is considered for validating and demonstrating the proposed reconstruction process. It is found that the present method, when employed in the NAH based on the inverse BEM, can be an efficient way in the source identification of highly complex source and in the reduction of the reconstruction error.

Noise Source Ranking in Automotive Vehicle Using the Inverse FRF Method
In-Youl Jeon, Korea Advanced Institute of Science and Technology (KAIST)

Identification of location and strength distribution of extended noise sources is important in practice. In this study, the inverse frequency response function (iFRF) method was adopted for this purpose and an example was given to the noise source ranking inside an automotive vehicle. The iFRF method is based on the array measurement of field pressure and the inverse estimation of source strength. The method was employed to predict the interior sound pressure with the change of sound insulation materials. As a result, the source contribution from vehicle panel segments was successfully identified which is far physically meaningful and precise than the window method. The sound pressure at the drivers ear position was predicted based on the obtained data: Compared with the measured result, the agreement in spectral trends was acceptable.

Factorization of Potential and Field Distributions without Addition Theorems
Alireza Baghi-Wadji, Institute for Industrial Electronics and Materials Science Vienna University of Technology, Austria

The construction of multipole expansions for potential- and field distribution functions relies on the existence of certain factorization formulae, which are conventionally formulated by the so-called addition theorem.

The main objective in this talk is the presentation of a recipe for obtaining factorization formulae without the need for the addition theorem.

Factorization, as used here, refers to the multiplicative decomposition of the potential (field) functions into functions, which depend on the co-ordinates of the source- and the observation point individually. The multiplicative decomposition is crucial in all those instances, where we need to, e.g. integrate over a large number of source points, while keeping the observation point constant. Such circumstances routinely arise in large scale engineering computations based on singular surface integrals (boundary element method applications).

Simply speaking, the basic idea behind the factorization is, to break down the Euclidean distance between the source- and the observation point, into terms, which depend on the co-ordinates of the source point position vector and the observation point position vector, separately. There are several possibilities to achieve this objective, which will be described in our presentation:

(1) Multipole expansions and the underlying addition theorem result in factorized forms with functions being in real space.

(2) An alternative approach for obtaining factorized forms consists of employing integral transforms with certain translational shift invariance properties; a prominent example being the Fourier transform, which results in spectral decompositions. It should be pointed out that obtaining factorized forms in spectral domain is based on the diagonalization of the underlying partial differential equations with respect to a distinguished direction in space.

(3) In this talk we will diagonalize the governing equations with respect to the radial direction and use field expansions in the real space. It will be shown that our diagonalization allows the factorization of the field distribution functions directly.

The presentation can be outlined as follows: we first briefly discuss the addition theorem. Then we show how to diagonalize the Poisson's equation in spherical co-ordinates. Finally, we provide a detailed discussion on how to factorize the field expansion functions. The derivation procedure should be appealing to engineers, who will easily recognize the wide range applicability of the proposed iterative method. It can shown that the proposed recipe is applicable to electrodynamic and elastodynamic problems involving isotropic, anisotropic, or bi-anisotropic materials. Furthermore, it can be shown that whenever a system of PDEs permits diagonalization, our procedure is applicable. In a recent contribution we formulated a conjecture claiming that every physically-realizable system of PDEs allows diagonalization, with respect to a certain distinguished direction in space. Based upon this conjecture it is reasonable to expect, that our formulation, applied to various problems, will result in generalizations of the expansion formulae appearing in the addition theorem. We conclude that our method as a pleasant byproduct, and perhaps ironically, can be instrumental in formulating new addition theorems.

Quantum Mechanical Treatment of Unperturbed and Perturbed Harmonic Oscillators
Alireza Baghi-Wadji, Institute for Industrial Electronics and Materials Science Vienna University of Technology, Austria

The quantum mechanical analysis of unperturbed and perturbed harmonic oscillators is undoubtedly one of the most beautiful treatments in the theory of operators. This fact alone would easily justify several seminars dedicated to the subject. However, in this talk our aim is to look at these two problems from a different, rather engineering prospective. We want to discuss these problems and their brilliant solution methodologies as two utmost interesting examples in order to convey several intriguing ideas and concepts: The eigenfunctions of the unperturbed harmonic oscillator (UHO) constitute a complete (overcomplete) system of orthogonal functions, which were originally named coherent states by Erwin Schroedinger. Coherent states have rich properties, which have been exploited for solving problems not only in quantum mechanics and quantum optics, but also in the analysis and approximation of functions in mathematics and engineering applications. In order to obtain the eigenfunctions of the UHO and, therefrom the eigenfunctions of the perturbed harmonic oscillator (PHO), and consequently of more complex operators a firm understanding of some basic tools in operator theory is required. The main objective of this presentation is the logical development of the underlying ideas in a manner, which is accessible to everybody with an engineering background. Thereby, numerous examples will help to highlight the details. The speaker is convinced that engineers, not only those active in physical electronics, ought to be literate in quantum physics. Engineers with an interest in field analysis, signal processing, pattern recognition, communication, and of course materials science and device simulation are expected to have a sound grasp of the underlying ideas and concepts in quantum physics. This presentation will show why this claim is true, and will demonstrate that the way to this end is not necessarily a stony one, provided we look at the problem from the right perspective, and use a good deal of pedagogical sophistication.

The presentation includes the following topics:

Unperturbed and perturbed harmonic oscillators
Commutative and anti-commutative operators, their properties and related topics
Commutator Gymnastics
Creation and annihilation operators and their properties
Algebraic approach to simple quantum systems
Coherent states
Operator factorization
Angular momentum
Eigenfunctions
Completeness
Funactional analysis and approximation theory
Applications
Prospectives

A comprehensive manuscript will be made available to the seminar participants.

Wavelets and Wavelet Transforms
Alireza Baghi-Wadji, Institute for Industrial Electronics and Materials Science Vienna University of Technology, Austria

Introduction: Upon hearing a piece of music we are able to identify, almost instantly, its pitch and volume. In terms of prewavelet era mathematics, the pitch corresponds to the frequency domain description of the sound signal, and the volume corresponds to the time domain description. Classical analysis techniques such as the Fourier transform enable one to describe a signal in either time or frequency domain. However, the uncertainty principle states that it is not possible to obtain a complete description of the signal in one domain from local information in the other domain. Therefore, it is not possible to determine all the frequency information from samples of the signal over a very short duration. Short-time (windowed) Fourier transform was the answer to only parts of the problems. Joint time-frequency transforms and wavelets were born to revolutionize the analysis and approximation of signals and functions. Wavelet analysis allows us to represent signals localized in both domains. Applications of the wavelets include the signal analysis in seismology and medicine, image processing, signal processing, data compression, and numerical methods. Another issue of great interest to engineers is the construction of problem-specific basis functions for solving boundary value problems, and the approximation of signals. It turns out that the wavelet theory can be instrumental in these areas as well.

This short course has been designed to familiarize the attendees with the basic ideas of wavelets, wavelet transform, and related topics. It provides a sound introduction to the underlying theory.

Objective: The main objective of this short course is to explain some of the most beautiful and uniquely-powerful properties of the wavelet theory and wavelet transforms in a way, which is accessible to anyone with an engineering background.

Prerequisites: There is no particular prerequisite to this short course. All the mathematical notions, tools and concepts, necessary for a coherent explanation of the theory, will be developed in the classroom. Since the explanation methodology is algebraic, most students and practitioners will find the course appealing.

Course Material: A comprehensive manuscript will be made available to the course participants.

Content: The short course comprises eight units, each 55 minutes long. The presentation includes eight of the following items as a minimum. Thereby, the course participants will play an active role in the selection of the topics. (In the following list, the occurrence of one notion in more than one occasion indicates a succession of analyses with increasing attention to the details.)

Introduction: Waves and wavelets, Fourier transform, continuous wavelet transform, Fourier series, wavelet expansions, orthogonal basis, non-orthogonal basis, distinct properties of wavelets, multiresolution analysis, scaling functions, wavelet functions, Haar wavelet, numerous examples.
The multiresolution formulation of wavelet systems: signal spaces, scalar products, span of a basis set, closure of a space, scaling functions, nested spaces, prototypes for the scaling functions, Haar scaling functions, triangular scaling functions, wavelet functions, prototypes for the wavelets, Haar wavelets, triangular wavelets, numerous examples.
Filter banks, Mallat's algorithms: dilation equation, low pass filter coefficients, high pass filter coefficients, expansion coefficients in terms of scaling functions, expansion coefficients in terms of wavelets, the notion of analysis (from fine scale to coarse scale), time-reversed filter coefficients, down-sampling, two-band analysis bank, iterating the filter bank, multi-stage two-band analysis tree, frequency response of digital filters, the notion of synthesis (from coarse scale to fine scale), filtering, up-sampling, examples and problems.
Necessary and sufficient conditions for the existence and orthogonality of integer-translates of the scaling function: preliminary considerations, signal spaces, L2(R)-space, L1(R)-space, Fourier transform of the dilation equation, refinement matrices M0 and M1, examples, necessary conditions in the real space, necessary conditions in frequency domain, sufficient conditions in real domain, sufficient conditions in frequency domain.
Wavelet system design: three approaches for the construction of wavelets, parametrization of the scaling coefficients, examples, length-2 Daubechies coefficients, length-4 Daubechies coefficients, length-6 Daubechies coefficients, calculation of the basic scaling function and wavelet, successive approximation, the cascade algorithm, dyadic expansion of the scaling function, properties of the scaling function and wavelets.
Construction of wavelets using Fourier techniques: axiomatic definition of multiresolution analysis, Meyer's wavelet, examples.
Wavelet bases for piecewise polynomial spaces: spline wavelets, definitions, examples, B-splines, orthogonalization procedure.
Orthonormal wavelets with compact support: the algebraic construction of Daubechies wavelets, trigonometric polynomials, moments of wavelets, N-order wavelets, Riesz's lemma.
Examples: Calculation of Daubechies scaling functions and wavelets, cubic B-spline wavelet, Morlet wavelet, Mexican hat wavelet, calculation of Daubechies wavelets, Lemarie and Meyer orthogonalization approach, Haar expansion of discrete-time signals, orthonormal wavelets for spline spaces, Battle and Lemarie wavelets, Daubechies-Grossman-Meyer wavelet.
Construction of problem-specific scaling functions and wavelets.
Construction of problem-specific orthonormal bases.
Coherent states
Advanced topics.

Recent Advances in Modeling and Simulation of High-Speed Interconnects
M. S. NAKHLA, Department of Electronics, Carleton University, Ottawa, Canada

The rapid growth in microwave and VLSI circuit technology coupled with the trend towards more complex/miniature devices is placing enormous demands on computer-aided design algorithms and tools. The design requirements are becoming very stringent, demanding higher operating speeds, sharper excitations, denser layouts and low power consumption. Consequently, traditional boundaries between circuit, EM and thermal design considerations are rapidly vanishing. Also, mixed frequency/time analysis is creating difficulty for traditional simulators, due to the emerging need for inclusion of high-speed models. The high-speed interconnect effects such as ringing, delay, distortion, crosstalk, attenuation and reflections, if not predicted accurately at early design stages, can severely degrade the system performance. Managing the modelling, simulation and design optimization in such a complex environment presents highly demanding challenges.

Recently proposed model-order reduction (MOR) techniques such as Asymptotic Waveform Evaluation, Complex -Frequency Hopping and Krylov space-based methods have proven useful in the analysis of large interconnect structures containing lossless and lossy high-speed interconnects with linear or nonlinear terminations. At a CPU cost of a little more than one DC analysis, these techniques are 2-3 orders of magnitude faster than conventional methods. MOR techniques have been successfully applied to wide-variety of problems including high-speed interconnects, RF circuits, thermal analysis, EM simulation and packaging characterization.

This tutorial presents an overview of interconnect modeling/simulation strategies with emphasis on diverse algorithms and applications of model-reduction techniques. Various interconnect models will be considered including RC/RLC lumped, distributed, full-wave, measured and EMI-based. The basic principles of model-reduction techniques will be described and also their extension to frequently encountered practical situations such as simulation of subcircuits characterized by measured S-parameters and frequency-dependent components (e.g. resulting from skin and proximity effects) will be described. The underlying basic concepts will be demonstrated by several practical examples.

Multiscale Modeling of Ductile Crystalline
Solids Alberto M. Cuitińo, Department of Mechanical Engineering, Rutgers University

We present a modeling approach to bridge the atomistic with macroscopic scales in crystalline materials. We show that the meticulous application of this paradigm renders truly predictive models of the mechanical behavior of complex systems. In particular we predict the hardening of Ta single crystal and its dependency for a wide range of temperatures, strain rates. The feat of this approach is that predictions from these atomistically informed models recover most of the macroscopic characteristic features of the available experimental data, without a priori knowledge of such experimental tests. This approach provides a procedure to forecast the mechanical behavior of material in extreme conditions where experimental data is simply not available or very difficult to collect.

The present methodology combines identification and modeling of the controlling unit processes at microscopic level with the direct atomistic determination of fundamental material properties. This modeling paradigm is used to describe the mechanical behavior of Ta single crystals at high-strain rate. In formulating the model we specifically consider the following unit processes: double-kink formation and thermally activated motion of kinks (Wang, Strachan, Cagin and Goddard, 2002); the close-range interactions between primary and forest dislocations, leading to the formation of jogs; the percolation motion of dislocations through a random array of forest dislocations introducing short-range obstacles of different strengths; dislocation multiplication due to breeding by double cross-slip; and dislocation pair annihilation.

The resulting atmostically-informed model is then used to predict the macroscopic response of structural solids subjected to complex loading scenarios by resorting large scale massive parallel computations. The considerable computing effort is distributed among processors via a parallel implementation based on mesh partitioning and message passing.

Multiscale Modeling of Heterogeneous Granular Systems
Solids Alberto M. Cuitińo, Department of Mechanical Engineering, Rutgers University

Powder compaction is the operation used to form ceramic parts, pulvimetallurgy components, and pharmaceutical tablets. Compaction is also important in many applications including soil stabilization, avalanche and slide control, and storage of mining and agricultural products, as well as in a number of geological processes. Besides its vast technological interest, the densification of powders by compaction affords valuable insights into the physics of the granular state.

Cohesive particles tend to form structures with relatively large voids during die filling. Upon application of a relatively small compaction load, the large voids disappear by a densification mechanism involving non-affine particle motion or by particle rearrangement. In this talk, experimental observations, theoretical predictions and numerical simulations will show that the rearrangement process does not yield to a spatially uniform density profile. Instead, two zones with different densities separated by a sharp density gradient band, the rearrangement front, are observed. This process proceeds as a system exhibiting phase transformation. This compaction front is nucleated at the moving (top) punch and moves away as the compaction proceeds until the front reaches the stationary (bottom) punch, ending the rearrangement process. Also, numerical studies will be presented to analyze the evolution of the rearrangement process and the subsequent consolidation where particle deformation dominates. For this regime a Granular Quasi-Continuum formulation is proposed to trace particle motion within a constrained displacement field. The predictions of this multiscale modeling and simulation approach compare well with the experimental observations.

Ultrascale Simulations of Nanosystems
Aiichiro Nakano, Collaboratory for Advanced Computing and Simulations, Department of Computer Science, Department of Physics & Astronomy, Department of Materials Science & Engineering, University of Southern California

Large multiscale simulations, which seamlessly combine billion-atom molecular dynamics (MD) simulations with quantum mechanical (QM) calculation based on the density functional theory (DFT) and continuum mechanics based on the finite element (FE) method, are performed on multi-Teraflop parallel computers as well as on a Grid of distributed computing resources and visualization platforms. I will present the application of these simulations to the study of: fracture of ceramic nanocomposites, environmental effects on fracture, nanoindentation, hypervelocity impact, high energy density nanomaterials, colloidal and epitaxial quantum dots, and surface switching of self-assembled monolayers.

Work supported by NSF, DOE, ARL, AFRL, DoD DURINT, and NASA.

Discrete Dislocation Plasticity: Method and Applications
W. A. Curtin, Division of Engineering, Brown University

Plasticity in crystalline materials is due to the formation and motion of dislocation defects. Continuum crystal plasticity does not treat dislocations explicitly, which can limit its ability to predict deformation phenomena, particularly at small scales (microns and below). Dislocations are difficult to handle computationally, however, because they interact via long-range fields, have a singular core, and have a discontinuity in displacement across the slip plane. The Discrete Dislocation (DD) methodology is aimed at overcoming these difficulties so that well-defined boundary value problems can be solved in the presence of mobile dislocations.¹ Here, the DD method is presented and applied to study fracture and fatigue phenomena. Both fracture and fatigue emerge from the DD method without the explicit input of crack growth or fatigue input, but rather follow from the underlying creation and response of the dislocations to the applied loading. Under fatigue loading in particular, the model predicts a range of phenomena in qualitative agreement with experimental observations.² To allow for the application of discrete dislocation (DD) plasticity to a wider range of thermo-mechanical problems with reduced computational effort, a new superposition method is presented.³ Problems involving regions of differing elastic and/or plastic behavior are solved by superposing the solutions to coupled sub-problems: (i) a standard DD model only for those regions of the structure where dislocation phenomena are permitted, subject either zero traction or displacement at every point on the boundary, and (ii) a standard elastic/cohesive-zone model of the entire structure, subject to all desired loading and boundary conditions. Such a decomposition with the generic boundary conditions of the DD sub-problem permit the DD machinery to be easily applied as a "black-box" constitutive material description in an otherwise standard computational formulation. The method is validated against prior results for crack growth along a plastic/rigid bimaterial interface. Preliminary results for crack growth along a metal/ceramic bimaterial interface and deformation under indentation of a ceramic-coated metal substrate are presented to show the power and generality of the method.

1. E. Van der Giessen and A. Needleman, Mod. Sim. Mater. Sci. Eng. 3, 689 (1995).
2. V. S. Deshpande, A. Needleman, and E. Van der Giessen, Acta Mater. 50, 831 (2002).
3. M. P. Oday and W. A. Curtin, to appear in J. Appl. Mech. (2004)

Coupled Atomistics/Discrete Dislocation Model and Applications
W. A. Curtin, Division of Engineering, Brown University

Efficient, accurate calculations of material behavior rely on multiscale methods to reduce the number of degrees of freedom in any computation. Two general approaches can be taken: information passing from smaller scales to larger scales and direct coupling of scales within a single framework. Information passing is more common (e.g. statistical mechanics), but can require insight into those critical features at the small scales that must be exported to the larger scales. Direct coupling is rapidly evolving¹ and permits all phenomena at the smaller scales to be retained, but only in regions where such detail is expected to be necessary.

Mechanical phenomena such as crack growth (by high loading, fatigue, or chemical embrittlement), dislocation nucleation, and grain boundary deformation, all require explicit retention of nanoscale details but are also strongly influenced by, for instance, dislocations and their motion (plasticity) at the micron and larger scales. To handle these multiple scales simultaneously, the Coupled Atomistic and Discrete Dislocation (CADD) method is introduced wherein atomistic and continuum regions communicate across a coherent boundary and, in 2d models, can exchange dislocations back and forth as dictated by the mechanics of the problem.² The atomistic region can experience any deformations that occur under the applied loading while the continuum region evolves according to discrete dislocation plasticity. CADD thus permits study of problems involving large numbers of dislocations that are too large for fully atomistic simulations while preserving accurate atomistic details where necessary. Comparisons of CADD against full atomistic simulations for a 2d nanoindentation problem validate the method. Preliminary applications to nanoindentation, crack and void growth, void growth, and grain boundaries are also presented.³

This new multiscale method applies to any mobile defect having a continuum representation, such as impurities and diffusion. It can also be used in the information passing mode to study nano/microscale phenomena and provide input to larger-scale models, such as cohesive zone models for fracture. Finally, the general concepts developed here can be applied to formulate new multiscale models at other scales. Examples include coupling discrete dislocations to continuum strain-gradient plasticity, coupling kinetic Monte-Carlo models of diffusion to continuum diffusion, coupling quantum mechanics to atomistic or continuum models, and parallel coupling of discrete dislocation models for polycrystals and bimaterial interfaces,⁴ all with applications to a wide range of issues in the design of structural and electronic materials.

1. W. A. Curtin and R. M. Miller, Atomistic/Continuum Coupling in Computational Materials Science, invited review for Modeling and Simulation in Materials Science (2003).
2. L. Shilkrot, R. M. Miller, and W. A. Curtin, Physical Review Letters 89, (2002).
3. L. Shilkrot, R. M. Miller, and W. A. Miller, J. Mech. Phys. Sol., to appear (2003/04); R. M. Miller, L. Shilkrot, and W. A. Curtin, Acta Mater., to appear (2003/04).
4. M. P. Oday and W. A. Curtin, submitted to J. Applied Mechanics.

Multiscale Modeling: Underlying Methodologies
W. A. Curtin, Division of Engineering, Brown University

Multiscale modeling involves connected, through direct or indirect means, models at different length and time scales. Here, the underlying single-scale methodologies are discussed to set the stage for multiscale models. Quantum mechanics, atomistic, discrete defect mesoscale, and continuum models are all introduced, and their advantages and limitations are highlighted. Classes of applications well-suited to each type of model and scale are noted, with some examples. The two approaches to multiscale modeling, information passing and direct coupling, are then introduced.

Information Passing: Cohesive Zone Models and Applications
W. A. Curtin, Division of Engineering, Brown University

Information passing is the most-common mode of multiscale modeling wherein key phenomena from smaller-scale models are incorporated into higher-scale models through appropriate constitutive parameters. The Cohesive Zone Model (CZM) is a rapidly-growing method for handling the difficult problem of crack initiation and propagation within a continuum mechanics framework while incorporating key smaller-scale aspects of the process of material separation. Here, the CZM approach is introduced. Important scaling aspects of the CZM are discussed, along with issues associated with practical implementation. Applications to fracture in Ti-Al intermetallics and stress-corrosion cracking are presented to demonstrate the power and versatility of the method.

Review of Atomistic/Continuum Coupling: Statics
W. A. Curtin, Division of Engineering, Brown University

There has been considerable progress in the direct coupling of atomistic and continuum models in recent years. Here, the basic difficulties of connecting a local continuum model to a non-local atomistic model are presented. Then, several recent methods, including the Quasicontinuum Model, the Coupling of Length Scales Model, and the FeAt model are discussed and compared within a common framework. A simple 1-d model is used to highlight the behavior of these models near the atom/continuum interface.

Atomistic/Continuum Coupling: Dynamics
W. A. Curtin, Division of Engineering, Brown University

The coupling of a finite-temperature and/or dynamic atomistic model, such as Molecular Dynamics, to an appropriate coarse-grained continuum model is a particular challenge. Difficulties not encountered in addressing static problems include accounting for entropy in coarse-scale models and avoiding artificial wave reflection at the atom/continuum interface. Here, a recent framework for performing multiscale thermodynamic calculations (finite temperature statics) is presented, and the key features and limitations are emphasized. Recent approaches to handling zero temperature dynamics are then presented, with a 1-dimensional model used to demonstrate the formal success of the models. Extensions to practical 2d or 3d systems are discussed.