Computational Methods in Large Scale Simulation

The following sections are included:

• Contents

• Foreword

• Preface

Multiscale modeling can be accomplished by either information passing across scales, using appropriate methods at each individual scale, or by coupled modeling wherein several scales are handled simultaneously in different regions of space. This chapter reviews some of the basic methods at each scale and some approaches to atomistic/continuum coupling in solid mechanics.

The Multiple Multipole Program (MMP) is an advanced boundary method for computational electromagnetics and optics. Its closeness to analytical 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 1) a special consideration of ill-conditioned matrices, 2) the so-called connection concept, 3) an advanced eigenvalue solver with a special eigenvalue tracing procedure, and 4) the Parameter Estimation Technique (PET). 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 modeling, visualization, and animation features is presented and it is demonstrated how this software is used for handling complicated projects. Thereafter, some typical applications of computational optics with a focus on photonic bandgap computations and photonic crystal structures are presented.

Periodic structures have a variety of important applications in modern technologies and engineering due to their unique electromagnetic properties. Commonly used periodic structures include frequency selective surfaces, optical gratings, phased array antennas, photonic bandgap materials, and various metamaterials. The analysis of periodic structures has always been an important topic in computational electromagnetics. In this chapter, we describe an accurate and efficient numerical analysis, based on a higher-order finite element method (FEM), for characterizing the electromagnetic properties of periodic structures. Based on the Floquet theory, periodic boundary conditions and radiation conditions are first derived for the unit cell of a periodic structure. The FEM is then applied to solve Maxwell's equations in the unit cell. To enhance the accuracy and efficiency of the FEM, curvilinear elements are employed to discretize the unit cell and higher-order vector basis functions are used to expand the electric field. The asymptotic waveform evaluation (AWE) is implemented to perform fast frequency and angular sweeps. To demonstrate the capability of the proposed FEM, we apply it to the analysis of periodic absorbers, frequency selective structures, and phased array antennas. For the antenna analysis, a rigorous waveguide port condition is developed to accurately model the antenna feed structures. In all the cases studied, satisfactory results are obtained.

The classical N–body problem consists of analyzing the state of a system of N interacting bodies, where the force exerted on each body in the system results from its interaction with all the other bodies in the system. There are typically O(N²) operations necessary to compute the forces involved. In solving Singular Surface Integrals (Boundary Element Method, Method of Moments), we encounter a similar problem: Discretizing the underlying surface integrals, and, thus, introducing N unknowns, we generally need to compute a large number of interactions requiring O(N²) operations. Thereby, the interaction coefficients depend on the relative Euclidean distance between the interacting elements in a rather complicated form. Hierarchical tree-structured algorithms have been developed to reduce the complexity of computations. The key to the majority of standard fast algorithms is the availability of potential functions in factorized form, which are obtained by applying the so-called Addition Theorem. A question arises as how to proceed if a factorized form is not available, or cannot be easily constructed. It is the prime purpose of this work to present a general recipe for factorizing potential- and field distribution functions without utilizing the Addition Theorem.

A virtualization-aware application framework is developed, based on data-locality principles, to perform hierarchical multiscale simulations of materials on a Grid of distributed computing and visualization platforms. The framework combines: linear-scaling algorithms based on space-time multiresolution techniques; topology-preserving computational-space decomposition with wavelet-based adaptive load balancing; and immersive and interactive visualization of billion-atom datasets. Multiscale simulations are performed on a Grid to seamlessly integrate quantum mechanical calculation based on the density functional theory and atomistic simulation based on the molecular dynamics method.

Various roles of molecular dynamics simulation are discussed, at first. Then, in order to bridge the macroscopic simulation and atomic level simulation, calculation methods of the local quantities, particularly local pressure, in the molecular dynamics simulation are discussed. It is necessary to integrate force between atoms along the path connecting the atoms within the local volume. This integration corresponds to the virial term used in the molecular dynamics calculation under the periodic boundary conditions.

The rapid increase in operating speeds, density and complexity of modern integrated circuits has made interconnect analysis a requirement for all state-of-the-art circuit simulators. Interconnect effects such as ringing, signal delay, distortion and crosstalk can severely degrade signal integrity. Interconnections can be from various levels of design hierarchy. As the frequency of operation increases, the interconnect lengths become a significant fraction of the operating wavelength, and conventional lumped models become inadequate in describing the interconnect performance and transmission line models become necessary. This chapter describes some of the recent advances in transmission line simulation techniques. Application of mode-order-reduction algorithms to high-speed interconnect analysis is also presented.

This article introduces a multiscale modeling approach for simulating a key aspect affecting the reliability in narrow interconnect lines: migration, growth and coalescence of voids driven by electromigration and stress gradients. This approach is based on the Monte Carlo method with a Hamiltonian that contains both short-range and long-range interactions. The short-range interactions are resolved at atomic scale while the contributions of the long-range interaction are computed by solving a subsidiary continuum boundary value problem for the elastic and electric field. Thus, this approach combines both discrete and continuum techniques to provide a numerical tool to follow the complex evolving topologies that are derived from void migration, growth and coalescence. The predicted results are in good agreement with key experimental observations and other numerical and theoretical estimates.