Fast Multipole Methods for Three-Dimensional N-body Problems

We are developing computational tools for the simulations of three-dimensional flows past bodies undergoing arbitrary motions. High resolution viscous vortex methods have been developed that allow for extended simulations of two-dimensional configurations such as vortex generators. Our objective is to extend this methodology to three dimensions and develop a robust computational scheme for the simulation of such flows. A fundamental issue in the use of vortex methods is the ability of employing efficiently large numbers of computational elements to resolve the large range of scales that exist in complex flows. The traditional cost of the method scales as Omicron (N(sup 2)) as the N computational elements/particles induce velocities at each other, making the method unacceptable for simulations involving more than a few tens of thousands of particles. In the last decade fast methods have been developed that have operation counts of Omicron (N log N) or Omicron (N) (referred to as BH and GR respectively) depending on the details of the algorithm. These methods are based on the observation that the effect of a cluster of particles at a certain distance may be approximated by a finite series expansion. In order to exploit this observation we need to decompose the element population spatially into clusters of particles and build a hierarchy of clusters (a tree data structure) - smaller neighboring clusters combine to form a cluster of the next size up in the hierarchy and so on. This hierarchy of clusters allows one to determine efficiently when the approximation is valid. This algorithm is an N-body solver that appears in many fields of engineering and science. Some examples of its diverse use are in astrophysics, molecular dynamics, micro-magnetics, boundary element simulations of electromagnetic problems, and computer animation. More recently these N-body solvers have been implemented and applied in simulations involving vortex methods. Koumoutsakos and Leonard (1995) implemented the GR scheme in two dimensions for vector computer architectures allowing for simulations of bluff body flows using millions of particles. Winckelmans presented three-dimensional, viscous simulations of interacting vortex rings, using vortons and an implementation of a BH scheme for parallel computer architectures. Bhatt presented a vortex filament method to perform inviscid vortex ring interactions, with an alternative implementation of a BH scheme for a Connection Machine parallel computer architecture

A measurement based comparison of full-wave and quasi-static methods for baseband modeling of plated through hole via structures

As signaling rates increase, the usable bandwidth of modern telecommunication and storage systems is bounded by parasitic elements within the signal path. To improve data throughput, system designers use cascaded combinations of equivalent circuit models obtained through component level simulation and measurement to evaluate the communications channel. The analytical methods used to create these models have frequency dependent limitations and restrict applications. To this end, a measurement based comparison of quasi-static and full-wave simulation methodologies was performed on plated through hole via structures in a printed circuit board over a frequency range of 0.1GHz to 20GHz. Test fixtures and calibration standards, which isolate the behavior of the component under test from the measurement set-up, are required to establish performance metrics used for the evaluation. In this study, a printed circuit board was designed and fabricated to contrast two different calibration methodologies as applied to plated through hole via structures. The results of this study will describe the usable bandwidth associated with equivalent circuit models derived from quasi-static electromagnetic simulations while demonstrating strong correlation between the full-wave electromagnetic models and measured scattering parameters of a plated through hole via structure

Modeling techniques and verification methodologies for substrate coupling effects in mixed-signal system-on-chip designs

The substrate noise coupling problems in today's complex mixed-signal system-on-chip (MS-SOC) brings a new set of challenges for designers. In this paper, we propose a global methodology that includes an early verification in the design flow as well as a postlayout iterative optimization to deal with substrate noise, and helps designers to achieve a first silicon-success of their chips. An improved semi-analytical modeling technique exploiting the basic behaviors of this noise is developed. This method significantly accelerates the substrate modeling, avoids the dense matrix storage, and, hence, enables the implementation of an iterative noise-immunity optimization loop working at full-chip level. The integration of the methodology in a typical mixed-signal design flow is illustrated and its successful application to achieve a single-chip integration of a transceiver is demonstrated

Modelling and analysis of crosstalk in scaled CMOS interconnects

The development of a general coupled RLC interconnect model for simulating scaled bus structures m VLSI is presented. Several different methods for extracting submicron resistance, inductance and capacitance parameters are documented. Realistic scaling dimensions for deep submicron design rules are derived and used within the model. Deep submicron HSPICE device models are derived through the use of constant-voltage scaling theory on existing 0.75µm and 1.0µm models to create accurate interconnect bus drivers. This complete model is then used to analyse crosstalk noise and delay effects on multiple scaling levels to determine the dependence of crosstalk on scaling level. Using this data, layout techniques and processing methods are suggested to reduce crosstalk in system

Characterisation of crosstalk defects in submicron CMOS VLSI interconnects

The main problem addressed in this research work is a crosstalk defect, which is defined as an unexpected signal change due to the coupling between signals or power lines. Here its characteristic under 3 proposed models is investigated to find whether such a noise could lead to real logic faults in IC systems. As a result, mathematical analysis for various bus systems was established, with 3 main factors found to determine the amount of crosstalk: i) how the input buffers are sized; ii) the physical arrangements of the tracks; and iii) the number of switching tracks involved. Minimum sizes of the width and separation lead to the highest crosstalk while increasing in the length does not contribute much variation. Higher level of crosstalk is also found in higher metal layers due mainly to the reduced capacitance to the substrate. The crosstalk is at its maximum when the track concerned is the middle track of a bus connected to a weak buffer while the other signal lines are switching. From this information, the worse-case analysis for various bus configurations is proposed for 0.7, 0.5 and 0.35 µ CMOS technologies. For most of conventional logic circuits, a crosstalk as large as about a half of the supply voltage is required if a fault is to occur. For the buffer circuits the level of crosstalk required depends very much on the transition voltage, which is in turn controlled by the sizing of its n and p MOS transistors forming the buffer. It is concluded that in general case if crosstalk can be kept to be no larger that 30% of the supply voltage the circuit can be said to be very reliable and virtually free from crosstalk fault. Finally test structures are suggested so that real measurements can be made to verify the simulation result

A fast algorithm for modelling multiple bubbles dynamics

Master'sMASTER OF ENGINEERIN

Integral-equation-based fast algorithms and graph-theoretic methods for large-scale simulations

In this dissertation, we extend Greengard and Rokhlin's seminal work on fast multipole method (FMM) in three aspects. First, we have implemented and released open-source new-version of FMM solvers for the Laplace, Yukawa, and low-frequency Helmholtz equations to further broaden and facilitate the applications of FMM in different scientific fields. Secondly, we propose a graph-theoretic parallelization scheme to map the FMM onto modern parallel computer architectures. We have particularly established the critical path analysis, exponential node growth condition for concurrency-breadth, and a spatio-temporal graph partition strategy. Thirdly, we introduce a new kernel-independent FMM based on Fourier series expansions and discuss how information can be collected, compressed, and transmitted through the tree structure for a wide class of kernels

Stability Problems for Stochastic Models: Theory and Applications II

Most papers published in this Special Issue of Mathematics are written by the participants of the XXXVI International Seminar on Stability Problems for Stochastic Models, 21­25 June, 2021, Petrozavodsk, Russia. The scope of the seminar embraces the following topics: Limit theorems and stability problems; Asymptotic theory of stochastic processes; Stable distributions and processes; Asymptotic statistics; Discrete probability models; Characterization of probability distributions; Insurance and financial mathematics; Applied statistics; Queueing theory; and other fields. This Special Issue contains 12 papers by specialists who represent 6 countries: Belarus, France, Hungary, India, Italy, and Russia