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Current industry accepted methods for predicting structural blast damage to pre-engineered metal buildings (PEMBs) subject to vapor cloud explosions are quick and cost-effective to perform. However, they also tend to be overly conservative; while costs may be saved on analysis, the results may prompt unnecessary mitigation. This paper investigates the effectiveness of using more advanced computational techniques with increasing levels of refinement and sophistication to produce more realistic estimates of structural damage and corresponding human injury. The goal is to reduce the amount of over-conservatism in results and avoid the degree of costly mitigation measures that may be unnecessary. In this study, a typical PEMB is selected and assessed for selected loads using a variety of approaches of increasing analytical sophistication. The extent of building damage is determined by each method and the results are compared to demonstrate the benefit of each analytical method. The use of refined analytical methods is shown to estimate significantly less damage in the PEMB for the loads considered.

Vapor cloud explosions (VCEs) can cause significant damage to nearby buildings, facilities and infrastructure with potential loss of life and significant business interruption, so the accuracy of predicting blast loads on facility buildings is critical in estimating these losses. Closely spaced buildings and process equipment outside of the congested region of a VCE provide a complicated flow field for an expanding blast wave. Their presence can channel and shield the blast, resulting in significant effects on the blast load magnitude and waveform shape. Currently, the most common way to estimate applied blast pressures resulting from VCE's is to use simplified methods that account for the total energy from the stoichiometric portion of the vapor cloud, fuel reactivity, and level of congestion and confinement, such as the TNO Multi-energy, equivalent TNT, CAM, and BST methods. These simplified tools assume an unobstructed line-of-site condition, which can overestimate and/or underestimate blast loads. This paper illustrates the use of a fast-running Computational Fluid Dynamics (CFD) approach that can account for channeling and shielding effects without having to use a turbulent combustion model. This approach provides a convenient tool for designers and process safety planners to more accurately quantify the hazard from postulated VCE hazards that include site-specific channeling and shielding effects. The accuracy of the approach is demonstrated via comparisons of CFD simulations to experimentally measured waveforms. Computed pressure and impulse are also compared to the BST predictions for unobstructed and obstructed sites.

Recent events involving the impact of large transport aircraft such as the Boeing 767 and 757 into the World Trade Center Towers and the Pentagon have revealed the vulnerability of such structures to terrorist attack. Incidents involving smaller general aviation aircraft have shown the damage that this class of plane can do beyond a protected perimeter. These incidents have elicited inquiries with regard to the effects of impacts of these aircraft types into other critical facilities including aboveground and below ground storage facilities, nuclear power plants, damns and other military and civilian installations. A significant capability to evaluate these threats has been developed during the past 10 years. Small medium and large aircraft have been impacted into buried and aboveground reinforced concrete and light steel frame storage facilities. Both explicit aircraft models and Riera functions (a simplified aircraft impact loading function) have been used to generate an extensive data base. The effects of engines impacting have been studied separately as penetrators. Illustrated in this paper is validation of computational tools for impacts into structures and the initial development of a generalized evaluation tool for rapid evaluation of threats and consequence of aircraft impact into protected facilities.

In this paper, Berenger’s Perfectly Matching Layer (PML) and Bettess’ Infinite Element (IE) scheme are combined to create a new type of element for unbounded acoustic wave problems. An assessment of this new element formulation is made through its use in the calculation of the acoustic modal response of a spherical radiator in the frequency domain. The performance of the PML+IE approach is contrasted with the IE only methodology by comparing them to the exact solution of this test problem in terms of the surface inertia and resistance in the near field. The results are encouraging and the PML+IE approach shows a marked improvement in performance, particularly at lower frequencies.Copyright © 2014 by ASME

We describe a nonlinear least squares inversion algorithm for obtaining elastic and electromagnetic properties for piezoelectric materials from measured impedances. Richard Brent's PRAXIS, a general unconstrained minimization code is used for the nonlinear least squares fit. No explicit derivatives of the goal functional are required by this code. Bound constraints are imposed in order to limit the variability of the parameters to physically meaningful values. Since PRAXIS is an unconstrained optimization code, these constraints are introduced via a novel change of independent variables. The forward modeling is achieved by using a coupled finite element time domain code for the elastic and electro-magnetic parts of the problem. We also describe how a linearized sensitivity analysis can be used to suggest a priori which parameters can be calculated from impedances measured on a given sample. Numerical results are included.

The detonation of an explosive charge within a building produces complex propagating blast pressures that are strongly influenced by the building’s room layout and construction of interior walls. This paper looks at the effects of internal blast on common, non-structural steel stud or wood stud walls and unreinforced CMU walls in various multi-room configurations. Their blast response is investigated through experimental and numerical models with the goal of quantifying the blast pressure propagation into rooms adjacent to the blast. Risk assessments to power generation facilities should consider the potential for an explosive event within control buildings or other support facilities. These events could be an accidental explosion or the result of a terrorist action. A better understanding of the failure mechanisms and pressure transmission characteristics of typical power generation facility structures will lead to improved vulnerability assessments of these types of structures and the critically important control equipment located within them.Copyright © 2013 by ASME

The computer simulation of an ultrasonic transducer is typically an idealized representation of the device and assumes perfectly parallel faces and spatially invariant material properties. In the real world however, these assumptions are not necessarily valid. Fabrication processes such as dicing may result in a quantifiable taper across the height and/or length of a transducer array element. Furthermore, material properties, in particular those for piezoelectric materials, may vary appreciably across the length of an array element. Both of these "parasitic" facts-of-life will impact a device's performance to some extent or another, and in some cases may serve to degrade device performance below acceptable limits. The purpose of this paper is to study and quantify the impact that non-idealized geometries and inhomogeneous material properties have on an ultrasonic transducer's resonant response.

The purpose of this study is to analyze the damage to the structure of each of the WTC Twin Towers due to the high speed impacts of the Boeing 767 airplanes and subsequent fires such as to elucidate why the Twin Towers stood for as long as they did, and why they ultimately collapsed. The Boeing 767 airplane attacks on WTC 1 and WTC 2 caused immediate and significant structural damage to the towers: In each case, exterior columns were severed and the floor system at the point of impact was damaged. The airplanes broke up during the impact and the resulting projectiles and fragments proceeded to inflict further damage to the core. Much of the impact damage to the exterior walls of the towers was evident. However, damage to the interior was not visible and cannot be quantified on the basis of the physical evidence. Dynamic nonlinear explicit finite element FLEX simulations coupled with independently validated airplane crash models were leveraged to understand and assess the structural states of damage to the tower interiors that could not be observed; this includes the degradation or loss of the load carrying capacity of columns and floor assemblies as well as the stripping of fireproofing from structural members. The impact damage to the structure was substantial but so were the reserve capacity and redundancy of the structure. Iterative analyses of the load redistribution in the impact damaged towers clearly indicate that the that the outer tube structure was very effective in developing Vierendeel action around the severed exterior columns and that the outrigger hat truss provided a substantial redundant load path away from the damaged core columns. Although not specifically designed for this purpose, the hat trusses served to delay the eventual collapse of the towers. These analyses also indicate that the damage to the corner of the core in WTC 2 left it in a state more vulnerable to subsequent thermal loads compared to WTC 1. This eccentric damage, more than the height of the airplane impact, resulted in a shorter time to collapse for WTC 2, considering that the fire environments in both towers were not meaningfully different. Further degradation or loss of the load carrying capacity of columns stripped of fireproofing by direct debris impact and heated by fire is shown to be the cause of each collapse. The examination of smoke flow from each building indicates that there were no floor collapses subsequent to the initial impact throughout the fire [1] and our 1 Chief Technology Officer, Weidlinger Associates Inc., 375 Hudson Street, New York, NY 10014. Phone: 212.367.3000. Email: abboud@wai.com. 2 Chairman, Weidlinger Associates Inc., 375 Hudson Street, New York, NY 10014. Phone: 212.367.3000. Email: levy@wai.com. 3 Weidlinger Associates, Inc., 4410 El Camino Real, Los Altos, CA 94022. Phone: 650.949.3010 4 Weidlinger Associates, Inc, 2525 Michigan Avenue, Santa Monica, CA 90404. Phone: 310.998.9154

A conceptual design for a plutonium air transport package capable of surviving a 'worst case' airplane crash has been developed by Sandia National Laboratories (SNL) for the Japan Nuclear Cycle Development Institute (JNC). A full-scale prototype, designated as the Perforated Metal Air Transport Package (PMATP) was thermally tested in the SNL Radiant Heat Test Facility. This testing, conducted on an undamaged package, simulated a regulation one-hour aviation fuel pool fire test. Finite element thermal predictions compared well with the test results. The package performed as designed, with peak containment package temperatures less than 80 C after exposure to a one-hour test in a 1000 C environment.

Three cats and 2 dogs were evaluated for ophthalmologic complications associated with teeth extraction procedures. Orbital penetration leading to ocular and. in one case, brain trauma was secondary to iatrogenic injury from a dental elevator. Outcomes included enucleation of the affected eye in 3 cases, and death from brain abscessation in 1 case. Early treatment or, preferably, referral to a veterinary ophthalmology specialist may prevent such outcomes. Awareness of the anatomical proximity of caudal maxillary tooth roof, and the orbit, appropriate interpretation of diagnostic intraoral dental radiographs, and technical proficiency in tooth extraction techniques will minimize these complications in veterinary dental practice.

Investigates the accuracy of existing 2D pseudospectral and k-space formulations for acoustic propagation in tissue or model scattering media. They are intended to provide insight into tissue-ultrasound interaction and a "test bed" for aberration correction schemes in medical imaging. Both methods employ FFT's to evaluate spatial derivatives to high accuracy on coarse grids. The primary difference lies in the approach to time integration. Scattering in large-scale, 2D, inhomogeneous media is included. The authors compare simulations against analytical solutions to illustrate spatial and temporal discretization required for acceptable solutions.

Nonlinear propagation is recognized as an important aspect of ultrasonic medical imaging. In particular, rigorous estimates of tissue bioeffects must include it. Regulatory standards rely on measurements in water to estimate effects in lossy tissue, but nonlinearity confuses the relationship. To help clarify the connection the authors complement laboratory hydrophone data with computer simulations of acoustic pulses in water and a tofu tissue mimic. A 2.25 MHz focused disk transducer is used instead of a rectangular medical array to facilitate modeling with a 2D pseudospectral solver that includes causal attenuation, inhomogeneity, multiple reflections, nonlinearity, and shock smoothing. Pressure scans near the transducer characterize the source and drive the wave solver. Measured and calculated nonlinear acoustic fields are compared over a 6 cm range in water and behind tofu cylinders. In the absence of high drive data the authors rely on nonlinear simulations to contrast water and tofu results, in anticipation of derating studies.

We describe 2D/3D model studies of resonance and radiation characteristics of diced matching layers for ultrasound transducers. Calculations are done with PZFlex, a time-domain, finite element, electromechanical code. Continuous, thin film, quarter-wave matching layers have, of course, been used routinely at optical interfaces for most of this century. A similar approach is often vital to achieving the acoustic performance required of ultrasound imaging transducers. However, the ultrasound problem is complicated by lateral propagation in the layer and crosstalk between transducer elements. This necessitates dicing the continuous layer into discrete resonators on the piezoelectric element(s), whence, crosstalk is minimized, but sometimes at the expense of anomalous local modes and compromised radiation patterns. To better understand multi-dimensional diced matching layer dynamics, a single, solid piezoceramic element and a multi-element composite are modeled. We examine beam pressure and mode shapes and include comparisons with experimental composite data and a coupled-mode design curve.

Second harmonic imaging systems transmit relatively low frequency pulses, e.g., 2.5 MHz, and image the frequency-doubled second harmonic generated by acoustic nonlinearity. Imaging the second rather than the first harmonic eliminates significant wavefront aberration and attenuation on the forward path, narrows the beam, and suppresses sidelobes. This technique is used successfully in commercial medical imaging systems and may become dominant in the near future. However, system optimization requires a better understanding of second harmonic generation by focused ultrasound pulses in tissue. Data and simulations are presented quantifying aberration and second harmonic generation by two-dimensional ultrasound beams in realistic tissue models. A pseudo-spectral solver is used to achieve very high accuracy over long paths through lossy, nonlinear abdominal wall and liver.

In all practical transduction systems-such as biomedical imaging arrays, underwater sonar systems or piezoelectric actuators and transformers, electromechanical losses impact overall system performance. Adverse effects of these losses include heat generation, sub-optimal electrical matching, and reduced operational efficiency. Consequently, it is imperative to fully understand the implications of loss mechanisms and incorporate them properly in numerical and analytical models. In this paper, time-domain electromechanical absorption mechanisms are studied in terms of their physical mechanisms and frequency-domain counterparts. We examine the mechanical and dielectric losses of some common piezoelectric materials and discuss some of the issues that arise in attempting to measure and model them.

Thermal management is an important consideration in ultrasound transducer design. It arises in satisfying regulatory and safety requirements in diagnostic and therapeutic ultrasound, as well as in sustaining performance in high power applications such as underwater sonar. A finite element modeling approach was developed to aid in the analysis of this coupled electro-mechanical-thermal problem. The finite element model tracks the damping losses in the electromechanical portion of the problem and converts the lost energy into a thermal dose which constitutes the "input" to the thermal portion of the problem. The resultant temperature spatial and temporal distribution is then solved for. This modeling approach was used to study several 1-3 piezocomposite high power transducers for which experimental data was available. Previous experimental evaluation has demonstrated that these devices can suffer from a degradation in performance due to significant temperature rises at power levels of approximately 2 W/cm/sup 2/ for continuous operation, whereas they can operate efficiently at power levels greater than 20 W/cm/sup 2/ when the duty cycle is reduced below 10%. A detailed thermal analysis of these transducers with respect to efficiency of the thermal dissipation within them is required with a view to understanding and consequently improving the high drive performance of these devices. The goal of this preliminary study is to evaluate the modeling approach and identify key parameters to which the solution is sensitive. Parameters so identified, be they material constants or modeling approaches, will be subject to more complete characterization in follow-up studies aimed at quantitative validation of computational modeling of thermal management in ultrasonic applications.

Large-scale simulations of ultrasonic waves in heterogeneous tissue models are useful in biomedical R&D for imaging and therapeutics. The scale of bioacoustic models is hundreds of wavelengths. Typical 2D wave solvers are not practical at this scale, and 3D is out of the question, because of numerical errors and/or computer limits. To achieve much higher performance we use the periodic pseudospectral (PS) method, where spatial derivatives are calculated from FFTs over Cartesian grids. With a 4th order explicit time integrator, the PS method yields the necessary accuracy and efficiency. However, the domain must be periodic. We show how to circumvent this intrinsic limitation with Berenger's perfectly matched layer (PML) on the boundaries. High accuracy, computational efficiency, and parallelism are demonstrated and a large-scale bioacoustic model is calculated. Generalizations of the method are described, including attenuation and nonlinearity.

A 8000 element, 3 MHz 2-D transducer array with integrated transmitter and receiver electronics was proposed as the Audio-Visual Converter (AVC) in a acoustic lens focused, diver-held sonar system. The array consists of 8000, 1.0 mm diameter elements spaced 1.75 mm apart in a square grid. The objective of the AVC acoustic subsystem development program was to demonstrate experimentally that a 10/spl times/10 element sub-section, or "tile", of the AVC could be successfully manufactured and meet the design parameters contained in the initial study phase of the development program. The primary goals of this development was to demonstrate that composite tiles could be manufactured reliably and with 100% yield, that the transducer structure would be high bandwidth, high efficiency, uniform, and produce near ideal beam patterns, that suitable electrical interconnect could be provided to associated integrated companion circuits, and that the structure could be produced at a reasonable cost.

The US Navy has an immediate need for large-scale, 3D, nonlinear simulations of broadband sonar projectors. The example considered here is a volume array of high-power, electrostrictive (PMN) flextensionals. Analysis and design of these complex arrays are clearly pushing the limits of simplified models. Hence, the analytical burden should be shifted from clever but overextended designers to computers. Very large-scale models, broadband response and nonlinearity favor explicit time-domain methods over implicit time- or frequency-domain methods. We demonstrate comprehensive finite element modeling of an icosahedral array of 12 flextensionals. Behavior of the PMN driver is illustrated with a 1D electrostrictive finite element (nonlinear harmonic oscillator) and generalized to a 3D element. Full-scale, SMP simulations are shown for individual flextensionals and the icosahedral array including tow-body structure.

The authors describe experimental finite element modeling of tissue ablation by focused ultrasound. Emphasis is on nonlinear coupling of high intensity sound temperature, and tissue properties. The numerical basis for modeling nonlinearity is an incrementally linear, time-domain, finite element algorithm solving the electromechanical and bioheat equations in 2D/3D inhomogeneous elastic and acoustic media. Nonstandard modeling issues examined include harmonic generation/absorption and focal "bubble" evolution with consistent sound and thermal redistribution. The nonlinear pressure-density relation generates harmonics that increase absorption and heating, particularly in the focal zone. In the tissues modeled, harmonic heating is negligible for peak focal intensities of a few kW/cm/sup 2/. As the focal hot spot ablates tissue it may also generate "bubbles". Prefocal growth of a bubbly region is modeled using a simple boiling threshold and strong coupling between the scattered ultrasound and temperature redistribution as the region spreads. Generally, these experiments are intended to develop a more comprehensive modeling basis for quantifying tissue ablation phenomenology.

Micromachined ultrasonic transducer (MUT) arrays are topical because, among other opportunities, they may supplant piezoelectric arrays in ultrasound medical imaging. However, imaging system companies need to explore performance and design issues thoroughly before making a serious commitment to these silicon-based electrostatic devices. To this end, a fully nonlinear, 3D MUT virtual prototyping capability was recently added to PZFlex. It is described here and applied to medical-type arrays. Simple representations of MUT plate vibration, including acoustic loading and radiation, are presented. Array cross-talk through the surface structure and silicon substrate is quantified using large-scale models. Additionally, an intuitive 1D electromechanical MUT model is developed for modeling and design guidance.

Experimental and theoretical evidence of silicon substrate ringing in microfabricated ultrasonic transducers is presented. This ringing is clearly observed in immersion transducers with a 650 /spl mu/m thick substrate at 7 MHz and harmonics. An analytical model of the ringing is introduced, and simulations based on the model are shown to agree with experimental observation. Experimental results are further compared to simulations carried out in time-domain, large-scale PZFlex models and qualitative agreement is demonstrated. The insights gained from the simulations and experiments are used to design and fabricate a device whose ringing mode is eliminated with a backing layer.

Major components of storage facilities and nuclear power plants are designed using reinforced concrete walls. Accidental or intentional impact of these structures by aircraft is a concern. The potential for penetration of these facilities by the aircraft or its components and the subsequent damage to the contents and release of toxic substances is a major concern. This paper focuses on analyzing the impact of jet engines into heavily reinforced concrete walls. These engines are among the stiffest and most massive components of an aircraft and the most likely to seriously damage and penetrate the reinforced concrete. We model both the engine and the reinforced concrete deformations using failure models for reinforced concrete and metals. Unlike many projectile impact problems, the impacting engine cannot be considered to be rigid. A large amount of energy is consumed in the plastic deformation and fracture of the engine components. The reinforced concrete is modeled using hexahedral elements for the concrete and beam elements for the rebar reinforcement. An advanced three invariant viscoplastic softening cap constitutive model describes the ductile and brittle rate-dependent characteristics of concrete. The rebar is modeled using a rate dependent, strain hardening von Mises formulation with failure controlled by fracture energy dissipation. A similar constitutive model is employed for the shell elements used to represent the engine components. These failure models are included in the FLEX large deformation finite element code which uses an explicit, central difference solution procedure with subcycling to solve the equations of motion. Element erosion using different criteria for concrete and metals is used to remove severely distorted and failed elements. Procedures used to mitigate the deleterious and unrealistic effects of hourglass control and viscoplasticity in the softening and failure regimes are discussed. The results from the computations are compared with experimental data generated by impacting a TF-30 engine into a two foot thick concrete wall.Copyright © 2002 by ASME

Typical ultrasonic NDI is done with single-element, mechanically scanned transducers. However, phased arrays are now finding their niche because of versatility offered by multiple beams and variable focus. Electromechanical finite element modeling developed for medical array design is likewise useful for NDI phased arrays. Called virtual prototyping, this class of modeling is illustrated with PZFlex examples covering the practical scope of calculations, their analysis and interpretation, validation in composite arrays, and new silicon electrostatic array options.

Finite element modeling is being adopted in the design of ultrasonic transducers and imaging arrays. Impetus is accelerated product design cycles and the need to push the technology. Existing designs are being optimized and new concepts are being explored. This recent acceptance follows the convergence of improvements on many fronts: necessary computer resources are more accessible, lean, specialized algorithms replacing general-purpose approaches, and better material characterization The basics of the finite element method (REM) for the coupled piezoelectric-acoustic problem are reviewed. We contrast different FEM formulations and discuss the implications of each: time-domain versus frequency domain, implicit versus explicit algorithms, linear versus nonlinear. Beyond discussions of the theoretical underpinnings of numerical methods, the paper also examines other modeling ingredients such as discretization, material attenuation, boundary conditions, farfield extrapolation, and electric circuits. Particular emphasis is placed on material characterization, and this is discussed through an actual "modelbuild-test" validation sequence, undertaken recently. Some applications are also discussed. Keywords: Arrays, Attenuation, Finite Element Method, Imaging, Piezoelectric, Transducer, Ultrasound

The current image-theoretical basis for phase shifting masks (PSMs) relies on the scalar and Kirchhoff approximations, which neglect vector wave and edge diffraction effects around the mask. In this paper we use EMFlex finite element modeling to quantify vector diffraction effects, and show a method for modeling broadband illumination using the code's transient (optical pulse) capability and the Fourier transform in time. Simulations indicate that: the Kirchhoff approximation applied to etched quartz PSMs can lead to unacceptable errors due to a dark boundary layer on the quartz sidewall; diffraction produces relatively strong vector wave fields near feature edges but their contribution to the lithographic image is negligible; and the paraxial partial coherence approximation is generally valid for 4x or 5x projection systems. We discuss examples illustrating needs for better PSM metrology and phase measurements.

Coupling to compact waveguides can have improved efficiency with use of a grating with non uniform teeth. A combined numerical and mathematical technique for designing this coupler and Finite Element Method results is presented. This paper is a preprint of the final paper which was published in the Integrated Photonics Research Technical Digest, Optical Society of America, 1994 * Currently at Integrated Photonic Systems, Inc. P.O. Box 717, Clarksburg, NJ 08510, (609) 259-1654, e-mail: lcw@intphsys.com NON UNIFORM GRATING COUPLERS FOR COUPLING OF GAUSSIAN BEAMS TO COMPACT WAVEGUIDES Lawrence C. West, Charles Roberts, and Jason Dunkel AT&T Bell Laboratories, Room 4G518, Holmdel, NJ 07733 (908) 949-8715 and Gregory Wojcik, John Mould, Jr., Weidlinger Associates, 4410 El Camino Real, Los Altos, CA 94022 (415) 949-3010 INTRODUCTION: The beams that emit from lasers are typically have a Gaussian spatial profile. Yet the field profile that emits from a uniform grating is one of a decaying exponential[1]. This fundamental mismatch prevents high efficiency (> 90%) coupling of light into compact waveguides. This limitation has been supported by the difficulty of making high precision waveguide gratings in the near infrared on semiconductor surfaces, especially when non-uniform. However, improved lithography techniques and use of practical devices in the mid-IR overcome these limitations. We discuss a method for understanding and designing grating couplers that may emit beam profiles other than a decaying exponential. Special attention is given to the Gaussian beam profile because of its importance to the laser community and it’s minimum diffraction spread. The basic technique is to vary the tooth width along the grating coupler so as to project the desired beam image. THE DESIGN METHOD: We use a numerical Finite Element Method[2] to measure the complex amplitude reflection, transmission, and scattered power coefficients for a single mode waveguide for various tooth depths, widths, and geometries. A single mode is injected into a waveguide and scattered off a single tooth within the waveguide. The resultant complex reflection and transmission coefficients into the single mode are measured. The energy difference of these and the original beam is assumed to be the scattered power. A table of these coefficients versus tooth parameters is created. For fabrication simplicity, we typically constrain ourselves to a constant depth tooth so the grating can be fabricated with a single mask and single etch. But actual lithography results, such as a change in etch hole profile for smaller teeth, can and should be taken into account by a substitution of those teeth with their respective coefficients. Figure 1: Geometry of scattering problem solved per tooth. The grating itself will be made of an ensemble of the individual teeth. The tooth spacing will remain roughly constant (this is not a chirped grating) but the tooth size will change. The spacing between teeth can be adjusted to correct for phase changes under the teeth so as to maintain a flat phase profile in the scattered beam, but this is a small percentage of the overall spacing. These Input Reflection Transmission Scattered Power GaAs Substrate Ge Waveguide tooth 0.6 m 1.75 m n = 4.0 n = 3.27 = 10.0 m 0 w reflection and transmission coefficients are calculated by subtracting the original mode as if there were no tooth and taking the remainder as a change in transmission or reflection caused by the tooth. Shown here is a chart of the coefficients for the tooth of Figure 1 with a 0.477 mm deep tooth. Of particular interest is the scattered power versus tooth width. Since the period of the grating is 3.0 micrometers to match the phase period of the waveguide mode, the maximum tooth size is 3.0 micrometers, at which point the waveguide is a smaller unconfined material. The scattered power is almost linear in tooth width with a proportionality of α. The reflection coefficient is almost a constant. We use these assumptions to derive a simple analytic form for the scattered light. A simple first order derivation of the desired tooth width can be found by assuming the grating is efficient, so that the power remaining in the grating is that which has not been scattered yet. Since the desired scattered power is proportional to the product of remaining power in the guide and the tooth width with constant α, we have the n’th tooth has a size of w p e e dz n z z z z z zc = ∫ α 2 0 2

As integrated optical devices become more sophisticated, so does the experimentation and analysis required to design them. By augmenting conventional experiments with rigorous computer modeling we can lower costs, shorten schedules, and provide faster, more accurate predictions. Discrete modeling codes using finite differences or finite elements are the most general, albeit expensive. Nonetheless, they are competitive today by virtue of simple, robust algorithms and modern workstations that put near-supercomputer capabilities on the desktop. In support of computer modeling this paper demonstrates the practicality of timedomain finite element codes for simulating 2D and 3D devices on UNIX workstations. We describe EMFlex, a finite element wave solver for large-scale electromagnetic simulations, and apply it to highly confining dielectric waveguides in 3D routing and 2D grating couplers. EMFlex was originally developed for optical lithography and metrology studies1'2, funded in part by the NSF and SEMATECH.

Optical modeling on the computer can aid R&D efforts to enhance metrology methods, and similarly for lithography, alignment, and particulate monitoring. However, full exploitation of optical modeling is hindered by the lack of appropriate benchmarks for verifying algorithms and evaluating approximations. To help remedy this situation we describe a preliminary set of scalar, 2-D numerical reference models (NRMs). These include isolated thin and thick lines, periodic lines, and an isolated trench. Scattered fields are compared for three different solution methods, based on time-domain finite elements, boundary integrals, and a waveguide model. Correlation is good in general, although important differences are seen in both code accuracy and performance. NRM generalizations are suggested that accommodate 3-D effects, imaging, and experimental verification.

This paper describes a three-dimensional computer modeling technique for alignment system simulation, and some example calculations. The technique has been developed to address issues of alignment and overlay accuracy for future generation VLSI technology. The analytical basis is a general finite element electromagnetic wave propagation code, EMFlex, that rigorously simulates light scattering from the 3-D alignment mark. Using the Nikon Laser Step Alignment (LSA) system as a model instrument, the overlay error and signal shape are simulated. Examples of an idealized asymmetric metal mark are studied. Preliminary results suggest that the rigorous simulations are substantially different from the one-dimensional Fresnel approximations that have been used previously.

IC fabrication problems grow as nominal feature sizes shrink, due in large part to fundamental optical diffraction limits. Currently, one of the most pressing needs is robust critical dimension measurement. However, optical methods must be refined for this scale of submicron metrology, particularly in the case of thick features. This paper examines the problem of reflected light microscopy for nominal 1 micron high lines on silicon using 2-D, time-domain finite element simulations. The experimental basis is a prototype line width standard that is characterized using optical, contact, and SEM measurements. Microscope and simulated images are compared for 1 and 3 micron wide lines. Good first-order correlation is found between real and synthetic images but model uncertainties need to be reduced and microscope aberrations need to be quantified before second-order differences can be eliminated. Numerical experiments are used to relate images to resonance patterns in the feature; determine the strength of evanescent waves near the line; and contrast isolated and periodic line images as a function of pitch.

In order to simulate focused ultrasound fields in nonlinear tissue, a full wave pseudospectral finite difference model has been developed. Unlike models based on the KZK equation, this model can predict wide‐angle propagation effects as well as tissue backscatter, essential for pulse‐echo medical imaging. This simulation program is capable of including broadband pulse propagation, causal attenuation, nonlinearity (B/A), and inhomogeneities. Harmonics beyond the solver’s Nyquist limit are removed numerically and quantified. To facilitate comparison to data and to expedite calculation, the model has been adapted for circularly symmetric transducers. As input to the program, waveforms are taken from hydrophone scans close to the faces of 2.25‐ and 5‐MHz focusing transducers. Simulations are compared to data taken in water and in front of and behind a tissue mimicking cylinders of tofu, which are sliced repeatedly to give transmitted and reflected pressure data versus thickness. Tissue inhomogeneities are simulated by voids in the tofu. Acoustic properties of the mimic are measured separately. B/A is extracted numerically from observed harmonic generation versus input pressure and range.

Broadband finite‐element modeling codes have proven effective in the hands of medical ultrasound transducer designers. Similar modeling advantages have not yet been realized by image processing engineers concerned with aberrating media. Their principal difficulty is the much larger physical scale of bioacoustic models for imaging, e.g., a 20‐cm round trip at 5 MHz is 600 wavelengths. Conventional (low order) finite‐element/finite‐difference wave solvers are inadequate at this scale because of cumulative numerical errors and computer limits. The solution is more accurate algorithms, based on spectral methods or high‐order finite differences, in conjunction with parallel processing. The modeling problem is described, starting from a detailed transducer section, coupled to the heterogeneous abdominal wall and deeper tissue, to the scattering object(s) and back. Computed examples illustrate fundamental modeling and phenomenological issues. Solving the bioacoustic equations in the time domain is essential, and comprehensive models include both frequency‐dependent absorption and second‐order nonlinearity (B/A parameter). Algorithmic issues involve balanced, high‐order space and time differential operators, treatment of discontinuities, radiation boundary conditions, and parallelization. The key is maintaining consistent, high‐order accuracy throughout the analysis. An overall capability is demonstrated. [Work supported by DARPA, ONR, NSF, and NIH.]

Most time-domain, wave modeling problems in geophysics are intractable by classical analysis methods, due principally to nonseparability and to a lesser extent material nonlinearity. Therefore discrete numerical solutions are often necessary for the simulation of realistic models. Applications in 2-D and 3-D geophysical modeling are the subject of this paper, particularly as solved on a CRAY-2 supercomputer. Implementation and performance differences between earlier CRAYs and the CRAY-2 are described, including the discrepancy between scalar fetch and vector processing speeds. Explicit finite element solvers are applied to applications involving 2-D simulation of a seismic refraction experiment across the state of Maine, 3-D simulation of near-source scattering experiments, and both linear and nonlinear axisymmetric source simulation. Results show that the CRAY-2 allows cost-effective 2-D simulations of truly large-scale models, but only begins to be effective in 3-D for models of interest in geophysics. The large memory (256 megawords) is adequate but a speed increase of at least an order of magnitude is necessary for cost-effective 3-D. True multiprocessor capability (rather than ‘multicomputer’) provides an alternative to raw speed but involves another set of difficulties.