Scalar Formulation Research Articles

Computational methods for the interior structural acoustics of small spaces

In some biomechanical systems and micro-electro-mechanical systems (MEMS), the interaction of a viscous compressible fluid confined in a space bounded in part by a flexible structure is of central importance. Two specific examples are MEMS microphones (condenser or piezoelectric) and the cochlea. In both the manmade and biological acoustical sensor, the interior space is typically smaller than an acoustic wavelength, and a successful design involves trade-offs between sensitivity, bandwidth, and noise (including thermal, mechanical, electrical, or channel generated noise); the latter two criteria depend critically on the viscous and thermal forces in the system. A direct numeric approach to modeling viscous and thermal effects is often prohibitively expensive, as boundary layers must be resolved in the mesh. In this talk, we will present approximate methods that enable the inclusion of viscothermal effects in a computational framework. In particular, a variational approach amenable to inclusion in a finite element based code will be presented. Results for this method, which retains the accuracy of a Navier-Stokes formulation with the computational cost of a standard scalar acoustic formulation, will be given along with the limitations of the method and a discussion of alternative numerical approaches.

Linear-Complexity Direct and Iterative Integral Equation Solvers Accelerated by a New Rank-Minimized ${\cal H}^{2}$-Representation for Large-Scale 3-D Interconnect Extraction

We develop a new rank-minimized H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -matrix-based representation of the dense system matrix arising from an integral-equation (IE)-based analysis of large-scale 3-D interconnects. Different from the H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -representation generated by the existing interpolation-based method, the new H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -representation minimizes the rank in nested cluster bases and all off-diagonal blocks at all tree levels based on accuracy. The construction algorithm of the new H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -representation is applicable to both real- and complex-valued dense matrices generated from scalar and/or vector-based IE formulations. It has a linear complexity, and hence, the computational overhead is small. The proposed new H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -representation can be employed to accelerate both iterative and direct solutions of the IE-based dense systems of equations. To demonstrate its effectiveness, we develop a linear-complexity preconditioned iterative solver as well as a linear-complexity direct solver for the capacitance extraction of arbitrarily shaped 3-D interconnects in multiple dielectrics. The proposed linear-complexity solvers are shown to outperform state-of-the-art H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -based linear-complexity solvers in both CPU time and memory consumption. A dense matrix resulting from the capacitance extraction of a 3-D interconnect having 3.71 million unknowns and 576 conductors is inverted in fast CPU time (1.6 h), modest memory consumption (4.4 GB), and with prescribed accuracy satisfied on a single core running at 3 GHz.

Vector analysis of depleted stimulated Brillouin scattering amplification in standard single-mode fibers with nonzero birefringence

Numerical and experimental investigations of stimulated Brillouin scattering signal amplification in standard single-mode fibers under pump depletion show that while gains are reduced, the same orthogonal input states of polarization that provide maximum and minimum gain in the undepleted case practically continue to do so in the depleted regime. Using a scaled Brillouin gain coefficient, the power distribution of these max/min polarizations along the fiber can be deduced from a scalar formulation.

Ice flow relations for stress and strain-rate components from combined shear and compression laboratory experiments

AbstractThe generalized (Glen) flow relation for ice, involving the second invariants of the stress deviator and strain-rate tensors, is only expected to hold for isotropic polycrystalline ice. Previous single-stress experiments have shown that for the steady-state flow, which develops at large strains, the tertiary strain rate is greater than the minimum (secondary creep) value by an enhancement factor which is larger for shear than compression. Previous experiments combining shear with compression normal to the shear plane have shown that enhancement of the tertiary octahedral strain rate increases monotonically from compression alone to shear alone. Additional experiments and analyses presented here were conducted to further investigate how the separate tertiary shear and compression strain-rate components are related in combined stress situations. It is found that tertiary compression rates are more strongly influenced by the addition of shear than is given by a Glen-type flow relation, whereas shear is less influenced by additional compression. A scalar function formulation of the flow relation is proposed, which fits the tertiary creep data well and is readily adapted to a generalized form that can be extended to other stress configurations and applied in ice mass modelling.

Simple and complete formulation to compute propagation constants of photonic crystal fibers and predict their total chromatic dispersion

Within the scalar framework, a simple and complete formulation for the normalized propagation constants of the infinite cladding region of a photonic crystal fiber (PCF) with triangular lattice of air-holes is presented, which is dependent only on the ratio of air-hole diameters and their separation. The accuracy of the proposed formulation is depicted by comparing our results with those obtained by Russell. Then the refractive indices of the fundamental space-filling mode (nFSM) in the cladding region of the PCF from Russell's equation and the proposed relations are evaluated and the two indices are observed to match quite excellently for different values of relative air-hole size and wavelength. An equivalence between the two approaches of Russell and Saitoh is also sought. Finally, in order to check the validity of the formulation in problems of practical interest, the proposed relations are applied to evaluate the total chromatic dispersion in a PCF, treating it as a conventional step index fiber having its core and cladding indices as those of silica and nFSM, respectively. On comparison with the available results of Saitoh, the results match nicely.

Optimal Power Allocation in Multi-Relay MIMO Cooperative Networks: Theory and Algorithms

Cooperative networking is known to have significant potential in increasing network capacity and transmission reliability. Although there have been extensive studies on applying cooperative networking in multi-hop ad hoc networks, most works are limited to the basic three-node relay scheme and single-antenna systems. These two limitations are interconnected and both are due to a limited theoretical understanding of the optimal power allocation structure in MIMO cooperative networks (MIMO-CN). In this paper, we study the structural properties of the optimal power allocation in MIMO-CN with per-node power constraints. More specifically, we show that the optimal power allocations at the source and each relay follow a matching structure in MIMO-CN. This result generalizes the power allocation result under the basic three-node setting to the multi-relay setting, for which the optimal power allocation structure has been heretofore unknown. We further quantify the performance gain due to cooperative relay and establish a connection between cooperative relay and pure relay. Finally, based on these structural insights, we reduce the MIMO-CN rate maximization problem to an equivalent scalar formulation. We then propose a global optimization method to solve this simplified and equivalent problem.

An Approach to Determine the Circulation of Magnetic Field in FEM Computation Code With Vector Potential Formulation

To detect an electrical fault in a generator magnetic core with the ELCID method, a circulation of the magnetic field between two teeth is determined. To model this problem with the finite element method, two potential formulations can be used. With the scalar formulation, such circulation can be easily determined; this is not the case with the vector potential formulation. In this paper, we propose to use a supplementary relation to compute a magnetomotive force in the case of vector potential formulation.

Effect of remote field electromagnetic boundary conditions on microwave-induced plasma torches

A flexible versatile electromagnetic model constructed with the PLASIMO platform is employed to explore electromagnetic features of microwave-induced plasma torches. The bases, formed by a full-vector formulation of the Maxwell equations, provide the possibility to formulate the boundary conditions in a natural way. Together with the use of a direct matrix solver this gives a convergence speed-up of more than a factor of 100 when compared with a scalar formulation on an azimuthal magnetic field that uses an iterative solver. As a result, this electromagnetic model is ready to act in future studies as part of the self-consistent description of plasma–electromagnetic coupling. With the electromagnetic model three types of configuration were studied: the closed, semi-open and open configurations, all three based on the same simplified model plasmas. It is found that the closed configuration, acting as a cavity for which (de)tuning is extremely sensitive for the plasma conditions, is less suitable for applications in which changes in plasma compositions can be expected. The semi-open configuration can be seen as a model for the practice often used in laboratories to place microwave-induced plasma torches in a grid that aims at protecting the environment against microwave electromagnetic radiation. Calculations show that this is good practice provided the radius of this cylindrical grid is in the order of 90 mm. For the most often used configuration, the open version, we found that the power balance as expressed by the coefficients of absorption, transmission and reflection depends on the electron density of the plasma. The reason is that the plasma acts as an antenna, which converts the electromagnetic waves from the coaxial structure to that of the expansion region, and that this antenna function depends on the electron density. The influence of various other antenna elements is investigated as well.

Above Threshold Estimation of Alpha (Henry) Parameter in Stripe Lasers Using Near- and Far-Field Intensity Measurements

Henry parameter, α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> , is a key wavelength and carrier density dependent parameter which is central in determining the dynamic behavior of semiconductor lasers. While it may be extracted from below threshold measurements, its characteristics are of great interest above threshold where the lasers operate. We show that α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> along with the lateral gain profile may be determined above threshold in stripe laser structures through recovery of the lateral near-field phase profile. The lateral phase profile of the wave-front at the facet is calculated using measured near- and far-field intensities of the laser based on a model under scalar Helmholtz equation formulation. It is shown that α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> attains its maximum value under or in the proximity of the center of the pumped stripe. We apply the method to AlGaAs (λ = 776 nm) lasers with stripe widths ranging from 6-50 μm and with two levels of lateral current spreading achieved by having two different p-side cladding layer thicknesses. This permits comprehensive interpretation and analysis of the measured near- and far-field characteristics.

Analysis of diffraction of a plane wave on a grating consisting of impedance bodies of revolution

The problem of wave scattering by a grating consisting of coaxial impedance bodies of revolution is solved. The efficient numerical algorithm based on the modified null field method is offered. The method is applied both to scalar and vector formulations of the problem. The numerical results are obtained for various geometries of the grating elements.

Modelling of the mechanical behaviour of amorphous glassy polymer based on the Quasi Point Defect theory—Part I: Uniaxial validation on polycarbonate

This paper is devoted to the modelling of the viscoelasto-viscoplastic behaviour of amorphous polymers based on the Quasi Point Defects (QPD) theory. A summarized and unified description of the main assumptions is presented in its scalar formulation to compute the complex viscoelasto-viscoplastic behaviour of an amorphous medium in the glassy state from linear to non-linear ranges. Based on an original scenario for the microstructural reorganization caused by an external thermomechanical stimulus and related to the two main relaxation processes, this theory allows a coherent description of the material response under large conditions of temperature and strain rate by using a single set of material parameters. A convincing validation is performed on a ductile bisphenol-A polycarbonate (PC-BPA) by comparison between computed results and experimental measurements.

Coupled Magneto-Thermal FEM Model of Direct Heating of Ferromagnetic Bended Tubes

Scalar and vector magnetic formulations have been applied to solve the current distribution in a 50 Hz direct resistance heating system of ferromagnetic tubes. The scalar formulation driven via an external circuit has been also applied to solve the Time-Harmonic EM part of the problem coupled with the thermal transient: the computed warm up curves have been compared with experimental data.

Solution of the problem of diffraction by a grating of bodies of revolution with the use of the modified null field method

The 3D problem of scattering by a grating consisting of coaxial impedance bodies of revolution is solved with the help of the modified null field method. A system of integral equations is derived, and numerical results are obtained for the scalar and vector formulations of the problem.

Modeling of aluminum particle combustion with emphasis on the oxide effects and variable transport properties

A simplified analytical modeling of single aluminum particle combustion was conducted. Ignition and quasi-steady combustion (QSC) were separately formulated and integrated. Both the heat transfer from the hot ambient gas and the enthalpy of heterogeneous surface reaction (HSR) served to cause the particle ignition. Conservation equations were solved for QSC parameters in conjunction with conserved scalar formulation and Shvab-Zeldovich function. Limit temperature postulate was formulated by a sink term pertinent to the dissociation of the aluminum oxide near the flame zone. Effective latent heat of vaporization was modified for the thermal radiation. Ignition and QSC of the aluminum particle were predicted and discussed with emphasis on the effect of the aluminum oxide and variable properties. The model was validated with the experiments regarding ignition delay time, burning rate, residue particle size, flame temperature, QSC duration, and stand-off distance of the envelop flame. Agreement was satisfactory and the prediction errors were limited within 10%.

Qualitative properties and approximation of solutions of Bingham flows: On the stabilization for large time and the geometry of the support

We study the transient flow of an isothermal and incompressible Bingham fluID. Similar models arise in completely different contexts as, for instance, in material science, image processing and differential geometry. For the two-dimensional flow in a bounded domain we show the extinction in a finite time even under suitable nonzero external forces. We also consIDer the special case of a threedimensional domain given as an infinitely long cylinder of bounded cross section. We give sufficient conditions leading to a scalar formulation on the cross section. We prove the stabilization of solutions, when t goes to infinity, to the solution $$ u_\infty $$ of the associated stationary problem, once we assume a suitable convergence on the right hand forcing term. We give some sufficient conditions for the extinction in a finite time of solutions of the scalar problem. We show that, at least under radially symmetric conditions, when the stationary state is not trivial, $$ u_\infty \ne 0 $$ , there are cases in which the stabilization to the stationary solution needs an infinite time to take place. We end the paper with some numerical experiences on the scalar formulation. In particular, some of those experiences exhibit an instantaneous change of topology of the support of the solution: when the support of the initial datum is formed by two disjoint balls, but closed enough, then, instantaneously, for any t > 0, the support of the solution u( ·, t) becomes a connected set. Some other numerical experiences are devoted to the study of the “profile” of the solution and its extinction time.

Qualitative properties and approximation of solutions of Bingham flows: On the stabilization for large time and the geometry of the support

We study the transient flow of an isothermal and incompressible Bingham fluID. Similar models arise in completely different contexts as, for instance, in material science, image processing and differential geometry. For the two-dimensional flow in a bounded domain we show the extinction in a finite time even under suitable nonzero external forces. We also consIDer the special case of a threedimensional domain given as an infinitely long cylinder of bounded cross section. We give sufficient conditions leading to a scalar formulation on the cross section. We prove the stabilization of solutions, when t goes to infinity, to the solution $$ u_\infty $$ of the associated stationary problem, once we assume a suitable convergence on the right hand forcing term. We give some sufficient conditions for the extinction in a finite time of solutions of the scalar problem. We show that, at least under radially symmetric conditions, when the stationary state is not trivial, $$ u_\infty \ne 0 $$ , there are cases in which the stabilization to the stationary solution needs an infinite time to take place. We end the paper with some numerical experiences on the scalar formulation. In particular, some of those experiences exhibit an instantaneous change of topology of the support of the solution: when the support of the initial datum is formed by two disjoint balls, but closed enough, then, instantaneously, for any t > 0, the support of the solution u( ·, t) becomes a connected set. Some other numerical experiences are devoted to the study of the “profile” of the solution and its extinction time.

Constitutive modeling of intrinsic silicon monocrystals in easy glide

Constitutive modeling of silicon materials is currently restricted to the very early stage of deformation. Uniaxial tensile testing of monocrystals oriented for single glide is traditionally simulated by a scalar model relying on the so-called machine equation. The present work uses a crystal plasticity framework to identify the role of secondary slip systems in the yield region. A three-dimensional finite element model of a tensile apparatus is validated by comparison of its outputs to the results yielded by a scalar formulation. Best fits of the constitutive model of Alexander and Haasen to experimental data reveal strong variations in its parameters with temperature. An improved constitutive model for intrinsic silicon monocrystals deformed in single slip is described. Its parameters are identified as analytical functions of temperature. We show its excellent agreement with the observed steady state of deformation in stage I.

Frequency-domain modeling of TM wave propagation in optical nanostructures with a third-order nonlinear response

An enhanced method is developed for analysis of third-order nonlinearities in optical nanostructures with a scalar magnetic field frequency-domain formulation; it is shown to produce fast and accurate results for 2D problems without a superfluous vector electric field formalism. While a standard TM representation using cubic nonlinear susceptibility results in an intractable implicit equation, our technique alleviates this problem. In addition to a universal approach, simpler, more efficient solutions are proposed for media having solely either a real (lossless Kerr-type medium) or an imaginary (nonlinear absorbing medium) nonlinearity. Combining these solutions with a finite-element method, we show simulation examples validated with alternative approaches.

Elimination of the water‐layer response from multi‐component source and receiver marine electromagnetic data

ABSTRACTThis paper presents the theory to eliminate from the recorded multi‐component source, multi‐component receiver marine electromagnetic measurements the effect of the physical source radiation pattern and the scattering response of the water‐layer. The multi‐component sources are assumed to be orthogonally aligned above the receivers at the seabottom. Other than the position of the sources, no source characteristics are required. The integral equation method, which for short is denoted by Lorentz water‐layer elimination, follows from Lorentz' reciprocity theorem. It requires information only of the electromagnetic parameters at the receiver level to decompose the electromagnetic measurements into upgoing and downgoing constituents. Lorentz water‐layer elimination replaces the water layer with a homogeneous half‐space with properties equal to those of the sea‐bed. The source is redatumed to the receiver depth.When the subsurface is arbitrary anisotropic but horizontally layered, the Lorentz water‐layer elimination scheme greatly simplifies and can be implemented as deterministic multi‐component source, multi‐component receiver multidimensional deconvolution of common source gathers. The Lorentz deconvolved data can be further decomposed into scattering responses that would be recorded from idealized transverse electric and transverse magnetic mode sources and receivers. This combined electromagnetic field decomposition on the source and receiver side gives data equivalent to data from a hypothetical survey with the water‐layer absent, with idealized single component transverse electric and transverse magnetic mode sources and idealized single component transverse electric and transverse magnetic mode receivers.When the subsurface is isotropic or transverse isotropic and horizontally layered, the Lorentz deconvolution decouples into pure transverse electric and transverse magnetic mode data processing problems, where a scalar field formulation of the multidimensional Lorentz deconvolution is sufficient. In this case single‐component source data are sufficient to eliminate the water‐layer effect.We demonstrate the Lorentz deconvolution by using numerically modeled data over a simple isotropic layered model illustrating controlled‐source electromagnetic hydrocarbon exploration. In shallow water there is a decrease in controlled‐source electromagnetic sensitivity to thin resistors at depth. The Lorentz deconvolution scheme is designed to overcome this effect by eliminating the water‐layer scattering, including the field's interaction with air.

Vectorization and constraint grouping to enhance optimization of marine structures

Vectorization converts classical scalar optimization formulation, which strictly separates the objective from constraints, into a vector-based optimization, transforming constraints into objectives. Effectively, the search is not conducted anymore for a single optimum, but for a set of Pareto optima between the original objective and transformed constraints. Constraint grouping enhances handling of multiple constraints for vectorized problems, by combining several constraints within a single-objective function, thus reducing the computational time and computational difficulties of high-dimensional spaces created by vectorization. This paper formulates and investigates these two concepts with respect to design of marine structures. It analyses their effects on the possibility to improve the flexibility of optimization in a practical environment, by implementing them within a simple genetic algorithm. Obtained results of vectorization applied to realistic weight optimization problem are encouraging when compared with the results of the classical scalar form optimization, showing a significant improvement in magnitude as well as in reduced computation time needed to reach the optimum.