Approximate Closed-form Equations Research Articles

Characterizing local noise in QAOA circuits

Recently Xue et al (2019 arXiv:1909.02196) demonstrated numerically that QAOA performance varies as a power law in the amount of noise under certain physical noise models. In this short note, we provide a deeper analysis of the origin of this behavior. In particular, we provide an approximate closed form equation for the fidelity and expected cost in terms of the noise rate, system size, and circuit depth. As an application, we show these equations accurately model the trade off between larger circuits which attain better expected cost values, at the expense of greater degradation due to noise.

Quantitative mechanical analysis of indentations on layered, soft elastic materials.

Atomic force microscopy (AFM) is becoming an increasingly popular method for studying cell mechanics, however the existing analysis tools for determining the elastic modulus from indentation experiments are unable to quantitatively account for mechanical heterogeneity commonly found in biological samples. In this work, we numerically calculated force-indentation curves onto two-layered elastic materials using an analytic model. We found that the effect of the underlying substrate can be quantitatively predicted by the mismatch in elastic moduli and the homogeneous-case contact radius relative to the layer height for all tested probe geometries. The effect is analogous to one-dimensional Hookean springs in series and was phenomenologically modeled to obtain an approximate closed-form equation for the indentation force onto a two-layered elastic material which is accurate for up to two orders of magnitude mismatch in Young's modulus when the contact radius is less than the layer height. We performed finite element analysis simulations to verify the model and AFM microindentation experiments and macroindentation experiments to demonstrate its ability to deconvolute the Young's modulus of each layer. The model can be broadly used to quantify and serve as a guideline for designing and interpreting indentation experiments into mechanically heterogeneous samples.

Overall Buckling and Ultimate Load of an H-Section Steel Member Under Combined Compression and Bending

This paper presents a theoretical method and FEM to study the overall buckling of an H-section steel member with narrow flanges. The theoretical study aims to elaborate the significant parameters involved. Based on the theoretical analysis, a parametric study is performed to identify the adequate coefficients for the design of an efficient and general evaluation technique to form the equation. As a result, the approximate closed-form equations are successfully proposed for the elastic critical moment of the H-section beam. Besides, the numerical analysis by FEM is performed to investigate the ultimate load, and the design equation for the ultimate load of an H-section column under combined compression and bending is recommended.

Analytical description of the dispersion relation for phase resonances in compound transmission gratings

An analytical approach is used to describe the dispersion of phase resonances occurring between waveguide cavity modes in compound transmission gratings. The strongly enhanced evanescent fields associated with the phase resonances are used to derive an approximate closed-form equation for the dispersion of phase resonances. The equation that is derived predicts the dependence of the frequencies and dispersion of the phase resonances on structural feature sizes, material parameters, and the momentum of the phase resonance. This analytical description is compared to results obtained by using a rigorous coupled wave algorithm. This equation allows one to design a compound transmission grating that supports phase resonances with particular properties, such as resonant frequencies and momenta. Applications of the phase resonances to high-finesse electromagnetic filters and corrugated surface antennas are discussed.

Closed-Form Equations for Contact Force and Moment in Elastic Contact of Rough Surfaces

It is reasonable to expect that, when two nominally flat rough surfaces are brought into contact by an applied resultant force, they must support, in addition to the compressive load, an induced moment. The existence of a net applied moment would imply noneven distribution of contact force so that there are more asperities in contact over one region of the nominal area. In this paper, we consider the contact between two rectangular rough surfaces that provide normal and tangential contact force as well as contact moment to counteract the net moment imposed by the applied forces. The surfaces are permitted to develop slight angular misalignment, and thereby contact moment is derived. Through this scheme, it is possible to also define elastic contribution to friction since the half-plane tangential contact force on one side of an asperity is no longer balanced by the half-plane tangential force component on the opposite side. The elastic friction force, however, is shown to be of a much smaller order than the contact normal force. Approximate closed-form equations are found for contact force and moment for the contact of rough surfaces.

Assessment of the flexural capacity of corroded steel pipes

Flexural capacity of corroded pipes can be determined analytically assuming a full plastic failure mode for the pipe. A set of generalized solutions for flexural capacity of the pipe can be developed if the shape of the corrosion is known a priori. The generalized solutions derived in this paper are able to account for the simultaneous action of internal pressure and axial force. For practical purposes, the generalized solutions thus derived are simplified into approximate closed-form equations using three idealized corrosion shapes, namely, constant-depth, elliptical, and parabolic corrosions. Numerical examples indicate that the closed-form approximate solutions provide good comparison with the generalized solutions. The closed-form approximate solutions are subsequently compared to experimental results from full-size tests of pipes with different corrosion depth and width. Parameter study conducted as part of this paper indicates that the shape of a corrosion defect has significant influence on the flexural capacity of the corroded pipes.

A Contact Model of Nominally Flat Rough Surfaces Based on a Visco-Elasto-Adhesive Interaction

Approximate closed-form equations are derived for normal contact force between nominally flat rough surfaces in dry contact. The formulation is based on the asperity-level interaction in which adhesive forces between two asperities and elastic and rate-dependent forces are included. The elastic and time rate-dependent portion of force is derived using a viscoelastic interaction of the two asperities. Statistical consideration of rough surfaces then furnishes the mathematical formulation of total normal force due to adhesion, elastic, and rate-dependent properties of the solids in contact. The probabilistic formulation of contact force leads to integral equations. From these are derived approximate closed-form expressions that relate the microscale properties of the surfaces to the macroscale behavior in the form of the total normal contact force between the surfaces. The approximate equations for visco-elasto-adhesive contact of rough surfaces illustrate the dependence of the contact force on the time rate of approach based on a combination of 1/6, 1/3, and 1 power laws.

Closed-Form Equations for Three Dimensional Elastic-Plastic Contact of Nominally Flat Rough Surfaces

Approximate closed-form equations governing the shoulder-shoulder contact of asperities are derived based on a generalization by Chang, Etsion, and Bogy. The work entails the consideration of asperity shoulder-shoulder contact in which the volume conservation is assumed in the plastic flow regime. Shoulder-shoulder asperity contact gives rise to a slanted contact force comprising tangential and normal components. Each force component comprises elastic and plastic terms, which upon statistical summation yields the force component for the elastic and plastic forces for the contact of two rough surfaces. Half-plane tangential force due to elastic-plastic contact is derived through the statistical summation of tangential force component along an arbitrary tangential direction. Two sets of equations are found. In the first set of equations the functional forms are simpler and provide approximation of contact force to within 9%. The second set is enhanced equations derived from the first set of approximate equations that achieve an accuracy of within 0.2%.

A Simple Expression of BER Performance in DPSK/OFDM Systems with Post-Detection Diversity Reception

In this letter, we propose a simple but accurate calculation method, that is, an approximate closed-form equation of average bit error rate in DPSK/OFDM systems with post-detection diversity reception over both time- and frequency-selective Rayleigh fading channels. The validity of the proposed method is verified by the fact that BER performances given by the derived equation coincide with those by Monte Carlo simulation.

Design equations for the assessment and FRP-strengthening of reinforced rectangular concrete columns under combined biaxial bending and axial loads

A simple procedure is proposed for the assessment of reinforced rectangular concrete columns under combined biaxial bending and axial loads and for the design of a correct amount of FRP-strengthening for underdesigned concrete sections. Approximate closed-form equations are developed based on the load contour method originally proposed by Bresler for reinforced concrete sections. The 3D failure surface is approximated along its contours, at a constant axial load, by means of equations given as the sum of the acting/resisting moment ratio in the directions of principal axes of the sections, raised to a power depending on the axial load, the steel reinforcement ratio, and the section shape. The method is extended to FRP-strengthened sections. Moreover, to make it possible to apply the load contour method in a more practical way, simple closed-form equations are developed for rectangular reinforced concrete sections with a two-way steel reinforcement and FRP strengthenings on each side. A comparison between the approach proposed and the fiber method (which is considered exact) shows that the simplified equations correctly represent the section interaction diagram.

On Elastic Interaction of Nominally Flat Rough Surfaces

A hybrid interactive/optimization technique is used to derive in approximate closed-form equations relating contact load to mean plane separation. Equations governing Hertz contact for the interaction of surface asperities are considered in which asperity shoulder-to-shoulder contact results in normal and tangential components of force. The normal component of asperity force is summed statistically to find total normal force between the two surfaces. The tangential force over a half-plane corresponding to a select direction is found accounting for the directionality of the tangential component of asperity forces. Two sets of approximate equations are found for each of the normal and half-plane tangential force components. The simplest forms of the approximate equations achieve accuracy to within 5% error, while other forms yield approximation error within 0.2%.

Measurement of dynamic properties of small volumes of fluid using MEMS

A model based on the response of a micro-rheometer which permits the measurement of the linear viscoelastic properties of small volumes of a fluid is described. The configuration involves a liquid being contained within a capillary bridge between two flat smooth parallel platens that are actuated sinusoidally using a compliant MEMS device. Approximate closed-form equations are derived to analyse the data taking account of both the capillary forces and those arising from viscoelastic flow. The approximate theory is compared to a full numerical simulation of the response of the MEMS rheometer and the validity is discussed.

A Kinetic Friction Model for Viscoelastic Contact of Nominally Flat Rough Surfaces

Approximate closed-form equations are derived for normal and tangential contact forces of rough surfaces in dry friction. Using an extension of the Greenwood and Tripp (1970, Proc, Inst. Mech. Eng., 185, pp. 625–633) model, in which the derivations permit asperity shoulder-to-shoulder contact and viscoelastic asperity behavior, mathematical formulae are derived for normal and tangential components of the contact force that depend not only on the proximity of the two surfaces but also the rate of approach and relative sliding. A statistical approach is forwarded in which dependence of the asperity tangential contact force on relative tangential velocity of two asperities can be cast as corrective factors in the mathematical description of tangential force. In this regard two corrective coefficients are derived: force directionality corrective coefficient and force–velocity directionality corrective coefficient. The results show that for a moderate to high load ranges the contact force can be analytically described to within 20% accuracy of that from a numerical integration of the contact equations, well below the uncertainties due to surface profile measurement.

Rotating the intensity distribution of Gaussian beams propagating through paraxial ABCD systems with parallelogram apertures

By using the method of expanding the aperture function into a finite sum of complex Gaussian functions, approximate closed-form propagation equations of Gaussian beams through a paraxial ABCD optical system with parallelogram apertures are derived and illustrated with numerical examples. It is found that the intensity distribution pattern rotates round the propagation axis during the propagation.

A Simple Method of BER Calculation in DPSK/OFDM Systems over Fading Channels

In orthogonal frequency division multiplexing (OFDM) systems with differential phase shift keying (DPSK), it is possible to apply differential modulation either in the time or frequency domain depending on the condition of fading channels, such as the Doppler frequency shift and the delay spread. This paper proposes a simple calculation method, that is, an approximate closed-form equation of the bit error rate (BER) in DPSK/OFDM systems mentioned above over both time and frequency selective Rician fading channels. The validity of the proposed method is demonstrated by the fact that the BER performances given by the derived equation coincide with those by Monte Carlo simulation.

Torsional stiffness of several variable rectangular cross-section flexure hinges formacro-scale and MEMS applications

The paper develops approximate closed-form equations for the torsional stiffness ofseveral variable rectangular cross-section flexure hinges for macro-scale and MEMSapplications. Specifically, corner-filleted, elliptic, parabolic and hyperbolic flexureconfigurations, either longitudinally symmetric or non-symmetric, are studied. Themodel gives the tools for a preliminary assessment of the static/modal responseof flexure-based devices that deform torsionally. Several numerical simulationsare conducted based on the model, which indicate that, for similar values of thegeometric parameters, the hyperbolic flexure is the stiffest, followed by the parabolic,corner-filleted and elliptic configurations. The modal response is studied for atwo-flexure torsional micro-mirror by sequentially considering four different pairs oflongitudinally symmetric flexure designs. The results of the simulation confirmthe stiffness predictions and are also in agreement with finite element analysisresults.

Approximate analytical propagation equations of Gaussian beams through hard-aperture optics

By using the method of expanding the aperture function into a finite sum of complex Gaussian functions, approximate closed-form propagation equations of Gaussian beams through a paraxial multi-apertured ABCD optical system are derived and illustrated with numerical examples. The numerical results find an agreement with those by straightforward integral of the Collins formula, but the computing time is strongly reduced.

Approximate propagation equations of flattened gaussian beams passing through a paraxial ABCD system with hard-edge aperture

The propagation of flattened Gaussian beams through a paraxial optical ABCD system with hard-edge aperture is studied. Approximate closed-form equations of FGBs for the apertured case are derived by means of the expansion of the circ function into a finite sum of complex Gaussian functions. The advantages of the method as compared with the straightforward integration of the Collins formula are discussed and illustrated with numerical examples.

Simulation of SEU transients in CMOS ICs

An efficient computer simulation algorithm set, SITA, predicts the vulnerability of data stored in and processed by complex combinational logic circuits to SEU. SITA is described in detail to allow researchers to incorporate it into their error analysis packages. Required simulation algorithms are based on approximate closed-form equations modeling individual device behavior in CMOS logic units. Device-level simulation is used to estimate the probability that ion-device interactions produce erroneous signals capable of propagating to a latch (or an output node), and logic-level simulation to predict the spread of such erroneous, latched information through the IC. Simulation results are compared to those from SPICE for several circuit and logic configurations. SITA results are comparable to this established circuit-level code, and SITA can analyze circuits with state-of-the-art device densities (which SPICE cannot). At all IC complexity levels, SITA offers several factors of 10 savings in simulation time over SPICE. >

Design Procedures for Fiber Composite Box Beams

Step-by-step procedures are described which can be used for the prelimi nary design of fiber composite box beams subjected to combined loadings. These proce dures include a collection of approximate closed-form equations so that all the required calculations can be performed using pocket calculators. Included is an illustrative example of a tapered cantilever box beam subjected to combined loads. The box beam is designed to satisfy strength, displacement, buckling, and frequency requirements.