Direct Analytical Solutions Research Articles

Strategy against electromigration-induced stress by passivation thickness design

This study proposes a strategy to determine an appropriate passivation thickness to prevent electromigration (EM) deterioration. Although it is widely recognized that depositing passivation can effectively saturate the interconnect resistance against EM deterioration, the effect of passivation thickness on the lifetime of interconnections against EM has not been thoroughly investigated. In this study, quantitative models were developed to show direct analytical solutions. First, the effects of passivation thickness on the interconnect resistance against EM were examined through experiments using an Al interconnect coated with tetraethyl orthosilicate passivation. Assuming realistic conditions, the behavior of passivation fracturing as a function of passivation thickness was modeled under elastic, elastic-plastic, and elastic-plastic with plastic zone conditions. The models considered the critical tensile stress in the circumferential direction at the inner wall of a long cylindrical tube subject to internal pressure. These three models explain the saturation mechanism of passivation against EM resistance. Based on these findings, an appropriate passivation thickness for the required resistance against passivation fracture due to EM can be determined. This research provides valuable insights for designing reliable interconnects in microelectronics.

Air-water interactions regulating water temperature of lakes: Direct observations (Agamon Hula, Israel) and analytical solutions

Air-water interactions regulate lake-water temperature by balancing the rate of change of water temperature (stored heat) with the incoming and outgoing heat fluxes, which are functions of water temperature and external forcing. Yet, there is a large knowledge gap in quantifying the thermoregulation of a lake, and especially managed lakes, which is hypothesized to be related to both external environmental forcing and management decisions on lake depth and water inflow. Here we explore the thermoregulation of a restored and managed Mediterranean lake (Agamon Hula, Israel), by direct measurements of all major heat fluxes. We interpret the results with a rigorous analysis of the thermoregulation equation, obtained by formulating the energy balance equation in terms of water temperature. The equation shows how water temperature of a lake shifts toward steady state, scaled by the lake’s response time. Where the steady-state temperature was reached, it was controlled by external environmental forcing and not by lake depth. Whereas the thermal response time to re-attain steady state, following an abrupt change in various environmental conditions, is controlled by water depth. The lake-temperature response to pre-defined oscillations of the environmental forcing showed that the amplitude of water temperature fluctuations and the time delay from steady-state are controlled by the ratio between the environmental forcing’s time period (diurnal, seasonal or other cycles) and the thermal response time of the lake. The summertime measured CO2 fluxes of Agamon Hula revealed the lake acts as a CO2 source to the atmosphere, overpassing similar water bodies from different climates.

A hyperbolic model for mechanical behaviour of marginal soil-geosynthetic interface

Locally available marginal soils are of great interest to use as backfill material in geosynthetic reinforced soil structures due to their cost-effectiveness. The advanced stress–strain analysis of these structures necessitates the correct evaluation and the accurate description of marginal soil-geosynthetic interface behaviour. Therefore, the primary objectives of this study are numerical and analytical analysis of the soil-geosynthetic interface behaviour with application of a hyperbolic nonlinear elastic perfectly plastic constitutive model. In this investigation a series of large-scale direct shear tests was performed to evaluate the shear stress-displacement interface behaviour considering two types of marginal soils in contact with five types of geosynthetic materials in different soil moisture conditions. Accordingly, the hyperbolic model was used to simulate the soil-geosynthetic interface behaviour. Moreover, a three-dimensional finite element model of the direct shear test was developed using ABAQUS software and a user-defined subroutine was implemented to consider the hyperbolic interface model. The results indicated a very good agreement between the experimental data and the predicted finite element simulations of the direct shear tests and analytical solutions. Finally, a numerical simulation of a pullout test is presented in this paper with application of the hyperbolic interface model.

PyBindingCurve, Simulation, and Curve Fitting to Complex Binding Systems at Equilibrium.

Understanding multicomponent binding interactions in protein–ligand, protein–protein, and competition systems is essential for fundamental biology and drug discovery. Hand-deriving equations quickly become unfeasible when the number of components is increased, and direct analytical solutions only exist to a certain complexity. To address this problem and allow easy access to simulation, plotting, and parameter fitting to complex systems at equilibrium, we present the Python package PyBindingCurve. We apply this software to explore homodimer and heterodimer formations culminating in the discovery that under certain conditions, homodimers are easier to break with an inhibitor than heterodimers and may also be more readily depleted. This is a potentially valuable and overlooked phenomenon of great importance to drug discovery. PyBindingCurve may be expanded to operate on any equilibrium binding system and allows definition of custom systems using a simple syntax. PyBindingCurve is available under the MIT license at https://github.com/stevenshave/pybindingcurve as the Python source code accompanied by examples and as an easily installable package within the Python Package Index.

High-order fully actuated system approaches: Part II. Generalized strict-feedback systems

An advantage of a high-order fully-actuated (HOFA) system is that there exists a controller such that a constant linear closed-loop system with an arbitrarily assignable eigenstructure can be obtained. In this paper, a generalised form of the conventional first-order strict-feedback systems (SFSs) is firstly proposed, and a recursive solution is proposed to convert equivalently the generalised SFS into a HOFA model. Then the second- and high-order SFSs are defined and their equivalent HOFA models are also derived. It is further shown that, under certain common conditions, the recursive solutions for converting the generalised SFSs into HOFA models can be rearranged into direct analytical explicit solutions. Such a high-order system approach is more direct and simpler than the first-order system approach since it avoids the process of converting firstly these second- and high-order SFSs into first-order ones for control, and can finally produce a constant linear closed-loop system. Particularly, it is more effective than the well-known method of backstepping since, for the generalised complicated SFSs with more subsystems, the method of backstepping may simply be not applicable due to more serious ‘differential explosion’ problem. Two examples are worked out to demonstrate the effect of the approach.

Rapid path planning for Dubins vehicles under environmental currents

This paper presents a rapid (real time) solution to the minimum-time path planning problem for Dubins vehicles under environmental currents (wind or ocean currents). Real-time solutions are essential in time-critical situations (such as replanning under dynamically changing environments or tracking fast moving targets). Typically, Dubins problem requires to solve for six path types; however, due to the presence of currents, four of these path types require to solve the root-finding problem involving transcendental functions. Thus, the existing methods result in high computation times and their applicability for real-time applications is limited. In this regard, in order to obtain a real-time solution, this paper proposes a novel approach where only a subset of two Dubins path types (LSL and RSR) are used which have direct analytical solutions in the presence of currents. However, these two path types do not provide full reachability. We show that by extending the feasible range of circular arcs in the LSL and RSR path types from 2π to 4π: (1) full reachability of any goal pose is guaranteed, and (2) paths with lower time costs as compared to the corresponding 2π-arc paths can be produced. Theoretical properties are rigorously established, supported by several examples, and evaluated in comparison to the Dubins solutions by extensive Monte-Carlo simulations.

Approximate factorization of positive matrices by using methods of tropical optimization

The problem of rank-one factorization of positive matrices with missing (unspecified) entries is considered where a matrix is approximated by a product of column and row vectors that are subject to box constraints. The problem is reduced to the constrained approximation of the matrix, using the Chebyshev metric in logarithmic scale, by a matrix of unit rank. Furthermore, the approximation problem is formulated in terms of tropical mathematics that deals with the theory and applications of algebraic systems with idempotent addition. By using methods of tropical optimization, direct analytical solutions to the problem are derived for the case of an arbitrary positive matrix and for the case when the matrix does not contain columns (rows) with all entries missing. The results obtained allow one to find the vectors of the factor decomposition by using expressions in a parametric form which is ready for further analysis and immediate calculation. In conclusion, an example of approximate rank-one factorization of a matrix with missing entries is provided.

Torque data on non-isothermal rarefied Cylindrical Couette flow

The torque exerted on the walls of a micro cylindrical Couette gas flow is studied using Direct Simulation Monte Carlo (DSMC) method and analytical solutions. An analytical solution for temperature filed under specified wall heat flux condition is proposed using a previously introduced power-law (PL) wall-scaling approach. The results of cylindrical Couette flow under specified wall heat flux condition are compared with those of isothermal flow. The effects of rarefaction, compressibility, tangential momentum accommodation coefficient (TMAC) and gas-surface heat exchange on the torque values are investigated for slip to free molecular flow regimes. The results indicate that the torque coefficient has a maximum value at the transitional flow regime for TMAC values less than unity. Lower TMAC values and higher Knudsen numbers lead to reduction and deviation of torque magnitudes from those of continuum limit. Thermal simulations of cylindrical Couette flow show that heating and cooling have significant effects on the torque values. Velocity profiles and normalized torque values demonstrate mixed increasing/decreasing behavior under heating/cooling processes for rarefied gas flow.

Laplace‐domain, high‐order homogenization for transient dynamic response of viscoelastic composites

SummaryThis paper presents a high‐order homogenization model for wave propagation in viscoelastic composite structures. Asymptotic expansions with multiple spatial scales are employed to formulate the homogenization model. The proposed multiscale model operates in the Laplace domain allowing the representation of linear viscoelastic constitutive relationship using a proportionality law. The high‐order terms in the asymptotic expansion of response fields are included to reproduce micro‐heterogeneity‐induced wave dispersion and formation of bandgaps. The first and second‐order influence functions and the macroscopic deformation are evaluated using the finite element method with complex coefficients in the Laplace domain. The performance of the proposed model is assessed by investigating wave propagation characteristics in layered and particulate composites and verified against direct numerical simulations and analytical solutions. The analysis of dissipated energy revealed that material dispersion may contribute significantly to wave attenuation in dissipative composite materials. The wave dispersion characteristics are shown to be sensitive to microstructure morphology. Copyright © 2015 John Wiley & Sons, Ltd.

Surface mapping for visualization of wall stresses during inhalation in a human nasal cavity

Airflow analysis can assist in better understanding the physiology however the human nasal cavity is an extremely complicated geometry that is difficult to visualize in 3D space, let alone in 2D space. In this paper, an anatomically accurate 3D surface of the nasal passages derived from CT data was unwrapped and transformed into a 2D space, into a UV-domain (where u and v are the coordinates) to allow a complete view of the entire wrapped surface. This visualization technique allows surface flow parameters to be analyzed with greater precision. A UV-unwrapping tool is developed and a strategy is presented to allow deeper analysis to be performed. This includes (i) the ability to present instant comparisons of geometry and flow variables between any number of different nasal cavity models through normalization of the 2D unwrapped surface; (ii) visualization of an entire surface in one view and; (iii) a planar surface that allows direct 1D and 2D analytical solutions of diffusion of inhaled vapors and particles through the nasal walls. This work lays a foundation for future investigations that correlates adverse and therapeutic health responses to local inhalation of gases and particles.

Semi-analytical finite element method for modeling of lamb wave propagation

Lamb wave propagation in plates is multi-modal, dispersive and highly dependent on the material properties. In homogeneous isotropic plates, Lamb modes can be grouped into symmetric and anti-symmetric modes, and they are decoupled from the shear modes. Due to the material isotropy, Lamb wave propagation behavior is independent from the propagation direction. This property can be used to find analytical solutions for the field of displacements in the frequency domain. However, in composite plates having anisotropic material properties, Lamb wave propagation behavior depends on the propagation direction. For this reason, the complexity of Lamb wave propagation modes increases, and no direct analytical solutions are available. Thus, numerical methods need to be used. In this paper, the semi-analytical finite element (SAFE) method is applied as it is suitable for both isotropic homogeneous plates and anisotropic composite laminated plates. Dispersion curves for these complex materials are calculated using SAFE. The effect of obstacles on the reflection and transmission of Lamb waves is considered. SAFE is also used for 2D point force response analysis. By applying the 2D point response analysis for perfectly bonded strip actuators, mode tuning behavior is calculated for general composite plates. A method to consider plate edge reflections in the 2D force response analysis is also presented.

Alternative Solutions for Horizontal Circular Curves by Noniterative Methods

The most common type of horizontal curves used to connect intersecting straight sections of highways and other infrastructures are circular curves. Given the radius and the deflection angle of a horizontal circular curve, the other curve elements can be explicitly determined. However, there are 10 practical situations in which these parameters are unknown, and they have to be determined from two other given elements. In the present paper, these cases are classified into three groups according to their solution type. In the first group, direct analytical solutions can be easily obtained using simple algebraic operations. In the second group, full analytical solutions with physical meaning are not available, and in the third group, existing methods rely on a tedious trial procedure for most cases. The paper develops noniterative exact solutions for the two cases of the second group and noniterative near-exact solutions for the four cases of the third group. The proposed near-exact solutions for the third group, which have a maximum percentage error less than 2.5×10−4%, facilitate the design and analysis of horizontal circular curves.

Direct solutions for design of grass-lined channels

For grass-lined channels, the Manning roughness coefficient varies with the hydraulic radius and the functional relationship is highly non-linear. For this reason, a trial procedure is required in the traditional design process to solve the Manning formula to determine the bottom width and flow depth of the channel cross-section. To eliminate the trial procedure, direct graphical solutions have been developed for side slope z = 2. This paper presents direct graphical and analytical solutions for bottom width and flow depth for any value of side slope in the practical range of z = 2–8. The solutions are based on new dimensionless forms of the Manning formula. The graphical and analytical solutions for bottom width and flow depth are either exact or nearly exact. Application of the proposed solutions is demonstrated using a practical example. The proposed solutions, which make the design of grass-lined channels easier and more efficient, should be of interest to the hydraulic engineering community.

Inverse Determination of Interfacial Wear Temperatures with a Receding Boundary and the Implications for Tribotesting

Frictionally generated interfacial temperatures are of great interest during tribotesting and candidate materials evaluations, as well as from actual components in service. Unfortunately, it is very difficult, if not impossible, to directly interrogate the interacting surfaces for these temperatures, especially when the wear surface is receding. Though remote measurements of temperature or strain are usually obtainable, they can only hint at the frictional temperatures unless they are just below the surface and, thus, quickly vulnerable to wear damage. In order to overcome these inherent difficulties, a least-squares enforcement of remote temperature data was used with a direct analytical solution for a cylindrical specimen with a receding surface to determine the interfacial temperatures as a function of time. Comparisons between the direct and inverse analytical solutions, as well as resulting interfacial predictions, based on existing data showed excellent agreement and verified the ability of the method to determine temperatures based on remote data. As expected, stray transfer from the lateral surface of the pin, specimen dimensions, as well as the recession velocity of the surface clearly influenced the interfacial temperature history and will therefore have important ramifications for tribotesting and the evaluation of the results.

Comment on the uncertainties in isotope patterns of molecules

Here we outline several misconceptions and caveats in the uncertainty evaluation of isotope patterns of molecules, most notably the failure of the traditional first-order error propagation models in certain situations. We also demonstrate that many of the questions regarding the uncertainty evaluation of molecular isotope patterns have direct analytical solutions that do not require elaborate computational approaches.

Direct Analytical Solutions for Uniaxial Bending Design of Reinforced Concrete Τ Cross Sections according to the Eurocode 2 Standard

Direct Analytical Solutions for Uniaxial Bending Design of Reinforced Concrete Τ Cross Sections according to the Eurocode 2 Standard

Further Development of the Three-Dimensional Hoek–Brown Yield Criterion

The original Hoek–Brown failure criterion was first introduced in 1980. Since then, it has been further developed and used in different rock mechanics applications. The main advantage of the Hoek–Brown criterion is that its input data are determined from the uniaxial compressive strength, mineralogy and structural properties of the rock. However, its application in the petroleum industry has been limited, partly because it does not take account of the intermediate principal stress in a three-dimensional stress state. In 2005, Priest introduced comprehensive and simplified versions of a new three-dimensional Hoek–Brown yield criterion, which was developed by combining the two-dimensional Hoek–Brown and three-dimensional Drucker–Prager criteria. Until now, direct analytical solutions have not been developed to solve the governing equations and numerical solution strategies have been applied. This paper presents and validates an explicit solution for the comprehensive three-dimensional Hoek–Brown yield criterion, which can be combined with numerical stress modelling codes.

An Exact Equation for the Free Surface of a Fluid in a Porous Medium

We study the problem of the evolution of the free surface of a fluid in a saturated porous medium, bounded from below by a flat impermeable bottom, and described by the Laplace equation with moving‐boundary conditions. By making use of a convenient conformal transformation, we show that the solution to this problem is equivalent to the solution of the Laplace equation on a fixed domain, with new variable coefficients, the boundary conditions. We use a kernel of the Laplace equation which allows us to write the Dirichlet‐to‐Neumann operator, and in this way we are able to find an exact differential‐integral equation for the evolution of the free surface in one space dimension. Although not amenable to direct analytical solutions, this equation turns out to allow an easy numerical implementation. We give an explicit illustrative case at the end of the article.

Gas dynamic calibration of a nano-Newton thrust stand

The ability to measure extremely low thrust levels with unusual precision is becoming more critical as attempts are made to characterize the performance of emerging micropropulsion systems. Many new attitude control concepts for nanospacecraft involve the production of thrust below 1 μN. A simple, but uniquely successful thrust stand has been developed and used to measure thrust levels as low as 86.2 nano-Newtons (nN) with an estimated accuracy of ±11%. Thrust levels in the range of 712 (nN) to 1 μN have been measured with an estimated accuracy of ±2%. Thrust is measured from an underexpanded orifice operating in the free molecule flow regime with helium, argon, and nitrogen propellants. The thrust stand is calibrated using results from direct simulation Monte Carlo numerical models and analytical solutions for free molecule orifice flow. The accuracy of the gas dynamic calibration technique, using free molecule orifice flow, has also been investigated. It is shown that thrust stand calibration using high Knudsen number helium flow can be accurate to within a few percent in the 80 to 1 μN thrust range for thin walled orifices when the stagnation pressure is accurately measured. The thrust stand and calibration technique exhibit significant improvement for accurate, low thrust measurements compared to currently published results.

Exact solution for an optimal impermeable parachute problem

In the paper there are solved direct and inverse boundary problems and analytical solutions are obtained for optimization problems in the case of some nonlinear integral operators. It is modeled the plane potential flow of an inviscid, incompressible and nonlimited fluid jet, witch encounters a symmetrical, curvilinear obstacle—the deflector of maximal drag. There are derived integral singular equations, for direct and inverse problems and the movement in the auxiliary canonical half-plane is obtained. Next, the optimization problem is solved in an analytical manner. The design of the optimal airfoil is performed and finally, numerical computations concerning the drag coefficient and other geometrical and aerodynamical parameters are carried out. This model corresponds to the Helmholtz impermeable parachute problem.