Piecewise Analytical Solutions Research Articles

Piece-wise analytic solution with double shocks for 3-dimensional isentropic irrotational flow

We study the generalized Riemann problem for the 3-dimensional irrotational isentropic flow with analytic data. The existence of local analytic solutions containing two shock fronts is established under minimal compatibility conditions.

ANALYTICAL SOLUTION OF THE DIFFERENTIAL EQUATION OF FORCED OSCILLATIONS OF A VIBRATION PROTECTION SYSTEM WITH A PIECEWISE LINEAR STATIC FORCE CHARACTERISTIC UNDER HARMONIC KINEMATIC EXCITATION OF BASE DISPLACEMENTS

The need to protect operators of mobile machines from vibrations determines the relevance of the development of mathematical models for studying the dynamic processes of forced vibrations of the vibration-protected mass of the seat with the operator. Analytical solutions have the advantage of being the most accurate. For the design scheme of a vibration protection system with one degree of freedom in the form of a linear oscillator with kinematic excitation, an analytical solution to the problem of forced harmonic oscillations was obtained for the given values of the parameters: the initial absolute displacements and speed of the vibration-protected mass, the initial phase of the given harmonic oscillations of the seat base. At the same time, the expression of the static power characteristic of the vibration protection system was a straight line with a vertical displacement, which made it possible to break the transient process into separate time intervals, within which the local coordinate, showing the deformation of the vibration protection mechanism, is within a separate rectilinear segment of the piecewise linear static characteristic. The results of the comparative analysis showed a significant discrepancy between the piecewise analytical solution and the solution obtained by the numerical integration method.

Switched electromechanical dynamics for transient phase control of brushed DC servomotor.

Robotic tasks often exceed the scope of steady-state or periodic behavior, which necessitates generally-applicable models of actuators intended to generate transient or aperiodic motion. However, existing electromechanical models of servomotors typically omit consideration of the switching power converter circuits required for directional, speed, or torque control. In this study, a multi-domain framework is established for switched electromechanical dynamics in servomotor systems for their analysis and control in general aperiodic tasks including transient phases. The switched electromechanical dynamics is derived from the individual models of the internal DC motor, gear train, and H-bridge circuit. The coupled models comprehensively integrate all possible distinct switching configurations of on-state, off-state, and dead time. A combination of cycle averaging with piecewise analytical solutions of the non-smooth dynamics is introduced to handle different temporal scales from high-frequency electrical to low-frequency mechanical variables. System parameters were estimated from experimental data using a dual-servomotor test platform. The model was validated for predictive accuracy against measured data in two distinct tasks-dynamic braking of a pendulum system and sinusoidal trajectory following. The model was also used to formulate the servomotor power consumption, which was implemented for optimal control demonstration and energy analysis. In particular, the servomotor power consumption model provided true optimality (minimization) when compared with the squared rotor torque and the positive rotor mechanical power that are commonly used as proxy models. While the focus of this work is on permanent-magnet, armature-controlled brushed DC servomotors, the approach is applicable to general electromechanical systems with switching-based control.

Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.

An explicit reduced-order model of Cu-Zeolite SCR catalyst for embedding in ECM

A piecewise analytical solution is developed as part of an explicit reduced-order model for a state-of-the-art Cu-SSZ-13 SCR catalyst in the degreened state (550 °C-4 h). Simplifying the species, coverage and energy balance equations by treating the monolithic catalyst as an array of CSTRs in series, along with additional approximations based on intrinsic catalyst behavior allow for the decoupling of the conservation equations. This yields analytical solutions for all relevant catalyst states and outputs, which can be integrated using a simple fixed step explicit Euler scheme. The reduced-order model utilizes the same reaction kinetics as the high-fidelity 1D plug flow reactor model, developed using NH3-temperature programmed desorption (TPD) and 4-step [40] reactor data on the degreened catalyst. SCR kinetics are modeled here using a global single-site approach, accounting for ammonia (NH3) and ammonium nitrate (AN) dynamics. Washcoat diffusion is considered using the overall mass transfer approach [39], and in the case of SCR catalysts, we demonstrate the applicability of the asymptotic internal Sherwood number for realistic reaction rates and washcoat diffusivities. Model results show the ability of the piecewise analytical solution to produce nearly identical results as the high-fidelity model, which in turn is validated with experimental reactor data. The reduced-order model is also translated to a full-size catalyst under transient engine drive cycle conditions. SCR outlet temperature, NH3 and AN coverage, along with outlet NOx and N2O slip predictions are in line with the high-fidelity model and experimental measurements. The simplified model presented here is generally applicable to all single-layer monolithic catalysts utilizing linear reaction rate expressions for solved species. This model allows for the tracking of catalyst states (such as NH3 coverage) and outputs (such as NOx) onboard the engine control module (ECM), enabling the development of model-based estimators and control strategies necessary to meet the upcoming emission and durability requirements.

Quantification of drainable water storage volumes on landmasses and in river networks based on GRACE and river runoff using a cascaded storage approach – first application on the Amazon

Abstract. The knowledge of water storage volumes in catchments and in river networks leading to river discharge is essential for the description of river ecology, the prediction of floods and specifically for a sustainable management of water resources in the context of climate change. Measurements of mass variations by the GRACE gravity satellite or by ground-based observations of river or groundwater level variations do not permit the determination of the respective storage volumes, which could be considerably bigger than the mass variations themselves. For fully humid tropical conditions like the Amazon the relationship between GRACE and river discharge is linear with a phase shift. This permits the hydraulic time constant to be determined and thus the total drainable storage directly from observed runoff can be quantified, if the phase shift can be interpreted as the river time lag. As a time lag can be described by a storage cascade, a lumped conceptual model with cascaded storages for the catchment and river network is set up here with individual hydraulic time constants and mathematically solved by piecewise analytical solutions. Tests of the scheme with synthetic recharge time series show that a parameter optimization either versus mass anomalies or runoff reproduces the time constants for both the catchment and the river network τC and τR in a unique way, and this then permits an individual quantification of the respective storage volumes. The application to the full Amazon basin leads to a very good fitting performance for total mass, river runoff and their phasing (Nash–Sutcliffe for signals 0.96, for monthly residuals 0.72). The calculated river network mass highly correlates (0.96 for signals, 0.76 for monthly residuals) with the observed flood area from GIEMS and corresponds to observed flood volumes. The fitting performance versus GRACE permits river runoff and drainable storage volumes to be determined from recharge and GRACE exclusively, i.e. even for ungauged catchments. An adjustment of the hydraulic time constants (τC, τR) on a training period facilitates a simple determination of drainable storage volumes for other times directly from measured river discharge and/or GRACE and thus a closure of data gaps without the necessity of further model runs.

New Analytical and Numerical Solutions of the Particle Breakup Process

Objective: In this work, we obtained the analytical and approximate solutions of the population balance equations (PBEs) involving the breakup process in batch and continuous flow by applying the Adomian decomposition method and piecewise continuous basis functions, respectively. Methods: The key to the advanced numerical method is to represent the number distribution function of the dispersed phase through the orthogonal Chebyshev basis polynomials. It is a straightforward and effective method that has the advantage of simultaneously giving the distribution and the different required moments. Therefore, it does not require the construction of the distribution from moments computations obtained by the transformation of the initial problem and the lost information. Results: The performance of this numerical approach is evaluated by solving breakup equation and comparison against analytical solutions obtained from the Adomian decomposition method, which generally allows the analysis of this approach. Conclusion: The numerical results obtained by the present numerical method were compared with the new analytical solutions of the PBE. It was found that both piecewise continuous basis functions and analytical solutions have comparable results.

A conservative method of lines for advection-reaction-diffusion equations

Purpose The purpose of this study is to develop a new method of lines for one-dimensional (1D) advection-reaction-diffusion (ADR) equations that is conservative and provides piecewise analytical solutions in space, compare it with other finite-difference discretizations and assess the effects of advection and reaction on both 1D and two-dimensional (2D) problems. Design/methodology/approach A conservative method of lines based on the piecewise analytical integration of the two-point boundary value problems that result from the local solution of the advection-diffusion operator subject to the continuity of the dependent variables and their fluxes at the control volume boundaries is presented. The method results in nonlinear first-order, ordinary differential equations in time for the nodal values of the dependent variables at three adjacent grid points and triangular mass and source matrices, reduces to the well-known exponentially fitted techniques for constant coefficients and equally spaced grids and provides continuous solutions in space. Findings The conservative method of lines presented here results in three-point finite difference equations for the nodal values, implicitly treats the advection and diffusion terms and is unconditionally stable if the reaction terms are implicitly treated. The method is shown to be more accurate than other three-point, exponentially fitted methods for nonlinear problems with interior and/or boundary layers and/or source/reaction terms. The effects of linear advection in 1D reacting flow problems indicates that the wave front steepens as it approaches the downstream boundary, whereas its back corresponds to a translation of the initial conditions; for nonlinear advection, the wave front exhibits steepening but the wave back shows a linear dependence on space. For a system of two nonlinearly coupled, 2D ADR equations, it is shown that a counter-clockwise rotating vortical field stretches the spiral whose tip drifts about the center of the domain, whereas a clock-wise rotating one compresses the wave and thickens its arms. Originality/value A new, conservative method of lines that implicitly treats the advection and diffusion terms and provides piecewise-exponential solutions in space is presented and applied to some 1D and 2D advection reactions.

A coupled heat transfer model of medium-depth downhole coaxial heat exchanger based on the piecewise analytical solution

To describe the heat transfer processes of medium-depth downhole coaxial heat exchangers (MDCHEs) effectively, an analytical solution that couples the transient heat conduction in surrounding soil (or rock) with the quasi-steady heat transfer process in borehole was developed in this study. The piecewise calculation method was adopted both on the time scale and in the depth dimension in this model. With MATLAB as the programming platform, the proposed model was validated with reported experiments and numerical simulations, and the heat extraction efficiency was defined for comprehensive evaluation of heat transfer performance of MDCHEs. Afterwards, the impacts of Reynolds number, geothermal temperature gradient and inlet fluid temperature on the heat transfer performance of MDCHEs were analyzed.It is found that the analytical model with the time-piecewise function of the heat flux through the borehole wall instead of the temperature at the borehole wall as the coupling interface yielded more accurate heat transfer rates and fluid temperatures. Besides, the concave curves of fluid temperature were in better accordance with the practicalities. It was also found that the heat transfer of MDCHEs can be improved by increasing Reynolds number, but its increments gradually decreased. The recommended value of Reynolds number ranges from 1 × 104 to 4 × 104 and the heat extraction efficiency is recommended to be greater than 70% in engineering applications. The decrease of inlet fluid temperature and the increase of geothermal temperature gradient enhance the heat transfer rate at a steady pace. The analytical solution facilitates convenient design and optimal operation of MDCHEs.

A conservative, piecewise–analytical, transversal method of lines for reaction–diffusion equations

PurposeThe purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.Design/methodology/approachA conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.FindingsThe transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.Originality/valueA new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.

Numerical Study of Dry Friction Vibration System with Oblique Friction Force

This study investigates a vibration system considering friction force and external excitation. In particular, when the direction of the friction force is at an oblique angle along the horizontal and vertical directions, the influences of dry friction on the system dynamics are discussed. The two-degree-of-freedom dynamical model with dry friction and external excitation is established. The piecewise analytical solution corresponding to different phases is deduced based on theoretical method, the complicated dynamic behaviors are investigated by numerical simulations. The results indicate that there is no vibration of the system when external excitation does not act in either the horizontal or vertical direction, but system vibration remains when external excitation acts in both the vertical direction and horizontal direction. As parameters change largely, there exist rich symmetric and asymmetric motion, quasi-periodic motion, and chaos, and the system stability is discussed when the system parameters change within a certain range based on the Lyapunov exponent method.

Friction-Induced Vibration Due to Mode-Coupling and Intermittent Contact Loss

A two degrees-of-freedom (2DOFs) single mass-on-belt model is employed to study friction-induced instability due to mode-coupling. Three springs, one representing contact stiffness, the second providing lateral stiffness, and the third providing coupling between tangential and vertical directions, are employed. In the model, mass contact and separation are permitted. Therefore, nonlinearity stems from discontinuity due to dependence of friction force on relative mass-belt velocity and separation of mass-belt contact during oscillation. Eigenvalue analysis is carried out to determine the onset of instability. Within the unstable region, four possible phases that include slip, stick, separation, and overshoot are found as possible modes of oscillation. Piecewise analytical solution is found for each phase of mass motion. Then, numerical analyses are used to investigate the effect of three parameters related to belt velocity, friction coefficient, and normal load on the mass response. It is found that the mass will always experience stick-slip, separation, or both. When separation occurs, mass can overtake the belt causing additional nonlinearity due to friction force reversal. For a given coefficient of friction, the minimum normal load to prevent separation is found proportional to the belt velocity.

BLUES function method in computational physics

We introduce a computational method in physics that goes ‘beyond linear use of equation superposition’ (BLUES). A BLUES function is defined as a solution of a nonlinear differential equation (DE) with a delta source that is at the same time a Green’s function for a related linear DE. For an arbitrary source, the BLUES function can be used to construct an exact solution to the nonlinear DE with a different, but related source. Alternatively, the BLUES function can be used to construct an approximate piecewise analytical solution to the nonlinear DE with an arbitrary source. For this alternative use the related linear DE need not be known. The method is illustrated in a few examples using analytical calculations and numerical computations. Areas for further applications are suggested.

A conservative, spatially continuous method of lines for one-dimensional reaction-diffusion equations

PurposeThe purpose of this paper is to develop a new finite-volume method of lines for one-dimensional reaction-diffusion equations that provides piece-wise analytical solutions in space and is conservative, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficient on wave propagation.Design/methodology/approachA conservative, finite-volume method of lines based on piecewise integration of the diffusion operator that provides a globally continuous approximate solution and is second-order accurate is presented. Numerical experiments that assess the accuracy of the method and the time required to achieve steady state, and the effects of the nonlinear diffusion coefficients on wave propagation and boundary values are reported.FindingsThe finite-volume method of lines presented here involves the nodal values and their first-order time derivatives at three adjacent grid points, is linearly stable for a first-order accurate Euler’s backward discretization of the time derivative and has a smaller amplification factor than a second-order accurate three-point centered discretization of the second-order spatial derivative. For a system of two nonlinearly-coupled, one-dimensional reaction-diffusion equations, the amplitude, speed and separation of wave fronts are found to be strong functions of the dependence of the nonlinear diffusion coefficients on the concentration and temperature.Originality/valueA new finite-volume method of lines for one-dimensional reaction-diffusion equations based on piecewise analytical integration of the diffusion operator and the continuity of the dependent variables and their fluxes at the cell boundaries is presented. The method may be used to study heat and mass transfer in layered media.

A solution for antiplane‐strain local problems using elliptic integrals of Cauchy type

Quasi‐periodic piecewise analytic solutions, without poles, are found for the local antiplane‐strain problems. Such problems arise from applying the asymptotic homogenization method to an elastic problem in a parallel fiber‐reinforced periodic composite that presents an imperfect contact of spring type between the fiber and the matrix. Our methodology consists of rewriting the contact conditions in a complex appropriate form that allow us to use the elliptic integrals of Cauchy type. Several general conditions are assumed including that the fibers are disposed of arbitrary manner in the unit cell, that all fibers present imperfect contact with different constants of imperfection, and that their cross section is smooth closed arbitrary curves. Finally, we obtain a family of piecewise analytic solutions for the local antiplane‐strain problems that depend of a real parameter. When we vary this parameter, it is possible to improve classic bounds for the effective coefficients. Copyright © 2017 John Wiley & Sons, Ltd.

Simple Time Averaging Current Quality Evaluation of a Single-Phase Multilevel PWM Inverter

This paper addresses theoretical calculation of a single-phase multilevel pulsewidth modulation (PWM) inverter current total harmonic distortion (THD). Analytical approach introduced in late 1980s for a two-level inverter is generalized for an arbitrary level count for a single-phase inverter. Although the obtained closed-form piecewise analytical solutions assume (infinitely) large ratio of switching and fundamental frequencies and pure inductive load, they are practically very accurate for the ratios of (apparent) switching and fundamental frequencies larger than 25–30 and inductance-dominated loads. Along with strict mathematical solutions, simple accurate Bessel function approximation and hyperbolic average trend one are suggested. The formulas clearly reveal a single-phase PWM inverter current THD dependence on modulation index for an arbitrary voltage level count and are easily modified to cover grid-connected cases.

Asymptotic Time Domain Evaluation of a Multilevel Multiphase PWM Converter Voltage Quality

Voltage total harmonic distortion and quadratic voltage ripple criteria are accepted figures of merit of a multilevel pulse width modulation (PWM) converter voltage quality. The dominating frequency domain approach does not deliver simple closed-form expressions and practically requires accounting up to 40-50 switching harmonics. Assuming infinitely high switching frequency, suggested time domain approach to a multilevel multiphase PWM voltage quality evaluation provides closed-form piece-wise analytical solutions. These simple formulas clearly reveal the dependence on modulation index for arbitrary voltage level and odd phase numbers. The results are applicable to any type of multilevel converters (including interleaved, open winding and other) operated with nearest level/nearest space vector modulation and relatively high switching frequency.

Monte Carlo Simulations of Product Distributions and Contained Metal Estimates

Estimation of product distributions of two factors was simulated by conventional Monte Carlo techniques using factor distributions that were independent (uncorrelated). Several simulations using uniform distributions of factors show that the product distribution has a central peak approximately centered at the product of the medians of the factor distributions. Factor distributions that are peaked, such as Gaussian (normal) produce an even more peaked product distribution. Piecewise analytic solutions can be obtained for independent factor distributions and yield insight into the properties of the product distribution. As an example, porphyry copper grades and tonnages are now available in at least one public database and their distributions were analyzed. Although both grade and tonnage can be approximated with lognormal distributions, they are not exactly fit by them. The grade shows some nonlinear correlation with tonnage for the published database. Sampling by deposit from available databases of grade, tonnage, and geological details of each deposit specifies both grade and tonnage for that deposit. Any correlation between grade and tonnage is then preserved and the observed distribution of grades and tonnages can be used with no assumption of distribution form.

Cosmological fluid dynamics in the Schrödinger formalism

We investigate the dynamics of a cosmological dark matter fluid in the Schr¨ odinger formulation, seeking to evaluate the approach as a potential tool for theorists. We find simple wave-mechanical solutions of the equations for the cosmological homogeneous background evolution of the dark matter field, and use them to obtain a piecewise analytic solution for the evolution of a compensated spherical overdensity. We analyse this solution from a ‘quantum mechanical’ viewpoint, and establish the correct boundary conditions satisfied by the wavefunction. Using techniques from multiparticle quantum mechanics, we establish the equations governing the evolution of multiple fluids and then solve them numerically in such a system. Our results establish the viability of the Schr¨ odinger formulation as a genuine alternative to

A new method of solution for one-dimensional quasi-neutral bounded plasmas

AbstractA new method is proposed for calculating the potential distribution Φ(z) in a one-dimensional quasi-neutral bounded plasma; Φ(z) is assumed to satisfy a quasi-neutrality condition (plasma equation) of the form ni{Φ(z)} = ne(Φ), where the electron density ne is a given function of Φ and the ion density ni is expressed in terms of trajectory integrals of the ion kinetic equation. While previous methods relied on formally solving a global integral equation (Riemann, Phys. Plasmas, vol. 13, 2006, paper no. 013503; Kos et al., Phys. Plasmas, vol. 16, 2009, paper no. 093503), the present method is characterized by piecewise analytic solution of the plasma equation in reasonably small intervals of z. As a first concrete application, Φ(z) is found analytically through order z4 near the center of a collisionless Tonks–Langmuir discharge with a cold-ion source.