Set Of Linear Equations Research Articles

New approach to interface crack problems in transversely isotropic materials

The novelty of the approach contains several aspects: the method of derivation of the governing equations, the form and number of equations, the optimal choice of elastic constants. While majority of published articles derives the governing equations from the Green’s function for the compound space and the use of the reciprocal theorem, we use the combination of Green’s functions for 2 different half-spaces and Fourier transform. The usual approach leads to 3 hypersingular integral equations, we arrive at 2 integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic elastic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications. The new form of the governing equations allows us to obtain an exact closed form solution for an axisymmetric interface crack problem by elementary means. The same method can be applied to obtain the exact solution for a non-axisymmetric problem. A comparison is made with the existing exact solutions and some are shown to be incorrect.

A new analytical solution of vibration response of orthotropic composite plates with two adjacent edges rotationally-restrained and the others free

For the first time, the double half-Sinusoidal series is adopted as integral kernel to construct integral transform pair for new precise vibration analysis of orthotropic rectangular thin plates, supported with two adjacent edges rotationally-restrained and the other edges are free. Impose the defined single/double transforms on both the vibration governing partial differential equation (PDE) and some of the boundaries investigated, which gives the Fourier coefficient for the deflection expressed by some specific unknown constants. Utilizing the inverse formular to satify the remaining boundaries leads to a set of linear algebraic simultaneous equations, which yields the solutions of the natural frequency parameter and the associated mode shapes. Moreover, plates under classical boundary conditions also can be investigated via changing the rotating fixed coefficients introduced. Finally, more than 500 comprehensive analytical solutions were well validated by numerical results presented by finite element method (FEM), which can be treated as new reference results for further studies. Since its straightforward and rational solution procedure, the presented approach is promising to be further extended to solve more complicated boundary value problems (BVPs) of plates.

Differential parameters uncertainty estimation via a PCA-based monte carlo sampling approach: IRT-4M fuel type as a case study

ABSTRACT In this work, a hybrid Monte Carlo/sensitivity-based uncertainty contribution estimation approach is derived and tested. The approach is based on using the statistical samples used in the Monte Carlo uncertainty quantification combined with the subspace analysis to define and solve a set of linear equations to estimate the first-order sensitivities whenever the adjoint and forward sensitivity analyses are not available. The proposed approach empowers the Monte Carlo uncertainty quantification method with a pathway to evaluate and rank the uncertainty contributors efficiently without prior access to the sensitivity coefficients. The proposed approach has been illustrated and verified via an IRT-4 M 8 tubes fuel assembly uncertainty quantification problem where the proposed approach is compared to the sensitivity-based adjoint method. Results indicate that both methods concluded the same key differential contributors to the integral uncertainty with good agreement in the values of the sensitivity coefficients.

Prediction Models Based on Regression and Artificial Neural Network for Moduli of Layers Constituted by Open-Graded Aggregates.

The impermeable cover in urban area has been growing due to rapid urbanization, which prevents stormwater from being naturally infiltrated into the ground. There is a higher chance of flooding in urban area covered with conventional concretes and asphalts. The permeable pavement is one of Low-Impact Development (LID) technologies that can reduce surface runoff and water pollution by allowing stormwater into pavement systems. Unlike traditional pavements, permeable pavement bases employ open-graded aggregates (OGAs) with highly uniform particle sizes. There is very little information on the engineering properties of compacted OGAs. In this study, the moduli of open-graded aggregates under various compaction energies are investigated based on the Plate Load Test (PLT) and Light-Weight Deflectometer (LWD). Artificial Neural Network (ANN) and Linear Regression (LR) models are employed for estimation of the moduli of the aggregates based on the material type and level of compaction. Overall, the moduli from PLT and LWD steeply increase until the number of roller passes reaches 4, and they gradually increase until the number of roller passes becomes 8. A set of simple linear equations are proposed to evaluate the moduli of open-graded aggregates from PLT and LWD based on the material type and the number of roller passes.

Quadratic hyper-surface kernel-free least squares support vector regression

We present a novel kernel-free regressor, called quadratic hyper-surface kernel-free least squares support vector regression (QLSSVR), for some regression problems. The task of this approach is to find a quadratic function as the regression function, which is obtained by solving a quadratic programming problem with the equality constraints. Basically, the new model just needs to solve a system of linear equations to achieve the optimal solution instead of solving a quadratic programming problem. Therefore, compared with the standard support vector regression, our approach is much efficient due to kernel-free and solving a set of linear equations. Numerical results illustrate that our approach has better performance than other existing regression approaches in terms of regression criterion and CPU time.

Free Vibration Analysis of Laminated Composite Double-Plate Structure System with Elastic Constraints Based on Improved Fourier Series Method

An analytical model of laminated composite double-plate system (LCDPS) is established, which efficiently analyzes the common 3D plate structure in engineering applications. The proposed model combines the first-order shear deformation theory (FSDT) and the classical delamination theory, and then the LCDPS’s vibration characteristics are investigated. In the process of analysis, the improved Fourier series method (IFSM) is used to describe the displacement admissible function of the LCDPS, which can remove the potential discontinuities at the boundaries. Five sets of artificial springs are introduced to simulate the elastic boundary constraints, and the restraints of the Winkler elastic layer can be adjustable. The improved Fourier series is substituted into the governing equations and boundary conditions; then, applying the Rayleigh–Ritz method, we take all the series expansion coefficients as the generalized coordinates. After that, a set of standard linear algebraic equations was obtained. On this basis, the natural frequency and mode shapes of the LCDPS can be obtained by solving the standard eigenvalue problem. By the discussion of numerical examples and the comparison with those of the reports in the literature, the convergence and the reliability of the present approach are validated. Finally, the parametric investigations of the free vibration with complex boundary conditions are carried out, including the influence of boundary conditions, lamination scheme, plate geometric parameters, and elastic coefficient between two plates.

A Feasible Delay Margin Sensitivity Analysis Method

This letter proposes a method for calculating the sensitivity of delay margin with respect to system parameters. According to the property that the delay margin is achieved when the rightmost eigenvalues of the close-loop system locate exactly on the imaginary axis, a characteristic equation with inclusion of the delay margin is established. By differentiating the characteristic equation and further introducing normalization condition for the eigenvector, the sensitivity of the delay margin with respect to any given parameters can be obtained by solving a set of sparse linear equations. Due to the general differential and algebraic equation used in the derivation, the proposed delay margin sensitivity analysis method applies to various power system models and control problems.

Accurately charge-conserving scheme of current assignment based on the current continuity integral equation for particle-in-cell simulations

In this paper we present a high-accuracy charge conservation scheme of current assignment for particle-in-cell (PIC) simulations in two dimensional space. The current continuity integral equation is applied to set up a linear equation set about the assigned currents on edges of the cells through which a particle moves during a time step. The currents on the edges can be very accurately assigned by solving the linear equation set. This scheme can be used in Cartesian and cylindrical geometry, and is not affected by the form-factor of quasi-particle. It has been applied to the PIC simulation of Z-pinch of the current-carrying rarefied deuterium plasma shell without resorting to the electric field correction by solving Poisson equation. The simulated results reveal that the relative and absolute errors in satisfying the current continuity integral equation and Poisson equation are very tiny.

A novel contact model for rough surfaces using piecewise linear interpolation and its application in gear wear

Wear is directly related to rough surface contact, and the accuracy of a wear prediction model depends on the accurate determination of contact pressure. In this work, a numerical model with high computational accuracy for rough surface contact is developed and applied to the wear prediction of spur gears. To acquire contact pressure, the integral equation that contact pressure should satisfy is discretized into a set of linear equations by piecewise linear interpolating contact pressure. Then, a gear wear model including rough surface contact is established. The proposed contact model is verified by comparing with smooth and rough surface contact, and the wear depths of a spur gear drive with geometry and pressure updates are analyzed. Result shows that a small roughness value among contact surfaces may cause a large change in contact pressure. The effects of main design parameters on gear wear are subsequently investigated. Gear wear is found to decrease with increased module and pressure angle but increases with increased transmission ratio and input torque. These findings indicate that reasonable parameter matching is beneficial to improving gear wear resistance.

Tight Bounds on the Simultaneous Estimation of Incompatible Parameters

The estimation of multiple parameters in quantum metrology is important for a vast array of applications in quantum information processing. However, the unattainability of fundamental precision bounds for incompatible observables has greatly diminished the applicability of estimation theory in many practical implementations. The Holevo Cramer-Rao bound (HCRB) provides the most fundamental, simultaneously attainable bound for multi-parameter estimation problems. A general closed form for the HCRB is not known given that it requires a complex optimisation over multiple variables. In this work, we develop an analytic approach to solving the HCRB for two parameters. Our analysis reveals the role of the HCRB and its interplay with alternative bounds in estimation theory. For more parameters, we generate a lower bound to the HCRB. Our work greatly reduces the complexity of determining the HCRB to solving a set of linear equations that even numerically permits a quadratic speedup over previous state-of-the-art approaches. We apply our results to compare the performance of different probe states in magnetic field sensing, and characterise the performance of state tomography on the codespace of noisy bosonic error-correcting codes. The sensitivity of state tomography on noisy binomial codestates can be improved by tuning two coding parameters that relate to the number of correctable phase and amplitude damping errors. Our work provides fundamental insights and makes significant progress towards the estimation of multiple incompatible observables.

Engineering fast high-fidelity quantum operations with constrained interactions

Understanding how to tailor quantum dynamics to achieve the desired evolution is a crucial problem in almost all quantum technologies. Oftentimes an otherwise ideal quantum dynamics is corrupted by unavoidable interactions, and finding ways to mitigate the unwanted effects of such interactions on the dynamics is a very active field of research. Here, we present a very general method for designing high-efficiency control sequences that are fully compatible with experimental constraints on available interactions and their tunability. Our approach relies on the Magnus expansion to find order by order the necessary corrections that result in a high-fidelity operation. In the end finding, the control fields are reduced to solve a set of linear equations. We illustrate our method by applying it to a number of physically relevant problems: the strong-driving limit of a two-level system, fast squeezing in a parametrically driven cavity, the leakage problem in transmon qubit gates, and the acceleration of SNAP gates in a qubit-cavity system.

Multistage Linear Gauss Pseudospectral Method for Piecewise Continuous Nonlinear Optimal Control Problems

This article aims at proposing a multistage linear Gauss pseudospectral method (MS-LGPM) for solving the piecewise continuous nonlinear optimal control problem (OCP) with interior-point constraints and terminal constraints. First, the first-order necessary conditions for the multistage nonlinear OCP are derived, and a typical multipoint boundary value problem is obtained. Second, the original problem can be solved iteratively by integral prediction and quasi-linearization. Then Lagrange interpolation polynomials are used to approximate the state, control, and costate variables, to transfer those differential equations into a set of linear algebraic equations. Therefore, the control update coming closer to the optimal solution can be derived in an analytical manner. Additionally, those linear algebraic equations are formulated in regular banded matrix forms to make the code as concise as possible. Finally, the proposed method is applied to the midcourse guidance design for a dual-pulse missile to evaluate its performance. Simulation results show that the proposed method performs well in providing the optimal control with high accuracy and computational efficiency. Furthermore, a comparison with other typical guidance method and Monte Carlo simulations are also provided. Results demonstrate, even with some random uncertainties, the proposed method still has strong robustness and superior performance.

An inverse approach to estimate bacterial loading into an estuary by using field observations and residence time

Pathogen, whose abundance is often measured by the concentration of fecal indicator bacteria, is listed as the top cause of waterbody impairments in the US. An accurate estimation of the bacterial loading from watershed is thus fundamentally important for water quality management. Despite advances in watershed modeling, accurate estimation of bacterial load is still very challenging due to large uncertainties associated with bacterial sources, accumulation, and removal in the watershed. We introduce an inverse method using field-measured bacterial concentrations and numerical model-calculated residence time to estimate the bacterial loading from the drainage basin. In this method, an estuary is divided into multiple segments. Water and bacterial fluxes between neighboring segments are computed from a set of linear equations derived based on mass balance equation and the relationship between residence time and water fluxes. Loading to each segment can then be estimated by combining the computed water fluxes and observed bacterial concentrations. The approach accounts for seasonal and interannual variations in hydrodynamics due to tide, river discharge, and estuarine circulations. The method was applied to Nassawadox Creek, a sub-estuary of Chesapeake Bay, where Fecal Coliform concentrations at 46 stations were continuously monitored. The method is verified by the high consistency between estimated loadings and presumably known input loadings in numerical experiments with either constant or time-varying input loadings. With sparse observational data, the inversely estimated loadings agree well with the loadings from a previously calibrated watershed model, demonstrating the reliability of the method. The inverse approach can be used to cross-check the result of watershed models and assess changes in watershed condition. The method is also readily applicable to other types of materials, such as inorganic nutrients.

A Study for MHD Boundary Layer Flow of Variable Viscosity over a Heated Stretching Sheet via Lie-Group Method

The present work deals with the study of Magnetohydrodynamic (MHD) boundary layer flow over a heated stretching sheet with variable fluid viscosity. The fluid viscosity is assumed to vary as a linear function of temperature in the presence of uniform transverse magnetic field. The fluid is assumed to be electric ally conducting. Lie-group method is applied for determining symmetry reductions for the MHD boundary-layer equations. Lie-group method starts out with a general infinitesimal group of tran sformations under which the given partial differential equations are invariant. The determining equations are a set of linear diffe rential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variable s. After the group has been determined, a solution to the given partial differe ntial equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the n umber of independent variables of the system. The effect of the Hartmann number (M ), the viscosity parameter ( A ) and the Prandtl number ( Pr ) on the horizontal and vertical velocities, temperature pr ofiles, wall heat transfer and the wall shear stress (skin friction) , have been studied and the results are plotted.

Conjugate Heat Transfer Analysis Using the Discrete Green's Function

Abstract Conjugate heat transfer problems generally require a coupled solution of the temperature fields in the fluid and solid domains. Implementing the boundary condition at the surface of the solid using a discrete Green's function (DGF) decouples the solutions. A DGF is determined first considering only the fluid domain with prescribed thermal boundary conditions, then the temperature distribution in the solid is calculated using standard numerical methods. The only compatibility requirement is that the DGF must be specified with the same discretization as the surface of the solid. The method is demonstrated for both steady-state and transient heating of a thin plate with laminar boundary layers flowing over both sides. The resulting set of linear algebraic equations for the steady-state problem or linear ordinary differential equations for the transient problem are easily solved using conventional scientific programming packages. The method converges with nearly second-order accuracy as the discretization resolution is increased.

A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system

We propose a stable scheme to solve numerically the Cahn–Hilliard–Hele–Shaw system in three-dimensional space. In the proposed scheme, we discretize the space and time derivative terms by combining with backward differentiation formula, which turns out to be both second-order accurate in space and time. Using this method, a set of linear elliptic equations are solved instead of the complicated and high-order nonlinear equations. We prove that our proposed scheme is uniquely solvable. We use a linear multigrid solver, which is fast and convergent, to solve the resulting discrete system. The numerical tests indicate that our scheme can use a large time step. The accuracy and other capability of the proposed algorithm are demonstrated by various computational results.

Thermal modelling of cutting tool temperatures and heat partition in orthogonal machining of high-strength alloy steel

Cutting temperatures and heat partition into the cutting tool are critical factors that significantly affect tool life and part accuracy during metal removal operations, especially in dry machining. Among many thermal modelling studies, uniform heat partition ratio, and/or uniform heat intensity along the tool-chip interface are frequently assumed. This assumption is not valid in actual machining and can lead to erroneous estimated results in the presence of sticking and sliding friction zones. Therefore, it is necessary to accurately predict the cutting tool temperature and heat partition during machining. This paper presents an analytical thermal modelling approach which considers the combined effect of the primary and the secondary heat sources and determines the temperature rise and non-uniform heat partition ratio along the tool-chip interface. Cutting tests were conducted on AISI/SAE 4140 high-strength alloy steel using carbide cutting tools over a wide range of cutting speeds. Cutting temperatures were measured experimentally using an infrared thermal imaging camera. Experimentally established sticking and sliding friction regions were used to evaluate non-uniform frictional heat intensity along the tool-chip interface. The temperature matching condition along the tool-chip interface leads to the solution of distributed non-uniform heat partition ratio by solving a set of linear equations through programming in MATLAB®. Experimental results show to be consistent well with those obtained from the thermal model, yielding a relative difference of predicted average tool-chip interface temperature from −0.8% to 6.3%. It is found that average heat partition into the cutting tool ( RT) varies from 35% down to 15% for the entire range of cutting speeds. These results suggest that, to address the thermal problem in metal cutting, the research and development of tooling should also focus on reducing friction on the tool rake face in addition to the contribution of the combined effect of primary and secondary heat sources on temperature rise at the tool-chip interface.

Simplified gasification simulation using zero-dimensional model of empty palm fruit bunch

As one of the most efficient biomass conversions, the gasification process is hard to understand because of its complexity. A simplified gasification process simulation using a zero-dimensional model was designed for an empty fruit bunch considering its potential in Indonesia. A stoichiometric equilibrium model was applied to simulate the whole gasification process. Global gasification reaction, mass, and energy balance with two independent chemical reactions are integrated to perform the model. Some sets of linear and nonlinear equations were solved with the Newton Raphson method in Matlab. The output of the model, like syngas composition and the heating value, was validated with available experimental data published. The zero-dimensional thermodynamic equilibrium model could be used to predict the maximum performance of the gasifier and make the gasification process more feasible.

Research on Enterprise ordering and Transportation Model based on Fuzzy Comprehensive Evaluation and Multi-objective Optimization

This paper makes reasonable assumptions about the ordering and transportation of materials in production enterprises, uses the relevant knowledge of statistics and operational research to establish a model, and reasonably formulates the optimization strategy of raw material ordering and transportation. Firstly, the influence model of fuzzy evaluation of enterprise production is established. This paper makes a comprehensive evaluation of the six factors affecting the production of the enterprise, evaluates the influence of the importance of the production of the enterprise, and then assigns the weight of the influencing factors, so as to establish a comprehensive evaluation model, according to which the strength of the supplier can be deduced. The supply intensity index and the types of suppliers evaluate the supply characteristics of suppliers. Finally, the weight is (0.3508, 0.3304, 0.3188). Then this paper first establishes the optimal ordering scheme model of modular multi-objective optimization, analyzes the constraints, establishes a set of linear programming equations to solve it, and adjusts it according to the actual situation. The optimal ordering scheme can be obtained.

Stationary Distributions of Continuous-Time Markov Chains: A Review of Theory and Truncation-Based Approximations

Computing the stationary distributions of a continuous-time Markov chain (CTMC) involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too lar...