Picard Iteration Method Research Articles

Hydro-mechanical modeling of hydraulic fracture propagation based on embedded discrete fracture model and extended finite element method

In this study, an efficient hydro-mechanical model of hydraulic fracture propagation is proposed using the embedded discrete fracture model in conjunction with the extended finite element method. Fracture is embedded into a common computational grid for both stress and pressure fields. A hybrid scheme of fixed stress split and Picard iterative method is presented to deal with the hydro-mechanical coupling and fracture propagation. The model is verified against other numerical solution of hydro-mechanical coupling problem and analytical solution of fracture propagation available from literature. Numerical results show that as matrix permeability increases, more fracturing fluid is required to get a desired length of fracture. The effect of Biot's coefficient is similar to that of matrix permeability. Finally, the model is extended to simulate simultaneous propagation of multiple hydraulic fractures. It's interesting to find that the effect of stress shadowing could be weakened by the poroelastic stress when accounting for hydro-mechanical coupling. The study indicates that the hydro-mechanical coupling could have significant influence on hydraulic fracturing, especially for multi-cluster fracturing of horizontal wells.

The Improvement of Coupling Method in TINTE by Fully Implicit Scheme

TINTE is a well-established code for the pebble-bed high-temperature gas-cooled reactor (HTR), including the complicated nuclear module and thermal-hydraulic module, which has been validated by experiments and widely used in the transient behavior simulation. However, only an operator splitting scheme is employed in TINTE to couple the neutronics and thermal hydraulics, and some physical quantities are not consistent in time. As a result, the accuracy and stability are limited by the additional error term derived from the unconverged physical term. In this paper, a fully implicit coupling method was investigated in which the coupled nonlinear fields at each time step are converged using Picard iterations. A physics-based preconditioning is proposed in the work here to further improve the computational performance of the fully implicit coupling method. Seven test problems are implemented based on a practical engineering model, rather than a simple model, to evaluate the performance of the Picard method. The numerical results show that the fully implicit Picard iteration method is more accurate and more stable, which permits longer time steps and a reduction of the computational burden for solving the coupled field equations. The computational efficiency is further enhanced when the physics-based preconditioning is utilized.

Optimization and Numerical Simulation of Outlet of Twin Screw Extruder

In view of the unreasonable design of non-intermeshing counter-rotating twin screw extruder die, the problem of productivity reduction was discussed. Firstly, the mathematical model of extruder productivity was established. The extruder die model was improved. Secondly, the force analysis of twin screw extruder physical model was carried out. Meanwhile, A combination of mechanical analysis and numerical simulation was adopted. The velocity field, pressure field and viscosity field were calculated by Mini-Element interpolation method, linear interpolation method and Picard iterative convergence method respectively. The influence of die model on the quantity of each field before and after improvement was analyzed. The results show that the improved model had increased the rheological parameters of the flow field, the leakage and reverse flow decreased. Through post-processing calculation, the productivity of the third dies extruder was 10% higher than before. The research results provide a theoretical basis for the design and optimization of die model of non intermeshing counter-rotating twin screw extruder.

Mittag-Leffler and Padé approximations to stiff fractional two point kinetics equations

The mathematical model of stiff two-point kinetics equations for a compact core with bulky reflector plays a basic role in kinetics schemes of reflector reactor in the nuclear reactor dynamic systems. This model is a non-conventional models describe the number of neutrons per volume and delayed precursor into reactors. To analyze reactor response, a novel mathematical treatment should be introduced to solve the stiff fractional differential equations in the matrix form. Mittag-Leffler functions have recently caught the interest, particularly with the models which using fractional calculus. In this work, a fractional formula of the coupled stiff two-point reactor kinetics model with I-groups of delayed neutrons are considered by merging Mittag-Leffler function with the developed Padé approximations. The validity of the proposed fractional formula is confirmed for various reactivity such as step and ramp reactivity in various fractional order. The estimated values confirm that the treatment for the problem under consideration using Mittag-Leffler function and the developed Padé approximations agree with Picard iteration method and the conformist techniques.

Efficacy of Coefficients on Rate of Convergence of Some Iteration Methods for Quasi-Contractions

By comparing rate of convergence for some iteration methods, we show that the coefficients have important role in rate of convergence for quasi-contractions in Picard, Mann, Ishikawa, Noor, Agarwal and Thakur iteration methods. Using the Matlab, we provide some numerical examples for our results.

Global existence and uniqueness of a classical solution to some differential evolutionary system

Theorems of global existence and uniqueness of a classical solution to a nonlinear differential evolutionary system with initial conditions are proved. This system is composed of one partial hyperbolic second-order equation and an ordinary subsystem with a parameter. In the proof of the theorems we use the Picard iteration method, the monotone method of lower and upper solutions, the integral form of the differential problem, weak differential inequalities and the Arzeli-Ascola lemma.

An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

Nonlinear Effects on the Convergence of Picard and Newton Iteration Methods in the Numerical Solution of One-Dimensional Variably Saturated–Unsaturated Flow Problems

Finite element discretization of the pressure head form of the Richards equation leads to a nonlinear model, which yields numerical convergence difficulties. When the numerical solution to this problem has either an extremely sharp moving front, infiltration into dry soil, flow domains containing materials with spatially varying properties, or involves time-dependent boundary conditions, the corrector iteration used in many time integrators can terminate prematurely, which leads to incorrect results. While the Picard and Newton iteration methods can solve this problem through tightening the tolerances provided to the solvers, there is a more efficient approach to overcome the convergence difficulties. Four tests examples are examined, and each test case is solved with five sufficiently small tolerances to demonstrate the effectiveness of convergence. The numerical results illustrate that the methods greatly improve the convergence and stability. Test experiments show that the Newton method is more complex and expensive on a per iteration basis than the Picard method for simulating variably saturated–unsaturated flow in one spatial dimension. Consequently, it is suggested that the resulting local and global mass balance is exact within the minimum specified accuracy.

Global existence and uniqueness of a classical solution to some differential evolutionary system

Theorems of global existence and uniqueness of a classical solution to a nonlinear differential evolutionary system with initial conditions are proved. This system is composed of one partial hyperbolic second-order equation and an ordinary subsystem with a parameter. In the proof of the theorems we use the Picard iteration method, the monotone method of lower and upper solutions, the integral form of the differential problem, weak differential inequalities and the Arzeli-Ascola lemma.

Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications

This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo‐type fractional derivative with respect to another function. Existence and uniqueness results for the problem are established by means of the some standard fixed point theorems. Next, we develop the Picard iteration method for solving numerically the problem and obtain results on the long‐term behavior of solutions. Finally, we analyze a population growth model and a gross domestic product model with governing equations being fractional differential equations that we have introduced in this work.

Feedback-Accelerated Picard Iteration for Orbit Propagation and Lambert’s Problem

This paper presents a new feedback-accelerated Picard iteration method for solving long-term orbit-propagation problems and perturbed Lambert’s problems. This method is developed by combining the collocation method and the variational iteration method over large time steps. The resulting iterative formulas are explicitly derived so that they can be directly adopted to solve problems in orbital mechanics. Several typical orbit regimes incorporating high-order gravity and air drag force are used to demonstrate the application of the proposed method in orbit propagation. Further, the feedback-accelerated Picard iteration method is used to solve perturbed orbit-transfer problems. The combination of it with a fish-scale-growing method successfully extends its convergence domain and provides a potential approach for solving long-duration two-point boundary-value problems in conservative systems. The numerical results show that the proposed method is highly precise and efficient.

A modified Picard iteration scheme for overcoming numerical difficulties of simulating infiltration into dry soil

Numerical models based on Richards’ equation are often employed to simulate the soil water dynamics. Among them, those Picard iteration models which use the head as primary variable are widely adopted due to their simplicity and capability for handling partially saturated flow conditions. However, it is well-known that those models are prone to convergence failure in some unfavorable flow conditions, especially when simulating infiltration into initially dry soils. Here we analyze the reasons that give rise to the numerical difficulty. Moreover, several modifications to the mass-conservative Picard iteration method are proposed so that numerical difficulty is avoided in these unfavorable flow conditions. Our proposed modifications do not degrade the simulated results, while they lead to more robust convergence performances and cost-effective simulations.

Three-dimensional modelling of coupled leachate and gas flow in bioreactor landfills

A three-dimensional two-phase flow model for leachate and gas transportation in bioreactor landfills has been developed based on OpenFOAM. Darcy’s law and the van Genuchten soil-water retention curve model are incorporated into the governing equations for leachate and gas flow. A modified Picard iteration method is introduced to solve the model. The developed model is effectively verified by published results. A three-dimensional conceptual landfill cell with leachate recirculation well and gas collection well is simulated. The results show that leachate recirculation is beneficial for gas collection, and the effect of gas on leachate recirculation can be neglected if the gas collection well is working.

Analysis of Carreau fluid flow between corrugated plates

The paper deals with a problem of Carreau fluid flow between corrugated plates. A meshless numerical procedure for solving nonlinear governing equation is constructed using the Picard iteration method in combination with the method of fundamental solutions and the radial basis functions. The dimensionless average velocity and the product of friction factor and Reynolds number are calculated for different values of the fluid model and parameters of the considered region.

Post-buckling behaviour of shear deformable functionally graded curved shell panel under edge compression

In this article, the post-buckling behaviour of functionally graded curved shell panels of different shell geometries (spherical, elliptical, cylindrical and hyperbolic) are investigated under the uniaxial and the biaxial edge compression. The inhomogeneity of the functionally graded material along the thickness direction is achieved using power-law distribution through Voigt's micromechanical model to obtain the effective material properties. The kinematic model is developed with the help of higher-order shear deformation theory in conjunction with Green-Lagrange geometrical nonlinear strains. The governing equation of the axially loaded functionally graded curved panel is derived using the minimum total potential energy principle. The nonlinear finite element steps have been employed to discretise the shell panel domain and solved with the help of Picard's iterative method to obtain the desired load parameter. Further, the effects of different parameters such as the amplitude ratios, the power-law indices, the curvature ratios, the thickness ratios, and the support conditions on the buckling and post-buckling responses of the functionally graded curved panels are demonstrated through suitable numerical illustrations.

An efficient modification of PIM by using Chebyshev polynomials

In this article, an efficient modification of the Picard iteration method (PIM) is presented by using Chebyshev polynomials. Special attention is given to study the convergence of the proposed method. The proposed modification is tested for some examples to demonstrate reliability and efficiency of the introduced method. A comparison between our numerical results against the conventional numerical method, fourth-order Runge-Kutta method (RK4) is given. From the presented examples, we found that the proposedmethod can be applied to wide class of non-linear ordinary differential equations.

A study of nonlinear biochemical reaction model

The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method (LWPM). Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge–Kutta method of order four (RK-4). Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.

On the approximate solution of a Kirchhoff type static beam equation

The paper deals with a boundary value problem for the nonlinear integro-differential equation u⁗−m(∫0lu′2dx)u″=f(x,u), m(z)≥α>0, 0≤z<∞, modelling the static state of the Kirchhoff beam. The problem is reduced to an integral equation which is solved using the Picard iteration method. The convergence of the iteration process is established and the error estimate is obtained.

Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems1

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix (STM) Φ(t,α), associated with the linear part of the equation, can be expressed in terms of the periodic Lyapunov–Floquét (L-F) transformation matrix Q(t,α) and a time-invariant matrix R(α) containing a set of symbolic system parameters α. Computation of Q(t,α) and R(α) in symbolic form as a function of α is of paramount importance in stability, bifurcation analysis, and control system design. In earlier studies, since Q(t,α) and R(α) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In 2009, an attempt was made by Butcher et al. (2009, “Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245) to compute the Q(t,α) matrix in a symbolic form using the Magnus expansions with some success. In this work, an efficient technique for symbolic computation of Q(t,α) and R(α) matrices is presented. First, Φ(t,α) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then, R(α) is computed using a Gaussian quadrature integral formula. Finally, Q(t,α) is computed using the matrix exponential summation method. Using mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four-dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

Conservative tightly-coupled simulations of stochastic multiscale systems

Multiphysics problems often involve components whose macroscopic dynamics is driven by microscopic random fluctuations. The fidelity of simulations of such systems depends on their ability to propagate these random fluctuations throughout a computational domain, including subdomains represented by deterministic solvers. When the constituent processes take place in nonoverlapping subdomains, system behavior can be modeled via a domain-decomposition approach that couples separate components at the interfaces between these subdomains. Its coupling algorithm has to maintain a stable and efficient numerical time integration even at high noise strength. We propose a conservative domain-decomposition algorithm in which tight coupling is achieved by employing either Picard's or Newton's iterative method. Coupled diffusion equations, one of which has a Gaussian white-noise source term, provide a computational testbed for analysis of these two coupling strategies. Fully-converged (“implicit”) coupling with Newton's method typically outperforms its Picard counterpart, especially at high noise levels. This is because the number of Newton iterations scales linearly with the amplitude of the Gaussian noise, while the number of Picard iterations can scale superlinearly. At large time intervals between two subsequent inter-solver communications, the solution error for single-iteration (“explicit”) Picard's coupling can be several orders of magnitude higher than that for implicit coupling. Increasing the explicit coupling's communication frequency reduces this difference, but the resulting increase in computational cost can make it less efficient than implicit coupling at similar levels of solution error, depending on the communication frequency of the latter and the noise strength. This trend carries over into higher dimensions, although at high noise strength explicit coupling may be the only computationally viable option.