Newton Iteration Method Research Articles

A kernel-free boundary integral method for the nonlinear Poisson-Boltzmann equation

This work proposes a boundary integral formulation based Cartesian grid method for the nonlinear Poisson-Boltzmann (PB) interface problem in biophysics. The method (a) does not have the limitation associated with the standard finite difference method for complex interfaces, (b) avoids the generation of any unstructured volume or surface grids needed by the finite element method, and (c) solves the nonlinear and variable coefficient PDE in the framework of boundary integral equations. The method solves the nonlinear PB equation with the Newton iterative method. It first reformulates the linearized PB equation in each Newton iteration as a Fredholm system of two boundary integral (BI) equations of the second kind, which is well-conditioned, and then solves the BI system with a Krylov subspace method. The evaluation of boundary and volume integrals involved in the solution of the BI system is done with a kernel-free method, which does not need to know Green's function associated with the variable coefficient PB equation. The kernel-free method evaluates volume and boundary integrals by solving equivalent simple interface problems on Cartesian grids with either a fast Fourier transform based Poisson solver or a geometric multigrid preconditioned conjugate gradient solver, either of which involves computational work essentially linearly proportional to the number of nodes on the Cartesian grid. Numerical examples in both two and three space dimensions are presented to demonstrate the efficiency and accuracy of the proposed numerical method.

Novel Results on Output-Feedback LQR Design

This paper provides novel developments in output-feedback stabilization for linear time-invariant systems within the linear quadratic regulator (LQR) framework. First, we derive the necessary and sufficient conditions for output-feedback stabilizability in connection with the LQR framework. Then, we propose a novel iterative Newton's method for output-feedback LQR design and a computationally efficient modified approach that requires solving only a Lyapunov equation at each iteration step. We show that the proposed modified approach guarantees convergence from a stabilizing state-feedback to a stabilizing output-feedback solution and succeeds in solving high dimensional problems where other, state-of-the-art methods, fail. Finally, numerical examples illustrate the effectiveness of the proposed methods.

Two‐level multiscale enrichment finite volume method based on the Newton iteration for the stationary incompressible magnetohydrodynamics flow

In this paper, we propose a stabilized finite volume method based on the Newton's type iteration for the steady incompressible magnetohydrodynamics (MHD) problem. In order to reduce the computational complexity, the lowest order mixed finite element pair ( ‐ ‐ ) is adopted to approximate the velocity, pressure and magnetic fields, and the multiscale enrichment method is introduced to overcome the restriction of discrete inf‐sup (LBB) condition. We firstly solve the incompressible MHD equations by the Newton iterations on a coarse mesh with the mesh size , stability and convergence results of numerical solutions are provided. Secondly, the combination of the two‐level method with the Newton iteration is used to approximate the considered problem, a coupled nonlinear problem is solved on the coarse mesh , then the Stokes and Maxwell equations are considered on a fine mesh with the mesh size , the uniform stability and optimal error estimates of two‐level Newton iterative method are provided. Theoretical findings show that the two‐level method has the same order as the one‐level method in ‐norm as long as the mesh sizes satisfy . However, the two‐level method involves much less work than the one‐level method. Finally, some numerical examples are presented to demonstrate the effectiveness of the considered numerical schemes.

Nonlinear dynamics of a drillstring system with PDC bit in curved wells based on the finite element method

In this paper, a nonlinear mathematics model for axial-torsional-lateral dynamics of drillstring system equipped with a PDC bit in curved wells is established and analyzed. In the proposed model, using the finite element method, the drillstring structure is spatially discretized by Euler-Bernoulli beam elements with twelve degrees of freedom. Based on rock surface morphology of bit-rock interaction, a more realistic cutting and friction model with state-dependent delay effects is introduced to represent the bit-rock interaction. Meanwhile, a new top boundary condition is proposed to simulate the desired weight on bit (WOB). The nonlinear dynamic model is discretized in time-domain employing the generalize-α method, and the modified Newton iteration method is employed to calculate displacement response at the while-time step. Finally, the parametric analysis is carried out for discussing the influences of weight on bit and rotating speed on the nonlinear vibration of the drillstring system, respectively.

Prediction of Refractive Index of Petroleum Fluids by Empirical Correlations and ANN

The refractive index is an important physical property that is used to estimate the structural characteristics, thermodynamic, and transport properties of petroleum fluids, and to determine the onset of asphaltene flocculation. Unfortunately, the refractive index of opaque petroleum fluids cannot be measured unless special experimental techniques or dilution is used. For that reason, empirical correlations, and metaheuristic models were developed to predict the refractive index of petroleum fluids based on density, boiling point, and SARA fraction composition. The capability of these methods to accurately predict refractive index is discussed in this research with the aim of contrasting the empirical correlations with the artificial neural network modelling approach. Three data sets consisting of specific gravity and boiling point of 254 petroleum fractions, individual hydrocarbons, and hetero-compounds (Set 1); specific gravity and molecular weight of 136 crude oils (Set 2); and specific gravity, molecular weight, and SARA composition data of 102 crude oils (Set 3) were used to test eight empirical correlations available in the literature to predict the refractive index. Additionally, three new empirical correlations and three artificial neural network (ANN) models were developed for the three data sets using computer algebra system Maple, NLPSolve with Modified Newton Iterative Method, and Matlab. For Set 1, the most accurate refractive index prediction was achieved by the ANN model, with %AAD of 0.26% followed by the new developed correlation for Set 1 with %AAD of 0.37%. The best literature empirical correlation found for Set 1 was that of Riazi and Daubert (1987), which had %AAD of 0.40%. For Set 2, the best performers were the models of ANN, and the new developed correlation of Set 2 with %AAD of refractive index prediction was 0.21%, and 0.22%, respectively. For Set 3, the ANN model exhibited %AAD of refractive index prediction of 0.156% followed by the newly developed correlation for Set 3 with %AAD of 0.163%, while the empirical correlations of Fan et al. (2002) and Chamkalani (2012) displayed %AAD of 0.584 and 0.552%, respectively.

Research on prediction of critical pressure of FGM cylindrical shells under hydrostatic pressure based on homogenization transformation method

This study uses the classical Flügge shell theory and homogenization transformation method to predict the critical pressure of functionally graded material (FGM) cylindrical shells under hydrostatic pressure. The method considers the similarity in physical properties and mechanical behavior between FGM and uniform materials. The vibration equation of the coupled system of an underwater FGM cylindrical shells is derived using the wave method, taking into account the influence of fluids. The natural frequency of the FGM cylindrical shell under hydrostatic pressure is obtained using the Newton iteration method. Based on the linear correlation between critical load and load level with zero natural frequency, the fitting curve method and the formula method after homogenization transformation are used to predict and analyze the critical pressure of FGM cylindrical shells under hydrostatic pressure. The effects of various parameters on the critical pressure are discussed. The results show that the material elastic modulus E, geometric dimensions h/R and L/R of FGM cylindrical shells, and different boundary conditions have a significant impact on the critical pressure. Through multiple comparative analyses of examples, the correctness and effectiveness of the proposed method are verified. The method has high prediction accuracy and low computational cost, providing a new exploration path for the analysis of non-uniform structural mechanical behavior.

Robust depletion shutdown guidance algorithm for long-range vehicles with a solid divert control system in large deviation conditions

To address the problem of depletion shutdown precise guidance for a long-range vehicle in large deviation condition, an online guidance algorithm based on Bezier curves and zero-range orientation (ZRO) is proposed under the conditions of thrust magnitude deviation, thrust direction deviation, mass flow rate deviation, and depletion shutdown time deviation of the solid divert control system. First, the mapping relationship between the apparent velocity to be gained (AVTG) and the impact deviation is analyzed based on the variational principle. The AVTG is solved with the Newton iterative method in the designed AVTG solution plane. Then, in the plane spanned by the AVTG and the ZRO, the key parameters of the Bezier curve are solved to obtain the Bezier energy management curve by taking the apparent acceleration direction and the angular velocity of the energy management terminal as the ZRO and zero, respectively, as constraints, which effectively reduces the influence of the solid divert control system performance parameter perturbation. Finally, the feedback linearization method is applied to track the Bezier energy management curve. A detailed simulation experiment is designed to analyze the advantages and computational complexity of the proposed algorithm. The Monte Carlo simulation results show that the proposed algorithm has high precision and good robustness, and the 3σ error of the proposed algorithm is 2.58 m. On the STM32F407 single-chip microcomputer, the proposed algorithm takes 25.38 ms, which demonstrates good real-time performance and a certain engineering application value.

Deep learning for Mean Field Games with non-separable Hamiltonians

This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs). We achieve this by using two neural networks to approximate the unknown solutions of the MFG system and forward–backward conditions. Our method is efficient, even with a small number of iterations, and is capable of handling up to 300 dimensions with a single layer, which makes it faster than other approaches. In contrast, methods based on Generative Adversarial Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We demonstrate the effectiveness of our approach by applying it to a traffic flow problem, which was previously solved using the Newton iteration method only in the deterministic case. We compare the results of our method to analytical solutions and previous approaches, showing its efficiency. We also prove the convergence of our neural network approximation with a single hidden layer using the universal approximation theorem.

The precipitation process for a rare earth leach solution with calcium oxide in the presence of ascorbate and calcination to oxide of high purity

Sulfate ions coprecipitate with rare earth metal ions to form alkaline rare earth sulfate (RE2(OH)4SO4) in the CaO precipitation process, leading to excessive content of sulfate in the mixed rare earth oxide. Due to the difference of coordination ability between ascorbate, sulfate, hydroxyl and rare earth metal ions, ascorbate is introduced into the pregnant leach solution (PLS) during the CaO precipitation process. It realizes the coprecipitation between ascorbate (C6H7O6−) and rare earth metal ions, reducing the formation of RE2(OH)4SO4 and improving the purity of mixed rare earth oxide. In this paper, the standard reaction Gibbs free energy changes of the possible reactions in the precipitation process were calculated using the group contribution method, to draw the species distribution equilibrium diagram with Newton's iteration method. When ascorbate exists, rare earth metal ions in the PLS tend to form the alkaline rare earth ascorbate (RE2(OH)4(C6H7O6)2) instead of RE2(OH)4SO4. Furthermore, the effect of the reaction conditions on the precipitation process were studied. When the molar ratio of rare earth metal ions to ascorbate was 1.17, the purity of the mixed rare earth oxide was 96.2 wt% and the sulfate content was 2.2 wt%. Finally, based on the results of the material balance calculation, FTIR, XRD and TG-DSC tests, it was further confirmed that ascorbate participated in the form of RE2(OH)4(C6H7O6)2, and effectively suppressed the generation of RE2(OH)4SO4. Consequently, the results facilitate the green and efficient extraction of rare earths from the ion-adsorption type ore.

A multi-dimensional nonlinear optimisation method for the TianQin constellation

TianQin is a geocentric space-based gravitational wave detecting mission that relies on three drag-free spacecraft with a mean inter-spacecraft distance of approximately 1.7 × 105 km. Considering the efficiency requirement of orbit optimisation, we apply a gradient descent method based on variational gradient calculation and the damped Newton iterative method to find the most stable solution of the constellation near the given initial position and velocity of the spacecraft. The efficiency was improved and the number of orbital integrals required for gradient calculation was significantly reduced by over 83%. Based on this optimisation method, we analysed the influence of perturbation on the stability of the TianQin constellation, and Moon’s three-body perturbation was the main factor. In addition, the influence of the orbit elements on the optimisation results was studied. The influence of the initial mean anomaly can be ignored, and the stability of the optimised constellation decreases with an increase in the semi-major axis. Working in the ‘3 months on +3 months off’ observation scheme for 2 years, the optimisation result is the ideal direction when the normal direction of the orbital plane of the TianQin constellation points to RXJ0806.3 + 1527.

The numerical treatment of fractal‐fractional 2D optimal control problems by Müntz–Legendre polynomials

AbstractIn this work, we introduce a method based on the Müntz–Legendre polynomials (M‐LPs) for solving fractal‐fractional 2D optimal control problems that the fractal‐fractional derivative is described in Atangana‐Riemann‐Liouville's sense. First, we obtain operational matrices of fractal‐fractional‐order derivative, integer‐order integration, and derivative of the M‐LPs. Second, the under study problem is converted into an equivalent variational problem. Then, by applying the M‐LPs, their operational matrices and Gauss–Legendre integration, the mentioned problem is converted to a system of algebraic equations. Finally, this system is solved by Newton's iterative method. Also, we introduce an error bound for the described method. Two examples are included to test the applicability and validity of the present scheme.

An Efficient Discretization Scheme for Solving Nonlinear Ill-Posed Problems

Abstract Information based complexity analysis in computing the solution of various practical problems is of great importance in recent years. The amount of discrete information required to compute the solution plays an important role in the computational complexity of the problem. Although this approach has been applied successfully for linear problems, no effort has been made in literature to apply it to nonlinear problems. This article addresses this problem by considering an efficient discretization scheme to discretize nonlinear ill-posed problems. We apply the discretization scheme in the context of a simplified Gauss–Newton iterative method and show that our scheme requires only less amount of information for computing the solution. The convergence analysis and error estimates are derived. Numerical examples are provided to illustrate the fact that the scheme can be implemented successfully. The theoretical and numerical study asserts that the scheme can be employed to nonlinear problems.

Fast iterative method for local steric Poisson–Boltzmann theories in biomolecular solvation

This work proposes a fast iterative method for local steric Poisson–Boltzmann (PB) theories, in which the electrostatic potential is governed by the Poisson's equation and ionic concentrations satisfy equilibrium conditions. To present the method, we focus on a local steric PB theory derived from a lattice-gas model, as an example. The advantages of the proposed method in efficiency are achieved by treating ionic concentrations as scalar implicit functions of the electrostatic potential, though such functions are only numerically achievable. The existence, uniqueness, boundness, and smoothness of such functions are rigorously established. A Newton iteration method with truncation is proposed to solve a nonlinear system discretized from the generalized PB equations. The existence and uniqueness of the solution to the discretized nonlinear system are established by showing that it is a unique minimizer of a constructed convex energy. Thanks to the boundness of ionic concentrations, truncation bounds for the potential are obtained by using the extremum principle. The truncation step in iterations is shown to be energy and error decreasing. To further speed-up computations, we propose a novel precomputing-interpolation strategy, which is applicable to other local steric PB theories and makes the proposed methods for solving steric PB theories as efficient as for solving the classical PB theory. Analysis on the Newton iteration method with truncation shows local quadratic convergence for the proposed numerical methods. Applications to realistic biomolecular solvation systems reveal that counterions with steric hindrance stratify in an order prescribed by the parameter of ionic valence-to-volume ratio. Finally, we remark that the proposed iterative methods for local steric PB theories can be readily incorporated in well-known classical PB solvers.

Efficient inversion of matrix ϕ-functions of low order

The paper is concerned with efficient numerical methods for solving a linear system ϕ(A)x=b, where ϕ(z) is a ϕ-function and A∈RN×N. In particular in this work we are interested in the computation of ϕ(A)−1b for the case where ϕ(z)=ϕ1(z)=ez−1z and ϕ(z)=ϕ2(z)=ez−1−zz2. Under suitable conditions on the spectrum of A we design fast algorithms for computing both ϕℓ(A)−1 and ϕℓ(A)−1b based on Newton's iteration and Krylov-type methods, respectively. Adaptations of these schemes for structured matrices are considered. In particular the cases of banded and more generally quasiseparable matrices are investigated. Numerical results are presented to show the effectiveness of our proposed algorithms.

A robust attitude estimation algorithm for seabed inverted ultra-short baseline

The ultra-short baseline(USBL) is fixed upside down on the seafloor to position an autonomous underwater vehicle, which is the inverted ultra-short baseline (i-USBL). Three attitude angles between the i-USBL coordinate system and the navigation coordinate system will be produced when the USBL is installed on the seabed. Significant positioning error will occur if the USBL attitude is undefined. Besides, it also affects the accuracy of attitude angles estimation that the outliers produced when using USBL. In this paper, we propose a robust singular value decomposition (RSVD) method for estimating the attitude angles. Specifically, we develop an attitude calibration model based on the i-USBL positioning mode. The USBL attitude angles is solved using singular value decomposition (SVD) according to the least squares method. The new attitude angles are used to calculate the residuals of the coordinates at each measurement point. The residuals are used to calculate the weights of each point by the IGG III weighting function. The recalculation of attitude angles is performed by using up-to-date weights and SVD. The performance of SVD, the conventional Gaussian Newton iterative method (GN), and RSVD are compared in the simulations. The simulations show that the RSVD has high calibration accuracy and robustness.

A new general analytical PBEM for solving three-dimensional transient nonlinear heat conduction problems with spatially-varying heat generation

In this paper, a polygonal boundary element method (PBEM) is presented, to solve the transient nonlinear heat conduction problems with spatially variable heat generation and temperature-dependent thermal physical properties. A new general method is developed to analytically compute the domain integrals regarding arbitrary nonlinearly-varying thermal conductivity, specific heat, and density, by which the computations of numerous Gaussian points in the numerical method can be circumvented and the efficiency of the PBEM can be significantly improved. The finite difference scheme is employed to evaluate the transient term, and the Newton iterative method is used to search the exact solution of the nonlinear system equation. Four numerical examples are designed to examine the property of the proposed method. The results indicate that the proposed analytical PBEM can accurately solve the three-dimensional transient nonlinear heat conduction problems with heat generation.

Application of Variable Gauge Rolling Technology for Plate to Reduce the Head Impact

The Variable Gauge Rolling (VGR) technology is adopted for plate rolling. Two passes are treated as a group, with reducing the head reduction, increasing the tail reduction, to reduce the peak impact torque of the head, and give full play to the equipment capacity in the stable rolling stage to increase the pass reduction. The prediction models of plate width and length, rolling force and torque in VGR process are established. The using strategy of VGR are determined, and two methods for increasing reduction are put forward. The main body and head and tail sections of the rolled plate are calculated separated. The reduction corresponding to the target rolling torque is calculated by Newton iterative method, and pass schedule distribution of VGR process is shown. The VGR technology is applied for plate rolling process. The practical application shows that the influence of head impact in the plate rolling process can be reduced by using VGR. The torque amplification coefficient decreases linearly with the increase of gauge variation, and rolling passes number can be reduced through reasonable pass schedule distribution. The practical application shows that VGR can reduce the influence of head impact in the plate rolling process.

A feedforward neural network based on Legendre polynomial for solving linear Fredholm integro-differential equations

This paper presents a method for solving linear Fredholm integro-differential equations using a feedforward neural network based on Legendre polynomials. Firstly, the Legendre polynomials are used to approximate the unknown function and the kernel function in the equations. Secondly, The roots of Legendre polynomials are obtained by using Newton iteration method to obtain Gaussian integration points, and the obtained Gaussian integration points are used as the input nodes of the neural network, and the corresponding weight is learned by the gradient descent method to obtain an approximate solution. Finally, through the numerical value Case analysis to verify the effectiveness of the method.

Underwater sound radiation from a Mindlin plate with an acoustic black hole

Acoustic black holes (ABHs) have been shown very efficient to reduce sound radiation in the air. However, their underwater performance is not clear. In this paper, we consider a Mindlin plate with an embedded circular ABH indentation. The Gaussian expansion method is used to compute the rotations and bending displacement at the mid-surface, with boundary conditions imposed via the nullspace method. The interaction between the solid and fluid is accounted for by the work done by the sound pressure. The theoretical method is validated against FEM and hybrid BEM-FEM simulations. Results show that the ABH is very effective for suppressing the underwater sound power between the cut-on frequency and the critical frequency. Yet at higher frequencies, the advantage of the ABH is not significant. The underlying physics is revealed using modal loss factor (MLF) that is obtained by solving the eigenvalue problem with the Newton iteration method. It is found that beyond the critical frequency, the radiation damping is very high, because of energy transmission from the plate to the water. Observation on the non-negative intensity (NNI) demonstrates that at higher frequencies the radiation area moves from the plate corners to the force point.

Solving Sturm–Liouville inverse problems by an orthogonalized enhanced boundary function method and a product formula for symmetric potential

We develop a fast convergence iterative method after transforming the Sturm–Liouville problem into the weak-form integral equations, and using sinusoidal functions as the testers as well as the bases of eigenfunction. Owing to the orthogonal property of these bases, we can obtain the expansion coefficients in closed-form. A new idea of orthogonalized and enhanced boundary function (OEBF) is introduced to compute eigenvalues. In the Sobolev space, we prove the closeness of the OEBF to the eigenfunction with their distance being reduced when the eigenvalue increases. Moreover, upon expressing the Rayleigh quotient in terms of OEBF, the unknown functions in the Sturm–Liouville operator can be recovered quickly, merely two boundary data of unknown functions and the first eigenvalue are sufficing. The OEBF is a promising method with a few iterations to obtain quite accurate estimations of the unknown potential and weight functions. Thanks to the symmetry property a symmetric matrix eigenvalue problem is derived to recover a symmetric potential. The characteristic equation endowing with a special coefficient matrix is decomposed into a product of two characteristic equations. The Newton iterative method is thus developed by relying on the product formula to reconstruct the unknown symmetric potential function with the aid of a few lower orders eigenvalues.