Modified Newton Iteration Research Articles

A note on approximate Jacobians of implicit Runge–Kutta methods and convergence of modified Newton iterations

A note on approximate Jacobians of implicit Runge–Kutta methods and convergence of modified Newton iterations

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.

Robustness and efficiency of phase stability testing at VTN and UVN conditions

Beyond the well-documented conventional phase equilibrium calculations at pressure and temperature (PT) specifications, other specifications, such as volume, temperature and moles (VTN) and internal energy, volume and moles (UVN) have received an increasingly interest in the last years. Whatever the selected set of independent variables, phase stability at UVN conditions can be asserted by solving simpler VTN or PT stability problems (a non-linear equation is solved only once at UVN specifications for temperature). In VTN stability, modified Newton iterations were previously used to achieve convergence, since successive substitution iterations (SSI) may exhibit problematic convergence. A mathematical analysis of convergence properties of the successive substitution (SSI) method for VTN stability reveals its potential instability, unlike for PT stability, in which SSI is very robust. This problem can be theoretically overcome by using a damping factor, but severe damping may lead to unacceptable slow convergence rates. It is shown that the SSI method should never be used alone, combined undamped SSI-standard Newton method is neither robust nor efficient as claimed previously and must be avoided; moreover, damped SSI can be used with caution only in early iteration stages. The convergence behavior of several methods for VTN stability is analyzed for a variety of mixtures and the numbers of iterations are compared with those reported in the literature. Several domains in the molar density-temperature plane were identified, where SSI is either systematically not detecting a phase split (converging to the trivial solution from all initial guesses) or systematically diverges. Our calculation procedures are highly robust and systematically faster than previous methods (for some test points up to one or even two orders of magnitude) and are able to find the global minimum of the TPD function in all test cases, unlike previously proposed SSI/Newton methods, which either are strongly attracted by local minima or they diverge. This paper gives for the first time a detailed mathematical and numerical analysis of the convergence behavior of SSI and combined SSI-Newton methods in VTN stability testing.

Development of a Modified Newton Iteration Algorithm for massive MIMO systems with precoding and its study in MATLAB environment

The aim of this work is to create an iterative precoding algorithm with a higher convergence rate compared to known algorithms. This paper considers the task of antenna system pattern formation using linear precoding for massive MIMO systems, including precoders with approximate matrix inversion methods, such as the use of the Newton Iteration Algorithm. A variant of the Modified Newton Iteration Algorithm with a variable adaptation step is also proposed. A mathematical description of signal processing in the downlink model for massive MIMO systems is given, as well as linear precoding procedures. The convergence of the algorithm has been analyzed from the initial parameters determining the adaptation step, from the number of user equipment served, as well as the angle of user equipment position.

Recurrent Neural Dynamics Models for Perturbed Nonstationary Quadratic Programs: A Control-Theoretical Perspective

Recent decades have witnessed a trend that control-theoretical techniques are widely leveraged in various areas, e.g., design and analysis of computational models. Computational methods can be modeled as a controller and searching the equilibrium point of a dynamical system is identical to solving an algebraic equation. Thus, absorbing mature technologies in control theory and integrating it with neural dynamics models can lead to new achievements. This work makes progress along this direction by applying control-theoretical techniques to construct new recurrent neural dynamics for manipulating a perturbed nonstationary quadratic program (QP) with time-varying parameters considered. Specifically, to break the limitations of existing continuous-time models in handling nonstationary problems, a discrete recurrent neural dynamics model is proposed to robustly deal with noise. This work shows how iterative computational methods for solving nonstationary QP can be revisited, designed, and analyzed in a control framework. A modified Newton iteration model and an improved gradient-based neural dynamics are established by referring to the superior structural technology of the presented recurrent neural dynamics, where the chief breakthrough is their excellent convergence and robustness over the traditional models. Numerical experiments are conducted to show the eminence of the proposed models in solving perturbed nonstationary QP.

An iteration solver for the Poisson–Nernst–Planck system and its convergence analysis

In this paper, we provide a theoretical analysis for an iteration solver to implement a finite difference numerical scheme for the Poisson-Nernst–Planck (PNP) system, based on the Energetic Variational Approach (EnVarA), in which a non-constant mobility H−1 gradient flow is formulated. In particular, the nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty. In the numerical design, the mobility function is explicitly updated, for the sake of unique solvability analysis. The logarithmic and the electric potential diffusion terms, which come from the gradient of convex energy functional parts, are implicitly computed. The positivity-preserving property for all the concentrations, an unconditional energy stability, and the optimal rate error estimate have been established in a recent work. A modified Newton iteration for the nonlinear and logarithmic part, combined with a linear iteration for the electric potential part, is proposed to implement the given numerical scheme, in which a non-constant linear elliptic equation needs to be solved at each iteration stage. A theoretical analysis is presented in this article, and a linear convergence is proved for such an iteration, with an asymptotic error constant in the same order of the time step size. A numerical test is also presented in this article, which demonstrates the linear convergence rate of the proposed iteration solver.

Modeling multi-fracturing fibers in fiber networks using elastoplastic Timoshenko beam finite elements with embedded strong discontinuities — Formulation and staggered algorithm

To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface. We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper.

A Modified Newton–Harmonic Balance Approach to Strongly Odd Nonlinear Oscillators

Since combinations of the Newton’s method and the harmonic balance (HB) method require, at each iteration step, calculating the first or the first- and second-order derivatives of the restoring force function, and expanding the function, its first- and second-order derivatives into Fourier series, the procedural costs are high and sometimes difficult to achieve algebraically. It is thus preferable to avoid expensive re-linearization or computation of the second-order derivative. A new approach is proposed to construct accurate analytical approximation solutions to strongly nonlinear conservative oscillators with odd nonlinearities. The approach is based on a combination of a modified Newton method and the HB method. For the modified Newton method, two simplified Newton steps are taken between each Newton step where only one linearization of the restoring force function is required. The resulting equations are solved by applying the HB method appropriately. Using only one modified Newton iteration step may achieve highly accurate analytical approximation solutions to the strongly nonlinear oscillators. Three examples with physical implications are used to illustrate the proposed method. Through the modified Newton iteration step, the multiple cumbersome linearizations of the restoring force function are replaced by only one linearization, and the corresponding governing equations can be properly solved by the HB method. The current work is expected to extend to the study of other nonlinear oscillations.

Inverse Analysis Approach to Identify the Loads on the External TBM Shield Surface and Its Application

Identifying the surrounding rock loads acting on the external shield surface is critical to the safe construction of deep and long tunnels excavated by TBMs (tunnel boring machines). However, it is rather difficult or even impossible to carry out monitoring outside the TBM shield to directly determine the surrounding rock loads. In the present paper, based on modified Newton iteration and numerical simulation, an inverse analysis approach to identify the surrounding rock loads is proposed. This inverse problem is solved by minimizing the objective function that incorporates the difference between the monitored and calculated strains of discrete points on the internal TBM shield surface. During the inverse analysis, the external TBM shield surface is discretized into rectangular regions, and the nodal loads of each region are taken as the inverse parameters. Then, the distributed loads are approximately calculated by the interpolation of the nodal loads of each region. The proposed approach is calibrated with two numerical examples and then applied to analyze in situ monitoring data. The results show that the proposed approach is robust and efficient for identifying the distribution and magnitude of surrounding rock loads, and it is possible to carry out timely countermeasures to avoid shield jamming according to the identified loads.

A Modified Newton Method for Multilinear PageRank

When studying the multilinear PageRank problem, a system of polynomial equations needs to be solved. In this paper, we propose a modified Newton method and develop a monotone convergence theory for a third-order tensor when $\alpha \lt 1/2$. In this parameter regime, the sequence of vectors produced by the Newton-like method is monotonically increasing and converges to the solution. When $\alpha \gt 1/2$ we present an always-stochastic modified Newton iteration. Numerical results illustrate the effectiveness of this method.

Piecewise linear mapping optimization based on the complex view

AbstractWe present an efficient modified Newton iteration for the optimization of nonlinear energies on triangle meshes. Noting that the linear mapping between any pair of triangles is a special case of harmonic mapping, we build upon the results of Chen and Weber [CW17]. Based on the complex view of the linear mapping, we show that the Hessian of the isometric energies has a simple and compact analytic expression. This allows us to analytically project the per‐element Hessians to positive semidefinite matrices for efficient Newton iteration. We show that our method outperforms state‐of‐the‐art methods on 2D deformation and parameterization. Further, we inspect the spectra of the per triangle energy Hessians and show that given an initial mapping, simple global scaling can shift the energy towards a more convex state. This allows Newton iteration to converge faster than starting from the given initial state. Additionally, our formulations support adding an energy smoothness term to the optimization with little additional effort, which improves the mapping results such that concentrated distortions are reduced.

A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

We propose a modified Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. When the tensor is irreducible, all nonnegative eigenpairs are known to be positive. We prove local quadratic convergence of the new iteration to any positive eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration. A big advantage of the modified Newton iteration is that it seems capable of finding a nonnegative eigenpair starting with any positive unit vector. Special attention is paid to transition probability tensors.

A volume-based approach to phase equilibrium calculations at pressure and temperature specifications

Phase equilibrium calculations at pressure and temperature specifications, consisting in the minimization of the Gibbs free energy with respect to mole numbers, are the most commonly used and are well documented in the literature. A very attractive alternative is given by the volume-based calculations, in which the volume and mole numbers, which are the natural variables for pressure explicit equations of state (EoS) are treated as independent variables; in this case pressure equality is an additional equation for each phase and there is no need to solve the EoS for volume. The problem is a bound- and linear inequality-constrained optimization problem; the objective function is a formal Gibbs free energy. A major disadvantage is that, unlike in conventional calculations, the successive substitution method cannot be used, thus robust modified Newton iterations are required. Using the block structure of the Hessian matrix, a link is established between conventional and volume-based Newton iterations and it is shown that the latter are inherently slower than the former due to a lower implicitness level in the Hessian. A modified Cholesky factorization (to ensure a descent direction) and a two-stage line search procedure (ensuring that iterates remain in the feasible domain and the objective function is decreasing between two iterations) are used in Newton iterations. Numerical experiments carried out on several test mixtures show that two-phase volume-based flashes initialized from the ideal equilibrium constants pose systematic convergence problems near phase boundaries, while the algorithm is rapid and robust when initial guesses from stability testing are used. The proposed algorithm is not dependent on the thermodynamic model and any pressure explicit EoS can be used, provided the required partial derivatives are available.

GPU-accelerated locally injective shape deformation

We present a highly efficient planar meshless shape deformation algorithm. Our method is based on an unconstrained minimization of isometric energies, and is guaranteed to produce C ∞ locally injective maps by operating within a reduced dimensional subspace of harmonic maps. We extend the harmonic subspace of [Chen and Weber 2015] to support multiply-connected domains, and further provide a generalization of the bounded distortion theorem that appeared in that paper. Our harmonic map, as well as the gradient and the Hessian of our isometric energies possess closed-form expressions. A key result is a simple-and-fast analytic modification of the Hessian of the energy such that it is positive definite, which is crucial for the successful operation of a Newton solver. The method is straightforward to implement and is specifically designed to harness the processing power of modern graphics hardware. Our modified Newton iterations are shown to be extremely effective, leading to fast convergence after a handful of iterations, while each iteration is fast due to a combination of a number of factors, such as the smoothness and the low dimensionality of the subspace, the closed-form expressions for the differentials, and the avoidance of expensive strategies to ensure positive definiteness. The entire pipeline is carried out on the GPU, leading to deformations that are significantly faster to compute than the state-of-the-art.

Volume-based phase stability testing at pressure and temperature specifications

Conventional phase equilibrium calculations at temperature and pressure specifications require the resolution of the equation of state (EoS). In volume-based calculations, the EoS must not be solved for volume, which is a primary variable. The tangent plane distance (TPD) function can be expressed in terms of mole numbers and volume or of component molar densities. The stationary points of the TPD function, as well as the location of the stability test limit locus (STLL) are different for the two formulations. A modified TPD function in terms of mole numbers and volume is proposed here. Using the block structure of the Hessian matrix, it is shown that Newton iterations in volume-based stability are inherently slower than those in the conventional PT stability, since the Hessian matrix is evaluated at a lower implicitness level. The minimization of several TPD functions (in volume-based and conventional stability testing) is analyzed using various sets of independent variables and scaling procedures. A modified Cholesky factorization and a two-stage line search procedure ensure a sequence of decreasing TPD functions in all cases. The proposed methods are tested for several mixtures, with emphasis on the vicinities of singularities (STLL and spinodal). With proper scaling, the modified Newton iterations are robust and converge very fast for most conditions and reasonably fast for very difficult conditions. The proposed algorithm is not model-dependent; any pressure explicit EoS can be used, provided the required partial derivatives are available.

A family of modified Newton iteration method for solving nonlinear algebraic equations

In this study, a modified Newton iteration version for solving nonlinear algebraic equations is formulated using a correction function derived from convergence order condition of iteration. If the second order of convergence is selected, we get a family of the modified Newton iteration method. Several forms of the correction function are proposed in checking the effectiveness and accuracy of the present iteration method. For illustration, approximate solutions of four examples of nonlinear algebraic equations are obtained and then compared with those obtained from the classical Newton iteration method.

Reversibility Iteration Method for Understanding Reaction Networks and for Solving Microkinetics in Heterogeneous Catalysis

Solving microkinetics of catalytic systems, which bridges microscopic processes and macroscopic reaction rates, is currently vital for understanding catalysis in silico. However, traditional microkinetic solvers possess several drawbacks that make the process slow and unreliable for complicated catalytic systems. In this paper, a new approach, the so-called reversibility iteration method (RIM), is developed to solve microkinetics for catalytic systems. Using the chemical potential notation we previously proposed to simplify the kinetic framework, the catalytic systems can be analytically illustrated to be logically equivalent to the electric circuit, and the reaction rate and coverage can be calculated by updating the values of reversibilities. Compared to the traditional modified Newton iteration method (NIM), our method is not sensitive to the initial guess of the solution and typically requires fewer iteration steps. Moreover, the method does not require arbitrary-precision arithmetic and has a higher ...

Blind Image Separation Based on an Optimized Fast Fixed Point Algorithm

An optimized fast fixed point algorithm based on modified Newton iteration method has been proposed. With good performance ofthe blind image separation, the optimized algorithm can improve the convergence speed greatly.We proposed a new adaptive enhancement parameter to enhance the separated images effectively. The experimental results demonstrate that the new algorithm is superior.

Efficient mesh optimization schemes based on Optimal Delaunay Triangulations

In this paper, several mesh optimization schemes based on Optimal Delaunay Triangulations are developed. High-quality meshes are obtained by minimizing the interpolation error in the weighted L 1 norm. Our schemes are divided into classes of local and global schemes. For local schemes, several old and new schemes, known as mesh smoothing, are derived from our approach. For global schemes, a graph Laplacian is used in a modified Newton iteration to speed up the local approach. Our work provides a mathematical foundation for a number of mesh smoothing schemes often used in practice, and leads to a new global mesh optimization scheme. Numerical experiments indicate that our methods can produce well-shaped triangulations in a robust and efficient way.

Numerical analysis of soft pH‐sensitive hydrogel: Effect of multivalent ionic compositions

AbstractInfluence of the ionic compositions of bathing solution with multivalences is investigated on the responsive behavior of pH‐sensitive hydrogel to environmental change in solution pH. It is based on a chemo‐electro‐mechanical formulation, previously termed the multieffect‐coupling pH‐stimulus (MECpH) model, which is improved in this article by incorporating the finite deformation theory into the mechanical equilibrium. The improved MECpH model consists of the coupled nonlinear partial differential equations, which are solved via a meshless numerical technique called the Hermite‐cloud method with the modified Newton iteration. After the MECpH model is examined by comparison between the computational results and experimental data available in literature, several case studies are numerically carried out to discuss the effects of the multivalent ionic compositions of surrounding solution on the variation of the deformation of the pH‐stimulus‐responsive hydrogel and the distributions of the diffusive ion concentrations and electric potential, when the hydrogel is placed in the buffered solutions. POLYM. ENG. SCI., 2010. © 2009 Society of Plastics Engineers