Newton Scheme Research Articles

A two-point Newton-like method of optimal fourth order convergence for systems of nonlinear equations

A two-step Newton-like method is proposed to efficiently solve the systems of nonlinear equations. Extending Newton scheme to a next step as weighted-Newton iteration, the proposed iteration scheme shows optimal fourth order of convergence. The primary objective in formulating the method is to keep the computational efficiency as high as possible. In this context, the efficiency analysis is thoroughly examined using a systematic approach, wherein the efficiency index of the new method is compared with those of existing methods of comparable complexity. Numerical experimentation is performed to investigate the computational efficacy of the developed method. Results indicate higher efficiency and numerical precision in comparison to the existing counterparts.

Breathing discrete nonlinear Schrödinger vortices

Breathing discrete vortices are obtained as numerically exact and generally quasiperiodic, localized solutions to the discrete nonlinear Schrödinger equation with cubic (Kerr) on-site nonlinearity, on a two-dimensional square lattice with nearest-neighbor couplings. We identify and analyze three different types of solutions characterized by circulating currents and time-periodically oscillating intensity distributions, two of which have been discussed in earlier works while the third being, to our knowledge, presented here for the first time. (i) A vortex-breather, constructed from the anticontinuous limit as a superposition of a single-site breather and a discrete vortex surrounding it, where the breather and vortex are oscillating at different frequencies. (ii) A charge-flipping vortex, constructed from an anticontinuous solution with an even number of sites on a closed loop, with alternating sites oscillating at different frequencies. (iii) A breathing vortex, constructed by continuation of a non-resonating linear internal eigenmode of a stationary discrete vortex. We illustrate by examples, using numerical Floquet analysis for solutions obtained from a Newton scheme, that linearly stable solutions exist from all three categories, at sufficiently strong discreteness.

Modeling and simulation of chemo-elasto-plastically coupled battery active particles

AbstractAs an anode material for lithium-ion batteries, amorphous silicon offers a significantly higher energy density than the graphite anodes currently used. Alloying reactions of lithium and silicon, however, induce large deformation and lead to volume changes up to 300%. We formulate a thermodynamically consistent continuum model for the chemo-elasto-plastic diffusion-deformation behavior of amorphous silicon and it’s alloy with lithium based on finite deformations. In this paper, two plasticity theories, i.e. a rate-independent theory with linear isotropic hardening and a rate-dependent one, are formulated to allow the evolution of plastic deformations and reduce occurring stresses. Using modern numerical techniques, such as higher order finite element methods as well as efficient space and time adaptive solution algorithms, the diffusion-deformation behavior resulting from both theories is compared. In order to further increase the computational efficiency, an automatic differentiation scheme is used, allowing for a significant speed up in assembling time as compared to an algorithmic linearization for the global finite element Newton scheme. Both plastic approaches lead to a more heterogeneous concentration distribution and to a change to tensile tangential Cauchy stresses at the particle surface at the end of one charging cycle. Different parameter studies show how an amplification of the plastic deformation is affected. Interestingly, an elliptical particle shows only plastic deformation at the smaller half axis. With the demonstrated efficiency of the applied methods, results after five charging cycles are also discussed and can provide indications for the performance of lithium-ion batteries in long term use.

A comparative study of sensitivity computations in ESDIRK-based optimal control problems

This paper compares the impact of iterated and direct approaches to sensitivity computation in fixed step-size explicit singly diagonally implicit Runge–Kutta (ESDIRK) methods when applied to optimal control problems (OCPs). We strictly use the principle of internal numerical differentiation (IND) for the iterated approach, i.e., reusing iteration matrix factorizations, the number of Newton-type iterations, and Newton iterates, to compute the sensitivities. The direct method computes the sensitivities without using the Newton schemes. We compare the impact of these sensitivity computations in OCPs for the quadruple tank system (QTS). We discretize the OCPs using multiple shooting and solve these with a sequential quadratic programming (SQP) solver. We benchmark the iterated and direct approaches against a base case. This base case applies the ESDIRK methods with exact Newton schemes and a direct approach for sensitivity computations. In these OCPs, we vary the number of integration steps between control intervals and evaluate the performance based on the number of SQP and QPs iterations, KKT violations, function evaluations, Jacobian updates, and iteration matrix factorizations. We also provide examples using the continuous-stirred tank reactor (CSTR), and the IPOPT algorithm instead of the SQP. For OCPs solved using SQP, the QTS results show the direct method converges only once, while the iterated approach and base case converges in all situations. Similar results are seen with the CSTR. Using IPOPT, both the iterated approach and base case converge in all cases. In contrast, the direct method only converges in all cases regarding the CSTR.

A Gauss-Newton method for iterative optimization of memory kernels for generalized Langevin thermostats in coarse-grained molecular dynamics simulations.

In molecular dynamics simulations, dynamically consistent coarse-grained (CG) models commonly use stochastic thermostats to model friction and fluctuations that are lost in a CG description. While Markovian, i.e., time-local, formulations of such thermostats allow for an accurate representation of diffusivities/long-time dynamics, a correct description of the dynamics on all time scales generally requires non-Markovian, i.e., non-time-local, thermostats. These thermostats typically take the form of a Generalized Langevin Equation (GLE) determined by a memory kernel. In this work, we use a Markovian embedded formulation of a position-independent GLE thermostat acting independently on each CG degree of freedom. Extracting the memory kernel of this CG model from atomistic reference data requires several approximations. Therefore, this task is best understood as an inverse problem. While our recently proposed approximate Newton scheme allows for the iterative optimization of memory kernels (IOMK), Markovian embedding remained potentially error-prone and computationally expensive. In this work, we present an IOMK-Gauss-Newton scheme (IOMK-GN) based on IOMK that allows for the direct parameterization of a Markovian embedded model.

Mixing neural networks, continuation and symbolic computation to solve parametric systems of non linear equations

We consider a square non linear parametric equations system F(P,X) = 0 which is constituted of n non differential equations in the n unknowns {x1,…,xn} that are the components of X while P={p1,…,pm} is a set of m parameters that play a role in the definition of the equations F. We assume that P is restricted to lie in a bounded region and we are interested in developing a solver for obtaining all real solutions exactly (a notion that is defined in the paper) for any parameter values within the bounded region. The starting point of the proposed approach is that we assume that a numerical methods has allowed us to determine the real solutions (but not necessarily all of them) for a very limited number of fixed P called the initial solution set. Starting from this set we show that we can create multiple pairs (parameters, solution) and that these pairs may be structured into coherent learning sets that will be used to train multi-layer perceptrons (MLP). The training process is specific: although it still uses a decrease of a loss function its main objective is to maximize the success rate i.e. the number of occurrences, expressed in percentage of number of samples of the training set, for which the Newton scheme, initialized with the MLP prediction, converges toward the expected solution. We then show that for a sufficiently large number of MLPs we may obtain a 100% success rate for all learning sets. The solver is obtained by running a set of local solvers each of which is based on a specific MLP whose prediction may lead to an exact solution of the system. This solver is tested on verification sets i.e. set of samples constituted of parameter values (all different from the samples in the learning set) and all the solutions of the corresponding system. We show that these sets may be automatically generated and that they may also be used in a self-learning process for improving the performance of the solver established from the initial solution set.This approach is illustrated on two engineering problems in robotics and chemistry and it is shown that the solver provides all solutions for any instance of P with a high probability although we cannot guarantee it. The time required to design the final solver is large but the solving time is extremely low so that this approach should be used when the system has to be solved for a sufficient number of occurrences of the P. Furthermore we will show that the computation time required for establishing the solver may be drastically reduced by using a distributed implementation.

Numerically efficient and robust interior-point algorithm for finite strain rate-independent crystal plasticity

The challenges of classic rate-independent crystal plasticity models regarding the determination of active slip systems and computation of solutions due to linear dependent slip systems are still a field of active research. In that regard, interior-point methods are promising approaches because a logarithmic barrier term naturally avoids the combinatorial active set search, and these methods are applicable even when the gradient of inequality constraints is linearly dependent. In this paper, a new interior-point algorithm suitable for rate-independent crystal plasticity at finite strains is developed. For that purpose, the postulate of maximum dissipation is recast as a constrained optimization problem whose optimality conditions are derived by utilizing interior-point approaches. This optimization problem is discretized in time by means of an exponential time integration of the flow rule, and the time-discrete equations are solved by a Newton scheme. The focus of this paper is the efficiency and robustness of the implementation of this approach. In particular, the effect of different line search procedures, favorable residual forms, preconditioning methods, barrier parameter updates and the consistent algorithmic tangent modulus are discussed and further elaborated. In addition, aspects regarding uniqueness and well-posedness of the algorithm are analyzed in a geometrically linearized setting. Efficiency and robustness of the developed algorithm are shown by numerical examples of an FCC crystal including finite element simulations without stabilizing hardening or rate-dependent effects. Furthermore, the results are compared to the well-established augmented Lagrangian formulation.

Application of Newton iteration to the numerical solution of variable-density groundwater flow in saturated-unsaturated coastal zones

When addressing the question of variable saturation and density groundwater flow in coastal zones, the highly nonlinear system of coupled water-salt equations may deserve more attention. The classical Picard scheme is associated with slow calculation speeds and low precision, which hardly meet the actual needs of users. Here, we developed a new numerical solution for coastal groundwater flow issues based on the Newton scheme and compared the advantages and disadvantages of different numerical methods by analyzing the cases of seawater intrusion. The simulation results indicated that the variable-density effect significantly extends the computation time of the model, but the Newton scheme still has the advantages of computational speed and better convergence compared with the Picard scheme, especially in conditions involving high-frequency and large-amplitude tidal fluctuations, steep aquifer slopes, and a coarse grid. Furthermore, the Newton-Picard method, based on the Newton and Picard schemes, improves the robustness of the Newton solution and optimizes the convergence of the Picard solution. This study has revealed the computational characteristics of the Newton scheme in addressing the issues of coastal variable saturation and density groundwater flow, providing new ideas and insights for numerical solutions to coastal groundwater flow problems.

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, our experiments showed that the relaxation by rank-one convexification delivering an approximation to the quasiconvex envelope prevents mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

Generalized multistep Steffensen iterative method. Solving the model of a photomultiplier device

It is well known that the Steffensen-type methods approximate the derivative appearing in Newton's scheme by means of the first-order divided difference operator. The generalized multistep Steffensen iterative method consists of composing the method with itself m times. Specifically, the divided difference is held constant for every m steps before it is updated. In this work, we introduce a modification to this method, in order to accelerate the convergence order. In the proposed, scheme we compute the divided differences in first and second step and use the divided difference from the second step in the following m−1 steps. We perform an exhaustive study of the computational efficiency of this scheme and also introduce memory to this method to speed up convergence without performing new functional evaluations. Finally, some numerical examples are studied to verify the usefulness of these algorithms.

Topological edge breathers in a nonlinear Su-Schrieffer-Heeger lattice

We show the existence of breathing edge modes in the Su-Schrieffer-Heeger model with cubic (Kerr) on-site nonlinearity, bifurcating from stationary edge solitons with propagation constant inside the topological gap of the linear model. These edge breathers are exact solutions to the nonlinear equations of motion, with time-periodic intensity oscillations and tails exponentially decaying from the edge. They bifurcate from two localized internal eigenmodes of the stationary edge soliton, having eigenfrequencies inside the topological gap and all higher harmonics above the linear spectrum. Numerical Floquet analysis for solutions obtained from a Newton scheme shows that edge breathers may be linearly stable even in regimes of large-amplitude oscillations, mainly manifested as time-periodic power exchange between the edge site and its next-nearest neighbor.

Certified Newton schemes for the evaluation of low-genus theta functions

Theta functions and theta constants in low genus, especially genus 1 and 2, can be evaluated at any given point in quasi-linear time in the required precision using Newton schemes based on Borchardt sequences. Our goal in this paper is to provide the necessary tools to implement these algorithms in a provably correct way. In particular, we obtain uniform and explicit convergence results in the case of theta constants in genus 1 and 2, and theta functions in genus 1: the associated Newton schemes will converge starting from approximations to N bits of precision for N = 60, 300, and 1600 respectively, for all suitably reduced arguments. We also describe a uniform quasi-linear time algorithm to evaluate genus 2 theta constants on the Siegel fundamental domain. Our main tool is a detailed study of Borchardt means as multivariate analytic functions.

A Trained Iterative Shrinkage Approach Based on Born Iterative Method for Electromagnetic Imaging

A trained-based Born iterative method (TBIM) is developed for electromagnetic imaging (EMI) applications. The proposed TBIM consists of a nested loop; the outer loop executes TBIM iteration steps, while the inner loop executes a trained iterative shrinkage thresholding algorithm (TISTA). The applied TISTA runs linear Landweber iterations implemented with a trained regularization network designed based on U-net architecture. A normalization process was imposed in TISTA that made TISTA training applicable within the proposed TBIM. The iterative utilization of the regularization network in TISTA is a bottleneck that demands high memory allocation through the training process. Therefore TISTA within each TBIM step was trained separately. The TISTA regularization network in each TBIM step was initialized using the weights from the previous TBIM step. The above approach achieved high-quality image restoration after running few TBIM steps while maintained low memory allocation through the training process. The proposed framework can be extended to Newton or quasi-Newton schemes, where within each Newton iteration, a linear ill-posed problem is optimized that differs from one example to another. The numerical results illustrated in this work show the superiority of the proposed TBIM compared to the conventional sparse-based Born iterative method (SBIM).

Finite Element Approximation of Large-Scale Isometric Deformations of Parametrized Surfaces

In this paper, the numerical approximation of isometric deformations of thin elastic shells is discussed. To this end, for a thin shell represented by a parametrized surface, it is shown how to transform the stored elastic energy for an isometric deformation such that the highest order term is quadratic. For this reformulated model, existence of optimal isometric deformations is shown. A finite element approximation is obtained using the Discrete Kirchhoff Triangle (DKT) approach and the convergence of discrete minimizers to a continuous minimizer is demonstrated. In that respect, this paper generalizes the results by Bartels for the approximation of bending isometries of plates. A Newton scheme is derived to numerically simulate large bending isometries of shells. The proven convergence properties are experimentally verified and characteristics of isometric deformations are discussed.

Direct Evaluation of the Sensitivity of the Nonlinear Critical Point to Geometrical Imperfections

AbstractA generalized path‐following method is proposed for the direct evaluation of the critical point sensitivity to geometrical imperfections expressed as combination of given shapes. After a finite element discretization, the critical point is defined by a system of nonlinear algebraic equations imposing equilibrium and critical condition according to the null vector method. An arc‐length equation is added to follow a path of critical points in the imperfection space. The solution of the extended system by a Newton scheme requires a Jacobian involving also the derivatives of the critical condition with respect to the FE DOFs and the derivatives of all the problem equations with respect to the imperfection parameters. An analytical and efficient computation of the Jacobian is achieved by a solid‐shell model and a strain‐based model of the imperfection. In addition, a mixed stress‐displacement iterative scheme is devised for a highly efficient and robust solution. Finally, the derivatives of the critical point with respect to the imperfection parameters are also obtained analytically and can be used to generate a gradient‐based critical path for a quick search of the worst‐case imperfection. A compressed cold‐formed lipped channel beam is analyzed as a case study.

Gradient-constrained model parametrization in 3-D compact full waveform inversion

SUMMARY Seismic full waveform inversion (FWI) can produce high-resolution subsurface models by using the complete information of the observed data, but its computational cost can be prohibitively large, particularly for realistically sized 3-D problems. Due to its relatively fast convergence rate, it would be beneficial when Gauss–Newton algorithms could be employed for such problems, but the approximate Hessian matrix required for Gauss–Newton schemes would be too large to be kept in computer memory. Therefore, compact FWI (CFWI) was introduced, with which the number of inversion model parameters can be reduced substantially, and thus the amenable properties of Gauss–Newton inversion schemes can be exploited. Here, we extend the CFWI technology to 3-D problems. Since the spatial coverage of sources and receivers is generally sparser in 3-D (compared with 2-D problems), the total number of model parameters can become very large, and adequate model parametrization is particularly important for 3-D problems. Furthermore, we introduce gradient constrained CFWI (GC-CFWI). This is a novel development that allows the number of model parameters to be further reduced significantly. CFWI employs hierarchical model parametrizations that can be, for example, based on spatial Fourier transforms or wavelet transforms. Only those parameters of such a hierarchical parametrization are retained that exhibit a sufficiently high formal resolution. With GC-CFWI, it is further checked which of these parameters are expected to be changed significantly during a single CFWI iteration. Only parameters with a potentially significant adjustment are retained in the inversion parameter space. We have performed numerical experiments to analyse the performance of CFWI and GC-CFWI for 3-D acoustic FWI problems. For that purpose, we have considered a crosshole geometry including four boreholes and a surface deployment of sources and receivers. As parametrizations, we have considered the Fourier-based Hartley transform and the Haar wavelet transform. For both set-ups, the number of inversion model parameters could be reduced to about 20 per cent for the crosshole model and to about 10 per cent for the surface-based acquisition using CFWI. With GC-CFWI, a further reduction of about 50 per cent for both experimental set-ups could be achieved. The different results for the crosshole and surface-based set-ups indicate that an optimal model parametrization is tightly coupled to the experimental layout.

Phase field modeling of brittle fracture in large-deformation solid shells with the efficient quasi-Newton solution and global–local approach

To efficiently predict the crack propagation in thin-walled structures, a global–local approach for phase field modeling using large-deformation solid shell finite elements considering the enhanced assumed strain (EAS) and the assumed natural strain (ANS) methods for the alleviation of locking effects is developed in this work. Aiming at tackling the poor convergence performance of standard Newton schemes, a quasi-Newton (QN) scheme is proposed for the solution of coupled governing equations stemming from the enhanced assumed strain shell formulation in a monolithic manner. The excellent convergence performance of this QN monolithic scheme for the multi-field shell formulation is demonstrated through several paradigmatic boundary value problems, including single edge notched tension and shear, fracture of cylindrical structure under mixed loading and fatigue induced crack growth. Compared with the popular alternating minimization (AM) or staggered solution scheme, it is also found that the QN monolithic solution scheme for the phase field modeling using enhanced strain shell formulation is very efficient without the loss of robustness, and significant computational gains are observed in all the numerical examples. In addition, to further reduce the computational cost in fracture modeling of large-scale thin-walled structures, a specific global–local phase field approach for solid shell elements in the 3D setting is proposed, in which the full displacement-phase field problem is considered at the local level, while addressing only the elastic problem at the global level. Its capability is demonstrated by the modeling of a cylindrical structure subjected to both static and fatigue cyclic loading conditions, which can be appealing to industrial applications.

A Shape Newton Scheme for Deforming Shells with Application to Capillary Bridges

We present a second order numerical scheme to compute capillary bridges between arbitrary solids by minimizing the total energy of all interfaces. From a theoretical point of view, this approach can be interpreted as the computation of generalized minimal surfaces using a Newton-scheme utilizing the shape Hessian. In particular, we give an explicit representation of the shape Hessian for functionals on shells involving the normal vector without reverting back to a volume formulation or approximating curvature. From an algorithmic perspective, we combine a resolved interface via a triangulated surface for the liquid with a level set description for the constraints stemming from the arbitrary geometry. The actual shape of the capillary bridge is then computed via finite elements provided by the FEniCS environment, minimizing the shape derivative of the total interface energy.

Double-diffusive natural convection energy transfer in magnetically influenced Casson fluid flow in trapezoidal enclosure with fillets

The prime motive of this disquisition is to deal with mathematical analysis of natural convection energy transport driven by combined buoyancy effects of thermal and solutal diffusion in a trapezoidal enclosure. Casson fluid rheological constitutive model depicting attributes of viscoelastic liquids is envisioned. The influence of the inclined magnetic field governed by Lorentz field law is also considered. To raise the essence of the presently used computational and physical domain, an innovative structural design called fillet is capitalized at the edges of the cavity. A numerical solution of the leading formulation is sought through Galerkin finite element discretization. Momentum, temperature, and concentration equations are interpolated by quadratic polynomials, whereas pressure distribution is approximated by a linear interpolating function. Domain discretized version is evaluated in view of triangular and rectangular elements. Newton's scheme is employed to resolve the non-linearly discretized system and a matrix factorization-based non-linear solver renowned as PARADISO is used. Validation of results is ascertained by forming an agreement with existing studies. Streamlines, isothermal and iso concentration contours patterns are portrayed to evaluate variation inflow distributions. Kinetic energy, local heat, and mass fluxes are also divulged in graphical and tabular formats.

Computation of Troesch Problem by a Modified Newton’s Approach

We introduce an improved Newton scheme to solve the nonlinear Troesch problem; which is a nonlinear parameter-sensitive differential equation applied to model plasma confinement in a column.. A predictor-corrector scheme based on a modified backward Euler method was adopted for the Newton’s update. The hyperbolic sine component of the governing equation was converted to its logarithmic analog in order to handle high gradients and discontinuities. This was also found to be useful when the initial position values are not very close to the projected equilibrium. The proposed algorithm is straightforward and offers a relatively high degree of accuracy for a remarkably wide range of values of the sensitivity parameter. The trade-off is a slightly more computation time than the classical Newton’s approach.