Block Newton Method Research Articles

Application of efficient algorithm based on block Newton method to elastoplastic problems with nonlinear kinematic hardening

PurposeThe purpose of this study is to present the effectiveness and robustness of a numerical algorithm based on the block Newton method for the nonlinear kinematic hardening rules adopted in modeling ductile materials.Design/methodology/approachElastoplastic problems can be defined as a coupled problem of the equilibrium equation for the overall structure and the yield equations for the stress state at every material point. When applying the Newton method to the coupled residual equations, the displacement field and the internal variables, which represent the plastic deformation, are updated simultaneously.FindingsThe presented numerical scheme leads to an explicit form of the hardening behavior, which includes the evolution of the equivalent plastic strain and the back stress, with the internal variables. The features of the present approach allow the displacement field and the hardening behavior to be updated straightforwardly. Thus, the scheme does not have any local iterative calculations and enables us to simultaneously decrease the residuals in the coupled boundary value problems.Originality/valueA pseudo-stress for the local residual and an algebraically derived consistent tangent are applied to elastic-plastic boundary value problems with nonlinear kinematic hardening. The numerical procedure incorporating the block Newton method ensures a quadratic rate of asymptotic convergence of a computationally efficient solution scheme. The proposed algorithm provides an efficient and robust computation in the elastoplastic analysis of ductile materials. Numerical examples under elaborate loading conditions demonstrate the effectiveness and robustness of the numerical scheme implemented in the finite element analysis.

Elastoplastic damage analysis with algebraic derivation of consistent tangent by block Newton method

Elastoplastic damage analysis with algebraic derivation of consistent tangent by block Newton method

Elastoplastic analysis of shells without any local iterative calculations by block Newton method

Elastoplastic analysis of shells without any local iterative calculations by block Newton method

Finite element analysis of hyperelastic shells without any local iterative calculations by block Newton method

Finite element analysis of hyperelastic shells without any local iterative calculations by block Newton method

Simultaneously iterative procedure based on block Newton method for elastoplastic problems

AbstractIn this article, the authors formulate elastoplastic problems as a coupled problem of the equilibrium equation and the yield condition at each material point, and develop a numerical procedure based on the block Newton method to solve the overall structure using the finite element discretization. For the integration of stress, the backward difference scheme is employed. In the conventional return mapping algorithm, the algorithmic tangent moduli are derived analytically so that it is consistent with local iterative calculation to determine internal variables. On the other hand, in the proposed block Newton method, the tangent moduli can be obtained algebraically by eliminating the internal variables, and the internal variables are also updated algebraically without local iterative calculation. The residual of the yield condition is incorporated into the linearized equilibrium equation. The proposed approach enables the errors of the equilibrium equation and the yield condition to decrease simultaneously. Some numerical examples show the validity and effectiveness of the proposed approach by comparing the results of the proposed approach with those of the conventional return mapping algorithm.

Application of block Newton method to elastoplastic problems under plane stress state

Application of block Newton method to elastoplastic problems under plane stress state

A comparison of three algorithms applied in thermal-hydraulics and neutronics codes coupling for lbe-cooled fast reactor

In this research, the Operator-Splitting semi-implicit (OSSI) method, fixed-point implicit (FPI) method and Approximate Block Newton (ABN) method have been applied to the coupling of thermal-hydraulics code COBRA and neutronics code SKETCH-N. In the OSSI method, there’s only one data exchange within a time step. In the FPI coupling, an iteration convergence has been added to the OSSI coupling framework to ensure the coupling results converged. And in the ABN method, a variant of Jacobian-free Newton Krylov method has been applied, which removes the Jacobian matrix construction and storage. Three temporal schemes have been validated through a PWR control rod bank ejection benchmark to show their applicability. Moreover, a LBE-cooled fast reactor control rods withdrawal accident applying two time steps has been simulated. And the results indicate that the ABN method applied in the mild transient generally outperforms the other two methods in both the convergence speed and the computational cost.

Block Preconditioning Matrices for the Newton Method to Compute the Dominant λ-Modes Associated with the Neutron Diffusion Equation

In nuclear engineering, the λ -modes associated with the neutron diffusion equation are applied to study the criticality of reactors and to develop modal methods for the transient analysis. The differential eigenvalue problem that needs to be solved is discretized using a finite element method, obtaining a generalized algebraic eigenvalue problem whose associated matrices are large and sparse. Then, efficient methods are needed to solve this problem. In this work, we used a block generalized Newton method implemented with a matrix-free technique that does not store all matrices explicitly. This technique reduces mainly the computational memory and, in some cases, when the assembly of the matrices is an expensive task, the computational time. The main problem is that the block Newton method requires solving linear systems, which need to be preconditioned. The construction of preconditioners such as ILU or ICC based on a fully-assembled matrix is not efficient in terms of the memory with the matrix-free implementation. As an alternative, several block preconditioners are studied that only save a few block matrices in comparison with the full problem. To test the performance of these methodologies, different reactor problems are studied.

Block hybrid multilevel method to compute the dominant [formula omitted]-modes of the neutron diffusion equation

The dominant λ-modes associated with a nuclear reactor configuration describe the neutron steady-state distribution and its criticality. Furthermore, they are useful to develop modal methods to study reactor instabilities. Different eigenvalues solvers have been successfully used to obtain such modes, most of them are implemented reducing the original generalized eigenvalue problem to an ordinary one. Thus, it is necessary to solve many linear systems making these methods not very efficient, especially for large problems. In this work, the original generalized eigenvalue problem is considered and two block iterative methods to solve it are studied: the block inverse-free preconditioned Arnoldi method and the modified block Newton method. All of these iterative solvers are initialized using a block multilevel technique. A hybrid multilevel method is also proposed based on the combination of the methods proposed. Two benchmark problems are studied illustrating the convergence and the competitiveness of the methods proposed. A comparison with the Krylov-Schur method and the Generalized Davidson is also included.

Spatial modes for the neutron diffusion equation and their computation

Different spatial modes can be defined for the neutron diffusion equation such as the λ,α and γ-modes. These modes have been successfully used for the analysis of nuclear reactor characteristics. In this work, these modes are studied using a high order finite element method to discretize the equations and also different methods to solve the resulting algebraic eigenproblems, are compared. Particularly, Krylov subspace methods and block-Newton methods have been studied. The performance of these methods has been tested in several 3D benchmark problems: a homogeneous reactor and several configurations of NEACRP reactor.

A Newton-based Jacobian-free approach for neutronic-Monte Carlo/thermal-hydraulic static coupled analysis

In the field of nuclear reactor analysis, multi-physics calculations that account for the bonded nature of the neutronic and thermal-hydraulic phenomena are of major importance for both reactor safety and design. So far in the context of Monte-Carlo neutronic analysis a kind of “serial” algorithm has been mainly used for coupling with thermal-hydraulics. The main motivation of this work is the interest for an algorithm that could maintain the distinct treatment of the involved fields within a tight coupling context that could be translated into higher convergence rates and more stable behaviour. This work investigates the possibility of replacing the usually used “serial” iteration with an approximate Newton algorithm. The selected algorithm, called Approximate Block Newton, is actually a version of the Jacobian-free Newton Krylov method suitably modified for coupling mono-disciplinary solvers. Within this Newton scheme the linearised system is solved with a Krylov solver in order to avoid the creation of the Jacobian matrix. A coupling algorithm between Monte-Carlo neutronics and thermal-hydraulics based on the above-mentioned methodology is developed and its performance is analysed. More specifically, OpenMC, a Monte-Carlo neutronics code and COBRA-EN, a thermal-hydraulics code for sub-channel and core analysis, are merged in a coupling scheme using the Approximate Block Newton method aiming to examine the performance of this scheme and compare with that of the “traditional” serial iterative scheme. First results show a clear improvement of the convergence especially in problems where significant coolant density gradients appear.

Adaptive robust time-of-arrival source localization algorithm based on variable step size weighted block Newton method

We propose a line-of-sight (LOS)/non-line-of-sight (NLOS) mixture source localization algorithms that utilize the weighted block Newton (WBN) and variable step size WBN (VSSWBN) method, in which the weighting matrix is determined in the form of the inverse of the squared error or as an exponential function with a negative exponent. The proposed WBN and VSSWBN algorithms converge in two iterations; thus, the required number of extra samples in the transient period is negligible. Also, we perform an analysis of the mean square error (MSE) of the weighted block Newton method. To verify the superiority of the proposed methods, the MSE performances are compared via extensive simulation.

Multilevel method to compute the lambda modes of the neutron diffusion equation

Abstract Determination of the reactor kinetic characteristics is very important for the design and development of a new reactor system. In this sense, the computation of lambda modes associated to a nuclear power reactor has interest since these modes can be used to analyze the reactor criticality and to develop modal methods to analyze transient situations in the reactor. In this paper, the lambda problem has been discretized using a high order finite element method to obtain a generalized algebraic eigenvalue problem. A multilevel method is proposed to solve this generalized eigenvalue problem combining a hierarchy of meshes with a Modified Block Newton method. The Krylov-Schur method is used to compare the efficiency of the multilevel method solving several benchmark problems.

A class of two-level implicit unconditionally stable methods for a fourth order parabolic equation

In this paper, we discuss two-level implicit methods for the solution of a special type of fourth order parabolic partial differential equation of the form uxxxx-2uxxt+utt=f(x,t,u), 0<x<1, t>0 subject to appropriate initial and Dirichlet boundary conditions by converting the original problem to a coupled system of two second order parabolic equations. We use only three spatial grid points and it is not required to discretize the boundary conditions. The proposed Crank-Nicolson type scheme is second order accurate in both the temporal and spatial dimensions while the compact Crandall's type scheme is second order accurate in temporal and fourth order accurate in spatial dimension. The methods do not require any fictitious nodes outside the solution domain for handling the boundary conditions. For a fixed mesh ratio parameter (∆t/∆x2), the proposed Crandall's type method behaves like a fourth order method in space. Using matrix stability analysis, the proposed methods are shown to be unconditionally stable. The resulting implicit difference formulas gives block tri-diaginal matrix structure which is solved efficiently using block Gauss-Seidel method or block Newton method depending on linear or nonlinear behaviour of the equations. The methods compute the numerical value of u and time-dependent Laplacian uxx−ut, simultaneously. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed methods.

Block Newton Method and Block Rayleigh Quotient Iteration for computing invariant subspaces of general complex matrices

We consider the Block Newton Method and a modification of it, the Block Rayleigh Quotient Iteration, for approximating a simple p-dimensional invariant subspace X=im(X) and the corresponding eigenvalues collected in the projection L=(XHX)−1XHAX of an arbitrary complex matrix A∈Cn×n. Both methods generate a sequence of bases {Uk} such that the subspaces im(Uk) approximate the target subspace X. The Block Newton Method is Newton's method applied to the extended system AX−XL=0, WHX=I with respect to (X,L) where W is a normalization matrix. We give a direct convergence analysis which, in contrast to the approach by Beyn/Kless/Thümmler [2001], explicitly exploits the product form of the second derivative and, therefore, leads to new and essentially sharper error recursions for each of the sequences {Uk} and {Θk} where Uk and Θk are the k-th X and L iterates, resp. Then, following Lösche/Schwetlick/Timmermann [1998] where the case of real symmetric A is considered, we modify the Block Newton Method as follows: In step k we set W=Uk and use only the X-update from the Newton method to improve Uk and then, with the improved basis, perform a Rayleigh–Schur process and take the orthonormal basis and the generalized Rayleigh quotient matrix obtained from it as new iterates. So the method can be considered as a Block Rayleigh Quotient Iteration. We show that for a sufficiently good initial approximation U0 the method converges in that the sequence {sin⁡ξk} with ξk=∡(im(Uk),X) converges Q-quadratically to zero provided that the invariant subspace X is simple, i.e., that the eigenvalues corresponding to X which are the eigenvalues of the projection L=(XHX)−1(XHAX) of A onto X are separated from those corresponding to X⊥ which are given by λ(A)−λ(L). The convergence results are directly proven – without exploiting standard Newton convergence results – by using the angles between the corresponding subspaces instead of working with the norm differences of the bases as done by Beyn/Kless/Thümmler [2001]. As a byproduct the method is shown to converge cubically if A=AH as in the case p=1.

Modified Block Newton method for the lambda modes problem

AbstractTo study the behaviour of nuclear power reactors it is necessary to solve the time dependent neutron diffusion equation using either a rectangular mesh for PWR and BWR reactors or a hexagonal mesh for VVER reactors. This problem can be solved by means of a modal method, which uses a set of dominant modes to expand the neutron flux. For the transient calculations using the modal method with a moderate number of modes, these modes must be updated each time step to maintain the accuracy of the solution. The updating modes process is also interesting to study perturbed configurations of a reactor. A Modified Block Newton method is studied to update the modes. The performance of the Newton method has been tested for a steady state perturbation analysis of two 2D hexagonal reactors, a perturbed configuration of the IAEA PWR 3D reactor and two configurations associated with a boron dilution transient in a BWR reactor.

Multi-scale crystal growth computations via an approximate block Newton method

Multi-scale and multi-physics simulations, such as the computational modeling of crystal growth processes, will benefit from the modular coupling of existing codes rather than the development of monolithic, single-application software. An effective coupling approach, the approximate block Newton approach (ABN), is developed and applied to the steady-state computation of crystal growth in an electrodynamic gradient freeze system. Specifically, the code CrysMAS is employed for furnace-scale heat transfer computations and is coupled with the code Cats2D to calculate melt fluid dynamics and phase-change phenomena. The ABN coupling strategy proves to be vastly more reliable and cost efficient than simpler coupling methods for this problem and is a promising approach for future crystal growth models.

A block Newton method for nonlinear eigenvalue problems

We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability.

An approximate block Newton method for coupled iterations of nonlinear solvers: Theory and conjugate heat transfer applications

A new, approximate block Newton (ABN) method is derived and tested for the coupled solution of nonlinear models, each of which is treated as a modular, black box. Such an approach is motivated by a desire to maintain software flexibility without sacrificing solution efficiency or robustness. Though block Newton methods of similar type have been proposed and studied, we present a unique derivation and use it to sort out some of the more confusing points in the literature. In particular, we show that our ABN method behaves like a Newton iteration preconditioned by an inexact Newton solver derived from subproblem Jacobians. The method is demonstrated on several conjugate heat transfer problems modeled after melt crystal growth processes. These problems are represented by partitioned spatial regions, each modeled by independent heat transfer codes and linked by temperature and flux matching conditions at the boundaries common to the partitions. Whereas a typical block Gauss–Seidel iteration fails about half the time for the model problem, quadratic convergence is achieved by the ABN method under all conditions studied here. Additional performance advantages over existing methods are demonstrated and discussed.

Block Newton法を用いた弾塑性体のミクロ－マクロ解析

Block Newton法を用いた弾塑性体のミクロ－マクロ解析