Full Newton-Raphson Research Articles

Numerical Simulation of Plasma Oscillation in 2-D Electron Gas Using a Periodic Steady-State Solver

The terahertz oscillation due to the plasma instability in the 2-D electron gas is numerically investigated using an in-house developed device simulator, G-Device. In order to overcome practical difficulties in the conventional transient simulation, a periodic steady-state (PSS) solver is implemented. The full Newton–Raphson scheme is applied to a set of discretized equations sampled at various time points during an oscillation period. Numerical results show that there is a threshold value of the injection velocity that allows the PSS oscillation. Moreover, the impact of the drain load resistance on the oscillation amplitude is estimated. It is found that considerable amplitude of the voltage oscillation can be achieved even with finite drain resistances. Although the growth rate of the voltage oscillation exhibits its maximum value with the infinite drain resistance, the maximum output power is obtained at a finite drain resistance.

Postbuckling Analysis Of A Rectangular Plate Loaded In Compression

Abstract The stability analysis of a thin rectangular plate loaded in compression is presented. The nonlinear FEM equations are derived from the minimum total potential energy principle. The peculiarities of the effects of the initial imperfections are investigated using the user program. Special attention is paid to the influence of imperfections on the post-critical buckling mode. The FEM computer program using a 48 DOF element has been used for analysis. Full Newton-Raphson procedure has been applied.

Effect of a Nonlinear Term in a Strain-Displacement Relationship on the Load-Displacement Path of Von Mises Truss

Von Misses truss is one of the best examples to explain different theoretical approaches, nature of non-linear solution, define the snap-through, illustrate interactive buckling, etc. The presented paper compares two nonlinear approaches to the problem. Effect of nonlinear terms in strain-displacement relationship on the load level in critical point of nonlinear solution is analyzed. To obtain the nonlinear equilibrium paths, the Newton-Raphson iteration algorithm is used. Corresponding levels of the total potential energy are defined. The peculiarities of the effects of the initial imperfections are investigated. Custom FEM computer program has been used for analysis. Full Newton-Raphson procedure, in which the stiffness matrix is updated at every equilibrium iteration, has been applied. Obtained results are compared with results of the nonlinear analysis using ANSYS system, element type BEAM3 is used. The arc-length method is chosen for analysis, the reference arc-length radius is calculated from the load increment. Only fundamental path of nonlinear solution has been presented.

Probabilistic Estimation of Voltage Unbalance in MV Distribution Networks With Unbalanced Load

Voltage unbalance negatively affects distribution network operators and customers by introducing overheating, reduced efficiency, and reduced lifetime of power system equipment and customer devices. The overall unbalance in the distribution network that leads to additional system losses is contributed to by every individual source of unbalance. This paper first analyzes the propagation of unbalance originating from the main cause of voltage unbalance—asymmetrical loading and then develops a methodology for probabilistic estimation of unbalance in the network. By considering daily loading curves, normally distributed load and employing three-phase full Newton-Raphson load flow, the Monte Carlo simulations give the possible ranges of levels of voltage unbalance for specified buses. The analysis enables distribution network operators to derive the level of unbalance in the network from either historical monitoring data or probabilistic estimation when full monitoring data are not available. Finally, the methodology is validated on a section of the real U.K. distribution network and the results of assessment compared with actual monitoring data.

Behaviour of Frictional Joints in Steel Arch Yielding Supports

Abstract The loading capacity and ability of steel arch supports to accept deformations from the surrounding rock mass is influenced significantly by the function of the connections and in particular, the tightening of the bolts. This contribution deals with computer modelling of the yielding bolt connections for different torques to determine the load-bearing capacity of the connections. Another parameter that affects the loading capacity significantly is the value of the friction coefficient of the contacts between the elements of the joints. The authors investigated both the behaviour and conditions of the individual parts for three values of tightening moment and the relation between the value of screw tightening and load-bearing capacity of the connections for different friction coefficients. ANSYS software and the finite element method were used for the computer modelling. The solution is nonlinear because of the bi-linear material properties of steel and the large deformations. The geometry of the computer model was created from designs of all four parts of the structure. The calculation also defines the weakest part of the joint’s structure based on stress analysis. The load was divided into two loading steps: the pre-tensioning of connecting bolts and the deformation loading corresponding to 50-mm slip of one support. The full Newton-Raphson method was chosen for the solution. The calculations were carried out on a computer at the Supercomputing Centre VSB-Technical University of Ostrava.

Evaluate the performance of an accelerated initial stiffness method in three dimensional finite element analysis

Newton’s method is a commonly used algorithm for elasto-plastic finite element analysis and has three common variations: the full Newton–Raphson method, the modified Newton–Raphson method and the initial stiffness method. The Newton–Raphson methods can converge to the solution in a small number of iterations when the system is stable; however, the methods can be quite computationally expensive in some types of problems, for example where the tangent stiffness matrix is unsymmetric or the plasticity is highly localized. The initial stiffness method is robust in those cases but requires a larger number of iterations. This prompted the formulation of many acceleration techniques in literature. In this paper, those techniques will be briefly discussed. This will be followed by the development of a modified acceleration technique for the initial stiffness method. The performance of the modified accelerated initial stiffness method will be examined in elasto-plastic analyses, using both direct and iterative matrix solvers. The results will be compared – in terms of the required number of iterations and the computation time – with an existing accelerated initial stiffness method, the non-accelerated initial stiffness method and the Newton–Raphson tangent stiffness method.

Efficient full Newton–Raphson technique for the solution of molecular integral equations – example of the SPC/E water-like system

It is shown how a full Newton–Raphson technique speeds up in impressive proportions the iterative resolution of molecular integral equations and makes it possible to reach quadratically complete convergence down to machine precision in a very few cycles. The technique generalises what has been originally proposed by Zerah and extensively used since then with great success for various fluids and mixtures of spherical objects. At each main iteration, the linearised cycle obtained by differentiating the Ornstein–Zernike and the integral equations is itself solved iteratively in terms of Δgmnlμν(r) projections. Its solution is reached very rapidly thanks to the powerful biconjugate gradient method and to the absence of any Euler angle manipulation. The virial equation is written in a shape formally different from the standard one, which allows a much higher numerical precision for the pressure without extra numerical work. The complete scheme is illustrated on the popular SPC/E water model.

Online Static Security Assessment Module Using Artificial Neural Networks

Fast and accurate contingency selection and ranking method has become a key issue to ensure the secure operation of power systems. In this paper multi-layer feed forward artificial neural network (MLFFN) and radial basis function network (RBFN) are proposed to implement the online module for power system static security assessment. The security classification, contingency selection and ranking are done based on the composite security index which is capable of accurately differentiating the secure and non-secure cases. For each contingency case as well as for base case condition, the composite security index is computed using the full Newton Raphson load flow analysis. The proposed artificial neural network (ANN) models take loading condition and the probable contingencies as the input and assess the system security by screening the credible contingencies and ranking them in the order of severity based on composite security index. The numerical results of applying the proposed approach to IEEE 118-bus test system demonstrate its effectiveness for online power system static security assessment. The comparison of the ANN models with the model based on Newton Raphson load flow analysis in terms of accuracy and computational speed indicate that the proposed model is effective and reliable in the fast evaluation of the security level of power systems. The proposed online static security assessment (OSSA) module realized using the ANN models are found to be suited for online application.

Simulation Analysis of Longitudinal Shear Bond Performance of Lightweight Aggregate Concrete Composite Slab

Because of the experiment can only reflect the overall performance of composite slabs under the various parameters combined action, and to ensure the materiality level of one single parameter, ANSYS, one of the finite element analysis software, was used in analogue simulation. According to selected reasonable elements and generated contact elements where between profiled steel sheet and lightweight aggregate concrete, appropriate mathematical model was been built. With full Newton-Raphson method to calculate nonlinear equation, the relevant data and curves were been obtained. Through the comparison between simulation analysis results and experiment results, the mathematical model was verified that it could reflect experiment correctly. These studies could provide assistance in the lightweight aggregate concrete composite slab project finally.

A Quadratically-Converging Nudged Elastic Band Optimizer.

Nudged elastic band (NEB) is a well established chain-of-states method to locate the minimum energy path in configuration space. Unfortunately, existing search algorithms suffer from slow convergence. We provide an analytic derivative of the nudged elastic band force, enabling a full Newton-Raphson optimization. For molecular systems, the components of the step belonging to translations and rotations are removed with an efficient algorithm. Minimization of the NEB force is ensured by reversing components for which the Newton-Raphson step would increase the force. We achieve quadratic convergence of this optimizer when applied to simple test cases where analytic Hessians are available: one analytic two-dimensional potential and a system of Lennard-Jones particles.

Contact Finite Element Analysis of Spindle-Toolholder Joint

A performance evaluation model for spindle-toolholder joint system is built by applying the contact finite element method (FEM). Contact detection is implemented at the Gaussian integration points of the contact elements. Contact state is determined by calculating normal distance and normal contact force between every two points on the contact surface. Friction state is determined by calculating tangential contact force. The determinations are introduced into the variable principle of nonlinear FEM based on pure penalty method. The updated FE equations are solved using the full Newton-Raphson method. Deformation and contact stress distribution are studied. The stiffness distribution is derived by calculating the displacement increments under the different forces, based on the stiffness definition. The proposed model can well illustrate and evaluate the joint performance.

Maximum likelihood estimation from interval censored recurrent event data

In this paper, we consider the recurrent failures of several repairable units, which can only be observed at periodic inspection times. A unit is not aging over the period between a failure and its detection. The failure times are interval censored by the periodic assessment times. The observed data consists of censoring intervals of failure times and the unobserved data are the actual ages of the units at the failure times. We formulate the likelihood function and use several iterative algorithms to find the maximum likelihood estimate (MLE) of the parameters. The complete Expectation–Maximization (EM) algorithm, the EM gradient, full Newton–Raphson (NR), and the Simplex method are used. We derive recursive equations to calculate the expected values required in the algorithms. We estimate the parameters for four failure datasets, assuming that the failures follow a non-homogeneous Poisson process (NHPP). Three datasets are obtained from a hospital for the components of general infusion pump, and the fourth dataset is simulated. Since the estimation could take a long time, we compare the performance of the algorithms in terms of the required number of iterations to converge, the total execution time, and the precision of the estimated parameters. We also use Monte Carlo and Quasi-Monte Carlo simulation as the substitutes for the recursive procedures in the Expectation step of the EM gradient and compare the results.

A nonlinear HHT- α method with elastic–plastic soil–structure interaction in a coupled SBFEM/FEM approach

This article presents a new method for the calculation of elastic–plastic building ground deformations and elastic–plastic building ground failure including wave propagation in the ground. The presented procedure is a hybrid method, based on several common calculation methods. Included is a nonlinear calculation with the finite element method (FEM), a nonlinear HHT-alpha method with full Newton–Raphson iteration and the scaled boundary finite element method (SBFEM). The presented method can be used as a tool for the accurate calculation of building ground deformations and the stability of the subsoil with included dynamic loading.

Hydraulic-Fracture Predictions With a Fully Coupled Geomechanical Reservoir Simulator

Summary A new geomechanical reservoir simulator has been developed that combines hydraulic fracture growth, multiphase/multicomponent Darcy/nonDarcy porous flow, heat convection and conduction, solids deposition, and poroelastic/poroplastic deformation in a single application. The equations for the different mechanisms such as fracture-width changes, laminar-channel flow in the fracture, porous flow in the reservoir, heat convection and conduction, and poroelastic/poroplastic deformations are combined to produce an implicit fully coupled formulation. The nonlinear system of equations is solved through use of a full Newton-Raphson expansion of all solution variables, which enhances solution stability and allows second-order convergence rates for the nonlinear iterations. The program contains two separate criteria that one can use to model fracture propagation. Fracture-growth computations can be based on critical stress-intensity factors or can use cohesive elements that exhibit strain-softening behavior. The critical stress-intensity factor is based on the asymptotic stress/strain state at the tip of a fracture and is limited to linear poroelastic applications or applications in which the plastic zone is small relative to the fracture length; while cohesive elements are based on energy-release rates and cohesive stresses and can be used for both poroelastic and poro-plastic applications. In addition to the fracture-propagation logic, the program allows a dry zone to develop at the fracture tip as a natural part of the solution process. It is shown in sample simulation that a dry zone may develop naturally at the tip of a propagating fracture when there is a large pressure drop down the fracture. The new geomechanical simulator is described and several examples are included to demonstrate the predictive capability of the application. Examples are also included to highlight the differences between the two fracture-propagation models and to illustrate when a dry zone may be expected to develop at the fracture tip. The examples also allow one to compare the program's predictions with analytic solutions validating the fracture propagation algorithms used in the application.

Performance of a new partitioned procedure versus a monolithic procedure in fluid–structure interaction

Fluid–structure interaction (FSI) can be simulated in a monolithic way by solving the flow and structural equations simultaneously and in a partitioned way with separate solvers for the flow equations and the structural equations. A partitioned quasi-Newton technique which solves the coupled problem through nonlinear equations corresponding to the interface position is presented and its performance is compared with a monolithic Newton algorithm. Various structural configurations with an incompressible fluid are solved, and the ratio of the time for the partitioned simulation, when convergence is reached, to the time for the monolithic simulation is found to be between 1/2 and 4. However, in this comparison of the partitioned and monolithic simulations, the flow and structural equations have been solved with a direct sparse solver in full Newton–Raphson iterations, only relatively small problems have been solved and this ratio would likely change if large industrial problems were considered or if other solution strategies were used.

Discussion of “Nonlinear Truss Analysis by One Matrix Inversion” by A. Fafitis

The author presents a method of analysis for truss structures with material and geometric nonlinearity. The method consists of a single factorization of the stiffness matrix, followed by its successive application to corrective force vectors in order to find structural equilibrium when nonlinearities arise due to strainhardening/softening, buckling, breaking, and stiffness degradation. The author points out that the stiffness matrix used for the iteration does not necessarily have to be the exact stiffness matrix of the structure, and the proposed analysis method treats proportional, nonproportional, and cyclic loadings uniformly. Although the developments are clear and the applications are practical, this method of analysis is not significantly different from the modified Newton–Raphson algorithm, where the stiffness matrix is held constant over the course of a load increment. The modified Newton–Raphson algorithm is discussed extensively in a reference Crisfield 1991 cited by the author; however, the author neither acknowledges this material nor cites any of the numerous references available in the mathematical literature on the convergence properties of the modified Newton– Raphson algorithm, such as Shamanskii 1967, Stoer and Bulirsch 1993, and Kelley 1995 to name but a few. Due to its slow convergence rate, the excessive number of iterations required to achieve equilibrium with the modified Newton–Raphson algorithm can far outweigh the computational savings afforded by a single matrix factorization for structures with a small to moderate number of degrees of freedom. For such structures, the computational cost of a matrix factorization is cheap relative to the cost of evaluating the corrective force vector, making the full Newton– Raphson algorithm more efficient. Furthermore, the use of equation solvers Mackay et al. 1991; Demmel et al. 1999; Davis 2003 that exploit the sparse matrix topology generated by a finite-element analysis can mitigate the computational expense of full Newton–Raphson for large structural systems. The discussers acknowledge that the full Newton–Raphson algorithm requires an exact stiffness matrix be computed at every iteration, which may not be possible when using complex numerical models of constitutive behavior. This is not the case, however, for closed-form scalar expressions, such as the bilinear stress-strain law and the Euler buckling formula Eq. 5 the author uses in the numerical examples. The author proposes holding an inexact stiffness matrix, Kinexact, constant during the equilibrium iteration. This approach can increase the number of iterations required to reach equilibrium. Consistent with the numerical properties of the modified Newton–Raphson algorithm, if the spectral radius of the matrix

Approach to determine voltage control areas considering impact of contingencies

This paper presents an efficient method for the formation of voltage control areas. The proposed method considers the effect of P–V and Q–δ coupling and changes in network topology. The full Newton Raphson load flow (NRLF) Jacobian sensitivities and voltage variations under contingencies have been used for the formation of the voltage control areas. These areas have been reduced to equivalent nodes using the REI reduction technique, and the reduced network has been validated for the voltage stability analysis. The effectiveness of the method has been established on the IEEE 14-bus system and a practical 75-bus Indian system.

F‐bar‐based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking

AbstractThis paper proposes a new technique which allows the use of simplex finite elements (linear triangles in 2D and linear tetrahedra in 3D) in the large strain analysis of nearly incompressible solids. The new technique extends the F‐bar method proposed by de Souza Neto et al. (Int. J. Solids and Struct. 1996; 33: 3277–3296) and is conceptually very simple: It relies on the enforcement of (near‐) incompressibility over a patch of simplex elements (rather than the point‐wise enforcement of conventional displacement‐based finite elements). Within the framework of the F‐bar method, this is achieved by assuming, for each element of a mesh, a modified (F‐bar) deformation gradient whose volumetric component is defined as the volume change ratio of a pre‐defined patch of elements. The resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near‐incompressibility. As the original F‐bar procedure, the present methodology preserves the displacement‐based structure of the finite element equations as well as the strain‐driven format of standard algorithms for numerical integration of path‐dependent constitutive equations and can be used regardless of the constitutive model adopted. The new elements are implemented within an implicit quasi‐static environment. In this context, a closed form expression for the exact tangent stiffness of the new elements is derived. This allows the use of the full Newton–Raphson scheme for equilibrium iterations. The performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two‐ and three‐dimensional numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.

Buried Pipe Modeling With Initial Imperfections

A formulation using the finite element method is presented in this paper to analyze stresses and displacements of underground pipelines with initial imperfections. The formulation includes both thermal expansion of the pipeline and internal pressure. The method is based on large deformation beam theory with the finite element formulation based on Euler-Bernoulli beam elements of constant cross-section. The pipe-soil interaction is modeled as a nonlinear elastic foundation in the problem formulation. The resulting nonlinear finite element model with appropriate boundary conditions is solved using full Newton-Raphson as the iterative procedure. Numerical examples are provided to show the application of the methodology and to demonstrate the effect of the initial imperfections in the stress distribution of a buried pipe. The results show that initial imperfections have a considerable influence on the stress distribution of buried pipelines, leading in some cases to stress levels above the allowable limit established by the design codes. The results also help to identify the critical temperature at which buckling of the buried pipe might occur.

An assessment of the average nodal volume formulation for the analysis of nearly incompressible solids under finite strains

AbstractThis paper provides an assessment of the average nodal volume methodology originally proposed by Bonet and Burton (Commun. Numer. Meth. Engng. 1998; 14:437–449) for the analysis of finitely strained nearly incompressible solids. An implicit version of the average nodal pressure formulation is derived by re‐casting the original concept in terms of average nodal volume change ratio within the framework of the F‐bar method proposed by de Souza Neto et al. (Int. J. Solids Struct. 1996; 33: 3277–3296). In this context, a linear triangle for implicit plane strain and axisymmetric analysis of nearly incompressible solids under finite strains is obtained. An exact expression for the corresponding element stiffness matrix is presented. This allows the use of the full Newton–Raphson algorithm, ensuring quadratic rates of asymptotic convergence in the global equilibrium iterations. The performance of the procedure is thoroughly assessed by means of numerical examples. The results show that the nodal averaging technique substantially reduces the volumetric locking tendency of the linear triangle and allows an accurate prediction of deformed shapes and reaction forces in situations of practical interest. However, the formulation is found to produce considerable checkerboard‐type hydrostatic pressure fluctuations which poses severe limitations on its range of applicability. Copyright © 2004 John Wiley & Sons, Ltd.