Newton-Raphson Iterative Scheme Research Articles

A novel sequentially linear approach based on exponential saw-tooth softening in fracture analysis of quasi-brittle material

The Newton-Raphson iterative scheme is a widely used technique for modeling the non-linear problems, but it is well known that the scheme is unstable and the convergence may be difficult to obtain in case of snap-back or bifurcation. The sequentially linear approach, which sets a critical integration point and uses a saw-tooth softening approximation, is an alternative approach to address these issues. However, the sequentially linear approach cannot fully solve some drawbacks such as mesh dependency of crack patterns, which is main shortcomings of smeared crack approach. In this paper, the saw-tooth curve based on exponential softening is newly developed and an approach combined it with local crack tracking algorithm is proposed. The crack tip point is chosen by crack tracking algorithm and the fracture behavior is analyzed by sequentially reducing the stiffness and strength in elements within crack path. Several benchmark examples are numerically solved to check the accuracy of crack path, ultimate load and capture of snap-back predicted by the proposed approach and the results are compared with those by traditional sequentially linear approach and experiments.

On mechanics of piezocomposite shell structures

This study presents an original and novel investigation into the mechanics of piezo-flexo-magneto-elastic nanocomposite doubly-curved shells (PFMDCSs) and the ability to detect the lower and higher levels of electro-magnetic fields. In this context, by utilizing the first-order shear deformation shell model, stresses and strains are acquired. By imposing Hamilton's principle and the von Kármán approach, the governing equations have been obtained. The intelligent shell model consists of size-dependent influences, viz., strain gradients. This will take place via Mindlin's strain gradient elasticity theory and the subsequent re-establishing of the mathematical framework by incorporating this concept. The strain gradient results in a flexoelectric/flexomagnetic effect. The converse effect of the magnetic field on the basis of a close circuit has been assumed. The developed bending equations have been transferred into the algebraic ones by substituting an analytical technique based on homogeneous immovable simple support for the four edges. The problem has been solved according to the Newton-Raphson iteration scheme, and transverse deflections have been computed. For researching the rightness and precision of the shell models together with the solution process, a comparison is prepared by the finite element method (FEM) results for simplified shells, and a good correlation has been observed. At last, by examining several factors governing the problem, the conditions under which the magnetic effects can be noticeable and dominant in doubly-curved shells have been sought. This study could serve as a benchmark reference for piezoceramic-DCSs, as the presented governing equations are original. The most interesting outcome of this research is that the electro-magnetic response of intelligent structures can be entirely geometry-dependent.

Non-Linear Behavior of Double-Layered Grids

Abstract This study presents a numerical approach to an analysis of the mechanical behavior of double-layered tensegrity grids. We present a comparative study on the behavior of tensegrity grids through geometric nonlinear analysis (GNA) and combined nonlinear analysis (CNLA) (geometric and material), considering the possible effect of evolution in the elasto-plastic domain of the cable elements. The effect of the relaxation of cable on the amplification of the displacement of these grids was taken into account. The updated Lagrangian formulation, which modifies the Newton-Raphson iterative scheme with incremental loading, was adopted. We have developed a numerical computational model specific to tensegrity structures that simulates the geometric and material nonlinear behavior. The reliability of the calculation tool developed has been validated. Additionally, the results of the application of the numerical model on a grid, which was generated based on demi-cuboctahedral tensegrity cells, are presented.

Active control of thermally induced vibrations of temperature-dependent FGM circular plate with piezoelectric sensor/actuator layers

This study investigates the novel application of active control methods in mitigating thermally induced vibrations of functionally graded material (FGM) circular plates subjected to thermal shock. The top surface of the FGM circular plate is exposed to rapid surface heating, resulting in thermally induced vibrations. To counteract these deflections, two piezoelectric layers, one functioning as a sensor and the other as an actuator, are strategically integrated into the plate opposite sides. This layered configuration effectively distances the direct rapid surface heating from piezoelectric layers. All thermos-electro-mechanical properties of the used materials are considered temperature-dependent. The research employs the Generalized differential quadrature element (GDQE) method and the Crank-Nicolson time marching procedure to solve the nonlinear Fourier-type heat conduction equations for the layered structure. After obtaining the transient temperature profile, the study calculates thermal moments, forces, and pyroelectric terms. These thermally induced effects are then integrated into the motion and electrostatic equations, which inherently exhibit nonlinearity due to the von Kármán large deformations theory. The plate displacements are approximated using the first-order shear deformation theory (FSDT) along the plate thickness, while a quadrature distribution is considered for approximating the electric potential within the piezoelectric layers. The highly coupled electro-mechanical equations are solved using the GDQ method and the Newmark approach, with the Newton-Raphson iterative scheme utilized to address nonlinearity. A significant contribution of this research lies in the exploration of three distinct active control strategies: proportional control, derivative control, and proportional-derivative control. The study thoroughly analyzes the impact of each control type on the plate response, considering various boundary conditions.

Stochastic analysis of thin beams and plates incorporating von-Kármán nonlinear strains using meshless method

The current paper proposes methods for stochastic meshless analysis of thin beams and plates, by considering von-Kármán nonlinear strains. Here, Young’s modulus is assumed to have a spatial variation over the domain and is modeled as a homogeneous random field. Further, it is discretized utilizing moving least square based shape functions. Gaussian and lognormal properties are presumed to model the randomness related to Young’s modulus. Element-free Galerkin method is used as the meshless tool. The present study suggests employing perturbation method if the quantities of interest are limited to the first or second probabilistic moments of the responses. The nonlinear equations in the proposed perturbation formulations are solved using Newton-Raphson iterative scheme for finding out the deterministic part of the response, as well as its initial two derivatives with respect to random variables. Additionally, if there is an interest in higher probabilistic moments, probability density functions, cumulative distribution functions, and so forth, the study proposes a method using high-dimensional model representation (HDMR) in the meshless framework. In HDMR, the nonlinear connection between input variables and the output response is represented in terms of hierarchical correlated function expansions. These hierarchical functions are approximated over the response values evaluated using deterministic analysis on selected number and combinations of control points of random variables involved. Lagrange interpolation method is employed for the construction of response surfaces. Unlike Monte Carlo simulation (MCS) which needs numerous deterministic nonlinear analyses to be performed during each simulation, HDMR requires only a few deterministic nonlinear analyses to construct the component functions. Hence, simulations performed on the reduced order models of high-dimensional response can be used to estimate the probabilistic moments as well as density functions and cumulative distributions of response, with high computational efficiency compared to MCS. The probabilistic moments and distributions produced for a range of coefficient of variation of input random field up to 30%, for both the normal and lognormal input random fields are compared with those provided by crude MCS on the system of equations and are observed to be matching well.

Nonlinear transient thermo-mechanical responses of porous graphene platelet-reinforced cylindrical panels under moving distributed loads

This paper is the first attempt, to the best of the authors’ knowledge, to explore the nonlinear temperature-dependent dynamic responses of a porous functionally graded graphene nanoplatelet-reinforced composite (FG-GPLRC) cylindrical panel subjected to a moving distributed load. The desired porous FG-GPLRC structure can be achieved by reasonably designing the inner pore size and GPL dispersion patterns. A temperature-dependent dynamic model is proposed by introducing the equivalent thermo-mechanical parameters of the porous FG-GPLRCs with the help of the Halpin–Tsai micromechanics model, extended rule of mixtures, and open-cell metal foam model. The nonlinear governing equations of motion of nanocomposite cylindrical panels are derived based on the first-order shear deformation theory and the standard Lagrange equation with the aid of von Kármán geometric nonlinearity. Additionally, a closed-form Navier-type solution is implemented to model the simply-supported edges of the structures. Finally, the nonlinear dynamic response is determined using the Newmark direct integration technique combined with the Newton–Raphson iterative scheme. A parametric analysis is conducted, and the results indicate that the present model can predicate the temperature-dependent buckling behaviors and transient dynamic responses of the porous FG-GPLRC cylindrical panel. It is also found that dispersing more GPLs and fabricating more internal pores near the mid-surface of the panel can greatly reduce the response amplitudes induced by the moving loads, and the moving distributed load with a shorter length can result in a higher response.

Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality

After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. (1) We determine an equation that governs the near-identity transformation underlying its self-similar solution. (2) We develop its approximation scheme and construct the first-order approximation. (3) We obtain the source-type solution numerically by the Newton–Raphson iteration scheme and find it to agree well with the first-order approximation. Implications of the source-type solution are given, regarding the possibility of linearisation of the hypoviscous Burgers equation. Finally we address the problems of the incompressible fluid equations in two dimensions, centering on the surface quasi-geostrophic equation with standard and hypoviscous dissipativity.

Barycentric Lagrange interpolation collocation method for solving the Sine–Gordon equation

This paper mainly discusses the numerical solution of Sine–Gordon (SG) equation which is widely used in engineering field. Different from the classical numerical calculation method, we choose the barycentric Lagrange interpolation collocation method (BLICM) to solve the SG equation. Firstly, the barycentric Lagrange interpolation is introduced and its differential matrix is given. Secondly, two linearized iterative schemes namely direct linearized iterative scheme and Newton linearized iterative scheme to solve SG equation are constructed, the matrix equation of the two iterative scheme is obtained. Thirdly, the Newton–Raphson iterative scheme for SG equation is also presented and the detailed solution process is given. Finally, several numerical examples with exact solution are given to show the accuracy of BLICM, the results of three iterative schemes with equidistant nodes and Chebyshev nodes are compared. Further, some examples with line and ring solitons are given to simulate by BLICM, the accurate numerical results are obtained to support our experiments.

Time-domain simulation of sum-frequency hydrodynamic performance of a TLP-type floating wind turbine in multidirectional seas

Previous studies (Ren et al., 2022) by the authors have confirmed that the second-order diffracted wave loads are very sensitive to the wave directional spreading of the so-called storm-swell mixed seas where the storm-driven local seas combined with long-period swells propagating from different angles. In this paper, particular attention is paid to estimate the sum-frequency hydrodynamic forces and motion responses of a TLP-type floating wind turbine (FWT) in such complex sea states. A time-domain potential flow model is developed for the second-order interaction between the multidirectional irregular waves and moored structures. The boundary value problems for the scattered potential are solved by a higher-order boundary element method (HOBEM). The fourth-order Adams–Bashforth–Moulton method is used to update the wave surface and the Newmark method combined with the Newton–Raphson iteration scheme is adopted to solve the body motion equations.For validation, a freely floating hemisphere in unidirectional bichromatic waves is investigated and good agreement is found by comparing with Kim and Yue's frequency-domain results. Numerical tests are conducted on the TLP-type FWT with different combinations of incident angles and spreading factors of the storm-swell mixed seas. The effects of the wave headings and directional spreading on the sum-frequency hydrodynamic response are discussed. It is noted that when the swells approach the storm seas from large relative angles, the sum-frequency vertical response of the FWT can be up to seven times greater than that in the collinear mixed seas. In addition, the present model can extend the rotor-floater-tether coupled dynamic simulator of FWT to more realistic seas such as storm-swell mixed seas.

Optimal design of geometrically nonlinear structures using topology optimization

This study deals with comparing the stiffness design of geometrically nonlinear structures and linear structures using topology optimization. Bi-directional Evolutionary Structures Optimization (BESO) is used for the design process. The geometrically nonlinear behavior of the structures is analyzed using a total Lagrangian finite element formulation and the equilibrium is achieved by Newton-Raphson iterative scheme. The topology optimization of linear and nonlinear modeling is performed. The sensitivity of the objective function is found with the adjoint method and the optimization problem is solved using BESO’s update method. Objective function of complementary work is evaluated. A special technique, the continuation method, is applied to eliminate the instability of nonlinear structure optimization. ANSYS APDL is also used to do FEA of optimal topology to verify the effectiveness of geometrically nonlinear modelling. The results show that differences in stiffness of structures optimized using linear and nonlinear modelling are generally small but it can be large in some cases, especially structure highly involving buckling behaviour.

A nonlinear time-domain element differential method for solving two-dimensional electro- and magneto- quasistatic problems

The transient quasi-static electromagnetic problems with nonlinear conductive effects are under consideration in this paper. The quasi-static electromagnetic problems include electro-quasistatic (EQS) and magneto-quasistatic (MQS) problems, which neglect the wave propagation effects, and are important in the electrical engineering. Numerical computation is an efficient manner for solving these problems, apart from all the existing numerical methods, a new efficient nonlinear time-domain element differential method (TD-EDM) is presented to solve these problems in this paper. In the method, the governing equations and boundary conditions are directly solved in a strong form. The isoparametric elements are used to discrete the geometries, and the first- and second- order derivatives of the shape functions with respect to global coordinates are analytically derived. And the time derivatives of the basic variable are discretized by implicit Euler scheme. In order to handle the nonlinear effects induced by the nonlinear conductivities in the final system of equations, the Newton-Raphson iterative scheme is constructed to solve the nonlinear system of equations at each time step. Finally, several numerical examples are employed to validate the correctness of the proposed method.

Numerical Integration Scheme for Coupled Elastoplastic–Viscoplastic Constitutive Law for Tunnels

The paper presents an efficient numerical integration scheme for coupled elastoplastic–viscoplastic constitutive behavior with internal-state variables standing for irreversible processes. In most quasi-static structural analyses, the solution to boundary value problems involving materials that exhibit time-dependent constitutive behavior proceeds from the equation integration, handled at two distinct levels. On the one hand, the first, or local, level refers to the numerical integration at each Gaussian point of the rate constitutive stress–strain relationships. For a given strain increment, the procedure of local integration is iterated for stresses and associated internal variables until convergence of the algorithm is achieved. On the other hand, the second, or global, level is related to structure equilibrium between internal and external forces achieved by the Newton–Raphson iterative scheme. A review of the elastoplastic and viscoplastic model is given, followed by the coupling between these models. Particular emphasis is given in this contribution to address the first level integration procedure, also referred to as the algorithm for stress and internal variable update, considering a general elastoplastic–viscoplastic constitutive behavior. The formulation is described for semi-implicit Euler schemes. The efficacy of the numerical formulation is assessed by comparison with analytical and numerical solutions derived for deep tunnels in coupled elastoplasticity–viscoplasticity. Finally, a parametric analysis is performed to show the importance that this model can have, in the long-term convergence profile, against other models. For the considered flow surfaces, potential functions, and properties, differences on the order of 23% to 52% are found in the long-term convergence profile.

Variable-kinematic finite beam elements for geometrically nonlinear dynamic analyses

This article investigates the dynamic nonlinear response of three-dimensional structures using variable-kinematics finite beam elements obtained with the Carrera Unified Formulation. The formalism enables one to consider the three-dimensional form of displacement–strain relations and constitutive law. The deformation mechanisms and the associated couplings are described consistently with the selected kinematic model. The Hilbert–Hughes–Taylor method and the iterative Newton–Raphson scheme are adopted to solve the motion equations derived in a total Lagrangian scenario. Various models have been obtained by using Taylor- and Lagrange-like expansions. The capabilities of the beam elements are assessed considering isotropic, homogeneous structures with compact and thin-walled sections.

Buckling of Nanowires Under Self-Weight and Tip Load Including Effect of Surface Stress

In this paper, buckling of a nanowire column subjected to self-weight and tip load is investigated. One end of the nanowire is free, while the other end is attached to a rotational spring support. Considering the equilibrium equations together with the Euler–Bernoulli beam theory, the governing differential equation describing the behavior of the column can be obtained. Effect of surface stress is also incorporated into the formulations in terms of transverse distributed loading. The differential equation has been solved analytically and the general solution can be presented in the terms of Bessel function of the first kind. Applying the boundary conditions, the characteristic equations influenced by surface stress and stiffness of the rotational spring at the support can be expressed and then the critical load can be determined using the Newton–Raphson iterative scheme. From the results, they reveal that the positive surface stress could strengthen the nanowire against the buckling. Fixity at the base is also influenced to the critical load where the increase of the stiffness of the spring results in the increase of critical load as well.

Static analysis of porous core functionally graded piezoelectric (PCFGP) sandwich plate resting on the Winkler/Pasternak/Kerr foundation under thermo-electric effect

This paper analyzed static bending and deflection of the porous core functionally graded piezoelectric (PCFGP) sandwich plate resting on the Winkler/Pasternak/ Kerr foundations under thermomechanical and electromechanical loading with different conventional and unconventional boundary conditions. First-order shear deformation theory (FSDT) based displacement field and von Karman geometric nonlinear strain are used for problem formulation. The governing equations of the PCFGP sandwich plate are obtained using Hamilton’s principle. A higher-order finite element method has been employed with a modified iterative Newton-Raphson scheme for solving the governing equations. Convergence and comparison studies demonstrate that the validation of the current model has suitably correct. It has been observed that an increase in the porous exponent in the core part of the sandwich plate reduces the stiffness of the plate with an increase in stress, deflection, and electric potential. The maximum stress, deflection, and electric potential are obtained for porosity dispersion in the Uniform pore distribution (UD) type, while minimum in Un-symmetric pore distribution (USD) type dispersions. Hence, the different embedding porosity in the core part of the sandwich plate plays a significant role in reducing plate structural weight and offering good stability in plate structural bending stiffness.

Constrained Smoothing Splines by Optimal Control

In this letter, we consider the problem of constructing constrained smoothing splines, which are of great importance in data science. The novelty of this work is to formulate the problem as an optimal control problem, and we mathematically analyze the optimal smoothing spline with intermediate constraints using first-order optimality condition from a nonstandard version of the Pontryagin maximum principle. We also propose a novel algorithm to compute the optimal constrained splines based on Newton-Raphson iterative scheme combined with stochastic approximation. A numerical example is shown to illustrate the advantages of the proposed method.

Three-dimensional dynamic behaviour of flexible catenary risers with an internal slug flow

This work analyses the three-dimensional dynamic response of flexible catenary risers when subjected to the effects of an internal slug flow. The interaction between the flexible riser and the internal slug flow is a rather complex phenomenon, and this is because the forces due to the acceleration of the internal fluid influence the riser dynamics, which, in turn, influences the slug flow motion in its interior. In order to carry out this analysis, a new computational tool called SLUGFLEX has been developed, which is composed by two computational codes: the code that calculates the structural dynamic response of the flexible riser, and the other one that predicts the development of the internal slug flow. In the structural analysis, the spatial discretization of the riser is performed using spatial Euler beam finite elements, which suffer large displacements, although with small deformations. The tangent stiffness matrix and the internal force vector of the structure are calculated using the corotational approach. The internal slug flow, whose key feature is the intermittence of two different regions: a liquid slug and a large bubble of gas, is mathematically modelled by using a new one-dimensional lagrangian slug tracking method, which has been based on a model developed for straight pipes, and over which the effects of the curvature of the riser and its dynamics were added. The dynamic response of the riser is calculated by using the constant average acceleration method, in combination with the incremental iterative Newton–Raphson scheme for follower loads. The numerical results show good agreement when compared to the experimental ones obtained from the literature.

Effect of three-body interaction on the topology of basins of convergence linked to the libration points in the R3BP

The modified circular restricted three-body problem is numerically investigated by exploring the effect of three-body interaction on the basins of convergence connected to the in-plane as well as out-of-plane equilibrium points. The evolution of the positions of the libration points and their stability are illustrated as a function of parameter k due to three-body interaction. It is observed that the number of equilibrium points strongly depends on the sign and the magnitude of the three-body interaction parameter and there exist at most seven libration points where four are non-collinear and three are collinear with the primaries. Moreover, in the Copenhagen case we have found seven collinear libration points. Moreover, the non-collinear as well as collinear libration points are stable for various combinations of mass parameter μ and k. The collinear as well as non-collinear libration points are stable for even those values of μ which are higher than the μcrit of the classical restricted three-body problem. The study of the regions of possible motion shows that as the value of the Jacobian constant decreases, the forbidden region decreases significantly. The attracting domains of the basins of convergence, on several types of two-dimensional planes, are unveiled by applying the multivariate Newton-Raphson iterative scheme. In an attempt to analyse the effect of parameter k on the topology of basins of convergence, a systematic and thorough investigations are presented. The degree of fractality is also unveiled by determining the basin entropy of the convergence plane.

Combined effect of small perturbations in the Coriolis and centrifugal forces and three-body interaction on the existence of stationary points in the R3BP

In the present manuscript, the effects of small perturbations in the centrifugal and Coriolis forces with additional effect of three-body interaction on the locations as well as stability of the stationary points (SPs), and on the topology of the basins of convergence (BoCs) linked with these SPs in the restricted three-body problem (R3BP) are discussed by using the multivariate version of the Newton-Raphson iterative scheme. It is found that the positions of the SPs are highly influenced by the centrifugal force but their stability is highly influenced by combinations of the centrifugal and Coriolis forces. In addition, the correlations betwixt the basin of convergence (BoC) connected to the SPs and the associated number of iterations needed to assure pre-assumed precision are analyzed. Our analysis suggests that the mass ratio μ, three-body interaction parameter k and the effect in centrifugal and the Coriolis forces controlled by parameters ϕ and φ are very influential parameters.

A closed-form solution for temperature-dependent elastoplastic problems using the Prandtl operator approach

Finite element simulations of the temperature-dependent stress-strain response in the elastoplastic region of a material usually involve incremental procedures based on the Newton–Raphson iterative scheme. Although essential to obtaining the correct result, iterations inherently extend the computational time of the simulations. In order to increase the computational effectiveness of such finite element simulations, a novel solution technique is presented here, which introduces a closed-form determination of the elastoplastic stress-strain response using the Prandtl operator approach. Using this solution, the iterative procedure is no longer required. The positions of the tensile-compressive and shear meridians of the Haigh–Westergaard coordinate space are first conveniently modified, which then enables the configuration of coordinate-independent play operators. These play operators connect the stress and the strain tensors in a unique closed-form solution that significantly increases the computational power of the simulations, while retaining both the vigorous stability of the procedure and the high accuracy of the results. The method is successfully validated on several load cases, consisting of variable tensile, shear and combined thermomechanical load histories. Limitations of the current version of the approach, that are a part of on-going research, include extremely low values of the third deviatoric strain-invariant increments, in which the directions of movement of the yield surfaces can be changed. Furthermore, the discretisation of the cyclic stress-strain curves plays an important role. The optimal positions of the yield strains are hence another important issue for future studies. Additionally, the consistent material Jacobian results in an unsymmetric form in e.g. Abaqus when engineering shear strains are provided in the simulations and thus the computational power is not fully used. Nevertheless, the results using the Prandtl operator approach, when compared to the results obtained using the conventional, Besseling material model, show excellent agreement, while substantially reducing the computational time by up to 45%.