Newton-Raphson Iteration Research Articles

A membrane theory for circular graphene sheets, based on a hyperelastic material model for large deformations

Large deflections relevant for suspended circular graphene sheets with simply supported boundaries are computed by a theory for 2D membranes subjected to several types of vertical axisymmetric forces, based on the principle of virtual power (PVP). Corresponding stress–strain relations are provided in the form of a nonlinear hyperelastic material model for graphene. When approximating the deflections through Fourier series, the PVP yields a nonlinear algebraic system of equations, which is solved by the iterative Newton–Raphson procedure. The new computational efficient method is validated through comparison of the numerical results it provides, with predictions obtained from experimental nanoindentaion measurements.

Fast algorithms for the quantile regression process

The widespread use of quantile regression methods depends crucially on the existence of fast algorithms. Despite numerous algorithmic improvements, the computation time is still non-negligible because researchers often estimate many quantile regressions and use the bootstrap for inference. We suggest two new fast algorithms for the estimation of a sequence of quantile regressions at many quantile indexes. The first algorithm applies the preprocessing idea of Portnoy and Koenker (Stat Sci 12(4):279–300, 1997) but exploits a previously estimated quantile regression to guess the sign of the residuals. This step allows for a reduction in the effective sample size. The second algorithm starts from a previously estimated quantile regression at a similar quantile index and updates it using a single Newton–Raphson iteration. The first algorithm is exact, while the second is only asymptotically equivalent to the traditional quantile regression estimator. We also apply the preprocessing idea to the bootstrap by using the sample estimates to guess the sign of the residuals in the bootstrap sample. Simulations show that our new algorithms provide very large improvements in computation time without significant (if any) cost in the quality of the estimates. For instance, we divide by 100 the time required to estimate 99 quantile regressions with 20 regressors and 50,000 observations.

Accurate iteration-free mixed-stabilised formulation for laminar incompressible Navier–Stokes: Applications to fluid–structure interaction

Stabilised mixed velocity–pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier–Stokes. In these formulations, the Newton–Raphson scheme is employed to solve the nonlinearity in the convection term. One fundamental issue with this approach is the computational cost incurred in the Newton–Raphson iterations at every load/time step. In this paper, we present an iteration-free mixed finite element formulation for incompressible Navier–Stokes that preserves second-order temporal accuracy of the generalised-alpha and related schemes for both velocity and pressure fields. First, we demonstrate the second-order temporal accuracy using numerical convergence studies for an example with a manufactured solution. Later, we assess the accuracy and the computational benefits of the proposed scheme by studying the benchmark example of flow past a fixed circular cylinder. Towards showcasing the applicability of the proposed technique in a wider context, the inf–sup stable P2–P1 pair for the formulation without stabilisation is also considered. Finally, the resulting benefits of using the proposed scheme for fluid–structure interaction problems are illustrated using two benchmark examples in fluid-flexible structure interaction.

Nonlinear supersonic flutter study of porous 2D curved panels including graphene platelets reinforcement effect using trigonometric shear deformable finite element

The nonlinear panel flutter behaviour of two-dimensional porous curved panel reinforced by graphene platelets exposed to a supersonic flow is investigated. A curved beam element developed based on the trigonometric shear deformation theory is employed. The formulation integrates the geometric nonlinearity with von Karman’s approximation. The effort to model the fluid–structure interaction is reduced by implementing the first-order form of piston theory aerodynamics to describe the flow and accounting for the influence of static aerodynamic load due to the inherent geometric curvature of the panel. The nonlinear governing equations are formulated adopting the Lagrangian formulation. The panel deflection under the static aerodynamic load is evaluated using the Newton–Raphson iteration method. The flutter behaviour is analysed with reference to the large deflection equilibrium state through an eigenvalue analysis and by tracing the complex eigenvalues and identifying the first coalescence of any two vibratory modes. The flutter dynamic pressure is also predicted iteratively using the eigenvalue approach for the selected range of limit cycle amplitudes. The influence of static aerodynamic load and vibration amplitude on the flutter characteristics is brought out for both isotropic and graphene reinforced composite panels with different boundary conditions. The pre-flutter static deflection shape of the panel is also examined. The material parameters such as porosity level in the metal foam and the graphene platelet content are assessed on the nonlinear flutter features of 2D panels.

Geometrically nonlinear isogeometric analysis of a partly wrinkled membrane structure

An isogeometric membrane element based on the non-uniform rational basis spline (NURBS) model is presented that accounts for membrane wrinkling based on tension-field theory. First, the element is validated by means of a benchmark problem involving a partly wrinkled membrane. It is then applied to the large deformation of a thin membrane structure using a two-stage procedure that combines dynamic relaxation and Newton–Raphson iteration. A simple technique is introduced that takes advantage of the geometrical symmetry of an isogeometric analysis model by using a “one-sided” open knot vector to treat the continuity of the membrane surface with respect to the symmetry plane. Because NURBS has many suitable features for representing complex geometries, it enriches the function space of the membrane element. Consequently, the characteristic mechanical responses of membranes, such as deep folding, are captured appropriately by the present isogeometric membrane element. In addition, a numerical example demonstrates that the convergence rate of the isogeometric membrane analysis with respect to refinement of the discretization is much better when tension-field theory is introduced in the analysis.

Consolidation responses of rheological aquitards to tide-induced groundwater fluctuations in coastal soft deposits

To investigate the consolidation responses of coastal rheological aquitards to tide-induced groundwater fluctuations, this study adopts a nonlinear 1-D consolidation model considering the rheological properties of the aquitard and non-Darcian flow. In this model, the depth-dependent property of the initial permeability and compressibility of the aquitard are considered using the continuous exponential functions of the vertical coordinate z. Moreover, the stress-dependent property of the permeability and compressibility during consolidation are described using the correlations of the void ratio with effective stress and permeability coefficient. The rheological nature of the aquitard is characterized with the e-log t curve for the secondary consolidation, while the flow in the aquitard is assumed to follow the exponential non-Darcian law. The numerical solutions for the model are obtained using finite volume method and Newton-Raphson iteration method. A comprehensive parametric study is conducted with regard to the mechanical and hydraulic parameters of the aquitard and its burial conditions. The analysis results reveal the generation, accumulation and oscillation mechanisms of pore pressure change and soil skeleton deformation in the aquitard. The proposed model can provide a theoretical foundation for predicting and controlling the deformation of engineering structures in coastal aquitards.

One-Dimensional Hydrodynamic Modeling of the Euphrates River and Prediction of Hydraulic Parameters

Forecasting techniques are essential in the planning, design, and management of water resource systems. The numerical model introduced in this study turns governing differential equations into systems of linear or non-linear equations in the flow field, thereby revealing solutions. This one-dimensional hydrodynamic model represents the varied unsteady flow found in natural channels based on the Saint-Venant Equations. The model consists of the equations for the conservation of mass and momentum, which are recognized as very powerful mathematical tools for studying an important class of water resource problems. These problems are characterized by time dependence of flow and cover a wide range of phenomena. The formulations, held up by the four-point implicit finite difference scheme, solve the nonlinear system of equations using the Newton-Raphson iteration method with a modified Gaussian elimination technique. The model is calibrated using data on the Euphrates River during the early spring flood in 2015. It is verified by its application to an ideal canal and to the reach selected at the Euphrates River; this application is also used to predict the effect of hydraulic parameters on the river’s flow characteristics. A comparison between model results and field data indicates the feasibility of our technique and the accuracy of results (R2 = 0.997), meaning that the model is ready for future application whenever field observations are available.

Effects of elastic foundation on the large-amplitude vibration analysis of functionally graded GPL-RC annular sector plates

The main purpose of this research is to investigate the impacts of the two-parameter elastic medium on the geometrically nonlinear free vibration of nanocomposite annular sector plates reinforced with graphene platelets. The modified Halpin–Tsai micromechanical model is implemented to determine the overall mechanical properties. With the aid of Hamilton’s principle and in according the first-order shear deformation plate theory in conjunction with geometric nonlinearity effects, the nonlinear sets of motion equations are derived for functionally graded nanocomposite annular sector plates. The 2D generalized differential quadrature GDQ method, the harmonic balance method and Newton–Raphson iteration technique are utilized to numerically solve the problem and obtain the nonlinear vibration response. Multiple comparative investigations are provided to express the efficiency of the model. The impacts of involved geometrical and material factors are also examined on the nonlinear vibration behavior of functionally graded nanocomposite annular sector plates.

Postbuckling analysis of multi-directional perforated FGM plates using NURBS-based IGA and FCM

Geometric modeling and numerical analysis of multi-directional FGM (Functionally Graded Material) plate, whose material properties grade continuously both in its thickness and in-plane directions, are increasingly required. In this work, postbuckling behavior of this type of plates with multiple cutouts is, for the first time, numerically investigated through the combination of NURBS-based IGA (IsoGeonetric Analysis) and FCM (Finite Cell Method). The nonlinear deformation of plate is determined by TSDT (Third-order Shear Deformation Theory) and von Kármán nonlinear assumptions without the requirement of SCFs (shear correction factors). Besides, the higher continuity advantage of NURBS basis functions can easily meet the C1-continuous requirement of the displacement field. The main contribution is introducing the FCM to deal with the influence of complex cutouts on the postbuckling characteristics. The geometric interfaces of the cutouts are approached and approximated by adaptive quadrature procedure in the distinguished cut elements. The advantage of this implementation is that the previously tricky process of representing the geometry of perforated plate with multiple NURBS patches can be eliminated, which naturally avoids the imposition of C1-continuity condition across the patch boundaries. The cylinder arc-length method combined with modified Newton–Raphson iteration algorithm, which takes into account of the initial geometric imperfections, is applied to implement geometrically nonlinear stability analysis and track the postbuckling paths. The effectiveness and reliability of the presented method are verified with available solutions of isotropic and conventional perfect FGM plates. Subsequently, a series of factors, including material volume fraction, length-to-thickness ratio, boundary condition, cutout size, etc., affecting the postbuckling responses of multi-directional perforated FGM plates are considered and investigated.

Simulation of Dynamics During Deployment of Foldable Origami Structures

Foldable origami-based structures are a type of deployable structures that are increasingly applied in the space and building industries. When folded, the small size of such structures facilitates transportation and storage. Meanwhile, the properties of their larger deployed state may be of interest to different applications. A stable working condition is established by locking the structure in its deployed state, as in the process of deployment, the driving forces may generate a dynamic effect, thus leading to instability of the system. Hence, the study of dynamic characteristics of such structures, including trajectory, duration, velocity, and acceleration is of paramount importance. In this paper, based on the general dynamic equation and Lagrange’s equations of the first kind, the finite element method is adopted to investigate the dynamic deployment of foldable plate structures in terms of the generalized nodal coordinates. The proposed geometric description of a quadrilateral plate element is based on a folding plate composed of refined triangular elements, which are used to approximate the real shells in the structure. Subsequently, a MATLAB framework is developed on the basis of the element using the Newmark integration and the Newton–Raphson iteration method to simulate the deployment process of the structure. Comparisons between MATLAB results and ADAMS results verify the reliability of the framework in analyzing the dynamic deployment of the foldable origami-based structures with sufficient accuracy.

Modeling Claim Frequency in Indonesia Auto Insurance Using Generalized Poisson-Lindley Linear Model

This paper will discuss the modeling of claim frequency from Indonesian auto insurance using the generalized Poisson-Lindley linear model. This modeling method assumes that the data of claim frequency are from populations that follow generalized Poisson-Lindley distribution. Generalized Poisson-Lindley linear model is an alternative to modeling count data that contains overdispersion. The parameters in the generalized Poisson-Lindley linear model can be estimated using the maximum likelihood estimation method through Newton Raphson's iteration numerical method. The data are the secondary data took from XYZ Company for the 2013 policy which is overdispersed. The data contains policyholder partial loss claims for comprehensive motor vehicle insurance products. From the research conducted it was concluded that the data is suitable to be modeled with generalized Poisson-Lindley linear models and produce better models than ordinary Poisson linear modeling because of produced the smaller AIC value. Of the 3 predictor variables that are modeled on the frequency of claims, 2 variables influenced they are the use variable and vehicle brand variable.

ГОЛОМОРФНОЕ ПОГРУЖЕНИЕ КАК АНАЛИТИЧЕСКИЙ МЕТОД РАСЧЕТА ЭЛЕКТРИЧЕСКИХ СЕТЕЙ НЕФТЯНЫХ И ГАЗОВЫХ МЕСТОРОЖДЕНИЙ

Для того чтобы обеспечить требуемый уровень надежного функционирования объектов нефтедобычи, необходимо обращать внимание на условия эксплуатации энергосистемы. Это важно, если принять во внимание, что для добычи нефти и газа необходимы достаточно мощные источники энергии, которые существенно влияют на режим работы электросети. При этом актуальным является задача расчета установившихся режимов электрической сети, питающей объекты нефтедобычи. Расчеты установившихся режимов имеют большое практическое значение для обеспечения эффективного и безопасного управления режимами работы нефтяных и газовых предприятий, являются важными при проектировании электрических сетей, питающих объекты нефтегазовых предприятий. Однако применение классических итерационных методов расчета установившихся режимов, таких как метод Гаусса–Зайделя и Ньютона–Рафсона, не всегда позволяет найти правильное решение системы нелинейных уравнений, описывающих установившиеся режимы работы сети, так как сходимость данных методов зависит от начальных приближений. В работе предлагается аналитический метод расчета установившихся режимов электрических сетей, описываемых нелинейными уравнениями. Метод основан на аппроксимации Паде и методе возмущений. Приводятся преимущества метода перед известным методом итераций Гаусса–Зайделя и Ньютона–Рафсона. Приводятся примеры решения задач электроэнергетических сетей, и обсуждаются недостатки предлагаемого метода. Рассматриваются проблемы устойчивости. Цель: применить аналитический метод голоморфного погружения для расчета двух и трехузловой энергетической схемы; сравнить возможности метода с другими альтернативными методами; исследовать ограничения метода голоморфного погружения и показать область его работы. Методы: разложение Тейлора, аналитическое продолжение, решения алгебраических уравнений рекуррентным методом, бесконечные дроби. Результаты. Получены примеры использования метода голоморфного погружения для двух и трех PQ узловых схем, и показаны недостатки метода голоморфного погружения. Проведено сравнение метода голоморфного погружения с альтернативными методами. Выводы. Аналитический метод голоморфного погружения обладает рядом преимуществ: физической наглядностью, простотой алгоритмической реализации, заключающейся в рекуррентных соотношениях для коэффициентов разложения искомой функции в ряд Тейлора. Разложенная в ряд функция является голоморфной, что позволяет осуществлять ее аналитическое продолжение и получить желаемую точность решения.

Nonlinear vibration analysis of third-order shear deformable functionally graded beams by a new method based on direct numerical integration technique

This study investigates the nonlinear vibration and dynamic response of a beam made of functionally graded material (FGM) within the framework of the improved third-order shear deformable theory. The beam is subjected to a concentrated moving load, and the included angle of the load direction and axial direction varies with time. Considering the von Karman geometric nonlinearity, the nonlinear formulations of the FGM beam is derived. The Newmark method in conjunction with the Newton–Raphson iteration is adopted to analyze the dynamic response of the beam. A new method, based on a direct numerical integration technique for the matrix form motion equation, is presented to study the nonlinear vibration of the FGM beam. The proposed method overcomes the problem of signal determination for the solution of the eigenvector equation that often leads to nonconvergence or a false result of the nonlinear eigenvalue equation. In relation to the numerical results, the effects of material property distribution, vibration amplitude on the nonlinear dynamic behavior of the FGM beams, and the energy transference phenomenon are discussed in this paper.

A Transition-Aware Method for the Simulation of Compliant Contact With Regularized Friction

Multibody simulation with frictional contact has been a challenging subject of research for the past thirty years. Rigid-body assumptions are commonly used to approximate the physics of contact, and together with Coulomb friction, lead to challenging-to-solve nonlinear complementarity problems (NCP). On the other hand, robot grippers often introduce significant compliance. Compliant contact, combined with regularized friction, can be modeled entirely with ODEs, avoiding NCP solves. Unfortunately, regularized friction introduces high-frequency stiff dynamics and even implicit methods struggle with these systems, especially during slip-stick transitions. To improve the performance of implicit integration for these systems we introduce a Transition-Aware Line Search (TALS), which greatly improves the convergence of the Newton-Raphson iterations performed by implicit integrators. We find that TALS works best with semi-implicit integration, but that the explicit treatment of normal compliance can be problematic. To address this, we develop a Transition-Aware Modified Semi-Implicit (TAMSI) integrator that has similar computational cost to semi-implicit methods but implicitly couples compliant contact forces, leading to a more robust method. We evaluate the robustness, accuracy and performance of TAMSI and demonstrate our approach alongside relevant sim-to-real manipulation tasks.

Numerical study of mixed bio-convection associated with a micropolar fluid

In the present study, the nanoliquid flow, heat and mass transfer over a vertical stretching sheet under the impact of motile microorganisms are investigated, numerically. Analysis is performed using the two-component four-equation non-homogeneous equilibrium nanoliquid model. At the same time, the nanoliquid is modeled as a micropolar non-Newtonian fluid. Formulated partial differential governing equations are transformed to ordinary differential equations and solved by the fourth-order Runge-Kutta method with the iterative Newton-Raphson technique. Analysis is conducted for a wide range of control parameters and heat transfer enhancement is obtained for high values of the Richardson number, bioconvection Lewis number and bioconvection Peclet number. It has been revealed that with increasing microorganisms concentration difference parameter (Ω), the angular velocity of fluid particles slightly increases and the temperature reduces and increases with increment in Ω and the bioconvection Rayleigh number (Rb), respectively. The results also showed that increasing bioconvection Lewis number (Lb) and Peclet number (Pe) leads to a decrease in the coupled stress; moreover, an increment in the vortex viscosity parameter (Δ) intensifies the penetration of microorganisms from the sheet to the boundary layer and amplifies the density number of motile microorganisms.

A new online modelling method for aircraft engine state space model

To overcome the drawbacks of current modelling method for aircraft engine state space model, a new method is introduced. The form of state space model is derived by using Talyor series to expand the nonlinear model that is implicit equations and involves many iterations. A partial derivative calculation method for iterations is developed to handle the influence of iterations on parameters. The derivative calculation and the aerothermodynamics calculations are combined in the component level model with fixed number Newton-Raphson (N-R) iterations. Mathematical derivation and simulations show the convergence ability of proposed method. Simulations show that comparing with the linear parameter varying model and centered difference based state space model, much higher accuracy of proposed online modelling method is achieved. The accuracy of the state space model built by proposed method can be maintained when the step amplitudes of inputs are within 2%, and the responses of the state space model can match those of the component level model when each input steps larger amplitudes. In addition, an online verification was carried out to show the capability of modelling at any operating point and that state space model can predict future outputs accurately. Thus, the effectiveness of the proposed method is demonstrated.

A single domain velocity - vorticity Fast Multipole Boundary Domain Element Method for three dimensional incompressible fluid flow problems, part II

In the present work, the Fast Multipole Boundary Element Method (FMM / BEM) for solving three-dimensional incompressible fluid flow problems governed by the Navier–Stokes equations is proposed. With the velocity-vorticity formulation the pressure gradient is eliminated from the equations. The kinematics equation, related to the velocity field satisfies continuity and provides a direct boundary condition for the vorticity equation. The single-domain approach is used for the discretization of the entire computational volume. The system of equations is compressed into two vectors and a preconditioner matrix, which is negligible in size. The involved unknowns are velocities, vorticities, and boundary vorticity fluxes. The two governing equations are coupled together in a convergent Newton–Raphson iteration scheme, successfully used for the solution of 3D fluid flow problems on a 32 GB memory computer. The degrees of freedom of the benchmark problems are above to 300000, which is an unreachable limit for the conventional single-domain BEM.

Finite volume modeling of coupled thermo-hydro-mechanical processes with application to brine migration in salt

The disposal of heat-generating nuclear waste in salt host rock generates a thermal gradient around the waste package that may cause brine inclusions in the salt grains to migrate toward the waste package. In this study, a dual-continuum model is developed to analyze such a phenomenon. In this model, fluid flow in terms of advective and diffusive fluxes in the interconnected pore space and diffusive and thermal-diffusive fluxes in the salt grains is considered. Due to the very distinct behavior of fluid flow in the interconnected pore space versus in the salt grains, this process is simulated based on a dual-continuum model. In the dual-continuum model, the mass balance of salt and water in the two continua is separately considered, and the coupling between the two continua is represented by flux associated with brine migration in one medium and out of another. The energy balance is simulated assuming thermal equilibrium among different components and phases in the whole system. For mechanical analysis, a new formulation (extended finite volume method, XFVM) is proposed and is applied with a Voronoi tessellated mesh. The coupling between the hydraulic and mechanical fields in terms of pore-volume effects is consistent with Biot’s theory, while thermal and mechanical fields are linked in terms of thermal expansion. The resulting fully coupled THM model is capable of modeling strongly nonlinear features, involving salt concentration effects on fluid mass associated with advection, and thermal effects on brine migration. A Newton-Raphson iteration formula is used to generate the linearized equations for this nonlinear problem. The model was verified step by step for each component of the coupling terms, including thermal-hydraulic (TH) and hydro-mechanical (HM) couplings, and was applied to analyze diffusion in single continuum and dual continua, small-scale brine migration, and large-scale brine migration induced by thermal gradient. The results show that the model is able to quantify brine under different conditions and thermal gradients, making it a valuable tool for performance assessment for nuclear waste disposal in salt.

A nonlinear vibration model of fiber metal laminated thin plate with amplitude dependent property

In this paper, a nonlinear vibration model of fiber metal laminated thin plate with amplitude dependent property considering the material nonlinearity is proposed. Based on the Jones-Nelson nonlinear theory and the complex modulus method, the Lagrange energy function has been derived. Then, combining the orthogonal polynomial method and the energy method, the homogeneous linear algebraic equations of the proposed modal are solved. By using the Newton-Raphson iteration method, the nonlinear natural frequencies and responses are obtained. And, the nonlinear modal damping ratios are acquired according to the complex modulus method. As an example to demonstrate the feasibility of the developed model, the experimental test of a TA2/TC500 laminated thin plate is implemented. The calculated and measured results show a good consistency.

Dynamics calculation of complex deep-sea cable system based on hybrid optimization algorithm

The deep-sea cable system, attached with an 11,000 m depth Autonomous Remotely Vehicle (ARV), is composed of two types of cables: Fiber Optical Micro-Cable (FOMC) and umbilical cable. The FOMC can be automatically released from a canister according to the tension of the cable. In the motion of ARV, the length, velocity, and tension of the FOMC are uncertain, which increases the complexity of dynamics calculation of the cable system. The nonlinear governing equations are set upon the basis of mechanical equilibrium. Owing to the complexity of the cable system, we employed the hybrid optimization method to search the best solution to the nonlinear equations, which makes full use of the globalization of Particle Swarm Optimization (PSO) searching and the rapidness of Newton-Raphson (NR) iteration. Numerical results of dynamics calculation verify the robustness of the hybrid optimization method and the feasibility of the ARV cable system.