Newton-Raphson Procedure Research Articles

Uniformization and bounded Taylor series in Newton-Raphsonmethod improves computational performance for a multistate transition model estimation and inference.

Multistate transition models (MSTMs) are valuable tools depicting disease progression. However, due to the complexity of MSTMs, larger sample size and longer follow-up time in real-world data, the computation of statistical estimation and inference for MSTMs becomes challenging. A bounded Taylor series in Newton-Raphson procedure is proposed which leverages the uniformization technique to derive maximum likelihood estimates and corresponding covariance matrix. The proposed method, namely uniformization Taylor-bounded Newton-Raphson, is validated in three simulation studies, which demonstrate the accuracy in parameter estimation, the efficiency in computation time and robustness in terms of different situations. This method is also illustrated using a large electronic medical record data related to statin-induced side effects and discontinuation.

Nonlinear vibration of sandwich beams made of FGM faces and FGP core under multiple moving loads using a quasi-3D theory

This study aims to investigate nonlinear dynamic behavior of sandwich beams constituted by functionally graded material (FGM) faces and functionally graded porous (FGP) core under the action of multiple moving loads. The energy equations based on a quasi-3D beam theory considering several functions of shear deformation were established for producing the nonlinear equations of motion used for describing dynamic responses of beams with various boundary conditions. By using the hybrid processes of the Newton-Raphson iteration procedure, the time-integration approach of Newmark-β, and the Gram-Schmidt-Ritz method, the new results of nonlinear analysis were explored and proposed. Several prime parameters such as the porous coefficient, sandwich thickness ratio, power law index and others that significantly affect the dynamic response of the beams were taken into investigations. It can be concluded that beams under one moving load have larger dynamic deflections compared to that under many moving loads. This is because the distance between the moving loads can produce a bending moment against the beam deformation.

A new approach for fracture process zone evaluation

The study presents a novel method for identifying and monitoring the fracture process zone and its impact on cracking parameters, such as damage rate and fracture energy. Concrete samples subjected to a Wedge Splitting Test were investigated. Digital Image Correlation, a full-field measurement technique, was utilized to measure the experimental displacement fields at various scales. The experimental measurements, optimized through a Newton-Raphson iterative adjustment procedure, revealed the presence of a damaged zone accompanying the cracking process. This analysis also enabled the estimation of the material damage rate and provided a more realistic assessment of cracking parameters by integrating the entire process of damage and macro-cracking into the calculation of fracture energy. The results suggest that the appearance of a fracture process zone correlates with a reduction in the material’s hardness downstream of the crack propagation front.

A hybrid neural ordinary differential equation model of the cardiovascular system.

In the human cardiovascular system (CVS), the interaction between the left and right ventricles of the heart is influenced by the septum and the pericardium. Computational models of the CVS can capture this interaction, but this often involves approximating solutions to complex nonlinear equations numerically. As a result, numerous models have been proposed, where these nonlinear equations are either simplified, or ventricular interaction is ignored. In this work, we propose an alternative approach to modelling ventricular interaction, using a hybrid neural ordinary differential equation (ODE) structure. First, a lumped parameter ODE model of the CVS (including a Newton-Raphson procedure as the numerical solver) is simulated to generate synthetic time-series data. Next, a hybrid neural ODE based on the same model is constructed, where ventricular interaction is instead set to be governed by a neural network. We use a short range of the synthetic data (with various amounts of added measurement noise) to train the hybrid neural ODE model. Symbolic regression is used to convert the neural network into analytic expressions, resulting in a partially learned mechanistic model. This approach was able to recover parsimonious functions with good predictive capabilities and was robust to measurement noise.

A New Class of Generalized Extreme Value Distribution and Application under Alpha Power Transformation Method

This paper presents an expansion of the generalized extreme value distribution to new distribution classes, specifically the Alpha Power Transformation Generalized Extreme Value (APTGEV) distribution. This extension is achieved by combining the Extreme Value theory and the alpha power transformation technique. We employ the maximum likelihood method in conjunction with the Newton-Raphson procedure to estimate the parameters in these proposed distributions. In the final stages of our research, we simulate these new distributions and apply them to real-world data. For this study, we have chosen extreme rainfall data from a weather station in the Si Samrong District Sukhothai Province of Thailand as our dataset. These extended distribution classes are designed to provide greater flexibility and adaptability in understanding complex data patterns, and their application to real-world data offers valuable insights into their effectiveness.

Numerical Analysis on the Flow Depth Behaviour in Dividing Open-Channel for Trapezoidal and Rectangular Cross-Sectional Channel

Open-channel with dividing flow refers to any side water withdrawals from streams, rivers or main channel. It has become an important issue, especially when measuring the flow rate and depth of water as part of environmental management schemes and many practical projects such as the system of irrigation and drainage, water and wastewater treatment plants and other water resources projects. Although the behaviour of water flow has been studied by the previous researchers, most studies have only been carried out in simulation and lab experiment for a straight prismatic main channel. The mathematical research has not yet been fully understood, particularly in relation to the behaviour of water flow. Therefore, in this study, the modified general equations of dividing open-channel flow for rectangular and trapezoidal in the form of non-linear polynomial equations are proposed to determine the upstream flow depth in rectangular and trapezoidal cross-sectional channel. The computations of water flow depth influenced by Froude number and discharge ratio are performed with Newton-Raphson procedure in Maple software. The numerical solution shows the good agreement with the experimental data as reported in the literature. When the discharge ratio is low, the flow depth in trapezoidal and rectangular channels exhibits similarity. However, as the discharge ratio increases, the flow depth in the rectangular channel surpasses that in the trapezoidal channel. Furthermore, as the Froude number increases, the rectangular channel experiences a greater flow depth in comparison to the trapezoidal channel.

Thermal Postbuckling Characteristics Of Angle-ply Laminated Truncated Circular Conical Shells

Here, the nonlinear thermoelastic pre- and post-buckling characteristics of angle-ply laminated composite conical shells subjected to uniform temperature rise are studied using semi-analytical finite element approach. The finite element formulation is based on first-order shear deformation theory and field consistency principle. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton-Raphson iteration procedure coupled with displace- ment control method to trace the prebuckling followed by postbuckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, ply-angles, number of layers and number of circumferential waves on the nonlinear prebuckling/postbuckling thermoelastic response of the laminated circular conical shells. The participation of axisymmetric and asymmetric modes in the total response of the shells is brought out through the deformation shape analysis.

Postbuckling of Laminated Composite Circular Conical Shells Under Combined Thermo-Mechanical Loads

In this paper, the postbuckling characteristics of cross- and angle-ply laminated composite conical shells subjected to combined thermo-mechanical loading are studied using shear deformable semi-analytical finite element. The nonlinear governing equations, considering geometric nonlinearity based on Sanders type of kinematic approximations, are solved using Newton-Raphson iteration procedure coupled with adaptive displacement control method to trace the postbuckling equilibrium path. It is brought out from the study that the inward displacement in the postbuckling region is significantly higher as compared to the outward one. The ratio of the lowest load in the postbuckling path and the bifurcation load increases with the increase in the pressure loading proportion and decreases with the increase in the axial loading proportions. The relative load carrying capacity of shells under different set of loading combinations changes with the increase in the maximum postbuckling displacement. It is also observed that the nonlinearity in the load interaction relation corresponding to lowest load in the postbuckling path is less compared to that for bifurcation load.

Size dependent pull-in instability of functionally graded microbeams using a finite element formulation

The size dependent static pull-in instability of functionally graded (FG) microbeams in micro-electromechanical systems (MEMS) is studied, considering the influence of the axial force. The material properties of the microbeams are varied in the beam thickness by a power-law function, and they are calculated by the rule of mixture. To account for the microsize effect, the classical Euler-Bernoulli beam theory is employed in combination with the modified couple stress theory to describe the microbeams deformation. Based on Von Kármán nonlinear relationship, a beam element is derived and employed to establish the discretized governing equation for the microbeams. Newton-Raphson iterative procedure is adopted to compute frequencies and pull-in voltages for the microbeam with clamped ends. Numerical result reveals that the pull-in voltage is increased by the increase of the power-law exponent and the microscale parameter. The effects of the material distribution, the axial force as well as the microstructural parameter on the pull-in instability of the FG microbeams are investigated in detail.

A study of the radiation thermal boundary conditions influence in three-dimensional thermomechanical contact problems

Any body, by the simple fact of having a certain temperature on its surface, emits heat in the form of electromagnetic radiation. The heat flux emitted is proportional to the fourth power of the temperature of the solid, increasing the inherent non-linearity of the thermomechanical problems. This energy transfer, for systems at high temperatures, has greater relevance than the possible exchange of heat between bodies in contact by conduction or convection processes with a surrounding fluid in the interstitial areas. For this reason, this work presents a robust formulation to study three-dimensional thermomechanical contact problems where radiation boundary and interstitial conditions are simulated. The Boundary Element Method is used to compute the thermomechanical influence coefficients, and a double iterative procedure is used to ensure the achievement of the thermomechanical contact conditions. The final non-linear thermomechanical system of equations is solved using a Newton-Raphson procedure. The proposed formulation is validated with a comparison with some benchmark problems found in the literature and afterward applied to solve different three-dimensional thermomechanical contact problems where the thermomechanical variables are greatly affected by radiation boundary and interstitial conditions.

Non-local integral-type damage combined to mean-field homogenization methods for composites and its parallel implementation

Modern lightweight materials are often decomposed into several distinct materials, e.g. a matrix material and some reinforcements. A manufacturing process for these materials often induces eigenstrains, both reversible and non reversible. This could lead to an initial damage of a composite material, and eventually result into fracture of the structure. This contribution deals with non-local integral-type damage of composites subjected to eigenstrains. It is well known that local damage models lead to mesh sensitive simulation results. To avoid this effect, non-local integral-type regularization of equivalent strains and damage variables are exploited. Effective macroscopic properties are obtained using mean-field homogenization methods. A regularization in each material phase within the framework of an iterative Newton–Raphson procedure leads to the update of a non-local tangent stiffness matrix at each equilibrium iteration. To reduce the computational time parallel non-local assembly and averaging procedures are introduced. In this regard, the performance of different representations of dynamic sparse matrices and the scalability of the parallel implementation are investigated. Various microstructural morphologies are distinguished in representative examples to show the capability of the proposed work. Furthermore, the transition from damage to fracture is performed by aid of an element deletion method. To achieve the best compromise between memory consumption and speed over a range of non-zero entries, an array of flatmaps should be used. It is shown that the introduced parallel algorithms enable a drastic reduction of computational time. Moreover, it is illustrated that for various volume fractions of an inclusion a mesh sensitivity is clearly reduced.

Nonlinear convolution finite element method for solution of large deformation elastodynamics

A new algorithm based on the convolution finite element method (CFEM) is proposed for the nonlinear wave propagation in elastic media. The formulation is developed in the context of the total Lagrangian framework, encompassing contributions due to both geometrical and material nonlinearities. As a basis, a counterpart of equations of motion – namely, the alternative field equations – is first established. The satisfaction of the alternative field equations is then realized in a weak sense. Next, the Newton–Raphson procedure and the consistent tangential matrix are applied to the weak formulation, where the CFEM is used as the linear solver in each iteration. Finally, several examples are carried out to examine the theoretical aspects and the feasibility of the proposed algorithm. In particular, problems of free vibration of Neo-Hookean and Saint Venant–Kirchhoff plates are explored. Also, a cantilever beam of the Neo-Hookean material is simulated for the case of forced vibrations. Conspicuously, in contrast to the existing time-step methods with finite order of accuracy, the new solution procedure obtains the accurate solution when the time-step size is increased.

Vibration of electrically actuated MEMS Timoshenko microbeams based on a hierarchical beam element

In this paper, vibration of Timoshenko microbeams with an axial force in micro-electromechanical systems (MEMS) is studied for the first time by using a nonlinear finite element procedure. Based on the von Kármán geometric nonlinearity and the modified couple stress theory (MCST), a beam element is formulated by employing hierarchical functions to interpolate the displacement field. Using the derived element, the discretized equation of motion for the microbeam is constructed and then solved by the Newton-Raphson iterative procedure in conjunction with the Newmark method. The natural frequencies, pull-in voltages and dynamic deflections are computed for a clamped-clamped microbeam under electrostatic actuation of a given direct current (DC) voltage. The numerical result reveals that the axial force and the microsize effect have a significant influence on the vibration, and the fundamental frequency of the microbeams is underestimated by ignoring the size effect. The effects of the axial force, the applied voltage and the material length scale parameter on the vibration of the beam are studied in detail and highlighted.

Pipeline element for upheaval buckling analysis of submarine pipelines with geometric-imperfections under high temperature high pressure

Submarine pipeline is used for transporting oil from wells to resource storage facilities. The oil inside the pipeline is heated and pressured for long-distance transportation, making the pipeline under high temperature and high pressure (HTHP). The pipeline is usually placed inside a trench where surrounding soil restraints its longitudinal expansion. The pipeline is not perfectly straight, but with geometric imperfections, so the upheaval buckling may be triggered when the accumulated compression force has reached its buckling strength. It is not easy to conduct a geometrically nonlinear analysis for the pipeline because of the nonlinear soil–pipeline interactions (SPIs). This paper proposes a new line element aiming for the geometrically nonlinear large deformation analysis of the pipeline under HTHP. The axial expansion force due to high working temperature and pressure is included in the element formulation. Furthermore, a Gauss–Legendre integral method is employed to consider continuously distributed nonlinear SPIs in both lateral and longitudinal directions. Since the pipeline might exhibit large deflections, the analysis adopts the Updated-Lagrangian approach to establish the equilibrium conditions. A Newton–Raphson procedure is developed to execute the analysis. Detailed element formulations are given. Six groups of examples are provided to examine the robustness of the numerical method.

Numerical simulation of the behavior of kinematically unstable slopes under dynamic influences

Introduction. The concept of estimating the dynamic parameters of the “base — weakened layer — block” system is proposed, taking into account the physical nonlinearity of the material and the kinematic method of excitation of vibrations. In accordance with this approach, the physical nonlinearity of the base and block material is considered using the Drucker- Prager model. The weakened layer is modeled by 3D spring finite elements. The verification procedure of the proposed methodology is carried out on the example of the dynamic calculation of the “base — weakened layer — slope” system.Materials and Methods. The computational experiments were performed using the ANSYS Mechanical software package in combination with a nonlinear solver based on the Newton-Raphson procedure. SOLID45 volumetric finite elements were used to discretize the computational domains. Combined elastic-viscous elements COMBIN14 were used to simulate the displacement of the block relative to the fixed base.Results. An engineering technique for the dynamic analysis of the stress-strain state of the “base — weakened layer — block” spatial system with kinematic method of excitation of vibrations is developed. The accuracy and convergence of the proposed method is investigated using specific numerical examples.Discussion and Conclusion. Based on the mathematic simulation performed, it is shown that the developed technique provides assessing the risks of the occurrence of real landslide processes caused by external non-stationary impacts.

Large strain Flory’s decomposition for Lagrangian modeling of viscoleastic solids and compressive fluids

The fundamental difference in the solution of solids and fluids relies on the respective constitutive laws. Based on the Rivlin-Saunders-Düster-Hartmann hyperelastic model and using the Flory’s strain decomposition, we present a new total Lagrangian viscoelastic constitutive model for both Kelvin-Voigt viscoelastic solids and free-surface compressive viscous isothermal fluids. A dissipative viscous virtual work is written as a function of the time rate of isochoric invariants and its relation with the viscous stress is derived. Local time derivatives are solved by backward finite difference, allowing a consistent tangent viscoelastic constitutive tensor. The virtual work principle is used to write the weak equilibrium equation and its position-based finite element counterpart. Dynamic time integration is carried out by the Newmark β method and the Newton-Raphson procedure is used to solve time steps. The formulation is validated against experimental and numerical literature results revealing good precision. Additional examples are shown in order to demonstrate the applicability and future possibilities of the technique.

On the identification of the probability density function of a dimension from data with measurement errors

Metrology tools are applicable to check whether dimensions lay within their tolerance intervals. However, measurement errors cannot be avoided. This paper develops a method to partially correct the effect of such errors in case their probability density function is known. We assume that the probability density function associated with the measured dimension is available, for instance identified after multiple parts are manufactured and their dimension is measured. In this context, the probability density function of the measured value is the convolution product between the probability density functions of the measurement error and of the true value, respectively. Therefore, characterizing the true value is a challenging task, as this involves a deconvolution. We propose a procedure based on: (i) the transformation of the integral involved in the convolution product into a sum using numerical integration; (ii) the definition of a set of non-linear equalities solved using Newton-Raphson's procedure. An application example is used to discuss the relevance of the proposed method and the influence of various parameters on the results.

A Solution Algorithm for a Modified Bouc-Wen Model Capable of Simulating Cyclic Softening and Pinching Phenomena

The paper illustrates an extension of a solution algorithm required to evaluate the output variable of a modified Bouc-Wen model, namely a differential model capable of simulating generalized force-displacement hysteresis loops exhibiting cyclic softening and/or pinching. The proposed solution scheme has been derived by adopting the Backward Euler Method in conjunction with the Newton-Raphson iterative procedure. In order to demonstrate the accuracy and the computational efficiency of the proposed algorithm, the results of numerical simulations are compared to those obtained by employing the widely used MATLAB ode45 solver.

SYSTEM IDENTIFICATION USING NEWTON–RAPHSON METHOD BASED ON SYNERGY OF HUBER AND PSEUDO–HUBER FUNCTIONS

In real situations the presence of outliers is unavoidable and that is why the distribution of a disturbance is non-Gaussian. A synthesis of an algorithm of identification based on the Newton-Raphson method is considered for this case. The method requires that the loss function should be twice differentiable. Huber loss function, relevant for the treatment of outliers, has just the first derivative. In order to overcome the problem, the pseudo- Huber loss function is introduced. This function behaves similarly to the Huber loss function and has derivatives of an arbitrary order. In this paper, the pseudo- Huber loss function is used for the second derivative of functional in the Newton-Raphson procedure. The main contributions of the paper are: (i) Design of a new robust recursive algorithm based on the synergy of Huber and pseudo – Huber functions; (ii) The convergence analysis.

Implicit coupling of heterogeneous and asynchronous time-schemes using a primal approach based on velocity continuity at the subdomain interface

In the field of structural dynamics, it can be particularly interesting to consider a different time integrator and time scale in a different part of a problem (i.e. in the case of multi-physics problems, non smooth contact mechanics, seismic engineering with impacts, soil-structure interaction problems, or multiscale models using macro-element systems with a dynamic internal equilibrium). This paper presents a primal coupling algorithm based on a velocity gluing at the interface between two subdomains in order to be able to take into account both heterogeneous (different time schemes) and asynchronous (different time steps) time integrations (HATI). This algorithm allows for an implicit nonlinear resolution in providing the exact algorithmic tangent operator to maintain quadratic convergence for Newton-Raphson procedures. It is not intrusive in the finite element code as it only requires an interface element. The algorithm is presented in this paper for coupling different time schemes stemming from both Newmark families and Euler+ θ integration schemes (which can be very attractive when dealing with hard contact non smooth mechanics using complementarity methods). The proposed primal approach, which is based on imposing velocity continuity at the interface, is a viable alternative to the classical dual approaches since it is highly suitable for multiscale and sub-structuring models relying on subdomains with internal time integration schemes as well as for problems using macro-element families. The stability analysis exhibits a second order accuracy of the proposed approach. A selection of numerical examples under linear and nonlinear assumptions and for multiple degree-of-freedom system is provided; these examples show that no energy is being dissipated at the interface and overlap very closely with reference solutions. • A primal coupling algorithm based on a velocity gluing at the interface for implicit problems is introduced. • The tangent algorithm operator is derived even for multi-time stepping (i.e. asynchronous) and for various integration schemes using different integration parameters and order approximation (i.e. heterogeneous). • The stability analysis exhibits a second order accuracy of the proposed approach.