Direct Iteration Technique Research Articles

A Direct Iterative Technique for Weibull Parameters Estimation Optimized for lowest Root Mean Square Error

A Direct Iterative Technique for Weibull Parameters Estimation Optimized for lowest Root Mean Square Error

Nonlinear isogeometric analysis of axially functionally graded graphene platelet-reinforced composite curved beams

The linear and nonlinear isogeometric finite element models of an axially functionally graded graphene platelet-reinforced composite (AFG-GPLRC) curved beam are established within the framework of the third-order shear deformation beam theory (TSDT) and von-Kármán’s nonlinear geometric relation. The AFG-GPLRC curved beams can be seen as composite structures in which the graphene platelets (GPLs) are continuously distributed in the matrix along the length direction of the curved beam according to different patterns. The modified Halpin-Tsai parallel model and the rule of mixture are implemented to predict the effective Young’s modulus and mass density as well as Poisson’s ratio, respectively. Hamilton's principle, TSDT, and von-Kármán’s strain-displacement relation are combined to derive the governing partial differential equation of motion and corresponding boundary conditions. Furthermore, the Non-Uniform Rational B-splines (NURBS)-based isogeometric analysis (IGA) approach together with a direct iterative technique are utilized to solve the nonlinear governing equation. The accuracy and efficiency of the proposed IGA framework are confirmed by comparing corresponding numerical solutions with other available results. The parametric investigations, such as the curved beam’s geometric parameters, boundary conditions, and GPL’s distribution patterns, on the nonlinear bending and vibration responses of the AFG-GPLRC curved beams are carried out by some illustrative examples.

Nonlinear Thermomechanical Static and Dynamic Responses of Bidirectional Porous Functionally Graded Shell Panels and Experimental Verifications

AbstractThe present article examines the nonlinear static/dynamic behavior of the functionally graded porous shell panel with variable geometrical shapes exposed to thermomechanical load. The higher-order shear deformation theory (HSDT) is employed to develop a finite element (FE)-based mathematical model. The geometric nonlinearity is incorporated using Green–Lagrange nonlinear strains (GLNS). Voigt's micromechanical model, in association with power-law (GT-I), sigmoid (GT-II) and exponential (GT-III) kinds of material grading patterns, is adopted to calculate the graded panel's effective properties. Also, even (PRT-I) and uneven (PRT-II) distributions of porosity are considered in the present work. The temperature-dependent (TD) properties are adopted in association with variable temperature fields, i.e., uniform (TD-I), linear (TD-II), and nonlinear (TD-III) for the computation of flexural responses. To compute the desired nonlinear responses, the direct iterative technique is utilized. Convergence is used to validate the established model's stability and correctness is further verified by comparing the current numerical data to published and experimental results. The experiment was carried out by fabricating a few natural fiber-reinforced linearly varying layerwise panels for the test run. The study is further extended to investigate the influence of design parameters on nonlinear static and transient data (flexural/stress) of the functionally graded curved/flat panel considering thermal environmental conditions.

Nonlinear static and dynamic (deflection/stress) responses of porous functionally graded shell panel and experimental validation

The static and dynamic characteristics of functionally graded (FG) structures have been investigated in this article, considering full geometrical nonlinearity. The finite element (FE) solutions are obtained for the graded panel by modelling through higher-order kinematics (HSDT) and Green-Lagrange strain-displacement relations (GLNST). The desired graded panel properties are obtained through Voigt’s micromechanical approach, considering different grading patterns (exponential, EPL; sigmoid, SGM and power law, POL). Additionally, to maintain the generality of the graded structure, different porosity distribution types (even and uneven, EVP and UEP) are incorporated into the proposed mathematical model. Further, the numerical solutions are obtained using Newmark’s method’s direct iterative technique and constant integration steps. The solution sensitivities are verified for the developed algorithm via adequate convergence and comparison tests. In addition, the experimental validation is performed using the layerwise fabricated luffa-fibre reinforced FG plates to validate the proposed theoretical model’s accuracy. Lastly, the responses are computed using the newly derived model for the variable design-dependent parameters associated with the FG structural geometry and properties. The final deliverables, that is the nonlinear deflection responses (NDFR)/stress values, are discussed under mechanical loadings (static and dynamic).

On the large amplitude vibration of shallow sandwich shells with FG-GNPRP core considering initial geometric imperfections

The present study investigates the influence of initial geometric imperfections on large amplitude vibration of shallow sandwich shells with functionally graded graphene nanoplatelets reinforced porous (FG-GNPRP) core embedded between two aluminium facets. The aluminium core is reinforced with graphene nano-platelets (GNP) and the effective properties such as elastic modulus, and mass density, etc. are estimated using Halpin-Tsai and Voigt models, respectively. The continuous functions are adopted to accomplish micro-structural gradation while creating pores which imparts the variation in material properties along thickness direction. The nonlinear governing equations based on higher order shear deformation theory and Sander’s approximation are derived using variational approach. The nonlinearity is introduced in von-Kármán sense and various imperfection modes such as sine, global and local types are considered in transverse direction. The nonlinear results are accessed using finite element method in conjunction with direct iterative technique. The influence of porosity, GNP weight fraction, imperfection amplitude, thickness, side, and radius ratios on the vibration response of the sandwich shells is explored in detail.

Solution of Steady Incompressible MHD Problems with Quasi-Least Square Method

A quasi-least-squares (QLS) mixed finite element method (MFE) based on the L2-inner product is utilized to solve an incompressible magnetohydrodynamic (MHD) model. These models are associated with the three unknown terms, i.e., fluid velocity, fluid pressure, and magnetic field. For the MHD-based models, common theories and algorithms for approximation of the solutions are not always applicable because of the choice of the functional spaces during the utilization of the weak formulation. It is well known that the spaces used for the approximation of the different unknowns, e.g., the spaces for the unknowns, cannot be chosen independently for the variational formulation, and may have to satisfy strict stability conditions such as the inf-sup, or Ladyzhenskaya–Babuska–Brezzi (LBB) condition. The dependency of the selection of the spaces for the unknowns are critical and always not applicable for some pair of unknowns. Because of this, the numerical or theoretical solutions must have to face some stability issue. The proposed scheme (L2-inner product) is introduced to circumvent this deficiency of the conditions (inf-sup or LBB) and obtained a well-posed solution theoretically. The model equations are nonlinear and highly coupled with the combination of Navier–Stokes and Maxwell relations. First, these nonlinear models are made linear around a specific state wherein the modified system represents an algebraic equation in a first-order symmetric form. Secondly, a direct iteration technique is applied to solve the nonlinearities and obtain a theoretical convergent rate for a general initial guess. Theoretical results show that only a single parameter with a single initial guess is sufficient to establish the well-posedness of the solution.

Geometrical nonlinear numerical frequency prediction of porous functionally graded shell panel under thermal environment

The nonlinear eigenfrequency responses of a functionally graded material (FGM) panel in a thermal environment are numerically estimated in the present article using the finite element method (FEM). The constituents of the FGM are considered as the function of the temperature. For the evaluation of material properties of the FGM panel, Voigt’s micromechanical model is used in conjunction with three distinct types of material distribution patterns, namely power-law (PL), sigmoid (SM), and exponential (EN). Also, two kinds of porosity distributions, i.e. even (PT-I) and uneven (PT-II) through the panel thickness are considered in the present work. HSDT kinematics and Green–Lagrange nonlinear strain terms are employed to prepare a mathematical model of the FG panel. The governing equation is obtained using Hamilton’s principle, and a direct iterative technique is used to compute the final vibration responses. The convergence and validation are performed to check the stability and accuracy of the proposed model. Afterwards, several numerical examples are solved to demonstrate the proposed model efficacy in terms of currently adopted input parameters on the frequency (nonlinear) responses deliberated in detail.

Nonlinear pyrocoupled dynamic response of functionally graded magnetoelectroelastic plates under blast loading in thermal environment

The nonlinear dynamic response of magnetoelectroelastic (MEE) functionally graded plates (MEEFG) subjected to blast loads and operating in a temperature environment is examined in this study. The goal of this research is to investigate the impact of pyrocoupling along with the blast loads on the coupled nonlinear behavior of smart MEEFG plates. The material properties of MEEFG plates are calculated using the well-known modified power law. The kinematics of the plate is supposed to follow Reddy's higher-order shear deformation theory (HSDT). Also, von-Karman's equations are used to bring nonlinearity into the model. The blast loads are calculated using an exponential function and are expected to be dispersed evenly across the plate's surface. The principle of virtual work is used to develop the governing equations using finite element (FE) methods, which is then solved using the direct iterative technique. The numerical examples are offered to show how gradient patterns, gradient index, and blast loading parameters affect the nonlinear dynamic response.

On linear and nonlinear bending of functionally graded graphene nanoplatelet reinforced composite beams using Gram-Schmidt-Ritz method

The current study examined the linear and nonlinear bending results of functionally graded graphene nanoplatelet reinforced composite beams. The equation system was constructed using Reddy's third order shear deformation theory and a von Kármán type nonlinear strain-displacement relationship. The nonlinear bending results of this system were solved using the Gram-Schmidt-Ritz method in conjunction with the direct iteration technique. By employing the Gram-Schmidt orthogonalization procedure for generating shape functions in the method, numerically stable functions for nonlinear bending analysis of beams were obtained. Our solutions were validated for accuracy by comparing them to some published results. The numerical results indicate that the geometrical nonlinearity of beams subjected to uniform distributed load is greater than that of beams subjected to sinusoidal, linear, or parabolic distributed loads. New results on several significant effects of reinforcement patterns, graphene nanoplatelet weight fraction, and different types of distributed loads are presented and discussed in this study, in which the findings can serve as a benchmark for future research.

Geometrically nonlinear static and dynamic analysis of CNT reinforced laminated composite plates: A finite element study

In this paper, a finite element study is conducted using the Green Lagrange strain field based on vonKarman assumptions for the geometric nonlinear static and dynamic response of the laminated functionally graded CNT reinforced (FG-CNTRC) composite plate. The governing equations for determining the nonlinear static and dynamic behavior of the FG-CNTRC plate are derived using the Lagrange equation of motion based on Reddy's higher order theory. Using the direct iteration technique, the nonlinear eigenvalues for analyzing the free vibration response are obtained and the nonlinear dynamic responses of the FG-CNTRC plate are encapsulated based on the nonlinear Newmark integration scheme. The impact of the amplitude of vibration on mode switching phenomena and the consequence of the duration of the pulse on the free vibration regime of the plate are outlined. Also, the effect of time dependent loads is studied on the normal stresses of the plate. Furthermore, the impact on the nonlinear static and dynamic response of the laminated FG-CNTRC plate of various parameters such as span-thickness ratio (b/h ratio), aspect ratio (a/b ratio), different edge constraints, CNT fiber gradation, etc. are also studied.

Nonlinear dynamic responses of layered skew sandwich composite structure and experimental validation

The time-dependent deflection responses of the mechanically excited layered skew sandwich shell panels are computed numerically via a generic model developed mathematically using the higher-order shear deformation theory including the effects of the large displacement. The model includes the large displacements associated with the structural distortion under the small strain regime through Green–Lagrange nonlinear strain kinematics. The derived nonlinear system governing equation is converted to a set of algebraic form with the help of finite element steps. Subsequently, the time-dependent displacement values are computed numerically through the direct iterative technique including Newmark’s integration scheme. The dynamic deflections of the sandwich structural component under the influence of the externally excited mechanical loading are obtained through a generic computer code (developed in MATLAB) via the nonlinear higher-order finite element model. Before the implementation of the proposed model for the sandwich analysis, the solution stability and accuracy have been established by solving different kinds of numerical example from the published domain. Additionally, a few layered sandwich plates of different face sheet layers have been fabricated and the experimental dynamic data are recorded for the comparison purpose with the help of available modal test rig. Finally, the influences of the different structure-dependent design parameters on the nonlinear dynamic responses are investigated using the presently developed numerical model of skew sandwich shell panel, which also reveals that the present results give more accurate results.

Numerical analysis of thermal post-buckling strength of laminated skew sandwich composite shell panel structure including stretching effect

The computational post-buckling strength of the tilted sandwich composite shell structure is evaluated in this article. The computational responses are obtained using a mathematical model derived using the higher-order type of polynomial kinematic in association with the through-thickness stretching effect. Also, the sandwich deformation behaviour of the flexible soft-core sandwich structural model is expressed mathematically with the help of a generic nonlinear strain theory i.e. Green-Lagrange type strain-displacement relations. Subsequently, the model includes all of the nonlinear strain terms to account the actual deformation and discretized via displacement type of finite element. Further, the computer code is prepared (MATLAB environment) using the derived higher-order formulation in association with the direct iterative technique for the computation of temperature carrying capacity of the soft-core sandwich within the post-buckled regime. Further, the nonlinear finite element model has been tested to show its accuracy by solving a few numerical experimentations as same as the published example including the consistency behaviour. Lastly, the derived model is utilized to find the temperature load-carrying capacity under the influences of variable factors affecting the soft-core type sandwich structural design in the small (finite) strain and large deformation regime including the effect of tilt angle.

Nonlinear deformation and stress responses of a graded carbon nanotube sandwich plate structure under thermoelastic loading

The large deformation and stresses (normal and shear) of the graded nanotube-reinforced sandwich structure are numerically investigated under the influence of mechanical loading and a uniform temperature field. A higher-order nonlinear finite element model in conjunction with the direct iterative technique has been adopted for the solution purpose. Also, the structural distortion was modeled via the full-scale geometrical nonlinearity (Green–Lagrange strain) in the framework of higher-order displacement functions. Further, to replicate the actual operational conditions, the temperature-dependent properties of the individual material constituents (i.e., carbon nanotube and polymer) have been implemented in the current material modeling steps. The final deflections and stress values are evaluated via an own developed computer code using the currently proposed nonlinear mathematical formulation. The model accuracy and solution stability are checked by comparing the responses (deflection and stress) with available published results. Lastly, a variety of numerical examples is solved for different design parameters and deliberated in detail.

Nonlinear bending of porous curved beams reinforced by functionally graded nanocomposite graphene platelets applying an efficient shear flexible finite element approach

In the present study, the nonlinear flexural bending of thick and thin porous curved composite beams reinforced and functionally graded by graphene platelets is carried out using a three-noded C1 continuous curved beam finite element developed introducing an efficient shear deformation theory based on trigonometric function. The nonlinearity through the strain–displacement relationship by introducing von Karman’s assumptions is considered. The nonlinear equilibrium equations resulting from minimum potential energy principle are numerically solved based on the direct iteration technique. The bending nonlinear features through the load–deflection relationship are presented by selecting various design parameters like long and short beams, shallow and deep curved cases, support conditions, the variation of graphene platelets in the metal foam and the existence of porosity. This investigation reveals that the level of nonlinearity gets noticeably affected by the depth coupled with the curved beam slenderness ratio.

Nonlinear Free Vibration Analysis of a Toroidal Pressure Vessel under Constrained Volume Condition

This paper focussed on the large displacement nonlinear free vibration of a toroidal pressure vessel in free space under constrained volume condition. Three states of the toroidal pressure vessel surface comprising the initial unstrained surface, reference surface, and vibrating surface were considered in the energy functional of the toroidal pressure vessel system based on the principle of virtual work. For the reference surface, it was assumed that the toroidal pressure vessel has a circular cross-section and is subjected to uniform internal pressure. Nonlinear numerical solutions were obtained utilizing the FEM and a modified direct iteration technique, and compared with those generated by ABAQUS. More importantly, this study showed that the fundamental nonlinear natural frequency of a toroidal pressure vessel under the constrained volume condition was lower than that of the published results for an inflated toroidal membrane and shell.

Large amplitude free flexural vibrations of functionally graded graphene platelets reinforced porous composite curved beams using finite element based on trigonometric shear deformation theory

In this paper, the large amplitude free flexural vibration characteristics of fairly thick and thin functionally graded graphene platelets reinforced porous curved composite beams are investigated using finite element approach. The formulation includes the influence of shear deformation which is represented through trigonometric function and it accounts for in-plane and rotary inertia effects. The geometric non-linearity introducing von Karman’s assumptions is considered. The non-linear governing equations obtained based on Lagrange’s equations of motion are solved employing the direct iteration technique. The variation of non-linear frequency with amplitudes is brought out considering different parameters such as slenderness ratio of the beam, curved beam included angle, distribution pattern of porosity and graphene platelets, graphene platelet geometry and boundary conditions. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and material parameters of the curved composite beam. Also, the degree of hardening behaviour increases with the weight fraction and aspect ratio of graphene platelet. The rate of change of nonlinear behaviour depends on the level of amplitude of vibration, shallowness and slenderness ratio of the curved beam.

Nonlinear thermal free vibration frequency analysis of delaminated shell panel using FEM

The current research reported the thermal free vibration characteristics of the debonded composite shell structure considering the large geometrical deformation. The delaminated composite panel structural model is derived using two different higher-order polynomial kinematics. The nonlinear structural geometry has been modeled via Green-Lagrange relations in conjunction with temperature loading. The separation between the adjacent layers of the composite has been incurred through a sub-laminate approach and the corresponding displacement continuity imposed at the boundaries (laminated and delaminated). Moreover, the isoparametric Lagrangian type of element (eighty-one and ninety degrees of freedom) is adopted for the discretization of the physical shell structure. The nonlinear governing equation of motion of heated shell (healthy and debonded) structure derived using Hamilton principle and the nonlinear frequency responses computed using the direct iterative technique for the computational purpose. The solution (numerical) stability and the accuracy have been established as a priori by solving a series of examples available in the literature. The conclusions regarding the model applicability for debonded (size, location and position) structural analysis represented with or without temperature effect.

Nonlinear transient analysis of delaminated curved composite structure under blast/pulse load

The nonlinear time-dependent displacement values of the curved (single/doubly) composite debonded shell structure are examined under different kinds of pulse loading in this research. The structural curved panel model is derived mathematically using the higher-order displacement theories containing the thickness stretching effect, whereas the sub-laminate approach is adopted for the inclusion of delamination between the subsequent layers. The structural geometry distortion under variable loading has been included in the current theoretical analysis through Green–Lagrange type of strain kinematics. Further, the governing differential equation order has been reduced with the help of 2D finite element formulation via the nine-noded isoparametric Lagrangian elements with variable degrees of freedom (eighty-one and ninety) for two different higher-order kinematics, respectively. The final equation of motion is solved computationally to evaluate the transient responses through an original computer code including the direct iterative technique and Newmark’s average acceleration method. The convergence criteria of the current numerical solution are established as a priori and the subsequent validity is demonstrated via comparing the current responses with available published data. Further, the comprehensive behavior of the debonded structure under the influence of the variable loads (time and area dependent) is evaluated by solving different numerical illustrations for variable geometrical configuration and described in detail.

Buckling and post-buckling analyses of functionally graded graphene reinforced piezoelectric plate subjected to electric potential and axial forces

The buckling and post-buckling behaviors are analyzed for the multilayer functionally graded graphene platelets reinforced piezoelectric (FG-GRP) plates. The FG-GRP plates are subjected to the external electric potential and axial forces, including the uniaxial loading and biaxial loading. The graphene platelets (GPLs) disperse uniformly and parallelly in each graphene platelets reinforced piezoelectric composite (GRPC) layer, but they spread grading across the thickness of the FG-GRP plates. The effective Young’s modulus of each layer for the FG-GRP plates is calculated by the Halpin-Tsai parallel model. The rule of the mixture is employed to predict the Poisson’s ratio, effective mass density and piezoelectric properties of each layer of the FG-GRP plates. The governing equations of motion for the FG-GRP plates are obtained by the first-order shear deformation plate theory, von Karman nonlinear theory and principle of virtual displacements. To obtain the buckling and post-buckling behaviors of the FG-GRP plates with different boundary conditions, the differential quadrature (DQ) method and a direct iterative technique are combined to solve the governing equations of motion for the FG-GRP plates. The impacts of the external electric voltage, distribution pattern, volume fraction, piezoelectric properties, length-to-thickness of the GPLs and geometry of the plates on the critical buckling load and post-buckling equilibrium paths of the FG-GRP plates are discussed in detailed. It is clearly illustrated that the GPLs have a significantly enhancing influence on the buckling and post-buckling strength of the FG-GRP plates.

Geometric nonlinear formulation of plate buckling structure

The analytical or exact mathematical formulation of stresses and displacements for plate buckling structure become impossible to develop if the plate geometry is so complicated. Numerical technique is one another approach to solve this problem and it is chosen in this study for plate structure under in-plane and out-plane load. The formulation of elastic stiffness matrix (ke) and geometric nonlinear stiffness matrix (kg) of the plate structure due to buckling is presented and based on virtual displacement principle. The geometric nonlinear stiffness matrix (kg) is found function of internal stresses. The direct iteration technique is applied to find nodal displacements. Under this technique, the Gauss points stresses are initialized as zeros, then the kg matrix is updated, and then a new nodal displacement vector is found for the next approximation of internal stresses. Iterative process is done until convergence of displacement is satisfied. The rectangular plate with one fixed edge supported is used to test the proposed nonlinear formulation and procedure. The compressive in-plane load and moment is considered and applied for the tested plate. The plate is discretized with appropriate number of triangular finite element mesh. It is found that, the convergence of displacement is satisfied by using direct iteration technique. The load - deflection curve shows nonlinear relationship and approach to critical load. This finding shows that the direct iteration method can be accepted for the analyzing of plate buckling by considering geometric nonlinear assumption.