Quadratic Extrapolation Research Articles

Meshfree approach for linear and nonlinear analysis of sandwich plates: A critical review of twenty plate theories

Present paper reviews twenty different theories used for analysis of multilayered plates. The mathematical formulation of the actual physical problem of the plate subjected to mechanical loading is presented using von Karman nonlinear kinematics. These non-linear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. A meshfree approach based on polynomial radial basis function is used for obtaining the solution. The results obtained for the sandwich plate are validated with other available results. It is observed that some theories under predicts the deflection by a reasonable amount. The effect varying core thickness on stresses and deflection of the sandwich plate is also presented.

Efficient and accurate modeling of rigid rods

Ten years ago, an original semi-recursive formulation for the dynamic simulation of large-scale multibody systems was presented by Garcia de Jalon et al. (Advances in Computational Multibody Systems, pp. 1–23, 2005). By taking advantage of the cut-joint and rod-removal techniques through a double-step velocity transformation, this formulation proved to be remarkably efficient. The rod-removal technique was employed, primarily, to reduce the number of differential and constraint equations. As a result, inertia and external forces were applied to neighboring bodies. Those inertia forces depended on unknown accelerations, a fact that contributed to the complexity of the system inertia matrix. In search of performance improvement, this paper presents an approximation of rod-related inertia forces by using accelerations from previous time-steps. Additionally, a mass matrix partition is carried out to preserve the accuracy of the original formulation. Three extrapolation methods, namely, point, linear Lagrange and quadratic Lagrange extrapolation methods, are introduced to evaluate the unknown rod-related inertia forces. In order to assess the computational efficiency and solution accuracy of the presented approach, a general-purpose MATLAB/C/C++ simulation code is implemented. A 15-DOF, 12-rod sedan vehicle model with MacPherson strut and multi-link suspension systems is modeled, simulated and analyzed.

Inelastic cross-sections and energy loss properties by non-relativistic heavy ions in zirconium dioxide

A formalism for the inelastic cross-section for electronic collisions of protons and heavier ions in a material is developed based on a quadratic extrapolation of the experimentally based dipole oscillator strength distribution (DOSD) of the material into the energy momentum plane. The approach is tested by calculating various energy loss properties in zirconium dioxide. Mean free path, stopping power and continuous slowing down approximation (csda) range are predicted as a function of ion energy for various incident ions, with the stopping powers compared to experimental data to assess the effectiveness of the methodology. The DOSD is straightforwardly obtained from the experimentally measured energy loss function data below 80eV and atomic photo-absorption cross-section data above 100eV. Agreement between the results of the calculation for stopping power and the experimental data is within 10% for all ions when compared for energies greater than the Bragg peak. The discrepancy is larger below the peak due to limitations in the methodology, especially the failure to make corrections for the Barkas and higher order effects and the lack of charge cycling cross-section data.

Two different technical criteria for determining the minimum miscibility pressures (MMPs) from the slim-tube and coreflood tests

In this paper, two different technical criteria, i.e., the oil recovery factor (ORF) and break-over pressure (BOP), are studied and compared to determine the minimum miscibility pressures (MMPs) of two light crude oil–CO2 systems. More specifically, five slim-tube tests with the live light crude oil–CO2 system and five coreflood tests with the dead light crude oil–CO2 system are conducted to measure the ORFs at different injection pressures and the actual reservoir temperature of Tres=53.0°C. The linear and quadratic extrapolation methods as well as the linear intersection method are applied by using the ORF criterion. Three different high threshold ORFs of 88% for the slim-tube tests or 87% for the coreflood tests, 90% and 95% are chosen to determine the MMPs by means of the linear and quadratic extrapolation methods. On the other hand, the linear intersection method is used to determine the MMP by finding a sudden slope change point in the measured ORF versus injection pressure curve for the slim-tube or coreflood tests. Moreover, the BOP criterion is based on the cubic regression of the measured ORF versus injection pressure data and used to determine the MMPs, which correspond to four different low threshold slopes or incremental ORFs per incremental injection pressure increase of 5, 3, 2, and 1%/MPa. It is found that different MMP ranges can be obtained from the same measured ORF versus injection pressure data if different MMP criteria, regression methods, and threshold values or numerical options are adopted. The determined MMP is better given in a small pressure range than specified as a definitive pressure value. Two MMP ranges of the live and dead light crude oil–CO2 systems are determined to be 15.2–15.4MPa and 12.4–12.9MPa at Tres=53.0°C, respectively.

Stress concentration factors and fatigue behavior of square bird-beak SHS T-joints under out-of-plane bending

In this study, fatigue tests of square bird-beak square hollow section (SHS) joints were conducted under out-of-plane bending. Nine orthogonally designed T-joints were fabricated. Static loading was first conducted, during which elastic strain distributions near the potential hot spots were measured. Stress concentration factors (SCFs) at weld toes were calculated using the quadratic extrapolation approach. Critical locations were also identified. Based on the high-cycle fatigue testing that followed, the fatigue crack initiation, propagation and fatigue failure mode were obtained. Variations of the surface crack length and out-of-plane rigidity against number of load cycles were also revealed. The cycle numbers that correspond to through-thickness cracks were observed to be greater than the numbers obtained from S–N curves provided by the IIW code, which indicates that the IIW curves/formulations for conventional joints could also be used to estimate the fatigue lives of square bird-beak joints if the hot spot stress ranges have been properly determined.

An Accelerated Two-level Multigrid Method for Markov Chains

Abstract Based on the quadratic extrapolation method and its generalization, this paper presents an accelerated two-level multigrid method for speeding up the numerical computation of the stationary probability vector of an irreducible Markov chain. It shows how to combine these vector extrapolation methods with the two-level multigrid method on the coarse level in detail. Numerical results on two Markov chain problems are provided to illustrate the effectiveness of our proposed method in terms of reducing the iteration counts and computing time.

A computational investigation for propagation of elasto-viscoplastic zones in the shock loaded circular plates

Purpose– The purpose of this paper is to introduce a computational time dependent modeling to investigate propagation of elastic-viscoplastic zones in the shock wave loaded circular plates.Design/methodology/approach– Constitutive equations are implemented incrementally by the Von-Kármán finite deflection system which is coupled with a mixed strain hardening rule and physical-base viscoplastic models. Time integrations of the equations are done by the return mapping technique through the cutting-plane algorithm. An integrated solution is established by pseudo-spectral collocation methodology. The Chebyshev basis functions are utilized to evaluate the coefficients of displacement fields. Temporal terms are discretized by the Houbolt marching method. Spatial linearizations are accomplished by the quadratic extrapolation technique.Findings– Results of the center point deflections, effective plastic strain and stress (dynamic flow stress) and temperature rise are compared for three features of the Von-Kármán system. Identifying time history of resultant stresses, propagations of the viscoplastic plastic zones are illustrated for two circumstances; with considering strain rate and hardening effects, and without them. Some of modeling and computation aspects are discussed, carefully. When the results are compared with experimental data of shock wave loadings and finite element simulations, good agreements between them are observed.Originality/value– This computational approach makes coupling the structural equations with the physical descriptions of the high rate deformation through step-by-step spectral solution of the constitutive equations.

Post-buckling analysis of skew plates subjected to combined in-plane loadings

Post-buckling behavior of laminated composite, sandwich and functionally graded skew plates is analyzed in the present work. The problem formulation is based on higher-order shear deformation theory and von Karman’s nonlinear kinematics. Linear mapping is used to transform the physical domain into the computational domain. Chebyshev polynomials are used for spatial discretization of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. The effect of the skew angle on the buckling and post-buckling response of the composite, sandwich and FGM-clamped skew plates is investigated for different combinations of in-plane compressive loadings.

Tests to Determine Stress Concentration Factors for Square Bird-Beak SHS Joints under Chord and Brace Axial Forces

AbstractBird-beak joints are a type of welded square hollow section (SHS) joint, the static strength of which is higher than that of conventional joints. Experimental analyses were conducted on the stress concentration factors (SCFs) of square bird-beak joints to obtain more information regarding their fatigue behavior. Twelve specimens, i.e., three conventional joints and nine square bird-beak joints, are separately tested by considering chord and brace axial force load cases. Both conventional and square bird-beak joints tested have a T-shape and are simply supported at their chord ends. Elastic strain distributions near the crown and saddle areas are measured, and strain concentration factors (SNCFs) at potential hot spots are calculated using an extrapolation approach. Quadratic extrapolation is suggested for the square bird-beak joints because linear extrapolation may underestimate SNCFs by up to 35%. Furthermore, critical locations are identified for square bird-beak joints subjected to chord and br...

A static PDE Approach for MultiDimensional Extrapolation Using Fast Sweeping Methods

A static partial differential equation (PDE) approach is presented for multidimensional extrapolation under the assumption that a level set function exists which separates the region of known values from the region to be extrapolated. Arbitrary orders of polynomial extrapolation can be obtained through solutions of a series of static linear PDEs. Fast sweeping methods of first and second orders are presented to solve the PDEs for constant, linear, and quadratic extrapolation. Numerical examples are presented to demonstrate the approach.

Incremental integrated modeling of dynamic viscoplastic responses of the annular sector plates exposed to shock wave loading

An incremental integrated modeling is presented to obtain high-rate dynamic viscoplastic behavior of annular sector plates. The large amplitude shock loads are imparted uniformly over the plate’s surface. Using the first-order shear deformation theory including the nonlinear Von-Kármán system, incremental differential equations are derived for nonaxisymmetric motion of the plate. A combined strain hardening law, in conjunction with special physical-based viscoplastic models, is applied to consider the material nonlinearity. Evaluation of the viscoplastic constitutive equations is accomplished by a semi-implicit scheme of the return mapping technique. An efficient algorithm is applied by the cutting-plane iteration to enforce plasticity admissibility during evolution of yield surface. A pseudo-spectral collocation methodology is implemented based on the Chebyshev polynomials in order to calculate displacement fields stepwise. Velocity and inertia terms are discrete by the Houbolt marching method. The nonlinear terms are eliminated by the quadratic extrapolation. The aliasing phenomenon caused by collocation treatment of nonlinear terms is controlled by applying an exponential filtering. A number of sector plates with different thicknesses, angularities, sector angles and boundary conditions are examined. The maximum transverse deflections, the effective plastic strain, dynamic yield stress, plastic works and temperatures are obtained. The present modeling is validated by comparing results with other approaches available in literature.

Non-linear static and dynamic analysis of skew sandwich plates

The present work deals with the flexural behavior of thin skew sandwich plates, with different types of facings, subjected to transverse static and dynamic loadings. Based on higher order shear deformation theory and von-Karman’s non-linearity, the equations of motion are derived using Hamilton’s principle. Transformation from physical to computational domain is carried out using linear transformation and chain rule of differentiation. Fast converging finite double Chebyshev series and Houbolt time marching scheme are used for spatial and temporal discretizations, respectively. Quadratic extrapolation technique is used for linearization of the equations of motion. The effect of skew angle, core thickness, lamination scheme and material properties on the static and dynamic behavior of skew sandwich plate is presented. Transient response of skew sandwich plates subjected to short duration pulse loadings is also obtained.

Post-buckling behavior of composite and sandwich skew plates

Present analysis deals with the buckling and Post-buckling behavior of laminated composite and sandwich skew plates. Higher order shear deformation theory and von-Karman's nonlinearity is considered in the problem formulation. The physical domain is mapped into computational domain using linear mapping. Finite degree double Chebyshev series method is employed for spatial discretizations of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. Post-buckling response of the skew plates is investigated for different types of in-plane compressive loadings. Effects of skew angle, lamination scheme, core thickness and face to core material property ratio of the sandwich plate on the buckling and Post-buckling response are studied in detail.

Real-time yaw rate prediction based on a non-linear model and feedback compensation for vehicle dynamics control

The yaw rate is a key state for vehicle dynamics stability control. However, the time delay from the steering input to the yaw rate measured by the yaw rate sensor changes with the frequency of the steering input and can also be affected by the road friction and the vehicle speed. In order to reduce the time delay effect on the vehicle dynamics stability controller, a new comprehensive yaw rate prediction method is proposed. A seven-degree-of-freedom non-linear vehicle model is adopted as the prediction model. A quadratic polynomial extrapolation method is used to compute the future steering-wheel input in the prediction horizon. A model-based yaw rate prediction algorithm is combined with a feedback compensator to guarantee the robustness of the algorithm. Meanwhile, a linear extrapolation method is utilized to minimize the defects of model-based prediction since the simplified vehicle model cannot fully model the hysteresis or other non-linearities of the vehicle systems. Experimental tests on the vehicle show that the combined algorithm is able to predict the yaw rate about 100 ms earlier than the sensor measurement is. Simulations show that the vehicle stability control strategy based on the proposed comprehensive prediction method has a better performance than the traditional control strategy does.

Geometrically nonlinear static and dynamic analysis of functionally graded skew plates

The present paper deals with nonlinear static and dynamic behavior of functionally graded skew plates. The equations of motion are derived using higher order shear deformation theory in conjunction with von-Karman’s nonlinear kinematics. The physical domain is mapped into computational domain using linear mapping and chain rule of differentiation. The spatial and temporal discretization is based on fast converging finite double Chebyshev series and Houbolt’s method. Quadratic extrapolation technique is employed to linearize the governing nonlinear equations. The spatial and temporal convergence and validation studies have been carried out to establish the efficacy of the present solution methodology. In case of dynamic analysis, the results are obtained for uniform step, sine, half sine, triangular and exponential type of loadings. The effect of volume fraction index, skew angle and boundary conditions on nonlinear displacement and moment response are presented.

Structural Stress Analysis of Trough to Deck Plate Weld in Orthotropic Steel Bridge Decks

A comparative study among five methods to calculate the structural stress of the trough to deck plate joint in orthotropic steel deck was carried out. The five methods used are surface stress extrapolation method (SSE), Haibach method, Xiao and Yamada method and CAB method. FEM analysis shows the five methods are mesh sensitive. Structural stress calculated using element type solid95 is larger than that of solid45. For fine and medium mesh model, structural stress calculated by Haibach method is larger than other methods, which means a lower fatigue life. Structural stress calculated by Yamada method is lower than other methods, which means a higher fatigue life than that calculated by other methods. For coarse mesh model, structure stress calculated by quadratic extrapolation method is larger than other methods, and, structural stress calculated by Yamada method is much lower than other methods.

Non-linear flexural and dynamic response of CNT reinforced laminated composite plates

The paper presents the non-linear flexural and dynamic response of CNT reinforced laminated composite plates using fast converging finite double Chebyshev polynomials. Halpin–Tsai model is used for evaluating the properties of matrix by dispersing CNT in it. Thereafter, CNT reinforced polymer matrix is treated as new matrix and then reinforced with E-Glass fiber in an orthotropic manner. Mathematical formulation of the laminated plate is based on first order shear deformation theory and von-Karman non-linear kinematics. Houbolt time marching scheme and quadratic extrapolation techniques are used for temporal discretization and linearization, respectively. The effects of CNT % and its aspect ratio on the non-linear flexural and dynamic response of the laminated composite plates are presented.

Nonlinear Dynamic Response of Elastically Supported Laminated Composite Plates

The large amplitude response of elastically supported shear deformable laminated composite plates subjected to uniform step loading in a hygrothermal environment is presented in this article. The problem is formulated utilizing a higher-order shear deformation theory and von-Karman non-linear kinematics. The elastic subgrade is modeled as shear deformable with cubic nonlinearity. The elastic and hygrothermal properties of the fiber composite material are considered to be dependent on temperature and moisture concentration. A fast converging finite double Chebyshev series is used to obtain the semi-analytical solution of nonlinear governing differential equations of motion along with Houbolt time marching scheme and Quadratic extrapolation technique. The effects of plate parameters, foundation parameters (Winkler and Pasternak type), temperature, and moisture on nonlinear dynamic response of laminated composite plates with classical and non-classical boundary conditions are studied.

Nonlinear flexural analysis of functionally graded plates under different loadings using RBF based meshless method

The nonlinear flexural response of functionally graded plates under different transverse loading is investigated using multiquadric radial basis function (MQRBF) method. The mathematical formulation of the actual physical problem of the plate subjected to mechanical loading is presented utilizing Levinson shear deformation theory and von Karman nonlinear kinematics. These non-linear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. The effect of the volume fraction exponent on the nonlinear flexural response of simply supported and clamped plates under uniformly distributed, sinusoidal, line and point loads is studied.

Large deformation flexural behavior of laminated composite skew plates: An analytical approach

The paper presents the large deformation flexural response of composite laminated skew plates subjected to uniform transverse pressure. Third order shear deformation theory (TSDT) and von-Karman’s nonlinearity is used for the analysis. Skew domain is mapped into a square domain and finite degree double Chebyshev series is used to discretize the space domain. No grid generation is required in the present solution technique. The nonlinear equations are linearized using quadratic extrapolation technique and the behavior of moderately thick laminated composite skew plates is studied. The effects of geometric nonlinearity, transverse shear, boundary conditions, aspect ratio and modular ratio on the behavior of laminated composite skew plates are discussed in detail.