Extended Kantorovich method for local stresses in composite laminates upon polynomial stress functions

The extended Kantorovich method is employed to study the local stress concentrations at the vicinity of free edges in symmetrically layered composite laminates subjected to uniaxial tensile load upon polynomial stress functions. The stress fields are initially assumed by means of the Lekhnitskii stress functions under the plane strain state. Applying the principle of complementary virtual work,the coupled ordinary differential equations are obtained in which the solutions can be obtained by solving a generalized eigenvalue problem. Then an iterative procedure is established to achieve convergent stress distributions. It should be noted that the stress function based extended Kantorovich method can satisfy both the traction-free and free edge stress boundary conditions during the iterative processes. The stress components near the free edges and in the interior regions are calculated and compared with those obtained results by finite element method（FEM）. The convergent stresses have good agreements with those results obtained by three dimensional（3D） FEM. For generality, various layup configurations are considered for the numerical analysis. The results show that the proposed polynomial stress function based extended Kantorovich method is accurate and efficient in predicting the local stresses in composite laminates and computationally much more efficient than the 3D FEM.

Nonlinear thickness-shear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales

This paper presents a nonlinear thickness-shear vibration model for onedimensional infinite piezoelectric plate with flexoelectricity and geometric nonlinearity.The constitutive equations with flexoelectricity and governing equations are derived from the Gibbs energy density function and variational principle.The displacement adopted here is assumed to be antisymmetric through the thickness due to the thickness-shear vibration mode.Only the shear strain gradient through the thickness is considered in the present model.With geometric nonlinearity,the governing equations are converted into differential equations as the function of time by the Galerkin method.The method of multiple scales is employed to obtain the solution to the nonlinear governing equation with first order approximation.Numerical results show that the nonlinear thickness-shear vibration of piezoelectric plate is size dependent,and the flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro-size thin plates.The geometric nonlinearity also affects the thickness-shear vibration frequencies greatly.The results show that flexoelectricity and geometric nonlinearity cannot be ignored in design of accurate high-frequency piezoelectric devices.

A-Posteriori Error Estimation for the Legendre Spectral Galerkin Method in One-Dimension

In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis shows that the postproeess improves the order of convergence. Consequently, we obtain asymptotically exact aposteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.

LEGENDRE-GAUSS-RADAU SPECTRAL COLLOCATION METHOD FOR NONLINEAR SECOND-ORDER INITIAL VALUE PROBLEMS WITH APPLICATIONS TO WAVE EQUATIONS

We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method.For more effective implementation,we propose a multi-interval LGR spectral collocation scheme,which provides us great flexibility with respect to the local time steps and local approximation degrees.Moreover,we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations.Numerical results show that the proposed methods have high accuracy and excellent long-time stability.Numerical comparison between our methods and several commonly used methods are also provided.

Legendre-Gauss Spectral Collocation Method for Second Order Nonlinear Delay Differential Equations

In this paper, we present and analyze a single interval Legendre-Gaussspectral collocation method for solving the second order nonlinear delay differentialequations with variable delays. We also propose a novel algorithm for the singleinterval scheme and apply it to the multiple interval scheme for more efficient implementation. Numerical examples are provided to illustrate the high accuracy ofthe proposed methods.