Strong Formulation Isogeometric Analysis (SFIGA) for laminated composite arbitrarily shaped plates

Numerical solution to linear bending analysis of circular plates is obtained by the method of harmonic differential quadrature (HDQ). In the method of differential quadrature (DQ), partial space derivatives of a function appearing in a differential equation are approximated by means of a polynomial expressed as the weighted linear sum of the function values at a preselected grid of discrete points. The method of HDQ that was used in the paper proposes a very simple algebraic formula to determine the weighting coefficients required by differential quadrature approximation without restricting the choice of mesh grids. Applying this concept to the governing differential equation of circular plate gives a set of linear simultaneous equations. Bending moments, stresses values in radial and tangential directions and vertical deflections are found for two different types of load. In the present study, the axisymmetric bending behavior is considered. Both the clamped and the simply supported edges are considered as boundary conditions. The obtained results are compared with existing solutions available from analytical and other numerical results such as finite elements and finite differences methods. A comparison between the HDQ results and the finite difference solutions for one example plate problem is also made. The method presented gives accurate results and is computationally efficient.

Free vibration analysis of laminated doubly-curved shells with arbitrary material orientation distribution employing higher order theories and differential quadrature method

A solution method based on a novel combination of the differential quadrature (DQ) method and the moving-least-squares (MLS) approximation is developed for solving the partial differential equations. The unknown variables in the domain are expressed in terms of the moving-least-squares approximation. The weight functions used in the differential quadrature method are calculated through a fast computation of the shape functions and their derivatives. The proposed method, called the moving least squares differential quadrature (MLSDQ) method, is applied to two-dimensional problems described by the Laplace and Poisson equations. Numerical experiments on MLSDQ method are presented for several example problems by considering the numerical accuracy, and h and p convergence. Sensitivity of the solution to the nodal patterns and effect of domains of influence are also carefully investigated.

Numerical simulation of chaotic dynamical systems by the method of differential quadrature

Bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method

Local radial basis function-based differential quadrature (RBF-DQ) method was recently proposed by us. The method is a natural mesh-free approach. Like the conventional differential quadrature (DQ) method, it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots, which may be distributed randomly. However, different from the conventional DQ method, the weighting coefficients in present method are determined by taking the radial basis functions (RBFs) instead of high order polynomials as the test functions. The method works in a similar fashion as conventional finite difference schemes but with truly mesh-free property. In this paper, we mainly concentrate on the multiquadric (MQ) radial basis functions since they have exponential convergence. The effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied, and how the value of optimal c varies with the number of local support knots is also numerically demonstrated. The proposed method is validated by its application to solve incompressible Navier-Stokes equations. Excellent numerical results are obtained on an irregular knot distribution.

A differential quadrature analysis of vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

The authors are presenting a novel formulation based on the Differential Quadrature (DQ) method which is used to approximate derivatives and integrals. The resulting scheme has been termed strong and weak form finite elements (SFEM or WFEM), according to the numerical scheme employed in the computation. Such numerical methods are applied to solve some structural problems related to the mechanical behavior of plates and shells, made of isotropic or composite materials. The main differences between these two approaches rely on the initial formulation – which is strong or weak (variational) – and the implementation of the boundary conditions, that for the former include the continuity of stresses and displacements, whereas in the latter can consider the continuity of the displacements or both. The two methodologies consider also a mapping technique to transform an element of general shape described in Cartesian coordinates into the same element in the computational space. Such technique can be implemented by employing the classic Lagrangian-shaped elements with a fixed number of nodes along the element edges or blending functions which allow an “exact mapping” of the element. In particular, the authors are employing NURBS (Not-Uniform Rational B-Splines) for such nonlinear mapping in order to use the “exact” shape of CAD designs.

Orthotropic laminated structures have attracted much interest in many applications because of their desirable properties such as improved thermal and mechanical properties, high strength to weight ratio, and high stiffness to weight ratio. In this paper, an improved version of the differential quadrature method, called the generalized differential quadrature (GDQ) method, is used for the first time to study the vibration of antisymmetric angle-ply laminated cylindrical panels with boundary conditions other than the simply supported type on its four edges. The GDQ method has shown computational advantages over the existing differential quadrature methods in that there is no restriction on the number of grid points in the approximation and the weighting coefficients are determined using a recurrence relation. The GDQ method has shown superb accuracy, efficiency and convenience, and great potential in solving the differential equations of structural problems. The present analysis is formulated using Donnell's shell theory and is validated by comparing results with those in the literature.

Free Vibration Analysis of Curvilinear Quadrilateral Plates by the Differential Quadrature Method

Out-of-plane static analysis of circular arches by DQM

Buckling analysis of thin rectangular plates under uniaxial or biaxial compressive point loads by the differential quadrature method

Differential quadrature element method for static analysis of Reissner–Mindlin polar plates

Strong Formulation IsoGeometric Analysis for the vibration of thin membranes of general shape

Application of localized meshless methods to 2D shallow water equation problems