A DXDR large deflection analysis of uniformly loaded square, circular and elliptical orthotropic plates using non-uniform rectangular finite-differences

A finite-difference analysis of the large deflection response of uniformly loaded square, circular and elliptical clamped and simply-supported orthotropic plates is presented. Several types of non-uniform (graded) mesh are investigated and a mesh suited to the curved boundary of the orthotropic circular and elliptical plate is identified. The DXDR method-a variant of the DR (dynamic relaxation) method-is used to solve the finite-difference forms of the governing orthotropic plate equations. The DXDR method and irregular rectilinear mesh are combined along with the Cartesian coordinates to treat all types of boundaries and to analyze the large deformation of non-isotropic circular/elliptical plates. The results obtained from plate analyses demonstrate the potential of the non-uniform meshes employed and it is shown that they are in good agreement with other results for square, circular and elliptical isotropic and orthotropic clamped and simply-supported plates in both fixed and movable cases subjected to transverse pressure loading.     

Dynamic relaxation method based on Lanczos algorithm

This paper tries to accelerate the convergence rate of the general viscous dynamic relaxation method. For this purpose, a new automated procedure for estimating the critical damping factor is developed by employing a simple variant of the Lanczos algorithm, which does not require any re‐orthogonalization process. All of the computational operations are performed by simple vector–matrix multiplication without requiring any matrix factorization or inversion. Some numerical examples with geometric nonlinear behavior are analyzed by the proposed algorithm. Results show that the suggested procedure could effectively decrease the total number of convergence iterations compared with the conventional dynamic relaxation algorithms. Copyright © 2017 John Wiley & Sons, Ltd.     

A new fictitious time for the dynamic relaxation (DXDR) method

This paper addresses the development of the DXDR method by introducing a modified fictitious time (MFT) increment. The MFT is determined by minimizing the residual force after each iteration. The rank of the convergence rate shows the advantage of the new method. The results obtained from plate and truss analyses demonstrate the potential of the new method. It is shown that, compared with a unit fictitious time, the MFT is more efficient, especially during the initial iterations. Moreover, MFT does not impose any additional constraints on the DXDR method. Copyright © 2007 John Wiley & Sons, Ltd.     

A new method of fictitious viscous damping determination for the dynamic relaxation method

This paper develops a new method for calculating the viscous fictitious damping of the dynamic relaxation (DR) method to overcome one of the most crucial difficulties in its application – the low convergence rate. The DR formulation was derived by error minimizations between two successive iterations to deduce an optimum fictitious mass and viscous damping with the aid of the Stodola iterative process. The efficiency of the new method was verified by its application to a wide range of typical structures with strong nonlinearity. The results show that compared to the conventional DR algorithm such as kinetic approach, the new method improves the convergence rate considerably.     

Improving the convergence rate of kinetic dynamic relaxation method with new R-point

This study deals with a new strategy for the restart phase in the kinetic dynamic relaxation (DR) method. First, the position of the restart point (R-point) is determined by the kinetic energy modeling as a quadratic function in successive DR iterations. Then, the displacement vector of the R-point is formulated based on the finite difference method. The proposed relation for the R-point is very simple and does not impose any additional calculations on the kinetic DR algorithm. For numerical evaluation, several truss, frame, and shell structures, with linear and nonlinear behaviors, are analyzed by different kinetic DR algorithms. The results show that the proposed R-point formulation increases the convergence rate of the kinetic DR method so that the average number of required iterations decreases by about 9% and 5% in linear and nonlinear analyzes, respectively.

     

A new model-dependent time integration scheme with effective numerical damping for dynamic analysis

This paper presents a new semi-explicit dissipative model-dependent time integration algorithm for solving structural dynamics problems. Motivated by the superior properties of the composite time-stepping scheme, the proposed method is designed, so that it fully inherits the numerical characteristics of its parent algorithm, namely the Bathe method. The algorithm design procedure is carried out by assuming unknown integration parameters for the proposed method. Afterwards, by time discretization of an SDOF model equation, the unknown parameters can be obtained explicitly by solving nonlinear system of equations. Some numerical examples are analyzed by the presented technique and comparisons are also made with two other dissipative model-dependent time integration algorithms as well as the Bathe method. Results demonstrate that the suggested technique can effectively damp out the spurious oscillations of the high-frequency modes, while the other schemes exhibit significant overshoot in the calculated responses. Furthermore, it is also observed that numerical results of the presented method totally coincide with the parent algorithm. While the Bathe method subdivides each time increment into two sub-steps, the proposed algorithm is single-step, non-iterative and does not involve any time-step subdividing.

     

Nonlinear bending analysis of variable cross-section laminated plates using the dynamic relaxation method

This paper tries to analyze the laminated plates with variable cross section, using the Dynamic relaxation method for solving the governing equations of the thin composite plate, obtained from the CPT theory. Comprehensive comparison and parametric studies prove the accuracy and efficiency of the utilized approach with interesting specifications such as fully vector calculations, independency to the lamina scheme (angle and number of plies) and boundary conditions so that the laminated plates with uniform, variable one direction and variable two direction of cross section could be analyzed. Results show that the behaviour of the composite plates depends on both lamina scheme (the stacking sequences and the number of the plies), and the cross section variation of the laminated plate. In this manner, utilizing the laminated plates with variable two direction of cross section could absorb more potential energy in comparison with uniform and variable one direction of cross section, so that they are useful for using in passive control mechanisms in which the kinetic energy should be removed from the structure by transforming to the potential energy.