Idimeshev S.   Golushko S.K.   Shapeev V.P.  

hp-Version of Collocation and Least Residuals Method in Mechanics of Laminated Composite Plates

Reporter: Idimeshev S.

Collocation and least residuals (CLR) method is projection numerical method for solving boundary value problems for partial differential equation. The authors suggest hp-version of CLR method that based on the use of polynomials of higher degrees [1, 2]. The developed hp-version has three major advantages. The method is based on the use of polynomials of higher degrees, written as a direct product of Chebyshev polynomials that helps to minimize the accumulation of rounding errors. Special choice of collocation points, using the roots of Chebyshev polynomials, provides fast convergence and high accuracy of numerical solutions for a wide class of functions. In the CLR method the solution of a corresponding linear algebra problem is defined as the minimum of the functional of residual norm. This brings linear algebra problem well-conditioned. These advantages of hp-version of CLR method allows to carry out calculations at a lower computational cost.
The developed numerical method was used to calculation of laminated composite plates. Mathematical modeling and the calculation of strength and stiffness characteristics of laminated constructions lead to difficult problems of computational mathematics [3]. Anisotropy and layered (heterogeneous) structure (Fig. 1) lead to a complex stress-strain state distribution in plates. Calculation of stress-strain state of laminated plates leads to ill-conditioned problems due to presence of small parameters arising from the small relative thickness and anisotropy of layers. The hp-version of CLR method does not requires special technics for solving differential equations of high order, that characterize the considered problem statements. And it is easy to implement the boundary conditions that contain linear combinations of derivatives of a higher order.