Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

A frequency accuracy study is presented for the isogeometric free vibration analysis of Mindlin–Reissner plates using reduced integration and quadratic splines,which reveals an interesting coarse mesh superconvergence.Firstly,the frequency error estimates for isogeometric discretization of Mindlin–Reissner plates with quadratic splines are rationally derived,where the degeneration to Timoshenko beams is discussed as well.Subsequently,in accordance with these frequency error measures,the shear locking issue corresponding to the full integration isogeometric formulation is elaborated with respect to the frequency accuracy deterioration.On the other hand,the locking-free characteristic for the isogeometric formulation with uniform reduced integration is illustrated by its superior frequency accuracy.Meanwhile,it is found that a frequency superconvergence of sixth order accuracy arises for coarse meshes when the reduced integration is employed for the isogeometric free vibration analysis of shear deformable beams and plates,in comparison with the ultimate fourth order accuracy as the meshes are progressively refined.Furthermore,the mesh size threshold for the coarse mesh superconvergence is provided as well.The proposed theoretical results are consistently proved by numerical experiments.

China Should Adopt Integration Policies to Reduce Energy Waste

China should pilot renewable energy integration policies similar to those adopted in Germany and Texas in the Beijing-Tianjin-Hebei region to transmit excess energy and reduce wasted wind power.Renewable energy integration will be fundamental to China?s transition to a low-carbon economy.Although China now leads the world in terms

An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions

In this paper,we discuss an algebraic multigrid(AMG)method for nearly incompressible elasticity problems in two-dimensions.First,a two-level method is proposed by analyzing the relationship between the linear finite element space and the quartic finite element space.By choosing different smoothers,we obtain two types of two-level methods,namely TL-GS and TL-BGS.The theoretical analysis and numerical results show that the convergence rates of TL-GS and TL-BGS are independent of the mesh size and the Young’s modulus,and the convergence of the latter is greatly improved on the order p.However the convergence of both methods still depends on the Poisson’s ratio.To fix this,we obtain a coarse level matrix with less rigidity based on selective reduced integration(SRI)method and get some types of two-level methods by combining different smoothers.With the existing AMG method used as a solver on the first coarse level,an AMG method can be finally obtained.Numerical results show that the resulting AMG method has better efficiency for nearly incompressible elasticity problems.

LINEAR FINITE ELEMENT APPROXIMATIONS FOR THE TIMOSHENKO BEAM AND THE SHALLOW ARCH PROBLEMS

Focuses on a study that discusses the linear finite element approximations for the Timoshenko beam and the shallow arch problems with shear dampening and reduced integration. Information on the Timoshenko beam; Details on the shallow arch problem; Factors that lead to the explicit formulation of the exact solution.