Quasi-optimal complexity of adaptive finite element method for linear elasticity problems in two dimensions

This paper introduces an adaptive finite element method （AFEM） using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.

Two-Level Hierarchical PCG Methods for the Quadratic FEM Discretizations of 2D Concrete Aggregate Models

The concrete aggregate model is considered as a type of weakly discontinuous problem consisting of three phases:aggregates which randomly distributed in different shapes,cement paste and internal transition zone(ITZ).Because of different shapes of aggregate and thin ITZs,a huge number of elements are often used in the finite element(FEM)analysis.In order to ensure the accuracy of the numerical solutions near the interfaces,we need to use higher-order elements.The widely used FEM softwares such as ANSYS and ABAQUS all provide the option of quadratic elements.However,they have much higher computational complexity than the linear elements.The corresponding coefficient matrix of the system of equations is a highly ill-conditioned matrix due to the large difference between three phase materials,and the convergence rate of the commonly used solving methods will deteriorate.In this paper,two types of simple and efficient preconditioners are proposed for the system of equations of the concrete aggregate models on unstructured triangle meshes by using the resulting hierarchical structure and the properties of the diagonal block matrices.The main computational cost of these preconditioners is how to efficiently solve the system of equations by using linear elements,and thus we can provide some efficient and robust solvers by calling the existing geometric-based algebraic multigrid(GAMG)methods.Since the hierarchical basis functions are used,we need not present those algebraic criterions to judge the relationships between the unknown variables and the geometric node types,and the grid transfer operators are also trivial.This makes it easy to find the linear element matrix derived directly from the fine level matrix,and thus the overall efficiency is greatly improved.The numerical results have verified the efficiency of the resulting preconditioned conjugate gradient(PCG)methods which are applied to the solution of several typical aggregate models.

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.

An adaptive p-FEM for three-dimensional concrete aggregate models

Numerical simulation for concrete aggregate models(CAMs)with different shape aggregates usually requires high accuracy and convergence near the material interfaces.But high memory usage will be needed for those traditional finite element methods such as the method by using mesh refinement throughout the domain.Thus,an adaptive p-version finite element method(p-FEM)is proposed in this paper for the solution of 3D CAM problems,and meanwhile the resulting adaptive computational algorithm and post-processing program are presented.We firstly focused two typical 3D weak discontinuity problems on the influence of different convergence criterions for the computational results of each point on the interface in order to verify the efficiency and convergence of the resulting p-FEM,and then this method is successfully applied to the numerical simulation of CAMs with different shape aggregates.In addition,an efficient hybrid realization method which combines ANSYS and Hypermesh software is also presented in order to quickly establish the geometric models of 3D CAMs.The numerical results have been shown that the proposed p-FEM can efficiently solve the concrete-like particle-reinforced composite problems and more accurate numerical results can be obtained under the case of fewer elements used in simulation of CAMs,even there being some elements with poor quality.