Higher order two-scale finite element error analysis for thermoelastic problem in quasi-periodic perforated structure

2017 ◽  
Vol 28 (07) ◽  
pp. 1750097
Author(s):  
Mingxiang Deng ◽  
Yongping Feng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Analysis ◽  
Thermal Behavior ◽  
Higher Order ◽  
Numerical Results ◽  
Thermoelastic Problem ◽  
Asymptotic Expressions ◽  
Element Error ◽  
Element Method

In this paper, one new coupled higher order two-scale finite element method (TSFEM) for thermoelastic problem in composites is proposed. Firstly, some new two-scale asymptotic expressions and homogenization formulations for the problem are briefly given. Next, some high–low coupled approximate errors corresponding to TSFEM are analyzed. Finally, some numerical results of the displacement and the increment of temperature are presented, which show that TSFEM is an effective method for predicting the mechanical and the thermal behavior of composites in quasi-periodic perforated structure.

A Numerical Analysis for Cracks Emanating from a Quarter-Spherical Cavity on the Edge of an Elastic Body

2012 ◽  
Vol 499 ◽  
pp. 243-247
Author(s):  
Long Hai Yan ◽  
Bao Liang Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Analysis ◽  
Elastic Body ◽  
Spherical Cavity ◽  
Numerical Results ◽  
Shielding Effect ◽  
Corner Crack ◽  
Element Method

This note is specifically concerned with cracks emanating from a quarter-spherical cavity on the edge in an elastic body (see Fig.1) by using finite element method. The numerical results show that the existence of the cavity has a shielding effect of the corner crack. In addition, it is found that the effect of boundaries parallel to the crack on the SIFs is obvious when.H/R≤3

scholarly journals MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

2021 ◽  
Author(s):  
Dong T.P. Nguyen ◽  
Dirk Nuyens
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Monte Carlo ◽  
Convergence Rates ◽  
Higher Order ◽  
Elliptic Pdes ◽  
Infinite Dimensional ◽  
Quasi Monte Carlo ◽  
Higher Order Convergence ◽  
Element Method

We introduce the \emph{multivariate decomposition finite element method} (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\bsy) = \sum_{j \ge 1} y_j \, \phi_j$ with $y_j \sim \calN(0,1)$ and a given sequence of functions $\{\phi_j\}_{j \ge 1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the \emph{multivariate decomposition method} (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using \emph{quasi-Monte Carlo} (QMC) methods, and for which we use the \emph{finite element method} (FEM) to solve different instances of the PDE.   We develop higher-order quasi-Monte Carlo rules for integration over the finite-di\-men\-si\-onal Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of \emph{anchored Gaussian Sobolev spaces} while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm.   Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-\dd/\lambda}) = O(\epsilon^{-(p^* + \dd/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-\dd/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $\dd = d \, (1+\ddelta)$ for some $\ddelta \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2 + \dd/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j \ge 1}$.

The Semi-Discrete Finite Element Method for the Cauchy Equation

2014 ◽  
Vol 668-669 ◽  
pp. 1130-1133
Author(s):  
Lei Hou ◽  
Xian Yan Sun ◽  
Lin Qiu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Results ◽  
Cauchy Equation ◽  
The Time Domain ◽  
Nicolson Scheme ◽  
Discrete Finite Element ◽  
Better Than ◽  
Element Method ◽  
Crank Nicolson

In this paper, we employ semi-discrete finite element method to study the convergence of the Cauchy equation. The convergent order can reach. In numerical results, the space domain is discrete by Lagrange interpolation function with 9-point biquadrate element. The time domain is discrete by two difference schemes: Euler and Crank-Nicolson scheme. Numerical results show that the convergence of Crank-Nicolson scheme is better than that of Euler scheme.

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