HAMILTONIAN AS ERROR INDICATOR IN THE P-VERSION OF FINITE ELEMENT METHOD

2010 ◽  
Vol 34 (2) ◽  
pp. 215-223
Author(s):  
Yong-Lin Kuo ◽  
William L. Cleghorn ◽  
Kamran Behdinan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Rates Of Convergence ◽  
Computational Time ◽  
Governing Equations ◽  
Accuracy Enhancement ◽  
Error Indicator ◽  
Energy Error ◽  
Interpolation Polynomials ◽  
Element Method

This paper presents the Hamiltonian-based error analysis applied to two-dimensional elasostatic problems. The accuracy enhancement is achieved by using the p-version of finite element method. The results show that the Hamiltonian error has faster rates of convergence at lower order of interpolation polynomials to compare with the energy error, and the Hamiltonian error clearly indicates great error reductions at a certain polynomial order. This can not only obtain an accurate enough solution but also save extra computational time. Another strategy is presented by computing the residual of the Hamiltonian-based governing equations. Relative values of residuals between elements can provide an index of selecting the best polynomial orders. Illustrative examples show the validities of the two approaches.

Coupled Thermoelasticity Analysis of Annular Laminate Disk Using Laplace Transform and Galerkin Finite Element Method

2014 ◽  
Vol 656 ◽  
pp. 298-304 ◽  
Author(s):  
S.M. Nowruzpour Mehrian ◽  
Amin Nazari ◽  
Mohammad Hasan Naei
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Thermal Shock ◽  
Outer Edge ◽  
Galerkin Finite Element Method ◽  
Governing Equations ◽  
Space Domain ◽  
Annular Disk ◽  
Galerkin Finite Element ◽  
Element Method

In this paper, a dynamic analysis of annular laminate disk under radial thermal shock is carried out by employing a Galerkin Finite Element (GFE) approach. The governing equations, including the equation of the motion and energy equation are obtained based on Lord-Shulman theory. These two equations are solved simultaneously to obtain the displacement components and temperature distributions. A simply support boundary condition through outer edge is assumed for the annular disk. The inner radius is subjected to thermal shock and free of any traction. The outer edge is keeping at a constant temperature. Using Laplace transfer technique to transfer the governing equations into the space domain, where the Galerkin Finite Element Method is employed to obtain the solution in space domain. The inverse of Laplace transfer is performed numerically to achieve the final solution in the real time domain. The results are validated with the known data reported in the literature.

scholarly journals Quasi-optimality of an Adaptive Finite Element Method for Cathodic Protection

2019 ◽  
Vol 53 (5) ◽  
pp. 1645-1665
Author(s):  
Guanglian Li ◽  
Yifeng Xu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cathodic Protection ◽  
Finite Element Approximation ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Energy Error ◽  
Element Method

In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2D cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We propose a standard adaptive finite element method involving the Dörfler marking and a minimal refinement without the interior node property. Furthermore, we establish the contraction property of this adaptive algorithm in terms of the sum of the energy error and the scaled estimator. This essentially allows for a quasi-optimal convergence rate in terms of the number of elements over the underlying triangulation. Numerical experiments are provided to confirm this quasi-optimality.

A Least-Squares/Galerkin Finite Element Method for Incompressible Navier-Stokes Equations

2008 ◽  
Author(s):  
Rajeev Kumar ◽  
Brian H. Dennis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Least Squares ◽  
Basis Functions ◽  
Galerkin Finite Element Method ◽  
Governing Equations ◽  
Galerkin Finite Element ◽  
First Order ◽  
Convection Dominated ◽  
Element Method

The least-squares finite element method (LSFEM), which is based on minimizing the l2-norm of the residual, has many attractive advantages over Galerkin finite element method (GFEM). It is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike GFEM. However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have prevented the application of LSFEM to large-scale practical problems, such as three-dimensional compressible viscous flows. A simple finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by pure a LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the leastsquares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equalorder basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated with preliminary results of several benchmark problems solved using low-order C0 continuous elements.

An efficient iterative algorithm for the natural convection equations based on finite element method

2018 ◽  
Vol 28 (3) ◽  
pp. 584-605 ◽  
Author(s):  
Lei Wang ◽  
Jian Li ◽  
Pengzhan Huang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Natural Convection ◽  
Iterative Method ◽  
Error Correction ◽  
Computational Time ◽  
Content Type ◽  
Natural Convection Problem ◽  
New Iterative Method ◽  
Element Method

Purpose This paper aims to propose a new highly efficient iterative method based on classical Oseen iteration for the natural convection equations. Design/methodology/approach First, the authors solve the problem by the Oseen iterative scheme based on finite element method, then use the error correction strategy to control the error arising. Findings The new iterative method not only retains the advantage of the Oseen scheme but also saves computational time and iterative step for solving the considered problem. Originality/value In this work, the authors introduce a new iterative method to solve the natural convection equations. The new algorithm consists of the Oseen scheme and the error correction which can control the errors from the iterative step arising for solving the nonlinear problem. Comparing with the classical iterative method, the new scheme requires less iterations and is also capable of solving the natural convection problem at higher Rayleigh number.

Evaluation of Relative Permeability Models in CO2/Brine System Using Mixed and Hybrid Finite Element Method

2016 ◽  
Vol 819 ◽  
pp. 401-405
Author(s):  
J.S. Pau ◽  
William K.S. Pao ◽  
Suet Peng Yong ◽  
Paras Qadir Memon
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Relative Permeability ◽  
Carbon Capture ◽  
Carbon Capture And Storage ◽  
Computational Time ◽  
Permeability Model ◽  
Hybrid Finite Element Method ◽  
Hybrid Finite Element ◽  
Element Method

The requirement to reduce 40% carbon emission in 2020 has lead Malaysia to adopt the carbon capture and storage (CCS) technology in 2009. In this research, the pressure and transport differential equation for CO2 – brine phases flow is discretized using mixed and hybrid finite element method (MHFEM) which ensures the local continuity of the finite elements. Result shows that CO2 flow radially outward from the injection well. Three relative permeability models are investigated and it was find out that the simplified relative permeability model (SRM) has reduced the computational time by 8.3 times (when compare to Brooks and Corey model) but it is accurate for 1 year preliminary prediction. For longer period of prediction, classical Brooks and Corey and van Genuchten models shall be used.

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