Application of 2D base force element method with complementary energy principle for arbitrary meshes

2014 ◽  
Vol 31 (4) ◽  
pp. 691-708 ◽  
Author(s):  
Yijiang Peng ◽  
Nana Zong ◽  
Lijuan Zhang ◽  
Jiwei Pu
Keyword(s):  
Lagrange Multiplier ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
Complementary Energy ◽  
Energy Principle ◽  
Content Type ◽  
Complementary Energy Principle ◽  
Force Element ◽  
2D Model ◽  
Element Method

Purpose – The purpose of this paper is to present a two-dimensional (2D) model of the base force element method (BFEM) based on the complementary energy principle. The study proposes a model of the BFEM for arbitrary mesh problems. Design/methodology/approach – The BFEM uses the base forces given by Gao (2003) as fundamental variables to describe the stress state of an elastic system. An explicit expression of element compliance matrix is derived using the concept of base forces. The detailed formulations of governing equations for the BFEM are given using the Lagrange multiplier method. The explicit displacement expression of nodes is given. To verify the 2D model, a program on the BFEM using MATLAB language is made and a number of examples on arbitrary polygonal meshes and aberrant meshes are provided to illustrate the BFEM. Findings – A good agreement is obtained between the numerical and theoretical results. Based on the studies, it is found that the 2D formulation of BFEM with complementary energy principle provides reliable predictions for arbitrary mesh problems. Research limitations/implications – Due to the use of Lagrange multiplier method, there are more basic unknowns in the control equation. The proposed method will be improved in the future. Practical implications – This paper presents a new idea and a new numerical method, and to explore new ways to solve the problem of arbitrary meshes. Originality/value – The paper presents a 2D model of the BFEM using the complementary energy principle for arbitrary mesh problems.

scholarly journals A 4-Mid-Node Plane Model of Base Force Element Method on Complementary Energy Principle

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yinghua Liu ◽  
Yijiang Peng ◽  
Lijuan Zhang ◽  
Qing Guo
Keyword(s):  
Complementary Energy ◽  
Plane Model ◽  
Energy Principle ◽  
Deformation State ◽  
Plane Element ◽  
Complementary Energy Principle ◽  
Elasticity Problems ◽  
Force Element ◽  
Base Forces ◽  
Element Method

Using the base forces as fundamental variables to describe the stress state and the displacement gradients that are the conjugate variables of the base forces to describe the deformation state for the two-dimensional elasticity problems, a 4-mid-node plane model of base force element method (BFEM) based on complementary energy principle is proposed. In this paper, the complementary energy of an element of the BFEM is constructed by using the base forces. The equilibrium conditions are released by the Lagrange multiplier method, and a modified complementary energy principle described by the base forces is obtained. The formulation of the 4-mid-node plane element of the BFEM is derived by assuming that the stress is uniformly distributed on each edge of the plane elements. A procedure of the BFEM on complementary energy principle is developed using MATLAB language. The numerical results of examples show that this model of the BFEM has high precision and is free from mesh sensitivity. This model shows good performances.

scholarly journals Application of Base Force Element Method on Complementary Energy Principle to Rock Mechanics Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Yijiang Peng ◽  
Qing Guo ◽  
Zhaofeng Zhang ◽  
Yanyan Shan
Keyword(s):  
Crack Propagation ◽  
Rock Mechanics ◽  
Safety Factor ◽  
Isoparametric Element ◽  
Complementary Energy ◽  
Plane Model ◽  
Energy Principle ◽  
Complementary Energy Principle ◽  
Force Element ◽  
Element Method

The four-mid-node plane model of base force element method (BFEM) on complementary energy principle is used to analyze the rock mechanics problems. The method to simulate the crack propagation using the BFEM is proposed. And the calculation method of safety factor for rock mass stability was presented for the BFEM on complementary energy principle. The numerical researches show that the results of the BFEM are consistent with the results of conventional quadrilateral isoparametric element and quadrilateral reduced integration element, and the nonlinear BFEM has some advantages in dealing crack propagation and calculating safety factor of stability.

scholarly journals The Pulley Element

2016 ◽  
Vol 16 (2) ◽  
pp. 161-164
Author(s):  
Hynek Štekbauer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Lagrange Multiplier ◽  
Penalty Method ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
The Finite Element Method ◽  
Cable System ◽  
Mechanical Advantage ◽  
Element Method

Abstract The pulley is used in a number of structures for the mechanical advantage it gives. This paper presents an approach for the calculation of a pulley-cable system using a special pulley element in the finite element method. The Lagrange Multiplier method and Penalty method are used to define the pulley element, as described in this paper. Both approaches are easy to implement in general FEM codes.

A combined reordering procedure for preconditioned GMRES applied to solving equations using Lagrange multiplier method

2014 ◽  
Vol 31 (7) ◽  
pp. 1283-1304
Author(s):  
Zixiang Hu ◽  
Zhenmin Wang ◽  
Shi Zhang ◽  
Yun Zhang ◽  
Huamin Zhou
Keyword(s):  
Iterative Method ◽  
Lagrange Multiplier ◽  
Large Scale ◽  
Minimum Degree ◽  
Coefficient Matrix ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
Content Type ◽  
Wide Range ◽  
Solving Equations

Purpose – The purpose of this paper is to propose a combined reordering scheme with a wide range of application, called Reversed Cuthill-McKee-approximate minimum degree (RCM-AMD), to improve a preconditioned general minimal residual method for solving equations using Lagrange multiplier method, and facilitates the choice of the reordering for the iterative method. Design/methodology/approach – To reordering the coefficient matrix before a preconditioned iterative method will greatly impact its convergence behavior, but the effect is very problem-dependent, even performs very differently when different preconditionings applied for an identical problem or the scale of the problem varies. The proposed reordering scheme is designed based on the features of two popular ordering schemes, RCM and AMD, and benefits from each of them. Findings – Via numerical experiments for the cases of various scales and difficulties, the effects of RCM-AMD on the preconditioner and the convergence are investigated and the comparisons of RCM, AMD and RCM-AMD are presented. The results show that the proposed reordering scheme RCM-AMD is appropriate for large-scale and difficult problems and can be used more generally and conveniently. The reason of the reordering effects is further analyzed as well. Originality/value – The proposed RCM-AMD reordering scheme preferable for solving equations using Lagrange multiplier method, especially considering that the large-scale and difficult problems are very common in practical application. This combined reordering scheme is more wide-ranging and facilitates the choice of the reordering for the iterative method, and the proposed iterative method has good performance for practical cases in in-house and commercial codes on PC.

An efficient preconditioned Krylov subspace method for large-scale finite element equations with MPC using Lagrange multiplier method

2014 ◽  
Vol 31 (7) ◽  
pp. 1169-1197
Author(s):  
Zixiang Hu ◽  
Shi Zhang ◽  
Yun Zhang ◽  
Huamin Zhou ◽  
Dequn Li
Keyword(s):  
Iterative Method ◽  
Lagrange Multiplier ◽  
Numerical Stability ◽  
Large Scale ◽  
Krylov Subspace ◽  
Direct Method ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
Memory Usage ◽  
Content Type

Purpose – The purpose of this paper is to propose an efficient iterative method for large-scale finite element equations of bad numerical stability arising from deformation analysis with multi-point constraint using Lagrange multiplier method. Design/methodology/approach – In this paper, taking warpage analysis of polymer injection molding based on surface model as an example, the performance of several popular Krylov subspace methods, including conjugate gradient, BiCGSTAB and generalized minimal residual (GMRES), with diffident Incomplete LU (ILU)-type preconditions is investigated and compared. For controlling memory usage, GMRES(m) is also considered. And the ordering technique, commonly used in the direct method, is introduced into the presented iterative method to improve the preconditioner. Findings – It is found that the proposed preconditioned GMRES method is robust and effective for solving problems considered in this paper, and approximate minimum degree (AMD) ordering is most beneficial for the reduction of fill-ins in the ILU preconditioner and acceleration of the convergence, especially for relatively accurate ILU-type preconditioning. And because of concerns about memory usage, GMRES(m) is a good choice if necessary. Originality/value – In this paper, for overcoming difficulties of bad numerical stability resulting from Lagrange multiplier method, together with increasing scale of problems in engineering applications and limited hardware conditions of computer, a stable and efficient preconditioned iterative method is proposed for practical purpose. Before the preconditioning, AMD reordering, commonly used in the direct method, is introduced to improve the preconditioner. The numerical experiments show the good performance of the proposed iterative method for practical cases, which is implemented in in-house and commercial codes on PC.

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