Performances of finite element solution for crack problem
Abstract
The elasticity problem on a cracked panel presents the most severe difficulties in computational structural mechanics because of the appearance of strong singularity of the solution around the tips. The crack problem is characterized as a boundary value problem. There are three approaches of the finite element method to computational mechanic problems: the h, p, and hp version. The h version uses loworder polynomials and achieves the accuracy by reducing the size h of elements. The p version uses coarse mesh and increases the order of p polynomials to achieve the desired accuracy. The hp version combines both, namely reduces the element size h and increases polynomial order to obtain the exponential rate of convergence even for the problems with strong singularity with respect to the number of degrees of freedom. It was found that numerical results coincide well with the theoretical asymptotic error analysis for the h, p, and hp version at the practical engineering range of accuracy. The accuracy of the finite element solution of the h and p version is severely governed by the singularity of the solution and converges very slowly. It is very costly and even practically impossible for the h and p version to achieve high accuracy. The p version converges twice as fast as the h. The hp version converges exponentially, and the desired high accuracy can only be achieved by the hp version if the geometric mesh is properly designed. The linear distribution of polynomial order p can reduce substantially the number of degrees of freedom comparing with uniform distribution p.
 Publication:

13th Canadian Congress of Applied Mechanics
 Pub Date:
 May 1991
 Bibcode:
 1991ccam.proc..386G
 Keywords:

 Computational Grids;
 Crack Tips;
 Dynamic Structural Analysis;
 Finite Element Method;
 Polynomials;
 Singularity (Mathematics);
 Boundary Value Problems;
 Convergence;
 Error Analysis;
 Plates (Structural Members);
 Structural Mechanics