摘 要

本文主要是探討有限元素法中的高階元素對求解裂縫端之應力強度因子（SIF）的效能，並與奇異元素作個比較。在這之前也對超收斂點的位置和靜態縮減的方法作一些的探討。
對能量誤差而言， 階Lagrange型態的元素在n*n個高斯積分點的位置上有(n+1)的超收斂率；且其收斂率和精確度方面亦較其它兩種元素來的好。若將高階Lagrange型態之元素在求解部分搭配靜態縮減的方法，將可大量的縮短求解時間，尤其是整體自由度越來越大時。
當使用位移外插法來求SIF時，二次外插法似乎較線性外插法來的合理，且有時其精確度亦較高。針對奇異性為1/sqrt(r)的問題時，奇異元素較高階元素有著更好的近似結果，特別是在裂縫端配置三角形奇異元素時
，然而高階元素亦有不錯的結果（誤差小於1%）。
針對相同總自由度的問題，若高階元素能搭配靜態縮減的方法，其不僅較低階元素求解時間快且精度亦較高。

關鍵字：有限元素法、超收斂率、高階元素、位移外插法、應力強度因
子、靜態縮減矩陣

Abstract

The purpose of this research is to study the effect of higher order elements
in solving the crack tip stress intensity factor (SIF), and to compare the results
with those obtained by using the singular elements. We also study the positions
of the superconvergent points as well as the method of static condensation
beforehand.

In energy norm,the n-th order Lagrange elements have the superconvergent
rate with the order of n+1,and their locations are on n by n Gauss integration
points. The accuracy and convergent rate of the Lagrange elements are better
than the other two kinds of elements which are Serendipity and Enriched
Serendipity. The method which using the static condensation in solving process
will save a large amount of time with lager and lager degrees of freedom
especially.

When using the method of displacement extrapolation to solve SIF, we
find that the second order interpolation method seems to be more reasonable
than the first order one.Singular elements have the better approximate results
than higher order elements with the problems of the singularity equaling to 1/sqrt(r), especially in the case of using triangular singular elements. But the
higher order elements also have satisfying results (the error less than 1% ).

For the problem of the same total degrees of freedom,the higher order
elements spend less time and have better accuracy than lower order elements,
if these higher order ones can collocate the method of static condensation in
solving process.

keywords : finite element method,superconvergent rate, higher order
elements, displacement extrapolation method,stress intensity
factor, static condensation

參考文獻
[1] W. E. Anderson, “An engineer views brittle fracture history”, Boeing rept., 1969.
[2] D. Broek, “Elementary Engineering Fracture Mechanics”, 4th Edition, Martinus Nijhoff Publishers, 1986.
[3] W. X. Zhu and D. J. Smith, “On the use of displacement extrapolation
to obtain crack tip singular stresses and stress intensity factors”,
Engng. Fractre Mech. 51, 391-400, 1995.
[4] G. V. Guinea, J. Planas and M. Elices, “ evaluation by the
displacement extrapolation technique”, Engng. Fracture Mech. 66,
243-255, 2000.
[5] L. S. Chen and J. H. Kuang, “A modified linear extrapolation formul
for determination of stress intensity factors”, Int. J. Fract. 54, R3-R8
, 1992.
[6] R. D. Henshell and K. G. Shaw, “Crack tip finite element are unnecessary”
, Int. J. Numer. Methods Engng. 9, 495-507, 1975.
[7] J. R. Rice, “A path independent integral and the approximate
analysis of strain concentration by notches and cracks”, J. Appl.
Mech. 35 , 379-386, 1968.
[8] L. Banks-Sills and D. Sherman, “Comparison of method for
calculating stress intensity factors with quarter-point elements”,
Int. J. Fract. Mech. 32, 127-140, 1986.
[9] D. M. Parks, “A stiffness derivative finite element technique for
determination of crack tip stress intensity factors”, Int. J. Fract. 10,
487-502, 1974.
[10] B. Moran and CF. Shih, “A general treatment of crack tip contour
integrals”, Int. J. Fract. 35, 295-310, 1987.
[11] J. N. Reddy, “An Introduction to the Finite Element Method”, 2nd
Edition, McGraw-Hill, Singapore, 1993.
[12] D. R. J. Owen and A. J. Fawkes, “Engineering Fracture Mechanics :
Numerical Methods and Applications”, Pineridge Press Ltd., Swansea
, U.K., 1983.
[13] R. S. Barsoum, “On the use of isoparametric finite elements in linear
fracture mechanics”, Int. J. Numer. Methods Engng. 10, 25-37, 1976.
[14] L. Banks-Sills and O. Einav, “On singular, nine-noded, distorted,
isoparametric elements in linear elastic fracture mechanics”,
Comput. Struct. 25, 445-449, 1987.
[15] F. C. M. Menandro, E. T. Moyer, JR and H. Liebowitz, “A new
optimal crack tip mesh”, Engng. Fracture Mech. 50, 703-711,
1995.
[16] F. C. M. Menandro, E. T. Moyer, JR and H. Liebowitz, “A
methodology for crack tip mesh design”, Engng. Fracture Mech. 50,
713-726, 1995.
[17] K. S. Woo and C. G. Lee, “P-version finite element approximations
of stress intensity factors for cracked plates including shear
deformation”, Engng. Fracture Mech. 52, 493-502, 1995.
[18] K. S. Woo and P. K. Basu, “Analysis of singular cylindrical shells
by p-version of FEM”, Int. J. Solids Struct. 25, 151-165, 1989.
[19] D. S. Burnett, “Finite Element Analysis From Concepts to
Applications”, Addison-Wesley, Canada, 1987.
[20] D. L. Logan, “A First Course in the Finite Element Method”, PWSKENT
, Boston, 1992.
[21] A. P. Boresi and K. P. Chong, “Elasticity In Engineering
Mechanics”, 2nd Edition, John Wiley & Sons, USA, 2000.
[22] H. T. Rathod and S. Kilari, “General complete lagrange family for
the cube in the finite element interpolations”, Comput. Methods
Appl. Mech. Engng. 181, 295-344, 2000.
[23]蔡文傑，”二維有限元素法之收斂性探討”，國立成功大學機械工程
研究所碩士論文，2001。
[24] H. Jean-Francois and B. Klaus-Jurgen, “On higher-order-accuracy
points in isoparametric finite element analysis and an application
to error assessment”, Comput. Struct. 79, 1275-1285, 2001.
[25]A. H. Stroud and D. Secrest, “Gaussian Quadrature Formulas”, Prentice
Hall, Englewood Cliffs, N. J., 1966.
[26] M. E. Laursen and M. Gellert, “Some criteria for numerically
integrated matrices and quadrature formulas for triangles”, Int. J.
Numer. Methods Engng. 12, 67-76, 1978.
[27] P. Hillion, “Numerical integration on a triangle”, Int. J. Numer.
Methods Engng. 11, 797-915, 1977.
[28] C. T. Reddy and D. J. Shippy, “Short communications alternative
integration formulae for triangular finite elements”, Int. J. Numer.
Methods Engng. 17, 133-153, 1981.
[29] Y. Murakami, “Stress Intensity Factors Handbook”, Pergamon
Press, 2000.
[30] L. N. Trefethen and D. Bau, “Numerical Linear Algebra”, SIAM,
USA, 1997.
[31]陳明凱，”平行有限元素法之負載平衡探討”，國立成功大學機械工
程研究所碩士論文，2001。