professor s.liu的經(jīng)典有限元實用教程special purpose elements

1、1Finite Element MethodSPECIAL PURPOSE ELEMENTSA Practical Course G. R. Liu and S. S. QuekCHAPTER 10: 2CONTENTSCRACK TIP ELEMENTSMETHODS FOR INFINITE DOMAINSInfinite elements formulated by mappingGradual damping elementsCoupling of FEM and BEMCoupling of FEM and SEMFINITE STRIP ELEMENTSSTRIP ELEMENT

2、METHOD3CRACK TIP ELEMENTSFracture mechanics singularity point at crack tipConventional finite elements do not give good approximation at/near the crack tip.4CRACK TIP ELEMENTSFrom fracture mechanics,(Near crack tip)(Mode I fracture)5CRACK TIP ELEMENTSSpecial purpose crack tip element with middle nod

3、es shifted to quarter position:6CRACK TIP ELEMENTSx = -0.5 (1-)x1 + (1+)(1-)x2 + 0.5 (1+) x3 u = -0.5 (1-)u1 + (1+)(1-)u2 + 0.5 (1+) u3 (Measured from node 1)Move node 2 to L/4 positionx1 = 0, x2 = L/4, x3 = L, u1 = 0 x = 0.25(1+)(1-)L + 0.5 (1+)L u = (1+)(1-)u2+0.5 (1+) u3 7CRACK TIP ELEMENTSSimpli

4、fying,x = 0.25(1+)2L u= (1+)(1-)u2+0.5u3 Along x-axis, x = rr = 0.25(1+)2L or u = 2(r/L) (1-)u2 + 0.5u3 Note: Displacement is proportional to rwhereTherefore,Note: Strain (hence stress) is proportional to 1/r8CRACK TIP ELEMENTSTherefore, by shifting the nodes to quarter position, we approximating th

5、e stress and displacements more accurately.Other crack tip elements:9METHODS FOR INFINITE DOMAINInfinite elements formulated by mapping (Zienkiewicz and Taylor, 2000)Gradual damping elementsCoupling of FEM and BEMCoupling of FEM and SEM10Infinite elements formulated by mappingUse shape functions to

6、approximate decaying sequence:In 1D:(Coordinate interpolation)11Infinite elements formulated by mappingIf the field variable is approximated by polynomial,Substituting will give function of decaying form, For 2D (3D):12Infinite elements formulated by mappingElement PP1QQ1RR1 :with13Infinite elements

7、 formulated by mappingInfinite elements are attached to conventional FE mesh to simulate infinite domain.14Gradual damping elementsFor vibration problems with infinite domainUses conventional finite elements, hence great versatilityStudy of lamb waves propagation15Gradual damping elementsAttaching a

8、dditional damping elements outside area of interest to damp down propagating waves16Gradual damping elementsStructural damping is defined asEquation of motion with damping under harmonic load:Since,Therefore,(Since the energy dissipated by damping is usually independent of )17Gradual damping element

9、sComplex stiffnessReplace E with E(1 + i) where is the material loss factor. Therefore,Hence,18Gradual damping elementsFor gradual increase in damping,Complex modulus for the kth damping element setInitial modulusInitial material loss factorConstant factorSufficient damping such that the effect of t

10、he boundary is negligible.Damping is gradual enough such that there is no reflection cause by a sudden damped condition. 19Coupling of FEM and BEMFEM used for interior and the BEM used for exterior which can be extended to infinity Liu, 1992Coupling of FEM and SEMFEM used for interior and the SEM us

11、ed for exterior which can be extended to infinity Liu, 200220FINITE STRIP ELEMENTSDeveloped by Y. K. Cheung, 1968Used for problems with regular geometry and simple boundary.Key is in obtaining the shape functions.21FINITE STRIP ELEMENTS(Approximation of displacement function)(Polynomial)(Continuous

12、series)Polynomial function must represent state of constant strain in x direction and continuous series must satisfy end conditions of the strip. Together the shape function must satisfy compatibility of displacements with adjacent strips.22FINITE STRIP ELEMENTSY(0) = 0, Y(0) = 0, Y(a) = 0 and Y(a)

13、= 0 am = , 2, 3, , m Satisfies23FINITE STRIP ELEMENTSTherefore, 24FINITE STRIP ELEMENTSorwherei = 1, 2, 3 ,4 The remaining procedure is the same as the FEM. The size of the matrix is usually much smaller and makes the solving much easier.25STRIP ELEMENT METHOD (SEM)Proposed by Liu and co-workers Liu et al. 1994, 1995; Liu and Xi, 2001Solving wave propagation in composite laminates.S