Presentation on theme: "SPECIAL PURPOSE ELEMENTS"— Presentation transcript:
1SPECIAL PURPOSE ELEMENTS Finite Element Methodfor readers of all backgroundsG. R. Liu and S. S. QuekCHAPTER 10:SPECIAL PURPOSE ELEMENTS
2CONTENTS CRACK TIP ELEMENTS METHODS FOR INFINITE DOMAINS Infinite elements formulated by mappingGradual damping elementsCoupling of FEM and BEMCoupling of FEM and SEMFINITE STRIP ELEMENTSSTRIP ELEMENT METHOD
3CRACK TIP ELEMENTSFracture mechanics – singularity point at crack tip.Conventional finite elements do not give good approximation at/near the crack tip.
4CRACK TIP ELEMENTS From fracture mechanics, (Near crack tip) (Mode I fracture)
5CRACK TIP ELEMENTSSpecial purpose crack tip element with middle nodes shifted to quarter position:
6CRACK TIP ELEMENTS Move node 2 to L/4 position x = -0.5 (1-)x1 + (1+)(1-)x (1+) x3u = -0.5 (1-)u1 + (1+)(1-)u (1+) u3(Measured from node 1)Move node 2 to L/4 positionx1 = 0, x2 = L/4, x3 = L, u1 = 0x = 0.25(1+)(1-)L + 0.5 (1+)Lu = (1+)(1-)u2+0.5 (1+) u3
7CRACK TIP ELEMENTS Simplifying, Along x-axis, x = r where Therefore, x = 0.25(1+)2Lu= (1+)[(1-)u2+0.5u3]Along x-axis, x = rr = 0.25(1+)2L orNote: Displacement is proportional to ru = 2(r/L) [(1-)u u3]whereNote: Strain (hence stress) is proportional to 1/rTherefore,
8CRACK TIP ELEMENTSTherefore, by shifting the nodes to quarter position, we approximating the stress and displacements more accurately.Other crack tip elements:
9METHODS FOR INFINITE DOMAIN Infinite elements formulated by mapping(Zienkiewicz and Taylor, 2000)Gradual damping elementsCoupling of FEM and BEMCoupling of FEM and SEM
10Infinite elements formulated by mapping Use shape functions to approximate decaying sequence:In 1D:(Coordinate interpolation)
11Infinite elements formulated by mapping If the field variable is approximated by polynomial,Substituting will give function of decaying form,For 2D (3D):
12Infinite elements formulated by mapping Element PP1QQ1RR1 :with
13Infinite elements formulated by mapping Infinite elements are attached to conventional FE mesh to simulate infinite domain.
14Gradual damping elements For vibration problems with infinite domainUses conventional finite elements, hence great versatilityStudy of lamb wave propagation
15Gradual damping elements Attaching additional damping elements outside area of interest to damp down propagating waves
16Gradual damping elements (Since the energy dissipated by damping is usually independent of )Structural damping is defined asEquation of motion with damping under harmonic load:Since,Therefore,
17Gradual damping elements Complex stiffnessReplace E with E(1 + i) where is the material loss factor.Therefore,Hence,
18Gradual damping elements For gradual increase in damping,Constant factorComplex modulus for the kth damping element setInitial modulusInitial material loss factorSufficient damping such that the effect of the boundary is negligible.Damping is gradual enough such that there is no reflection cause by a sudden damped condition.
19Coupling of FEM and BEM Coupling of FEM and SEM The FEM used for interior and the BEM for exterior which can be extended to infinity [Liu, 1992]Coupling of FEM and SEMThe FEM used for interior and the SEM for exterior which can be extended to infinity [Liu, 2002]
20FINITE STRIP ELEMENTS Developed by Y. K. Cheung, 1968. Used for problems with regular geometry and simple boundary.Key is in obtaining the shape functions.
21FINITE STRIP ELEMENTS (Approximation of displacement function) (Polynomial)(Continuous series)Polynomial function must represent state of constant strain in the x direction and continuous series must satisfy end conditions of the strip.Together the shape function must satisfy compatibility of displacements with adjacent strips.
24FINITE STRIP ELEMENTS or where i = 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, 2001].Solving wave propagation in composite laminates.Semi-analytic method for stress analysis of solids and structures.Applicable to problems of arbitrary boundary conditions including the infinite boundary conditions.Coupling of FEM and SEM for infinite domains.