SPECIAL PURPOSE ELEMENTS Finite Element Method for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 10: SPECIAL PURPOSE ELEMENTS

CONTENTS CRACK TIP ELEMENTS METHODS FOR INFINITE DOMAINS Infinite elements formulated by mapping Gradual damping elements Coupling of FEM and BEM Coupling of FEM and SEM FINITE STRIP ELEMENTS STRIP ELEMENT METHOD

CRACK TIP ELEMENTS Fracture mechanics – singularity point at crack tip. Conventional finite elements do not give good approximation at/near the crack tip.

CRACK TIP ELEMENTS From fracture mechanics, (Near crack tip) (Mode I fracture)

CRACK TIP ELEMENTS Special purpose crack tip element with middle nodes shifted to quarter position:

CRACK TIP ELEMENTS Move node 2 to L/4 position  x = -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 position x1 = 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

CRACK TIP ELEMENTS Simplifying, Along x-axis, x = r  where Therefore, x = 0.25(1+)2L u= (1+)[(1-)u2+0.5u3] Along x-axis, x = r r = 0.25(1+)2L or  Note: Displacement is proportional to r u = 2(r/L) [(1-)u2 + 0.5u3] where Note: Strain (hence stress) is proportional to 1/r Therefore,

CRACK TIP ELEMENTS Therefore, by shifting the nodes to quarter position, we approximating the stress and displacements more accurately. Other crack tip elements:

METHODS FOR INFINITE DOMAIN Infinite elements formulated by mapping (Zienkiewicz and Taylor, 2000) Gradual damping elements Coupling of FEM and BEM Coupling of FEM and SEM

Infinite elements formulated by mapping Use shape functions to approximate decaying sequence: In 1D: (Coordinate interpolation) 

Infinite elements formulated by mapping If the field variable is approximated by polynomial, Substituting  will give function of decaying form, For 2D (3D):

Infinite elements formulated by mapping Element PP1QQ1RR1 : with

Infinite elements formulated by mapping Infinite elements are attached to conventional FE mesh to simulate infinite domain.

Gradual damping elements For vibration problems with infinite domain Uses conventional finite elements, hence great versatility Study of lamb wave propagation

Gradual damping elements Attaching additional damping elements outside area of interest to damp down propagating waves

Gradual damping elements (Since the energy dissipated by damping is usually independent of ) Structural damping is defined as Equation of motion with damping under harmonic load: Since, Therefore,

Gradual damping elements Complex stiffness Replace E with E(1 + i) where  is the material loss factor. Therefore, Hence,

Gradual damping elements For gradual increase in damping, Constant factor Complex modulus for the kth damping element set Initial modulus Initial material loss factor Sufficient 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.

Coupling 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 SEM The FEM used for interior and the SEM for exterior which can be extended to infinity [Liu, 2002]

FINITE STRIP ELEMENTS Developed by Y. K. Cheung, 1968. Used for problems with regular geometry and simple boundary. Key is in obtaining the shape functions.

FINITE 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.

FINITE STRIP ELEMENTS Y(0) = 0, Y’’(0) = 0, Y(a) = 0 and Y’’(a) = 0 Satisfies m = , 2, 3, …, m

FINITE STRIP ELEMENTS Therefore,

FINITE 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.

STRIP 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.