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

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Finite Element Method by G. R. Liu and S. S. Quek 2 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

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Finite Element Method by G. R. Liu and S. S. Quek 3 CRACK TIP ELEMENTS Fracture mechanics – singularity point at crack tip. Conventional finite elements do not give good approximation at/near the crack tip.

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Finite Element Method by G. R. Liu and S. S. Quek 4 CRACK TIP ELEMENTS From fracture mechanics, (Near crack tip) (Mode I fracture)

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Finite Element Method by G. R. Liu and S. S. Quek 5 CRACK TIP ELEMENTS Special purpose crack tip element with middle nodes shifted to quarter position:

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Finite Element Method by G. R. Liu and S. S. Quek 6 CRACK TIP ELEMENTS x = 0.5 (1- )x 1 + (1+ )(1- )x (1+ ) x 3 u = 0.5 (1- )u 1 + (1+ )(1- )u (1+ ) u 3 (Measured from node 1) Move node 2 to L/4 position x 1 = 0, x 2 = L/4, x 3 = L, u 1 = 0 x = 0.25(1+ )(1- )L (1+ )L u = (1+ )(1- )u (1+ ) u 3

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Finite Element Method by G. R. Liu and S. S. Quek 7 CRACK TIP ELEMENTS Simplifying, x = 0.25(1+ ) 2 L u= (1+ )[(1- )u u 3 ] Along x-axis, x = r r = 0.25(1+ ) 2 L or u = 2( r/ L) [(1- )u u 3 ] Note: Displacement is proportional to r where Therefore, Note: Strain (hence stress) is proportional to 1/ r

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Finite Element Method by G. R. Liu and S. S. Quek 8 CRACK TIP ELEMENTS Therefore, by shifting the nodes to quarter position, we approximating the stress and displacements more accurately. Other crack tip elements:

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Finite Element Method by G. R. Liu and S. S. Quek 9 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

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Finite Element Method by G. R. Liu and S. S. Quek 10 Infinite elements formulated by mapping Use shape functions to approximate decaying sequence: In 1D: (Coordinate interpolation)

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Finite Element Method by G. R. Liu and S. S. Quek 11 Infinite elements formulated by mapping If the field variable is approximated by polynomial, Substituting will give function of decaying form, For 2D (3D):

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Finite Element Method by G. R. Liu and S. S. Quek 12 Infinite elements formulated by mapping Element PP 1 QQ 1 RR 1 : with

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Finite Element Method by G. R. Liu and S. S. Quek 13 Infinite elements formulated by mapping Infinite elements are attached to conventional FE mesh to simulate infinite domain.

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Finite Element Method by G. R. Liu and S. S. Quek 14 Gradual damping elements For vibration problems with infinite domain Uses conventional finite elements, hence great versatility Study of lamb wave propagation

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Finite Element Method by G. R. Liu and S. S. Quek 15 Gradual damping elements Attaching additional damping elements outside area of interest to damp down propagating waves

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Finite Element Method by G. R. Liu and S. S. Quek 16 Gradual damping elements Structural damping is defined as Equation of motion with damping under harmonic load: Since, Therefore, (Since the energy dissipated by damping is usually independent of )

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Finite Element Method by G. R. Liu and S. S. Quek 17 Gradual damping elements Complex stiffness Replace E with E(1 + i ) where is the material loss factor. Therefore, Hence,

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Finite Element Method by G. R. Liu and S. S. Quek 18 Gradual damping elements For gradual increase in damping, Complex modulus for the k th damping element set Initial modulus Initial material loss factor Constant 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.

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Finite Element Method by G. R. Liu and S. S. Quek 19 Coupling of FEM and BEM 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]

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Finite Element Method by G. R. Liu and S. S. Quek 20 FINITE STRIP ELEMENTS Developed by Y. K. Cheung, Used for problems with regular geometry and simple boundary. Key is in obtaining the shape functions.

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Finite Element Method by G. R. Liu and S. S. Quek 21 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.

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Finite Element Method by G. R. Liu and S. S. Quek 22 FINITE STRIP ELEMENTS Y(0) = 0, Y(0) = 0, Y(a) = 0 and Y(a) = 0 a m =, 2, 3, …, m Satisfies

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Finite Element Method by G. R. Liu and S. S. Quek 23 FINITE STRIP ELEMENTS Therefore,

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Finite Element Method by G. R. Liu and S. S. Quek 24 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.

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Finite Element Method by G. R. Liu and S. S. Quek 25 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.

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