EN175-23

EN175-23 - EN0175 11 / 29/ 06 Chap. 9 Finite element method...

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EN0175 11 / 29/ 06 Chap. 9 Finite element method (Read Chap 7 of Prof Bower’s notes) Principle of virtual work: + = T A i i V i i V ij ij S u t V u f V d d d δ εδσ Principle of minimum potential energy = T A i i V i i V ij ij S u t V u f V V d d d 2 1 Min εσ i t We can represent the displacement by interpolation through its values at a network of nodes (discretization), () () () a a n a a a u x N u x N x u v v v v v v = = = 1 where is the number of FEM nodes, n a u v is the nodal displacement, and are the interpolation functions. () x N a v = = nodes other all at , 0 if , 1 a a x x x N v v v Recall such interpolation function for 1D element as follows. a a N 1
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EN0175 11 / 29/ 06 More non-local forms of have been proposed in meshless/element-free FEM methods. a N FEM formulation based on principle of virtual work Consider a virtual displacement field: ( ) ( ) b b u x N x u v v v v δ = (Note summation convention over repeated indices). Inserting () a a u N x u v v v = , () ( ) b b u x N x u v v v v = into Principle of Virtual Work, the left side of the equation becomes b i a k aibk V b i a k j b l a ijkl V j i l k ijkl V ij kl ijkl V ij ij u u K V u u x N x N C V u u C V C V εδε εδσ = = = = d d d d , , The right side of the equation becomes b i b i b i A b i V b i A i i V i i u F u S N t V N f S u t V u f T T = + = + d d d d The principle of virtual work becomes ( ) 0 = b i b i a k aibk u F u K Since this must be true for any , b i u 0 = b i a k aibk F u K In matrix form, this is a set of linear algebraic equations: ( ) 1 3 2 3 2 2 2 1 1 3 1 2 1 1 1 3 2 3 2 2 2 1 1 3 1 2 1 1 3 3 3 , 3 3 , 3 2 , 3 1 , 3 3 , 2 23 22 21 3 , 1 13 12 11 × × × =
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EN175-23 - EN0175 11 / 29/ 06 Chap. 9 Finite element method...

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