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Outline Introduction to Finite Element Formulations

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1 Outline Introduction to Finite Element Formulations
Direct Stiffness Method for Linear Elasticity Problems Element Shape Functions, Tetrahedron T4 Implementation Generalized Finite Element Formulations (Weighted Residuals Methods and Variational Methods) Tetrahedron T10 Implementation “Hierarchical” Shape Functions, Tetrahedron TP2 Implementation

2 differential equations
Engineering Mathematics Structural analog substitution Direct continuum elements Trial functions for differential equations Variational methods Weighted residuals Present-Day Finite Element Method

3 u: unknown displ. in element e N: prescribed element shape functions ae: unknown nodal displ. for element e For a particular location (x, y) in element e y i e j k x i 1.0 x y ?? 1.0 0.0

4 e: strains : suitable differential operator where In 2D linear elasticity:

5 s: stresses e: strains D: constitutive matrix containing proper material properties In 2D linear elasticity (plane stress): s D

6 Boundary Conditions qe: equivalent nodal forces e i j k (qy)i (qx)i (qy)j (qx)j i y e k j x

7 Impose an arbitrary (virtual) nodal displacement
Internal Virtual Work = External Virtual Work Linear algebra problem For the whole domain

8 Tetrahedron T4 Implementation (Linear Shape Function)
2 N0 = 1 when r=1 N0 = 0 when r=0 3 1 N0 = r s r

9 Tetrahedron T4 Implementation
*T4Shape::shape_functions shape->at(0) = r shape->at(1) = s shape->at(2) = t shape->at(3) = u N0 = r N1 = s N2 = t N3 = 1.0-r-s-t = u *T4Shape::shape_derivatives deriv->at(0,0) = 1.0 deriv->at(0,1) = 0.0 deriv->at(0,2) = 0.0 deriv->at(0,3) = -1.0 dN0/dr= 1.0 dN1/dr= 0.0 dN2/dr= 0.0 dN3/dr= -1.0 etc. etc.

10 Generalization of FEM Concept
Differential equations Direct Stiffness Method Approximate Cast in an integral form (weak form) by: Weighted Residual Methods Variational Methods Leads to Ka=q if A and B are linear differential operators

11 Weighted Residual Methods
Differential equations t Elasticity Problems Residuals Galerkin Method Find weighting functions w so: Integration by parts If w=N, we have Galerkin Method

12 Variational Methods Differential equations t Elasticity Problems S V
Define a functional Solution is a function u that makes P stationary w.r.t. small changes (Principal of Min. Potential Energy)

13 Tetrahedron T10 Implementation (Quadratic Shape Function)
End node 0 N0 = 1 when r=1 N0 = 0 at other nodal locations 2 7 5 6 3 N0 = r(2r-1) 9 8 1 Midside node 4 N4 = 1 when r=1/2 and s=1/2 N4 = 0 at other nodal locations s 4 r N4 = 4rs

14 Tetrahedron TP2 Implementation (Hierarchical Quadratic Shape Function)
End node 0 N0 = 1 when r=1 N0 = 0 at other nodal locations 2 7 5 6 3 N0 = r 8 9 1 Edge 4 Any quadratic polynomial of r and s that yields 0 at node 0 (r=1, s=0) and node 1 (s=1, r=0). s 4 r N4 = 4rs

15 Hierarchical Shape Function (Polynomial Order = 2)
N0 = r N1 = s P=1 N2 = t N3 = u N4 = 4rs P=2 N5 = 4st N6 = 4rt N7 = 4tu N8 = 4su N9 = 4ru

16 What to study next? Finite Element Books Finite Element Programs
Zienkiewicz, O.C. and Taylor, R.L.: The Finite Element Method, Volume and Volume 2, 1991 Bathe, K.J.: Finite Element Procedures, 1996 Hughes, T.J.R.: The Finite Element Method (Linear Static and Dynamic Finite Element Analysis), 1987 Cook, R.D. et al.: Concepts and Applications of Finite Element Analysis, 1989 Szabo and Babuska: Finite Element Analysis, 1991 Finite Element Programs Software developed at CFG Commercial codes: ABAQUS, ANSYS, NASTRAN, DIANA Proprietary codes: WARP3D, STAGS Internet resource:


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