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Higher Order Elements - Introduction to Finite Elements - Lecture Slides

Slides, Finite Element Method

Post: May 7th, 2013
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The lecture slides of the Introduction to Finite Elements are very helpful and interesting the main points are:Higher Order Elements, Properties of Shape Functions, Triangular Elements, Area Coordinates, Lagrange Family, Rectangular Elements, Serendipity Family, Kronecker Delta Property, Polynomial Completeness, Boundary Conditions
The lecture slides of the Introduction to Finite Elements are very helpful and interesting the main points are:Higher Order Elements, Properties of Shape Functions, Triangular Elements, Area Coordinates, Lagrange Family, Rectangular Elements, Serendipity Family, Kronecker Delta Property, Polynomial Completeness, Boundary Conditions
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Contents
MANE 4240 & CIVL 4240 Introduction to Finite Elements Reading assignment: Lecture notes Summary: • Properties of shape functions • Higher order elements in 1D • Higher order triangular elements (using area coordinates) • Higher order rectangular elements Lagrange family Serendipity family Higher order elements Recall that the finite element shape functions need to satisfy the following properties 1. Kronecker delta property ⎧ 1 at node i Ni = ⎨ ⎩0 at all other nodes 2. Polynomial completeness If u ≈ α1 + α 2 x + α 3 y Then ∑N i i i =1 i Inside an element u = N1u1 + N 2 u 2 + ... ∑N x i i i =x =y ∑N y i At node 1, N1=1, N2=N3=…=0, hence u node 1 = u1 Facilitates the imposition of boundary conditions Docsity.com 1 Higher order elements in 1D 3-noded (quadratic) element: x1 x3 x − x2 x1 − x 2 x − x1 x 2 − x1 2-noded (linear) element: N1 = x2 x 2 1 3 x1 1 x2 2 x (x − x 2 )(x − x3 ) (x1 − x 2 )(x1 − x3 ) (x − x1 )(x ..

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