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2-D Bilinear Lagrange Elements
2
1
1
u = n un
1
2
0
1
1
u
y
0
1
1
2D interpolation functions can be
constructed by defining two local
coordinates (ξ1, ξ2) and multiplying the
1D interpolation functions together:
x
x = n xn
y = n yn
u(1,  2 )   1 (1,  2 )u1   2 (1,  2 )u2   3 (1,  2 )u3   4 (1,  2 )u4
where
j1(x1,x2 ) = j1(x1)j1(x2 ) = (1- x1)(1- x2 )
j 2 (x1,x2 ) = j 2 (x1)j1(x2 ) = x1(1- x2 )
j 3 (x1,x2 ) = j1(x1)j 2 (x2 ) = (1- x1)x2
j 4 (x1,x2 ) = j 2 (x1)j 2 (x2 ) = x1x2
1 – 4 are the 2-D
bilinear Lagrange basis
functions
3-D Trilinear Lagrange Elements
The trilinear element has 8 nodes with basis functions formed
from products of of 1-D linear Lagrange functions 1=1- , and
2= as functions of three local coordinates ( 1, 2, 3
3
7
8
5
2
6
3
4
1
2
1
u   i 1,  2 , 3  ui
8
i 1
1-D Quadratic Lagrange Elements
Use three nodal parameters u1, u2 and u3
u    u11    u22    u33  
where
1  2   1  0.5 
φ2
φ1
1.0
1.0
2  4 1   
3  2   0.5 
0
1.0
0.5


0
0.5
1.0
φ3
1.0
0
0.5
1.0

1, 2, 3 are the 1-D quadratic Lagrange basis functions
Computing Strain Distributions
Install Continuity
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If running Mac OSX, first ensure that Xcode is installed and if not, install it from
the CD that shipped with your Mac or from the AppStore for Mac OSX Lion.
Go to http://www.continuity.ucsd.edu/Continuity/Download
Run installer if it doesn’t run automatically. Accept license and install in default
location.
Windows install will also install Python if not found; on Mac OS, Python 2.5 will
be installed
Run Continuity, click on Register Now.
Complete the Registration Form
making sure to use your ucsd.edu
email address, and your own key
will be emailed to you
Click OK, then File>Exit
Make and Deform a Simple 3-D Mesh
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Work through Biomechanics Tutorial 1 at:
http://www.continuity.ucsd.edu/Continuity/Documentation/Tutorials
You will make a 3-D bilinear Lagrange mesh with two elements (blue) and then
deform it (red)