CE 595 Section 1 - Purdue University
Download
Report
Transcript CE 595 Section 1 - Purdue University
Section 4: Implementation of Finite Element
Analysis – Other Elements
1. Quadrilateral Elements
2. Higher Order Triangular Elements
3. Isoparametric Elements
Implementation of FEA:
Other Elements
-1-
Section 4.1: Quadrilateral Elements
Refers in general
to any four-sided,
2D element.
We will start by
considering rectangular
elements with sides
parallel to coordinate
axes. (Thickness = h)
Implementation of FEA:
Other Elements
-2-
4.1: Quadrilateral Elements (cont.)
Normalized Element Geometry –
“Standard” setting
for calculations:
Mapping between real and normalized coordinates:
x xc
y yc
Implementation of FEA:
;
x a xc ; y b yc . -3Other Elements
a
b
4.1: Quadrilateral Elements (cont.)
First Order Rectangular Element (Bilinear Quad):
4 nodes; 2 translational
d.o.f. per node.
Displacements interpolated as follows:
u x , y a1 a2 a3 a4
v x , y a5 a6 a7 a8
Implementation of FEA:
Other Elements
“Bilinear terms” – implies that all
shape functions are products of
linear functions of x and y.
-4-
4.1: Quadrilateral Elements (cont.)
Shape Functions:
N1 ,
1 1 ; N 2 , 14 1 1 ;
N 3 , 14 1 1 ; N 4 , 14 1 1 .
Implementation of FEA:
Other Elements
N k , 14 1 k 1 k .
1
4
-5-
4.1: Quadrilateral Elements (cont.)
Displacement interpolation becomes:
d
u x, y N1 ,
0
N 2 ,
0
N 3 ,
0
N 4 ,
0
1
.
v
x
,
y
0
N
,
0
N
,
0
N
,
0
N
,
1
2
3
4
d
8
" N x "
Need to compute [B] matrix:
B x N x
x
0
y
0
N1 ,
0
N 2 ,
0
N 3 ,
0
N 4 ,
0
? .
y
0
N
,
0
N
,
0
N
,
0
N
,
1
2
3
4
x
Implementation of FEA:
Other Elements
-6-
4.1: Quadrilateral Elements (cont.)
Chain rule:
Nk
x
Nk
y
, N
N
,
k
x
Nk
k
y
Nk
0
1 k ;
a
1
1
1
*
k b
y
4 k
4 ab 1 k .
x
14 k 1 k * a1
bk
4 ab
k
0
Resulting [B(x)] matrix:
0
b 1
0
b 1
0
b 1
0
b 1
0
a 1
0
a 1
0
a 1
0
a 1
B x 41ab
a 1 b 1 a 1 b 1 a 1 b 1 a 1 b 1
Recall general expression for [k]:
k h *
Implementation of FEA:
Other Elements
area
B x C B x dA
T
Express in terms of and !
-7-
4.1: Quadrilateral Elements (cont.)
Can show that
1 1
dx
dy
dA dxdy
d
d ab * d d dA ab * d d .
d
d
area
1 1
k h * ab *
1 1
B CB d d.
T
Everything in terms of and !
1 1
Can also show that
f N x b x dV
T
Ve
h * ab *
1 1
1
1
1
h * b
Implementation of FEA:
Other Elements
T
Ae ,
1 1
h * b
N x t x dA
1
N , b , d d h * a
T
N 1, t 1, d h * a
T
1
1
1
1
N , 1 t , 1 d
T
N , 1 t , 1 d
T
N 1, t 1, d .
T
-8-
4.1: Quadrilateral Elements (cont.)
Gauss Quadrature:
Let’s take a closer look at one of the integrals for the
element stiffness matrix (assume plane stress):
1 1
E
h
2
2
2
1 2
k22
*
a
1
b
1
1
2 d d.
2
1 16ab 1 1
Can be solved exactly, but for various reasons FEA
prefers to evaluate integrals like this approximately:
Historically, considered more efficient and reduced coding errors.
Only possible approach for isoparametric elements.
Can actually improve performance in certain cases!
Implementation of FEA:
Other Elements
-9-
4.1: Quadrilateral Elements (cont.)
Gauss Quadrature:
Idea: approximate integral by a sum of function values at
predetermined points with optimal weights –
1
n
1
i 1
weights = known constants, depend on n
1D case: d Wi i .
Gauss points = known locations, depend on n
n = order of quadrature; determines accuracy of integral.
(Note: any polynomial of order 2n-1 can be integrated exactly using nth order Gauss quadrature.)
Implementation of FEA:
Other Elements
-10-
4.1: Quadrilateral Elements (cont.)
Gauss Quadrature:
Have tables for weights and Gauss points:
2D case handled as two 1D cases:
1 1
, d d WW , .
1 1
Implementation of FEA:
Other Elements
n
n
j 1 i 1
i
j
i
j
-11-
4.1: Quadrilateral Elements (cont.)
Higher Order Rectangular Elements
More nodes; still 2 translational d.o.f. per node.
“Higher order” higher degree of complete polynomial
contained in displacement approximations.
Two general “families” of such elements:
Implementation of FEA:
Other Elements
Serendipity
Lagrangian
-12-
4.1: Quadrilateral Elements (cont.)
Lagrangian Elements:
Order n element has (n+1)2 nodes arranged in square-
symmetric pattern – requires internal nodes.
Shape functions are products of nth order polynomials
in each direction. (“biquadratic”, “bicubic”, …)
Bilinear quad is a Lagrangian element of order n = 1.
Implementation of FEA:
Other Elements
-13-
4.1: Quadrilateral Elements (cont.)
Lagrangian Shape Functions:
Uses a procedure that automatically satisfies the
Kronecker delta property for shape functions.
Consider 1D example of 6 points; want function = 1 at 3 0.3
and function = 0 at other designated points:
0 1;
1 .75;
2 .2;
3 .3;
4 .6;
5 1.
Implementation of FEA:
Other Elements
L(5)
3
0 1 2 4 5 .
3 0 3 1 3 2 3 4 3 5
-14-
4.1: Quadrilateral Elements (cont.)
Lagrangian Shape Functions:
Can perform this for any number of points at any
designated locations.
( m)
k
L
m
0 1 k 1 k 1 m
i
.
k 0 k 1 k k 1 k k 1 k m i0 k i
ik
No -k term!
Implementation of FEA:
Other Elements
Lagrange
polynomial
of order m
at node k
-15-
4.1: Quadrilateral Elements (cont.)
Lagrangian Shape Functions:
Use this procedure in two directions at each node:
(3)
6
V
H
(3)
6
5 7 8
6 5 6 7 6 8
Implementation of FEA:
Other Elements
2 10 14
6 2 6 10 6 14
N 6 , H 6(3) V6(3) .
-16-
4.1: Quadrilateral Elements (cont.)
Notes on Lagrangian Elements:
Once shape functions have been identified, there are no
procedural differences in the formulation of higher order
quadrilateral elements and the bilinear quad.
Pascal’s triangle for the Lagrangian quadrilateral elements:
Implementation of FEA:
Other Elements
3x3
nxn
-17-
4.1: Quadrilateral Elements (cont.)
Serendipity Elements:
In general, only boundary nodes – avoids internal ones.
Not as accurate as Lagrangian elements.
However, more efficient than Lagrangian elements and
avoids certain types of instabilities.
Implementation of FEA:
Other Elements
-18-
4.1: Quadrilateral Elements (cont.)
Serendipity Shape Functions:
Shape functions for mid-side nodes are products of an
nth order polynomial parallel to side and a linear function
perpendicular to the side.
E.g., quadratic serendipity element:
Implementation of FEA:
Other Elements
N6 12 1 1 2 ; N7 12 1 2 1 .
-19-
4.1: Quadrilateral Elements (cont.)
Shape functions for corner nodes are modifications of
the shape functions of the bilinear quad.
Step #1: start with appropriate bilinear quad shape function, Nˆ 1 .
Step #2: subtract out mid-side shape function N5 with appropriate
weight Nˆ 1 node #5 12
Step #3: repeat Step #2 using mid-side shape function N8 and weight Nˆ 1 node #8 12
Implementation of FEA:
Other Elements
Nk
1
4
1 k 1 k k k 1 ; k 1, 2,3, 4.
-20-
4.1: Quadrilateral Elements (cont.)
Notes on Serendipity Elements:
Once shape functions have been identified, there are no
procedural differences in the formulation of higher order
quadrilateral elements and the bilinear quad.
Pascal’s triangle for the serendipity quadrilateral elements:
Implementation of FEA:
Other Elements
3x3
mxm
-21-
4.1: Quadrilateral Elements (cont.)
Zero-Energy Modes (Mechanisms; Kinematic Modes) –
Instabilities for an element (or group of elements) that
produce deformation without any strain energy.
Typically caused by using an inappropriately low order of
Gauss quadrature.
If present, will dominate the deformation pattern.
Can occur for all 2D elements except the CST.
Implementation of FEA:
Other Elements
-22-
4.1: Quadrilateral Elements (cont.)
Zero-Energy Modes –
Deformation modes for a bilinear quad:
#1, #2, #3 = rigid body modes; can be eliminated by proper constraints.
#4, #5, #6 = constant strain modes; always have nonzero strain energy.
#7, #8 = bending modes; produce zero strain at origin.
Implementation of FEA:
Other Elements
-23-
4.1: Quadrilateral Elements (cont.)
Zero-Energy Modes –
Mesh instability for bilinear quads using order 1 quadrature:
“Hourglass modes”
Implementation of FEA:
Other Elements
-24-
4.1: Quadrilateral Elements (cont.)
Zero-Energy Modes –
Element instability for quadratic quadrilaterals using 2x2
Gauss quadrature:
“Hourglass modes”
Implementation of FEA:
Other Elements
-25-
4.1: Quadrilateral Elements (cont.)
Zero-Energy Modes –
How can you prevent this?
Use higher order Gauss quadrature in formulation.
Can artificially “stiffen” zero-energy modes via penalty functions.
Avoid elements with known instabilities!
Implementation of FEA:
Other Elements
-26-
Section 4: Implementation of Finite Element
Analysis – Other Elements
1. Quadrilateral Elements
2. Isoparametric Elements
3. Higher Order Triangular Elements
Note: any type of geometry can be used for isoparametric elements;
we will only look at quadrilateral elements.
Implementation of FEA:
Other Elements
-27-
Section 4.2: Isoparametric Elements
For various reasons, need elements that do not “fit” the
standard geometry.
Curved boundaries
Implementation of FEA:
Other Elements
Transition regions
-28-
4.2: Isoparametric Elements (cont.)
Problem: How do you map a general quadrilateral onto
the normalized geometry?
x, y F , , F1 x, y ; F ?
Implementation of FEA:
Other Elements
-29-
4.2: Isoparametric Elements (cont.)
Idea: Approximate the mapping using “shape functions”.
x N1* , x1 N*2 , x2 N*3 , x3
N*n , xn ;
y N1* , y1 N*2 , y2 N*3 , y3
N*n , yn .
*
Require N k , to have Kronecker delta property.
*
N
k , not required to be the actual shape functions of
the element; n can be as large or as small as you want.
Implementation of FEA:
Other Elements
-30-
4.2: Isoparametric Elements (cont.)
Approximate “serendipity element” shown using bilinear
quad shape functions and approximation points at corners
1 1 x1 14 1 1 x2
14 1 1 x3 14 1 1 x4 .
x
Implementation of FEA:
Other Elements
1
4
-31-
4.2: Isoparametric Elements (cont.)
For an isoparametric element, the number of approximation
points equals the actual number of nodes for the element;
also, the approximation functions are the actual shape
functions for the element:
x N1 , x1 N 2 , x2 N 3 , x3 N n , xn ;
y N1 , y1 N 2 , y2 N 3 , y3
N n , yn ;
u x, y N1 , u1 N 2 , u2 N 3 , u3
N n , un ;
v x, y N1 , v1 N 2 , v2 N 3 , v3
N n , vn ;
If # of approx. pts. > # of nodes, element is called
superparametric; if # of approx. pts. < # of nodes,
element is called subparametric.
Implementation of FEA:
Other Elements
-32-
4.2: Isoparametric Elements (cont.)
Formulating an Isoparametric Element:
Recall the formulation of “standard” bilinear quad:
1 1
dx
dy
dA dxdy
d
d ab * d d dA ab * d d .
d
d
area
1 1
k h * ab *
1 1
1 1
Implementation of FEA:
Other Elements
B CB d d.
T
How does this work for an
isoparametric element?
-33-
4.2: Isoparametric Elements (cont.)
Formulating an Isoparametric Element:
Calculating the [B] matrix (assume isoparametric bilinear
quad element):
d
u x, y N1 ,
0
N 2 ,
0
N 3 ,
0
N 4 ,
0
1
.
v
x
,
y
0
N
,
0
N
,
0
N
,
0
N
,
1
2
3
4
d8
" N x "
x
B x 0
y
0
N1 ,
0
N 2 ,
0
N3 ,
0
N 4 ,
0
? .
y
0
N
,
0
N
,
0
N
,
0
N
,
1
2
3
4
x
Need to apply the chain rule!
Implementation of FEA:
Other Elements
-34-
4.2: Isoparametric Elements (cont.)
Formulating an Isoparametric Element:
Chain rule: compute inverse rule first –
N k N k x N k y N k N k x N k y
;
.
x y
x y
Using the approximate mapping:
n
n
N i
N i
x
x
x N i , xi
, xi ; , xi .
i 1
i 1
i 1
n
n
n
N i
N i
y
y
Similarly,
, yi ; , yi .
i 1
i 1
Implementation of FEA:
Other Elements
-35-
4.2: Isoparametric Elements (cont.)
Formulating an Isoparametric Element:
Put all of this together –
N k
N k
x
x
y N k
x
y N k
y
n N i
, xi
i 1
n
N i
, xi
i 1
Implementation of FEA:
Other Elements
N i
N k
,
y
i
i 1
x
n
N i
N k
, yi y
i 1
n
The Jacobian matrix [J] of the mapping.
.
-36-
4.2: Isoparametric Elements (cont.)
Formulating an Isoparametric Element:
Can now compute the regular chain rule –
N k
N k
N k
x
J
N k
y
N k
x
N k
y
n N i
, yi
1 i 1
1
J
J n N i
, xi
i 1
Implementation of FEA:
Other Elements
N k
1
J
N k
.
N i
, yi
i 1
; J det J .
n
N i
, xi “Jacobian” of
i 1
n
the mapping
-37-
4.2: Isoparametric Elements (cont.)
Formulating an Isoparametric Element:
J is a (nonconstant) scaling factor that relates area in
original geometry to area in normalized geometry; can
show that dxdy J * d d.
For a well-defined mapping, J must have same sign at
all points in normalized geometry.
Large variations in J imply highly distorted mappings –
leads to badly formed elements.
Implementation of FEA:
Other Elements
-38-
4.2: Isoparametric Elements (cont.)
Formulating an Isoparametric Element:
x
u N x d ε y B x d ;
xy
Calculating the [B] matrix:
ux ux
x 1 0 0 0 u u
1
J
y y
y 0 0 0 1 v ; v
0 1 1 0 x x 0
v v
xy
y y
1 0 0 0
1
B 0 0 0 1
J
0 1 1 0
n N
iy
i 1 i
n Ni
xi
i 1
0
0
Implementation of FEA:
Other Elements
n N
iy
i
i 1
n N
ix
i
i 1
0
0
u u N1
0 u u N1
;
1 v v
J 0
v v
0
0
0
n N
iy
i
i 1
n N
ix
i
i 1
0
n N
iy
i
i 1
n N
i
xi
i 1
0
N1
N1
0
0
0
0
N 2
N 2
0
0
N3
N3
0
0
N 4
N 4
N1
N1
0
0
N 2
N 2
0
0
N3
N3
0
0
0
0
N 2
N 2
0
0
N3
N3
0
0
N 4
N 4
N1
N1
0
0
N 2
N 2
0
0
N3
N3
0
0
d1
d2 .
N 4
N 4
d
8
0
0
0
0
N 4 .
N 4
-39-
4.2: Isoparametric Elements (cont.)
Calculating the element stiffness matrix:
1 1
k h * B CB * J , d d.
T
1 1
area scaling factor –
polynomial function of (,)
Note: [B] is proportional to J-1:
polynomials functions of ,
.
B
J ,
new polynomials functions of ,
k h *
d d .
J ,
1 1
1 1
In general, you are integrating ratios of polynomial functions, which typically
don’t have exact integrals use Gauss quadrature to evaluate!
Implementation of FEA:
Other Elements
-40-
4.2: Isoparametric Elements (cont.)
Calculating the element nodal forces:
f N x b x dV
T
Ve
N x t x dA ?
T
Ae ,
Body force contribution:
N x b x dV h *
T
Ve
What do you do with this?
1 1
1 1
N , b x, y * J , d d.
T
Surface traction contribution:
N x
T
t x dA
Ae ,
all edges
h*
edge # k
N ,
edge
#
k
T
t x, y
edge # k
d
edge # k
What do you do with these?
Implementation of FEA:
Other Elements
-41-
4.2: Isoparametric Elements (cont.)
Converting body force and surface tractions:
Idea #0: If body force = constant and/or surface traction on
edge #k = constant, do nothing!
Idea #1: Use the isoparametric mapping to modify force functions:
n
n
x N i , xi , y N i , yi b x, y b N i , xi , N i , yi .
i 1
i 1
i 1
i 1
" bˆ , "
1 1
T
T
N x b x dV h * N , bˆ , * J , d d .
n
n
1 1
Ve
Idea #2: Make an isoparametric approximation for the forces:
n
b x, y N , * b x , y
1 1
i 1
i
i
i
T
n
N x b x dV h * N i , * N , b xi , yi * J , d d .
Ve
1 1 i 1
T
Implementation of FEA:
Other Elements
-42-
4.2: Isoparametric Elements (cont.)
Converting dℓ on edge #k:
In general:
d
dx dy
2
2
; dx
x
x
y
y
d
d and dy
d
d .
On the given edge #k, d 0 :
2
d
edge # k
2
2
2
x
y
n Ni
n Ni
d
d
, 1 xi , 1 yi .
edge # k edge # k
i 1
i 1
Implementation of FEA:
Other Elements
L
-43-
4.2: Isoparametric Elements (cont.)
Thus, the contribution from surface tractions on edge #k is:
h*
edge # k
N ,
edge # k
T
t x, y
1
edge # k
d
edge # k
Idea #1!
n
n
h * N , 1 t N i , 1 xi , N i , 1 yi * L d .
i 1
i 1
1
T
Note: Ni , 1 0 unless i = k or i = k+1 !
Implementation of FEA:
Other Elements
-44-
4.2: Isoparametric Elements (cont.)
Example: Formulating an Isoparametric Bilinear Quad –
0.4* 8 y
t x, y
ksi
0
Given: 4-node plane stress element has E = 30,000 ksi, = 0.25, h =
0.50 in, no body force, and surface traction shown.
Required: Find [k] and (f). Use 2 x 2 Gauss quadrature for [k].
Implementation of FEA:
Other Elements
-45-
4.2: Isoparametric Elements (cont.)
Solution:
Isoparametric mapping:
x 14 1 1 x1 14 1 1 x2 14 1 1 x3 14 1 1 x4
14 1 1 * 4 14 1 1 *8 14 1 1 *11 14 1 1 * 2
= 254 134 14 54 ;
y 14 1 1 *3 14 1 1 * 4 14 1 1 *10 14 1 1 *8
= 254 34 114 14 ;
Jacobian matrix and Jacobian:
x
J x
Implementation of FEA:
Other Elements
y
134 54
y 14 54
14
35
27
1
;
J
det
J
4
8
8 .
11
1
4 4
3
4
-46-
4.2: Isoparametric Elements (cont.)
Solution:
[B] matrix:
1 0 0 0
1
B 0 0 0 1
J
0 1 1 0
n N
iy
i 1 i
n Ni
xi
i 1
0
0
n N
iy
i
i 1
n N
ix
i
i 1
0
0
n N
iy
i
i 1
n N
ix
i
i 1
0
0
11 1
1 0 0 0 4 4
1 5
8
=
0 0 0 1 4 4
0
70 27
0 1 1 0 0
3 1
4 4
13 5
4 4
0
0
11 1
4 4
1 5
4 4
4 6 2
1
=
0
70 27 63 9
Implementation of FEA:
Other Elements
0
63 9
4 6 2
0
n N
iy
i
i 1
n N
ix
i
i 1
0
7 5 2
0
7 2 9
0
0
0
7 2 9
7 5 2
N1
N1
0
0
N 2
N 2
0
0
N1
N1
1 1
4 4
0 1 1
4 4
3 1 0
4 4
13 5 0
4 4
45
0
6 2 4
N 2
N 2
0
0
0
0
0
6 2 4
45
0
0
1 1
4 4
1 1
4 4
0
0
1 1
4 4
1 1
4 4
1 1
4 4
1
1
4 4
0
N3
N3
0
0
7 6
0
7 3 4
0
0
0
0
N3
N3
N 4
N 4
0
0
1 1
4 4
1 1
4 4
0
0
1 1
4 4
1 1
4 4
0
0
0
0
N 4
N 4
1 1
4 4
1 1
4 4
0
0
7 3 4
76
0
-47-
0
1 1
4 4
1 1
4 4
0
4.2: Isoparametric Elements (cont.)
Solution:
[k] matrix:
0
32000 8000
C 8000 32000 0 ksi;
0
0
12000
1 1
k 0.5 in B CB * Jd d
T
1 1
31.25* 236276 196 315 2 354 275 2
8
*
70 27
sym
1 1
1 1
31.25* 70+231 203 90 2 231 43 2
31.25* 539588 490 180 228 131
2
2
d d
k ,
Implementation of FEA:
Other Elements
-48-
4.2: Isoparametric Elements (cont.)
Solution:
2 x 2 Gauss quadrature:
Wi W j 1; i, j 1, 2.
1
1
1 1 1
1 1
1
k WW
i
j * k i , j k 3 , 3 k 3 , 3 k 3 , 3 k 3 , 3
2
2
i 1 j 1
Implementation of FEA:
Other Elements
7028.9
k
1260.6
7136.6
Note: k exact
1263.9
1260.6
kips/in.
8489.9
1263.9
kips/in.
8499.0
-49-
4.2: Isoparametric Elements (cont.)
Solution:
Element nodal forces:
Implementation of FEA:
Other Elements
-50-