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MANE 4240 & CIVL 4240Introduction to Finite Elements
Convergence of analysis results
Prof. Suvranu De
Reading assignment:
Lecture notes
Summary:
• Concept of convergence• Criteria for monotonic convergence :
completeness (rigid body modes + constant strain) +
compatibility • Incompatible elements and the patch test• Rate of convergence
Errors that affect finite element solution results
Type of error Source
1. Discretization error Use of FE interpolations for geometry and solution variables
2. Numerical integration Evaluation of FE element matrices and vectors using numerical integration
3. Round off This error is due to the finite precision arithmetic used in digital computers
What is “convergence”?
Physical system
Mathematical model
FE model
“Convergence” of FE solution results to the exact solution of the mathematical model
FE scheme exhibits convergence if the Discretization error →0 as the mesh is made infinitely fine (i.e., element size →0)
Mesh refinement
h-refinementp-refinement
h=element sizep=polynomial order
Convergence in energy and displacementu : exact displacement solution to a problem that makes the potential energy of the system a minimum corresponding stress and strain Exact strain energy of the body
uh : FE solution (‘h’ refers to the element size) corresponding stress and strain Approximate strain energy of the body
)u()u(
V
T dVU 2
1
)u( hh)u( hh
V h
Thh dVU
2
1
Calculation of strain energies
Example:
80cm
1 2 2
( ) 140
xA x sqcm
Consider a linear elastic bar with varying cross section
xThe governing differential (equilibrium) equation
( ) 0 (0,80)d du
E A x for xdx dx
Boundary conditions
Analytical solution3 1
( ) 12 1
40
exactu xx
Eq(1)E: Young’s modulus
P=3E/80
80
( 0) 0
3
80x cm
u x
du EEA Pdx
The exact strain energy of the system is
280 80
0 0
1 1 ( ) 3 39
2 2 160 2080
exact
x x
du x E EU Adx EA dx
dx
If we discretize the problem using a single linear finite element, the stiffness matrix is
80
02
( ) 1 1
1 180
1 113
1 1240
xE A x dx
K
E
The strain energy of the FE system is
80
0
1 1 27sin 0 9 /13
2 2 2080T T
h h hx
EU Adx d Kd ce d
Note hU U
Convergence in strain energy
0 hasUU h
Monotonic convergenceNonmonotonic convergence
Convergence in displacement
00v-vu-uuu 2h
2h0
hasdVV
h
Monotonic convergenceNonmonotonic convergence
Criteria for monotonic convergence
1. COMPLETENESS2. COMPATIBILITY
© 2002 Brooks/Cole Publishing / Thomson Learning™
CONDITION 1. COMPLETENESSThis requires that the displacement interpolation functions must be chosen so that the elements can represent
1. Rigid body modes
2. Constant strain states
Rigid body modes
The # of rigid body modes of an element = # of zero eigenvalues of the element stiffness matrix
Constant strain states
x
Actual variation of strainStrain computed using linear finite elements
Mathematical implication of the two conditions (rigid body modes + constant strain state)
( ) ( )i ii
u x N x uInside a finite element (of any order) in 1D
but this is just a polynomial…2
0 1 2( )u x a a x a x
Hence
20 1 2
20 1 2
1
20 1 2
( ) ( ) ( )
( ) ( ) ( )
( )
i i i i ii i
i i i i ii i i
x
i ii
u x N x u N x a a x a x
a N x a N x x a N x x
a a x a N x x
The requirement for completeness in 1D is that the displacement approximation be at least a linear polynomial of degree (k=1), ie any 2 node element and higher is complete
Mathematical implication of the two conditions (rigid body modes + constant strain state)
( ) ( , )i ii
u x N x y uInside a finite element (of any order) in 2D
but this is just a polynomial…
0 1 2( , )u x y a a x a y
Hence
0 1 2
0 1 2
1
0 1 2
( , ) ( , ) ( , )
( , ) ( , ) ( , )
i i i i ii i
i i i i ii i i
x
u x y N x y u N x y a a x a y
a N x y a N x y x a N x y y
a a x a y
The requirement for completeness in 1D is that the displacement approximation be at least a linear polynomial of degree (k=1).
Mathematical implication of the two conditions (rigid body modes + constant strain state)The element displacement approximation must be at least a COMPLETE polynomial of degree one
2
1
x
x
22
1
yxyx
yx
1D 2D
k=1
In 2D, the minimum displacement assumption needs to be
yxv
yxu
321
321
Translation along x1
1
1 2 1 3 3 2
0 0
0 0
0 0 0
all other coeffs
all other coeffs
and but
Translation along y
Rigid body rotation about z-axis
CONDITION 2. COMPATIBILITYThe assumed displacement variations are continuous within elements and across inter-element boundaries
Ensures that strains are bounded within elements and across element boundaries. If ‘u’ is discontinuous across element boundaries then
the strains blow up in-between elements and this leads to erroneous contributions to the potential energy of the structure
Physical meaning: no gaps/cracks open up when the finite element assemblage is loaded
Nonconforming elements and the patch testConforming = compatibleNonconforming = incompatibleIdeal: Conforming elementsObservation: Certain nonconforming elements also give good results, at the expense of nonmonotonic convergence
Nonconforming elements:
• satisfy completeness• do not satisfy compatibility• result in at least nonmonotonic convergence if the element assemblage as a whole is complete, i.e., they satisfy the PATCH TEST
PATCH TEST:
1. A patch of elements is subjected to the minimum displacement boundary conditions to eliminate all rigid body motions2. Apply to boundary nodal points forces or displacements which should result in a state of constant stress within the assemblage3. Nodes not on the boundary are neither loaded nor restrained.4. Compute the displacements of nodes which do not have a prescribed value5. Compute the stresses and strains
The patch test is passed if the computed stresses and strains match the expected values to the limit of computer precision.
NOTES:
1. This is a great way to debug a computer code2. Conforming elements ALWAYS pass the patch test3. Nodes not on the boundary are neither loaded nor restrained.4. Since a patch may also consist of a single element, this test may be used to check the completeness of a single element5. The number of constant stress states in a patch test depends on the actual number of constant stress states in the mathematical model (3 for plane stress analysis. 6 for a full 3D analysis)
CONVERGENCE RATE
This is a measure of how fast the discretization error goes to zero a the mesh is refined
Convergence rate depends on the order of the complete polynomial (k) used in the displacement approximation
3223
22
1
yxyyxx
yxyx
yx k=1
k=2
k=3
It can be shown that for (1) a sufficiently refined mesh and (2) for problems whose analytical solution does not contain singularities
2khU U C h
Convergence in strain energy : order 2k
Convergence in displacements : order p=k+1
110
u u kh C h
C and C1 are constants independent of ‘h’ but dependent on1. the analytical solution2. material properties3. type of element used
hUU log
slope = 2
Large C shifts curve up
hlog
Ex: for a domain discretized using 4 node plane stress/strain elements (k=1)
210
2
uu hC
hCUU
h
h
Important property of finite element solution:
When the conditions of monotonic convergence are satisfied (compatibility and completeness) the finite element strain energy always underestimates the strain energy of the actual structure
Strain energy of mathematical model
Strain energy of FE model