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