method of finite elements ii · chair of structural mechanics | 05.10.2016 | 3 fundamental case...

||Chair of Structural Mechanics

Prof. Dr. Eleni ChatziInstitute of Structural Engineering (IBK)Department of Civil, Environmental, and Geomatic Engineering (DBAUG)ETH Zürich

05.10.2016Finite Elements II - Prof. Dr. Eleni Chatzi 1

Method of Finite Elements II13.10.2016


RecapWhere do linear systems end?

05.10.2016 2

Fundamental Case Study – The axial bar

Finite Elements II - Prof. Dr. Eleni Chatzi

ObjectivesGiven application of a load tR at point O, calculate the corresponding displacement.

O

What if the problem were linear?

Force Stress

t t ta bR A Aσ σ+ = ,

,

ta bt

a b Eσ

ε =

Strain

, ( )t

t t ta b

o

u u f RL

ε = ⇒ =

Displacement

linear

||Chair of Structural Mechanics 05.10.2016 3



ObjectivesGiven application of a load tR at point O, calculate the corresponding displacement.

O

What if the material is nonlinear (plasticity)?

Force Stress

t t ta bR A Aσ σ+ =

Strain

( )N tu f R=

Displacement

nonlinear

( )t t tK u u R=

or

path-dependent stiffness


tR

(0)tF

u(0)tu (1)tu(1)uδ05.10.2016 4

How to solve a nonlinear equation?


( ) ( ) ( ) 0

t

t t t t t t t

externalforceinternal

force F

K u u R r u K u u R= ⇒ = − =

r: residual

Iterate until convergencefor load step t

Use the Newton-Raphson (iterative) method:Tangent StiffnessMatrix

(0)(1)

(1)

t tt R FK

uδ−

=

(1)tK(0)tK


tR

(0)tF

u

(0)tK(0)tK (0)tK

(0)tu ( )t nu(1)tu

05.10.2016 5



( ) ( ) ( ) 0

t

t t t t t t t

externalforceinternal

force F

K u u R r u K u u R= ⇒ = − =

r: residual


or use the modified Newton-Raphson (iterative) method:Tangent StiffnessMatrix

( 1)t t it

i

R FKuδ

−−=

tKcalculating

can be computationally costly

(1)uδ




( ) ( ) ( ) 0t t t t t t t

externalforceintF

K u u R r u K u u R= ⇒ = − =

r: residual


Use the Newton-Raphson (iterative) method (NR):( 1) : t i i t t iK u R F −∆ = − out of balance load vector

( 1) : displacement incrementt i t i iu u u−= + ∆

Specify Initial Conditions

(0) (0) (0), ,t t t t t t t t tu u K K F F−∆ −∆ −∆= = =

or use the modified Newton-Raphson (iterative) method (m-NR):

( 1) : t t i t t iK u R F−∆ −∆ = − out of balance load vector




O

Tangent Stiffness

,,

section elastic section plastic

t t ta b

tta b

a b

t

T

K K Kwhere

CAK withL

EC

E

= +

=

=

( 1) ( 1)( ) ( )t t i t t i t ia b a bK K u R F F− −+ ∆ = − −

( 1)t i t i iu u u−= + ∆

(0) (0) (0), ,t t t t t t t t tu u K K F F−∆ −∆ −∆= = =

m-NR Iterations

with Initial Conditions




Tangent Stiffness

,,


t t ta b

tta b

a b

t

T

K K Kwhere

CAK withL

EC

E

= +

=

=

0 0 1 1 1 0 1 0 1 0( )a b a b aK K u R F F R F+ ∆ = − − = − 0bF−

( )4

1 3

7

2 10 6.67 101 110 10 5

u −⋅⇒ ∆ = = ⋅

+

Load Step 1: t = 1, R=2x104, Iteration 1: i = 1

In order to proceed to the next iteration the internal forces should be calculated, initiating from strains:

1 1 1 0 1 36.67 10u u u −= + ∆ = ⋅therefore:

1 11 1 4

1 11 1 3

6.67 10

1.33 10

a Ya

b Yb

uLuL

ε ε

ε ε

−

−

= = ⋅ <

= = ⋅ <

elastic section

elastic section




As expected, convergence is achieved in a single iteration!

1 1 1 3

1 1 1 4

6.67 10

1.33 10a a

b a

F EAF EA

ε

ε

= = ⋅

= = ⋅

Internal Force Calculation

Residual Calculation

1 1 1 0 1 0 0a br R F F= − − =

1 3 2 06.67 10u u−= ⋅ =

u

RLoad Step 1: t = 1, R=2x104, Iteration 1: i = 1

36.67 10−⋅

42 10⋅




Tangent Stiffness

,,


t t ta b

tta b

a b

t

T

K K Kwhere

CAK withL

EC

E

= +

=

=

( )

1 1 1 2 2 0 2 0 2 1 1

4 3 41 3

7

( ) ( ) ( )

4 10 6.67 10 1.33 10 6.67 101 110 10 5

a b a b a bK K u R F F R F F

u −

+ ∆ = − − = − −

⋅ − ⋅ − ⋅⇒ ∆ = = ⋅

+

Strain Calculations

2 1 2 0 1 21.33 10u u u −= + ∆ = ⋅therefore:

2 12 1 3

2 12 1 3

1.33 10

2.67 10

a Ya

b Yb

uLuL

ε ε

ε ε

−

−

= = ⋅ <

= = ⋅ >

elastic section

plastic section!

Load Step 2: t = 2, R=4x104, Iteration 1: i = 1




{ }2 2 1 4

2 2 1 4

1.33 10

( ) 2.01 10a a

b T a Y Y

F AE

F A E

ε

ε ε σ

= = ⋅

= − + = ⋅

Internal Force Calculation

Residual Calculation2 2 2 0 2 0

1 1 2 2 2 0 2 0 2 3

0

( ) 2.2 10a b

a b a b

r R F FK K u R F F u −

= − − ≠ ⇒

+ ∆ = − − ⇒ ∆ = ⋅u

RLoad Step 2: t = 2, R=4x104, Iteration 1: i = 2

36.67 10−⋅

Iterations are repeated until the residual drops below a user-defined tolerance (threshold)

2 2 2 1 2 21.55 10u u u −= + ∆ = ⋅

21.55 10−⋅

42 10⋅

43.34 10⋅




210u −⋅

RLoad Step 1: 2 = 1, R=4x104, Iteration 1: i >2

The procedure is repeated and the results of successive iterations are tabulated:

42 10⋅

43.34 10⋅i Δui 2ui

2 1.55E-02

3 1.45E-03 1.70E-02

4 9.58E-04 1.79E-02

5 6.32E-04 1.86E-02

6 4.17E-04 1.90E-02

7 2.76E-04 1.93E-02

When to stop? Depends on engineering judgment, i.e., you may specify a tolerance level when the calculated displacement increment ΔU is small enough to terminate the iteration process and subsequently continue with the next load step, t = 3.

42 10⋅

1.70

1.55

0.66

7

1.79


Nonlinear Solution Methods


The NR and m-NR methods offer an attractive solution for algorithmic implementation. Let’s recall is the pseudo-code for the Linear FE procedure.

• Physics of the problem Strong Form• Derive the Weak Form of the problem (integral equation)• Use Galerkin's method (weight functions) to approximate the solutionImplement the FE method:- Shape Functions Ni , i = 1…#dofs- Element Stiffness Matrices Ke , i = 1…#elements- Element Load Vectors fe , i = 1…#elements- Assemble the global stiffness and force vector K, F

• Finally, solve the problem equation:

KU = F

to obtain the unknown displacement vector U.

||Chair of Structural Mechanics 05.10.2016 14Finite Elements II - Prof. Dr. Eleni Chatzi

Pseudo-code for the Linear FE procedure




As demonstrated, the NR and m-NR methods offer an attractive solution for algorithmic implementation, by iterating for setting the following residual to zero:

What changes when material nonlinearity is added?

We now have to solve the problem equation:

to obtain the unknown displacement vector U.

nonlinear

( ) ( )= -r U K U U F

( ) =K U U F

For this, the tangent stiffness, let’s now note it as T, has to be defined as:

( )( )( ) ( )( )dd dT

d d d= = =

-K U U Fr U K UK U + UU U U

Then, the Newton iteration formula becomes:

( )11 ( )k k k k kδ −+ = + = −U U U U T U r U

||Chair of Structural Mechanics 05.10.2016 16Finite Elements II - Prof. Dr. Eleni Chatzi

Pseudo-code for the Nonlinear FE procedure


tR

(0)tF

u

(0)tK(0)tK (0)tK

(0)tu ( )t nu(1)tu

05.10.2016 17



Challenge #1: For multiple degrees of freedom, T is a matrix and its inversion comes with significant computational cost

Where do the computational challenges appear?

Remedy: The modified NR method offers a solution to this, by only calculating this once in the beginning of the analysis




Challenge #2: How to follow the nonlinear path, particularly if this comprises fast variations? Implementation of the NR method provides the displacement that would correspond to a specific load level.

Incremental Force Method (IFM)In order to track the evolution of the structure's response as the load level evolves, implement the load in sufficiently small increments Δ.

The NR is implemented for every load increment:The stiffness matrix K as well as the tangent matrix T, should be calculated using the “global” displacement and force values (U,F), since their value depends on the actual (global) state of the element.

However, the Newton-Raphson equation is solved for each load increment i:ΔF=fi=Kui, where ui , fi, denote “incremental” quantities within each load step.




Depends on the type of the problem, or the degree of nonlinearity- Mild nonlinearities: will require few IFC increments- Rough Nonlinearities: will require a higher number of IFC increments, wth the

increment size not necessarily uniform

NR or m-NR?- NR works better for mildly nonlinear problems, but with possibly tough

convergence criteria- m-NR is more convenient when computation is expensive, the increment size is

small, and convergence criteria are more loose.

How many IFC increments?

(Nam Ho Kim, 2015)

A number of FE programs will provide automatic stiffness updating options on the basis of convergence criteria or material properties.




Challenge #3: Inversion at Limit & Turning Points

Standard solution techniques lead to instability near the limit points and also present problems in case of snap-through and snap-back points, failing to predict the complete load-displacement response.

(due to the derivative)

Challenge #4: How to follow the complete response path?


r

u

05.10.2016 21



Challenge #5: Alternating sign of the residual r(U) between iterations

(Nam Ho Kim, 2015)Line search can help improve converge in this case




Remedy I: Single Displacement Control Method (Argyris,1965; Batoz & Dhatt,1979)

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.com

/

Force Control: gradually increase applied forces and find equilibrium configurationDisplacement Control: gradually increase prescribed displacements

Applied load can be calculated as a reaction More stable than force

control Useful for softening, contact,

snap-through limit points

However, it fails to converge in snap-back problems.

It is usually used in conjunction with other solution schemes in order to solve general nonlinear problems.



Nikolaos Vasios, Nonlinear Analysis of Structures, 2015

(a) A system that is unstable under load control (Snap-Through instability)(b) A system that is unstable under displacement control (Snap-Back Instability), (c) A system that is unstable under both displacement and load control




Remedy I: Arc-Length Method (Wempner,1971; Riks,1979; Ramm,1980)

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Unlike the Newton-Method, the Arc Length method employs asimultaneous variation in both the displacement Δu and the load increment ΔF




3 are the most commonly specified convergence criteria:

(a) Work, (b) displacement, (c) load (residual)

commonly, satisfaction of at least 2 of these criteria is required.

Choose a Load or Displacement-based criterion?

How is convergence specified?

(Nam Ho Kim, 2015)

use load criterion

use displacement criterion

load

displacement




Demo

Demonstrations of Newton Raphson and Arc-length methodby Zheng (MathWorks)

Matlab Source Files are Available Online

https://ch.mathworks.com/matlabcentral/fileexchange/48643-demonstrations-of-newton-raphson-method-and-arc-length-method

https://ch.mathworks.com/matlabcentral/profile/authors/1251764-zheng

method of finite elements ii · chair of structural mechanics | 05.10.2016 | 3 fundamental case...

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