implicit timestepping methods

IMPLICIT TIMESTEPPING

METHODS

Discontinuous Deformation Analysis

Contact Dynamics

OVERVIEW OF DEM SOFTWARES

Quasi-static metods

From an initial approximation of the equilibrium state searched for,

the displacements u are to be determined taking the system to the equilibrium

(assumption: time-independent behaviour, zero accelerations!!!)

Kishino, 1988

Bagi-Bojtár, 1991

Time-stepping methods

simulate the motion of the system along small, but finite t timesteps

Explicit timestepping methods:

UDEC

BALL-type models, e.g. PFC

Implicit timestepping methods:

DDA („Discontinuous Deformation Analysis”)

contact dynamics models2 / 24

" "K u f 0

" ( ) ( , ( ), ( ))"M a f u vt t t t

an equilibrium state is searched for

a process in time is searched for

circular, perfectly rigid elemets,

deformable contacts

deformable polyhedral elements, deformable contacts

rigid elements, deformable contacts

deformable polyhedral elements

rigid elements, non-deformable contacts

DISCONTINUOUS DEFORMATION ANALYSIS

„DDA”: Gen-Hua Shi (1988), Berkeleythen many others applied or developed research softwares!!!

1. The elements: The unknowns and the reduced loads

2. Contacts

3. The equations of motion: Potential energy minimization

4. Numerical solution of the equations of motion

Applications

3 / 24

DISCONTINUOUS DEFORMATION ANALYSISRepetition:

1.The elements: polyhedral [Deformable without subdivision]

displacement vector of the p-th element:

(reference point;

rigid-body translation and rotation;

the uniform strain of the element)

The translation of another point in the element in e.g. in 2D:

in 3D:4 / 24

u

p

x

p

y

p

z

p

x

p

y

p

zp

p

x

p

y

p

z

p

yz

p

zx

p

xy

u

u

u

x

z

y

rp

r

( )1 0 ( ) ( ) 0

2

( )0 1 ( ) 0 ( )

2

( , )

( ,

p

p p

p

p p

p

xp

y

p

x zp

y xp

y

p

xy

y yy y x x

x xx x y y

u

u

u x y

u x y

( , , ) ( , , )u T u

px y z x y z

DISCONTINUOUS DEFORMATION ANALYSISRepetition:

„reduced load” belonging to the p-th element:

f

p

x

p

y

p

z

p

x

p

y

p

zp

p p

x

p p

y

p p

z

p p

yz

p p

zx

p p

xy

f

f

f

m

m

m

V

V

V

V

V

V

5 / 24


How to reduce a force acting at (x, y) to the reference point of the element:

e.g. in 2D:

Illustration of its meaning:

6 / 24

x

z

y

rp r

1 0

0 1

( ) ( )( , )

( ,( ) 0

0 ( )

( ) ( )

2 2

( , ) ( , )

p p

x

yp

p

p p

T

y y x x f x y

f x yx x

y y

y y x x

x y x y

T f

p

x

p

y

p

zp

p p

x

p p

y

p p

xy

f

f

m

A

A

A

f



e.g. in 2D:

7 / 24

1 0

0 1

( ) ( )( , )

( ,( ) 0

0 ( )

( ) ( )

2 2

( , ) ( , )

p p

x

yp

p

p p

T

y y x x f x y

f x yx x

y y

y y x x

x y x y

T f

p

x

p

y

p

zp

p p

x

p p

y

p p

xy

f

f

m

A

A

A

f

x

y

Af

Bf

A B Q f f

Ay

By

x

y

0 0;A B

Q Q

f f

;A p B p

A By y y y y y

p p

yA

p

yy x y Q

( ) ( )A p A B p By yy y f y y f

( ) ( )A By Q y Q ( )A By y Q y Q

p

y

Q

x

0p p

xA

0

( ) / 2 ( ) / 2 0p p A p A B p B

xy y yA x x f x x f



the reduced force in 3D:

Remark: Higher-order polynomials also possible! [strains/stresses]

(e.g: M. MacLaughlin) 8 / 24

x

z

y

rp r

( , , ) ( , , )T x y z x y zT f


2.The contacts: (material point) with (material point)

in 2D: Node – to – Edge contacts

in 3D: Node – to – Face contacts:

Edge – to – Edge contacts:

Mechanical model:

originally: infinitely rigid contacts, Coulomb-friction

recent codes: deformable contacts included

+ other friction conditions, cohesion etc.

Remark: infinitely rigid contact: „penalty function”:

linearly elastic in normal and in tangential directions9 / 24

; ( )N N N T T TF k u dF k d u

„first entrance position”

contact deformation:

normal & tangential, perhaps sliding

direction of the contact:

the normal vector of the face

???? for edge-to-edge contact

;N Tu u


3. The equations of motion: „Potential energy” stationarity principle

„Potential” of the system:

deformed springs

external pot.

strain energy

inertial forces

velocity-proportional damping

initial stress

prescribed displacement history

or: ( ) ( , ( ), ( ))M a f u vt t t t

0 for all ,p

i

p iu

10 / 24

blocks contacts

( ) ( ) ( ) ( , ( ), ( ))extt t t t t t M a C v K u f v ugeneralized

displacement

incrementstiffness

matrix + etcdamping + etc

inertia

more exactly: „ Hamilton principle ”


4. Numerical solution of the equations of motion:

(ti, ti+1) time interval:

at ti : known ui, vi, f(ti, ui, vi); satisfy the eqs. of motion

Find ui+1, vi+1, ai+1 so that the eqs of motion would be satisfied at ti+1

Remember: Newmark’s –method:

[stability: 2 ½ ]

DDA: Newmark’s –method, with :

let

11 / 24

1 1 1 1 1 1 1( , , ) ( , , ) 0i i i i i i it t r u v f u v M a

1 1u u ui i i

1 12

1( )

/ 2a u vi i it

t

1 1v v ai i it

1/ 2 ; 1

2

1 1

1 1

(1 2 ) 22

: (1 )

i i i i i

i i i i

tt

t t

u u v a a

v v a a

2

1 1

1 1

2:

i i i i

i i i

tt

t

u u v a

v v a

1 1

2 2( )i i i i it

t t

v u v u v


4. Numerical solution of the equations of motion :

Determine ui+1, so that the residual

would be sufficiently close to zero!

Newton-Raphson:the Jacobian of the residual:

this matrix can be compiled from elementary calculations at ti: contains the stiffness matrix contains the inertia, contact forces,

geometric characteristics etc.

the residual can also be compiled from elementary calculations at ti: contains the external forces, inertia effects,

prescribed displacements, damping etc.12 / 24

( ) ( , ( ), ( ))M a f u vt t t t 1 1 1 1 1 1 1( , ( , ( )) ( )i i i i i i it 0 f u u v u M a u

1 1 1 1 1 1 1 1 1( , ) ( , ( ), ( )) ( )i i i i i i i i it t r u f u u v u M a u

( , )( , )

r uu

u

d tt

d


4. Numerical solution of the equations of motion :

Analysis of a time interval:

initial estimation for ui+1 :

k+1-th estimation for ui+1 :

then continue until becomes sufficiently small

„Open – close iterations”: at the end of t: check the topology and the forces;

modify the topology if necessary (e.g. new contacts, sliding, contact loss)

with the new topology, repeat: Newton-Raphson to find another ui+1

if acceptable topology not found: decrease timestep t to 1/3 of its previous length

CONVERGENCE WITHIN A TIME STEP ??? 13 / 24

( , )( , )

r uu

u

d tt

d

(0)

1 :u 0i

( 1) ( ) ( ) 1 ( )

1 1 1 1 1 1: ( , ) ( , )u u u r uk k k k

i i i i i it t

1

1 1( , )r uk

i it

1 1 1 1 1 1 1 1 1( , ) ( , ( ), ( )) ( )i i i i i i i i it t r u f u u v u M a u

( ) ( , ( ), ( ))M a f u vt t t t 1 1 1 1 1 1 1( , ( , ( )) ( )i i i i i i it 0 f u u v u M a u


Comparison to UDEC:

Main differences from UDEC:

→ basic unknowns: also the components of ;

uniform stress and strain field inside the elements;

→ numerical integration: implicit

→ stiffness matrix included artificial damping not necessary

advantages to UDEC: implicit numerical stability;

fast convergence if topology does not change

no artificial damping required

disadvantages: no commercial software inconvenient

(several research codes; e.g. ask from Gen-Hua Shi)

too simple mechanics of the elements and of the contacts

large storage requirements & longer computations

open-close iterations: convergence is not ensured if topology changes

14 / 24


Remark 1:

Comparison to UDEC:

M.S. Kahn (2010)

NOT EFFICIENT IN CASES IF

SIGNIFICANT TOPOLOGY MODIFICTIONS OCCUR !!!

15 / 24


Remark 2: Importance of experimentally verified material parameters

e.g. D.B. Mazor (2011): in DDA-3D,

Aim: Simulate shaking table experiments

What is the proper friction angle?

measurement of friction angle:

= 36

16 / 24


Remark 2: Importance of experimentally verified material parameters

e.g. D.B. Mazor (2011): in DDA-3D,

Aim: Simulate shaking table experiments

What is the proper friction angle?

experience: „friction degradation” during dynamic sliding

importance of contact penalty stiffness17 / 24


Applications

arches, arch bridges

underground

excavations

masonry structures

e.g. Fractured rock for earthquake and thermal effects18 / 24


Y.H.Hatzor, 1999-2004: Masada-hill, Israel (King Herod’s Palace)

Herod the Great built his fortress a hundred kilometers away from

Jerusalem, partly as a retreat, but more as a place of refuge for himself.

It happened at Masada seventy years after Herod's death that the

Jewish Revolt against the Roman occupation was well underway. As

the story goes, a last brave group of Jewish rebels fled Jerusalem to

Masada, and withstood a two-year long siege by a 15,000-strong

Roman Army. The crafty Romans finally built a ramp up the mountain

slope.

On their last night in Masada before the Romans broke through, the 967

rebels opted for mass suicide rather than surrender.

„Masada must not fall again”19 / 24



20 / 24



experience: huge blocks released, or already fallen out

the aim of the analysis: is it stable for an earthquake? „Masada must not fall again”

DDA model, earthquake simulation;

21 / 24



experience: huge blocks released, or already fallen out

the aim of the analysis: is it stable for an earthquake? „Masada must not fall again”

DDA model, earthquake simulation; minor earthquake: considerable damages can be expected; 7,1 earthquake (1995, Sinai-desert): huge blocks would fall out

proposals for strengthening

22 / 24


Mazor (2011): 3D investigations

23 / 24

QUESTIONS

1. How the displacement vector of an element looks like in DDA?

(i.e., what are the degrees of freedom of an element in DDA?)

2. Introduce the mechanical model of the contacts in DDA.

3. How does the equation of motion looks like in DDA? Write it,

and tell the name of the terms in it.

4. What kind of time integration method is used in DDA? What

values of the numerical control parameters is chosen in it, and

what is the mechanical meaning of this choice?

5. Explain the meaning of the expression "open-close iteration"

and how it is used in DDA.

6. Tell five differences between DDA and 3DEC.

implicit timestepping methods

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