computational methods in transient dynamics

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MECALOG Computational Methods in Transient Dynamic

Computational Methods in

Transient Dynamics

H. Shakourzadeh

Training Manager

September 2003

Overview

Mechanics

Theoretical

Applied

Computational

Solids & Structures

Fluids

Multiphysics

Nano & Micromechanics

Statics

Dynamics

Linear

Nonlinear

Dynamic Terminology

Periodic motion :is a steady-state oscillatory motion of sustained amplitude and constant period

For linear systems :

• The principal of superposition holds,

• The mathematical techniques are available and well-developed.

For nonlinear systems :

•All systems tend to become nonlinear with increasing amplitude of oscillation.

• Techniques are less well known and difficult to apply

Free vibration :takes place when a system oscillates under the action of forces inherent in the systemitself.

Forced vibration :takes place under the excitation of external forces

Harmonic excitation:Common sources of harmonic excitation are unbalance in rotating machines.

Transient vibration /dynamic :A dynamical system is excited by a suddenly applied nonperiodic excitation F(t).

A transient motion is a nonsteady-state oscillatory motion of diminishing amplitude.

Free Vibration of a Spring-mass System

Equation of motion for a spring without damping :

Equilibrium of forces leads to :

General solution :

Evaluating the constants from initial conditions and

Special case with and :

0 kxxm

tBtAx nn cossin m

00 x 00 V x

cos0sin0

Damping effect on the Oscillation

Free vibration of a mass-spring system

Free vibration without damping (C = 0.)

Free vibration with critical damping ( C = Ccr )

Free vibration with underdamping ( C = 0.2 Ccr)

Free vibration with overdamping ( C = 5Ccr)

Displacement

KmC cr 2

Impulse ExcitationImpulse is the time integral of the force :

The impulse of a very large magnitude force acting for a very short time is

finite.

Unit impulse when Fimp = 1 :

The response to the unit impulse is of

importance to the transient problemsand designed by h(t).

dttFFimp )(

ttF )( is delta function

0 t tall for

t tfor 10

Impulse Excitation

mF mdvFdt t

mdvFdt V mFimp

cos0sin0

x(0) = 0

imp)0(

kn with

1impF tm

Newton law :

The impulse Fimp acting on the mass will result in a sudden change in its velocity.

Using the general equation of a mass-spring system :

Response for a unit impulse :

Having the response h(t) to a unit impulse excitation, it is possible to establish

the response of the system excited by an arbitrary force f(t).

Arbitrary ExcitationAn arbitrary force is considered

as series of impulses.

Each impulse contributes to the

response at time t =

where h is the unit impulse function

For a linear system the principle of superposition holds :

Another form of the last equation :

ttFF imp

)( thttFx

dtthtFx

d dtthentLetting )(

Arbitrary impulse equation

Impulse ExampleResponse of a single degree of freedom

system to the following step excitation

Damping is neglected

Solution :

Response for a unit impulse :

Substituting into the arbitrary impulse equation

sin1)(

)(sin0

)cos1(0 tm

The pick response to the step excitation of

magnitude F0 is equal to twice the statical deflection

Example : Dropping Body

Physical problem :

How far a body can be dropped without incurring damage?

Applications :

Landing of airplanes

Transportation of packaged articles

Analytical solution :Consider a spring-mass system dropped through a height h :

The differential equation of motion as long as the spring remains in contact with the floor :

General solution of the differential equation of motion :

Initial conditions to satisfied :

mgkxxm

C tCosBtSinAx

At t=0

tSingh

Maximum displacement and acceleration :

Static solution :

Maximum dynamic acceleration :

Remark :

The deceleration is always greater than g and grows nonlinearly with respect

to the height h.

Example : Dropping Body

mgK st 2

0xWhen 12

Numerical Integration in Time

When the differential equation cannot be integrated in closed form :

Dynamic force cannot be expressed by simple analytic functions

The system is nonlinear

Finite difference method :

The continuous variable t is replaced by the discrete variable t i

The differential equation is solved progressively in time increments t

The solution is approximate :

Sufficiently small time increment results good accuracy.

Central difference method :

Any differential equation of motion for a dynamic system relates acceleration,

velocity and displacement with time variable t.

Based on the Taylor expansion of xi+1 and xi-1 about the pivotal point i :

0,,, txxxf

iii xx

tx )2(

iiii xxx

Industrial Problems

High numbers of degrees of freedom

Highly nonlinear behavior due to :

Large displacements and deformation

Large rotations in space

Nonlinear material :

Plasticity, viscosity, damage, rupture, etc.

Dynamic nonlinear solicitation in time

Contact between several parts

Time-dependent response

Each unknown variable varying in timeand in space

Need of numericalcomputations

Computer methods

Need of discretization intime and in space

Computational Mechanics

Linear

Nonlinear

Finite Element Method

Finite Difference Method

Boundary Element Method

Finite Volume Method

Spectral Method

Mesh-Free Method

Nonlinear Softwares

Implicit Softwares

SAP by Ed Wilson

(Berkeley 1960)

NONSAP

(Nonlinear)ADINA

Brown UniversityMARC

(1969)

ABAQUS

(1972)

Westinghouse(Nuclear)

ANSYS(1969)

Hydro-codes HEMP

Wilkins(1964)

Costantino

(1967)

SAMSON (1969)

US Air Force

WRECKER

(1972)WHAMS

(1975)

SADCAT(1975)

(1975) PRONTO

DYNA(1976)

PAMCRASH

(1980)

LS-DYNA(1989)

RADIOSS

(1987)

Explicit Softwares

Application Fields

Structural Mechanics

Fluid-Structure interaction

Material characterization

A Physical Problem

Crushing of thin-walled members

Impact of a tubular thin-walled beam on a rigid wall

Thin-walled beam

Initial velocity

Rigid wall

M’ , V0

Contributory mass

Crushing of a tubular beam

Initial velocity field

High pressure region

Wave front

Wave front traveling

backward at material

wave speed

High pressure region

V0 Crimpling in the

high pressure region

Variable

contact

Plastic

Global buckling phase

Modeling of a Physical Problem

1. Geometry (Physical model)

1D, 2D or 3D ? Beam, Shell or Solid ?

2. Physical laws (conservation)

Physical laws (conservation)

Mass conservation

Energy conservation

Momentum conservation (equilibrium)

3. Space Discretisation:

Finite Difference (FD)

Finite Volume (FV)

Finite Element (FE)

Contact : use of specific elements (spring)

4. Time Discretisation:

Newmark scheme

5. Formulation:Choice of time and space discretisations

Explicit formulation

Implicit formulation

simplifications

FormulationsHow to combine time and space discretisations?

1. Lagrangian Formulation

– The mesh points coincide with the material points

– Elements are deformed with material

– Element deformation = Material deformation

Higher the mesh deformation, lower the quality of results2. Eulerian Formulation

– Nodes fixed in space, Material goes through the mesh

– Fixed nodes No degradation of mesh in large deformation problems

– Problems in solid mechanics:

Boundary conditions application Constitutive laws

3. Arbitrary Lagrangian Eulerian Formulation (ALE)– Between two previous formulations

– Internal nodes move to minimize element distortion

– Boundary nodes remain on the boundary of domain

More complicated formulation

How to describe the kinematic of geometrically nonlinear structures?

What tracking process ? With respect to which reference state?

Total Lagrangian Description : The FEM equations are formulated with respect

to a fixed reference (initial) configuration.

Updated Lagrangian Description : The reference configuration is the last

accepted solution (not fixed) => Used in implicit codes

Corotational Description : The motion is splitted into two parts :

Rigid body motion

Deformational motion

Lagrangian Descriptions

Initial undeformed configuration C0 Current configuration Cn Following configuration Cn+1

TL and UL descriptions

Lagrangian Formulations

Corotaional formulation :

CR configuration is obtained as a rigid body motion of the base configuration

A CR reference for each element

The coordinate system is Cartesian

The element deformations are measured with respect to the corotated configuration.

Initial undeformed configuration C0

Corotational configuration (CR)

Current

Corotational formulation

Finite Element Formulation

Equilibrium equation for a continuum : Differential equation

with boundary conditions for applied forces and velocities.

Variational form by Galerkin method : Integral form

Discrete form by Finite Element Method :

b : Body force

: Material density

v : velocity

: Virtual nodal velocitiesiv

0extbodyint

F F F t vM

Or : bodyintext F F F t

: External applied load vector

: Internal force vector

: Body force vector

M : Mass matrix

General Discrete Equations

Overall equation of motion for translational velocities with anti-hourglass

and contact forces :

Overall equation of motion for rotational velocities with anti-hourglass

moments :

conthgrbodyintext F F F F F t

hgrF : Anti-hourglass resistant forces

contF : Contact forces

hgrintext M M M t

: Externally applied moments

: Internal moments

: Anti-hourglass resistant forces

Explicit Method

Express the equilibrium equation at time step « n »:

Only [M] has to be inverted.

[M] is diagonal with lumped mass approach

Easy to invert [M] (not expensive)

Every D.O.F. is treated separately

)( next nn t F X K X M

)()( int1

nnext n t F t F M X

dvBX K t F nT nnn )(intwith

MECALOG

Example 1 : Time integration

System mass-spring without damping

1 d.o.f.

Equilibrium without damping:

k is a function of x (nonlinear spring)

Determine :

Explicit scheme with Central Difference Method

f ext(t)

)(t f kxxm ext

MECALOG

Example 1 : Solution

Explicit time discretisation:

Express the equilibrium at moment tn

ttn-1 tn

knownunknown

)( next nn t f kxxm

kxt f x nnext n

t f t f x nnext n

)()( int

MECALOG

Example 1 : (continued)

Centeral Differences:

Constant derivative

ttn-1 tn tn+1

t xxx nnn

t xxxnnn 211

xn+1 is obtained with a precision 2t

MECALOG

Example 2 :Shocked bar

Propagation of a shockwave in a bar

Small strain assumption

(elastic behavior)

Discretized Model :f ext(t)

Ni-1 Ni Ni+1

: Equilibrium

E : Hypo-Elastic Material

MECALOG

Example 2 : Solution

1. Loop over nodes : ( Central Difference Calculation)

2. Loop over elements : ( Stress Calculation)

With one integration point on element

Hypo-elasticity

t f t f x nnext n

)()( int t xxx nnn

t xxxnnn 211

nnE with

t E nnn 211

MECALOG

Example 2 : Continued

3. Nodal Forces :

4. Resultant Forces at nodes :

5. Loop over nodes

f 1 f 2

njni t f t f )()( 11

f j f j+1

t f t f x nnext n

)()( 1int11

Go to the first step

MECALOG

Remarks

1. Explicit method Element by Element Approach

No need of global resolution

2. Direct calculation of , and for linear or nonlinear stiffness

3. The expression of x is quadratic in terms of

t f t f x nnext n

)()( int

t xxx nnn

t xxx nnn 211

kxt f )(intwith

linear or nonlinear

MECALOG C i l h d i i i

Explicit Flow Chart

Time integration

extF •Loop over elements

)( ijij f

t t t t ijijij

•Assemble hrgF F ,int cont F

iii mF v

computational methods in transient dynamics

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