Explicit_Dynamics_Chapter 1_Intro_to_Exp_Dyn.ppt

8/10/2019 Explicit_Dynamics_Chapter 1_Intro_to_Exp_Dyn.ppt

1/38

1-1ANSYS, Inc. Proprietary

2009 ANSYS, Inc. All rights reserved.February 27, 2009

Inventory #002665

Chapter 1

Introduction toExplicit Dynamics

ANSYS Explicit Dynamics


2/38

In t rodu ct ion to Expl ic i t Dynamics



Inventory #002665

Training ManualWelcome!

Welcome to the ANSYS Expl ici t Dynamics introductory trainingcourse!

This training course is intended for all new or occasional ANSYSExpl ic i t Dynamics users, regardless of the CAD software used.

Course Objectives:

Introduction to Explicit Dynamics Analyses. General understanding of the Workbench and Explicit

Dynamics (Mechanical) user interface, as related to geometry

import and meshing.

Detailed understanding of how to set up, solve and post-

process Explicit Dynamic analyses. Utilizing parameters for optimization studies.

Training Courses are also available covering the detailed use of otherWorkbench modules (e.g. DesignModeler, Meshing, Advanced

meshing , etc.).


3/38




Inventory #002665

Training ManualCourse Materials

The Training Manual you have is an exact copy of the

slides.

Workshop descriptions and instructions are included inthe Workshop Supplement.

Copies of the workshop files are available on the ANSYS

Customer Portal (www.ansys.com).

Advanced training courses are available on specifictopics. Schedule available on the ANSYS web page

http://www.ansys.com/ under Solutions> Services and

Support> Training Services.
http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/


4/38




Inventory #002665

Training ManualA. About ANSYS, Inc.

ANSYS, Inc.

Developer of ANSYS family of products Global Headquarters in Canonsburg, PA - USA (south of Pittsburgh) Development and sales offices in U.S. and around the world

Publicly traded on NASDAQ stock exchange under ANSS

For additional company information as well as descriptions and

schedules for other training courses visit www.ansys.com


5/38




Inventory #002665

Training ManualCourse Overview

Chapter 1: Introduction to Explicit Dynamics

Chapter 2: Introduction to Workbench

Chapter 3: Engineering Data

Chapter 4: Explicit Dynamics Basics

Chapter 5: Results Processing

Chapter 6: Explicit Meshing

Chapter 7: Body Interactions Chapter 8: Analysis Settings

Chapter 9: Material Models

Chapter 10: Optimization Studies


6/38




Inventory #002665

Training ManualWhy Use Explicit Dynamics?

Implicit and Explicit refer to two types of time integrationmethods used to perform dynamic simulations

Explicit time integration is more accurate and efficient for simulationsinvolving

Shock wave propagation

Large deformations and strains

Non-linear material behavior

Complex contact

Fragmentation

Non-linear buckling

Typical applications Drop tests

Impact and Penetration


7/38




Inventory #002665

Training Manual

Solution Impact Velocity

(m/s)

Strain Rate (/s) Effect

Implicit 12000 > 108Vaporization of colliding

solids

Impact Response of Materials

Why Use Explicit Dynamics?


8/38




Inventory #002665

Training Manual

VELOCITY LOW HIGH

Deformation Global Local

Response Time ms - s s - ms

Strain 50%

Strain Rate < 10 s-1

> 10000 s-1

Pressure < Yield Stress 10-100 x Yield Stress

Typical Values for Solid Impacts

Why Use Explicit Dynamics?


9/38




Inventory #002665


Electronics Applications


10/38




Inventory #002665


Aerospace Applications


11/38




Inventory #002665


Applications in Nuclear Power safety


12/38




Inventory #002665


Applications in Homeland Security


13/38




Inventory #002665


Sporting Goods Application


14/38




Inventory #002665

Training ManualExplicit Solution Strategy Solution starts with a mesh having assigned material

properties, loads, constraints and initial conditions.

Integration in time, produces motion at the mesh nodes Motion of the nodes produces deformation of the elements

Element deformation results in a change in volume and densityof the material in each element

Deformation rate is used to derive strain rates (using variouselement formulations)

Constitutive laws derive resultant stresses from strain rates Stresses are transformed back into nodal forces (using various

element formulations)

External nodal forces are computed from boundary conditions,loads and contact

Total nodal forces are divided by nodal mass to produce nodalaccelerations

Accelerations are integrated Explicitly in time to produce newnodal velocities

Nodal velocities are integrated Explicitly in time to produce newnodal positions

The solution process (Cycle) is repeated until the calculationend time is reached


15/38




Inventory #002665

Training ManualBasic FormulationImplicit Dynamics

The basic equation of motion solved by an implicit transient dynamic analysis is

where mis the mass matrix, cis the damping matrix, kis the stiffness matrix

and F(t)is the load vector

At any given time,t, this equation can be thought of as a set of "static" equilibrium equations that also take

into account inertia forces and damping forces. The Newmark or HHT method is used to solve theseequations at discrete time points. The time increment between successive time points is called theintegration time step

For linear problems: Implicit time integration is unconditionally stable for certain integration parameters.

The time step will vary only to satisfy accuracy requirements.

For nonlinear problems: The solution is obtained using a series of linear approximations (Newton-Raphson method), so each

time step may have many equilibrium iterations.

The solution requires inversion of the nonlinear dynamic equivalent stiffness matrix.

Small, iterative time steps may be required to achieve convergence.

Convergence tools are provided, but convergence is not guaranteed for highly nonlinear problems.

)(tFkxxcxm


16/38




Inventory #002665

Training ManualBasic FormulationExplicit Dynamics The basic equations solved by an Explicit Dynamic analysis express the conservation of mass, momentum

and energy in Lagrange coordinates. These, together with a material model and a set of initial andboundary conditions, define the complete solution of the problem.

For Lagrange formulations, the mesh moves and distorts with the material it models, so conservation ofmass is automatically satisfied. The density at any time can be determined from the current volume of thezone and its initial mass:

The partial differential equations which express the conservation of momentum relate the acceleration tothe stress tensor ij:

Conservation of energy is expressed via:

For each time step, these equations are solved explicitly for each element in the model, based on inputvalues at the end of the previous time step

Only mass and momentum conservation is enforced. However, in well posed explicit simulations, mass,momentum and energy should be conserved. Energy conservation is constantly monitored for feedback onthe quality of the solution (as opposed to convergent tolerances in implicit transient dynamics)

V

m

V

V

00

zyxbz

zyxby

zyxbx

zzzyzxz

yzyyyx

y

xzxyxxx

zxzxyzyzxyxyzzzzyyyyxxxxe

2221


17/38




Inventory #002665

Training ManualBasic FormulationExplicit Dynamics

The Explicit Dynamics solver uses a central difference time integration scheme (Leapfrogmethod). After forces have been computed at the nodes (resulting from internal stress,contact, or boundary conditions), the nodal accelerations are derived by dividing force by

mass:

where xiare the components of nodal acceleration (i=1,2,3),Fiare the forces acting on the nodes, biarethe components of body acceleration and mis the mass of the node

With the accelerations at time n - determined, the velocities at time n + are found from

Finally the positions are updated to time n+1 by integrating the velocities

Advantages of using this method for time integration for nonlinear problems are:

The equations become uncoupled and can be solved directly (explicitly). There is no requirement foriteration during time integration

No convergence checks are needed since the equations are uncoupled

No inversion of the stiffness matrix is required. All nonlinearities (including contact) are included in theinternal force vector

ii

i bm

Fx

nni

ni

ni txxx

2121

21211

nn

i

n

i

n

i txxx


18/38




Inventory #002665

Training ManualStability Time Step To ensure stability and accuracy of the solution, the size of the time step used in Explicit time

integration is limited by the CFL (Courant-Friedrichs-Levy[1]) condition. This condition implies that thetime step be limited such that a disturbance (stress wave) cannot travel further than the smallest

characteristic element dimension in the mesh, in a single time step. Thus the time step criteria forsolution stability is

where tis the time increment, fis the stability time step factor (= 0.9 by default), his the characteristicdimension of an element and cis the local material sound speed in an element

The element characteristic dimension, h,is calculated as follows:

[1] R. Courant, K. Friedrichs and H. Lewy, "On the partial difference equations of mathematical physics",

IBM Journal, March 1967, pp. 215-234

min

c

hft

Hexahedral /Pentahedral The volume of the element divided by the square of the longest diagonal and

scaled by 2/3

Tetrahedral The minimum distance of any element node to itsopposing element face

Quad Shell The square root of the shell area

Tri Shell The minimum distance of any element node to itsopposing element edge

Beam The length of the element


19/38




Inventory #002665

Training ManualStability Time Step

The time steps used for explicit timeintegration will generally be much smaller

than those used for implicit time integration

e.g. for a mesh with a characteristic

dimension of 1 mm and a material sound

speed of 5000 m/s. The resulting stability

time step would be 0.18 -seconds. To solve

this simulation to a termination time of 0.1

seconds will require 555,556time steps

The minimum value of h/cfor all elementsin a model is used to calculate the time

step. This implies that the number of time

steps required to solve the simulation is

dictated by the smallest element in the

model.

Take care when generating meshes for

Explicit Dynamics simulations to ensure that

one or two very small elements do not

control the time step

h

min

c

hft


20/38




Inventory #002665

Training ManualStability Time Step and Mass Scaling

The maximum time step that can be used in explicit time integration is inversely proportionalto the sound speed of the material and therefore directionally proportional to the square rootof the mass of material in an element

where Ci jis the material stiffness (i=1,2,3), is the material density, mis the material massand Vis the element volume

Artificially increasing the mass of an element can increase the maximum allowable stabilitytime step, and reduce the number of time increments required to complete a solution

Mass scaling is applied only to those elements which have a stability time step less than aspecified value. If a model contains relatively few small elements, this can be a usefulmechanism for reducing the number of time steps required to complete an Explicit simulation

Mass scaling changes the inertial properties of the portions of the mesh to which scaling isapplied. Be careful to ensuring that the model remains representative for the physicalproblem being solved

iiii VC

m

Cct

11


21/38




Inventory #002665

Training ManualWave Propagation

Explicit Dynamics computes wave propagation in solids and liquidsAverage Velocity

Velocity at Gauge 1

Constant pressure applied to left surface for 1 ms

Rarefaction

Shock


22/38




Inventory #002665

Training ManualElastic Waves

Different types of elastic waves can propagate in solids depending on how the motion of points in the solidmaterial is related to the direction of propagation of the waves [Meyers].

The primary elastic wave is the longitudinal wave. Under uniaxial stress conditions (i.e. an elastic wavetravelling down a long slender rod), the longitudinal wave speed is given by:

For the three-dimensional case, additional components of stress lead to a more general expression for thelongitudinal elastic wave speed

The secondary elastic wave is the distortional or shear wave and its speed can be calculated as

Other forms of elastic waves include surface (Rayleigh) waves, Interfacial waves and bending (or flexural)waves in bars/plates [Meyers]

Meyers M A, (1994) Dynamic behaviour of Materials, John Wiley & Sons, ISBN 0 -471-58262-X

Ec 0

GKcP

34

GcS


23/38




Inventory #002665

Training ManualPlastic Waves

Plastic (inelastic) deformation takes place in a ductile metal when the stress in the material exceeds theelastic limit. Under dynamic loading conditions the resulting wave propagation can be decomposed into

elastic and plastic regions [Meyer]. Under uniaxial strain conditions, the elastic portion of the wave travelsat the primary longitudinal wave speed whilst the plastic wave front travels at a local velocity

For an elastic perfectly plastic material, it can be shown [Zukas] that the plastic wave travels at a slowervelocity than the primary elastic wave, so an elastic precursor of low amplitude often precedes the stronger

plastic wave

Meyers M A, (1994) Dynamic behaviour of Materials, John Wiley & Sons, ISBN 0-471-58262-X

Zukas J A, (1990) High velocity impact dynamics, John Whiley, ISBN 0-471-51444-6

dd

cplastic

Kcplastic

I d i E l i i D i


24/38




Inventory #002665

Training ManualShock Waves

Typical stress strain curves for a ductile metal

Uniaxial Stress Uniaxial Strain

Under uniaxial stress conditions, the tangent modulus of the stress strain curve decreases with strain. Theplastic wave speed therefore decreases as the applied jump in stress associated with the stress waveincreasesshock waves are unlikely to form under these conditions

Under uniaxial strain conditions the plastic modulus (AB) increases with the magnitude of the applied jumpin stress. If the stress jump associated with the wave is greater than the gradient (OZ), the plastic wave willtravel at a higher speed than the elastic wave. Since the plastic deformation must be preceded by theelastic deformation, the elastic and plastic waves coalesce and propagate as a single plastic shock wave

x

x

z

o

A

B

C

I t d t i t E l i i t D i


25/38




Inventory #002665

Training ManualShock Waves

A shock wave is a discontinuity in material state (density (), energy (e), stress (), particle velocity(u) ) which propagates through a medium at a velocity equal to the shock velocity (U

s)

Relationships between the material state across a shock discontinuity can be derived using theprincipals of conservation of mass, momentum and energy The resulting Hugoniotequations are

given by:

1

e11

u1

0

e00

u0

Us



26/38




Inventory #002665

Training ManualShock and Rarefaction Waves

Rarefaction

Shock

Elastic precursor

Shock (compression) and

rarefaction (expansion) waves

generated by a pressure

discontinuity



27/38




Inventory #002665

Training ManualSpatial Discretization

Geometries (bodies) are meshed into a (large) number of smaller elements All elements use in Explicit Dynamics have Lagrange formulations i.e. elements follow the deformation of the bodies

Advanced Explicit Dynamics (AUTODYN) allows other formulations to beused

Euler (Multi-material, Blast)

Particle free (SPH)



28/38




Inventory #002665

Training Manual

Element formulations for Explicit Dynamics

Solid elements

HexahedralExact volume integration

Approximate Gauss volume integration

Pentahedral

Automatically converted to a degenerate hex

Tetrahedral

SCP (Standard Constant Pressure)

ANP (Average Nodal Pressure)

Shell elements

Quadrilateral Triangular

Beam (Line) element

Element Formulations

1

2

3

4

1

2

3



29/38




Inventory #002665

Training Manual

Hexahedral Solid Elements

Two Formulations:

8 node, exact volume integration, constantstrain element

Single quadrature point with hourglass

stabilization

8 node, approximate Gauss volumeintegration element

LS-DYNA formulation (Hallquist)

Some accuracy is lost for fastercomputation

Single quadrature point with hourglassstabilization




30/38




Inventory #002665

Training ManualElement Formulations

Tetrahedral Solid Elements

Two formulations:

SCP (Standard Constant Pressure)

Textbook 4 noded iso-parametric tetelement

Designed as filler element for hex-dominant meshes

Exhibits volume locking if over constrainedor during plastic flow

ANP (Average Nodal Pressure)

Enhanced 4 noded iso-parametric tetelement (Burton, 1996)

Overcomes volume locking problems

Can be used as a majority mesh element

SCP Tet

ANP Tet



31/38




Inventory #002665


Tetrahedral Solid Elements

Pull-out test simulated using both

hexahedral elements (top) and ANP

tetrahedral elements (bottom).

Similar plastic strains and material

fracture are predicted for both element

formulations used.



32/38




Inventory #002665

Training Manual

Shell Elements

Quadrilateral shell element Belytschko-Tsay, with Chang-Wong correction

Co-rotational formulation, bi-linear, 4 noded

Single quadrature point with hourglass stabilization

Isotropic and layered orthotropic formulations

Number of through thickness integration points can be specified

Triangular shell element

C0 Triangular Plate Element (Belytscho, Stolarski and Carpenter1984)

Should be used in quad-dominant meshes

Thickness is a parameter (not modelled geometrically)

Actual thickness can be rendered

Time step is controlled by the element length, not by thickness

1

2

3

E

1

2

3

4

E




33/38




Inventory #002665


Hourglass Control (Damping) for Hexahedral Solid and Quad Shell Elements

For the hexahedral and quad element formulations, the expressions for strain rates and forces involve onlydifferences in velocities and / or coordinates of diagonally opposite corners of the element

If an element distorts such that these differences remain unchanged there is no strain increase in the element and

therefore no resistance to this distortion

On the left, the two diagonals remain the same length even though the element distorts. If such distortions occur

in a region of several elements, a pattern such as that shown on the right occurs and the reason for the name

hourglass instability is easily understood

In order to avoid such hourglass instabilities, a set of corrective forces are added to the solution

Two formulations are available for hexahedral solid elements

AD standard (default)

Most efficient option in terms of memory and speed

Flanagan-Belytschko

Invariant under rotation

Improved results for large rigid body rotations

21

3 4

21

3 4

2D 3D



34/38




Inventory #002665


Beam (Line) Elements

2 noded Belytschko-Schwer resultant

beam formulation

Extended to allow large axial strains

Resultant plasticity implemented for

range of cross section types Cross-section is a parameter (not

modelled geometrically)

Actual cross section can be rendered

Time step is controlled by the elementlength, not by dimensions of cross-

section



35/38




Inventory #002665

Training Manual

X

Y

Z

11

22

33

Node #1

Node #222

Local 11-Direction

Always defined from node #1 to node#2

Local 22-Direction

Defined by user for Rectangular, I-Beam andGeneral Sections

User defines initial unit vector 22at cyclezero. This should lie in plane 11-22

Local 33-Direction

Orthogonal to Local directions 11 and 22

Rin

Rout

aa

A

A ab

A

B

22

a

A

B

22

tw

tf

22

33


Beam cross-sections



36/38




Inventory #002665

Training ManualElement Usage

What is required for meshing Explicit Applications?

Uniform element size (in finest zoned regions).

Smallest element size controls the time step used to advance the solution in time. Explicit analyses compute dynamic stress waves that need to be accurately modeled as they

propagate through the entire mesh.

Element size controlled by the user throughout the mesh.

Not automatically dependent on geometry.

Implicit analyses usually have static region of stress concentration where mesh is refined(strongly dependent on geometry).

In explicit analyses, the location of regions of high stress constantly change as stress wavespropagate through the mesh.

Mesh refinement is usually used to improve efficiency.

Mesh transitions should be smooth for maximum accuracy.

Hex-dominant meshing preferred.

More efficient. Sometimes more accurate for slower transients.

Chapter 6 will cover Explicit Meshing in more detail



37/38




Inventory #002665

Training ManualMaterial Modeling

Class of Material Material Effects

Metals Elasticity

Plasticity

Isotropic Strain HardeningKinematic Strain Hardening

Isotropic Strain Rate Hardening

Isotropic Thermal Softening

Ductile Fracture

Brittle Fracture (Fracture Energy based)

Dynamic Failure (Spall)

Concrete / Rock Elasticity

Porous Compaction

PlasticityStrain Hardening

Strain Rate Hardening in Compression

Strain Rate Hardening in Tension

Pressure Dependent Plasticity

Lode Angle Dependent Plasticity

Shear Damage / Fracture

Tensile Damage / Fracture

Soil / Sand Elasticity

Porous CompactionPlasticity

Pressure Dependent Plasticity

Shear Damage / Fracture

Tensile Damage / Fracture

Rubbers / Polymers Elasticity

Viscoelasticity

Hyperelasticity

Orthotropic Orthotropic Elasticity

In general, materials have a complex

response to dynamic loading,particularly when the loading is rapid,intense and distructive

The Material models available forExplicit Dynamics simulations facilitatethe modeling of a wide range ofmaterials and material behaviors, as

shown in the table

Chapter 3 will explain how materialdata can be created or retrieved fromlibraries using Engineering Data

The actual material models available

for Explicit Dynamics analyses arepresented at length in Chapter 6



38/38


Training ManualBasic Formulation

Models available forExplicit Dynamics

Chapter 9 willcover thesematerial models inmore detail

Explicit_Dynamics_Chapter 1_Intro_to_Exp_Dyn.ppt

Documents