ANSYS Explicit Dynamics€¢ The Explicit Dynamics solver uses a central difference time integration scheme (Leapfrog method). After forces have been computed at the nodes (resulting

1-1ANSYS, Inc. Proprietary© 2009 ANSYS, Inc. All rights reserved.

February 27, 2009Inventory #002665

Chapter 1

Introduction to Explicit Dynamics

ANSYS Explicit Dynamics




Training ManualWelcome!• Welcome to the ANSYS Explicit Dynamics introductory training

course!

• This training course is intended for all new or occasional ANSYS Explicit 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 other Workbench modules (e.g. DesignModeler, Meshing, Advanced meshing, etc.).




Training ManualCourse Materials

• The Training Manual you have is an exact copy of the slides.

• Workshop descriptions and instructions are included in the Workshop Supplement.

• Copies of the workshop files are available on the ANSYS Customer Portal (www.ansys.com).

• Advanced training courses are available on specific topics. Schedule available on the ANSYS web page http://www.ansys.com/ under “Solutions> Services and Support> Training Services”.




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




Training ManualCourse Overview• Chapter 1: Introduction to Explicit Dynamics• Chapter 2: Introduction to Workbench• Chapter 3: Engineering Data• Chapter 4: Explicit Dynamics Basics

– Workshop 1: Taylor Test (Cylinder Impact)• Chapter 5: Results Processing

– Workshop 2: Processing results of Taylor Test• Chapter 6: Explicit Meshing

– Workshop 3: Can Crush• Chapter 7: Body Interactions

– Workshop 4: Drop test on reinforced concrete– Workshop 5: Circuit Board Drop Test (breakable bonded contact / spot welds)

• Chapter 8: Analysis Settings– Workshop 6: Oblique Projectile impact (with erosion) – Workshop 7: Add mass scaling to Can Crush

• Chapter 9: Material Models– Workshop 8: 1D Shock propagation (time histories / profile plots)

• Chapter 10: Optimization Studies– Workshop 9: Taylor Test What-if Study




Training ManualWhy Use Explicit Dynamics?• “Implicit” and “Explicit” refer to two types of time integration

methods used to perform dynamic simulations

• Explicit time integration is more accurate and efficient for simulations involving

– Shock wave propagation– Large deformations and strains– Non-linear material behavior– Complex contact– Fragmentation– Non-linear buckling

• Typical applications– Drop tests– Impact and Penetration




Training Manual

Explicit

Implicit

Solution

Vaporization of colliding solids> 108> 12000

Hydrodynamic (pressure many times material strength)

106 - 1083000 - 12000

Primarily Plastic (pressure equals or exceeds material strength)

105 - 1061000 - 3000

Elastic-Plastic (material strength significant)10-1 - 10150 -1000

Elastic10-5 - 10-1< 50

Static / Creep<10-5

EffectStrain Rate (/s)Impact Velocity (m/s)

Impact Response of Materials

Why Use Explicit Dynamics?




Training Manual

10-100 x Yield Stress< Yield StressPressure

> 10000 s -1< 10 s -1Strain Rate

>50%<10%Strain

µs - msms - sResponse Time

LocalGlobalDeformation

HIGHLOWVELOCITY

Typical Values for Solid ImpactsWhy Use Explicit Dynamics?




Training ManualWhy Use Explicit Dynamics?

• Electronics Applications





• Aerospace Applications





• Applications in Nuclear Power safety





• Applications in Homeland Security





• Sporting Goods Application




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 density of the material in each element

• Deformation rate is used to derive strain rates (using various element 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 nodal accelerations

• Accelerations are integrated Explicitly in time to produce new nodal velocities

• Nodal velocities are integrated Explicitly in time to produce new nodal positions

• The solution process (Cycle) is repeated until the calculation end time is reached




Training ManualBasic Formulation – Implicit Dynamics• The basic equation of motion solved by an implicit transient dynamic analysis is

where m is the mass matrix, c is the damping matrix, k is 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 these equations at discrete time points. The time increment between successive time points is called the integration 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 =++ &&&&




Training ManualBasic Formulation – Explicit 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 and boundary conditions, define the complete solution of the problem.

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

• The partial differential equations which express the conservation of momentum relate the acceleration to the stress tensor σij:

• Conservation of energy is expressed via:

• For each time step, these equations are solved explicitly for each element in the model, based on input values 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 on the quality of the solution (as opposed to convergent tolerances in implicit transient dynamics)

Vm

VV

=00ρ

zyxbz

zyxby

zyxbx

zzzyzxz

yzyyyxy

xzxyxxx

∂∂

+∂

∂+

∂∂

+=

∂

∂+

∂

∂+

∂

∂+=

∂∂

+∂

∂+

∂∂

+=

σσσρ

σσσρ

σσσρ

&&

&&

&&

( )zxzxyzyzxyxyzzzzyyyyxxxxe εσεσεσεσεσεσρ

&&&&&&& 2221+++++=




Training ManualBasic Formulation – Explicit Dynamics• The Explicit Dynamics solver uses a central difference time integration scheme (Leapfrog

method). 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 xi are the components of nodal acceleration (i=1,2,3), Fi are the forces acting on the nodes, bi are the components of body acceleration and m is the mass of the node

• With the accelerations at time n - ½ determined, the velocities at time n + ½ 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 for iteration 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 the

internal force vector

ii

i bmF

x +=&&

nni

ni

ni txxx ∆+= −+ &&&& 2121

21211 +++ ∆+= nni

ni

ni txxx &




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 the time 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 for solution stability is

where ∆t is the time increment, f is the stability time step factor (= 0.9 by default), h is the characteristic dimension of an element and c is 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⎥⎦⎤

⎢⎣⎡∗≤∆chft

The length of the elementBeam

The minimum distance of any element node to it’s opposing element edgeTri Shell

The square root of the shell areaQuad Shell

The minimum distance of any element node to it’s opposing element faceTetrahedral

The volume of the element divided by the square of the longest diagonal and scaled by √2/3

Hexahedral /Pentahedral




Training ManualStability Time Step• The time steps used for explicit time

integration 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 µ-seconds. To solve this simulation to a termination time of 0.1 seconds will require 555,556 time steps

• The minimum value of h/c for all elements in 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⎥⎦⎤

⎢⎣⎡∗≤∆chft




Training ManualStability Time Step and Mass Scaling• The maximum time step that can be used in explicit time integration is inversely proportional

to the sound speed of the material and therefore directionally proportional to the square root of the mass of material in an element

where Cij is the material stiffness (i=1,2,3), ρ is the material density, m is the material mass and V is the element volume

• Artificially increasing the mass of an element can increase the maximum allowable stability time 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 a specified value. If a model contains relatively few small elements, this can be a useful mechanism 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 is applied. Be careful to ensuring that the model remains representative for the physical problem being solved

iiii VCm

Cct ==∝∆

ρ

11




Training ManualWave Propagation• Explicit Dynamics computes wave propagation in solids and liquids

Average Velocity

Velocity at Gauge 1

Constant pressure applied to left surface for 1 ms

Rarefaction

Shock




Training ManualElastic Waves

• Different types of elastic waves can propagate in solids depending on how the motion of points in the solid material 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 wave travelling 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 the longitudinal elastic wave speed

• The secondary elastic wave is the distortional or shear wave and it’s 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

ρEc =0

ρ

GKcP

34+

=

ρGcS =




Training ManualPlastic Waves

• Plastic (inelastic) deformation takes place in a ductile metal when the stress in the material exceeds the elastic 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 travels at 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 slower velocity 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-XZukas J A, (1990) “High velocity impact dynamics”, John Whiley, ISBN 0-471-51444-6

ρε

σd

dcplastic =

ρKcplastic =




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. The plastic wave speed therefore decreases as the applied jump in stress associated with the stress wave increases – shock waves are unlikely to form under these conditions

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

εx

σ σ

εx

z

o

A

B

C




Training ManualShock Waves• A shock wave is a discontinuity in material state (density (ρ), energy (e), stress (σ), particle velocity

(u) ) which propagates through a medium at a velocity equal to the shock velocity (Us)

• Relationships between the material state across a shock discontinuity can be derived using the principals of conservation of mass, momentum and energy The resulting Hugoniot equations are given by:

•

ρ1e1σ1u1

ρ0e0σ0u0

Us




Training ManualShock and Rarefaction Waves

Rarefaction

Shock

Elastic precursor

Shock (compression) and rarefaction (expansion) waves

generated by a pressure discontinuity




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

• ANSYS AUTODYN allows other formulations to be used– Euler (Multi-material, Blast)– Particle free (SPH)




Training Manual

• Element formulations for Explicit Dynamics

– Solid elements• Hexahedral

– Exact 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




Training ManualElement Formulations• Hexahedral Solid Elements

– Two Formulations:

• 8 node, exact volume integration, constant strain element

– Single quadrature point with hourglass stabilization

• 8 node, approximate Gauss volume integration element

– LS-DYNA formulation (Hallquist)

– Some accuracy is lost for faster computation





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 constrained or 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




Training ManualElement Formulations• 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.




Training Manual

• Shell Elements

• Quadrilateral shell element

– Belytschko-Tsay, with Chang-Wong correction

– Co-rotational formulation, bi-linear, 4 noded


– Isotropic and layered orthotropic formulations

– Number of through thickness integration points can be specified

• Triangular shell element

– C0 Triangular Plate Element (Belytscho, Stolarski and Carpenter 1984)

– 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 thickness1

2

3

E

1

2

3

4

E





Training ManualElement Formulations• Shell Elements - Examples

Snap-throughShear Buckling




Training ManualElement Formulations• 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 only differences 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




Training ManualElement Formulations• 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 element length, not by dimensions of cross-section




Training Manual

X

Y

Z

11

22

33

Node #1

Node #2 22’

Local 11-Direction Always defined from node #1 to node#2

Local 22-Direction Defined by user for Rectangular, I-Beam and General Sections User defines initial unit vector 22’ at cycle zero. This should lie in plane 11-22

Local 33-Direction Orthogonal to Local directions 11 and 22

Rin

Rout a

a

A

A a b

A

B

22

a

A

B

22

tw

tf

22

33


• Beam cross-sections




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 waves propagate 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




Training ManualMaterial Modeling

Orthotropic ElasticityOrthotropic

ElasticityViscoelasticityHyperelasticity

Rubbers / Polymers

ElasticityPorous CompactionPlasticityPressure Dependent PlasticityShear Damage / FractureTensile Damage / Fracture

Soil / Sand

ElasticityPorous CompactionPlasticityStrain HardeningStrain Rate Hardening in CompressionStrain Rate Hardening in TensionPressure Dependent PlasticityLode Angle Dependent PlasticityShear Damage / FractureTensile Damage / Fracture

Concrete / Rock

ElasticityPlasticityIsotropic Strain HardeningKinematic Strain HardeningIsotropic Strain Rate HardeningIsotropic Thermal SofteningDuctile FractureBrittle Fracture (Fracture Energy based)Dynamic Failure (Spall)

Metals

Material EffectsClass of Material

• In general, materials have a complex response to dynamic loading, particularly when the loading is rapid, intense and distructive

• The Material models available for Explicit Dynamics simulations facilitate the modeling of a wide range of materials and material behaviors, as shown in the table

• Chapter 3 will explain how material data can be created or retrieved from libraries using Engineering Data

• The actual material models available for Explicit Dynamics analyses are presented at length in Chapter 6




Training ManualBasic Formulation

• Models available for Explicit Dynamics

– Chapter 9 will cover these material models in more detail

Documents

ANSYS Explicit Dynamics€¢ The Explicit Dynamics solver uses a central difference time integration scheme (Leapfrog method). After forces have been computed at the nodes (resulting