1. CHAPTER 2 - EXPLICIT TRANSIENT DYNAMIC ANALYSYS

2. CONTENTS
Explicit Solution Technique
Explicit Timestep
Explicit Versus Implicit Technology
Applicability of Explicit Techniques

3. EXPLICIT SOLUTION TECHNIQUE
General Technique
Problem in space solved by FEM methods
Problem in time solved by explicit time integration
- many small time increments
Implementation in MSC.Dytran
Problem in space solved by:
Lagrange-Finite Element Technology
Euler-Finite Volume Technology
Problem in time solved by:
Central difference integration

4. EXPLICIT vs. IMPLICIT TIMESTEP
The timestep size is usually set by the requirements to maintain stability of the central difference integration
The timestep must subdivide the shortest natural period of the elements in the mesh.
Every transient is automatically resolved
IMPLICIT
The solution is unconditionally stable, so the timestep size is dictated by the required accuracy
The timestep must subdivide the shortest natural period of interest in the structure
High transients (e.g. pressure spikes) may not be resolved
IMPLICIT TIMESTEP >> EXPLICIT TIMESTEP
The timestep for an implicit analysis will normally be 10 to 100 times greater than that for an explicit analysis

5. EXPLICIT TIMESTEP
Timestep must Subdivide Smallest Natural Period of the Mesh
The timestep used by MSC.DYTRAN must be smaller than the smallest natural period of the mesh. Imagine doing an eigenvalue analysis with the same mesh and extracting every possible mode. The timestep must be smaller than the period associated with the highest natural frequency given. The mode shape associated with this eigenvalue is typically one grid point oscillating on the stiffness of the elements to which it is attached.
Courant Criterion
Since it is impossible to do a complete eigenvalue analysis every cycle to calculate the timestep, an approximate method, known as the Courant Criterion, is used. This is based on the minimum time for a stress wave to cross on elements.
It depends on the smallest element dimension, L
where c is the speed of sound through the element material and S is the timestep Scale Factor (<1).
For 1-D elements:

6. TIME INTEGRATION L: smallest element length t: time stepD
t: time step
c: sound speed t L c F
Force 
Implicit Integration
Explicit Integration
Explicit: -
Small time step
Implicit: -
Bigger time step
-No big matrices and -
Big matrices and
matrix inversion required
matrix inversion by
having a diagonal -
Solution procedure
complicated with
increasing degree of
nonlinearities
matrix (lumped mass)
-Robust solution
procedure even for
high degree of nonlinearities

7. TIMESTEP EXAMPLE Consider a rectangular steel cantilever beam:
EXPLICIT
Minimum element dimension, L = 3.33 mm
Speed of sound, c = 5113 m/s
Timestep = 0.9 L/c = microseconds
IMPLICIT
Assume interest up to the third mode
Frequency = 3643 rad/s; period = milliseconds
For accuracy, subdivide the period by 20
Timestep = 86.2 microseconds
IMPLICIT TIMESTEP = 147 x explicit timestep

8. EXPLICIT VS IMPLICIT TECHNOLOGY
EXPLICIT codes are relatively more efficient for problems with the following characteristics:
Short duration
Computational cost increases linear with problem time, but already small problem times need a lot of time integration steps
Large number or extent of non-linearities
Computational cost remains the same, while for an implicit method the CPU time increases exponentially
Large problem size
Computational cost increases linear with problem size. CPU time increases by a factor of two when number of elements is doubled

9. COMPUTATIONAL EFFICIENCY
IMPLICIT VS EXPLICIT

10. APPLICABILITY OF EXPLICIT TECHNIQUES
NONLINEAR
Large Displacement
The model can undergo large translations and rotations. There is no small displacement option in MSC.Dytran, but small displacement analysis is not precluded
Contact and Coupling
These interactions allow the simple modeling of complex interactions between two or more separately meshed bodies
Plasticity
A large number of material models are available to model a wide range of material behavior.
Materials as diverse as metals, alloys, plastics and composites can be simulated

11. APPLICABILITY OF EXPLICIT TECHNIQUES (Cont’d)
Large strain formulation
Most elements have large strain formulation. An option allows the shell elements to get thinner due to membrane straining
Failure
Material flow
In cases where extreme material deformation takes place Euler with strength can be used
TRANSIENT AND DYNAMIC
MSC.Dytran is most suitable for short term events such as explosions and high speed impact
It is intended for dynamic events. Static problems can be analyzed quasi-statically, but the technique is only cost effective if the problem incorporates significant non-linearities