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Chapter 15 Explicit Dynamics
1
Chapter 15
Explicit Dynamics
15.1 Basics of Explicit Dynamics
15.2 Step-by-Step: High-Speed Impact
15.3 Step-by-Step: Drop Test
15.4 Review
Chapter 15 Explicit Dynamics
Section 15.1 Basics of Explicit Dynamics
2
Section 15.1
Basics of Explicit Dynamics
Key Concepts
•
Implicit Integration Methods
•
Explicit Integration Methods
•
Solution Accuracy
•
Integration Time Steps
•
Automatic Mass Scaling
•
Static Damping
Chapter 15 Explicit Dynamics
Section 15.1 Basics of Explicit Dynamics
3
Implicit Integration Methods
{ }
+
{ }
+
⎦
{ }
=
{ }
⎦

⎦
M
C
K
⎣
⎣
⎣
•
<Transient Structural> solves the above equation using the following algorithm:
D
n
+
1
=
D
n
+ Δ
t
γ

D
n
+
1
+
(1
− γ
)

D
n
⎡
⎣
⎤
⎦
D
n
+
1
=
D
n
+ Δ
t D
n
+
1
2
β

D
n
+
1
+
(1
−
2
β
)

D
n
Δ
t
2
⎡
⎣
⎤
⎦
2
•
The parameters
γ
and
β
are chosen to control characteristics of the algorithm such as
accuracy, numerical stability, etc.
•
It is called an
implicit method
because the response at the current time step depends on
not only the historical information but also the current information; iterations are
needed in a single time step.
Chapter 15 Explicit Dynamics
Section 15.1 Basics of Explicit Dynamics
4
Explicit Integration Methods
{ }
+
{ }
+
⎦
{ }
=
{ }
⎦

⎦
M
C
K
⎣
⎣
⎣
•
<Explicit Dynamics> solves the above equation using the following algorithm:
D
n
+
2
=
D
n
−
2
+

D
n
Δ
t
D
n
+
1
=
D
n
+
D
n
+
2
Δ
t
•
It is called
explicit methods
because the response at the current time can be calculated
explicitly; no iterations within a time step is needed.
 Chapter 15 Explicit Dynamics
Section 15.1 Basics of Explicit Dynamics
5
Solution Accuracy
•
<Explicit Dynamics> uses the
principle of conservation of energy
to monitor the solution
accuracy.
(Reference Energy) + (Work Done)
Reference
→
Current
=
(Current Energy)
•
It calculates overall energy at each cycle. If the energy error reaches a threshold, the
solution is regarded as unstable and stops. The default threshold is 10%.
•
The
Energy Error
is defined by
Energy Error
=
(Current Energy)-(Reference Energy) - (Work Done)
Reference
→
Current
max Current Energy , Reference Energy , Kinetic Energy
(
)
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