2.3 Computational analysis
21
element stiffness are predefined and cannot be altered. Therefore, the time step can
only be increased by increasing the mass of the elements. This process is called mass
scaling and enables to reduce the computation time of the simulation. However, increasing the mass also increases the kinetic energy, which may destabilize the solution leading
to invalid results. In addition, the time step directly affects the quality of the solution.
Large time increments as well as a great number of time steps lead to increasing error.
This is illustrated in Figure 18, where the vertical lines along with the capital letters A to
D indicate the time increments. Therefore, applying explicit integration generally requires some preliminary sensitivity studies to determine a suitable time step.
Actual curve
progression
Explicit
solution
Current system
behaviour
Time
Function value
Large time increments
Small time increments
Function value
Time
A
B
C
D
Error
Figure 18 Explicit time integration for large and small time increments
Implicit time integration
Unlike the explicit solution, the implicit integration scheme attempts to solve Eq. (2.2) at
time ∆∆ based on assumptions regarding the unknown system quantities of this following time step. Therefore, the implicit method requires equilibrium iterations, which
approximate the solution until convergence is achieved. This process requires to invert
the stiffness matrix K making the equation solving computationally expensive, while the
equilibrium iterations may require multiple solver runs for each time increment (i.e.
A 1 – A n in Figure 19). This is compensated by the fact, that the implicit method enables
large time increments if compared to the explicit method (factor 100 to 1000 larger)
[Liu03]. Furthermore, the implicit method is unconditionally stable regardless the time
increment length, yet the increment length affects the quality of the solution and may
depend on the convergence conditions. Figure 19 illustrates the implicit integration
scheme for short and long time increments. Due to the mandatory equilibrium iterations,
the implicit integration may encounter convergence problems, which prevent the solver
from finding a solution.
21
element stiffness are predefined and cannot be altered. Therefore, the time step can
only be increased by increasing the mass of the elements. This process is called mass
scaling and enables to reduce the computation time of the simulation. However, increasing the mass also increases the kinetic energy, which may destabilize the solution leading
to invalid results. In addition, the time step directly affects the quality of the solution.
Large time increments as well as a great number of time steps lead to increasing error.
This is illustrated in Figure 18, where the vertical lines along with the capital letters A to
D indicate the time increments. Therefore, applying explicit integration generally requires some preliminary sensitivity studies to determine a suitable time step.
Actual curve
progression
Explicit
solution
Current system
behaviour
Time
Function value
Large time increments
Small time increments
Function value
Time
A
B
C
D
Error
Figure 18 Explicit time integration for large and small time increments
Implicit time integration
Unlike the explicit solution, the implicit integration scheme attempts to solve Eq. (2.2) at
time ∆∆ based on assumptions regarding the unknown system quantities of this following time step. Therefore, the implicit method requires equilibrium iterations, which
approximate the solution until convergence is achieved. This process requires to invert
the stiffness matrix K making the equation solving computationally expensive, while the
equilibrium iterations may require multiple solver runs for each time increment (i.e.
A 1 – A n in Figure 19). This is compensated by the fact, that the implicit method enables
large time increments if compared to the explicit method (factor 100 to 1000 larger)
[Liu03]. Furthermore, the implicit method is unconditionally stable regardless the time
increment length, yet the increment length affects the quality of the solution and may
depend on the convergence conditions. Figure 19 illustrates the implicit integration
scheme for short and long time increments. Due to the mandatory equilibrium iterations,
the implicit integration may encounter convergence problems, which prevent the solver
from finding a solution.