20
2 State of the art
being investigated while at the same many simultaneous non-linear effects have to be
considered. This is also evident in the reviewed literature, where both methods can be
found for the failure prediction of sandwich structures. There are various commercial
finite element software packages available for each of the two methods. However, due
to the fundamental differences, the available codes specialize in either one of the methods and finite element models are generally implemented for a specific integration
scheme. Therefore, the integration method cannot be readily changed for an existing
model. This requires at least some degree of model adaption. As a result of this, the
applied integration method should be defined carefully prior to the model development.
The following paragraphs briefly introduce the theory behind the two methods.
Non-linearity
Static
Dynamic
Velocity
Elastic
Plastic
Buckling
Damage
Rupture
Explicit
Implicit
Computational effort
Complexity
Non-linear dynamic
Static / Elastic
Explicit
Implicit
a)
b)
Figure 17 Comparison of implicit and explicit integration schemes; a) suitability depending on type
of problem, b) Computational effort depending on type of problem according to Altair [Alt12]
Explicit time integration
Using the explicit integration Eq. (2.2) is solved at time ∆∆ by extrapolating the state
of equilibrium at time . The extrapolation is commonly based on the central difference
method, which makes assumptions regarding the relationship between displacement,
velocity and acceleration. Using this method, only the damping and mass matrix have to
be inverted, making the equation solving computationally inexpensive as long as under
integrated elements are implemented. However, the method is conditionally stable and
requires time steps below a critical time step in order to give plausible results. The critical
time step ∆∆ equals the time a stress wave takes to cross the smallest element in the
mesh [Liu03]. This time depends on the speed of sound of the material and the characteristic element length
, while relates to the density and modulus of the
material.
∆∆ ∆ ∆∆
; ;
2.33
Therefore, the maximum allowable time step depends on the mesh size as well as the
mass and stiffness of the elements. Generally, a large time step is desirable, since it reduces the total number of increments to be solved. However, often the mesh size and
2 State of the art
being investigated while at the same many simultaneous non-linear effects have to be
considered. This is also evident in the reviewed literature, where both methods can be
found for the failure prediction of sandwich structures. There are various commercial
finite element software packages available for each of the two methods. However, due
to the fundamental differences, the available codes specialize in either one of the methods and finite element models are generally implemented for a specific integration
scheme. Therefore, the integration method cannot be readily changed for an existing
model. This requires at least some degree of model adaption. As a result of this, the
applied integration method should be defined carefully prior to the model development.
The following paragraphs briefly introduce the theory behind the two methods.
Non-linearity
Static
Dynamic
Velocity
Elastic
Plastic
Buckling
Damage
Rupture
Explicit
Implicit
Computational effort
Complexity
Non-linear dynamic
Static / Elastic
Explicit
Implicit
a)
b)
Figure 17 Comparison of implicit and explicit integration schemes; a) suitability depending on type
of problem, b) Computational effort depending on type of problem according to Altair [Alt12]
Explicit time integration
Using the explicit integration Eq. (2.2) is solved at time ∆∆ by extrapolating the state
of equilibrium at time . The extrapolation is commonly based on the central difference
method, which makes assumptions regarding the relationship between displacement,
velocity and acceleration. Using this method, only the damping and mass matrix have to
be inverted, making the equation solving computationally inexpensive as long as under
integrated elements are implemented. However, the method is conditionally stable and
requires time steps below a critical time step in order to give plausible results. The critical
time step ∆∆ equals the time a stress wave takes to cross the smallest element in the
mesh [Liu03]. This time depends on the speed of sound of the material and the characteristic element length
, while relates to the density and modulus of the
material.
∆∆ ∆ ∆∆
; ;
2.33
Therefore, the maximum allowable time step depends on the mesh size as well as the
mass and stiffness of the elements. Generally, a large time step is desirable, since it reduces the total number of increments to be solved. However, often the mesh size and