2.3 Computational analysis
19
2.3.1 Finite Element Method
The Finite Element Method enables to analyze the mechanical properties of complicated
structures by discretizing the geometry into smaller segments also known as elements,
which can be analyzed using basic engineering mechanics. The elements are linked via
nodes while the relationship between the nodes is described by shape functions. The
mechanical quantities of the system are determined by calculating the distribution of
the nodes depending on the boundary conditions including external loads. The system is
described with a set of differential equations, which can be written as the following equilibrium equation ([Nas15])
M ∙ u C ∙ u K ∙ u u F
2.11
Here, M refers to the mass matrix, C to the damping matrix, K is the stiffness matrix and
F defines a set of external loads. , , and are the acceleration, velocity and displacement of the nodes respectively. The equation above describes a linear static system.
Modelling progressive damage mechanisms requires transient and non-linear relationships. Therefore, the displacement and its derivatives are time dependent while the stiffness and damping matrices as well as the external loads may depend on the displacement and velocity of the nodes (Eq 2.2)
M ∙ u t Cu, u ∙ u t Ku, u ∙ ut FFu, u
2.22
Solving such transient systems requires discretizing the time into increments ∆∆, while
the system is solved for each point in time. There are two main types of time integration
methods, the implicit and explicit one [Liu03]. Both methods have their pro’s and con’s
and hence their specific areas of application depending on the problem to be solved.
Generally, explicit methods are computationally more efficient for rapid and highly nonlinear phenomena such as impact or explosion. Implicit methods are advantageous for
quasi-static problems with limited degree of non-linearity. This issue is illustrated in
Figure 17 with the help two graphs. Figure 17 a) illustrates the suitable area of application for both methods depending on the velocity and degree of non-linearity of the investigated problem. Figure 17 b) displays the relationship between computational effort
and complexity of the problem for both methods, while the complexity refers to the
combination of velocity and degree of non-linearity. The computational effort largely
corresponds to the computing time. However, the computing time not only depends on
the model setup but also on the available computing resources. Therefore, computational effort is used as general term, which includes both aspects, computing time and
computational resources. The computational effort is of particular interest in industrial
applications where the simulation results are required to make design decisions in tightly
scheduled development projects. From the graphs in Figure 17, it can be seen that there
is a transition zone with applications where both methods are equally suitable. Virtual
testing of sandwich structures often falls into this transition zone, with quasi-static tests
19
2.3.1 Finite Element Method
The Finite Element Method enables to analyze the mechanical properties of complicated
structures by discretizing the geometry into smaller segments also known as elements,
which can be analyzed using basic engineering mechanics. The elements are linked via
nodes while the relationship between the nodes is described by shape functions. The
mechanical quantities of the system are determined by calculating the distribution of
the nodes depending on the boundary conditions including external loads. The system is
described with a set of differential equations, which can be written as the following equilibrium equation ([Nas15])
M ∙ u C ∙ u K ∙ u u F
2.11
Here, M refers to the mass matrix, C to the damping matrix, K is the stiffness matrix and
F defines a set of external loads. , , and are the acceleration, velocity and displacement of the nodes respectively. The equation above describes a linear static system.
Modelling progressive damage mechanisms requires transient and non-linear relationships. Therefore, the displacement and its derivatives are time dependent while the stiffness and damping matrices as well as the external loads may depend on the displacement and velocity of the nodes (Eq 2.2)
M ∙ u t Cu, u ∙ u t Ku, u ∙ ut FFu, u
2.22
Solving such transient systems requires discretizing the time into increments ∆∆, while
the system is solved for each point in time. There are two main types of time integration
methods, the implicit and explicit one [Liu03]. Both methods have their pro’s and con’s
and hence their specific areas of application depending on the problem to be solved.
Generally, explicit methods are computationally more efficient for rapid and highly nonlinear phenomena such as impact or explosion. Implicit methods are advantageous for
quasi-static problems with limited degree of non-linearity. This issue is illustrated in
Figure 17 with the help two graphs. Figure 17 a) illustrates the suitable area of application for both methods depending on the velocity and degree of non-linearity of the investigated problem. Figure 17 b) displays the relationship between computational effort
and complexity of the problem for both methods, while the complexity refers to the
combination of velocity and degree of non-linearity. The computational effort largely
corresponds to the computing time. However, the computing time not only depends on
the model setup but also on the available computing resources. Therefore, computational effort is used as general term, which includes both aspects, computing time and
computational resources. The computational effort is of particular interest in industrial
applications where the simulation results are required to make design decisions in tightly
scheduled development projects. From the graphs in Figure 17, it can be seen that there
is a transition zone with applications where both methods are equally suitable. Virtual
testing of sandwich structures often falls into this transition zone, with quasi-static tests