smart technologies for safety engineering

230 Smart Technologies for Safety Engineering
Figure 6.6 Structural remodeling in elastoplastic structures, modifications of material properties (numerical
example): virtual distortion forces modeling the assumed density modifications
with respect to element modified cross-sections Âi; derivatives with respect to other types of
modifications can be treated in an analogous way.
Direct differentiation of Equation (28) yields the following linear system, which can be used
iteratively in successive time steps to compute the derivatives of the virtual distortions:
S ∂x0
F
∂ Âl
= ∂b
∂ Âl
− ∂FS
x
∂ Âl
0
The principal matrix of Equation (29) stays the same as in Equation (28), which simplifies
the optimization process. The vector ∂x 0 /∂ Âl collects the derivatives of the virtual
Plastic distortion 0 [-]
0.0016
0.0012
0.0008
0.0004
0
–0.0004
VDM
ANSYS
Element 5
Element 2
Element 3
Element 1
Element 4
0 0.002 0.004 0.006 0.008 0.01
Tims [s]
Relative error of plastic strains [%]
0.2
0
–0.2
–0.4
–0.6
–0.8
(29)
Element 1
Element 2
Element 3
Element 4
Element 5
0 0.002 0.004 0.006 0.008 0.01
Time [s]
Figure 6.7 Structural remodeling in elastoplastic structures, modifications of material properties
(numerical example): (left) computed plastic distortions compared to the reference ANSYS results;
(right) errors of computed plastic distortions relative to the reference ANSYS results