1.17.1 Design sensitivity analysis for cantilever beam

Products: ABAQUS/Standard  ABAQUS/Design  

Design sensitivity analysis in ABAQUS is performed using the semianalytical method. The issue of obtaining accurate sensitivities with respect to design shape parameters using this method has been discussed extensively in the literature (for example, Pedersen et al., 1989; Barthelemy and Haftka, 1990; Fenyes and Lust, 1991; and Van Keulen and De Boer, 1998). The difficulty is that the accuracy of the sensitivities can depend on the number of elements. This dependency is not seen with either analytical sensitivity analysis or with the overall finite difference method. A canonical example is a cantilever beam with an applied tip load, where the sensitivity of the tip displacement to the length of the beam is sought. This example demonstrates the effectiveness of the default perturbation sizing algorithm used by ABAQUS/Design in obtaining accurate tip displacement sensitivities. In addition, a sensitivity analysis is carried out to obtain the sensitivities of natural frequencies.

Problem description

Results and discussion

Input files

References

Tables

Table 1.17.1–1 ABAQUS tip displacement sensitivity results.

Number of elements along the lengthPerturbation size chosen by ABAQUS for the dominant elementPercentage error
CPS8S4RB31CPS8S4RB31
501.5e–061.5e–061.5e–08–0.004–0.002–0.002
1001.5e–061.5e–071.5e–080.008–0.002–0.002
2001.5e–061.5e–071.5e–080.009–0.002–0.002
4001.5e–071.5e–071.5e–080.009–0.002–0.002

Table 1.17.1–2 Comparison of eigenvalue and frequency sensitivities obtained with ABAQUS and other methods for CPS8 element.

Bending ModeMode NumberSensitivity w.r.t. beam length ABAQUS (default) ABAQUS (SIZING FREQ=1)Overall central differencing schemeAnalytic
11Eigenvalue–3.460e–03–3.460e–03–3.460e–03–3.461e–03
Frequency–9.362e–04–9.362e–04–9.350e–04–9.363e–04
22Eigenvalue–1.355e–01–1.355e–01–1.353e–01–1.353e–01
Frequency–5.860e–03–5.860e–03–5.856e–03–5.859e–03
33Eigenvalue–1.057e–00–1.057e–00–1.052e–00–1.058e–00
Frequency–1.630e–02–1.636e–02–1.630e–02–1.637e–02

Table 1.17.1–3 Comparison of eigenvalue and frequency sensitivities obtained with ABAQUS and other methods for S4R element.

Bending ModeMode NumberSensitivity w.r.t. beam length ABAQUS (default) ABAQUS (SIZING FREQ=1)Overall central differencing schemeAnalytic
14Eigenvalue–3.460e–03–3.460e–03–3.460e–03–3.460e–03
Frequency–9.362e–04–9.362e–04–9.362e–04–9.362e–04
210Eigenvalue–1.357e–01–1.357e–01–1.355e–01–1.353e–01
Frequency–5.860e–03–5.860e–03–5.856e–03–5.859e–03
317Eigenvalue–1.062e–00–1.062e–00–1.060e–00–1.058e–00
Frequency–1.640e–02–1.640e–02–1.640e–02–1.637e–02

Table 1.17.1–4 Comparison of eigenvalue and frequency sensitivities obtained with ABAQUS and other methods for B31 element.

Bending Mode Mode NumberSensitivity w.r.t. beam length ABAQUS (default) ABAQUS (SIZING FREQ=1)Overall central differencing schemeAnalytic
12Eigenvalue–3.460e–03–3.460e–03–3.457e–03–3.461e–03
Frequency–9.366e–04–9.360e–04–9.370e–04–9.363e–04
24Eigenvalue–1.353e–01–1.353e–01–1.354e–01–1.353e–01
Frequency–5.854e–03–5.854e–03–5.850e–03–5.859e–03
37Eigenvalue–1.055e–00–1.055e–00–1.058e–00–1.058e–00
Frequency–1.634e–02–1.634e–02–1.639e–02–1.637e–02


Figures

Figure 1.17.1–1 Variation of error in tip displacement sensitivity with respect to the perturbation size.

Figure 1.17.1–2 Variation of error in tip displacement sensitivity with mesh refinement for a perturbation size giving 0.1% error in for coarsest meshes (taken from Figure 1.17.1–1).

Figure 1.17.1–3 Variation of error in tip displacement sensitivity with mesh refinement for a perturbation of 1 × 10–9.