Add the Two-Dimensional Reliability Test Function Example 1 from Grandhi and Wang (1999)

[damar-wicaksono]

The first reliabity test function from Grandhi and Wang (1999)¹ is defined as follows:

$$ \mathcal{M}(\boldsymbol{x}; a, b) = 1.0 + \left( \frac{x_1 + x_2}{a} \right)^2 - \left( \frac{x_1 - x_2}{b} \right)^2, $$

where $\boldsymbol{x} = \{ x_1, x_2 \}$ is the vector of input variables modeled as two independent standard normal random variables.
The function is parametrized by the parameters $a$ and $b$. In the original paper $a$ is fixed to $2.0$ while different $b$ values were considered ($\pm 0.5$, $\pm 2$, $\pm 4$).

Footnotes

1. R. V. Grandhi and L. Wang, “Higher-order failure probability calculation using nonlinear approximations,” Computer Methods in Applied Mechanics and Engineering, vol. 168, no. 1–4, pp. 185–206, Jan. 1999, doi: 10.1016/S0045-7825(98)00140-6. ↩