The two-dimensional reliability test function from Au and Beck (1999)[^Au] features a highly concave limit state curve; the function is defined as follows (see Eq.(27), Example 1d) :
where $\boldsymbol{x} = \{ x_1, x_2 \}$ is the vector of input variables modeled as two independent standard normal random variables.
[^Au]: S. K. Au and J. L. Beck, “A new adaptive importance sampling scheme for reliability calculations,” Structural Safety, vol. 21, no. 2, pp. 135–158, Jun. 1999, doi: 10.1016/S0167-4730(99)00014-4.
The two-dimensional reliability test function from Au and Beck (1999)[^Au] features a highly concave limit state curve; the function is defined as follows (see Eq.(27), Example 1d) :
$$ \mathcal{M}(\boldsymbol{x}) = 3 - x_2 + \left( 4 x_1 \right)^2, $$
where $\boldsymbol{x} = \{ x_1, x_2 \}$ is the vector of input variables modeled as two independent standard normal random variables.
[^Au]: S. K. Au and J. L. Beck, “A new adaptive importance sampling scheme for reliability calculations,” Structural Safety, vol. 21, no. 2, pp. 135–158, Jun. 1999, doi: 10.1016/S0167-4730(99)00014-4.