Second-Order Taylor’s Expansion

Adding the second-order terms, the Taylor’s series expansion for a performance response, $Y$ , is

\begin{matrix} (8) & Y (x) = y + \frac{d Y}{d x} Δ x + \frac{1}{2} Δ x^{T} \frac{d^{2} Y}{d x^{2}} Δ x \end{matrix}

The mean of this performance response is then obtained by taking the expectation of both sides of the expansion:

\begin{matrix} (9) & μ_{Y} = Y (μ_{X}) + \frac{1}{2} \sum_{i}^{n} \frac{d^{2} Y}{d x_{i}^{2}} σ_{x_{i}}^{2} \end{matrix}

and the standard deviation of $Y (x)$ is given by

\begin{matrix} (10) & σ_{Y} = \sqrt{\sum_{i = 1}^{n} {(\frac{\partial Y}{\partial x_{i}})}^{2} {(σ_{x_{i}})}^{2} + \frac{1}{2} \sum_{i}^{n} \sum_{j}^{n} {(\frac{\partial^{2} Y}{\partial x_{i} \partial x_{j}})}^{2} {(σ_{x_{i}})}^{2} {(σ_{x_{i}})}^{2}} \end{matrix}

where $σ_{x i}$ is the standard deviation of the $i$ th parameter and $n$ is the standard deviation of the $j$ parameter (Hsieh and Oh, 1992).

Once the performance response standard deviation is estimated with the necessary sensitivities, robustness or quality can be assessed (sigma level, percent variation, probability, or defects per million parts, based on defined specification limits).

Because second-order derivatives of responses, including crossed terms, with respect to random variables are needed and the mean value point is needed in the equation above, the second-order Taylor’s expansion estimates require $(n + 1) (n + 2) / 2$ analyses for evaluation. Because of this expense, cross terms are not usually included (only pure second-order deviations) and the expense is reduced to $2 n + 1$ . This approach is then twice as computationally expensive as the first-order Taylor’s expansion and usually will be more expensive than DOE. For low numbers of uncertain parameters, this approach can still be more efficient than Monte Carlo simulation but becomes less efficient with increasing $n$ . (Monte Carlo simulation is not dependent on the number of parameters.) The second-order Taylor’s expansion approach is recommended when there is curvature and the number of uncertain parameters is not excessive.

The Mean Value reliability method uses the Taylor’s series expansion of failure functions $g (X)$ at the mean values $μ x$ . The mean-value reliability index is then calculated as a function of the mean and standard deviation of $g (X)$ .