Alpha shape based design space decomposition for island failure regions in reliability based design

Abstract

Treatment of uncertainties in structural design involves identifying the boundaries of the failure domain to estimate reliability. When the structural responses are discontinuous or highly nonlinear, the failure regions tend to be an island in the design space. The boundaries of these islands are to be approximated to estimate reliability and perform optimization. This work proposes Alpha (α) shapes, a computational geometry technique to approximate such boundaries. The α shapes are simple to construct and only require Delaunay Tessellation. Once the boundaries are approximated based on responses sampled in a design space, a computationally efficient ray shooting algorithm is used to estimate the reliability without any additional simulations. The proposed approach is successfully used to decompose the design space and perform Reliability based Design Optimization of a tube impacting a rigid wall and a tuned mass damper.

Appendix A: error metrics

• 1. R²:

  The coefficient of multiple determinations is defined as:

  $$ {R}^2=1-\frac{{\displaystyle \sum_{i=1}^n{\left({y}_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle \sum_{i=1}^n{\left({y}_i-\overline{y}\right)}^2}} $$

  (A.1)

  where y _i is the actual value at the i ^th design point, ŷ _i is the predicted value at the i ^th design point, and $\overline{y}$ the mean of the actual response. R ² is a measure of the amount of reduction in the variability of y obtained by using the response surface. 0 ≤ R ² ≤ 1. A larger value of R ² is desirable for a good response surface. But, a larger R ² does not necessarily guarantee a good response surface. Thus, this estimate should be used in conjunction with other error estimates to gauge the quality of the response surface. R ² continuously increases with addition of terms irrespective of whether the additional term is statistically significant.

• 2. Adjusted R²:

  The adjusted coefficient of multiple determinations is defined as

  $$ {R}_{adj}^2=1-\frac{n-1}{n-p}\left(1-{R}^2\right) $$

  (A.2)

  where n is the number of design points, and p is the number of regression coefficients.

  Unlike R ², R ² _adj decreases when unnecessary terms are added. Hence, R ² _adj along with R ² can be used to comment on the quality of response surface and the presence of unnecessary terms in the response surface.

• 3. Root-Mean-Square Error (RMSE):

  The root-mean-square error, RMSE, and the predicted RMS errors are defined, respectively, as

  $$ \mathrm{R}\mathrm{M}\mathrm{S}=\sqrt{\frac{{\displaystyle \sum_{i=1}^n{\left({y}_i-{\widehat{y}}_i\right)}^2}}{n}} $$

  (A.3)

• 4. Prediction Error Sum of Squares (PRESS):

  The prediction error sum of squares provides error scaling. To estimate the PRESS, an observation is removed at a time and a new response surface is fitted to the remaining observations. The new response surface is used to predict the withheld observation. The difference between the withheld observation and the computed response value gives the PRESS residual for that observation. This process is repeated for all the observations and the PRESS statistic is defined as the sum of the squares of the n PRESS residuals. When polynomial response surfaces are used, the repetitive estimate of PRESS residuals can be obviated by using the following expression:

  $$ \mathrm{PRESS}={{\displaystyle \sum_{i=1}^n\left(\frac{e_i}{1-{E}_{ii}}\right)}}^2 $$

  (A.4)

  where E = X(X ^T X)^− 1 X ^T and X is the Grammian matrix (ŷ = Xb), and b is the coefficient vector. Data points at which E _ii are large will have large PRESS residuals. These observations are considered high influence points. That is, a large difference between the ordinary residual and the PRESS residual will indicate a point where the model fits the data well, but the model built without that point has a poor prediction. A RMS version of PRESS allows us to compare the PRESS_RMS with the RMS errors. This permits us to explore the influence that few points might have on the entire fit. The PRESS_RMS is expressed as:

  $$ \mathrm{PRESS}\_\mathrm{R}\mathrm{M}\mathrm{S}=\sqrt{\frac{\mathrm{PRESS}}{n}} $$

  (A.5)

  PRESS can be used to estimate an approximate R ² for prediction as:

  $$ {R}_{pred}^2=1-\frac{PRESS}{{\displaystyle \sum_{i=1}^n{\left({y}_i-\overset{\_}{y}\right)}^2}} $$

  (A.6)

  The denominator in (A.6) is referred to as total sum of the squares.