The extreme values of the equivalent stress are 218

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Unformatted text preview: y Figure 4; Principle of geometric uncertainty model The points of the surface are assumed to be randomly distributed around the mean feature. As a first approximation, a Gaussian distribution was assumed. For such distribution the geometrical errors can be characterized by their variance σ2 r. The variance-covariance matrix Cov( â ) of the parameters can also be calculated. Next, this data can be propagated to each point Mi of any feature. This allows computing the variance in a given direction (ni ): ˆ Cov (M i ) = J (M i ).Cov (a).J t (M i ) var (M i / n i ) = n i .Cov (M i ).ni t In the case of a line of a 2D plot (figure 4) the following relations are obtained: ⎛ var a 0 Cov (a 0 , a 1 ) = ⎜ ⎜ cov( a ,a ) ⎝ 01 u(Mi/n i ) = var (Mi/n i ) cov( a0 ,a1 ) ⎞ ⎟ var a 1 ⎟ ⎠ var(M i / n i ) = var(a 0 ) + 2.λi . cov(a 0 , a 1 ) + λ2 . var(a 1 ) i The error bar of the propagated uncertainty at point Mi can finally be estimated in the direction ni for any level of risk α : U(Mi/ni)=k(α).u(Mi/ni) . The localization of the mean surface...
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