Fig. 1 The model setting illustrates a three-point bending setup of a linear elastic Timoshenko beam, fixed at one end, and simply supported at the other. The beam has a square cross section with significant dimensions and properties shown. More about this image found in The model setting illustrates a three-point bending setup of a linear elast...

Fig. 2 Distribution of analytical model for different sample sizes with coefficient of variation of 2% using Monte Carlo method. The sample sizes are ( a ) 10, ( b ) 20, ( c ) 50, ( d ) 100, ( e ) 500, ( f ) 1,000, and ( g ) 10,000. Each plot shows the standard deviation ( σ ) of the distribut... More about this image found in Distribution of analytical model for different sample sizes with coefficien...

Fig. 3 Distribution of standard deviation estimated from Monte Carlo (crosses), propagation of error (triangle), and tolerance design (circles) methods for different sample sizes (horizontal axis) with coefficient of variation of ( a )1%, ( b ) 2%, ( c ) 5%, ( d ) 10%, and ( e ) 20% for the input ... More about this image found in Distribution of standard deviation estimated from Monte Carlo (crosses), pr...

Fig. 4 Estimated standard deviation versus coefficient of variation using propagation of error and different tolerance design arrays. The standard deviations are normalized against the value obtained from Monte Carlo calculations with N  = 10,000. The line indicates the ratio of the standard devi... More about this image found in Estimated standard deviation versus coefficient of variation using propagat...

Fig. 5 The model setting illustrates an aluminum plate with a notch on the longer edge under tensile displacement. The plate is fixed on one end, and a constant displacement of 0.1167 mm is applied at half of the top surface, which is the side closer to the notch. The significant dimensions and pr... More about this image found in The model setting illustrates an aluminum plate with a notch on the longer ...

Fig. 6 ( a ) X–Y plane view of the finite element model of aluminum plate, ( b ) deformed model with a scale factor of 100 in y -direction, ( c ) perspective view of the plate, and ( d ) model view of the plate with the boundary conditions. The fixed constraint is created at the initial step at... More about this image found in ( a ) X–Y plane view of the finite element model of aluminum plate, ( b )...

Fig. 7 The crack opening displacement of different mesh sizes of an arbitrarily chosen model from among the Monte Carlo models. The parameter values have a coefficient of variation of 5% of the nominal value. More about this image found in The crack opening displacement of different mesh sizes of an arbitrarily ch...

Fig. 8 Example of node path of six nodes around the crack with an exaggerated deformation in the y -direction with a scale factor of 100. The numbers denote the node number of the corresponding node. Six nodes are chosen to ensure that there are enough number of nodes to capture the actual crack ... More about this image found in Example of node path of six nodes around the crack with an exaggerated defo...

Fig. 9 Histogram of the crack opening distance of MC model. This distribution with two peaks is slightly skewed to the right. The most frequent bin ranges from 4.00 × 10 − 5  m to 4.05 × 10 − 5  m with a count of 600 out of 10,000 cases. The median is 4.20 × ... More about this image found in Histogram of the crack opening distance of MC model. This distribution with...

Fig. 10 Relative standard deviation of tolerance design method at each array design to Monte Carlo model with N  = 10,000 (stars). The line indicates the ratio of the standard deviation of Monte Carlo model to itself as a reference. More about this image found in Relative standard deviation of tolerance design method at each array design...

Fig. 1 Grid convergence index procedure for practical applications (formal order of accuracy: p f  = 2) More about this image found in Grid convergence index procedure for practical applications (formal order o...

Fig. 2 Least squares procedure for practical applications (formal order of accuracy: p f  = 2) More about this image found in Least squares procedure for practical applications (formal order of accurac...

Fig. 3 Illustration of discretization error of ϕ i from the LS procedure. The solid circle represents solution at each grid set ( ϕ i ). The hollow circle denotes the solution estimate from the fit ( ϕ ̂ i ). More about this image found in Illustration of discretization error of ϕ i from the LS procedure. The s...

Fig. 4 Computational domain used for the simulations ( D  = the depth of the cylinder) More about this image found in Computational domain used for the simulations ( D  = the depth of the cylin...

Fig. 5 Three types of grids: ( a ) structured grid (G1), ( b ) unstructured grid (Delaunay, G2), and ( c ) unstructured grid (Advancing Front, G3) More about this image found in Three types of grids: ( a ) structured grid (G1), ( b ) unstructured grid (...

Fig. 6 Variation of solutions as a function of governing equation tolerance for a representative case More about this image found in Variation of solutions as a function of governing equation tolerance for a ...

Fig. 7 Grid convergence ratios from Grids 1, 2, and 3 and from Grids 2, 3, and 4 (G1 and Tr-SST (blue (dark gray) hollow); G2 and Tr-SST (green (light gray) hollow); G3 and Tr-SST (red (medium gray) hollow); G1 and SST (blue (dark gray) solid); G2 and SST (green (light gray) solid); G3 and SST (re... More about this image found in Grid convergence ratios from Grids 1, 2, and 3 and from Grids 2, 3, and 4 (...

Fig. 8 Observed p from LS and GCI procedures (G1 and Tr-SST (blue (dark gray) hollow); G2 and Tr-SST (green (light gray) hollow); G3 and Tr-SST (red (medium gray) hollow); G1 and SST (blue (dark gray) solid); G2 and SST (green (light gray) solid); G3 and SST (red (medium gray) solid)) More about this image found in Observed p from LS and GCI procedures (G1 and Tr-SST (blue (dark gray) ho...

Fig. 9 Distribution of σ fit/ Δ ϕ obtained from the LS procedure for 36 cases More about this image found in Distribution of σ fit/ Δ ϕ obtained from the LS procedure for 36 cases...