Parametric modeling method for assembly systems

Assembly parameters are interrelated and mutually restricted. Consequently, the correlation between each parameter should be considered to establish the mapping relations between every design parameter, as well as the mapping relations between design parameters and mating-surface parameters, to ensure that the mating relations between the parts remain valid when the assembly model is updated. Mating between the parts of the assembly is achieved by matching the features of the parts, and a contact surface can be considered as a match of multiple features between two parts. To avoid conflicting design variables caused by part modeling when updating the assembly, a parametric modeling method for assembly oriented systems was devised.

First, the design variables of each part were determined, and the corresponding features of each design variable were determined accordingly. Subsequently, the decoupling relations between the design features and design variables were completed by transforming the design-associated matrix to achieve mutual independence of the design variables. Therefore, if a certain design variable is changed, it does not cause conflicts with other design variables. Finally, the assembly was transformed based on features from the contact relations between parts to the contact relations between design variables. In this manner, the mating between parts of the assembly can be controlled using design variables, as shown in Figure 1.

Fig. 1Full-screen View

Parametric modeling method for assembly-oriented systems

The nuclear fuel assembly top-connection structure, as a typical assembly model, contains three contact faces: the contact between the top nozzle and connection-sleeve outer surface, the contact between the connection-sleeve inner surface and the locking-thimble outer surface, and the contact between the guide-thimble and connection-sleeve inner surfaces. Parameterization was performed for these contact faces and the design variables of each part were determined, as shown in Figure 2.

Fig. 2Full-screen View

Schematics of the top-connection structure design parameters

Each part of the top-connection structure contained one parameter, and the design variables were independent of each other. Therefore, the mating surfaces of the top-connection structure correspond to the design parameters of each part, specifically D1–D2, R2–R6, R3–R5, D3–D4, and D6–D3. The links and equation constraints among the design parameters were established, as listed in Table 1, to ensure that the mating relationships between the parts remained valid when the top-connection structure model was updated.

Seven part design parameters were obtained by mapping the relations: D1, H2, R2, D5, D3, R6, and D7. The remaining eight parameters were obtained from the seven parameters by mapping relationships.

Contact surface gap and penetration value correction method for assembly systems

Errors occur between the mating surfaces during the assembly process; these have direct effects on the gap and penetration values between the contact surfaces. In addition, the quality of the modeling mesh affects the contact surface gap and penetration values of an assembly. In this study, a correction method for the initial penetration and gap of the contact surfaces was developed. This may be utilized in the following circumstances: When there is an initial gap between the contact surfaces of the assembly, the gap can be supplemented to facilitate surface contact or, in the case of slight penetration between the two contact surfaces, the common penetration area can be removed by adjusting the contact surfaces. The adjustment method is shown in Figure 3, in which the penetration area is marked in red.

Fig. 3Full-screen View

(Color online) Contact surface gap and penetration correction method

The top-connection structure of a nuclear fuel assembly contains three contact surfaces, of which two are nonlinear: the contact surface between the top nozzle and the connection-sleeve outer surface and that between the connection-sleeve inner surface and the outer surface of the locking-thimble outer surface. The other is linear: the contact surface between the guide thimble and the inner surface of the connection sleeve. Therefore, for the nonlinear contact surfaces of the top-connection structure, the gap and penetration values under different mesh sizes can be established by analyzing the penetration and gap variation patterns between each contact surface under different mesh sizes, as listed in Table 2.

As presented in Table 2, the direct penetration value of the contact surface decreases as the mesh size between each contact surface gradually decreases. The gap between each contact surface also decreases with a decrease in the mesh size. Corrections for the gap and penetration values of the contact surfaces for different mesh sizes were performed, and the results are listed in Table 3.

As presented in Table 3, all corrected gap values were 0. It is assumed that the contact surfaces are tightly matched with no gaps. In nonlinear contact analysis, this facilitates the complete transmission of force on each contact surface, is beneficial for the convergence of the results, and ensures the accuracy of the results. The corrected penetration values for different mesh sizes were all <10⁻¹³. Consequently, the penetration values between the contact surfaces can be considered negligible. This will assist in the adoption of a suitable mesh size to reduce the computational resources required for the assembly, as well as converge the analysis to ensure the accuracy of the results.

Pre-experimental design of top-connection structure variables

In the experimental design for assembly, unlike generic parts, the design variable spans of parts tend to overlap. Therefore, conducting a pre-experimental design can not only verify the plausibility of multiple design variable spans of the assembly but also reduce the number of design variables and preserve those with substantial influence on the reliability response. In the mechanical analysis of the top-connection structure of the nuclear fuel assembly under transportation and lifting conditions, the loads included the weights of the fuel assembly and control-rod assembly and axial acceleration. The total weight of the payload was ~7500 N and the axial acceleration was 4 g. Based on the ASME code, the third strength theory was used to evaluate the results of this study [23]. The simulation results for the top-connection structure under the transport and lifting conditions are shown in Fig. 4.

Fig. 4Full-screen View

(Color online) Stress distribution diagrams of each part. (a) Mating surface overall stress distribution diagram; (b) Top nozzle stress distribution diagram; (c) Lucking thimble stress distribution diagram; (d) Connection sleeve stress distribution diagram

Figure 4 shows that the overall stress of the assembly under the transportation and lifting conditions is mainly distributed near the contact surface. The maximum stresses of each part are at the contact surface, being 142.78 MPa for the connection sleeve, 94.992 MPa for the locking thimble, and 88.78 MPa for the top nozzle. The maximum stress value of each part was considered as the output response, and the maximum output response of each part was parameterized to realize a closed loop between the input and output parameters. The number of top-connection structure design variables was finally set to seven: D1, H2, R2, D5, D3, R6, and D7. The output response was set as the maximum stress value of each part from the simulation analysis, namely, Stress1, Stress2, and Stress3, corresponding to the maximum stress values of the top nozzle, locking thimble, and connection sleeve, respectively.

To ensure a uniform distribution of sample points in the design space and improve design efficiency, the Latin hypercube sampling method was deployed for the pre-experimental design of the design variables [24]. The design matrix contained 36 sample points, and the results of the pre-experimental design are presented in Appendix 1.

Sensitivity analysis of top-connection structure design variables

The number of assembly design variables is typically large, with each design variable exerting various degrees of influence on the output response. Therefore, reserving all design variables for analysis imposes a considerable burden on computational resources. Through a sensitivity analysis, the trend of the influence of each design variable on the output response, as well as the specific value of the influence, was obtained. To improve the computational efficiency, a sensitivity analysis was performed on the assembly design variables, and the design variables with a greater influence on the output response were selected to establish a reliability approximation model.

A sensitivity analysis was performed on the design variables of the top-connection structure. The extent of influence of design variables D1, H2, R2, D5, D3, R6, and D7 on the output responses Stress1, Stress2, and Stress3 are shown in Fig. 5, and the specific sensitivity values of each design parameter are listed in Table 4.

Fig. 5Full-screen View

(Color online) Extent of the influence of each parameter on the output response

Figure 5 shows that D1 demonstrated a negative correlation for both Stress1 and Stress3 and a positive correlation for Stress2. Among the design parameters, D1, D5, and D3 had the greatest influence on Stress3, Stress2, and Stress1, respectively. The extent of influence of each design parameter on Stress1 was ranked as D3 > D7 > D5 > H2 > R2 > R6 > D1, the extent of influence on Stress2 was ranked as D5 > D3 > H2 > D7 > R6 > D1 > R2, and the extent of influence on Stress3 was ranked as D1 > D5 > R6 > H2 > R2 > D7 > D3.

The construction of the response surface of each parameter versus the output response provides a straightforward understanding of the sensitivity of each parameter to the output response. Figure 6 shows the response surface of each parameter for the output response. From the response surface of each parameter to the output, the parameters did not exhibit monotonically increasing or decreasing effects on the output response, and a mutual influence existed between the two design variables. By taking H2 and D7 Stress1 as a reference, when D7 = 9 mm, the output response decreases monotonically with the increase in H2; when D7 = 10 mm, the output response decreases first with the increase in H2 and then increases; and, when D7 = 10.5 mm, the output response monotonically increases again with the increase in H2. When H2 = 4.5 mm, the output response increases monotonically with an increase in D7; when H2 = 5 mm, the output response decreases and then increases with an increase in D7; when H2 = 5.5 mm, the output response increases monotonically with an increase in D7.

Fig. 6Full-screen View

(Color online) Response surface of each parameter to the output response

Combining the response surfaces of each design parameter for Stress1, Stress2, and Stress3 reveals that all seven design variables of the top-connection structure have a significant impact on the output response. Therefore, there is no need to scale down the design variables.

Approximation model construction for top-connection structure

The pre-experimental design and design variable sensitivity analysis identified the design variables that influence the output response. Subsequently, an experiment was conducted to identify the design variables and obtain a more accurate approximation model of reliability. Appendix 2 presents the experimental design results for the top-connection structural variables.

The 120 generated sample points were divided into two groups, one serving as learning points to establish an approximation model and the other as verification points to assess the accuracy of the model. The principle of this division was to provide a sufficient number of learning points and reserve a certain number of verification points. Therefore, the sample points were divided in a ratio of 3:1 [25], with sample points 1–90 as the learning points and sample points 91–120 as the verification points.

The Kriging model, as an efficient interpolation method, can yield not only the predicted value of a prediction point but also the variance of that point. Because of this advantage, this method was used to construct a reliability approximation model in this study.

Assume that the actual relationship between the response and design variables can be expressed as $y(x)=F{(x)}^T\beta +z(x)，$ (1) where F(x) is a linear combination of polynomial functions, β is a linear regression coefficient, and z(x) is a Gaussian stochastic process with a mean of 0. The covariance between the two points x_i and x_j is $\mathrm{Cov}\left[z(x_i),z(x_j)\right]={\sigma }^{2}R\left(x_i,x_j;\theta \right)$ (2) where $\sigma$ is process variance, $R\left(x_i,x_j;\theta \right)$ is the correlation function at two points x_i and x_j, usually a Gaussian correlation function: $R\left(x_i,x_j;\theta \right)={\displaystyle \prod _{m=1}^M\exp }\left[-{\theta }_m{\left(x_i^m\text{-}{x}_j^m\right)}^2\right]$ (3) where $\theta$ is a parameter vector, m is the mth-dimensional vector of the input vector, and M is the total dimensionality of the input vector. $\theta$ can be estimated by using the maximum likelihood method as $\stackrel{*}{\theta }=\arg \max \left(-N_0\ln ({\sigma }^2)-\ln \left|R\right|\right).$ (4) A Kriging model was constructed based on n training points x_i (i = 1, 2, …, n). Let Y represent the vector of responses at n training points. The vectors of the regression coefficients and process variance can be estimated as $\stackrel{*}{\beta }={\left(F^T{R}^{-1}F\right)}^{-1}{F}^T{R}^{-1}Y,$ (5) ${\sigma }^2=\frac{1}{N_0}{\left(Y-F\stackrel{*}{\beta }\right)}^T{R}^{-1}\left(Y-F\stackrel{*}{\beta }\right),$ (6) $\begin{array}{l}r(x)={\left[R\left(x,x_1;\theta \right),R\left(x,x_2;\theta \right),\mathrm{...},R\left(x,x_n;\theta \right)\right]}^T,\\ \mu =F{\left(x\right)}^T\stackrel{*}{\beta }+\text{r}{\left(x\right)}^T{R}^{-1}\left(Y-F\stackrel{*}{\beta }\right).\end{array}$ (7)

Approximation model accuracy verification

The accuracy of the approximation model is crucial to the accuracy of the reliability analysis results, and it can be verified via validation points. The metrics for the accuracy verification of the approximation model include the relative maximum absolute error (RMAE), root mean square error (RMSE), and coefficient of determination R² [26,27]. RMAE is given by $\mathrm{RMAE}=\frac{\underset{i=1:N}{\max }\{|y_i-{\widehat{y}}_i|\}}{\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(y_i-{\overline{y}}_i\right)}^2}}}$ (8) where N is the number of verification points, $y_i$ is the true value of the ith verification point, ${\widehat{y}}_i$ is the predicted value of the ith verification point, and $\overline{y}$ is the mean value of the verification points. The RMAE was used to characterize the absolute maximum residual value relative to the standard deviation of the output value of the sample points. The closer the value of RMAE is to 0, the higher the accuracy of the approximation model.

RMSE is given by $\mathrm{RMSE}=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}}.$ (9) The RMSE was used to characterize the dispersion of the sample points. The closer the RMSE value is to 0, the higher the accuracy of the approximation model.

Finally, R² is given by $R^2=1-\frac{{\displaystyle \sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^N{\left(y_i-\overline{y}\right)}^2}},$ (10) R² was used to characterize the degree of agreement between the predicted and real values. The closer the $R^2$ value is to 1, the higher the accuracy of the approximation model.

Figure 7 shows the relationship between the predicted and true values, and the errors for each method are listed in Table 5.

Fig. 7Full-screen View

(Color online) Kriging goodness-of-fit graphs

As presented in Table 5, the results verified by each method meet the requirements and, consequently, prove that the Kriging model constructed according to this sample point meets the accuracy requirements.

Multivariate reliability analysis based on MCS

MCS is a robust stochastic simulation method for structural reliability analysis [28-30]. It is an excellent contemporary tool for structural reliability analysis because its calculation accuracy and convergence speed are not affected by the system complexity [31-33]. However, the computational efficiency of this method is limited and the required number of sample points is large. In light of this, to address the limited computational efficiency of MCS combining the assembly-oriented reliability approximation model, a Monte Carlo reliability analysis method based on an approximation model is proposed in this study. The specific steps are as follows:

Step 1: Determine the random variables and designate their probability distribution and probability distribution types.

Step 2: Define the maximum number of MCS simulation runs, N.

Step 3: Generate a sequence of uniformly distributed random numbers.

Step 4: Convert the generated random number sequence into corresponding random variable values.

Step 5: Invoke the assembly approximation model to compute the response to the current value.

Step 6: Repeat Steps 3–5 until the maximum number of simulations, N, is reached.

Probability distribution statistics of random variables for the top-connection structure

In machining the top-connection structure of nuclear fuel assemblies, the working accuracy varies because of the accuracy of the machine tools and instruments, as well as the proficiency of the machining operators. Therefore, reliability analysis of the top-connection structure requires considering each critical dimension as a random variable. Furthermore, the influence of material properties and the unevenness of loading on the structural function should also be considered, with the modulus of elasticity E of the material and loading force F as random variables. Under normal conditions, the dimensionality caused by machining errors conforms to a normal distribution. The part dimensions in the top-connection structure as an assembly have upper and lower limits, thus conforming to a truncated normal distribution. The modulus of elasticity E of the material follows a normal distribution, whereas the force load F fluctuates within a certain range, making it conform to a truncated normal distribution. The statistical properties of each random variable for the top-connection structure are listed in Table 6.

Limit-state function determination for the top-connection structure

The structural output response of the top connection must satisfy the ASME code, with a membrane stress of <138 MPa and a membrane plus bending stress of <207 MPa. The failure mode of the top-connection structure is a typical strength failure, and the following structural limit-state function can be established [34]: $G\left(x_1,\mathrm{...},x_n\right)=R\left(x_1,\mathrm{...},x_n\right)-S\left(x_1,\mathrm{...},x_n\right),$ (11) where the variables $x_1,\mathrm{...},x_n$ are random variables affecting the function, $R(x_1,\mathrm{...},x_n)$ is the rigidity random variable, and $S(x_1,\mathrm{...},x_n)$ is the stress random variable. When G > 0, the structure is in a safe state, whereas, when G < 0, it is in a failure state.

The top-connection structure, which is an assembly composed of several parts, fails when one part fails. Therefore, they can be regarded as a system. For the top-connection structure, the specific structure limit-state function is $G(X)=207-\max \left\{Stress1,Stress2,Stress3\right\}=0.$ (12) The probability of an individual part failure can be expressed as $P_{f_i}=P\left(G\left(x_1,\cdot \cdot \cdot ,x_n\right)<0\right)$; the probability of the entire top-connection structure failure is $P_f=P_1\cap \cdot \cdot \cdot \cap P_i$, where i is the number of system parts and the reliability is $P_s=1-P_f$.

Reliability assessment method for the top-connection structure

In this study, the MCS method was deployed, invoking the Kriging model to calculate the reliability of the top-connection structure. The maximum number of runs of the MCS was set to 10,000. The distribution types and data for each random parameter are listed in Table 8. The results for each response distribution are shown in Figure 9.

Fig. 9Full-screen View

(Color online) Cumulative distribution of each response

Figure 9 shows that the stress distributions of Stress1 are all within 130 MPa, concentrated in the range of 65–110 MPa, and do not exceed 207 MPa; the stress distributions of Stress2 are concentrated in the range of 40–110 MPa, with very few exceeding 200 MPa; and the stress distributions of Stress 3 are concentrated between 100 and 170 MPa, with a few exceeding 200 MPa. The failure probability of each part is $P_f=\frac{N}{N_0},$ (13) where N is the number of sample points with G(x) < 0 in the MCS and N₀ is the total number of MCS runs. The failure probability of each part can be obtained from Figure 9 and is listed in Table 7.

According to Table 7, the reliabilities of the top nozzle, locking thimble, and connection sleeve are 100%, 99.85%, and 98.15%, respectively. This corresponds to a real-world failure scenario. The connection sleeve, placed between the top nozzle and locking thimble, was directly loaded by the entire fuel assembly during operation. Compared to the top nozzle and locking thimble, it exhibited the highest probability of failure during operation in the fuel assembly [13]. In general, the structural reliability of the top connection remained sound even when considering the randomness of each variable.