Multi-Fidelity Modeling-Based Structural Reliability Analysis with the Boundary Element Method

In this work, a method for the application of multi-¯delity modeling to the reliability analysis of 2D elastostatic structures using the boundary element method (BEM) is proposed. Reliability analyses were carried out on a rectangular plate with a center circular hole subjected to uniaxial tension using Monte Carlo simulations (MCS), the ¯rst-order reliability method (FORM), and the second-order reliability method (SORM). Two BEM models were investi- gated, a low-¯delity model (LFM) of 20 elements and a high-¯delity model (HFM) of 100 elements. The response of these models at several design points was used to create multi- ¯delity models (MFMs) utilizing second-order polynomial response surfaces and their reliability, alongside that of the LFM and the HFM, was evaluated. Results show that the MFMs that directly called the LFM were signi¯cantly superior in terms of accuracy to the LFM, achieving very similar levels of accuracy to the HFM, while also being of similar computational cost to the LFM. These direct MFMs were found to provide good substitutes for the HFM for MCS, FORM, and SORM.


Introduction
All engineering parameters have uncertainties in their values, it is therefore important that these are considered during the design process. Failure to consider uncertainties can lead to the design failing or becoming overengineered, requiring large safety factors. As stated by Huang et al., 1 adequate performance cannot be guaranteed when there is uncertainty. Therefore, a probability of success that the design meets certain criteria needs to be de¯ned, this is referred to as reliability.
There are several methods that can be used to evaluate reliability, such as Monte Carlo simulations (MCS), the¯rst-order reliability method (FORM), and the second-order reliability method (SORM). Details of these methods can be seen in Refs. 2 and 3. In this work the reliability of a 2D elastostatic boundary element method (BEM) model is evaluated for various levels of uncertainty in design parameters such as the applied stress and the dimensions of the model.
The BEM is a numerical computational method for structural analysis that has found increasing use in the engineering industry over the past few decades since its introduction in the late 1960s. It has proven itself to be an e®ective tool for structural reliability analysis, with previous applications to 2D elastostatic structures, 1,4 fatigue crack growth, 5,6 and plate bending. 7 It has developed as an e®ective alternative to the¯nite element method (FEM) and has several features that make it an attractive choice. One feature is that the formulations used for the BEM result in a signi¯cantly smaller number of equations being obtained than with the FEM, the BEM is also able to obtain a similar level of accuracy to the FEM while using a much coarser mesh. 8 These advantages allow the BEM to be especially useful for structural reliability analysis. A comprehensive overview of the application of the BEM to structural analysis can be found in Ref. 8.
There have been several publications relating to structural reliability analysis with the BEM. One notable example is Huang and Aliabadi 1 where MCS and FORM were used to assess the reliability of a 2D elastostatics problem involving a rectangular plate with a center circular hole subjected to uniaxial tension. The implicit di®erentiation method (IDM) was employed to calculate the¯rst-order derivatives for FORM and reliability was evaluated for various levels of uncertainty in material strength and center hole radius. Results showed that MCS and FORM obtained similar results. Huang and Aliabadi 6 also addressed a more complex problem involving the reliability analysis of 2D cracked structures using the dual boundary element method (DBEM). 8 The sensitivity of the stress intensity factors with respect to several random variables, including crack length and fracture toughness, was calculated using the IDM and compared to those obtained from the¯nite di®erence method (FDM) with both the DBEM and the FEM and also with MCS. Results showed good agreement between these methods. Lionel and Venturini 5 built upon this work and assessed the reliability of a problem involving the modeling of crack growth in a 2D perforated panel using MCS and FORM. The failure condition of the structure was based on the number of cycles required for the structure to fail calculated from fatigue crack growth analyses performed using the DBEM. Reliability analyses were carried out using two methods. The¯rst, the direct method, involved the direct coupling of FORM with the DBEM model and approximating the derivatives for FORM using the FDM. The second, the indirect method, involved approximating the response of the DBEM model using a polynomial response surface. Results showed that the indirect coupling proved to be less accurate and more computationally expensive than the direct method.
Several recent publications 4,7 have involved the use of a stochastic spline¯ctitious boundary element method (SFBEM). The SFBEM is a modi¯ed indirect BEM and its use makes it unnecessary to create an explicit expression of the limit state function (LSF), improving the e±ciency of the reliability analyses. One example of its use is Su et al. 4 where it was used in conjunction with the advanced¯rst-order second moment (AFOSM) method, a derivative of FORM, to conduct reliability analyses of 2D elastostatic structures. Results showed that the AFOSM with the SFBEM can obtain good agreement with MCS. Su and Xu 7 later applied the SFBEM to a more complex problem involving the structural reliability analysis of Reissner plate bending problems using the AFOSM. The proposed method was validated through comparison with a stochastic FEM method and MCS. Results showed close agreement between the three methods, with the SFBEM proving to be the most e±cient, achieving computation times 80% lower than the stochastic FEM method. As well as the BEM, reliability analyses have also been conducted using the FEM 9,10 and experimental data. 11 Although there have been several publications regarding the use of the BEM for structural reliability analysis, 1,4-7 there have been none involving the use of multidelity modeling with the BEM, whether for structural reliability analysis or otherwise. The application of multi-¯delity modeling to the topic of structural reliability analysis has also not been investigated thoroughly by the research community, although there have been several instances where surrogate models have been employed successfully. 5,10,12 The development of reliability analysis formulations incorporating the use of multi-¯delity modeling with the BEM, with its many attractions, is therefore an area of great interest and has the potential to provide signi¯cant improvements in e±ciency to the employed reliability analysis procedures, especially since assessing the reliability of a structure can be very computationally expensive, particularly for complex problems. With regards to MCS, for relatively simple problems with only two random variables the number of simulations required could be 5 Â 10 5 (Ref. 1), however, for more complex problems, for example with 10 random variables, it could be much higher at 1 Â 10 8 . 9 By substituting a multi-¯delity model (MFM) for the high-¯delity model (HFM) used with MCS, similar accuracy to the HFM could be achieved at a fraction of the cost. Similar improvements could also be seen with FORM and SORM.
In the context of this work, multi-¯delity modeling, related to surrogate modeling or metamodeling, refers to the process of approximating the response of a HFM over a particular domain. The approximation is created by evaluating the response of the HFM and a low-¯delity model (LFM) at several points, called design points, within this domain. Types of metamodels that have been employed in the past include polynomials, 9,10,13,14 Kriging, [15][16][17] and radial basis neural networks (ANNs). 17 The aim is to provide a MFM that is of similar accuracy to the HFM over the required domain, but at much less computational cost. A recent review of multi-¯delity modeling was undertaken by Simpson et al. 18 There have been many publications on the topic of multi-¯delity modeling due to its many potential bene¯ts. One notable example is Vitali et al. 14 where a multidelity approach was taken toward the calculation of the stress intensity factor in the 0 plies in a composite sti®ened panel with a center crack. Coarse and¯ne FEM meshes were used as the LFM and HFM, respectively and the ratio and the di®erence between the stress intensity factors calculated for these models were used to create linear correction response surfaces with the ply thicknesses as independent variables. Results showed that the MFMs were of similar accuracy to the HFM. Sun et al. 13 used a similar approach for the optimization of a sheet-metal forming process. FEM models were used for the LFM and the HFM with the ratio and di®erence of the responses between these models being approximated as second-order polynomial correction response surfaces. To avoid calling the LFM for each multi-¯delity approximation, the response of the LFM was approximated using moving least square modeling, reducing computation time. The most e®ective MFM obtained a low absolute percentage error of 1.06% with respect to the HFM.
The main objective of this work concerns the development of reliability analysis formulations incorporating the use of multi-¯delity modeling with the BEM. In this work, the computational cost of performing reliability analyses on 2D elastostatic structures using the BEM through the application of multi-¯delity modeling is investigated. The aim is to use MFMs to obtain similar accuracy to the HFM but at much less computational cost, greatly improving the e±ciency of the reliability analysis procedures using MCS, FORM, and SORM. A second objective involves quantifying this improvement in e±ciency for each of these three methods, determining the extent to which each of these methods bene¯t from the application of multi-¯delity modeling.

BEM Models
The BEM models used in this work consist of a 2D rectangular plate with a center circular hole, similar to that employed by Huang et al. 1 The plate has a width W of 1 unit, a length L of 2 units, and a radius R of 0.25 units. The material properties are the Young's modulus, E ¼ 1 unit, and the Poisson's ratio, ¼ 0:3 units. The plate is subjected to a uniform far-¯eld remote tensile stress, T , of 1 unit along its top and bottom edges. The plate has roller constraints along the left and right edges and pinned supports at the middle of these edges. A diagram of the plate used in this work can be seen in Fig. 1.
It is necessary to de¯ne the failure condition of the plate in order to perform reliability analyses; this can be achieved through the use of a LSF. The LSF in this work is based on the maximum stress in the plate, which occurs as a tangential stress at the edge of the circular hole, and is marked as point A in Fig. 1. Let, Z denote a vector of random variables, Z ¼ ð c ; R; T ; ; L; W Þ ¼ ðZ 1 ; Z 2 ; Z 3 ; Z 4 ; Z 5 ; Z 6 Þ, and let X (X Z) denote the random variables that in°uence the stress at point A, X ¼ ðR; T ; ; L; W Þ. The LSF, gðZÞ, in this case is given by: where c is material strength and A is the tangential stress at point A. The structure fails when A > c (or gðZÞ < 0Þ. The random variables were given the probability distributions as seen in Table 1. They were assigned parameters based on their expected uncertainty, with c and R having the most uncertainty in their values. To test the performance of the various models with MCS, FORM, and SORM under varying levels of uncertainty, the coe±cient of variation (COV ¼ =, where and are the standard deviation and mean, respectively) of the center hole radius, COV R , was varied uniformly in the closed interval ½0; 0:1 ) fCOV R 2 R : 0 Creating the MFMs requires the de¯nition of a LFM and a HFM. The LFM is a coarse model of the plate as seen in Fig. 1, consisting of 20 quadratic elements, while the HFM is a¯ne model of 100 elements of the same type. The BEM models can be R A  seen in Fig. 2. The ratio of the number of elements on the outer boundary to the number of elements on the hole was kept as 3/2.

Multi-Fidelity Modeling with the BEM
In this work, response surface methodology (RSM) was used to create response surfaces that would be used to approximate the HFM using surrogate models. The ordinary least squares (OLS) method 19 is used to approximate the unknown coe±cients of the response surfaces. The response surfaces used in this work were second-order polynomial functions, with independent variables X, of the following form:ŷ whereŷðXÞ is the response to be approximated and f ðXÞ is a vector of p terms and is composed of the powers and cross-powers of the independent variables X: is a vector of p coe±cients: In this work, p ¼ 21, this includes six linear terms,¯ve squared terms, and 10 mixed terms.
Two types of MFMs were used to approximate the maximum stress in the HFM, A;HF ðXÞ. The direct type which involved directly calling the LFM, A;LF ðXÞ, each time a multi-¯delity approximation was required, and the indirect type which involved approximating the LFM as a polynomial response surface, res A;LF ðXÞ. In this work, obtaining these MFMs involved obtaining correction response surfaces res ðXÞ and res ðXÞ, where is the ratio between A;HF ðXÞ and A;LF ðXÞ and is the difference between A;HF ðXÞ and A;LF ðXÞ. and were used in the MFMs instead of derivative-based approximations due to their ability to be applied to larger areas within the design domain and their ability to smooth out noise. 14 and were calculated at several design points X i ði ¼ 1; 2; . . . ; NÞ to obtain res ðXÞ and res ðXÞ. The design points were obtained from a Latin hypercube design 6,13,14 of 30 design points (N ¼ 30), requiring a computation time of 18.3 s. In this work, the Latin hypercube design was modi¯ed to improve its application to reliability analysis; the design points were obtained by uniformly sampling from each random variable's cumulative distribution function (CDF) between the probabilities 2.5% and 97.5%. This ensured that the design points were highly concentrated in the highly-likely areas of each variable's domain, and less concentrated in the less-likely areas. The response surfaces were created with the parameters in Table 1 with COV R ¼ 0:05 to provide good accuracy over the entire range of COV R values investigated. A¯fth surrogate model for the HFM was obtained by approximating A;HF ðXÞ as a response surface, res A;HF ðXÞ. The details of the seven models are shown in Table 2.

Model performance evaluation
To test the robustness of these response surfaces, they were run with the design points from a second Latin hyper cube design, the \test" dataset. This dataset consisted of 100 design points sampled from each variable's CDF between a wider range of probabilities, 0.5% to 99.5%. The \design" dataset consisted of the 30 design points used to create each response surface. Various statistical parameters evaluated for the response surfaces res ðXÞ, res ðXÞ, res A;LF ðXÞ, and res A;HF ðXÞ with the design and test datasets can be seen in Table 3. These were the coe±cient of determination R 2 , the adjusted coe±cient of determination R 2 adj , the standard error (SE), and the mean absolute percentage error (MAPE).
The data in Table 3 suggests that the created response surfaces are able to provide highly accurate approximations for the responses of the LFM and the HFM over the required domain. res ðXÞ proved to be more accurate than res ðXÞ, this suggests that MFMs using res ðXÞ could prove to be more accurate.
To determine the performance of these surrogate models they were compared to the performance of several BEM models of a range of¯delities. A was calculated at the mean values of the parameters seen in Table 1 for each of the models. The average computation time for each was calculated over 1000 runs. The results are shown in Table 4.
It can be seen from Table 4 that the indirect MFMs utilizing res A;LF have similar accuracy to res A;HF . The most accurate MFMs involved directly calling the LFM, and of those the most accurate approximations were those that utilized res , while those  utilizing res proved to be less accurate. This agrees well with observations made of the data presented previously in Table 3. The best approximation was given by the multi-¯delity model MF1, which achieved the lowest MAPE of 6:38 Â 10 À4 %. This model is 4.81 times more accurate than the 80-element mesh, A;E80 ; while also having a CPU wall-time 2.65 times less. res A;HF proved to be the least accurate approximation, achieving similar accuracy to the 60-element mesh, A;E60 , but with a computation time about 1032 times less.
Results suggest that MF1 and MF2 both act as good substitutes for the HFM, achieving very similar levels of accuracy and with computation times similar to that of the LFM. On the other hand, the indirect models, MF3 and MF4, as well as HFM res proved to be less accurate, but showed much lower computation times.

Multi-Fidelity Modeling-Based Structural Reliability Analysis
In this section, multi-¯delity modeling is used in combination with reliability analyses performed using MCS, FORM, and SORM. The models shown in Table 2 are investigated, this includes the two BEM models, the LFM and the HFM, the four MFMs (MF1-MF4), and the response surface approximation of the high-¯delity BEM model, HFM res .
The reliability, P R , of a structure can be determined by evaluating the following integral: where f Z ðZÞ is the joint probability density function of Z, and P F is the probability of failure. P F and P R involve integrating f Z ðZÞ over the regions de¯ned by gðZÞ < 0 (the failure region) and gðZÞ > 0 (the safe region), respectively. All the random variables are assumed to be mutually independent. The direct evaluation of the above integral is usually very di±cult since it can be multidimensional if many random variables are involved. The integration boundary gðZÞ ¼ 0 can also be multidimensional and is usually a nonlinear function. The LSF, gðZÞ, may also be a black-box model (in this case a BEM model) and so it may be computationally expensive to evaluate. There are several methods that can be used to evaluate this integral; MCS, which involves sampling from each random variable's distribution, and FORM and SORM, which attempt to simplify the integration.

Monte Carlo simulation (MCS)
MCS involves randomly sampling from known probability distributions to determine an unknown probability distribution. 3 If Z ¼ ðZ 1 ; Z 2 ; . . . ; Z n Þ is a vector of known random variables, and if Y ¼ hðZÞ, where Y is a random variable with an unknown probability distribution and h is some process or function, then we can randomly sample the variables in Z per their probability distributions, input them into h and obtain a value for Y . This is repeated many times until a histogram can be created of Y , allowing us to estimate its probability distribution.
MCS can be used to evaluate the LSFs in Table 2 to obtain the reliability of the structure with the random variables Z ¼ ðZ 1 ; Z 2 ; . . . ; Z 6 Þ ¼ ð c ; R; T ; ; L; W Þ described in Table 1. For each simulation, the variables in Z are randomly sampled from their probability distributions, with the samples of the last¯ve variables ðR; T ; ; L; W Þ being used as inputs to a BEM model to evaluate A . If gðZÞ < 0 ð A ðR; T ; ; L; W Þ > c Þ for a simulation, then the structure is assumed to have failed. After a certain number of simulations have been carried out, N MCS , the total number of simulations in which the structure failed, N F , can be determined and the failure probability P F can be calculated as: Reliability can be calculated as: Because MCS is a simple brute-force sampling method, it can be used as a benchmark to validate the results obtained from the LSF approximation methods, FORM and SORM.

First-order reliability method (FORM)
FORM used in this work refers to the AFOSM method for nonlinear LSFs. AFOSM is an improvement over previous FORM methods such as the mean value¯rst-order second-moment (MVFOSM) method and the Hasofer-Lind (HL) AFOSM method in that it can be used with better accuracy with non-normal random variables and nonlinear LSFs. 2 There are two steps that FORM takes to make the integration in Eq. (5) more manageable. The¯rst step involves simplifying f Z ðZÞ such that its contours are more regular and symmetric; this is achieved by transforming the random variables from Z-space into U-space, the standard normal space. The second step involves approximating the integration boundary gðUÞ ¼ 0 as a¯rst-order Taylor expansion: where U is created by transforming Z into U-space, U Ã is the expansion point, and rgðU Ã Þ is: To minimize the accuracy lost by this approximation, it is necessary to expand gðUÞ at the point U Ã that contributes the most to the integrations as seen in Eq. (5) and so it will be the point that corresponds to the highest probability density. This point is termed as the most probable point (MPP) 3 and is the point along the integration boundary gðUÞ ¼ 0 that is closest to the origin of U-space. The distance between the MPP and the origin of U-space is termed the reliability index . is related to the probability of failure P F and reliability P R in the following manner: where È denotes the standard normal CDF. In this work, a Newton-Raphson type recursive algorithm developed by Rackwitz and Fiessler 20 is used to determine the MPP.

Second-order reliability method (SORM)
The SORM involves approximating gðUÞ ¼ 0 as a second-order Taylor expansion: where HðU Ã Þ is the Hessian matrix evaluated at U Ã and contains the second-order derivatives of the LSF with respect to the random variables in U-space. Since SORM is of a higher-order than FORM, it is expected that it will more accurately approximate nonlinear LSFs, such as those involved in this work. The SORM used in this work is Breitung's asymptotic approximation 21 : where FORM is the converged value of the reliability index calculated from FORM and i (i ¼ 1; . . . ; n À 1Þ are the principal curvatures of the LSF at the converged MPP location U Ã from FORM. This approximation is accurate only for large values of , which is the case for practical high-reliability problems. 2,22

Numerical example
FORM and SORM require the derivatives of the LSF under investigation, which can be directly calculated for the response surfaces res ; res ; res A;LF , and res A;HF . The calculation of the derivatives of A;LF and A;HF is more complex as they require the di®erentiation of the response of the BEM models. Methods used in the past include the IDM 1,23 and the FDM. 5 In this work, the FDM is used with the¯rst-order forward scheme and second-order central di®erence scheme. The step size in U-space, ÁU, was set as 1 Â 10 À5 for FORM and 1 Â 10 À3 for SORM as these were found to provide accurate results. The¯rst-order derivatives were evaluated in the following manner (for i 6 ¼ 1): For i ¼ 1: where z i is the standard deviation of the i'th random variable and ÁZ i ¼ z i ÁU. Z Ài is a vector that includes all the random variables in Z that in°uence A but excludes Z i . The second-order derivatives were calculated as (for i; j 6 ¼ 1 and i ¼ j): For i; j 6 ¼ 1 and i 6 ¼ j: where Z Ài;Àj is a vector that includes all the random variables in Z that in°uence A but excludes Z i and Z j . For i ¼ 1 or j ¼ 1: The reliability of each of the seven models was evaluated using FORM and SORM at 101 uniformly-spaced values of COV R in the closed interval ½0; 0:1. For MCS, 11 uniformly-spaced values were investigated. A total of 2:2 Â 10 9 MCS simulations were carried out for each of the seven models, 2 Â 10 8 for each of the 11 reliability indices corresponding to the 11 values of COV R . Figure 3 shows the reliability indices obtained from MCS, FORM, and SORM for the LFM and the HFM. It can be seen from this¯gure that SORM follows MCS much more closely than FORM, suggesting that SORM provides a much more accurate approximation for the failure domain than FORM. This is because SORM approximates each of these nonlinear LSFs in Table 2 using a second-order Taylor expansion, taking into account the curvature of the LSFs, therefore providing a more accurate evaluation of the failure domain. The consequence of this is that SORM is more computationally expensive than FORM, as can be seen in Fig. 4. For a low level of uncertainty (COV R ¼ 0), the MAPE between MCS and FORM for the HFM is 9:80 Â 10 À4 % while it is smaller at 2:13 Â 10 À4 % for SORM. In Fig. 4 and Given the data shown in Table 4 it was expected that the direct MFMs (MF1 and MF2) would prove to be the most accurate surrogate models for reliability analysis. This seems to be the case when looking at Figs. 5-8 where they are shown to be signi¯cantly superior in accuracy to the LFM while also being of similar computational cost, suggesting that they could act as very good substitutes for the HFM. The indirect MFMs (MF3 and MF4) and the response surface approximation of the HFM, HFM res , by comparison were shown to be less accurate than the direct models but signi¯cantly faster in terms of computation time, achieving wall-times that were around 10-20 times less than the direct MFMs for MCS, with HFM res proving to be the fastest at 70 times less. Despite this, the accuracy of the indirect MFMs is similar to that of the direct MFMs for low levels of uncertainty (COV R < 0:05), but worsens after this. This is due to the fact that the response surfaces were designed at COV R ¼ 0:05 in order to minimize the average error over the range of COV R values investigated. A method to mitigate this would be to design a set of response surfaces for each COV R under investigation. The most accurate MFM proved to be MF1, which provided the greatest accuracy improvement over the LFM for MCS, FORM, and SORM. This was expected from the results shown previously in Table 4. Although it provides very high accuracy over the entire range of COV R values investigated, achieving a very small MAPE for MCS with respect to the HFM of 2:7 Â 10 À3 % for COV R ¼ 0:10, it is most e®ective when used for low levels of uncertainty. MF1 provided very similar computation times to the LFM, being only 7.95% slower for MCS and 9.27% for FORM. MF1 is signi¯cantly slower than the LFM for SORM, providing computation times that are 295.47% greater, this is because the second derivative of its LSF contains six terms and requires three calls to the LFM in the case where i ¼ j and in the case i 6 ¼ j 8 terms need to be evaluated and six calls to the LFM are required. In contrast, the second derivative of the LFM's LSF only contains one term that requires two calls for i ¼ j or four calls for i 6 ¼ j. From Fig. 5, we can see that all the MFMs perform similarly well with FORM. This can also be seen from the sensitivities in Table 6 which are used by FORM. These sensitivities quantify the in°uence that a change in each of the random variables in X has on A . It is also worth mentioning that the sensitivities are in-line with what is expected, with R and T having the largest in°uence on A , and and L having the least amount of in°uence. The MFMs with SORM in Fig. 6, by comparison, perform very di®erently, with the indirect models proving to be less accurate than the direct models, especially at high levels of uncertainty (COV R > 0:05). This can be explained by the fact that the response surfaces used in this work are second-order polynomials, their second-order derivatives are therefore constant. The e®ect of this is maximized in the case of the indirect models since they are purely constructed of response surfaces. A means of mitigating this would be to use higherorder polynomials, possibly third or fourth order, at the expense of increased computation time. From Fig. 8, we can see that the reliability analysis method that showed the greatest improvement in accuracy over the LFM when used in conjunction with the MFMs was FORM, which experienced a 1011 times reduction in 3.08 3.08 3.08 3.08 3.08 3.08 À8.19 Â 10 À1 À7.86 Â 10 À1 À8.08 Â 10 À1 À7.64 Â 10 À1 À7.62 Â 10 À1 À7.74 Â 10 À1 À7.85 Â 10 À1 L À1.93 Â 10 À3 À1.42 Â 10 À3 À1.76 Â 10 À3 À3.77 Â 10 À3 À3.75 Â 10 À3 À1.01 Â 10 À3 À1.37 Â 10 À3 W À1.46 À1.42 À1.45 À1.44 À1.44 À1.43 À1.42 error when used with MF1. SORM and MCS experienced an improvement that was less pronounced but still signi¯cant, with MF1 providing an average MAPE that was 192 and 268 times less for SORM and MCS respectively than the LFM.

Results and discussion
In conclusion, the MFMs were found to provide superior accuracy to the LFM for each of the reliability analysis methods investigated with computation times very similar to those of the LFM, or in the case of the indirect models signi¯cantly less. The direct MFMs provided reliability indices that were of the order of 50-100 times more accurate than the indirect MFMs and the response surface approximation of the high-¯delity model, HFM res . The best overall performance was achieved by MF1 which involved coupling the LFM with the ratio response surface res ðXÞ. Results suggest that MF1 could act as a very good substitute for the HFM when used for reliability analysis with MCS, FORM, and SORM.

Conclusions
In this work, a method for the application of multi-¯delity modeling to the reliability analysis of 2D elastostatic structures using the BEM is proposed. The reliability of several MFMs created using second-order polynomial response surfaces was evaluated alongside that of a LFM and a HFM using MCS, FORM, and SORM. Results show that, MF1, the MFM that directly coupled the LFM with a ratio correction response surface proved to be the most accurate, providing errors relative to the HFM that were on average 1011, 192, and 268 times smaller than the LFM for FORM, SORM, and MCS, respectively. It was also of similar computational cost, being only 7.95% slower in the case of MCS. These results suggest that the multidelity models involving directly calling the LFM could act as good substitutes for the HFM for reliability analysis using MCS, FORM, and SORM.
Future work will seek to apply the proposed methodology to more complex structures, such as anisotropic plates and shells. The IDM will also be investigated.