Multimodal ellipsoid model for non-probabilistic structural uncertainty quantification and propagation

The traditional ellipsoid convex set is a kind of basic non-probabilistic model to measure uncertainties. However, it is difficult or inaccurate to quantify the uncertainties of variables with multimodal distributed samples. In this paper, a more generalized non-probabilistic ellipsoid model named multimodal ellipsoid model is proposed to effectively deal with the multimodal distributed samples. The samples with one or more similar properties are clustered together, and the principal directions of the samples and characteristic matrix are appropriately found through the Gaussian mixture model. Then, the multimodal ellipsoid model can be constructed by using the elliptical contour features of the Gaussian model to measure the uncertainties of variables. The proposed multimodal ellipsoid model can not only establish traditional ellipsoid model, but also establish multi-ellipsoid model for uncertain variables with multimodal samples. Furthermore, combining with the multimodal ellipsoid model and performance measure approach, the uncertain propagation results of system are obtained accurately. Three numerical examples and one engineering application are provided to demonstrate the effectiveness and accuracy of the proposed multimodal ellipsoid model.

the probabilistic model is the most mature and frequently used measurement tool for structural uncertainties (Liu et al. 2015;Liu et al. 2018a;Du et al. 2000;Meng et al. 2017a;Wu et al. 2019). However, for some complex engineering problems, it is quite difficult to obtain sufficient samples and obtain accurate probability density function (PDF) of uncertain variables since the limitations of experimental conditions or cost. For this reason, many kinds of nonprobabilistic methods have been developed to measure uncertainties, which include evidence theory (Cao et al. 2018Li et al. 2016;Liu et al. 2020), probability-box (Ferson et al. 2002;Xiao et al. 2016;Chen et al. 2016), information theory (Klir. 2004), fuzzy sets (Dubois et al. 1989;Shi et al. 2018;Simoen et al. 2015), and convex model (Ben-Haim. 1994;Jiang et al. 2011;Zeng et al. 2018;Elishakoff et al. 2016;Truong et al. 2017).
As an effective supplement to the probabilistic model, Ref. (Ben-Haim. 1994) proposed a non-probabilistic convex model under the situation of limited samples, which has been gradually applied in academic and engineering fields. Generally, the nonprobabilistic convex model is established based on the assumption that the uncertain parameters fall into a convex set. Interval model is an early-developed nonprobabilistic convex model, which can be treated as a hypercube for multi-dimensional uncertain parameters (Du et al. 2005;Degrauwe et al. 2010). Obviously, the required information to construct such a hypercube model is only the upper and lower bounds of each uncertain variable, which is relatively easy to obtain in practical engineering. In the past two decades, a series of significant researches on the interval model (Moore. 1979;Jiang et al. 2007a;Pouresmaeeli et al. 2018) has been carried out. Ref. (Qiu et al. 2005a) proposed a parameter vertex solution algorithm to compute the lower and upper bounds of the structural eigenvalues due to the uncertain-but-bounded variables. Ref. (Kang et al. 2016) proposed a semi-definite programming (SDP) formulation method for construction of ellipsoid model. Ref. (Li et al. 2018) used the nonprobabilistic interval model to describe uncertain parameters in model validation. Ref. (Jiang et al. 2007b(Jiang et al. , 2008 suggested a series of uncertain optimization methods to solve the interval number programming problem with uncertain variables in nonlinear objective functions and constraints. Ref. (Liu et al. 2018b) applied the interval model to the uncertain inverse problem to identify the dynamic loads acting on uncertain structures. Ref. (Meng et al. 2020a, b) proposed an experimental data-driven exponential convex model for reliability assessment.
With the deepening and development of research, the correlation problem in uncertain variables gradually attracts many researchers' attentions. The interval model is limited to represent the correlations of uncertain parameters, because it can only provide the upper and lower bounds. In order to solve this problem, the ellipsoid model (Ben-Haim. 1994) was proposed and adopted in engineering as its ability to deal with variable correlation. Similar to the interval model, only the ellipsoid bound is required to construct an ellipsoid model, which can surround all the limited samples of uncertain variables, instead of their precise probability distribution. Further, the fluctuations of uncertain variables are assumed to fall into a single ellipsoidal domain. Many researches on how to construct the most suitable ellipsoidal convex model have been published. With the help of hypercube model, Ref. (Qiu et al. 2005a) proposed a modeling method to construct hyper ellipsoid with the minimum volume. And yet, as the axes of hypercube model are parallel to the axes of original coordinate system, the constructed hyper ellipsoid model for uncertain variables with different correlations possibly have a same spindle direction. Ref. (Zhu et al. 1996) applied the experimental samples to establish the ellipsoid model using the minimum volume, but the process was complicated. One most appropriate interval model was constructed, and then the minimal circumscribed ellipsoid was treated as an uncertainty measurement model. Ref. (Kumar et al. 2005) proposed an ellipsoid modeling method using experimental samples without the construction of hypercube. According to the different properties of variables, another ellipsoid convex model was proposed by Ref. ). In addition, Ref. (Jiang et al. 2011) proposed a novel correlation analysis technique to construct multidimensional ellipsoid for uncertain structures. Ref. (Meng et al. 2018) created a mathematical foundation for non-probabilistic super parametric convex model to perform the reliability analysis and reliability-based design optimization. With the development of convex set theory, some related fields such as non-probabilistic uncertainty propagation (Qiu et al. 2003;Liu et al. 2018c) and uncertain topology optimization Luo et al. 2009) have been gradually investigated. Based on the convex model, an uncertainty propagation algorithm was developed to compute the structural dynamic response bounds . Using a quantified measure for non-probabilistic reliability based on the ellipsoid convex model, a topology optimization of continuum structures ) was investigated considering uncertainbut-bounded variables. Ref. (Meng et al. 2016) presented a decoupled strategy for non-probabilistic reliability-based design optimization to the efficiency. Ref. (Meng et al. 2017b) proposed a new directional stability transformation method for first order reliability analysis. Ref. (Wang et al. 2019) developed a non-probabilistic time-variant reliability-based optimization strategy for closed-loop controller design of vibration reduction issues.
Although the uncertainty modeling approaches based on the non-probabilistic convex set theory have achieved considerable developments, most of the above-mentioned ellipsoid models only use single convex model to surround the limited samples of uncertain variables as well as a lack of consideration regarding on the multimodal status of samples. Actually, the single ellipsoid model is only suitable for the situation that there is one simple peak of the distribution. However, in many instances, although the samples for establishing an ellipsoid convex model would be pretty limited, their distributions may have multiple distinct separate or intersecting multimodal. The multimodal situation is similar to the multi-peak situation of PDF in the probability theory Meng et al. 2020a, b). Obviously, the traditional single ellipsoid model is not suitable for practically descript the multimodal of samples. More importantly, in such case, if the single ellipsoid model is still adapted to surround the samples with multimodal situation and quantify the uncertainty, the established ellipsoid domain will be relatively large and the corresponding uncertainty analysis or design results would be conservative. In order to establish a more reasonable ellipsoid model for the uncertain variables with single or multimodal, a novel multimodal ellipsoid model (MEM) is proposed in this study through the multimodal analysis based on Gaussian mixture model (GMM). For the non-probabilistic convex model to measure the variables' uncertainties with limited samples, the determination of the appropriate bounder of non-probabilistic model to surround the samples is primary. Although the GMM is commonly applied to model the probabilistic uncertainties, here even if the samples are limited, it still can be adopted to find the principal directions of the samples and it is beneficial to obtain the characteristic matrix of the non-probabilistic ellipsoid model for appropriately surrounding the samples. Meanwhile, the proposed MEM is a more generalized non-probabilistic ellipsoid model for uncertainty measurement, which can not only realize the traditional single ellipsoid modeling, but also can effectively establish a multi-ellipsoid for the uncertain variables with multimodal samples.
In this paper, according to the inherent characteristics of uncertain variables' samples, the single or multimodal sample with similar property are gathered through Gaussian mixture model. For each cluster, the corresponding MEM can be established by using the compact elliptical contour feature of the Gaussian mixture model. Furthermore, combining the proposed MEM and performance measure approach (PMA) (Tu et al. 1999), an efficient and accurate uncertainty propagation method is proposed. The rest of this paper is organized as follows. In Sect. 2, two kinds of existing ellipsoid modeling methods are simply introduced. In Sect. 3, the proposed MEM is established based on the GMM and the ellipsoidal contour feature of the Gaussian mixture model. In Sect. 4, an efficient uncertainty propagation method based on the MEM and PMA is proposed. Section 5 presents three numerical examples and one engineering application to demonstrate the effectiveness of the proposed MEM and corresponding uncertainty propagation method. Some conclusions are summarized in Sect. 6.

Traditional ellipsoid model
Ellipsoid model is one of the most widely used nonprobabilistic convex models for uncertainty measurement, in which the variations of uncertain variables are assumed to fall into an ellipsoidal convex set. During the modeling process of ellipsoid, the minimum volume of model is commonly adopted for most convex models. At present, the minimum volume optimization method (Zhu et al. 1996) and the correlation approximation method (Jiang et al. 2011) are two kinds of frequently-used ellipsoid modeling methods. Generally speaking, the minimum volume optimization method is used to search an ellipsoid with the minimum volume which can surround all the samples of uncertain variables, and the correlation approximation method is used to search a minimum volume ellipsoid for a given interval of uncertain variables.

Minimum volume optimization method
For n-dimensional uncertain variables X ¼ ½X 1 ; X 2 ; . . .; X n T , assume that there exists m experimental samples X r ð Þ ; r ¼ 1; 2; Á Á Á; m in the ndimensional space. In order to measure the uncertainties, an ellipsoid model surrounding all the samples can be established as (Zhu et al. 1996), where E X denotes the ellipsoid domain of the uncertain variables X. The superscripts C and T represent the midpoint of ellipsoid domain and the transpose of a matrix, respectively. X is a characteristic matrix of the n-dimensional ellipsoid model, which is a symmetric positive definite matrix as Here, Eq.
(2) can determine the size and shape of the n-dimensional ellipsoid. As shown in Eq. (1), once the midpoint of ellipsoid domain X C and the characteristic matrix X are determined, then the ellipsoid model that can be obtained to measure the uncertainties of variables. Actually, it seems that there exist plenty of ellipsoids can be used to surround these m samples. However, the ellipsoid with a minimum volume is considered as the most appropriate one. Theoretically, such an ellipsoid model can be obtained by solving the following optimization problem, where k i denotes the half-length of the ith axis of the ellipsoid. It can be obtained through the eigenvalue decomposition of the characteristic matrix X, and the volume of the ellipsoid can be proportional to the product of all the k i . The optimization variables of the above problem are all the elements g ij in the characteristic matrix X and the midpoint X C i . Unfortunately, if the dimension of uncertain variables and sample numbers are large enough, it seems hard to establish a minimum volume model (MVM) for uncertain variables.

Correlation approximation method
The basic idea of the correlation approximation method (CAM) (Jiang et al. 2011) is to establish a minimum volume ellipsoid to surround the samples for a given hypercube of uncertain variables. Different with the minimum volume optimization method, the CAM is defined as searching an ellipsoid in each twodimensional sub-space. This ellipsoid should have two characteristics. (1) Ellipsoid and uncertain variables should be tangent. (2) Ellipsoid has a minimum volume in the premise of first characteristic. These two characteristics can be guaranteed through the correlation model reported in Ref. (Jiang et al. 2011). Then, all the ellipsoids, which are established in twodimensional sub-space, are assembled together in ndimension space.
For the convenience of expression, firstly the uncertainty modeling problem with two-dimensional variables is discussed. Assume that all the possible values of X 1 and X 2 located in the marginal interval where the superscripts I, L and U represent the interval, lower bound and upper bound, respectively. And the midpoint X C i and radius X w i of the interval are defined as, where D X i À Á is the defined variance which represents the deviation degree of uncertain parameter from midpoint. Further, the covariance Cov X 1 ; X 2 ð Þ and non-probabilistic correlation coefficient q X 1 ; X 2 ð Þ of uncertain variables are defined as, where h À45 o h 45 o ð Þis the rotation angle of the ellipse in the normal coordinate system. Therefore, the characteristic matrix X of the ellipsoid model can be obtained through the inverse operation of the covariance matrix R of uncertain variables, For an n-dimensional uncertain problem, the multidimensional ellipsoid can be established by assembling any two-dimensional ellipsoid model as the above-mentioned. For more details on the correlated ellipsoid model, one can refer to (Jiang et al. 2011).
In Eq. (6), the non-probabilistic correlation coefficient q X 1 ; X 2 ð Þ or rotation angle h needs to be determined to establish the ellipsoid model. Similar to Eq. (3), minimizing the area of the inscribed ellipse in the region of the two-dimensional marginal intervals can be applied to obtain a suitable rotation angle and the corresponding correlation coefficient. Fortunately, this optimization process avoids the shortcomings of the minimum volume optimization method as the above-mentioned. Through the CAM, the high dimensional optimization problem is transformed into a series of two-dimensional optimization problems, which greatly simplifies the optimization process. Meanwhile, in the two-dimensional space, the optimization variable is only the correlation coefficient or rotation angle, and this is beneficial to ensure the positive definite property of ellipsoidal characteristic matrix during each optimization iteration. However, there are also some disadvantages in the CAM. Since the central point of ellipsoid model is previously specified as the midpoint of interval model, the established ellipsoid model is probably not the one with the minimum volume. Additionally, a few number of samples may be outside the established ellipsoid model (Jiang et al. 2011).

The proposed multimodal ellipsoid model
As the above-mentioned, the two traditional ellipsoid modeling methods have the ability to achieve the uncertainty measurement of uncertain variables, but they do not fully consider the distribution and clustering status of the samples. In general, the number of samples to establish the non-probabilistic uncertainty measurement model is limited, but for many cases, the sample distribution still may display an obvious multimodal phenomenon, which is quite similar to the special PDF with multi-peak feature in the probability theory. Obvious, for such cases, these two above-mentioned ellipsoid modeling methods are not entirely suitable, as they can only establish a single ellipsoid model to surround the samples of uncertain variables. In order to effectively deal with various sample distributions, a more generalized ellipsoid model, namely MEM, based on cluster and GMM will be proposed in this section. The samples of uncertain variables are analyzed by using the GMM and then a single or multimodal samples can be obtained according to the distribution and clustering status of samples. Then, an appropriate elliptical contour lines are derived from the GMM to form the ellipsoid for surrounding all the samples.

Multimodal analysis for the samples of uncertain variables
Multimodal analysis belongs to the category of unsupervised machine learning method (Hofmann et al. 2001), which is developed to determine the multimodal of the samples (Fraley et al. 1998). It has been widely applied for data analysis in various fields. Usually, the pattern feature of samples is constructed to analyze the similarity between samples in multimodal analysis. The traditional pattern feature is a function with respect to distance, such as Block distance, Euclidean distance, Minkowski distance, which means each sample is assigned to a different cluster according to the distance between samples and the given center point of each cluster (Kanungo et al. 2002). Another important pattern feature can be represented by a mathematical model about probability distribution model. Its purpose is to achieve the optimal matching between the samples and probability distribution model through fitting method (Celeux et al. 1995;Mcnicholas et al. 2008). In this section, the Gaussian clustering method will be introduced to realize the determinations of the principal directions of the samples and the characteristic matrix of the nonprobabilistic multi-ellipsoid model.

Gaussian clustering
Gaussian clustering model is also called as GMM, which is a powerful framework for clustering analysis (Deng et al. 2013). Through observation, we find that the characteristic matrix of the ellipsoid model in Eq. (6) has the same formula with exponential term of the GMM, so it is promising to introduce it for the ellipsoid modeling. Even if the samples are limited, when the samples are shown the obvious multimodal, the advantages of the GMM still can be utilized to reasonably obtain the principal directions of the samples and to properly construct a non-probabilistic single or multiple ellipsoid model. When the samples are limited and the distribution of samples appears to have one or more multimodal, one or more Gaussian functions (Fraley et al. 1998;Er. 1997Er. , 1998 can be adopted to descript their distribution as follows Þis the Gaussian function for uncertain variables X with parameters o, K is a cluster number, a k (0\a k \1 and P K k¼1 a k ¼ 1) is the weighted value or mixture ratio of the kth cluster, and U X l k ; R k j ð Þ denotes the Gaussian function with mean vector l k and covariance matrix R k for the kth cluster. For the non-probabilistic convex model to measure the variables' uncertainties with limited samples, in fact, the distribution of the samples is unconcerned, and the focus is to determine the appropriate convex bounder to compactly surround the samples. Therefore, although the GMM is less accurate to descript their distribution under the limited samples, it still can be utilized to find the principal directions of the samples, especially for the situation that the samples exist multimodal. And the determined principal directions by GMM will form the foundation to establish the appropriate ellipsoid model.
For the GMM in Eq. (7), 3 K unknown parameters o can be determined by the maximum likelihood estimation (MLE) method. Assume that all the samples of uncertain variables X r ð Þ ; r ¼ 1; 2; Á Á Á ; m are independent, and then the probability of the event Then, in order to obtain the optimal values of parameters o, the following optimization problem based on the MLE is built, where L o ð Þ is the likelihood function of the event n X . As the number of parameters to be estimated may be large, an efficient expectation-maximization (EM) algorithm (Dempster et al. 1997) is adopted.

Expectation-maximization algorithm for the GMM
The EM algorithm mainly consists of two steps, i.e. E-step and M-step, which is a general approach to implement the MLE effectively. The key of the EM algorithm is Jensen's inequality (Kass. 1990), and the Lemma about Jensen's inequality is introduced as follow.
Lemma If function f(Á) is a convex function on its actual definition domain, it should satisfy the Jensen's inequality denotes the expectation function of independent variable x.
(1) E-step The aim of E-step is to construct the expectation of likelihood function L o ð Þ. The estimated conditional probability of uncertain variables X r ð Þ from the kth component can be expressed as, where b k X r ð Þ À Á denotes the conditional probability.â,R andl represent the estimated parameters. Then, combing with the Jensen's inequality and the concave character of likelihood function, we can obtain the following likelihood function formula, The essence of Eq. (11) is the expectation function of the conditional probability The aim of M-step is to calculate the lower bound of L o ð Þ, which is the maximum expectation. Based on the MLE, once the likelihood function L o ð Þ is equal to the maximum expectation in Eq. (11), the corresponding parameterŝ can be treated as the optimal values of GMM, Considering the necessary condition of the extreme value theorem and the constraint conditions P K k¼1 a k ¼ 1, make the partial derivatives of Eq. (12) with respect to the parameters a k , l k and R k equal to zero. Then, the optimal estimation of the parameters of the GMM can be obtained as follows, To sum up, the EM algorithm can refine partitions when started sufficiently close to the optimal value, which are the maximum likelihood estimation of the characteristic parameters of GMM on Eq. (7), by E-step of Eq. (11) and M-step of Eq. (13). The established GMM according to the samples of uncertain variables will be used to construct the proposed MEM in the next section.

Multimodal ellipsoid modeling
As each sub-item of the GMM in Eq. (7) indicates a sample cluster with the similar distribution feature for uncertain variables, the corresponding multi-ellipsoid model for uncertainty measurement can be derived from each Gaussian sub-model to surround the samples fallen in the same cluster. In this way, a single ellipsoid or multi-ellipsoid model can be reasonably constructed according to the determined principal directions and characteristic matrix by GMM.

Single-MEM
When all the samples of n-dimensional uncertain variables only display one cluster, the GMM can be simply expressed as, in Eq. (14), R is the covariance matrix and |R| denotes the determinant of R. Parameters l and R of Gaussian model can be appropriately determined by clustering the samples based on the above-mentioned EM algorithm. As the characteristic matrix of the ellipsoid model has the same formula with exponential term of the GMM, to treat the parameters l and R as the center point and characteristic matrix of ellipsoid, the bounds of a series of ellipsoids with respect to the uncertain variables can be derived through the contour surface of Gaussian model, where R 2 denotes the scale ratio of the contour ellipsoid. The optimalR 2 can be determined through the following steps.
Step 1 Give an initial value of R and set a step size DR.
Step 2 Generate the contour ellipsoid by using the current R.
Step 3 If all the samples are enveloped by the current contour ellipsoid as Eq. (15), the current R is considered as the optimal R andR 2 ¼ R 2 . Otherwise, R ¼ R þ DR and go back to Step 2.
Ensuring that all the samples can fall into the ellipsoid, and the corresponding single-MEM can be constructed as Taking two-dimensional problem as example, the established single-MEM for uncertainty measurement and the corresponding GMM for clustering samples are shown in Fig. 1. It is indicated that the samples are compactly surrounded by the constructed MEM. More importantly, the correlation of the uncertain variables can also be properly considered through the principal directions of samples obtained by GMM. Meanwhile, the proposed MEM avoids the disadvantages of the minimum volume method, such as large computational efforts. And for the multi-dimensional problems, the corresponding MEM can also be conveniently constructed without decomposing it into multiple two-dimensional models as the CAM (Jiang et al. 2011).

Multi-MEM
The main disadvantage of the single ellipsoid model is that the intrinsic distribution status of samples is ignored, and all the samples are simply surrounded by a single ellipsoid. Essentially, it is not appropriate to describe the uncertainty of variables with multimodal sample using the single ellipsoid. However, the proposed MEM can effectively analyze the samples with a similar distribution for establishing a more reasonable multi-ellipsoid model to measure the uncertainties of variables.
For the convenient expression, it can be assumed that all the samples of uncertain variables have two clusters. Then, the corresponding GMM built by the EM algorithm can be expressed as, In Eq. (17), each sub-item is a single Gaussian ellipsoid model, and thus, a multi-MEM can be established by the union operation of these two single ellipsoids, It can be ensured that the ellipsoids surround the relevant clustering samples through the appropriate scale ratiosR 2 1 andR 2 2 . Additionally, according to the sample distribution and the established GMM, there are two typical multi-MEMs. For the Type I, there is no intersection region for the single ellipsoids as shown in Fig. 2, while for the Type II, there exists an intersection region between these two ellipsoids as shown in Fig. 3. Obviously, the two ellipsoids provide less blind region. As the multi-MEM is the union of several single ellipsoid models, the uncertainty propagation method will be presented only for the single ellipsoid model. The system function of uncertainty propagation can be expressed as, where y and z(Á) denote the system response and system function.
Firstly, the space of uncertain variables (i.e. X space) is transformed into a set of regularized parameter space (i.e. U space), Then the initial uncertain domain E X can be transformed into a regularized convex domain E U , where X U denotes a characteristic matrix of the transformed convex model in the U space.The eigenvalue decomposition is performed for the characteristic matrix, where Q is an orthogonal matrix formed by the normalized eigenvectors, K is a diagonal matrix formed by the eigenvalues of X U and I is an identity matrix. Then another linear transformation is subsequently introduced, By using Eqs. (22) and (22), E U can be further transformed into a unit sphere domain E d in the d space, Therefore, the corresponding system function in the d space can be expressed as following, where Z(Á) denotes the system function in the d space.
Assume that the degree of nonlinearity of system function is not high, and according to the linear programming theory (Jiang et al. 2008), the optimal response of Eq. (21) will certainly occur on the boundary of the ellipsoid E X (Wang et al. 2008). As the space transformation of uncertain variables is linear, obviously, the optimal response also occurs on the boundary of the ellipsoid E d in the d space. For a two dimensional problem, the extreme state of system response in d space is shown in Fig. 4, where d 1 * and d 2 * denotes the extreme value points in the d space.
Then, the Lagrange-multiplier method (Kadapa et al. 2016) can be used to determine the analytic optimal solutions, where l and s denote the Lagrange function and Lagrange multiplier, respectively. Considering the necessary condition of the extreme value theorem, the extreme value point of Eq. (26) can be obtained when the first order differential equation of l equals to zero, i.e.
where rZ Á ð Þ denotes the partial derivative of system function in the d space. Combining Eq. (28) and E d ¼ d T d ¼ 1, the Lagrange multiplier s can be obtained as, where Á k k denotes the norm of a vector. In Eq. (30), it is clearly that the vector direction of extreme value point to origin point and the gradient direction are collinear. Hence, once the extreme value point is obtained in the d space, the bounds of system response can be calculated by the inverse space transformation.
Performance measure approach is an effective method to calculate the most suitable limit-state function by giving a reliability index. As we know, the reliability index is the minimum distance from the origin to the limit-state function. To sum up, the space of uncertain variables transformed from the X space to the d space, so the propagation process in the above section can be considered as the computation of corresponding most suitable limit-state function or the extreme value of system response as the precondition of a given reliability index. Noted that the multimodal ellipsoids have been divided into several ellipsoids and the design space of each ellipsoid is convex. Therefore, the performance measure approach can still be used in multimodal ellipsoids problems. According to Ref. (Youn et al. 2004), the optimization model can be constructed as follows, where c is the angle between d and rZ d ð Þ. The optimization process is summarized in Fig. 5.
Step 1 Specify an initial point d i , and set d i ¼ d i ; Step 2 Calculate the gradient rZ d i ð Þ; Step 3 If c i e, where e is a small angle, then d Ã ¼ d i . Otherwise, go to Step 4.
Step 4 Update the search direction The uncertainty quantification and propagation procedure by MEM is summarized in Table 1 Fig. 4 Extreme state of system response in unit sphere space In this example, 39 samples of three independent uncertain variables X ¼ ½X 1 ; X 2 ; X 3 T are considered, the values of which are listed in Appendix. Firstly, the proposed MEM is applied to quantify the uncertain variables under these samples. The optimal parameters of the single MEM can be obtained as follows, In order to make a comparison, we can make normalization for the MEM in Eq. (34). And, the results are compared with the CAM, MVM and SDP, which are shown in Table 2 and Fig. 6. From the compared results, it can find that the established single Step 1. Apply GMM to construct the non-probabilistic ellipsoid model by Gaussian Step 2. Construct the MLE functions L o ð Þ and substitute the samples into L o ð Þ Step 3. Implement E-step to construct the expectation of likelihood functions by substituting the samples into MLE functions Step 4: Implement M-step to calculate the lower bound of L o ð Þ. Then obtain the parameterŝ o¼â 1 ; Á Á Á ;â K ;l 1 ; Á Á Á ;l K ;R 1 ; Á Á Á ;R K À Á of the Gaussian function; Step 5: Based on GMM, determine each optimal R 2 of the contour ellipsoid that can envelop the clustered samples, and then obtain the multimodal ellipsoids; Uncertainty propagation procedure Input: Multimodal ellipsoid E X Step 1: Transform the uncertain domain of E X into regularized unit sphere domain . And the transformed system function can be expressed as Y ¼ Z d ð Þ; Step 2: Adopt Lagrange-multiplier method to determine the analytic optimal solutions l¼Z d ð Þþs d T dÀ1 À Á ; Step 3: Differentiate l with respect to Z and let the differential equation equal to zero, yielding d Ã ¼ À rZ d Ã ð Þ 2s ; Step 4: Construct the optimization problem min c ¼ cos À1 dÁrZ d ð Þ d k kÁ rZ d ð Þ k k ; s:t: d T d ¼ 1; Step 5: Specify an initial point d i , and set d i ¼ d i ; Step 6: Calculate the gradient rZ d i ð Þ; Step 7: If c i e, where e is a small angle, then d Ã ¼ d i . Otherwise, go back to Step 4. and then go back to Step 2; Step 8: Calculate the maximum and minimum values of Z d Ã ð Þ; Step 9: Combine the propagated result under each ellipsoid and obtain the upper and lower bounds of the responses MEM is closed to MVM and CAM, which indicates that the proposed uncertainty modeling method is effective. The SDP provides a smaller volume but some of the points may not be enveloped. Meanwhile, the modeling process of MEM based on maximum likelihood estimation is more convenient than SDP, MVM (optimization for all the parameters of characteristic matrix) and CAM (decomposition into several two dimensional optimizations). In addition, in order to compare the hyper-volumes of the ellipsoids by four modeling methods, the volume indexes defined by the products of the three half lengths of ellipsoidal axes are calculated, namely the minimum volume, which are equal to Herein, the superscripts G, J, M and S represent the MEM, CAM, MVM and SDP, respectively. The results indicate that the hypervolumes by MEM is larger than that by CAM and SDP, but smaller than that by MVM. However, as above-mentioned the CAM is an approximate method, and there exists a situation that a very few of samples may be outside of the constructed ellipsoid model. However, the proposed MEM can ensure that all the samples can be completely surrounded. As for the computational efficiency, the MEM uses only 20 iterations to obtain the response, which is close to the other methods when dealing with single ellipsoid problems.

Numerical example 2
The bivariate uncertain variables X¼ ½X 1 ; X 2 T are considered in the second example, and the system response function is defined as, where t and x denote the time and angular frequency. Through the scatter plot of the 60 samples as shown in Fig. 7, it can be clearly found that the samples have two distinct discrete multimodal and the values of samples are listed in Appendix. Therefore, in order to more accurately and reasonably measure the uncertainty of variable samples, the non-intersecting multi-MEM can be adopted. The optimal parameters of multi-ellipsoid are calculated by MLE and listed in Table 3. Then, with the estimation of the optimal parameter and the normalization process, the following multi-MEM for the samples of uncertain variables can be constructed, By comparison, the CAM and MVM for the samples are constructed as Eq. (37), and the compared results are shown in Fig. 8. Obviously, as the distribution and clustering phenomenon of the samples are not considered in CAM and MVM, they only provide Based on the characteristic matrices of multi-MEM, CAM and MVM, the minimum volumes are Fig. 6 The comparisons of three kinds of ellipsoid models Fig. 7 Scatter plot of the samples in example 2 These results quantitatively evaluate the volumes of the three methods and also indicate that the proposed MEM can provide more compact ellipsoid bounds for the uncertainty measurement of the samples with multimodal. Furthermore, the system response bounds under uncertain variables are calculated by using the uncertainty propagation methods of PMA and MCS with 10 6 samples, and the results are shown in Fig. 9. It can be found that the upper and lower bounds of system uncertain response under MEM are very closed to those by MCS, while the response bounds under CAM and MVM are more inaccurate and wide. This illustrates again that the proposed MEM is more suitable to measure uncertainties than CAM and MVM when the samples of uncertain variables display the phenomenon of the obvious multimodal. The MEM uses only 35 iterations to obtain the response, which is very closed to the other methods.

Numerical example 3
Another bivariate input variables X¼ ½X 1 ; X 2 T are considered in this example, and the system response function is given as where the K denotes the coefficient. The uncertain responses will be analyzed under different K. The samples of uncertain parameter are listed in Appendix.
Through the scatter plot of the samples shown in Fig. 10, it can be found that the samples have two distinct overlap sets. Therefore, we can adopt the intersecting MEM of the type II to measure the uncertainty of variable samples.
The optimal parameters for the construction of multi-ellipsoid are calculated by GMM and MLE and listed in Table 4.
Then, by the normalization of scale ratio, the MEM for uncertainty measurement in this example is built as Eq. (39) and shown in Fig. 11 By comparison, the constructed CAM and MVM are represented as Eq. (40) and also shown in Fig. 11. In view of this, for the samples of uncertain variables with two intersecting sets, the proposed MEM can reasonably and effectively model this kind of uncertainty. However, the CAM and MVM are more conservative and has not the ability to describe internal structure of uncertain samples in detail.
Based on the characteristic matrices of MEM, CAM and MVM, the minimum volume are equal to  It should be noted that in order to calculate volume index of MEM, the overlap region of the two component ellipsoids has been counted twice. Nevertheless, the total volume index of MEM is still much smaller than that of CAM and MVM. Then, the system response bounds are computed by using PMA and MCS with 10 6 samples, and the results are shown in Fig. 12. It also can be found that for different system coefficient, the propagated uncertain responses under MEM are all almost close to MCS, and more accurate than those under CAM and MVM.

Engineering application
The thin-walled beam connected by point-welding is a major structure of vehicle body for load bearing and energy absorption. The crashworthiness performance is an important consideration for its design and manufacture. In Fig. 13, the closed-hat beam (Jiang and Han 2007) is impacted on a rigid wall with the initial velocity of 10 m/s. The beam contains a hat beam and a web plate. The web plate is connected by some uniformly distributed point-welding points along the two rims of the hat beam. The material properties of the beam are Young's modulus E ¼ 2:0 Â 10 5 MPa, density q ¼ 7:85 Â 10 À6 Kg=mm 3 , yield stress r s ¼ 310MPaand tangent modulus E t ¼ 763MPa, respectively. Considering the manufacturing and measuring errors, the plate thickness t, the round radius r, the dimensions L 1 , L 2 and L 3 are treated as uncertain parameters. The marginal intervals of the five uncertain parameters are L I 1 ¼ 16; 20 ½ mm, L I 2 ¼ 65; 75 ½ mm, L I 3 ¼ 55; 65 ½ mm, t I ¼ 1; 2 ½ mm and r I ¼ 3; 5 ½ mm, respectively. We assume that the samples have two clusters and based on GMM and MEM, the values of the samples are listed in Appendix, and the optimal parameters for modeling the multi-ellipsoid are estimated and listed in Table 5. Therefore, we can construct the normalized MEM as follows, For the thin-walled beam in a vehicle body, the axial impact force is a key index to evaluate the crashworthiness performance, and it is closely related to the occupant security. Thus, the maximal axial impact force is treated as the system performance function. The corresponding FEM model shown in Fig. 14 is built to compute the maximal uncertain axial impact force under the multi-MEM of Eq. (41). Based on the PMA, the uncertain response of axial impact forces can be effectively solved, and its bounds are [15.9032, 48.0221] kN. By comparison, the result of the MCS with 10 5 samples is provided and the bounds are [13.6581, 55.4738]kN. This result shows that the proposed method can obtain a relatively closed propagation result even with insufficient samples. In this paper, a more generalized non-probabilistic convex model named multimodal ellipsoid model is proposed to reasonably measure the uncertainty of uncertain variables using a single or multimodal samples. The intrinsic distribution of samples is fully considered and the corresponding Gaussian mixture model is constructed. Through the concepts, the samples with one or more similar properties are clustered together, and the principal directions of the samples and characteristic matrix can be appropriately found. Then, the ellipsoidal model of each cluster is modeled by using the elliptical contour features of the Gaussian model. The proposed MEM is a more general uncertainty modeling method, which is suitable for the multi-ellipsoid modeling. In addition, the single-ellipsoid can be regarded as a special case of the multi-ellipsoid. The proposed MEM has many advantages compared with the previous ellipsoid model by the minimum volume optimization method and correlation approximation method. (1) The MEM can provide a suitable modeling process for the nonprobabilistic uncertain measurement, which has a rigorous mathematical theoretical foundation. (2) The MEM has a compact structure and the correlation of the MEM is fully based on the relevant samples. (3) The MEM can achieve the multi-ellipsoid modeling, which is impossible for the traditional ellipsoidal convex modelling method. Additionally, a non-probabilistic uncertainty propagation method based on the MEM and PMA is also developed for uncertain structures, in which the MEM is used to quantify the uncertain parameters and the PMA is employed to calculate the bounds of system responses.