Computer Methods in Applied Mechanics and Engineering
A sampling-based approach for probabilistic design with random fields
Introduction
Despite substantial research efforts and recent improvements, probabilistic design still faces major challenges. First, it is well known that the initial assumptions for the representation and quantification of uncertainties are of prime importance. For instance, for a problem with spatial variability (e.g., sheet metal thickness distribution), one should choose to describe the problem with random fields as they provide a more realistic representation than uncorrelated random variables. These assumptions are as important as the process used to propagate uncertainties. Second, in a simulation-based context, the nature of the problem might severely restrict the use of traditional algorithms. Of particular interest are problems with non-smooth and discontinuous responses, prohibitive computational costs, or disjoint failure spaces. Computational design for crashworthiness is an example which encompasses these difficulties.
The probabilistic design literature is mostly dominated by approaches and applications where uncertainties are quantified as independent random variables. Techniques such as Monte Carlo simulations (MCS), and first and second order reliability methods (FORM and SORM) [1], [2] are used to perform reliability assessment using assumed probability density functions (PDFs). These reliability assessment techniques are also embedded within optimization problems to carry out reliability based design optimization (RBDO) [3], [4], [5]. Many studies have also been dedicated to the reduction of computational costs associated with these reliability assessment and RBDO techniques. Approaches based on designs of experiments (DOE) and surrogate models (response surfaces and metamodels [6], [7]) are common.
Recently, the authors have introduced the notion of explicit design space decomposition [8], [9], [10] whereby the LSFs are constructed explicitly in terms of the design variables. The LSF construction is based on a SVM which allows one to define the boundaries of failure regions that can be disjoint and non-convex. The approach allows for a straightforward calculation of a probability of failure using MCS. In addition, because this technique does not approximate responses but rather classify them as failed or safe, it naturally handles discontinuities. The construction of explicit LSF is also complemented by an adaptive sampling scheme which minimizes the number of function evaluations and refines the LSF approximation [10]. Therefore, the explicit design space decomposition technique is aimed at handling the difficulties due to discontinuities, complex failure domains and computational costs.
In the case of random fields, which is the focus of this article, the literature revolves around stochastic finite elements (SFE). SFE enable the propagation of uncertainties to obtain the distribution of the system’s responses using polynomial chaos expansion (PCE) [11]. In order to represent a random field, it is approximated using a Karhunen–Loeve expansion [11]. The coefficients of the expansion are considered as random variables and the response is expanded on a specific polynomial basis (Hermite, Legendre etc.) depending on the assumed types of probability distributions.
Several implementations of SFE are available in literature. The early approaches required the modification of the equilibrium equation to account for the uncertainty in the stiffness matrix and the load vector [11]. This approach is by construction highly intrusive and required specific codes. Newer methods developed recently overcome this limitation and allow for the determination of PCE coefficients using deterministic “black-box” function evaluations (e.g., finite element analysis). Therefore, these methods can be used with available commercial simulation packages without modifying the code [12], [13], [14].
Most studies with SFE typically assume a prior distribution of the random field. However, in practical situations, such as a random field generated by a manufacturing process, the characteristics of the random field are not known a priori. Therefore, the only way to characterize a random field with a certain level of confidence is from experimental observations. In addition, another limitation of existing approaches is that the expansion of responses on a polynomial basis hampers the use of PCE for problems with discontinuities.
In this article, an alternate non-intrusive approach is proposed, which provides a combined solution to the difficulties of realistically representing random fields, handling discontinuous responses, and efficiently calculating a probability of failure. This is achieved by combining the explicit design space decomposition approach with a proper orthogonal decomposition (POD) for the characterization of random fields.
Based on a limited number of observations, referred to as snapshots, POD is used to extract the important features of a random field in the form of eigenvectors of its covariance matrix [15]. The eigenvalues provide an indication of the importance of the corresponding features, thus allowing one to gauge their individual contributions to the random field. This technique is similar to the one found in pattern recognition [16].
Once the random field is characterized with the important features, the corresponding eigenvectors form a basis that is used to generate various random field configurations. This is required for design purposes as an initial set of experimental snapshots may not be sufficient. The random field is modified by varying the coefficients of the eigenvectors in the POD expansion. For this purpose, the response of the system is studied with a DOE [17], [18] with respect to the coefficients of the expansion. At this stage, the actual PDFs of the coefficients are not considered, and they are assumed uniformly distributed. This is done in order to extract as much information as possible over the whole design space.
The responses, generated for each sample of the DOE, are classified into failure and non-failure using a threshold value or a clustering technique such as K-means [19]. Clustering is used in the case of discontinuous responses. These two classes are then separated in the design space using an (explicit) SVM LSF [9]. In addition, in order to refine the LSF using a limited number of samples, an adaptive sampling technique is used [10]. The sampling strategy is based on the generation of samples that maximize the probability of misclassification of the SVM while avoiding redundancy of information.
The coefficients of the POD expansion are random variables and their distributions, obtained from the snapshots, are found through basic PDF fitting techniques. A similar approach was used in [20] for the probabilistic design of turbine blade engine using POD expansion for turbine blade random geometries.
In the proposed approach, probabilities of failure are efficiently calculated using MCS. This simplicity, and this is the novelty of the proposed approach, is due to the fact that the limit state function is defined explicitly in terms of the coefficients of the POD. As mentioned previously, it is noteworthy that the accuracy of the limit state function is improved through adaptive sampling [10] to limit the number of function evaluations.
The proposed approach for the calculation of probabilities of failure is applied to two problems. The first problem consists of a three dimensional arch structure whose thickness is considered as a random field. A failure criterion is defined based on a threshold value on the critical load factor for linear buckling. The second problem involves the impact of a tube on a rigid wall. The planarity of the tube walls is modified by a random field, which leads to a global buckling (considered failure) or crushing of the tube.
Section snippets
Summary of the proposed approach
For the sake of clarity, this section summarizes the main steps of the approach, which are subsequently described in the following sections. The stages of the approach are (Fig. 1):
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Collection of snapshots and construction of the covariance matrix.
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Selection of the main features of the random field.
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Expansion of the field on a reduced basis. Sampling of the coefficients using a uniform design of experiments (DOE).
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Construction of an explicit LSF using SVM in the space of coefficients.
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Refinement of
Data collection and covariance matrix
The first step in the characterization of a random field is the collection of several observations of the random process output (e.g., a metal sheet after forming). The process generates a scalar random field , function of the position . M samples, outputs of this process, are obtained. On each sample, N measurements are performed at distinct positions. An example of observations, referred to as snapshots, is provided in Fig. 2. The snapshots can be condensed in the following matrix:
Sampling-based coefficient selection and response estimation
The characterization of a random field, and coefficient distributions is accomplished using the data from the snapshots. However, the mere characterization of the random field is not sufficient to account for uncertainties in the design process. For this purpose, several instances of random fields (other than the snapshots) are created by using different linear combinations of the eigenvectors.
The combinations are defined by selecting the POD coefficients using a DOE. The bounds of the DOE are
Explicit limit state function
SVM is a classification tool that belongs to the class of machine learning techniques. The main feature of SVM lies in its ability to explicitly define complex decision functions that optimally separate two classes of data samples. Thus, once the coefficient samples are categorized into two classes, SVM can provide an explicit decision function (the limit state function) separating the distinct classes. The equation of the SVM LSF is given by equating the quantity s in Eq. (13) equal to zero
Probability of failure calculation – MCS
The explicit LSF allows one to efficiently calculate the probability of failure using MCS. The PDFs of the coefficients, as found in Section 3.3, are used to generate Monte-Carlo samples. Predicting failure or non-failure at these samples involves calculating the sign of the analytical expression of the LSF (Eq. (13)). An example of calculation of the probability of failure is depicted in Fig. 6.
Linear buckling of an arch structure
This section provides an example of the effect of a random field on the critical load factor of an arch structure. The structure is subjected to a unit pressure load on the top surface. The thickness of the arch should ideally be constant over the entire surface; however it may vary due to uncertainties in the manufacturing processes. These variations are represented, for this study, by an artificial analytical random field (as opposed to real experimental data). The arch has a radius of R = 200
Conclusion
A technique for reliability assessment using random fields is proposed. A new sampling-based method is used for constructing various potential random field configurations. The method overcomes the need for assumption on the random field distribution by using experimental data and Proper Orthogonal Decomposition. In addition the SVM-based method of constructing explicit LSFs enables one to address discontinuous system responses, which is successfully shown in the case of the tube impact problem.
Acknowledgement
The support of the National Science Foundation (Award CMMI-0800117) is gratefully acknowledged.
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