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Crashworthiness analysis remains an important concern for the design of safety structures. In this context, uncertainties play an essential role in the response of a crash problem with non linear behavior. With this statement at hand, in this work it is presented a review of uncertainty quantification (UQ) techniques, with intrusive and non-intrusive approaches in stochastic finite element methods for crashworthiness. The well-known deterministic finite element solver VPS/Pamcrash is used to illustrate the currently available methods, developing a comparative analysis of these techniques in crashworthiness UQ. Finally, relevant non-intrusive methods are applied to analyze the behavior of a specific quantity of interest in a dynamic crash model.

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The Inertia Relief (IR) technique is widely used by industry and produces equilibrated loads allowing to analyze unconstrained systems without resorting to the more expensive full dynamic analysis. The main goal of this work is to develop a computational framework for the solution of unconstrained parametric structural problems with IR and the Proper Generalized Decomposition (PGD) method. First, the IR method is formulated in a parametric setting for both material and geometric parameters. A reduced order model using the encapsulated PGD suite is then developed to solve the parametric IR problem, circumventing the so-called curse of dimensionality. With just one offline computation, the proposed PGD-IR scheme provides a computational vademecum that contains all the possible solutions for a predefined range of the parameters. The proposed approach is nonintrusive and it is therefore possible to be integrated with commercial finite element (FE) packages. The applicability and potential of the developed technique is shown using a three-dimensional test case and a more complex industrial test case. The first example is used to highlight the numerical properties of the scheme, whereas the second example demonstrates the potential in a more complex setting and it shows the possibility to integrate the proposed framework within a commercial FE package. In addition, the last example shows the possibility to use the generalized solution in a multi-objective optimization setting. K E Y W O R D Sinertia relief, nonintrusive, proper generalized decomposition, reduced order model, shape optimization INTRODUCTIONUnconstrained structures are widespread in the automotive, aerospace and naval industry. As is well known, due to the singularity of the stiffness matrix, conventional static analyses cannot be performed if the system undergoes rigid body motions. At the same time, imposing dummy constraints in order to make a free-body system statically determinate leadsThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Uncertainty Quantification (UQ) is a booming discipline for complex computational models based on the analysis of robustness, reliability and credibility. UQ analysis for nonlinear crash models with high dimensional outputs presents important challenges. In crashworthiness, nonlinear structural behaviours with multiple hidden modes require expensive models (18 hours for a single run). Surrogate models (metamodels) allow substituting the full order model, introducing a response surface for a reduced training set of numerical experiments. Moreover, uncertain input and large number of degrees of freedom result in high dimensional problems, which derives to a bottle neck that blocks the computational efficiency of the metamodels. Kernel Principal Component Analysis (kPCA) is a multidimensionality reduction technique for non-linear problems, with the advantage of capturing the most relevant information from the response and improving the efficiency of the metamodel. Aiming to compute the minimum number of samples with the full order model. The proposed methodology is tested with a practical industrial problem that arises from the automotive industry.