A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures

Collaboration with University Southern California, Funding Agency: MURIInternational audienceIn this paper, we address the construction of a prior stochastic model for non-Gaussian deterministically-bounded positive-definite matrix-valued random fields in the context of mesoscale modeling of heterogeneous elastic microstructures. We first introduce the micromechanical framework and recall, in particular, Huet's Partition Theorem. Based on the latter, we discuss the nature of hierarchical bounds and define, under some given assumptions, deterministic bounds for the apparent elasticity tensor. Having recourse to the Maximum Entropy Principle under the constraints defined by the available information, we then introduce two random matrix models. It is shown that an alternative formulation of the boundedness constraints further allows constructing a probabilistic model for deterministically-bounded positive-definite matrix-valued random fields. Such a construction is presented and relies on a class of random fields previously defined. We finally exemplify the overall methodology considering an experimental database obtained from EBSD measurements and provide a simple numerical application

Validation of a probabilistic model for mesoscale elasticity tensor of random polycrystals

International audienceIn this paper, we present validation of a probabilistic model for mesoscale elastic behavior of materials with microstructure. The linear elastic constitutive matrix of this model is described mathematically as a bounded random matrix. The bounds reflect theoretical constraints consistent with the theory of elasticity. We first introduce a statistical characterization of an experimental database on morphology and crystallography of polycrystalline microstructures. The resulting statistical model is used as a surrogate to further experimental data, required for calibration and validation. We then recall the construction of a probabilistic model for the random matrix characterizing the apparent elasticity tensor of a heterogeneous random medium. The calibration of this coarse scale probabilistic model using an experimental database of microstructural measurements and utilizing the developed microstructural simulation tool is briefly discussed. Before using the model as a predictive tool in a system level simulation for the purpose of detection and prognosis, the credibility of the model must be established through evaluating the degree of agreement between the predictions of the model and the observations. As such, a procedure is presented to validate the probabilistic model from simulated data resulting from subscale simulations. Suitable quantities of interest are introduced and predictive accuracy of the model is studied by comparing probability density functions of response quantities of interest. The validation task is exercised under both static and dynamic loading condition. The results indicate that the probabilistic model of mesoscale elasticity tensor is adequate to predict the response quantity of interest in the elastostatic regime. The scatter in the model predictions is found to be consistent with the fine scale response. In the case of elastodynamic, the model predicts the mean behavior for lower frequency for which we have a quasistatic regime

Operational Strategies for Increasing Secondary Materials in Metals Production Under Uncertainty

Increased use of secondary raw materials in metal production offers several benefits including reduced cost and lowered energy burden. The lower cost of secondary or scrap materials is accompanied by an increased uncertainty in elemental composition. This increased uncertainty for different scraps, if not managed well, results in an increased risk that the elemental concentrations in the final products fall outside customer specifications. Previous results show that incorporating this uncertainty explicitly into batch planning can modify the potential use of scrap materials while managing risk. Chance-constrained formulations provide one approach to uncertainty-aware batch planning; however, typical formulations assume normal distributions to represent the compositional uncertainty of the materials. Compositional variation in scrap materials has been shown to have a skewed distribution, and therefore, the performance of these models, in terms of their ability to provide effective planning, it may then be heavily influenced by the structure of the compositional data used. To address this issue, this work developed several approximations for skewed distributional forms within chance-constrained formulations. We explored a lognormal approximation based on Fenton’s method; a convex approximation based on Bernstein inequalities; and a linear approximation using fuzzy set theory. Each of these methods was formulated and case studies executed using compositional data from an aluminum remelter. Results indicate that the relationship between the underlying structure/distribution of the compositional data and how these distributions are formulated in batch planning can modify the use of secondary raw materials.National Science Foundation (U.S.) (Award 1133422

A Methodology for Robust Comparative Life Cycle Assessments Incorporating Uncertainty

We propose a methodology for conducting robust comparative life cycle assessments (LCA) by leveraging uncertainty. The method evaluates a broad range of the possible scenario space in a probabilistic fashion while simultaneously considering uncertainty in input data. The method is intended to ascertain which scenarios have a definitive environmentally preferable choice among the alternatives being compared and the significance of the differences given uncertainty in the parameters, which parameters have the most influence on this difference, and how we can identify the resolvable scenarios (where one alternative in the comparison has a clearly lower environmental impact). This is accomplished via an aggregated probabilistic scenario-aware analysis, followed by an assessment of which scenarios have resolvable alternatives. Decision-tree partitioning algorithms are used to isolate meaningful scenario groups. In instances where the alternatives cannot be resolved for scenarios of interest, influential parameters are identified using sensitivity analysis. If those parameters can be refined, the process can be iterated using the refined parameters. We also present definitions of uncertainty quantities that have not been applied in the field of LCA and approaches for characterizing uncertainty in those quantities. We then demonstrate the methodology through a case study of pavements