Research

The Shields Uncertainty Research Group conducts methodological research in uncertainty quantification and stochastic methods for applications in computational mechanics and computational materials science. Below are a sample of some recent research projects conducted by the group.

Sample Projects

Methodologies for stochastic hierarchical multiscale modeling - In collaboration with Army Research Lab

Collaborators: Dr. Jarek Knap, Computational and Information Sciences Directorate, Army Research Lab

Group Members: Dr. Susmita Naskar

In the context of hierarchical multi-scale material modeling, we are working to develop reduced order stochastic surrogate models to represent the constitutive response of materials learned from lower scale models. Specifically, in the numerical solution of problems in continuum solid mechanics, multiscale methods will often invoke a "lower-scale" model to estimate relevant quantities (e.g. temperature, pressure) at each integration point. This can be computationally infeasible given the high expense of repeated calls to expensive lower-scale models at many integration points. The problem is exacerbated with the lower-scale model is stochastic. An alternative approach, proposed by Dr. Knap and colleagues [1], is to adaptively train a surrogate model for the constitutive response based on the lower-scale solution. In [1], this has framework has been developed for problems where the lower-scale model is deterministic. In this project, we are developing novel reduced-order stochastic surrogate models that utilize nonlinear manifold projections [2] and advanced interpolation schemes such as Gaussian process regression.

Relevant Publications:

  1. K. W. Leiter, B. C. Barnes, R. Becker, and J. Knap, “Accelerated scale-bridging through adaptive surrogate model evaluation,” J. Comput. Sci., vol. 27, pp. 91–106, 2018.
  2. D. G. Giovanis and M. D. Shields, “Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations,” J. Comput. Phys., vol. 364, 2018.

An initial investigation of structural reliability from sparse data - Office of Naval Research

Group Members: Dr. Dimitris Giovanis

Reliability analysis involves the estimation of small failure probabilities and is often computationally very expensive to achieve. To mitigate these challenges, simplifying assumptions such as linearity of the limit surface (as in the First Order Reliability Method - FORM) are often made or advanced Monte Carlo-type solutions such as subset simulation are employed that aim to quickly concentrate samples near the limit surface. The problem is made more challenging when data for quantifying the probability distribution of random parameters are lacking. Consequently, while the limit surface is clearly defined in the parameter space, the probability associated with the failure domain defined by that limit surface is imprecise. In this work, we are leveraging a novel new multimodel UQ methodology developed by our group [1, 2] to quantify the uncertainty in probability of failure estimates from both FORM and subset simulation under these conditions.

Relevant Publications:

  1. J. Zhang and M. D. Shields, “On the quantification and efficient propagation of imprecise probabilities resulting from small datasets,” Mech. Syst. Signal Process., vol. 98, pp. 465–483, Jan. 2018.
  2. J. Zhang and M. D. Shields, “The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets,” Comput. Methods Appl. Mech. Eng., 2018.

NSF CAREER: Higher-order methods for nonlinear stochastic structural dynamics - National Science Foundation

Group Members: Lohit Vandanapu

Numerous stochastic process models have been developed for stochastic excitation for structural dynamics systems. Most of these stochastic models are inherently second-order, matching only the mean and the covariance function of the stochastic process. Some models, such as Grigoriu's translation process theory are capable of modeling non-Gaussian processes but are still only capable of matching second-order moments of the process. Recently, our group developed a novel higher-order stochastic process expansion model that generalizes the Spectral Representation Method for third-order (skewed, non-Gaussian) processes [1]. In this project, we are working to expand this generalization to match higher-order characteristics that are representative of true stochastic processes observed in nature (e.g. turbulent wind velocity time histories).

This is important because nonlinear structural systems may be highly sensitive to variations in the stochastic excitation. Seemingly minor changes in the properties of the process may have profound impacts on the response of both geometrically and materially nonlinear structures. Using the advanced stochastic models developed here, we are studying the influence of modeling assumptions and model-form on the response of the system. By understanding the effect of stochastic modeling on these nonlinear systems, we can then further improve the standards of modeling practice for nonlinear structures and incorporate these practices into novel structural optimization and design standards.

Relevant Publications:

  1. M. D. Shields and H. Kim, “Simulation of higher-order stochastic processes by spectral representation,” Probabilistic Eng. Mech., vol. 47, pp. 1–15, 2017.

Efficient stochastic simulation-based computational modeling for structural design, reliability, and life-cycle assessment - Office of Naval Research Young Investigator Program

Group Members: Dr. Dimitris Giovanis, Mohit Chauhan, Dr. Jiaxin Zhang

Stochastic simulation is an important tool for uncertainty quantification in the computational modeling of naval structures. In this work, we are developing simulation-based approaches to address two critical challenges:

  1. Efficient propagation of uncertainty through adaptive sampling
  2. Propagation of uncertainties characterized by imprecise probabilities resulting from small datasets.

Toward the first challenge, we have developed novel adaptive Monte Carlo methods that are optimized for nonlinear systems [1], adaptive stochastic collocation methods for nonlinear systems with very high-dimensional response [2], and adaptive nonlinear manifold projection based surrogate modeling techniques for UQ of very high-dimensional problems [3]. We are also currently developing new adaptive stochastic reduced order models to efficiently propagate uncertainty through a surrogate representation.

Toward the second challenge, we have developed a new multimodel uncertainty quantification framework, which characterizes imprecise probabilities using a multimodel set of probability distributions identified through information theoretic or Bayesian multimodel inference and Bayesian parameter estimation [4]. The result is a set of candidate probability models, each having known probability. We further developed a methodology to efficiently propagate this set of candidate models using an importance sampling reweighting scheme that reduces a multi-loop uncertainty propagation problem to a single-loop one that can be performed at greatly reduced computational cost (see Figure above). We have further investigated the effect of prior probabilities on the results of this multimodel approach and shown that one should exercise great care in the selection of prior probability distributions as bias can be introduced even with only mildly incorrect priors [5].

Relevant Publications:

  1. M. D. Shields, “Adaptive Monte Carlo analysis for strongly nonlinear stochastic systems,” Reliab. Eng. Syst. Saf., vol. 175, pp. 207–224, Jul. 2018.
  2. D. Giovanis and M.D. Shields, "Variance-based simplex stochastic collocation with model order reduction for high-dimensional response." Int'l J. for Num. Meth. in Eng. (In Review).
  3. D. G. Giovanis and M. D. Shields, “Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations,” J. Comput. Phys., vol. 364, 2018.
  4. J. Zhang and M. D. Shields, “On the quantification and efficient propagation of imprecise probabilities resulting from small datasets,” Mech. Syst. Signal Process., vol. 98, pp. 465–483, Jan. 2018.
  5. J. Zhang and M. D. Shields, “The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets,” Comput. Methods Appl. Mech. Eng., 2018.

Connecting Atomistic and Continuum Amorphous Solid Mechanics via Non-Equilibrium Thermodynamics - National Science Foundation

Collaborators: Prof. Michael Falk, Dept. of Materials Science and Engineering, Johns Hopkins University; Prof. Chris Rycroft, School of Engineering and Applied Sciences, Harvard University

Group Members: Dr. Adam Hinkle, Dihui Ruan, Darius Alix-Williams, Dr. Dimitris Giovanis

Many efforts are underway to understand and model the deformation mechanisms of amorphous solids. In this work, we are investigating the inelastic deformation in metallic glasses using molecular dynamics (MD) simulations and continuum-scale modeling that employs the Shear Transformation Zone (STZ) theory of plasticity. A major point of emphasis in this study is to understand how to leverage atomistic scale information to initialize and parameterize the STZ model. To this effect, we have developed a novel coarse-graining procedure that converts the averaged atomic potential energies into an effective disorder temperature that serves as a measure of configurational disorder in the atomic structure of the glass [1]. Using this coarse-graining method, we observe very similar response in MD and continuum STZ simulations. To build on this, we are currently working to optimize the coarse-graining calibration procedure through an efficient global optimization routine and further, working to characterize the initial effective temperature using a calibrated stochastic field model that can be deployed the quantify variability in the inelastic deformation of amorphous solids.

Comparison of plastic deformation in coarse-grained MD simulations (left) and continuum STZ simulations (right) for a CuZr metallic glass.

Relevant Publications:

  1. A. R. Hinkle, C. H. Rycroft, M. D. Shields, and M. L. Falk, “Coarse graining atomistic simulations of plastically deforming amorphous solids,” Phys. Rev. E, vol. 95, no. 5, p. 053001, 2017.

Stress-strain curves for uniaxial tension tests of Al-6061-T651 from 9 different material batches at 6 different temperatures.

GOALI: Improving the Reliability of Aluminum Structures During Fire Through Computational Modeling - National Science Foundation

Collaborators: Pawel Woelke, Principle, Thornton Tomasetti

Group Members: Aakash Bangalore Satish

Aluminum, as a structural metal, is particularly vulnerable to the effects of elevated temperature due to both its low melting point and high thermal conductivity. Moreover, structural aluminum exhibits signficant variability in its stress-strain behavior. In this work, we have conducted approximately 200 tests of Al 6061-T651 coupons in both uniaxial tension and plane strain tension from 9 different batches of material sourced from different manufacturers/suppliers at 6 different temperatures, in an effort to quantify the variability in temperature-dependent material behavior. This data is now being used to calibrate an advanced damage plasticity based material model in order to conduct probabilistic modeling of aluminum structures during fire to assess their reliability and ultimately improve design standards for aluminum structures.

Relevant Publications:

  1. B.S. Aakash, JP Connors, and M.D. Shields, "Variability in the thermo-mechanical behavior of structural aluminum," Int'l J. of Solids and Str. (In Review).