Modeling material stress using integrated Gaussian Markov random fields

Peter W Marcy; Scott A Vander Wiel; Curtis B Storlie; Veronica Livescu; Curt A Bronkhorst

doi:10.1080/02664763.2019.1686131

Modeling material stress using integrated Gaussian Markov random fields

J Appl Stat. 2019 Nov 8;47(9):1616-1636. doi: 10.1080/02664763.2019.1686131. eCollection 2020.

Authors

Peter W Marcy¹, Scott A Vander Wiel¹, Curtis B Storlie², Veronica Livescu³, Curt A Bronkhorst⁴

Affiliations

¹ Statistical Sciences Group (CCS-6), Los Alamos National Laboratory (LANL), Los Alamos, NM, USA.
² Mayo Clinic, Rochester, MN, USA.
³ Materials Science in Radiation and Dynamics Extremes (MST-8), LANL, Los Alamos, NM, USA.
⁴ Fluid Dynamics and Solid Mechanics Group (T-3), LANL, Los Alamos, NM, USA.

Abstract

The equations of a physical constitutive model for material stress within tantalum grains were solved numerically using a tetrahedrally meshed volume. The resulting output included a scalar vonMises stress for each of the more than 94,000 tetrahedra within the finite element discretization. In this paper, we define an intricate statistical model for the spatial field of vonMises stress which uses the given grain geometry in a fundamental way. Our model relates the three-dimensional field to integrals of latent stochastic processes defined on the vertices of the one- and two-dimensional grain boundaries. An intuitive neighborhood structure of the said boundary nodes suggested the use of a latent Gaussian Markov random field (GMRF). However, despite the potential for computational gains afforded by GMRFs, the integral nature of our model and the sheer number of data points pose substantial challenges for a full Bayesian analysis. To overcome these problems and encourage efficient exploration of the posterior distribution, a number of techniques are now combined: parallel computing, sparse matrix methods, and a modification of a block update strategy within the sampling routine. In addition, we use an auxiliary variables approach to accommodate the presence of outliers in the data.

Keywords: Bayesian analysis; Gaussian Markov random field; blind deconvolution; large-scale inverse problem; materials science; process convolution; robust regression.

Grants and funding

This work was performed at Los Alamos National Laboratory and funded through the Laboratory Directed Research and Development program via projects 20170033DR and 20150594ER.