Active subspace analysis and uncertainty quantification for a polydomain ferroelectric phase-field model

Abstract

We perform parameter subset selection and uncertainty analysis for phase-field models that are applied to the ferroelectric material lead titanate. A motivating objective is to determine which parameters are influential in the sense that their uncertainties directly affect the uncertainty in the model response, and fix noninfluential parameters at nominal values for subsequent uncertainty propagation. We employ Bayesian inference to quantify the uncertainties of gradient exchange parameters governing 180° and 90° tetragonal phase domain wall energies. The uncertainties of influential parameters determined by parameter subset selection are then propagated through the models to obtain credible intervals when estimating energy densities quantifying polarization and strain across domain walls. The results illustrate various properties of Landau and electromechanical coupling parameters and their influence on domain wall interactions. We employ energy statistics, which quantify distances between statistical observations, to compare credible intervals constructed using a complete set of parameters against an influential subset of parameters. These intervals are obtained from the uncertainty propagation of the model input parameters on the domain wall energy densities. The investigation provides critical insight into the development of parameter subset selection, uncertainty quantification, and propagation methodologies for material modeling domain wall structure evolution, informed by density functional theory simulations.

A ferroelectric phase field study of nanoindentation contact mechanics

“…The parameter values used in the phase field analysis can be found elsewhere. 5 The use of variational methods leads to three governing equations: linear momentum, the Ginzburg-Landau equation for polarization evolution, and Gauss' law. These three equations can be written as…”

“…The first relation in (5) is linear momentum which is written in terms of the Cauchy stress, σ ij = ∂ ψ ∂ϵij . The second equation is the Ginzburg-Landau equation where η i = ∂ ψ ∂Pi and ξ ji = ∂ ψ ∂Pi,j .…”