Bayesian inference of elastic properties with resonant ultrasound spectroscopy

Abstract

Bayesian modeling and Hamiltonian Monte Carlo (HMC) are utilized to formulate a robust algorithm capable of simultaneously estimating anisotropic elastic properties and crystallographic orientation of a specimen from a list of measured resonance frequencies collected via Resonance Ultrasound Spectroscopy (RUS). Unlike typical optimization procedures which yield point estimates of the unknown parameters, computing a Bayesian posterior yields probability distributions for the unknown parameters, and HMC is an efficient way to compute this posterior. The algorithms described are demonstrated on RUS data collected from two parallelepiped specimens of structural metal alloys. First, the elastic constants for a specimen of fine-grain polycrystalline Ti-6Al-4 V with random crystallographic texture and isotropic elastic symmetry are estimated. Second, the elastic constants and crystallographic orientation for a single crystal Ni-based superalloy CMSX-4 specimen are accurately determined, using only measurements of the specimen geometry, mass, and resonance frequencies. The unique contributions of this paper are as follows: the application of HMC for sampling the Bayesian posterior of a probabilistic RUS model, and the procedure for simultaneous estimation of elastic constants and lattice-specimen misorientation. Compared to previous approaches these algorithms demonstrate superior convergence behavior, particularly when the initial parameterization is unknown, and enable substantially simplified experimental procedures.

Elastic Properties of Novel Co- and CoNi-Based Superalloys Determined through Bayesian Inference and Resonant Ultrasound Spectroscopy

“…Analysis of RUS data is substantially simplified when both the single crystal elastic constants and the arbitrary crystal reference frame rotation are determined simultaneously. [16] Otherwise, the specimen must be precisely cut from a parent single crystal of known orientation (typically measured with X-ray diffraction techniques) such that the rotation between the crystal and specimen reference frames is nearly (ideally) zero. [21] 2.…”

“…COMPUTATIONAL METHODS As there is no direct solution to the general inverse problem of determining elastic properties from RUS measurements, elastic properties must be determined indirectly through least-squares optimization [21,27,28] or statistical inference [16] methods. Both of these inversion methods ultimately rely on numerous forward problem evaluations, [16,21] therefore, computational methods begins by introducing the forward problem for calculating resonance frequencies of a specimen in Section III-A. Next, Section III-B details inversion with a short summary on traditional optimization-based methods followed by the distinguishing characteristics of a Bayesian approach.…”

“…In Bales et al, [16] the arbitrary rotation of the crystal reference frame was parameterized as a passive unit quaternion. A unit quaternion is expressed in four dimensions (w,x,y,z) but resides on a 3D manifold characterized by w 2 þ x 2 þ y 2 þ z 2 ¼ 1.…”

“…In a prior publication, [16] we demonstrated the first full Bayesian approach to determining elastic constants from simple and non-destructive resonant ultrasound spectroscopy (RUS) measurements. While simultaneously evaluating elastic constants and crystal orientation, this new approach is remarkably robust in comparison to traditional optimization algorithms only estimating elastic constants.…”

“…The ability to evaluate elastic constants of a parallelepiped specimen cut with arbitrary alignment of the crystallographic axes greatly simplifies specimen preparation requirements and eliminates a major potential source of measurement error. [16] The goals of this paper are threefold. First, provide an introduction to RUS Bayesian inference including the basic theory, the distinguishing characteristics from traditional optimization methods, and the nomenclature necessary for discussing and achieving the remaining goals.…”