Bayesian inversion in resin transfer molding

Abstract

We study the Bayesian inverse problem of inferring the permeability of a porous medium within the context of a moving boundary framework motivated by Resin Transfer Molding (RTM), one of the most commonly used processes for manufacturing fiber-reinforced composite materials. During the injection of resin in RTM, our aim is to update our probabilistic knowledge of the permeability of the material by inverting pressure measurements as well as observations of the resin moving domain. We consider both one-dimensional and two-dimensional forward models for RTM. Based on the analytical solution for the one-dimensional case, we prove existence of the sequence of posteriors that arise from a sequential Bayesian formulation within the infinite-dimensional framework. For the numerical characterisation of the Bayesian posteriors in the one-dimensional case, we investigate the application of a fully-Bayesian Sequential Monte Carlo method (SMC) for high-dimensional inverse problems. By means of SMC we construct a benchmark against which we compare performance of a novel regularizing ensemble Kalman algorithm (REnKA) that we propose to approximate the posteriors in a computationally efficient manner under practical scenarios. We investigate the robustness of the proposed REnKA with respect to tuneable parameters and computational cost. We demonstrate advantages of REnKA compared with SMC with a small number of particles. We further investigate, in both the one-dimensional and two-dimensional settings, practical aspects of REnKA relevant to RTM, which include the effect of pressure sensors configuration and the observational noise level in the uncertainty in the log-permeability quantified via the sequence of Bayesian posteriors.It has been extensively recognized [11,12,28,27,34,38] that imperfections in 73 during its fabrication and packing in the molding cavity can lead to variability in fib 74 results in a heterogenous highly-uncertain preform permeability. In turn, these unkn 75 in permeability of the preform give rise to inhomogeneous resin flow patterns which 76 detrimental e↵ect on the quality of the produced part, reducing its mechanical prope 77 leading to scrap. To limit these undesirable e↵ects arising due to uncertainties, 78 are used which lead to heavier, thicker and, consequently, more expensive material 79 performance being compromised. Clearly, the uncertainty quantification of material p 80 µ J n−1 (u) ≡ 1 J J j=1

Transform-based particle filtering for elliptic Bayesian inverse problems

“…A natural approximation that arises from the adaptive SMC framework described in subsection 3.1 involves ensemble Kalman inversion (EKI) [8]. More specifically, let us assume that at the n − 1 iteration level, we approximate µ n−1 with a Gaussian µ n−1 = N(m n−1 , C n−1 ) where the mean m n−1 and covariance C n−1 are the empirial mean and covariance of the particles (assumed with equal weights) at the current iteration level.…”

Quantifying uncertainty in thermophysical properties of walls by means of Bayesian inversion

“…The proposed Bayesian approach is embedded in a computational algorithm that uses an ensemble Kalman methodology [26] to merge in-situ measurements with computer simulations of heat fluxes, and generate an ensemble of realisations of the thermal properties that approximate the posterior distribution in a sequential fashion. The Kalman-based methodology at the core of the proposed algorithm is derived from a Sequential Monte Carlo (SMC) approach which, in contrast to the standard all-at-once existing Bayesian approaches [16,17], enables us to update our probabilistic knowledge of the thermal properties as new in-situ measurements are collected.…”

“…In Appendix B we dicuss how we adapt REnKA to the present application. The algorithm (see Algorithm 3), can be used in a black-box fashion; for further details of this numerical scheme the reader is referred to [26].…”

Ensemble-marginalized Kalman filter for linear time-dependent PDEs with noisy boundary conditions: application to heat transfer in building walls

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