Abstract
We study a Bayesian inverse problem arising in the context of resin transfer molding (RTM), which is a process commonly used for the manufacturing of fiber-reinforced composite materials. The forward model is described by a moving boundary problem in a porous medium. During the injection of resin in RTM, our aim is to update, on the fly, our probabilistic knowledge of the permeability of the material as soon as pressure measurements and observations of the resin moving domain become available. A probabilistic on-the-fly characterisation of the material permeability via the inversion of those measurements/observations is crucial for optimal real-time control aimed at minimising both process duration and the risk of defects formation within RTM. We consider both one-dimensional (1D) and two-dimensional (2D) forward models for RTM. Based on the analytical solution for the 1D 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 1D 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 1D and 2D 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. The results of this work are also useful for other applications than RTM, which can be modelled by a random moving boundary problem.
Highlights
In this paper we study the Bayesian inverse problem within the moving boundary setting motivated by applications in manufacturing of fiber-reinforced composite materials
In this paper we propose the application of the Bayesian approach to inverse problems [46] in order to infer the logarithm of the permeability u(x) = log κ(x), from observations {yn}Nn=1 collected at some prescribed measurement/observation times {tn}Nn=1 during the resin injection in Resin Transfer Molding (RTM)
In this work we studied the Bayesian inverse problem that arises from inferring physical properties in a setting for porous media flow with a moving boundary
Summary
In this paper we study the Bayesian inverse problem within the moving boundary setting motivated by applications in manufacturing of fiber-reinforced composite materials. The very extensive review published in 2010 [41] reveals that most conventional methods for measuring permeability assume that (i) the material permeability tensor is homogenous and (ii) the flow is one-dimensional (including 2D radial flow configurations) Under these assumptions the resin injection in RTM can be described analytically, via expressions derived from Darcy’s law, which enable a direct computation of the permeability in terms of quantities that can be measured before or during resin injection. They do not account for the heterogenous structure of the preform permeability, and they provide an estimate of an effective permeability, this does not enable the prediction of the potential formation of voids and dry spots Those conventional methods compute the permeability in an off-line fashion (i.e before RTM) with specific mold designs that satisfy the aforementioned assumptions intrinsic to those methods (e.g. rectangular flat molds). It is clear from our literature review that the estimation of permeability of preform during resin injection deserves substantial attention from an inverse problems perspective capable of quantifying uncertainty inherent to the fabrication and packing of the preform
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