Proper Generalized Decomposition model reduction in the Bayesian framework for solving inverse heat transfer problems

In this paper, the proper generalized decomposition (PGD) is used for model reduction in the solution of an inverse heat conduction problem within the Bayesian framework. Two PGD reduced order models are proposed and the approximation Error model (AEM) is applied to account for the errors between the complete and the reduced models. For the first PGD model, the direct problem solution is computed considering a separate representation of each coordinate of the problem during the process of solving the inverse problem. On the other hand, the second PGD model is based on a generalized solution integrating the unknown parameter as one of the coordinates of the decomposition. For the second PGD model, the reduced solution of the direct problem is computed before the inverse problem within the parameter space provided by the prior information about the parameters, which is required to be proper. These two reduced models are evaluated in terms of accuracy and reduction of the computational time on a transient three-dimensional two region inverse heat transfer problem. In fact, both reduced models result on substantial reduction of the computational time required for the solution of the inverse problem, and provide accurate estimates for the unknown parameter due to the application of the approximation error model approach.     

Grain growth prediction based on data assimilation by implementing 4DVar on multi-phase-field model

We propose a method to predict grain growth based on data assimilation by using a four-dimensional variational method (4DVar). When implemented on a multi-phase-field model, the proposed method allows us to calculate the predicted grain structures and uncertainties in them that depend on the quality and quantity of the observational data. We confirm through numerical tests involving synthetic data that the proposed method correctly reproduces the true phase-field assumed in advance. Furthermore, it successfully quantifies uncertainties in the predicted grain structures, where such uncertainty quantifications provide valuable information to optimize the experimental design.     

A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition

In this paper, a non‐intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine‐scale analysis. To validate the developed reduced‐order model, the method is implemented to: (1) the stochastic steady‐state heat diffusion in a square slab; (2) the incompressible, two‐dimensional laminar boundary‐layer over a flat plate with uncertainties in free‐stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi‐random sequence is used to generate the sample points. The numerical results of the three test cases show that the non‐intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non‐intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.     

Prediction of apparent properties with uncertain material parameters using high‐order fictitious domain methods and PGD model reduction

This contribution presents a numerical strategy to evaluate the effective properties of image‐based microstructures in the case of random material properties. The method relies on three points: (1) a high‐order fictitious domain method; (2) an accurate spectral stochastic model; and (3) an efficient model‐reduction method based on the proper generalized decomposition in order to decrease the computational cost introduced by the stochastic model. A feedback procedure is proposed for an automatic estimation of the random effective properties with a given confidence. Numerical verifications highlight the convergence properties of the method for both deterministic and stochastic models. The method is finally applied to a real 3D bone microstructure where the empirical probability density function of the effective behaviour could be obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Uncertainty quantification of a three-dimensional in-stent restenosis model with surrogate modelling

In-stent restenosis is a recurrence of coronary artery narrowing due to vascular injury caused by balloon dilation and stent placement. It may lead to the relapse of angina symptoms or to an acute coronary syndrome. An uncertainty quantification of a model for in-stent restenosis with four uncertain parameters (endothelium regeneration time, the threshold strain for smooth muscle cell bond breaking, blood flow velocity and the percentage of fenestration in the internal elastic lamina) is presented. Two quantities of interest were studied, namely the average cross-sectional area and the maximum relative area loss in a vessel. Owing to the high computational cost required for uncertainty quantification, a surrogate model, based on Gaussian process regression with proper orthogonal decomposition, was developed and subsequently used for model response evaluation in the uncertainty quantification. A detailed analysis of the uncertainty propagation is presented. Around 11% and 16% uncertainty is observed on the two quantities of interest, respectively, and the uncertainty estimates show that a higher fenestration mainly determines the uncertainty in the neointimal growth at the initial stage of the process. The uncertainties in blood flow velocity and endothelium regeneration time mainly determine the uncertainty in the quantities of interest at the later, clinically relevant stages of the restenosis process.     

Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency

A rigorous computational framework for the dimensional reduction of discrete, high‐fidelity, nonlinear, finite element structural dynamics models is presented. It is based on the pre‐computation of solution snapshots, their compression into a reduced‐order basis, and the Galerkin projection of the given discrete high‐dimensional model onto this basis. To this effect, this framework distinguishes between vector‐valued displacements and manifold‐valued finite rotations. To minimize computational complexity, it also differentiates between the cases of constant and configuration‐dependent mass matrices. Like most projection‐based nonlinear model reduction methods, however, its computational efficiency hinges not only on the ability of the constructed reduced‐order basis to capture the dominant features of the solution of interest but also on the ability of this framework to compute fast and accurate approximations of the projection onto a subspace of tangent matrices and/or force vectors. The computation of the latter approximations is often referred to in the literature as hyper reduction. Hence, this paper also presents the energy‐conserving sampling and weighting (ECSW) hyper reduction method for discrete (or semi‐discrete), nonlinear, finite element structural dynamics models. Based on mesh sampling and the principle of virtual work, ECSW is natural for finite element computations and preserves an important energetic aspect of the high‐dimensional finite element model to be reduced. Equipped with this hyper reduction procedure, the aforementioned Galerkin projection framework is first demonstrated for several academic but challenging problems. Then, its potential for the effective solution of real problems is highlighted with the realistic simulation of the transient response of a vehicle to an underbody blast event. For this problem, the proposed nonlinear model reduction framework reduces the CPU time required by a typical high‐dimensional model by up to four orders of magnitude while maintaining a good level of accuracy. Copyright © 2014 John Wiley & Sons, Ltd.