Multi-fidelity uncertainty quantification for homogenization problems in structure-property relationships from crystal plasticity finite elements

Crystal plasticity finite element method (CPFEM) has been an integrated computational materials engineering (ICME) workhorse to study materials behaviors and structure-property relationships for the last few decades. These relations are mappings from the microstructure space to the materials properties space. Due to the stochastic and random nature of microstructures, there is always some uncertainty associated with materials properties, for example, in homogenized stress-strain curves. For critical applications with strong reliability needs, it is often desirable to quantify the microstructure-induced uncertainty in the context of structure-property relationships. However, this uncertainty quantification (UQ) problem often incurs a large computational cost because many statistically equivalent representative volume elements (SERVEs) are needed. In this paper, we apply a multi-level Monte Carlo (MLMC) method to CPFEM to study the uncertainty in stress-strain curves, given an ensemble of SERVEs at multiple mesh resolutions. By using the information at coarse meshes, we show that it is possible to approximate the response at fine meshes with a much reduced computational cost. We focus on problems where the model output is multi-dimensional, which requires us to track multiple quantities of interest (QoIs) at the same time. Our numerical results show that MLMC can accelerate UQ tasks around 2.23x, compared to the classical Monte Carlo (MC) method, which is widely known as the ensemble average in the CPFEM literature

A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger

We present an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with coefficients that are random fields. A classical technique for sampling from these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm is based on the adaptive algorithm used in sparse grid cubature as introduced by Gerstner and Griebel (2003), and automatically chooses the number of terms needed in this expansion, as well as the required spatial discretizations of the PDE model. We apply the method to a simplified model of a heat exchanger with random insulator material, where the stochastic characteristics are modeled as a lognormal random field, and we show consistent computational savings

An overview of p-refined Multilevel quasi-Monte Carlo Applied to the Geotechnical Slope Stability Problem

[EN] Problems in civil engineering are often characterized by significant uncertainty in their material parameters. Sampling methods are a straightforward manner to account for this uncertainty, which is typically modeled as a random field. A popular sampling method consists of the classic Multilevel Monte Carlo method (h-MLMC). Its most distinctive feature consists of a hierarchy of h-refined meshes, where most of the samples are taken on coarse and computationally inexpensive meshes, and few are taken on finer but computationally expensive meshes. We present an improvement upon the classic Multilevel Monte Carlo, called the prefined Multilevel quasi-Monte Carlo method (p-MLQMC). Its key features consist of a mesh hierarchy constructed from a p-refinement scheme combined with a deterministic set of samples points (quasi-Monte Carlo points). In this work we show how the uncertainty needs to be accounted for and present results comparing the total computational cost of the h-ML(Q)MC and p-MLQMC method. Specifically, we present two novel approaches in order to account for the uncertainty in case of p-MLQMC. We benchmarking the different multilevel methods on a slope stability problem, and find that p-MLQMC outperforms h-MLMC up to several orders of magnitude.The authors gratefully acknowledge the support from the Research Council of KU Leuven through project C16/17/008 “Efficient methods for large-scale PDE-constrained optimization in the presence of uncertainty and complex technological constraints”. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government – department EWI.Blondeel, P.; Robbe, P.; François, S.; Lombaert, G.; Vandewalle, S. (2022). An overview of p-refined Multilevel quasi-Monte Carlo Applied to the Geotechnical Slope Stability Problem. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 25-35. https://doi.org/10.4995/YIC2021.2021.12236OCS253

Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial

This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an in-depth survey of a similar title by the same authors, with an accompanying software package which is also briefly discussed here. Embedded in this article is a step-by-step tutorial of the required analysis for the setting known as the uniform case with first order QMC rules. The aim of this article is to provide an easy entry point for QMC experts wanting to start research in this direction and for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods.Comment: arXiv admin note: text overlap with arXiv:1606.0661