Abstract
Concurrent multiscale finite element analysis (FE2) is a powerful approach for high-fidelity modeling of materials for which a suitable macroscopic constitutive model is not available. However, the extreme computational effort associated with computing a nested micromodel at every macroscopic integration point makes FE2 prohibitive for most practical applications. Constructing surrogate models able to efficiently compute the microscopic constitutive response is therefore a promising approach in enabling concurrent multiscale modeling. This work presents a reduction framework for adaptively constructing surrogate models for FE2 based on statistical learning. The nested micromodels are replaced by a machine learning surrogate model based on Gaussian Processes (GP). The need for offline data collection is bypassed by training the GP models online based on data coming from a small set of fully-solved anchor micromodels that undergo the same strain history as their associated macroscopic integration points. The Bayesian formalism inherent to GP models provides a natural tool for online uncertainty estimation through which new observations or inclusion of new anchor micromodels are triggered. The surrogate constitutive manifold is constructed with as few micromechanical evaluations as possible by enhancing the GP models with gradient information and the solution scheme is made robust through a greedy data selection approach embedded within the conventional finite element solution loop for nonlinear analysis. The sensitivity to model parameters is studied with a tapered bar example with plasticity and the framework is further demonstrated with the elastoplastic analysis of a plate with multiple cutouts and with a crack growth example for mixed-mode bending. Although not able to handle non-monotonic strain paths in its current form, the framework is found to be a promising approach in reducing the computational cost of FE2, with significant efficiency gains being obtained without resorting to offline training.
Highlights
There is a growing demand for high-fidelity numerical techniques capable of describing material behavior across spatial scales
We introduce the Bayesian regression approach used to construct surrogate constitutive models, beginning from parametric versions of the surrogate model S — i.e. by encapsulating the constitutive information in D into a set of parameters w — and eventually moving to a non-parametric model based on Gaussian Processes (GP) that uses the data in D directly in order to make predictions
This can be circumvented by either allowing φ to change in shape during training or by explicitly choosing a kernel k xp, xq and adopting a distribution over functions instead of over weights, as in Gaussian Process regression models
Summary
There is a growing demand for high-fidelity numerical techniques capable of describing material behavior across spatial scales. Resulting in models of distinct natures, both approaches rely on the existence of an observation database on the behavior of the original micromodel that is usually obtained offline (before deployment on a multiscale setting) and should cover every possible scenario the surrogate is expected to approximate online. Building such a database of model snapshots can be a challenging task (see [20,22] for interesting approaches based on Design of Experiments and [14] for a data-driven training framework based on Bayesian Optimization). Depending on how complex the microscopic behavior is, M can range from having a relatively simple form (e.g. linear elasticity) to being next to impossible to formulate explicitly
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