/pdf/interfacing-finite-elements-with-deep-neural-operators-for-1p06lobc.pdf

Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems

This Note deals with approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of “periodization” and other ”cut-off” procedures. For instance in the case of periodic approximation, we consider a cubic sample (0, ) of the random medium, extend it periodically in and use the effective coefficients of the obtained periodic operators as an approximation of the effective coefficients of the original random operator. It is shown that these approximations converge a.s. as → ∞ and give back the effective coefficients of the original random operator. Moreover, under additional mixing conditions on the coefficients, the rate of convergence can be estimated by some negative power of which only depends on the dimension, the ellipticity constant and the rate of decay of the mixing coefficients. Similar results are established for approximations in terms of appropriate Dirichlet and Neumann problems localized in a cubic sample (0, ).

https://papers.ssrn.com/sol3/Delivery.cfm/S1574035804700489.pdf?abstractid=3133595&mirid=1

Approximations of Effective Coefficients in Stochastic Homogenization

http://manuscript.elsevier.com/S0045782522003450/pdf/S0045782522003450.pdf

Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN)

A robust solution of a statistical inverse problem in multiscale computational mechanics using an artificial neural network

Experimental solid mechanics is at a pivotal point where machine learning (ML) approaches are rapidly proliferating into the discovery process due to significant advances in data storage and processing capabilities. Much of the ML that is being adopted by the mechanics community was initially developed for application outside of science and engineering, and has the potential to produce results of questionable physical validity. To ensure that these data-driven approaches are trustworthy, there is a clear need to embed physical principles into their architectures, to evaluate and compare ML frameworks against benchmark datasets, and to test their broader extensibility. Frameworks must be grounded in a clear objective, quantifiable error, and a well-defined scope of extensibility. These principles enable ML models with a wide range of architectures to be meaningfully categorized, compared, evaluated, and extended to broader experimental and computational frameworks. Application of these principles are demonstrated through an investigation of ML models in two different use cases, acoustic emission and resonant ultrasound spectroscopy, along with a discussion of outlooks for the future of trustworthy ML in experimental mechanics. 

Perspective: Machine learning in experimental solid mechanics

The aim of this work is to efficiently and robustly solve the statistical inverse problem related to the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneous anisotropic materials with a complex microstructure that usually cannot be properly described in terms of their mechanical constituents at microscale. Within the context of linear elasticity theory, the apparent elasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field. A general methodology using multiscale displacement field measurements simultaneously made at both macroscale and mesoscale has been recently proposed for the identification the hyperparameters of such a prior stochastic model by solving a multiscale statistical inverse problem using a stochastic computational model and some information from displacement fields at both macroscale and mesoscale. This paper contributes to the improvement of the computational efficiency, accuracy and robustness of such a method by introducing (i) a mesoscopic numerical indicator related to the spatial correlation length(s) of kinematic fields, allowing the time-consuming global optimization algorithm (genetic algorithm) used in a previous work to be replaced with a more efficient algorithm and (ii) an ad hoc stochastic representation of the hyperparameters involved in the prior stochastic model in order to enhance both the robustness and the precision of the statistical inverse identification method. Finally, the proposed improved method is first validated on in silico materials within the framework of 2D plane stress and 3D linear elasticity (using multiscale simulated data obtained through numerical computations) and then exemplified on a real heterogeneous biological material (beef cortical bone) within the framework of 2D plane stress linear elasticity (using multiscale experimental data obtained through mechanical testing monitored by digital image correlation).

https://hal-upec-upem.archives-ouvertes.fr/hal-02879369/document

Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements.

The aim of this work is to efficiently and robustly solve the statistical inverse problem related to the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneous anisotropic materials with a complex microstructure that usually cannot be properly described in terms of their mechanical constituents at microscale Within the context of linear elasticity theory, the apparent elasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field A general methodology using multiscale displacement field measurements simultaneously made at both macroscale and mesoscale has been recently proposed for the identification the hyperparameters of such a prior stochastic model by solving a multiscale statistical inverse problem using a stochastic computational model and some information from displacement fields at both macroscale and mesoscale This paper contributes to the improvement of the computational efficiency, accuracy and robustness of such a method by introducing (i) a mesoscopic numerical indicator related to the spatial correlation length(s) of kinematic fields, allowing the time-consuming global optimization algorithm (genetic algorithm) used in a previous work to be replaced with a more efficient algorithm and (ii) an ad hoc stochastic representation of the hyperparameters involved in the prior stochastic model in order to enhance both the robustness and the precision of the statistical inverse identification method Finally, the proposed improved method is first validated on in silico materials within the framework of 2D plane stress and 3D linear elasticity (using multiscale simulated data obtained through numerical computations) and then exemplified on a real heterogeneous biological material (beef cortical bone) within the framework of 2D plane stress linear elasticity (using multiscale experimental data obtained through mechanical testing monitored by digital image correlation)

Tianyu Zhang

Papers

A robust solution of a statistical inverse problem in multiscale computational mechanics using an artificial neural network

Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements.

Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements.