Congratulations again to Nick! 

We (first author = Bahador Bahmani) introduce an isometric manifold embedding data-driven paradigm designed to enable model-free simulations with noisy data sampled from a constitutive manifold. The proposed data-driven approach iterates between a global optimization problem that seeks admissible solutions for the balance principle and a local optimization problem that finds the closest point projection of the Euclidean space that isometrically embeds a nonlinear constitutive manifold.

​To de-noise the database, a geometric autoencoder is introduced such that the encoder first learns to create an approximated embedding that maps the underlying low-dimensional structure of the high-dimensional constitutive manifold onto a flattened manifold. Unlike conventional auto-encoder where the dimension is reduced by reducing the number of neurons in the downstream layers, we reduce the dimension in a geometric sense through the flattening process. We then obtain the noise-free constitutive responses by projecting data onto a de-noised latent space that is completely flat by assuming that the noise and the underlying constitutive signal are orthogonal to each other, leveraging the conformal mapping between the de-noised latent space and the reconstructed de-manifold. 

Consequently, a projection from the conservative manifold onto this de-noised constitutive latent space enables us to complete the local optimization step of the data-driven paradigm. Our results show that the isometry constraint may prevent the decoder reintroducing noise. Numerical examples are used to both validate the implementation and demonstrate the accuracy, robustness, and limitations of the proposed paradigm.

Preprint available [URL]. 
​

​

Authors: Ran Ma, WaiChing Sun, Catalin R. Picu,  Tommy Sewell
Preprint: [URL]
Abstract: Heterogeneous energetic materials (EMs) subjected to mechanical shock loading exhibit complex thermo-mechanical processes which are driven by the high temperature, pressure, and strain rate behind the shock. These lead to spatial energy localization in the microstructure, colloquially known as ``hotspots'',
where chemistry may commence possibly culminating in detonation. Shock-induced pore collapse is one of the dominant mechanisms by which localization occurs. In order to physically predict the shock sensitivity of energetic materials under these extreme conditions, we formulate a multiplicative crystal plasticity model with key features inferred from molecular dynamics (MD) simulations. Within the framework of thermodynamics, we incorporate the pressure dependence of both monoclinic elasticity and critical resolved shear stress into the crystal plasticity formulation. Other fundamental mechanisms, such as strain hardening and pressure-dependent melting curves, are all inferred from atomic-scale computations performed across relevant intervals of pressure and temperature. To handle the extremely large deformation and the evolving geometry of the self-contact due to pore collapse, we leverage the capabilities of the Material Point Method (MPM) to track the interface via the Lagrangian motion of material points and the Eulerian residual update to avoid the mesh distortion issue. This combination of features enables us to simulate the shock-induced pore collapse and associated hotspot evolution with a more comprehensive physical underpinning, which we apply to the monoclinic crystal beta-HMX. Treating MD predictions of the pore collapse as ground truth, head-to-head validation comparisons between MD and MPM predictions are made for samples with identical sample geometry and similar boundary conditions, for reverse-ballistic impact speeds ranging from 0.5 to 2km per second. Comparative studies are performed to reveal the importance of incorporating a frictional contact algorithm, pressure-dependent elastic stiffness, and non-Schmid type critical resolved shear stress in the mesoscale model.

Case 1: impact velocity = 320 meter per second 

Generally speaking, reduced order modeling and constitutive modeling are considered different disciplines. In Part II of our research on geometric learning for computational Mechanics (see Part I here), we attempt to conduct these two very different ideas by using a graph isomorphism network to learn the low-dimensional representation of finite element solutions of microstructures.

​Then, instead of reconstructing the low-dimensional dynamics directly via a black-box approach, we use the macroscopic plasticity theory to create additional constraints (e.g., yield function consistency, plastic flow direction) such that the low-dimensional dynamics can be compatible with the macroscopic observations. ​

On the other hand, the autoencoder of the graph isomorphism network also gives us a chance to recast the element of the latent space as the internal variables of the plasticity model and therefore gives more direct and interpretable relations between microstructural deformation patterns and macroscopic plasticity. ​

Our results also indicate that the geometric learning approach may enable us to more explicitly understand the geometry of the data manifested by the shape of the yield function as a function of both stress and elements of latent space of plastic deformation of RVE. This understanding, in return, gives us the opportunity to recast the reduced order modeling problem that can be updated via a classical constitutive law that is now with internal variables that can b be encoded as a snapshot of finite element mesh. The resultant models is found to perform robustly in forward predictions with loading paths different from the training data, in thew sense that there is no spurious patterns that might exhibit in RNN or 1D convolutional NN (see below). 

Preprint posted at ResearchGate here. Part III of this series is under progress and will submit soon. 

In this work, we first use MD data to train a deep neural network to create the "first draft" of the continuum surrogate model, then use well known physics constraints from continuum mechanics (e.g. material frame indifference, material symmetry, growth condition, rank-one convexity) to further fine-tune the neural network. The resultant framework is used to predict the finite strain elasticity of Nitroamine high explosive (HMX), an application that requires robustness and consistency on the forecast quality. 

Link to research article:
https://onlinelibrary.wiley.com/doi/10.1002/nme.6992
​
Link to the cover:
https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.7087

When predicting plastic responses of complex microstructures, we often propose mechanisms to explain the physics of the yielding, then propose mathematical expressions to recapture what we describe in words, then propose algorithms to generate the constitutive updates. However, what if the materials are so complex that we cannot easily find a single equation to express them precisely? What if our symbolic regression skill is not sufficient to recover the surface to which those data points belong? The yield function or damage criterion of a material is a common example where our abilities to compose equations precisely and accurately are often put to test. A yield function may take many different types of variables (stress invariants, strain, sometimes also other descriptors such as volume fraction).  A simple solution we proposed is to not propose the yield surface as a function in the parametric space but directly regard it as a manifold.

In this work (first author = Mian Xiao), Mian and I explore the use of geometric prior to generating the yielding manifold based on point cloud data obtained from direct numerical simulations or experiments. By modifying the geometric approach by Williams, et al. CVPR 2019 to incorporate plastic flow information to regularize the yield surface, we have successfully recovered a highly complex yield surface through the construction of a collection of coordinate charts and the atlas, a task that is difficult to complete via training a single neural network. Meanwhile, we also show that the availability of local patches also enables us to overcome the longstanding slow convergence issue commonly exhibited in classical non-smooth plasticity models and leads to a very robust reconstruction of yield surface even with noisy data. Preprint available via ResearchGate. [PDF]

Link to open-access article: https://onlinelibrary.wiley.com/doi/full/10.1002/nag.3380
Link to the cover: https://onlinelibrary.wiley.com/doi/10.1002/nag.3429

​This article (see [preprint]) presents a multi-phase-field poromechanics model that simulates the growth and thaw of ice lenses and the resultant frozen heave and thaw settlement in multi-constituent frozen soils. In this model, the growth of segregated ice inside the freezing-induced fracture is implicitly represented by the evolution of two phase fields that indicate the locations of segregated ice and the damaged zone, respectively. The evolution of two phase fields is induced by their own driving forces that capture the physical mechanisms of ice and crack growths respectively, while the phase field governing equations are coupled with the balance laws such that the coupling among heat transfer, solid deformation, fluid diffusion, crack growth, and phase transition can be observed numerically. Unlike phenomenological approaches that indirectly capture the freezing influence on the shear strength, the multi-phase-field model introduces an immersed approach where both the homogeneous freezing and the ice lens growth are distinctively captured by the freezing characteristic function and the driving force accordingly. Verification and validation examples are provided to demonstrate the capacities of the proposed models. Support provided by US Army Research Office and National Science Foundation is gratefully acknowledged. 

​The first author of this paper,  the 6th graduated PhD of our group, Dr. Hyoung Suk Suh (see picture below) has received the DongJu Lee '03 Memorial Award in recognition of "his superior achievement and in honor of the integrity, curiosity, and creativity exhibited as a student at SEAS of Columbia." Congratulations, Hyoung Suk! I am looking forward for more great news in the upcoming year! 

This paper (see the preprint available here) introduces a manifold embedding data-driven paradigm to solve small- and finite-strain elasticity problems without a conventional constitutive law. ​

​

Traditionally, data-driven paradigm often replace a constitutive law with a search that select an experimental data point that is " the closest" to the balance principle. However, how distance or length is measured remains ambiguous. Often time, an arbitrarily energy norm is chosen, but such a practice has shown to be affected which closest data point is getting selected. In particular, in a nonlinear manifold, the shortest distance between two points and the Euclidean distance can be quite different when they are far apart. Furthermore, the resultant model-free simulation could be sensitive to the chosen norm used to measure distance (see figure below). ​

Authors: 
Qing Yin, Edward Andò, Gioacchino Viggiani, WaiChing Sun

Abstract:
This paper presents a combined experimental-modeling effort to interpret the coupled thermo-hydro-mechanical behaviors of the freezing soil, where an unconfined, fully saturated clay is frozen due to a temperature gradient. By leveraging the rich experimental data from the microCT images and the measurements taken during the freezing process, we examine not only how the growth of ice induces volumetric changes of the soil in the fully saturated specimen but also how the presence and propagation of the freezing fringe front may evolve the anisotropy of the effective media of the soil-ice mixture that cannot be otherwise captured phenomenologically in the isotropic saturation-dependent critical state models for plasticity. The resultant model is not only helpful for providing a qualitative description of how freezing
affects the volumetric responses of the clayey material, but also provide a mean to generate more precise predictions for the heaving due to the freezing of the ground. [PDF]

Congratulations to Dr. Hyoung Suk Suh for successfully defending his PhD dissertation (see URL). We are very grateful to other committee members, Professor Ronaldo Borja, Professor George Deodatis, Professor Majid Manzari and Professor Haim Waisman for their suggestions and feedbacks, and the Army Research Office and National Science Foundation for providing the financial support for his PhD study. 

Hyoung Suk joined  my research group in 2018 from Yonsei University. He has published the following works during his tenure at Columbia. His PhD work focuses on  the microporomechanics of geomaterials at extreme temperature with implications on how climate changes may affect the freeze-thaw action in frozen soil in Alaska.  In addition to his research accomplishment, Hyoung Suk is also a beloved TA and had also been nominated as the candidate for the Presidential awards for outstanding teaching at Columbia twice. 

Dr. Suh, thank you for the 4 years of hard work and dedications! It is a privilege to serve as your PhD advisor and colleague! 

​Published work:

1. H.S. Suh, W.C. Sun, An open source FEniCS implementation of a phase field fracture model for micropolar continua, International Journal of Multiscale Computational Engineering, doi:10.1615/IntJMultCompEng.2020033422, 2019. [open source code]
2. H.S. Suh, D. O'Conner, W.C. Sun, A phase field model for cohesive fracture in micropolar continua, Computer Methods in Applied Mechanics and Engineering, accepted, 2020. 
3. H.S. Suh, W.C. Sun, An immersed phase field model for microporomechanics of fracture-induced leakage, Physics of Fluids (Editor's pick), doi:10/1063/5.0035602, 2021. [PDF]
4. Y. Heider", H.S. Suh, W.C. Sun, An offline multi-scale unsaturated poromechanics model enabled by self-designed/self-improved neural network, International Journal for Numerical and Analytical Methods in Geomechanics, doi:10.1002/nag.3196, 2021. 
5. H. Suh, W.C. Sun, Asynchronous phase field fracture model for porous media with thermally non-equilibrated constituents, Computer Methods in Applied Mechanics and Engineering, doi:10.1016/j.cma.2021.114182, 2021.
6. H. Suh, W.C. Sun, Multi-phase-field microporomechanics model for simulating ice lens growth and thaw in frozen soil, under review. 
7. H.S. Suh, W.C. Sun, An immersed phase field fracture model in fluid-infiltrating porous media with evolving Beavers-Joseph-Saffman condition, 2nd International Conference on Energy Geotechnics, La Jolla, California, USA, 2021. 

After 2 years of delay due to pandemic, we are back to the roads now for conferences. Below are two talks, one on phase field modeling of ice lens, one on immersed phase field for fluid-driven fracture with Darcy-Stokes flow. 

Congratulations to Dr. Kun Wang who has been selected from around 200 applicants by ExxonMobil! He will join  Computational Physics Section at EMRE’s Corporate Strategic Research Laboratories as a computational physicist to develop computational methods aimed at solving large-scale physical problems pertaining to the energy industry, with focus in the areas of flow in porous media, multi-scale phenomena, PDE-constrained optimization and uncertainty quantification.

Kun has a distinguished career at both Columbia University and Los Alamos National Laboratory. His recent work published in Nature Communication and PNAS has been featured at Economist. During his time at Columbia, he has published 12 papers with the research group, as listed below. 
​
Published Work:

1. K. Wang, W.C. Sun, A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain, Computer Methods in Applied Mechanics and Engineering,  doi:10.1016/j.cma.2016.02.020, 2016. [DRAFT]
2. K. Wang, W.C. Sun, Anisotropy of a tensorial Bishop's coefficient for wetted granular materials, Journal of Engineering Mechanics, doi:10.1061/(ASCE)EM.1943-7889.0001005, 2015. [DRAFT] [Bibtex]
3. K. Wang, W.C. Sun, S. Salager, S. Na, G. Khaddour, Identifying material parameters for a micro-polar plasticity model via X-ray micro-CT images: lessons learned from the curve-fitting exercises, accepted, International Journal of Multiscale Computational Engineering, 2016. [DRAFT]
4. K. Wang, W.C. Sun, A unified variational eigen-erosion framework for interacting fractures and compaction bands in brittle porous media, doi:10.1016/j.cma.2017.01.017, Computer Methods in Applied Mechanics and Engineering, 2017. [DRAFT]
5. K. Wang, W.C. Sun,   A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning​, Computer Methods in Applied Mechanics and Engineering, 334(1):337-380, doi:10.1016/j.cma.2018.01.036, ​2018. 
6. R. Gupta, S. Salager, K. Wang, W.C. Sun, Open-source support toward validating and falsifying discrete mechanics models using synthetic granular materials Part I: Experimental tests with particles manufactured by a 3D printer, Acta Geotechnica, doi:10.1007/s11440-018-0703-0, 2018. 
7. K. Wang, W.C. Sun, An updated Lagrangian LBM-DEM-FEM coupling model for dual-permeability porous media with embedded discontinuities, Computer Methods in Applied Mechanics and Engineering, 344:276-305, doi:10.1016/j.cma.2018.09.034, 2019. [PDF][Bibtex]
8. K. Wang, W.C. Sun, Meta-modeling game for deriving theory-consistent, micro-structure-based traction-separation laws via deep reinforcement learning, Computer Methods in Applied Mechanics and Engineering, accepted, 346:216-241,  doi:10.1016/j.cma.2018.11.026, 2019. [PDF][Bibtex]
9. K. Wang, W.C. Sun, Q. Du, A cooperative game for automated learning of elasto-plasticity knowledge graphs and models with AI-guided experimentation, Computational Mechanics, special issue for Data-Driven Modeling and Simulations: Theory, Methods and Applications, doi:,10.1007/s00466-019-01723-1, 2019. 
10. Y. Heider, K. Wang, W.C. Sun, SO(3)-invariance of graph-based deep neural network for anisotropic elastoplastic materials, Computer Methods in Applied Mechanics and Engineering, doi:10.1016/j.cma.2020.112875, 2019.
11. K. Wang, W.C. Sun, Q. Du, A non-cooperative meta-modeling game for automated third-party training, validating, and falsifying constitutive laws with adversarial attacks, Computer Methods in Applied Mechanics and Engineering, doi:10.1016/j.cma.2020.113514, 2020. [arxiv][Video]
12. A. Fuchs, Y. Heider, K. Wang, W.C. Sun, M. Kaliske, DNN2: A hyper-parameter reinforcement learning game for self-design1neural network elasto-plastic constitutive laws, Computer and Structures, accepted, 2021.

While there are many neural network constitutive laws published, there are relatively little work on focusing on the validation against the physical constraints to make the predictions admissible. In this work, we train the neural network with multiple steps where it first fits the data, then corrections are made through transfer learning to introduce additional physical constraints, such as material frame indifference, material symmetry, as well as checking the ellipticity, growth conditions, stability and uniqueness (through examining the acoustic tensor). Another interesting point is that we are using the gradient and the Hessian data to back calculate the underlying scalar elasticity functional. 

The neural network model is used to predict the responses of a monoclinic organic molecular crystal, beta-HMX inferred from MD simulations. Accuracy of the blind predictions are tested in the parametric space. Preprint manuscript available at  [URL].

Conventionally, neural network constitutive laws for path-dependent elasto-plastic solids are trained via supervised learning performed on recurrent neural networks, with the time history of strain as input and the stress as input. However, training neural networks to replicate path-dependent constitutive responses requires significantly more data due to the path dependence. This demand on diversity and abundance of accurate data, as well as the lack of interpretability to guide the data generation process, could become major roadblocks for engineering applications. In this work, we attempt to simplify these training processes and improve the interpretability of the trained models by breaking down the training of material models into multiple supervised machine learning programs for elasticity, initial yielding and hardening laws that can be conducted sequentially. To predict pressure-sensitivity and rate dependence of the plastic responses, we reformulate the Hamliton-Jacobi equation such that the yield function is parametrized in a product space spanned by the principal stress, the accumulated plastic strain and time. To test the versatility of the neural network meta-modeling framework, we conduct multiple numerical experiments where neural networks are trained and validated against (1) data generated from known benchmark models (2) data obtained from physical experiments and (3) data inferred from homogenizing sub-scale direct numerical simulations of microstructures. The neural network model is also incorporated into an offline FFT-FEM model to improve the efficiency of the multiscale calculations. Preprint available here [URL]

If the injected fluid is much hotter or colder than the host matrix of the porous media and the specific heat capacities of the solid and fluid constituents are sufficiently different, then the assumption that the two constituents will have the same temperature at the continuum scale may not be correct. In this work, we formulate a dual-heat-transfer theory (in analog to the dual-permeability poromehanics theory) to examine how these local temperature difference affect the fracture patterns and the path-dependent responses at the small time scale and under what condition the one temperature theory is sufficient. To address the issue of the time scale difference of the coupled heat transfer problem, we introduce an asynchronous time integrator for the operator-split algorithm to improve the efficiency of the solver. Preprint available here [URL]. 

I am excited that the thesis committee (Professor George Deodatis, Professor JS Chen, Professor Richard Regueiro, Professor Marco Giometto and myself) have approved our team member Nick Vlassis's PhD dissertation "Towards Trustworthy Geometric Deep Learning for Elastoplasticity". Nick will continue to collaborate with us on the DOE NNSA project "Center for Micromorphic Multiphysics Porous and Particulate Materials Simulations with Exascale Computing Workflows (MSC)" (by led University of Colorado Boulder) as a Postdoctoral Research Scientist .

Nick's work focuses on formulating geometric learning tasks to create meta-models that generate interpretable constitutive laws from MD, DNF, DEM and experimental data across different length scales, often with physical constraints that often involves higher-order derivatives (see list of publication below). During PhD study, Nick has been awarded the Mindlin Scholarship by the Fu Foundation School of Engineering and Applied Science and a few NSF travel fellowships to conferences. 

Congratulations for the well-deserved distinction, Nick! We are looking forward for your outstanding contribution to the DOE NNSA project! 

Publications: 

1. N. Vlassis, W.C. Sun, Sobolev training of thermodynamic-informed neural network for interpretable elasto-plasticity models with level set hardening, Computer Methods in Applied Mechanics and Engineering, doi:10.1016/j.cma.2021.113695, 2021. [Video][preprint]
2. N. Vlassis, R. Ma, W.C. Sun, Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity, Computer Methods in Applied Mechanics and Engineering, doi:10.1016/j.cma.2020.113299, 2020. [PDF]
3. X. Sun, B. Bahmani, N. Vlassis, W.C. Sun, Y. Xu, Data-driven discovery of interpretable causal relations for deep learning material laws with uncertainty quantification, Granular Matter, accepted, 2021. [arxiv]
4. C. Cai, N. Vlassis, L. Magee, R. Ma", Z. Xiong, B. Bahmani, T-F Wong, Y. Wang, W.C. Sun, Equivariant geometric learning for digital rock physics. Part I: Estimating formation factor and effective permeability tensors, under review. [arxiv]
5. N. Vlassis, W.C. Sun, Interpretable machine learning paradigm for discovering new plasticity theories, under review. ​

​

Abstract: 
We propose a material point method (MPM) to model the evolving multi-body contacts due to crack growth and fragmentation of thermo-elastic bodies.  By representing particle interface with an implicit function, we adopt the gradient partition techniques introduced by Homel and Herbold 2017 to identify the separation between a pair of distinct material surfaces. This treatment allows us to replicate the frictional heating of the evolving interfaces and predict the energy dissipation more precisely in the fragmentation process. By storing the temperature at material points, the resultant MPM model captures the thermal advection-diffusion in a Lagrangian frame during the fragmentation, which in return affects the structural heating and dissipation across the frictional interfaces. The resultant model is capable of replicating the crack growth and fragmentation without requiring dynamic adaptation of data structures or insertion of interface elements. A staggered algorithm is adopted to integrate the displacement and temperature sequentially. Numerical experiments are employed to validate the diffusion between the thermal contact, the multi-body contact interactions and demonstrate how these thermo-mechanical processes affect the path-dependent behaviors of the multi-body systems.

Preprint:
Available via ResearchGate: [URL]

Video Summary:
https://youtu.be/pg535F6bqtQ

Starting next month, the PI of Sun Group will join the editorial board of Acta Geotechnica: 
https://www.springer.com/journal/11440/editors