Continuum Mechanics, Numerical Methods, Data-driven Approach: Next Steps

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Short-term goals

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I. Extension to 3D simulations

 

Both the numerical improvements and the multi-scale modeling above were developed in 2D, and we can extend them to 3D.

 

BMGs have richer and more complicated plastic deformation behaviors in 3D. For example, the formation of shear bands in different directions and at different angles inside the material can interact with each other. However, they are poorly understood and difficult to analyze in dimensions greater than one. Boffi et al. [1] extended the original projection method to a 3D implementation, and investigated fine-scale and uniquely three-dimensional features of shear-banding in BMGs.

 

Extending our numerical improvements to 3D will help us obtain highly accurate solutions more efficiently and in fewer timesteps, which is highly beneficial. This is because the computation is very expensive for 3D large-scale and high-resolution simulations, and these simulations are necessary for researchers to study fine-scale shear-banding phenomenon in the material.

 

Extending the multi-scale modeling methodology to 3D would also be highly beneficial. The original 2D model we developed is a simplified model. In the real world, engineering materials are all three-dimensional. Therefore, it would be very impactful to be able to model realistic and stochastic plastic deformation in 3D. Furthermore, our methods are not restricted to BMGs. Their frameworks and procedures can be applied to other elastoplastic materials. Therefore, in the future, we can study plastic deformation for a wide range of materials using the methods developed.

 

II. Arbitrarily high-order temporal accuracy formulation of the quasi-static elastoplasicity projection method

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I would also like to further improve the numerical methods. Specifically, I want to derive an arbitrarily high-order numerical formulation of the elastoplastic projection method.

 

In a recent work, Minion et al. [2] formulated the fluid’s projection method with spectral deferred correction, and developed a numerical scheme that can reach arbitrarily high order of temporal accuracy for the incompressible Navier-Stokes equations. In essence, it performs several iterations of each timestepping, and use substepping of a timestep in each iteration. After each substep in time, it corrects the pressure term via a projection.

 

Drawing inspiration from the method, I would like to formulate a spectral deferred correction version of the elastoplastic projection method. By iteratively correcting the velocity term in the elastoplastic problem in each timestep, we can achieve high-order numerical accuracy.

Long term goals

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I. Apply our methods to other elastoplastic materials

I would like to further explore data-driven methods and multi-scale modeling. Specifically, the numerical methods and the multi-scale modeling methodology developed in my previous work is not restricted to BMGs. The framework and procedure are appropriate for other elastoplastic materials. Therefore, I would like to apply the method in future research to a broad range of elastoplastic materials.

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II. Fluid dynamics, solid deformation, fluid-solid interaction in multi-material multi-physics settings

Furthermore, I am extremely interested in simulating the rich dynamics of fluid flows, and fluid-solid interaction in multi-material multi-physics settings. In all of these simulations, I can incorporate data-driven models and strive to make the simulations physically accurate, reproducing all the intricate statistical phenomenon happening in nature for these systems.