Structure Function Relation of Porous 2D Material via SGCMC Simulation and Statistical Models

Abstract

To improve the design for structural resilience and reduced environmental impact, we need to make the structure function relation of concrete more accurate, accessible, and cost-effective. First, we formulate and implement the Semi-Grand Canonical Monte Carlo (SGCMC) simulation for fracture mechanics, which is a stochastic method that is capable of capturing both the initiation and the propagation of fractures in a medium. We then optimize the performance of our SGCMC simulation to reduce its time complexity from O(n²·³⁸) to O(n¹·²⁴) and its space complexity from O(n²) to O(n). The key step to performance optimization is exploiting the sparsity of the stiffness matrix. We also deploy our code to run multiple simulations concurrently on a super-computing infrastructure to achieve scalability. Then, we try to achieve an even more accessible and cost-effective structure function relation by applying statistical modeling to predict the strength of a two-dimensional porous material without running the simulation. We generate samples by randomly placing circular pores with radii drawn from a log-normal distribution until we reach the target porosity and run our SGCMC simulations on the generated samples to create a data set to train our statistical models. We defined several parameters, including the two-point correlation function, the multi-scale disorder index, the distribution of pore radius as recovered by Circle Hough Transformation (CHT), and the area moments of the pores to parameterize the porous geometry of the samples beyond the porosity, which is a well-known and very important parameter. We found our best model to be a Gradient Boosting Decision Trees (GBDT) regression model, whose out-of-sample R2 is 0.904, as opposed to the baseline model of linear regression with the porosity, whose out-of-sample R2 is 0.752.