The team shows how knowledge of one randomly sampled location in a material can be leveraged to infer surrounding microstructure with increasing accuracy, akin to targeted guesses in the game Battleship. Their approach solves the Poisson model's multipoint correlation functions, a result rarely available for materials with randomness at the microscale.
"With this study, we've solved the famous Poisson model for heterogenous materials," said lead author Alec Shelley, a PhD student in applied physics in Daniel Tartakovsky's lab at the Stanford Doerr School of Sustainability. "Our result could have a broad impact on several areas of science, because heterogenous materials are common and their models almost never have exact solutions."
Because many useful properties emerge from microstructural arrangements, the findings could help engineers design stronger and cheaper composites. "Using his approach, you could design a composite material to your specifications and obtain certain properties based on the proper mixture of components," said Tartakovsky, professor of energy science and engineering.
For concrete, the method could guide optimization of voids and admixtures such as fly ash, slag, or biochar to cut cement content, lower manufacturing emissions, and improve strength while reducing costs. Beyond construction, applications span fractured and porous media central to groundwater modeling, geothermal systems, carbon sequestration, and nuclear waste disposal.
The Poisson model, rooted in 19th century statistics of independent events, represents space partitioned by independently rendered straight boundaries. As a microstructural model, it reproduces diverse heterogeneous systems, from ice fragments on a frozen lake to marbling in meat, offering realistic mosaics for analysis.
Shelley describes generating realizations by drawing random lines to form regions, then assigning colors. Knowing the color at one or more points and applying the solved multipoint correlations allows prediction of the hidden mosaic with growing certainty. "It's like we've created the perfect Battleship player for guessing colors in this model," he said.
Deriving the solution required tools from stochastic geometry. Shelley began with pen-and-paper calculations for two-point cases, expanded to 128 terms for three points, and ultimately used computer simulations for four-point evaluations. The work was supported by an Oak Ridge Institute for Science and Education Fellowship and Sandia National Laboratories.
Research Report:Multipoint Correlations in Poisson Media
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