High-order multiscale method for elastic deformation of complex geometries

Computational methods, such as finite elements, are indispensable for modeling the mechanical compliance of elastic solids. However, as the size and geometric complexity of the domain increases, the cost of simulations becomes prohibitive. One example is the microstructure of a porous material, such as a piece of rock or bone sample, captured by an X-ray $\mu$CT image. The solid geometry consists of numerous grains, cavities, and/or channels, with the domain large enough to allow inferring statistically converged macroscale properties. The pore-level multiscale method (PLMM) was recently proposed by the authors to reduce the associated computational cost through a divide-and-conquer strategy. The domain is decomposed into subdomains via watershed segmentation, and local basis and correction functions are built numerically, then assembled to obtain an approximate solution. However, PLMM is limited to domains corresponding to microscale porous media, incurs large errors when modeling loading conditions that generate significant bending/twisting moments locally, and it is equipped with only one mechanism to control approximation errors during a simulation. Here, we generalize PLMM into a high-order variant, called hPLMM, that removes these drawbacks. In hPLMM, appropriate mortar spaces are defined at subdomain interfaces that allow improving the boundary conditions used to solve local problems on the subdomains, thus the accuracy of the approximation. Moreover, errors can be reduced by a second mechanism wherein an upfront cost is paid prior to a simulation, useful if basis functions can be reused many times, e.g., across loading steps. Finally, the method applies not just to pore-scale, but also Darcy-scale and non-porous domains. We validate hPLMM against a range of complex 2D/3D geometries and discuss its convergence and algorithmic complexity. Implications for modeling failure and nonlinear problems are discussed.