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 μ 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.

计算方法，例如有限元法，对于模拟弹性固体的力学柔度是必不可少的。然而，随着研究区域的尺寸和几何复杂性增加，模拟成本变得过高。一个例子是通过X射线微计算机断层扫描（μCT）图像获取的多孔材料（如一块岩石或骨样本）的微观结构。固体几何形状由众多颗粒、孔洞和/或通道组成，其区域足够大，能够推断出统计上收敛的宏观性质。作者最近提出了孔隙级多尺度方法（PLMM），通过分治策略来降低相关的计算成本。通过分水岭分割将区域分解为子区域，数值构建局部基函数和修正函数，然后组装以获得近似解。然而，PLMM仅限于对应于微观多孔介质的区域，在模拟局部产生显著弯曲/扭转力矩的加载条件时会产生较大误差，并且在模拟过程中仅配备一种控制近似误差的机制。在此，我们将PLMM推广为一种高阶变体，称为hPLMM，它消除了这些缺陷。在hPLMM中，在子区域界面处定义了适当的 mortar空间，这可以改进用于解决子区域上局部问题的边界条件，从而提高近似的准确性。此外，可以通过第二种机制降低误差，即在模拟之前付出前期成本，这在基函数可以多次重复使用（例如在加载步骤中）时是有用的。最后，该方法不仅适用于孔隙尺度，也适用于达西尺度和无孔区域。我们针对一系列复杂的二维/三维几何形状验证了hPLMM，并讨论了其收敛性和算法复杂性。还讨论了对模拟失效和非线性问题的影响。