Multiphysics embedding localized orthogonal decomposition for thermomechanical coupling problems

Multiscale thermomechanical problems in highly heterogeneous media are challenging because the elastic, thermal, and coupling coefficients may vary on unresolved spatial scales. We propose a multiphysics-embedding localized orthogonal decomposition (ME-LOD) method in which displacement and temperature correctors are generated by a coupled static operator. The corrector problems are localized to coarse-grid patches and solved in the kernel of a projective quasi-interpolation operator. We prove uniform inf-sup stability on the global fine-scale kernel and on all zero-extension patch kernels, establish exponential decay of the coupled correctors and the resulting multiscale basis functions, and derive spatial approximation and fully discrete reduction estimates. Numerical experiments demonstrate that, for the tested periodic, random, and high-contrast coefficient fields, ME-LOD attains smaller errors than the comparison method at the same coarse resolution and patch size and can reach a prescribed accuracy with fewer oversampling layers. Although each coupled local corrector is more expensive than a decoupled corrector, the improved localization yields a favorable overall accuracy-to-cost balance in the reported tests.