A computational macroscopic model of piezomagnetoelectric materials using Generalized Multiscale Finite Element Method

Generalized macroscale model for Cosserat elasticity using Generalized Multiscale Finite Element Method

“…Such online adaptivity is suggested and examined mathematically in [11]. To be more precise, at the current nth Picard iterative step, if the local residual associated with some coarse neighborhood w i is big (refer to Section 6.2), a new basis function 3 (defined in Section 6.1.2) can be constructed at the online stage (using the equipped norm ∥ • ∥ V i defined in (6.12)), and added to the multiscale basis functions space. It is further demonstrated that the online basis construction yields an efficient approximation of the fine-scale solution u h if the offline space has offline basis functions that contain sufficient information.…”

“…For this reason, they are also called spectral basis functions. However, we need the multiscale partition of unity [25,3] to ensure their conformality.…”

“…Since the considered model is coupled, this partition of unity functions is created in a coupled way [3]. For this purpose, we solve the following local problems on coarse-grid cells.…”

“…Second approach to compute the dual norm. Let us recall the function space 3 in the given coarse neighborhood ω i , where V h was defined in (4.1). At the Picard iteration nth, with ν = (v, Ψ) ∈ V i , using the operator a n defined in (4.5), we equip the space V i with the energy norm ∥ν∥ V i :…”

“…Since this model is coupled, the multiscale basis functions will be established in a coupled way. The resulting coupled basis functions, on average, can provide better accuracy (in comparison with the uncoupled multiscale basis functions) due to considering the mutual influence of the solution fields [52,51,2,3]. The algorithm to compute the GMsFEM solution is presented in Section 5.2.3.…”