In this paper, we construct and analyze a fully discrete two-grid Galerkin finite element method (FEM) that is unconditionally optimal and α-robust, utilizing two time-stepping schemes for nonlinear time-fractional nonlocal diffusion equations on graded meshes. To address the problem of the solution’s initial singularity, we apply a cohesive nonuniform approach that merges the L1 and L2-1σ formulas in the time direction. We integrate a highly efficient and unconditionally optimal two-grid Galerkin FEM, characterized by its α-robust features, to reduce computational costs in the spatial dimension. In the linearized step of the two-grid algorithm, one Newton-like iteration is employed to handle the nonlinear term f(u) based on a coarse-grid solution. We demonstrate the existence and uniqueness of solutions alongside the unconditional convergence. Additionally, we perform an α-robust analysis to examine stability and provide optimal error estimates in both L2 and H1 norms for the two fully discrete time-stepping schemes operating on graded meshes, derived from Galerkin FEM and its associated two-grid Galerkin FEM, respectively. The suggested two-grid approach eliminates the limitations associated with the temporal-spatial grid ratio, easing the constraints on f(u) to a locally Lipschitz continuous function. The analysis of unconditional convergence is conducted through an innovative analytical approach, whose key point lies in the application of the inverse inequality when N−μ≤hr+1 and the use of the discrete Sobolev embedding inequality when N−μ≥hr+1. Here, r represents the polynomial degree, and h indicates the mesh size. Additionally, this innovative analysis integrates the cutoff functional technique alongside the auxiliary variable method. The resulting error bounds remain bounded as α→1−. To our knowledge, there is no prior study in the literature that has explored two-grid techniques for nonlinear time-fractional nonlocal diffusion equations. Numerical simulations corroborate the theoretical results and illustrate the unconditional convergence, α-robustness, accuracy, and efficiency of the proposed two-grid Galerkin FEM methodology.
Stress-driven design of lightweight structures plays a crucial role in advanced engineering applications. However, it remains highly challenging owing to the complexity of accurately characterizing multiscale coupling relationships and conducting reliable stress analysis across disparate length scales. In this work, a phase-field-based multiscale topology optimization framework is presented for stress-driven design of composite structures. Successive two-stage strategy is established to obtain the optimized material distributions in different scales. Coupled multiscale energy formulation is established, in which microstructural effects are incorporated into the macro response via material homogenization. The corresponding sensitivity analysis for both macro and micro design is derived. The Method of Moving Asymptotes is utilized for the multiscale optimization issues. Owing to the improvement of the phase-field formulation, the proposed approach supports multifunctional designs by accommodating both free evolution and target-oriented evolution mechanisms. Several two- and three-dimensional numerical examples under different loading and design scenarios are conducted to demonstrate the effectiveness of the proposed method.
