A posteriori modeling error estimates in the optimization of two-scale elastic composite materials

The a posteriori analysis of the discretization error and the modeling error is studied for a compliance cost functional in the context of the optimization of composite elastic materials and a two-scale linearized elasticity model. A mechanically simple, parametrized microscopic supporting structure is chosen and the parameters describing the structure are determined minimizing the compliance objective. An a posteriori error estimate is derived which includes the modeling error caused by the replacement of a nested laminate microstructure by this considerably simpler microstructure. Indeed, nested laminates are known to realize the minimal compliance and provide a benchmark for the quality of the microstructures. To estimate the local difference in the compliance functional the dual weighted residual approach is used. Different numerical experiments show that the resulting adaptive scheme leads to simple parametrized microscopic supporting structures that can compete with the optimal nested laminate construction. The derived a posteriori error indicators allow to verify that the suggested simplified microstructures achieve the optimal value of the compliance up to a few percent. Furthermore, it is shown how discretization error and modeling error can be balanced by choosing an optimal level of grid refinement. Our two scale results with a single scale microstructure can provide guidance towards the design of a producible macroscopic fine scale pattern

Impermeability through a perforated domain for the incompressible 2D Euler equations

We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $\varepsilon$ separated by distances $d_\varepsilon$ and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of $\frac{d_{\varepsilon}}\varepsilon$ when $\varepsilon$ goes to zero. If $\frac{d_{\varepsilon}}\varepsilon\to \infty$, then the limit motion is not perturbed by the porous medium, namely we recover the Euler solution in the whole space. On the contrary, if $\frac{d_{\varepsilon}}\varepsilon\to 0$, then the fluid cannot penetrate the porous region, namely the limit velocity verifies the Euler equations in the exterior of an impermeable square. If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}}$ where $\gamma\in (0,\infty]$ is related to the geometry of the lateral boundaries of the obstacles. If $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}} \to \infty$, then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions. In particular it is equal to $\varepsilon^3$ for balls

Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions

We consider the Stokes system in a thin porous medium Ωε of thickness ε which is perforated by periodically distributed solid cylinders of size ε. On the boundary of the cylinders we prescribe non-homogeneous slip boundary conditions depending on a parameter γ. The aim is to give the asymptotic behavior of the velocity and the pressure of the fluid as ε goes to zero. Using an adaptation of the unfolding method, we give, following the values of γ, different limit systems.Junta de AndalucíaMinisterio de Economía y Competitividad (MINECO). Españ