A posteriori error estimates for sequential laminates in shape optimization (Q727511)

The authors consider a 2D linear elastic shape optimization problem which consists to minimize a cost functional among mixtures between hard and soft elastic materials with a given amount of hard material in a circumjacent domain \(D\subset \mathbb{R}^{2}\): \(\int_{D}\chi dx=\Theta \), where \(\chi \in L^{\infty }(D,\{0,1\})\). The hard material fills a subdomain \(\Omega \subset D\), the material is held fixed on \(\Gamma _{D}\subset \partial \Omega \cap \partial D\) and tractions \(g\) are applied on \(\Gamma _{N}\subset \partial \Omega \cap \partial D\), leading to the set \(\Sigma =\{\sigma :D\rightarrow \mathbb{R}_{sym}^{2\times 2}\mid div\{\sigma \}=0\) in \(D\), \(\sigma n=g \) on \(\Gamma _{N}\), \(u=0\) on \(\Gamma _{D}\), \(\sigma =C[q]\varepsilon [ u]\}\) of feasible stresses. The cost functional is taken as \(J[u[\chi ];\chi ]=\int_{\Gamma _{N}}g\cdot u[\chi ]da(x)=l(u[\chi ])=a(q;u[\chi ],u[\chi ])\) where \(l\) is a Lagrange multiplier associated to the amount of hard material and \(a(q;u,\varphi )=\int_{D}C[q]\varepsilon [ u]:\varepsilon [ \varphi ]dx\), with \(C[q]=\chi A[q]+(1-\chi )B[q]\), \( A[q]\) and \(B[q]\) being the elasticity tensors associated to the hard and soft materials, respectively. It is known that this shape optimization problem may be ill-posed with the onset of microstructures. The authors thus consider the relaxed problem \(\min_{\sigma \in \Sigma }\int_{D}\min_{0\leq \theta \leq 1}\min_{C^{\ast }[q]\in G_{\theta }^{B}}C^{\ast }[q]^{-1}\sigma :\sigma +l\theta dx\), where \(G_{\theta }^{B}\) is the set of homogenized tensors. They consider sequential laminate microstructures with a main laminate direction \(\alpha (x)\), a ratio of material spent in the second lamination stage \(m(x)\) and the overall local material density \(\theta (x)\). They give the expressions of the homogenized elasticity tensors in this case. The main purpose of the paper is to prove a posteriori estimates for a finite element discretization of this shape optimization problem. The authors introduce a triangulation of a polygonally bounded domain \(D\) and piecewise bi-quadratic and continuous vector-valued functions with vanishing trace on \(\Gamma _{D}\). The authors first prove that the local elastic energy is invariant with respect to \(\alpha (x)\). The main result of the paper proves an estimate for the difference \(\left| J[q;u]-J[q_{h};u_{h}]\right| \), and the paper ends with the presentation of numerical simulations.