We consider two-phase composites whose microstructures are two-dimensional and generated by the periodic replication of a convex polygonal cell containing a single inclusion embedded in a matrix. Adopting the framework of nonlinear conductivity, we address the problem of finding the inclusion shape that optimizes the effective energy. A conceptually simple but numerically effective approach is presented, in which the inclusion shape is parameterized by the Fourier coefficients of a scalar periodic function f that defines its polar representation. Truncating the Fourier expansion to a finite order turns the shape optimization problem into a finite-dimensional constrained optimization problem that can be solved using a numerical algorithm of choice. Explicit expressions of the function to optimize and its gradient are provided and can easily be evaluated from a finite-element model. The proposed approach is applied to perfectly conducting inclusions in a power law matrix. Results for the three types of regular tessellations (square, hexagonal and triangular) are presented and compared with the Vidgergauz [23, 24] microstructures giving the extremal inclusions in the linear case. The proposed method gives very simple representations of the extremal inclusions, which could useful for manufacturing the microstructures considered. The obtained nonlinear effective conductivities are compared with known Hashin–Shtrikman-type nonlinear bounds, which contributes to shed some light on the optimality of those bounds.

我们考虑了两相复合材料，其微观结构是二维的，由凸多边形单元的周期性复制生成，该单元包含嵌入基质中的单个夹杂物。采用非线性电导率框架，我们解决了寻找优化有效能量的夹杂物形状的问题。提出了一种概念上简单但数值有效的方法，其中包含形状由标量周期函数f的傅立叶系数参数化这定义了它的极坐标表示。将傅立叶展开式截断为有限阶将形状优化问题转变为可以使用选择的数值算法解决的有限维约束优化问题。提供了要优化的函数及其梯度的显式表达式，并且可以很容易地从有限元模型中进行评估。所提出的方法适用于在幂律矩阵中完美地进行包含。给出了三种规则镶嵌（正方形、六边形和三角形）的结果，并与 Vidgergauz [23, 24] 微观结构进行了比较，在线性情况下给出了极值夹杂物。所提出的方法给出了极值夹杂物的非常简单的表示，这可用于制造所考虑的微观结构。