We present an efficient multilevel Monte Carlo (MLMC) method for the topology optimization of flexoelectric structures. A flexoelectric composite consisting of flexoelectric and purely elastic building blocks is investigated. The governing equations are solved by Non-Uniform Rational B-spline (NURBS)-based isogeometric analysis (IGA) exploiting its higher order continuity. Genetic algorithms (GA) based integer-valued optimization is used to obtain the optimal topological design. The uncertainties in the material properties and the volume fraction of the constituents are considered to quantify the uncertainty in the electromechanical coupling effect. Then, a multilevel hierarchy of computational meshes is obtained by a uniform refinement according to a geometric sequence. We estimate the growth rate of the simulation cost, in addition to the rates of decay in the expectation and the variance of the differences between the approximations over the hierarchy. Finally, we determine the minimum number of simulations required on each level to achieve the desired accuracy at different prescribed error tolerances. The results show that the proposed method reduces the computational cost in the numerical experiments without loss of the accuracy. The overall computation saving was in the range 2.0–3.5.

我们提出了一种用于柔性电结构拓扑优化的高效多级蒙特卡罗 (MLMC) 方法。研究了由挠曲电和纯弹性构件组成的挠曲电复合材料。控制方程通过基于非均匀有理 B 样条 (NURBS) 的等几何分析 (IGA) 利用其高阶连续性来求解。基于遗传算法 (GA) 的整数值优化用于获得最佳拓扑设计。材料特性的不确定性和成分的体积分数被认为是量化机电耦合效应的不确定性。然后，根据几何序列通过均匀细化得到多级层次的计算网格。我们估计模拟成本的增长率，除了期望的衰减率和层次结构上近似值之间差异的方差之外。最后，我们确定每个级别所需的最小模拟次数，以在不同的规定误差容限下达到所需的精度。结果表明，所提出的方法在不损失精度的情况下降低了数值实验中的计算成本。总体计算节省在 2.0-3.5 范围内。结果表明，所提出的方法在不损失精度的情况下降低了数值实验中的计算成本。总体计算节省在 2.0-3.5 范围内。结果表明，所提出的方法在不损失精度的情况下降低了数值实验中的计算成本。总体计算节省在 2.0-3.5 范围内。