The uncertainties in material and other properties of structures are usually spatially correlated. We introduce an efficient technique for representing and processing spatially correlated random fields in robust topology optimisation of lattice structures. Robust optimisation considers the statistics of the structural response to obtain a design whose performance is less sensitive to the specific realisation of the random field. We represent Gaussian random fields on lattices by leveraging the established link between random fields and stochastic partial differential equations (SPDEs). It is known that the precision matrix, i.e. the inverse of the covariance matrix, of a random field with Mat\'ern covariance is equal to the finite element stiffness matrix of a possibly fractional PDE with a second-order elliptic operator. We consider the discretisation of the PDE on the lattice to obtain a random field which, by design, considers its geometry and connectivity. The so-obtained random field can be interpreted as a physics-informed prior by the hypothesis that the elliptic SPDE models the physical processes occurring during manufacturing, like heat and mass diffusion. Although the proposed approach is general, we demonstrate its application to lattices modelled as pin-jointed trusses with uncertainties in member Young's moduli. We consider as a cost function the weighted sum of the expectation and standard deviation of the structural compliance. To compute the expectation and standard deviation and their gradients with respect to member cross-sections we use a first-order Taylor series approximation. The cost function and its gradient are computed using only sparse matrix operations. We demonstrate the efficiency of the proposed approach using several lattice examples with isotropic, anisotropic and non-stationary random fields and up to eighty thousand random and optimisation variables.

翻译：材料及其他结构属性中的不确定性通常具有空间相关性。我们提出了一种高效的技术，用于在桁架结构的稳健拓扑优化中表示和处理空间相关随机场。稳健优化考虑了结构响应的统计特性，以获得对随机场具体实现敏感性较低的设计。我们利用随机场与随机偏微分方程之间的已知联系，在桁架上表示高斯随机场。已知具有马特恩协方差的随机场的精度矩阵（即协方差矩阵的逆）等于具有二阶椭圆算子的（可能为分数阶）偏微分方程的有限元刚度矩阵。我们考虑在桁架上离散化该偏微分方程，从而获得一个随机场，该设计固有地考虑了桁架的几何形状与连接性。由此获得的随机场可被解释为一种物理信息先验，其假设椭圆随机偏微分方程模拟了制造过程中发生的物理过程（如热扩散和质量扩散）。虽然所提方法具有普适性，但我们将其应用于由销接桁架建模的桁架结构，其中杆件杨氏模量存在不确定性。我们以结构柔顺度期望值与标准差的加权和作为代价函数。为计算期望值、标准差及其相对于杆件横截面的梯度，采用一阶泰勒级数近似。代价函数及其梯度仅通过稀疏矩阵运算计算。通过多个各向同性、各向异性及非平稳随机场的桁架实例（涉及多达八万个随机变量和优化变量），我们验证了所提方法的效率。