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：结构材料及其他属性中的不确定性通常具有空间相关性。本文提出一种高效技术，用于在格点结构鲁棒拓扑优化中表示和处理空间相关随机场。鲁棒优化考虑结构响应的统计特性，以获取对随机场具体实现不敏感的设计方案。通过利用随机场与随机偏微分方程（SPDEs）之间已知的联系，我们在格点上表示高斯随机场。已知具有Matérn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分数阶偏微分方程的有限元刚度矩阵。我们考虑在格点上离散化该偏微分方程，从而得到一种能自然考虑其几何形状与连接性的随机场。通过假设椭圆型SPDE可模拟制造过程中的物理过程（如热传导与质量扩散），所得随机场可被解释为物理信息先验。尽管所提方法具有普适性，我们以构件杨氏模量存在不确定性的销接桁架格点为例进行演示。采用结构柔度期望与标准差的加权和作为成本函数。为计算期望、标准差及其关于构件截面的梯度，我们使用一阶泰勒级数近似。成本函数及其梯度仅通过稀疏矩阵运算实现。通过各向同性、各向异性及非平稳随机场示例（含多达八万个随机变量与优化变量）验证了所提方法的高效性。