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The uncertainties in material and other properties of structures are usuallyspatially correlated. We introduce an efficient technique for representing andprocessing spatially correlated random fields in robust topology optimisationof lattice structures. Robust optimisation considers the statistics of thestructural response to obtain a design whose performance is less sensitive tothe specific realisation of the random field. We represent Gaussian randomfields on lattices by leveraging the established link between random fields andstochastic partial differential equations (SPDEs). It is known that theprecision matrix, i.e. the inverse of the covariance matrix, of a random fieldwith Mat\'ern covariance is equal to the finite element stiffness matrix of apossibly fractional PDE with a second-order elliptic operator. We consider thediscretisation of the PDE on the lattice to obtain a random field which, bydesign, considers its geometry and connectivity. The so-obtained random fieldcan be interpreted as a physics-informed prior by the hypothesis that theelliptic SPDE models the physical processes occurring during manufacturing,like heat and mass diffusion. Although the proposed approach is general, wedemonstrate its application to lattices modelled as pin-jointed trusses withuncertainties in member Young's moduli. We consider as a cost function theweighted sum of the expectation and standard deviation of the structuralcompliance. To compute the expectation and standard deviation and theirgradients with respect to member cross-sections we use a first-order Taylorseries approximation. The cost function and its gradient are computed usingonly sparse matrix operations. We demonstrate the efficiency of the proposedapproach using several lattice examples with isotropic, anisotropic andnon-stationary random fields and up to eighty thousand random and optimisationvariables.

Robust topology optimisation of lattice structures with spatially correlated uncertainties