Robust topology optimisation of lattice structures with spatially correlated uncertainties

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

On the data-driven description of lattice materials mechanics

In the emerging field of mechanical metamaterials, using periodic lattice structures as a primary ingredient is relatively frequent. However, the choice of aperiodic lattices in these structures presents unique advantages regarding failure, e.g., buckling or fracture, because avoiding repeated patterns prevents global failures, with local failures occurring in turn that can beneficially delay structural collapse. Therefore, it is expedient to develop models for computing efficiently the effective mechanical properties in lattices from different general features while addressing the challenge of presenting topologies (or graphs) of different sizes. In this paper, we develop a deep learning model to predict energetically-equivalent mechanical properties of linear elastic lattices effectively. Considering the lattice as a graph and defining material and geometrical features on such, we show that Graph Neural Networks provide more accurate predictions than a dense, fully connected strategy, thanks to the geometrically induced bias through graph representation, closer to the underlying equilibrium laws from mechanics solved in the direct problem. Leveraging the efficient forward-evaluation of a vast number of lattices using this surrogate enables the inverse problem, i.e., to obtain a structure having prescribed specific behavior, which is ultimately suitable for multiscale structural optimization problems

Cuaderno de prácticas de Resistencia de Materiales y Elasticidad

Cuaderno de prácticas de laboratorio de la asignatura de Resistencia de Materiales y Elasticidad, de segundo curso de Grado en Ingeniería Aeroespacial de la Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio

Cuaderno de ejercicios de resistencia de materiales y elasticidad : curso 2021/2022

Boletín de problemas que cubre el temario de la asignatura de elasticidad y resistencia de materiales del curso 2021_2022

Herramienta de aprendizaje online para la resolución interactiva de problemas de vigas y pórticos en resistencia de materiales

Herramienta de aprendizaje Online a través de Matlab Grader, que resuelve de forma interactiva problema de mecánica de sólidos en vigas y pórticos para las asignaturas de Resistencia de Materiale