• discrete differential geometry
• geometric deep learning
• machine learning
• architectural geometry

Design tools to assist the user in modelling geometric shapes for architecture taking into account statics (together with many possible targets as manufacturing, material economy and other aspects related with buildability and cost) are a known research goal: in this work the attention is all on gridshells, three-dimensional frame structures in which loads are entirely born by edges, or beams as they are known in technical jargon.

Main result is the development of a computational method which, given an input gridshell provided by the designer, slightly changes the input to ensure good static performances. The changing is induced by structure node repositioning. If the gridshell is represented as a surface mesh, the problem boils down to finding a proper vertex displacement. The vertex displacement should strike a happy medium between structure rigidity, with load deformation as low as possible, and structure resistance, preventing stress-caused breaks.

A definition of the problem: gridshell modification is framed as mesh vertex displacement
learning. Given a triangular mesh modelling the structure we want to minimize a loss function which represents mean strain energy over edges. Minimizing the loss ensures the above claimed static-aware criteria of structure rigidity and resistance.

My first contribution is the finding of strain energy as a suitable loss function and its implementation in a differentiable form with respect to mesh vertex coordinates. The loss function is fast to compute together with its derivatives and optimized to high parallelization on GPUs.

Then, I exploited such chance of loss differentiation with respect to mesh vertex coordinates to build an optimizer which solves the problem through gradient descent methods, by minimizing loss with respect to displacements.

Finally, I added the power of graph neural network models to the previously developed optimizer. I defined a vertex feature space that induces an underlying graph structure on the mesh which is exploited for learning purposes. In terms of the above optimization problem, we switched differentiation variables from displacements to learnable parameters inside the net model.
The adopted neural networks input an augmented representation of meshes which encapsulates
geometric information on vertices, the so called input features: I sifted different possibilities among intrinsic notions (geodesic distances, discrete Laplace-Beltrami eigenfunctions) and extrinsic ones (vertex coordinates, normals, principal curvatures).