  Modern digital engineering design process commonly involves expensive
repeated simulations on varying three-dimensional (3D) geometries. The
efficient prediction capability of neural networks (NNs) makes them a suitable
surrogate to provide design insights. Nevertheless, few available NNs can
handle solution prediction on varying 3D shapes. We present a novel deep
operator network (DeepONet) variant called Geom-DeepONet, which encodes
parameterized 3D geometries and predicts full-field solutions on an arbitrary
number of nodes. To the best of the authors' knowledge, this is the first
attempt in the literature and is our primary novelty. In addition to expressing
shapes using mesh coordinates, the signed distance function for each node is
evaluated and used to augment the inputs to the trunk network of the
Geom-DeepONet, thereby capturing both explicit and implicit representations of
the 3D shapes. The powerful geometric encoding capability of a sinusoidal
representation network (SIREN) is also exploited by replacing the classical
feedforward neural networks in the trunk with SIREN. Additional data fusion
between the branch and trunk networks is introduced by an element-wise product.
A numerical benchmark was conducted to compare Geom-DeepONet to PointNet and
vanilla DeepONet, where results show that our architecture trains fast with a
small memory footprint and yields the most accurate results among the three
with less than 2 MPa stress error. Results show a much lower generalization
error of our architecture on unseen dissimilar designs than vanilla DeepONet.
Once trained, the model can predict vector solutions, and speed can be over
$10^5$ times faster than implicit finite element simulations for large meshes.
