  This work highlights an approach for incorporating realistic uncertainties
into scientific computing workflows based on finite elements, focusing on
applications in computational mechanics and design optimization. We leverage
Mat\'ern-type Gaussian random fields (GRFs) generated using the SPDE method to
model aleatoric uncertainties, including environmental influences, variating
material properties, and geometric ambiguities. Our focus lies on delivering
practical GRF realizations that accurately capture imperfections and variations
and understanding how they impact the predictions of computational models and
the topology of optimized designs. We describe a numerical algorithm based on
solving a generalized SPDE to sample GRFs on arbitrary meshed domains. The
algorithm leverages established techniques and integrates seamlessly with the
open-source finite element library MFEM and associated scientific computing
workflows, like those found in industrial and national laboratory settings. Our
solver scales efficiently for large-scale problems and supports various domain
types, including surfaces and embedded manifolds. We showcase its versatility
through biomechanics and topology optimization applications. The flexibility
and efficiency of SPDE-based GRF generation empower us to run large-scale
optimization problems on 2D and 3D domains, including finding optimized designs
on embedded surfaces, and to generate topologies beyond the reach of
conventional techniques. Moreover, these capabilities allow us to model
geometric uncertainties of reconstructed submanifolds, such as the surfaces of
cerebral aneurysms. In addition to offering benefits in these specific domains,
the proposed techniques transcend specific applications and generalize to
arbitrary forward and backward problems in uncertainty quantification involving
finite elements.
