  This work presents a scalable computational framework for optimal design
under uncertainty with application to multi-material insulation components of
building envelopes. The forward model consists of a multi-phase
thermo-mechanical model of porous materials governed by coupled partial
differential equations (PDEs). The design parameter (material porosity) is an
uncertain and space-dependent field, resulting in a high-dimensional
PDE-constrained optimization under uncertainty problem after finite element
discretization. The robust design framework uses a risk-averse formulation
consisting of the mean and variance of the design objective to achieve target
thermal and mechanical performances and mitigate uncertainty. To ensure the
efficiency and scalability of the solution, a second-order Taylor approximation
of the mean and variance and the low-rank structure of the preconditioned
Hessian of the design objective are leveraged, which uncovers the low effective
dimension of the high-dimensional uncertain parameter space. Moreover, a
gradient-based optimization method is implemented using the Lagrangian
formalism to derive expressions for the gradient and Hessian with respect to
the design and uncertain parameters. Finally, approximated $\ell_0$
regularization functions are utilized via a continuation numerical scheme to
promote sparsity in the designed porosity. The framework's accuracy,
efficiency, and scalability are demonstrated with numerical examples of a
building envelope insulation scenario.
