  We present a methodical procedure for topology optimization under uncertainty
with multi-resolution finite element models. We use our framework in a
bi-fidelity setting where a coarse and a fine mesh corresponding to low- and
high-resolution models are available. The inexpensive low-resolution model is
used to explore the parameter space and approximate the parameterized
high-resolution model and its sensitivity where parameters are considered in
both structural load and stiffness. We provide error bounds for bi-fidelity
finite element (FE) approximations and their sensitivities and conduct
numerical studies to verify these theoretical estimates. We demonstrate our
approach on benchmark compliance minimization problems where we show
significant reduction in computational cost for expensive problems such as
topology optimization under manufacturing variability while generating almost
identical designs to those obtained with single resolution mesh. We also
compute the parametric Von-Mises stress for the generated designs via our
bi-fidelity FE approximation and compare them with standard Monte Carlo
simulations. The implementation of our algorithm which extends the well-known
88-line topology optimization code in MATLAB is provided.
