  We present the sliding basis computational framework to automatically
synthesize heterogeneous (graded or discrete) material fields for parts
designed using constrained optimization. Our framework uses the fact that any
spatially varying material field over a given domain may be parameterized as a
weighted sum of the Laplacian eigenfunctions enabling efficient design space
exploration with the weights as a small set of design variables. We further
improve computational efficiency by using the property that the Laplacian
eigenfunctions form a spectrum and may be ordered from lower to higher
frequencies. This approach allows greater localized control of the material
distribution as the sliding window moves through higher frequencies. The
approach also reduces the number of optimization variables per iteration, thus
the design optimization process speeds up independent of the domain resolution
without sacrificing analysis quality. Our method is most beneficial when the
gradients may not be computed easily (i.e., optimization problems coupled with
external black-box analysis) thereby enabling optimization of otherwise
intractable design problems. The sliding basis framework is independent of any
particular physics analysis, objective and constraints, providing a versatile
and powerful design optimization tool for various applications. We demonstrate
our approach on graded solid rocket fuel design and multi-material topology
optimization applications and evaluate its performance.
