The application of heterogeneous materials has become common in modern product design such as composites and porous media. Computational design tools for such materials, with higher complexity than the traditional homogeneous ones, will be a critical component in the realization of the heterogeneity systematically. It is foreseen that computer-aided design (CAD) systems will include computer-aided materials design modules in future so that the design of functional materials and structures can be integrated for optimal product design. The traditional CAD systems model three-dimensional (3D) geometry at macro-scales with boundary representation (B-Rep), whereas computer-aided materials design is concerned with the specification of material composition at scales ranging from nano-, meso-, to micro-. Thus, multi-scale CAD systems are desirable for the integration of product and materials information. The existing B-Rep based modeling scheme needs to be extended to incorporate heterogeneous material compositions. The new modeling scheme should also support seamless zoom-in and zoom-out operations in multi-scale CAD systems. Recently, a multi-scale model, dual-Rep, was proposed to represent geometry and material property distribution implicitly. The core part of dual-Rep is a new basis function called surfacelet. Surfacelet is able to represent boundary information more efficiently than the traditional wavelets, while keeping a unified form with wavelets so that the role exchange of boundary and internal structures during zooming operations is enabled. A surfacelet transform is able to represent microstructure distributions in 3D images with surfacelet coefficients. In this dissertation, three enabling techniques for surfacelet-based heterogeneous materials modeling are developed. First, a method of inverse surfacelet transform is developed such that the original images can be reconstructed from the surfacelet coefficients. The surface integrals of voxel (i.e., volumetric pixel) values are obtained from the surfacelet coefficients using the one-dimensional inverse wavelet transform. The images are then reconstructed by solving linear equations from discretized surface integrals. The prior knowledge of material properties and distributions is applied to solve the under-constrained problems. Second, composite surfacelets with the combinations of different types of primitive surfacelets are created to increase the flexibility of the surfacelet transform with potentially fewer surfacelets and improved reconstruction accuracy. Third, a multi-scale materials modeling method is proposed to support interactive design and visualization of material microstructures at multiple levels of details. It has the capability to support seamless zoom-in and zoom-out. This method provides a feature-based design approach based on the surfacelet basis.
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