Mechanical and Civil Engineering Seminar

In problems involving solidification, fracture, or electromigration, phase boundaries evolve driven by phase transformation, mechanical forces, or diffusion respectively. Similarly, in optimal design, boundary shape is deliberately evolved to conform to design objectives. The geometric description of evolving phase boundaries may be classified as being implicit (mathematical form of the surface that defines the phase interface is unknown, but sign of points relative to the interface is known) or explicit (mathematical form of the surface that defines the interface is known, but sign relative to interface is unknown). Correspondingly, the approximated interface behavior may be classified as being implicit if no explicit boundary exists in the model. In general, the accuracy of the behavioral approximation on the interface is tied isoparametrically to the geometrical approximation, with both converging only in the limit of refinement. Therefore, to accurately capture the interface geometry and behavior, implicit methods such as level-sets and phase field need to use a mesh that spans several orders of magnitude in lengthscale. In this talk, I will describe a geometrically explicit, but behaviorally implicit approximation strategy that decouples the accuracy of the geometric approximation from the behavioral approximation used over the domain while assuring exactness of tangents, normals and curvatures in imposing the interface behavior condition. Such a strategy enables embedding of arbitrary parametric surfaces within a domain as well as enriching the surface with appropriate interface conditions. The strategy is inherently "narrow band" in nature since additional unknowns describing interface behavior are isoparametrically defined on the interface geometry. Explicit parametric geometry requires the solution to fundamental CAD problems of point classification, distance estimation and point projection from three-dimensional parametric surfaces. I will describe purely algebraic solutions to the above problems. I will then demonstrate the developed procedure on representative solidification, fracture propagation and diffusion-driven phase evolution problems.