Polar plots of diamond surface energy

Diamond surface energy sigma_hkl determines crystal habit. We discuss three aspects of a paper by Terentiev (1991). Firstly, we compare Terentiev's algorithm for exact sigma_hkl with the analytic solution for h <= k <= l and h + k < l. Secondly, we show that the general formula given by Terentiev should be interpreted probabilistically in order to be self consistent. Finally, we replicate in principle the simulation results for sigma_hkl in a nickel melt using nothing more than Matlab routines

B\'ezier curves that are close to elastica

We study the problem of identifying those cubic B\'ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B\'ezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B\'ezier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input B\'ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure

Svendborg Ny Kirkegaard: Østre Kirkegaard

Intet resumé

Practical isogeometric shape optimization: Parameterization by means of regularization

International audienceShape optimization based on Isogeometric Analysis (IGA) has gained popularity in recent years. Performing shape optimization directly over parameters defining the CAD geometry, such as for example the control points of a spline parametrization, opens up the prospect of seamless integration of a shape optimization step into the CAD workflow. One of the challenges when using IGA for shape optimization is that of maintaining a valid geometry parametrization of the interior of the domain during an optimization process, as the shape of the boundary is altered by an optimization algorithm. Existing methods impose constraints on the Jacobian of the parametrization, to guarantee that the parametrization remains valid. The number of such validity constraints quickly becomes intractably large, especially when 3D shape optimization problems are considered. An alternative, and arguably simpler, approach is to formulate the isogeometric shape optimization problem in terms of both the boundary and the interior control points. In order to ensure a geometric parametrization of sufficient quality a regularization term, such as the Winslow functional, is added to the objective function of the shape optimization problem. We illustrate the performance of these methods on the optimal design problem of electromagnetic reflectors and compare their performance. Both methods are implemented for multipatch geometries, using the IGA library G+Smo and the optimization library Ipopt. We find that the second approach performs comparably to a state of the art method with respect to both the quality of the found solutions and computational time, while its performance in our experience is more robust for coarse discretizations