Curve fitting and optimal interpolation for CNC machining under confined error using quadratic B-splines

In CNC machining, fitting the polyline machining tool path with parametric curves can be used for smooth tool path generation and data compression. In this paper, an optimization problem is solved to find a quadratic B-spline curve whose Hausdorff distance to the given polyline tool path is within a given precision. Furthermore, adopting time parameter for the fitting curve, we combine the usual two stages of tool path generation and optimal velocity planning to derive a one-step solution for the CNC optimal interpolation problem of polyline tool paths. Compared with the traditional decoupled model of curve fitting and velocity planning, experimental results show that our method generates a smoother path with minimal machining time.     

Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors. This work was supported by DARPA/AFOSR Grants FA9550-05-1-0114 and FA-9550-07-1-0425,the Singapore-MIT Alliance,the Pappalardo MIT Mechanical Engineering Graduate Monograph Fund,and the Progetto Roberto Rocca Politecnico di Milano-MIT.We acknowledge many helpful discussions with Professor Yvon Maday of University Paris6.