Optimization of Tool Path for Uniform Scallop-Height in Ultra-precision Grinding of Free-form Surfaces

Free-form surfaces have been widely used in complex optical devices to improve the functional performance of imaging and illumination quality and reduce sizes. Ultra-precision grinding is a kind of ultra-precision machining technology for fabricating free-form surfaces with high form accuracy and good surface finish. However, the complexity and variation of curvature of the free-form surface impose a lot of challenges to make the process more predictable. Tool path as a critical factor directly determines the form error and surface quality in ultra-precision grinding of free-form surfaces. In conventional tool path planning, the constant angle method is widely used in machining free-form surfaces, which resulted in non-uniform scallop-height and degraded surface quality of the machined surfaces. In this paper, a theoretical scallop-height model is developed to relate the residual height and diverse curvature radius. Hence, a novel tool-path generation method is developed to achieve uniform scallop-height in ultra-precision grinding of free-form surfaces. Moreover, the iterative closest-point matching method, which is a well-known algorithm to register two surfaces, is exploited to make the two surfaces match closely through rotation and translation. The deviation of corresponding points between the theoretical and the measured surfaces is determined. Hence, an optimized tool-path generator is developed that is experimentally verified through a series of grinding experiments conducted on annular sinusoidal surface and single sinusoidal surface, which allows the realization of the achievement of uniform scallop-height in ultra-precision grinding of free-form surfaces

T-spline based unifying registration procedure for free-form surface workpieces in intelligent CMM

With the development of the modern manufacturing industry, the free-form surface is widely used in various fields, and the automatic detection of a free-form surface is an important function of future intelligent three-coordinate measuring machines (CMMs). To improve the intelligence of CMMs, a new visual system is designed based on the characteristics of CMMs. A unified model of the free-form surface is proposed based on T-splines. A discretization method of the T-spline surface formula model is proposed. Under this discretization, the position and orientation of the workpiece would be recognized by point cloud registration. A high accuracy evaluation method is proposed between the measured point cloud and the T-spline surface formula. The experimental results demonstrate that the proposed method has the potential to realize the automatic detection of different free-form surfaces and improve the intelligence of CMMs

Hyperboloidal evolution of test fields in three spatial dimensions

We present the numerical implementation of a clean solution to the outer boundary and radiation extraction problems within the 3+1 formalism for hyperbolic partial differential equations on a given background. Our approach is based on compactification at null infinity in hyperboloidal scri fixing coordinates. We report numerical tests for the particular example of a scalar wave equation on Minkowski and Schwarzschild backgrounds. We address issues related to the implementation of the hyperboloidal approach for the Einstein equations, such as nonlinear source functions, matching, and evaluation of formally singular terms at null infinity.Comment: 10 pages, 8 figure

Covariant un-reduction for curve matching

The process of un-reduction, a sort of reversal of reduction by the Lie group symmetries of a variational problem, is explored in the setting of field theories. This process is applied to the problem of curve matching in the plane, when the curves depend on more than one independent variable. This situation occurs in a variety of instances such as matching of surfaces or comparison of evolution between species. A discussion of the appropriate Lagrangian involved in the variational principle is given, as well as some initial numerical investigations.Comment: Conference paper for MFCA201

Steady General Relativistic Magnetohydrodynamic Inflow/Outflow Solution along Large-Scale Magnetic Fields that Thread a Rotating Black Hole

General relativistic magnetohydrodynamic (GRMHD) flows along magnetic fields threading a black hole can be divided into inflow and outflow parts, according to the result of the competition between the black hole gravity and magneto-centrifugal forces along the field line. Here we present the first self-consistent, semi-analytical solution for a cold, Poynting flux-dominated (PFD) GRMHD flow, which passes all four critical (inner and outer, Alfven and fast magnetosonic) points along a parabolic streamline. By assuming that the dominating (electromagnetic) component of the energy flux per flux tube is conserved at the surface where the inflow and outflow are separated, the outflow part of the solution can be constrained by the inflow part. The semi-analytical method can provide fiducial and complementary solutions for GRMHD simulations around the rotating black hole, given that the black hole spin, global streamline, and magnetizaion (i.e., a mass loading at the inflow/outflow separation) are prescribed. For reference, we demonstrate a self-consistent result with the work by McKinney in a quantitative level.Comment: 13 Pages, incliding 2 tables and 5 Figures; accepted by Ap

Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Mode-matching without root-finding: Application to a dissipative silencer

This article presents an analytic mode-matching approach suitable for modelling the propagation of sound in a two-dimensional, three-part, ducting system. The approach avoids the need to the find roots of the characteristic equation for the middle section of the duct (the component) and is readily applicable to a broad class of problems. It is demonstrated that the system of equations, derived via analytic mode-matching, exhibits certain features which ensure that they can be re-cast into a form that is independent of the roots of the characteristic equation for the component. The precise details of the component are irrelevant to the procedure; it is required only that there exists an orthogonality relation, or similar, for the eigenmodes corresponding to the propagating wave-forms in this region. The method is applied here to a simple problem involving acoustic transmission through a dissipative silencer of the type commonly found in heating ventilation and air-conditioning (HVAC) ducts. With reference to this example, the silencer transmission loss is computed, and the power balance for the silencer is investigated and is shown to be an identity that is necessarily satisfied by the system of equations, regardless of the level of truncation