Sculptured surface machining using triangular mesh slicing

In this paper, an optimized procedure for tool path generation in regional milling is presented. The proposed procedure computes tool paths by slicing a CL-surface (Cutter Location surface), which is a triangular, mesh containing invalid triangles. Tool path generation consists of two steps: firstly, it obtains a set of line segments by slicing the triangular mesh with two-dimensional geometric elements (slicing elements), and, secondly, it extracts a valid tool path from the line segments by removing invalid portions. Two algorithms based on the slicing elements are presented: a ‘line projection’ algorithm based on the plane sweeping paradigm, which works efficiently by using the characteristics of a monotone chain; and a ‘curve projection’ algorithm for the projection of curves, which transforms the curve projection problem into a line projection problem by mapping the XYZ-space of the cylinder surface to the TZ-plane of the unfolded cylinder. The proposed procedure has been implemented and applied to tool path generation in regional milling. Performance tests show the efficiency of the proposed procedure.     

A new curve-based approach to polyhedral machining

This paper presents a new approach to three-axis NC tool path generation for sculptured surfaces. In the proposed curve-based approach, the gouge-free tool paths are generated from a polyhedral model of the STL (stereolithography) format. The polyhedral model is offset by a local-offsetting scheme. Then, the offset elements such as triangular facets, trimmed cylinders, and trimmed spheres are sliced by a series of drive planes. The curve segments on a drive plane are sorted, trimmed and linked, while the concave gouge is removed during the trimming process. The method is implemented on a PC, and some illustrative examples are provided in this paper. The main advantage of the proposed method is that the tool path can be generated from a polyhedral model without any concave and convex gouge, especially on an NC machine that supports NURBS interpolation. Other advantages and disadvantages of the proposed model are also discussed.     

Non-asymptotic end-to-end performance bounds for networks with long range dependent fBm cross traffic

Fractional Brownian motion (fBm) emerged as a useful model for self-similar and long-range dependent aggregate Internet traffic. Asymptotic, respectively, approximate performance measures are known for single queueing systems with fBm through traffic. In this paper end-to-end performance bounds for a through flow in a network of tandem queues under open-loop fBm cross traffic are derived. To this end, a rigorous sample path envelope for fBm is proven that complements previous approximate results. The sample path envelope and the concept of leftover service curves are employed to model the remaining service after scheduling fBm cross traffic at a queuing system. Using composition results for tandem systems from the stochastic network calculus end-to-end statistical performance bounds for individual flows in networks under fBm cross traffic are derived. The discovery is that these bounds grow in O(n(logn)^1/(2-2H)) for n systems in series where H is the Hurst parameter of the cross traffic. Explicit results on the impact of the variability and the burstiness of through and cross traffic on network performance are shown. Our analysis has direct implications on fundamental questions in network planning and service management.     

A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ? _k ³ ?? _k?1 elements whereas the magnetic part of the equations is approximated by discontinuous ? _k ³ ?? _k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments.     

An analytical C3 continuous tool path corner smoothing algorithm for 6R robot manipulator

Tool path smoothness is important to guarantee good dynamic and tracking performance of robot manipulators. An analytical C³ continuous tool path corner smoothing algorithm is proposed for robot manipulators with 6 rotational (6R) joints. The tool tip position is smoothed directly in the workpiece coordinate system (WCS). The tool orientation is smoothed after transferring the tool orientation matrix as three rotary angles. Micro-splines of the tool tip position and tool orientation are constructed under the constraints of the maximum deviation error tolerances in the WCS. Then the tool orientation and tool tip position are synchronized to the tool tip displacement with C³ continuity by replacing the remaining linear segments using specially constructed B-splines. Control points of the locally inserted micro-splines are all evaluated analytically without any iterative calculations. Simulation and experimental results show that the proposed algorithm satisfies constraints of the preset tool tip position and the tool orientation tolerances. The proposed corner smoothing algorithm achieves smoother and lower jerks than C² continuous corner smoothing algorithm. Experimental results show that the tracking errors associated to the execution of the C³ continuous tool path are up to 10% smaller than C² continuous path errors.     

3D mode I crack analysis of piezoelectrics

The finite-part integral concept is first used to prove rigidly hypersingular integral equations for a mode I crack embedded an infinite transversely isotropic piezoelectric solid. With three-dimensional linear piezoelectricity theory, investigations on the singular electroelastic fields in the vicinity of the crack are thereafter made by the dominant-part analysis. Finally, numerical solutions of several typical planar cracks with hypersingular integral equation method are performed to give the electroelastic intensity factor K-fields and the energy release rate G. Accuracies of results are found to be very high.     

Amortized efficiency of generating planar paths in convex position

Let S be a set of n?3 points arranged in convex position in the plane and suppose that all points of S are labeled from 1 to n in clockwise direction. A planar path P on S is a path whose edges connect all points of S with straight line segments such that no two edges of P cross. Flipping an edge on P means that a new path P^′ is obtained from P by a single edge replacement. In this paper, we provide efficient algorithms to generate all planar paths. With the help of a loopless algorithm to produce a set of 2-way binary-reflected Gray codes, the proposed algorithms generate the next planar path by means of a flip and such that the number of position changes for points in the path has a constant amortized upper bound.     

Light-assisted A⁎ path planning

This paper introduces a novel variant of the A^? path planning algorithm, which we call Light-assisted A^? (or LA^? for short). The LA^? algorithm expands less nodes than A^? during the search process, especially in scenarios where there are complex-shaped obstacles in the path between the start and goal nodes. This is achieved using the concept of (virtual) light which identifies and demotes dead-end paths blocked by obstacles, thus ensuring that the search stays focused on promising paths. Three path planning problems are used to test the performance of LA^?. These include path finding in a grid cluttered by randomly placed obstacles, robot navigation in a map containing multiple solid walls, and finally mazes. The results of these experiments show that LA^? can achieve orders of magnitude improvement in performance over A^?. In addition, LA^? results in near-optimal solutions that are very close to the optimal path obtained by the conventional A^? algorithm.     

A numerical computation of a 3-D stokes problem in the description of intestinal peristaltic waves

The mathematical aspects of a physical model of intestinal peristaltic waves involve numerical methods for calculating and simulating the solution. A finite element method is applied to Stokes' equations in R³ in order to calculate the velocity-pressure couple at each vertex of the pentahedron elements of the geometrical model domain, for viscous and non-compressible fluid. By using special computational programs, the domain may be stitched to create a grid domain in R³. The finite reference element of the grid is a pentahedron. The resultant linear system may be solved by applying Gauss' direct method to obtain velocities and pressures at each grid point. Graphical representations of velocity profiles in R³, show positive and negative zones for the output, with positions that vary from one geometrical model to another. The calculated velocity and pressure values are shown to change according to the applied vector efforts. The numerical results obtained are in good agreement with experimental data. This suggests that the finite element method is useful for solving Stokes' equations to describe intestinal peristalsis waves.     

A Kirchhoff-mode method for C0 bilinear and serendipity plate elements

A new formulation for the C⁰ quadrilateral and Serendipity plate elements is presented. It is an extension of the mode-decomposition approach that has been successfully applied in evaluating the C⁰ triangular linear plate element. In contrast to the triangular element, the concept of an equivalent discrete Kirchhoff configuration is utilized. The elements are applied to several examples and the performance of these elements is shown to be quite good.     

A smooth spiral tool path for high speed machining of 2D pockets

We introduce a new algorithm for generating a spiral tool path for high-speed machining of pockets without islands. The pocket may be an arbitrary simply-connected 2D shape bounded by straight-line segments and circular arcs. The tool path is generated by interpolating growing disks placed on the medial axis of the pocket. It starts inside the pocket and spirals out to the pocket boundary. The tool path is guaranteed to be free of self-intersections and allows machining of the pocket without tool retractions. The start point of the spiral may be chosen freely by the user anywhere within the pocket. Most importantly, the spiral tool path complies with a user-specified maximum cutting width. The output of our algorithm is a G¹-continuous spiral path. However, in a post-processing step, a properly adapted variant of the recent “PowerApx” package [Heimlich M, Held M. Biarc approximation, simplification and smoothing of polygonal curves by means of Voronoi-based tolerance bands. International Journal of Computational Geometry & Applications 2008;18(3):221-50] can be used to boost the tool path to C²-continuity. Our new algorithm was implemented and tested successfully on real-world data. We conclude our paper by an analysis of sample tool paths produced by our implementation.     

A probabilistic analysis of an error-correcting algorithm for the towers of Hanoi puzzle

Let S be a set of n horizontal and vertical segments on the plane, and let s, t ∈ S. A Manhattan path (of length k) from s to t is an alternating sequence of horizontal and vertical segments, s = r₀, r₁,…,r_k = t, such that r_i intersects r_i+1, 0 ? i < k. An O(n log n) time O(n log n) space algorithm is presented which, given S and t, finds a tree of shortest Manhattan paths from all s ∈ S to t. The algorithm relies on a new data structure which makes it possible to find in O(log n + p) time all p segments currently in S which intersect a given s ∈ S, and which support a deletion of any segment from S in O(log n) time, where we assume that the cost of these operations is accumulated over the whole algorithm. The structure makes use of the recently discovered Gabow and Tarjan's linear time version of the union-find algorithm on consecutive sets. We prove an Ω(n log n) lower bound on the complexity of deciding whether there is a Manhattan path between two given segments, under the linear decision tree model. Finally, some applications of the Manhattan path algorithm are indicated.     

Fracture mechanics based fatigue analysis of steel bridge decks by two-level cracked models

An alternative fatigue assessment is introduced for orthotropic highway bridge decks, based on fracture mechanics. Crack growth simulation is carried out by the numerical integration of the Paris formula, using K factors obtained from two-level cracked models of the bridge deck. Prior to the simulation, unit wheels are applied to the 3D-shell model of the deck, and the translations of characteristic points are transferred to the 2D plane strain or plane stress finite element sub-models of fatigue-sensitive areas. The K factor is computed by numerical extrapolation, introducing cracks in the sub-models. Fixed values of input parameters (initial- and critical crack length, crack growth rate data, and the error of K factor computation) are used for single simulations as well as parametric studies for sensitivity analysis. Reliability assessment is performed by repeated crack growth analysis, using histograms to generate input parameters by Monte-Carlo simulation at the start of each “virtual experiment”.     

Optimal arc spline approximation

We present a method for approximating a point sequence of input points by a G¹$G^1$-continuous (smooth) arc spline with the minimum number of segments while not exceeding a user-specified tolerance. Arc splines are curves composed of circular arcs and line segments (shortly: segments). For controlling the tolerance we follow a geometric approach: We consider a simple closed polygon P and two disjoint edges designated as the start s and the destination d. Then we compute a SMAP (smooth minimum arc path), i.e. a smooth arc spline running from s to d in P with the minimally possible number of segments. In this paper we focus on the mathematical characterization of possible solutions that enables a constructive approach leading to an efficient algorithm.     

Finite element mesh for a complete solution of a problem with a singularity

It is well known that the stress field at the tip of a crack in an elastic body exhibits a singularity. In this paper, a complete elastic solution for a center cracked plate loaded under uniform tension is obtained by a finite element plane-stress analysis using constant strain elements. No a priori assumptions about the form of the stress singularity are required. The numerical results are compared with the exact analytic solution. It is shown that a particular mesh arrangement in which the size of the elements decreases in a geometric series as the crack tip is approached yields stress and strain fields which are accurate over the entire plate, even at distances very close to the crack tip. The effects of changes in mesh arrangements on the accuracy of the solution are considered. The computations are carried out on the CRAY-1 computer and the advantages of vectorization are discussed for this problem.     

One-sided offset approximation of freeform curves for interference-free NURBS machining

Generating valid tool path curves in NURBS form is important in realizing an efficient NURBS machining. In this paper, a method for computing one-sided offset approximations of freeform curves with NURBS format as tool paths is presented. The approach first uses line segments to approximate the progenitor curve with one-sided deviations. Based on the obtained line approximating curve and its offsets, a unilateral tolerance zone (UTZ) is constructed subsequently. Finally, a C¹-continuous and completely interference-free NURBS offset curve is generated within the UTZ to satisfy the required tolerance globally. Since all of the geometric computations involved are linear, the proposed method is efficient and robust. Interference-free tool path generation thus can be achieved in NURBS based NC machining.     

The use of optimization techniques in the analysis of cracked members by the finite element displacement and stress methods

Mathematical programming is applied to the two-dimensional stationary crack problem of a body composed of nonlinear elastic incompressible material. Fully admissible displacement as well as stress formulations are used to discretize the problem. Crack tip singularity is introduced in the displacement formulation by enriched elements for plane stress and, in certain cases, by superposition for plane strain. Pointwise incompressibility is obtained through constrained displacement functions. For three crack geometries Rice's J integral is evaluated by the energy difference method for different values of the hardening index. The numerical results, which are also applicable to secondary creep problems, appear to suggest a bounding character.     

X-FEM a good candidate for energy conservation in simulation of brittle dynamic crack propagation

This paper is devoted to the simulation of dynamic brittle crack propagation in an isotropic medium. It focuses on cases where the crack deviates from a straight-line trajectory and goes through stop-and-restart stages. Our argument is that standard methods such as element deletion or remeshing, although easy to use and implement, are not robust tools for this type of simulation essentially because they do not enable one to assess local energy conservation. Standard cohesive zone models behave much better when the crack’s path is known in advance, but are difficult to use when the crack’s path is unknown. The simplest method which consists in placing the cohesive segments along the sides of the finite elements leads to crack trajectories which are mesh-sensitive. The adaptive cohesive element formulation, which adds new cohesive elements when the crack propagates, is shown to have the proper energy conservation properties during remeshing. We show that the X-FEM is a good candidate for the simulation of complex dynamic crack propagation. A two-dimensional version of the proposed X-FEM approach is validated against dynamic experiments on a brittle isotropic plate.     

Application of the Feynman-Kac path integral method in finding excited states of quantum systems

Group theory considerations and properties of a continuous path are used to define a failure tree procedure for finding eigenvalues of the Schrödinger equation using stochastic methods. The procedure is used to calculate the lowest excited state eigenvalues of eigenfunctions possessing anti-symmetric nodal regions in configuration space using the Feynman-Kac path integral method. Within this method the solution of the imaginary time Schrödinger equation is approximated by random walk simulations on a discrete grid constrained only by symmetry considerations of the Hamiltonian. The required symmetry constraints on random walk simulations are associated with a given irreducible representation and are found by identifying the eigenvalues for the irreducible representation corresponding to symmetric or antisymmetric eigenfunctions for each group operator. The method provides exact eigenvalues of excited states in the limit of infinitesimal step size and infinite time. The numerical method is applied to compute the eigenvalues of the lowest excited states of the hydrogenic atom that transform as Γ² and Γ⁴ irreducible representations. Numerical results are compared with exact analytical results.     

An Analytical Method for Decoupled Local Smoothing of Linear Paths in Industrial Robots

The linear-format path is widely adopted to approximate the continuous contour in robot controllers. The tangential discontinuity of the linear paths usually causes the discontinuity of the joint velocity. To comply with the joint kinematics limits, the robots have to stop at each corner point, resulting in a great loss of efficiency. To achieve a smooth motion, this paper presents an analytical decoupled C³ continuous local path smoothing method for industrial robots. The tool position path is smoothed in the reference frame while the tool orientation is smoothed in the rotation parametric space based on the exponential coordinates of rotations. The quintic B-splines are inserted at the corners of the linear segments to achieve the G³ continuity of the tool position path and tool orientation path. The orientation smoothing error is constrained analytically. By reparameterization of the remaining linear segments using specially constructed B-splines, the C³ continuity of the tool position path and tool orientation path is achieved. Then, the synchronization of the tool orientation path and tool position path can be guaranteed by sharing the same curve parameter. Besides, to improve the smoothness of the angular motion on the remaining linear segments during parameter synchronization, the transition lengths of the inserted B-splines are optimized. The proposed local smoothing method guarantees that the generated smooth orientation path in the rotation space is invariant with the selection of the reference frame and the orientation of the tool frame, and ensures the jerk-continuous motion with a smoother angular motion on the remaining linear segments, which could improve the motion smoothness of the tool path and tracking accuracy. The effectiveness of the proposed method is validated through simulation and experiments.