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1.Tool path

1.1 Tool path generation method for Multi axis Machining

In the paper Presented by Young-Keun Choi & all, a tool path generation method for multi-axis machining of free-form surfaces using Bezier curves and surfaces. The tool path generation includes two core steps. In which first is

the forward-step function that determines the maximum distance, called forward step, between two cutter contact

(CC) points with a given tolerance. The second component is the side step function which determines the maximum

distance, called side step, between two adjacent tool paths with a given scallop height. Using the Bezier curves and

surfaces, we generate cutter contact (CC) points for freeform surfaces and cutter location (CL) data files for post

processing. a Several objects are machined using a multi-axis milling machine. As part of the validation process, the

tool paths generated from Bezier curves and surfaces are analyzed to compare the machined object and the desired

object.[1]

1.2 Tool path pattern

In the Paper presented by Avisekh Banerjee & all they proposes a process planning for machining of a Floor which

is the most prominent elemental machining feature in a 2D pocket machining . Traditionally, process planning of

2D pocket machining is posed as stand-alone problem involving either tool selection, tool path generation or

machining parameter selection, resulting in sub-optimal plans. For this reason, tool path generation and feed

selection is proposed to be integrated with an objective of minimizing machining time under realistic cutting force

constraints for given pocket geometry and cutting tool. A morphed spiral tool path consisting of G1 continuous bi-

arc and arc spline is proposed as a possible tool path generation strategy with the capability of handling islands in

pocket geometry. Proposed tool path enables a constant feed rate and consistent cutting force during machining in

typical commercial CNC machine tool. The constant feed selection is based on tool path and cutting tool geometries

as well as dynamic characteristics of mechanical structure of the machine tool to ensure optimal machining

performance. The proposed tool path strategy is compared with those generated by commercial CAM software. The

calculated tool path length and measured dry machining time shows the considerable advantage of the proposed tool

path. For optimal machining parameter selection, the feed per tooth is iteratively optimized with a pre-calibrated

cutting force model, under a cutting force constraint to avoid tool rupture. The optimization result shows around

32% and 40% potential improvement in productivity with one and two feed rate strategies respectively.[2]

1.3 Tool path

An analytical curvature-continuous path-smoothing algorithm is developed for the high speed machining of a linear

tool path. The algorithm can be used in a post-processing stage or NC unit. Every segment junction of the linear tool

path, which is the point of tangent discontinuity, is blended by inserting two cubic Bezier spiral curves. A tool path,

which is composed of cubic Bezier curves and lines, is then obtained to replace the linear tool path. The new tool

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path is everywhere curvature-continuous, and both the tangent and curvature discontinuities are avoided. The feed

motion will be more stable since the discontinuities are the most important sources of feed fluctuation. In the

blending algorithm, the approximation error at the segment junction is accurately guaranteed and the control points

of the two cubic Bezier transition curves are all analytically computed. The maximal curvature in every Be zier

transition pair, which is critical for velocity planning, is also analytically computed. The analytical expressions

provide a way to optimize the curvature radii of the transition curves to pursue the high feed speed. The path-

smoothing methods for the post-processing stage and NC unit are both developed. The computational examples

confirm the validity of the algorithm. The transition algorithm has been integrated into an open NC system. Cutting

experiments show that the curvature-continuous tool path generates smoother feed and consumes shorter machining

time than the original linear tool path[6]

Bezier curve for Metamodeling of Simulation output

M any design optimization problems rely on simulation models to obtain feasible solutions. Even with substantialimprovement in the computational capability of computers, the enormous cost of computation needed for simulation

makes it impractical to rely on simulation models. The use of metamodels or surrogate approximations in place of

actual simulation models makes analysis realistic by reducing computational burden. There are many popular

metamodeling techniques such as Polynomial Regression, Multivariate Adaptive Regression Splines, Radial Basis

Functions, Kriging and Artificial Neural Networks. This research proposes a new metamodeling technique

that uses Bezier curves and patches. The Bezier curve method is based on interpolation like Kriging and Radial

Basis Functions. In this research the Bezier Curve method will be used for output modeling of univariate and

bivariate output modeling. Results will be validated using comparison with some of the most popular meta modeling

techniques.[7]

Intersection of Bezier Curves

We give the first complete subdivision algorithm for the intersection of two Bezier curves F,G, possibly with

tangential intersections. Our approach to robust subdivision algorithms is based on geometric separation bounds, and

using a criterion for detecting non-crossing intersection of

curves. Our algorithm is adaptive, being based only on exact big float computations. In particular, we avoid

manipulation of algebraic numbers and resultant computations. It is designed to be competitive with current

algorithms on nice inputs. All standard algorithms assume F,G to be relatively prime our algorithm needs a

generalization of this.[4]

B-spline curve

Pottmann et al. propose an iterative optimization scheme for approximating a target curve with a B-spline curve

based on square distance minimization, or SDM. The main advantage of SDM is that it does not need a

parameterization of data points on the target curve. Starting with an initial B-spline curve, this scheme makes an

active B-spline curve converge faster towards the target curve and produces a better approximating B-spline curve

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than existing methods relying on data point parameterization. However, SDM is sensitive to the initial B-spline

curve due to its local nature of optimization. To address this, we integrate SDM with procedures for automatically

adjusting both the number and locations of the control points of the active spline curve. This leads to a method that

is more robust and applicable than SDM used alone. Furthermore, it is observed that the most time consuming part

of SDM is the repeated computation of the foot-point on the target curve of a sample point on the active B-spline

curve. In our implementation, we speed up the foot-point computation by pre-computing the distance field of the

target curve using the Fast Marching Method. Experimental examples are presented to demonstrate the effectiveness

of our method.[9]

B-splines technique

Three-dimensional (3D) anthropometry based on the laser scanning technique not only provides one-dimensional

measurements calculated in accordance with the landmarks which are pre-located on the human body surface

manually, but also the 3D shape information between the landmarks. This new technique used in recent ergonomic

research has brought new challenges to resolving the application problem that was generally avoided byanthropometric experts in their researches. The current research problem is concentrating on how to shift and

develop one-dimensional measurements (1D landmarks) into three-dimensional measurements (3D land-surfaces).

The main purpose of this paper is to test whether the function of B-splines can be used to fit 3D scanned human

heads, and to for further study to develop a computer aided ergonomic design tool (CAED). The result shows that B-

splines surfaces can effectively reconstruct 3D human heads based on the laser scanning technique[10]

Hermite b-spline approximation

Free-Form Deformation Techniques (FFD) are commonly used to generate animations, where a polygonal

approximation of the final object suffices for visualization purposes. However, for some CAD/CAM applications,

we need an explicit expression of the object, rather than a collection of sampled points. If both object and

deformation are polynomial, their composition yields a result that is also polynomial, albeit very high degree,

something undesirable in real applications. To solve this problem, we transform each curve or surface composing

the object, usually expressed in the Bernstein basis, to a modified Newton form. In this representation, the two-point

analogue of Taylor expansions, the composition admits a simple expression in terms of discrete convolutions, and

degree reduction corresponding to Hermite approximation is trivial by dropping high-degree coefficients.

Furthermore, degree-reduction can be incorporated into the composition. Finally, the deformed curve or surface is

converted back to the Bernstein form. This method extends to general non-polynomial deformation, such as bending

and twisting, by computing a polynomial approximant of the deformation[11]

B-Spline Surfaces

Creating freeform surfaces is a challenging task even with advanced geometric modeling systems. Laser range

scanners offer a promising alternative for model acquisitionthe 3D scanning of existing objects or clay maquettes.

The problem of converting the dense point sets produced by laser scanners into useful geometric models is referred

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to as surface reconstruction. In this paper, we present a procedure for reconstructing a tensor product B-spline

surface from a set of scanned 3D points. Unlike previous work which considers primarily the problem of fitting a

single B-spline patch, our goal is to directly reconstruct a surface of arbitrary topological type. We must therefore

define the surface as a network of B-spline patches. A key ingredient in our solution is a scheme for automatically

constructing both a network of patches and a parametrization of the data points over these patches. In addition, we

define the B-spline surface using a surface spline construction, and demonstrate that such an approach leads to an

efficient procedure for fitting the surface while maintaining tangent plane continuity. We explore adaptive

refinement of the patch network in

order to satisfy user-specified error tolerances,[12]

B-spline approximation in boundary face method

In this paper, basis functions generated from B-spline or Non-Uniform Rational B-spline (NURBS), are used for

approximating the boundary variables to solve the 3D linear elasticity Boundary Integral Equations (BIEs). The

implementation is based on the BFM framework in which both boundary integration and variable approximation areperformed in the parametric spaces of the boundary surfaces to keep the exact geometric information in the BIEs. In

order to reduce the influence of tensor product of B-spline and make the discretization of a body surface easier, the

basis functions defined in global intervals are translated into local form. B-spline fitting function built with the local

basis functions is converted into an interpolation type of function in which the nodal values of the boundary

variables are used for control points. Numerical tests for 3D linear elasticity problems show that the BFM with B-

spline basis functions outperforms that with the well-known Moving Least Square (MLS) approximation.[ Error!

Reference source not found.]

Hierarchical bases of spline spaces

The prospect of applying spline spaces over T-subdivisions to adaptive isogeometric

analysis is an exciting one. One major issue with spline spaces over T-subdivisions is in

providing proper bases (shape functions) for finite element analysis. In this paper, we

propose a method for the construction of hierarchical bases of a spline space with highest

order smoothness over a consistent hierarchical T-subdivision. Our method is induced by

the surjection condition, and this set of basis functions is hierarchically adaptive. We also

present a concrete set of non-negative hierarchical bases over a T-subdivision and apply

them in adaptive finite element analysis.[14]

T-spline based isogeometric analysis

Isogeometric analysis has been recently introduced as a viable alternative to the standard, polynomialbased finite

element analysis. Initially, the isogeometric approach has been developed using the NURBS and although it has

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been shown that it can outperform the classical finite element method in many aspects, there are several drawbacks,

namely related to the handling trimmed geometries and to the refinement of the adopted discretization. These may

be overcome by extending the concept of isogeometric analysis to so-called T-splines which are a generalization of

NURBS. This paper presents how the isogeometric analysis based on T-spline can be integrated within an object

oriented finite element environment. The class hierarchy and corresponding methods are designed in such a way,

that most of nthe existing functionality of the finite element code is reused. The missing data and algorithms are

developed and implemented in such a way that the object oriented features are fully retained. The performance of

the implemented T-spline based isogeometric analysis methodology is presented on a simple example.[15]

hierarchical b-spline finite element method

A novel technique is presented to facilitate the implementation of hierarchical b-splines and their interfacing with

conventional finite element implementations. The discrete interpretation of the two-scale relation, as common in

subdivision schemes, is used to establish algebraic relations between the basis functions and their coefficients ondifferent levels of the hierarchical b-spline basis. The subdivision projection technique introduced allows us first to

compute all element matrices and vectors using a fixed number of same-level basis functions. Their subsequent

multiplication with subdivision matrices projects them, during the assembly stage, to the correct levels of the

hierarchical b-spline basis. The proposed technique is applied to convergence studies of linear and geometrically

nonlinear problems in one, two and three space dimensions.[16]

B-splines and NURBS

This paper presents a B-splines and NURBS based finite element method for self-consistent solution of the Kohn

Sham equations [1,2] for electronic structure modeling of semiconducting materials. A Galerkin formulation is

developed for the Schrdinger wave equation (SWE) that yields a complex-valued generalized eigenvalue problem.

The nonlinear SWE that is embedded with a non-local potential as well as the nonlinear Hartree and exchange

correlation potentials is solved in a self-consistent fashion. In the self consistent solution procedure, a Poisson

problem is integrated and solved as a function of the electron density that yields the local pseudopotential (for

pseudopotential formulation) and the Hartree potential for SWE. Accuracy and convergence properties of the

method are assessed through test cases and the superior performance of higher-order B-splines and NURBS basis

functions as compared to the corresponding Lagrange basis functions is highlighted. Self-consistent solutions for

semiconducting materials, namely, Gallium Arsenide (GaAs) and graphene are presented and results are validated

via comparison with the plane wave solutions.[17]

Generalized B-splines as a tool in isogeometric analysis

The concept of isogeometric analysis has been proposed in [13], where NURBS are considered as basis of the

analysis, thanks to their ability to construct an exact geometric model in several practical applications and to their

popularity in commercial CAD systems. In this paper we propose an alternative to the rational model presenting an

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isogeometric analysis approach based on generalized B-splines. Geometric models exactly represented by

generalized B-splines include those generated by NURBS. Moreover, generalized B-splines possess all fundamental

properties of algebraic B-splines (and NURBS) including classical refinement processes as hpk refinements.

Finally, since generalized B-splines are not confined to rational functions, they behave completely similar to

algebraic B-splines with respect to differentiation and integration. This seems to be of interest in the treatment of

some relevant problems.[18]

A research tool for Isogeometric Analysis of PDEs

GeoPDEs (http://geopdes.sourceforge.net) is a suite of free software tools for applications on Isogeometric Analysis

(IGA). Its main focus is on providing a common framework for the implementation of the many IGA methods for

the discretization of partial differential equations currently studied, mainly based on B-Splines and Non-Uniform

Rational B-Splines (NURBS), while being flexible enough to allow users to implement new and more general

methods with a relatively small effort. This paper presents the philosophy at the basis of the design of GeoPDEs and

its relation to a quite comprehensive, abstract definition of IGA.[19]

References-

1. Young-Keun Choi a,*, A. Banerjee b, Jae-Woo Lee Tool path generation for free form surfaces usingBezier curves/surfaces Computers & Industrial Engineering 52 (2007) 486501

2. Avisekh Banerjee , Hsi-Yung Feng, Evgueni V. Bordatchev, Process planning for Floor machining of 2Dpockets based on a morphed spiral tool path pattern Computers & Industrial Engineering xxx (2012) xxx

xxx

3. Qing Zhen Bi, Yong Qiao Jin, Yu Han Wang, Li Min Zhu, Han Ding An analytical curvature-continuousBezier transition algorithm for high-speed machining of a linear tool path International Journal of Machine

Tools & Manufacture 57 (2012) 5565

4. Mohamed Azaouzi a,, Nadhir Lebaal Tool path optimization for single point incremental sheet formingusing response surface method Simulation Modeling Practice and Theory 24 (2012) 4958

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5. QiMing Tian *, YuPin Luo, DongCheng Hu Spiral-fashion embroidery path generation in embroidery CADsystems Computer-Aided Design 38 (2006) 125133

6. Qing Zhen Bi, Yong QiaoJin ,Yu Han Wang, Li Min Zhu, Han Ding n An analytical curvature-continuousBezier transition algorithm for high-speed machining of a linear tool path

7. Harish Kingre Nagpur University, December, 2004.8. J.M. Carnicer, J. Delgado , J.M. Pea Progressive iteration approximation and the geometric algorithm

Computer-Aided Design 44 (2012) 143145

9. Huaiping Yang Wenping Wang Jiaguang Sun Control point adjustment for B-spline curve approximationComputer-Aided Design 36 (2004) 639652

10. B. Zhang*, J.F.M. Molenbroek Representation of a human head with bi-cubic B-splines technique based onthe laser scanning technique in 3D surface anthropometry Applied Ergonomics 35 (2004) 459465

11. J. Snchez-Reyes , J.M. Chacn Hermite approximation for free-form deformation of curves and surfacesComputer-Aided Design 44 (2012) 445456.

12. Matthias Eck Hugues Hoppe Automatic Reconstruction of B-Spline Surfaces of Arbitrary TopologicalType

13. Jinliang Gu,JianmingZhang , XiaominSheng,GuanyaoLi B-spline approximation in boundary face methodfor three-dimensional linear elasticity Engineering Analysis with Boundary Elements 35 (2011) 11591167

14. Meng Wu, Jinlan Xu, Ruimin Wang, Zhouwang Yang Hierarchical bases of spline spaces with highestorder smoothness over hierarchical T-subdivisions Computer Aided Geometric Design 29 (2012) 499509

15. Daniel Rypl , Borek PatzakObject oriented implementation of the T-spline based isogeometric analysisAdvances in Engineering Software 50 (2012) 137149

16. P.B. Bornemann, F. Cirak_ A subdivision-based implementation of the hierarchical b-spline finite elementmethod Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ,

U.K.

17. Arif Masud , Raguraman Kannan B-splines and NURBS based finite element methods for KohnShamequations Comput. Methods Appl. Mech. Engrg. 241244 (2012) 112127

18. Carla Manni a, , Francesca Pelosi a, M. Lucia Sampoli Generalized B-splines as a tool in isogeometricanalysis Comput. Methods Appl. Mech. Engrg. 200 (2011) 867881

19. C. de Falco a,, A. Reali b,c,d, R. Vzquez c GeoPDEs: A research tool for Isogeometric Analysis of PDEsAdvances in Engineering Software 42 (2011) 10201034

20.