Optimization of Geometrically Trimmed B-Spline Surfaces

2005 ◽  
Author(s):  
Xinyu Zhang ◽  
Yaohang Li ◽  
Arvid Myklebust ◽  
Paul Gelhausen
Keyword(s):  
Optimization Method ◽  
Objective Functions ◽  
Global Optimization Method ◽  
Initial Value ◽  
B Spline ◽  
Original Surface ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Approximation Errors ◽  
High Degree

Unlike the visual trimming of B-spline surfaces, which hides unwanted portions in rendering, the geometric trimming approach provides a mathematically clean representation without redundancy. However, the process may lead to significant deviation from the corresponding portion on the original surface. Optimization is required to minimize approximation errors and obtain higher accuracy. In this paper, we describe the application of a novel global optimization method, so-called hybrid Parallel Tempering (PT) and Simulated Annealing (SA) method, for the minimization of B-spline surface representation errors. The high degree of freedom within the configuration of B-spline surfaces as well as the “rugged” landscapes of objective functions complicate the error minimization process. The hybrid PT/SA method, which is an effective algorithm to overcome the slow convergence, waiting dilemma, and initial value sensitivity, is a good candidate for optimizing geometrically trimmed B-spline surfaces. Examples of application to geometrically trimmed wing components are presented and discussed. Our preliminary results confirm our expectation.

B2: An Interactive Quality Visualization and Improvement Technique for B-Spline Surfaces

1998 ◽  
Author(s):  
Yifan Chen ◽  
Klaus-Peter Beier
Keyword(s):  
Surface Quality ◽  
Control Points ◽  
Small Surface ◽  
B Spline ◽  
Original Surface ◽  
Interactive Technique ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Surface Irregularities

Abstract A new interactive technique for B-spline surface quality visualization and improvement, called the B2 method, is presented. This method interpolates the control points of a given B-spline surface using a second B-spline surface. If small irregularities exist in the control points of the original surface, they will be magnified through the second B-spline and demonstrated as large distortions in its control points. This facilitates the detection of small surface irregularities. Subsequently, the surface may be improved through direct and interactive adjustment of the second B-spline’s control polyhedron.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

Sequential B-Spline Surface Construction Using Multiresolution Data Clouds

2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Yuan Yuan ◽  
Shiyu Zhou
Keyword(s):  
Control Points ◽  
Finite Sample ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Filtering Technique ◽  
Engineering Practices ◽  
Asymptotical Convergence ◽  
Surface Construction ◽  
Fitting Error

B-spline surfaces are widely used in engineering practices as a flexible and efficient mathematical model for product design, analysis, and assessment. In this paper, we propose a new sequential B-spline surface construction procedure using multiresolution measurements. At each iterative step of the proposed procedure, we first update knots vectors based on bias and variance decomposition of the fitting error and then incorporate new data into the current surface approximation to fit the control points using Kalman filtering technique. The asymptotical convergence property of the proposed procedure is proved under the framework of sieves method. Using numerical case studies, the effectiveness of the method under finite sample is tested and demonstrated.

scholarly journals Hierarchical Genetic Algorithm for B-Spline Surface Approximation of Smooth Explicit Data

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
C. H. Garcia-Capulin ◽  
F. J. Cuevas ◽  
G. Trejo-Caballero ◽  
H. Rostro-Gonzalez
Keyword(s):  
Genetic Algorithm ◽  
Geometric Modeling ◽  
Data Set ◽  
Surface Approximation ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Surface Dimension ◽  
Hierarchical Genetic Algorithm

B-spline surface approximation has been widely used in many applications such as CAD, medical imaging, reverse engineering, and geometric modeling. Given a data set of measures, the surface approximation aims to find a surface that optimally fits the data set. One of the main problems associated with surface approximation by B-splines is the adequate selection of the number and location of the knots, as well as the solution of the system of equations generated by tensor product spline surfaces. In this work, we use a hierarchical genetic algorithm (HGA) to tackle the B-spline surface approximation of smooth explicit data. The proposed approach is based on a novel hierarchical gene structure for the chromosomal representation, which allows us to determine the number and location of the knots for each surface dimension and the B-spline coefficients simultaneously. The method is fully based on genetic algorithms and does not require subjective parameters like smooth factor or knot locations to perform the solution. In order to validate the efficacy of the proposed approach, simulation results from several tests on smooth surfaces and comparison with a successful method have been included.

Filling N-Sided Holes With Trimmed B-Spline Surfaces Based on Energy-Minimization Method

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Xiaodong Liu
Keyword(s):  
Energy Minimization ◽  
Computer Aided Design ◽  
Control Points ◽  
Minimization Method ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Key Issues ◽  
Complex Constraints ◽  
Aided Design

Using a trimmed rectangular B-Spline surface to fill an n-sided hole is a much desired operation in computer aided design (CAD), but few papers have addressed this issue. Based on an energy-minimization or variational B-Spline technique, the paper presents the technique of using one single trimmed rectangular B-Spline surface to fill an n-sided hole. The method is efficient and robust, and takes a fraction of a second to fill n-sided holes with high-quality waterproof B-Spline surfaces under complex constraints. As the foundation of filling n-sided holes, the paper also presents the framework and addresses the key issues on variational B-Spline technique. Without any precalculation, the variational B-Spline technique discussed in this paper can solve virtually any B-Spline surface with up to 20,000 control points in real time, which is much more efficient and powerful than previous work in the variational B-Spline field. Moreover, the result is accurate and satisfies CAD systems' high-precision requirements.

A New Method for Deformation of B-Spline Surfaces

2010 ◽  
Vol 139-141 ◽  
pp. 1260-1263
Author(s):  
Xian Guo Cheng ◽  
Wei Jun Liu
Keyword(s):  
Optimization Problem ◽  
Local Deformation ◽  
Geometric Constraints ◽  
Surface Patch ◽  
The Finite Element Method ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Single Force ◽  
External Forces

This paper presents an efficient method for deforming B-spline surfaces, based on the surface energy minimization. Firstly, using an analogy between the B-spline surface patch and the thin-plate element of the finite element method, and applying external forces on the surface with some given geometric constraints, the forces can locate on part of the surface or the surface. Then, the energy of the B-spline surface can change with the change of the forces. Finally, a new B-spline surface is generated by solving an optimization problem of change of the energy. The forces can be a single force, a distributed force and set of isolated force. The method can accomplish easily local deformation and total deformation of the B-spline surface.

Low-degree approximation of high-degree B-spline surfaces

1993 ◽  
Vol 9 (4) ◽  
pp. 198-209 ◽  
Author(s):  
S. T. Tuohy ◽  
L. Bardis
Keyword(s):  
B Spline ◽  
Spline Surfaces ◽  
Low Degree ◽  
High Degree

Filling N-Sided Holes Using Trimmed B-Spline Surfaces

2012 ◽  
Author(s):  
Xiaodong Liu
Keyword(s):  
Energy Minimization ◽  
Control Points ◽  
Variational Technique ◽  
High Quality ◽  
B Spline ◽  
Cad Systems ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Key Issues ◽  
Complex Constraints

Using one single trimmed B-Spline surface to fill an n-sided hole is a much desired operation in CAD, but few papers have addressed this issue. The paper presents the method of using trimmed B-Spline surfaces to fill n-sided holes based on energy minimization or variational technique. The method is efficient and robust, and takes less than one second to fill n-sided holes with high quality B-Spline surfaces under complex constraints. As the foundation of filling n-sided holes, some key issues on variational B-Spline technique are also discussed. The variational technique discussed is significantly much more efficient and powerful than previous research, and the result is very accurate to satisfy CAD systems’ high-precision requirements. We demonstrate that, without any pre-calculation, the discussed technique is efficient enough to solve a B-Spline surface with up to 20,000 control points in real time while satisfying an arbitrary combination of point and curve constraints.

An Efficient Sensing Localization Algorithm for Free-Form Surface Digitization

2008 ◽  
Vol 8 (2) ◽  
Author(s):  
Yunbao Huang ◽  
Xiaoping Qian
Keyword(s):  
Search Time ◽  
Divide And Conquer ◽  
Free Form ◽  
Point Problem ◽  
Localization Algorithm ◽  
B Spline ◽  
Divide And Conquer Algorithm ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Free Form Surface

We present a divide-and-conquer method that efficiently finds a near-optimal distribution of sensing locations for free-form surface digitization. We formulate a next-best-point problem and transform the uncertainty of a B-spline surface into a higher-dimensional B-spline surface. This technique allows the use of the convex hull and subdivision properties of B-spline surfaces in the divide-and-conquer algorithm. It thus greatly reduces the search time for determining the next best sensing location.

scholarly journals Digital Images with 3D Geometry from Data Compression by Multi-scale Representations of B-Spline Surfaces

2011 ◽  
Vol 1 (3) ◽  
pp. 240-250 ◽  
Author(s):  
K. Koch
Keyword(s):  
Data Compression ◽  
Laser Scanning ◽  
3D Model ◽  
Digital Images ◽  
Three Dimensional Model ◽  
B Spline ◽  
Multi Scale ◽  
Spline Surface ◽  
Spline Surfaces ◽  
3D Geometry

Digital Images with 3D Geometry from Data Compression by Multi-scale Representations of B-Spline SurfacesTo build up a 3D (three-dimensional) model of the surface of an object, the heights of points on the surface are measured, for instance, by a laser scanner. The intensities of the reflected laser beam of the points can be used to visualize the 3D model as range image. It is proposed here to fit a two-dimensional B-spline surface to the measured heights and intensities by the lofting method. To fully use the geometric information of the laser scanning, points on the fitted surface with their intensities are computed with a density higher than that of the measurements. This gives a 3D model of high resolution which is visualized by the intensities of the points on the B-spline surface. For a realistic view of the 3D model, the coordinates of a digital photo of the object are transformed to the coordinate system of the 3D model so that the points get the colors of the digital image. To efficiently compute and store the 3D model, data compression is applied. It is derived from the multi-scale representation of the dense grid of points on the B-spline surface. The proposed method is demonstrated for an example.

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