Variationally Improved Bézier Surfaces with Shifted Knots

The Plateau-Bézier problem with shifted knots is to find the surface of minimal area amongst all the Bézier surfaces with shifted knots spanned by the admitted boundary. Instead of variational minimization of usual area functional, the quasi-minimal Bézier surface with shifted knots is obtained as the solution of variational minimization of Dirichlet functional that turns up as the sum of two integrals and the vanishing condition gives us the system of linear algebraic constraints on the control points. The coefficients of these control points bear symmetry for the pair of summation indices as well as for the pair of free indices. These linear constraints are then solved for unknown interior control points in terms of given boundary control points to get quasi-minimal Bézier surface with shifted knots. The functional gradient of the surface gives possible candidate functions as the minimizers of the aforementioned Dirichlet functional; when solved for unknown interior control points, it results in a surface of minimal area called quasi-minimal Bézier surface. In particular, it is implemented on a biquadratic Bézier surface by expressing the unknown control point P 11 as the linear combination of the known control points in this case. This can be implemented to Bézier surfaces with shifted knots of higher degree, as well if desired.

Generalized Developable Cubic Trigonometric Bézier Surfaces

This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bézier (GDCT-Bézier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-Bézier surfaces are made by means of GDCT-Bézier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-Bézier surfaces are designed by maintaining the classic Bézier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bézier surface. The conditions appropriate and suitable for G1, Farin–Boehm G2, and G2 Beta continuity in two adjacent GDCT-Bézier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering.

A Parametric Modeling Approach for Prediction of Load Distribution due to Fluid Structure Interaction on Aircraft Structures

The Fluid Structure Interaction (FSI) is a critical multi-physics phenomenon in the aerospace applications for computing loads. Including the FSI effects on the analysis requires high computational cost. A computationally efficient framework is presented in this study for predicting the FSI effects. The high-fidelity structural model is reduced on the elastic axis by using an efficient structural idealization technique. A parametric model generation process is developed by using Bezier surface control vertices (CVs) to estimate the changing load distribution under deformation. The aircraft wing outer surface is created by using Bezier surface modeling method for this purpose. The CVs of the surfaces are perturbed to predict the effect of the deformed shape on the load distribution. This method allows to predict the load distribution by using a few CVs instead of using all grid points. The Aerodynamic Influence Coefficients (AIC) matrix is generated based on the predicted loads based on this parametric modeling approach instead of conducting computationally expensive fluid flow analysis. The presented framework is implemented for an aircraft wing design to show its efficacy.

Automatic Spray Trajectory Optimization on Bézier Surface

The trajectory optimization of automatic spraying robot is still a challenging problem, which is very important in the whole spraying work. Spray trajectory optimization consists of two parts: spray space path and end-effector moving speed. A large number of spraying experiments have proved that it is very important to find the best initial trajectory of spraying. This paper presents an automatic spray trajectory optimization that is based on the Bézier surface. Spray the workpiece for Bezier triangular surface modeling and find the best initial trajectory of the spraying robot, establish the appropriate spraying model, plan the appropriate space path, and finally plan the trajectory optimization along the specified painting path. The validity and practicability of the method presented in this paper are proved by an example. This method can also be extended to other applications.