Surface Design Based On Triharmonic Equations

The design of curves and surfaces is one of the main roles of computational geometry,computer aided geometric design(CAGD)and computer aided design(CAD).Surface design based on partial differential equation(PDE)has been widely used in many fields and becomes a hot topic in recent years.In practical application,we hope to design a surface which interpolates given boundary conditions and satisfies a certain PDE,which is called PDE surface.In this way,through the adjustment of boundary conditions,we can design the physical-based PDE surfaces for interactive design requirements.At present,there are many researches on harmonic surfaces and biharmonic surfaces.Harmonic equation is related to minimal surfaces and have been widely used in physics,such as electromagnetism and fluid flows,etc.The biharmonic equation can be applied to the tension in elastic membranes,stress and strain.Triharmonic PDE is a sixth order elliptic partial differential equation.Surfaces is called triharmonic surfaces if it satisfies triharmonic equation with boundary condition.In fluid mechanics,the triharmonic equation can be used to describe the two-dimensional slowly rotating high viscous fluid flow in a small cavity.It provides curvature boundary conditions and more shape control parameters for surface design.The research on triharmonic surface is not only extends the results of the harmonic and biharmonic surfaces,but also deepens the basic theory of higher order polynomial solutions of PDEs.In this paper,we mainly study the design of triharmonic surfaces.For triharmonic tensor product and triangular B(?)zier surfaces which satisfy the given boundary conditions,the uniqueness of the surfaces is analyzed,and the algorithms for surface design under each condition also are given.For a special sixth order PDE equation,the bicubic B-spline surface solution satisfying the given boundary and its design algorithm are given.The structure and main work of this paper are as follows:In Chapter 1,firstly,the related backgrounds of parametric curves and surfaces,and PDE surface are briefly reviewed.Secondly,the research progress of harmonic and biharmonic surfaces are introduced.The definition of triharmonic surface and the detailed proof that triharmonic surface being quadratic functional extremum surface is also provided.Finally,the main work of this paper is briefly introduced.In Chapter 2,Firstly,the dimension of the solution space of n × n degree polynomial satisfying triharmonic equation is given.The idea of proof is to transform the parameters in real number field into complex number field,and then obtain the dimension of solution space in complex field.For tensor product B(?)zier surface satisfying triharmonic equation,we propose three kinds of boundary conditions.Starting from different boundary conditions,by the transformation between Bernstein basis and power base,the unknown control points of resulting surfaces are determined uniquely for every boundary condition by the linear system satisfying the triharmonic equation.And design algorithm of the triharmonic B(?)zier surface satisfying the boundary conditions are proposed.In Chapter 3,we study the construction of triharmonic triangular B(?)zier surfaces.We provide three kinds of boundary conditions,and propose method to construct triharmonic triangular Bézier surfaces uniquely.Representative examples are given for three kinds of boundary conditions.And we indicate a kind of boundary condition is more suitable for surface design,namely symmetric boundary condition,through comparison of examples.In Chapter 4,the design of bicubic B-spline surfaces satisfying a special sixth order PDE is proposed.We give a closed boundary conditions,and obtain the numerical stable linear system and the corresponding algorithm by moving the extended 4 × 4 mask on the control points.In general,it is impossible to construct a single piece of bicubic B-spline surface to satisfy this kind of sixth order PDE.By using the algorithm,we can construct piecewise bicubic B-spline surfaces with C1/G1 continuity,satisfying the specific sixth order PDE and interpolating the given boundary curves.The effectiveness of the method is verified by some examples.In Chapter 5,we conclude the whole thesis and point out the future works.