| As the requirement of the geometry modeling technologies is increasing in the pro-duction industry,the requirement of the methods for geometric modelling also grows in the geometry designation area.On the promotion of these requirements,more new re-search topics spring out,especially the modelling methods driven by physical properties have become an important topic in this research area.Curve construction is the most kernel problem in planar geometric processing.S-moothness and Continuity degree are common requirements of curve quality.In order to realize a more flexible adjustment to curves,parametric spline becomes the most im-portant method to represent curves.Rational Quadric Bezier curve is a kind of common splines,and it is a length of conic arc in essence.The traditional research works about such splines are always based on their parameterizations.But the optimization results are constrained by the fixed parameters at knots and the complexity of curvature ex-pression.How to sufficiently utilize the adjustment capacity from freedom degree of Rational Quadratic Bezier Splines to get splines with higher continuity and smoothness is an sensible problem in planar geometric modelling.Surface construction is an important problem in 3-D modelling.The discrete rep-resentation based on triangular mesh earns a wider application than splines due to its advantage in terms of flexibility.In the nature,the surfaces shaped by the surface ten-sion(such as water droplet,soap film and bubble)have a very good smoothness,so they are widely used in the building designation and art works.However,the current tension-determined surfaces constructed with triangular meshes are not able to get both high accuracy and high mesh quality.So searching for a new method for such genera-tion has both theoretical sense and application value.Self-Supporting Surface is a kind of smooth surfaces with special shapes,which can exactly ensure the balance between the gravity of materials and the stress along the tangent direction,such that the shear stress and bending moment are avoided.Be-cause this distribution can enhance the firmness of the building,self-supporting sur-faces have extremely high value in architectures.But the equilibrium equations about self-supportness are very complex and contain a troublesome unknown function,which causes great difficulties to find an objection function for their construction,not to men-tion constructing self-supporting surfaces with true sense via numerical computations.So finding an objective function to construct self-supporting surfaces in theory is an extremely vital research topic in architectures and elastic mechanics.In the finite element analysis,structural quadrilateral meshes have advantages in the speed of calculation and accuracy of results.Some methods for automatically gen-erating quadrilateral mesh have been found,but the output mesh quality cannot be en-sured.Parametrization method based on cross field is the most effective method to generate quadrilateral meshes by now.However,there are two unsolved problems in this method:One is how to optimize the positions and quantity of singularities to make the distribution of singularities and parameters be more rational;another is how to make the edges of the output mesh be parallel to the directions of the cross field to reduce the angular distortion and avoid defect.The two problems are vital to improve the mesh quality,so finding their solutions has a great applicative sense.Above all,in geometric modeling problems,the constructions of curves and sur-faces may bring multiple constraints,especially after the physical properties are ap-pended.These constraints not only bring conflicts,but also have complex forms,so that the optimum solutions are difficult to derive.Solving such kind of problems is very challenging.Due to the above discussions,this paper makes deep researches aboutthe conic spline interpolation with high continuity and smoothness,the construction of tension determined surfaces and self-supporting surfaces with high accuracy and meshquality and the generation method about quadrilateral meshes,and proposed new theory and algorithms to address these problems.The concrete research works and results are listed as follows:1.A new method to interpolate a sequence of samples with conic splines is pro-posed,such that the continuities at the joints can reach C3.The number of cur-vature extrema is also reduced to improve the smoothness of splines.The con-struction method is based on basic geometric elements to avoid the constraints from parameters,so that the degree of freedom of the splines is sufficiently uti-lized.In the new method,the weights in a Rational Quadratic Bezier Curve are translated into an equivalent form,called Chord-Tangent Ratio.The new method represents the curvatures and the curvature change rates at the terminals with the arguments of the tangents and Chord-Tangent Ratios,so that the geometric con-struction problem is converted to a conditional extremum problem,and derives the unknowns by solving such problem.Compared to the existing methods,the new method has 3 advantages:(1)All the splines get through the samplings which is beneficial for accurate control.(2)The new method can construct the splines with G3 continuity degree,which is the highest degree of the capacity of conics,while the existing method can only get G2.(3)The number of extrema on the curves is reduced to a minimum,so the curves are better faired.2.We propose a method based on triangular mesh to construct tension determined surfaces that can generate such surfaces with both high accuracy and high mesh quality.In the new method,the vectors of mean curvature flow are projected to the normals and the movements of CVT optimization are constrained to the tangent planes,then they are combined to generate the vertex adjustment without con-flict between tension determined energy and mesh optimization function.Both of the 2 objective functions are also utilized sufficiently.Compared to the exist-ing methods,the new method has the following advantages:(1)The constructed Tension-Determined Surfaces have very low error on mean curvature,so the con-struction has a very high accuracy.(2)The shape can be flexibly controlled.The filled volume can be strictly controlled;some other constraint can also be append-ed.(3)The mesh quality of the constructed surfaces is very high under the high accuracy condition.3.We discover a correspondence between a kind of self-supporting surfaces and the rotational minimal super surfaces in 4-dimension space,and apply it to con-struct self-supporting surfaces.The new method proves that the generatrix of a rotational minimal super surface in 4-dimension space is a self-supporting sur-face in 3-dimension space.An objective function for constructing self-supporting surfaces is proposed based on the above relation.The method for constructing self-supporting surfaces based on the above theory has the following advantages:(1)The objective function is accurate and can be tested via finite element method.(2)The surfaces can be accurately controlled by preset volume;and other forms of load can be appended flexibly.(3)The generation is automatic which is much more convenient than the traditional interactive construction.4.We propose new methods to construct quadrilateral meshes with high quality.The improvement of the mesh quality lies to 2 facets:One is the positions and number of singularities are optimized;another is the parametrization problem is convert-ed to a constrained least squared problem,so that the angular distortion of the mesh is reduced.In the construction of cross field,the new method derives the correspondence between cross field and static electric field with Gauss-Bonnet formula,and utilizes the mapping relation between the singularities and point charges in static electric field for adjusting the positions of singularities along the direction of static electric force.In the conversion from the cross field to pa-rameters,the new method subjects the parametrization to a constrained minimum problem and solves this problem with improved conjugate gradient method,so that the parameters with more rational distribution is derived.Compared to the existing method,the new method has advantages as follows:(1)Both the posi-tions and the number of singularities are optimized,so that the singularities are reduced and posed at more rational points.(2)The edges of quadrilateral meshes are more loyal to the cross field,so that the defect from the angular distortion is avoided.(3)The improvement of conjugate gradient method is not only fit for optimizing the parameters,but also has a good adaption in other constrained least square problems. |
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