Suitable For The Construction Of Degenerate Smooth Interpolated Surfaces Patches In Isogeometric Analysis

The mathematical description of the geometric shape of industrial products is the core problem of computer-aided geometric design.With the continuous development of the advanced manufacturing industry,in the fields of computer-aided engineering and computational mechanics,physical simulation analysis and optimization of products are often required.The most classical method is the finite element method.From a mathematical point of view,the core ideas behind the numerical solving of partial differential equations are "numerical approximation" and "discretization".However,due to the fact that the finite element method takes a long time to discretize the CAD model grid,the calculation accuracy is low,and some defects such as the separation between the geometric model and the analysis model during the analysis process,prompting researchers to find a new method to solve this problem.In 2005,Hughes et al.proposed the idea of isogeometric analysis.This method directly combines the geometric model in CAD,and uses the basis function of the geometry expressed in CAD for the analysis process,which achieves seamless integration of product design and analysis process.For the calculation domain of a given CAD model to construct the parameter spline of the entire model called a parameterization.When the surface has a complex geometric structure,singularities will inevitably appear in the parameterization of the surface.In the isogeometric analysis,the singularity will affect the regularity of the test functions.For instance,when we solve a second-order partial differential equation,the test function must satisfy the regularity.In this paper,by referring to regularity conditions,an interpolation operator under singular regular parameterizations in the isogeometric analysis is constructed,and the relevant results of degenerated surface patch D-patch are quoted to construct a smooth function.The smoothness of the interpolated surface is theoretically proved and numerical examples are given.In Chapter 1,the dissertation reviews the background,research status,and comparison between the finite element analysis and the isogeometric analysis.In Chapter 2,we list the related knowledge and theories used in this dissertation,including the framework of isogeometric analysis.Chapter 3 discusses the smoothness of interpolated surfaces under singular parameterization for one patch.We call the parameterization is degenerated at the given point if the partial derivatives are trivial at this point.The case discussed in this dissertation is the singular point caused by thevanishing of tangent vector.By constructing an interpolation operator,an interpolation type surface parameterized in the physical domain that meets the regularity condition is also defined.It is proved that the surface is a degenerate surface,which meets the definition of D-patch,and is a general D-patch.In Chapter 4,numerical examples are also provided,which show that the obtained interpolation surface is degraded and smooth at(0,0).Chapter 5 summarizes the work of this dissertation and its future research topics.