Research On High-order Visualization Method For Isogeometric Analysis

Isogeometric analysis (IGA for short) using NURBS (non-uniform rationalB-Spline), the industry standard, as the uniform geometric representation in the designand analysis processes, analysis and calculation can be performed directly on theNURBS-based parametric surfaces and volumes, without generating meshedanalysis-suitable geometries. It provides a new way to realize the seamless integrationof CAD/CAE.Before isogeometric analysis, the common processes of technologies thatextracting and rendering counter isoline/isosurface is to mesh the scalar fields at first,then extract the approximation of counter isoline/isosurface based on the interpolationin the grid. The traditional process cannot represent the high-order counterisoline/isosurface, neither represent counter isoline/isosurface by mathematicalfomulas. While geometries in isogeometric analysis must be represented exactly, thetraditional process cannot meet the requirement of isogeometric analysis. With thedevelopment of isogeometric analysis, it is necessary to propose a new visualizationmethod which can be used in isogeometric analysis.Firstly we present the background and research situation of isogeometricanalysis and visualization in scientific computing, respectively. And we present someconcepts that we will used in this paper, related to isogeometric analysis.Secondly, in order to realize the high-accuracy extraction and rendering ofcounter isoline on surfaces, we proposed a high-order extraction method of counterisoline on B-spline surface by functional composition theory. At first, the B-splinesurface and its parametric domain are subdivided by knot insertion method, then thecorresponding part of counter isoline in the parametric domain is obtained accordingto the value at the sampling points by marching squares method. After convertingeach sub-patch on each knot interval into Bézier surface, the high-order representationof the counter isoline on B-spline surface can be obtained by functional compositionmethod. Several examples, such as high-order extraction of iso-curvatrue lines and theextraction of isotherms lines in the heat conduction problems are shown to illustratethe efficiency of the proposed method.Subsequently, the proposed method is extended to three-dimensional scalar field,to realize the high-accuracy extraction and rendering of isosurface on volumes. Atfirst, the B-spline volume and its parametric domain are subdivided by knot insertionmethod, then the corresponding part of isosurface in the parametric domain isobtained according to the value at the sampling points by marching cubes method.After converting each sub-volume on each knot interval into Bézier volume, thehigh-order representation of the isosurface on B-spline volume can be obtained byfunctional composition method. Several examples are shown to illustrate theefficiency of the proposed method. On the basis of the open-source modeling softwareplatform AXEL, by using C++and OpenGL, we developed the Isovalue Tools Plugin of AXEL to achieve high-accuracy visualization results in this paper.Finally we summarize the whole research work in this paper, and future researchdirections and technical improvements related to this field are also discussed.