Constructing Trivatiate B-splines With Positive Jacobian By Pillow Operation And Geometric Iterative Fitting

Non-uniform rational B-spline(NURBS)curve and surface are the basic representation of curve and surface in CAD,while the traditional finite element analysis method is based on linear basis function.When using the finite element method to analyze the CAD model,we need to convert the CAD model into a linear grid.However,the mesh transformation process is very time-consuming.Therefore,Hughes et.al.proposed the isometric analysis(IGA)method,where the NURBS is used as the base function of finite element analysis.IGA represents the geometric model and analysis model in a unified expression,thus avoiding the complicated process of converting CAD models into linear grids.The traditional geometric design mainly studies the curve and surface design.However,the IGA method needs to deal with the solid model,and requires that every point in the solid model must be valid,that is,the Jacobian value of each point is positive.Therefore,it is an urgent problem to generate a solid model with positive Jacobian value.In recent years,with the development of geometric solid modeling techniques,some trivariate spline model generation methods are developed.However,due to the complexity of the geometrical conditions of ensuring that the Jacobian value of each point in the spline solid is positive,most of the existing methods are difficult to produce a valid spline model.In this paper,we propose a method to guarantee the generation of a trivariate B-spline solid with positive Jacobian.The input to the algorithm is a tetrahedral mesh model in which the surface is divided into six sub-regions.We first partition the tetrahedral mesh model into seven sub-volumes by the pillow operation.And a valid spline model is constructed as an initial spline for geometric iterative fitting.Since the geometric condition for ensuring that the validity of solid model is highly non-linear in theory,we decompose this problem into three steps,i.e.,the generation of valid B-spline boundary curves,the generation of valid boundary patches,and finally,the generation of valid trivariate spline solids.At each step,the geometric iteration method is employed to solve an energy minimization problem with hard validity constraints,thus ensuring that the generated trivariate B-spline solid is valid.Moreover,by optimizing the smoothness objective function,the smoothness between adjacent sub-volumes is improved.