دانلود رایگان مقاله ساخت سلسله مراتبی شبکه مش LR دو بعدی

1. Introduction

In the last decade the use of spline spaces has spread from the field of applied geometry, in particular Computer Aided Design (CAD), to that of numerical analysis of Partial Differential Equations (PDEs). This is largely due to the influence of the seminal paper by Hughes et al. (2005). The use of B-spline generated spaces in Galerkin methods was attempted before by Höllig (2003), but Hughes et al. (2005) recognized it as a possible way to remove the compatibility layer that is in-between the CAD tools and the Finite Element Method (FEM). The compatibility layer contains the mesh generation process and, in some cases, can be computationally more expensive than the simulation itself (Hughes et al., 2005). The method that reduces the compatibility layer proposed by Hughes et al. (2005) is called IsoGeometric Analysis (IGA) and is based on the isoparametric approach: the solution fields of the PDE are in the same B-spline or NURBS space used for the parametrization of the geometry. IGA sprouted new research in numerical methods due to the availability of basis functions with higher smoothness and with strong algebraic properties that allow for new numerical schemes like compatible discretizations. It had the same effect in the applied geometry field: the numerical simulation of PDEs requires high quality parametrizations of the domain while in CAD it is common to parametrize only the boundary and to allow both for small gaps and singularities.

8. Conclusions

We restrict our attention to the subset of bi-variate box meshes that have the N2S-property. We describe two subdomains of where respectively vertical and horizontal refinement preserves the N2S property. Using this knowledge we provide an explicit construction that is based on a hierarchy of tensor spaces and domains. The LR-space associated to the constructed mesh H has the N2S property, i.e. it has a basis of locally linearly independent functions. Moreover LR(H) is the whole space S(H) of piecewise polynomials associated to the mesh.The fact that our construction guarantees the N2S property is based on the results from Bressan (2013) that apply to the n-variate case. It seems thus reasonable that the construction can be generalized to n-variate case with similar definitions and proofs. Future work will be devoted to the generalization of Lemmas 6 and 7 to this situation. The proof of completeness (Theorem 12), however, relies on the results from Mourrain (2014) that are available for bi-variate splines only. It is expected that more effort is required to extend the completeness proof to n-variate constructions. Some results concerning the homology term for 3D meshes are reported in Berdinsky et al. (2014).