NURBS-based parametric mesh-free methods
NURBS-based parametric mesh-free methods
复制标题
DOI：
10.1016/j.cma.2007.11.024
复制
发表时间：
2008-03
影响因子：
7.2
通讯作者：
A. Shaw;D. Roy
中科院分区：
文献类型：
--
作者：
A. Shaw;D. Roy
本文提出了一种无网格误差再现核方法（ERKM）。基于nurbs的误差再现核方法及其在固体力学中的应用，计算机学报。机械，40(1)(2007)127-148]。ERKM基于非均匀有理b样条（NURBS）对目标函数的初始逼近，然后通过一系列非NURBS基函数再现误差。然而，非nurbs基础函数的窗口支持规范仍然是一个棘手的问题。此外，高维（>1）的NURBS通常是在矩形（3D中的立方体）网格结构上定义的，因此，在许多实际问题中，该领域的几何复杂性将阻止在ERKM中使用NURBS。目前，我们开发了一个纯粹使用NURBS的ERKM参数化重新表述。这种重新表述允许该方法适用于二维或更高维度的非矩形（3D中的非立方体）物理域，而无需显式指定窗口的支持尺寸。该开发的一个关键特征是几何映射，它提供了物理域和矩形（立方体）参数域之间的局部双射。然后在参数域上构造形状函数及其导数，从而在物理域上满足多项式再现和插值性质，同时保留几何映射。本文还提出了一些新的格式，使形状函数具有插值特性，从而能够精确地施加狄利克雷边界条件。本文以若干具有工程意义的线性和非线性边界和初值问题的强/弱解为例，说明了参数化无网格法。
A mesh-free error reproducing kernel method (ERKM) has recently been proposed by [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Comput. Mech. 40(1) (2007) 127–148]. The ERKM is based on an initial approximation of the target function by non-uniform-rational-B-splines (NURBS) followed by reproduction of the error via a family of non-NURBS basis functions. However, specifications of the window supports of non-NURBS basis functions remain a tricky issue. Moreover, NURBS in higher dimensions (>1) is generally defined over rectangular (cuboidal in 3D) grid structures and thus, in many problems of practical interest, the geometric complexity of the domain would prevent making use of NURBS in the ERKM. Presently, we develop a parametric reformulation of the ERKM purely using NURBS. This reformulation allows the method to be applicable to non-rectangular (non-cuboidal in 3D) physical domains in two or still higher dimensions without a need to explicitly specify the support size of the window. A key feature of this development is a geometric map that provides a local bijection between the physical domain and rectangular (cuboidal) parametric domain. The shape functions and their derivatives are then constructed over the parametric domain so that polynomial reproduction and interpolation properties are satisfied over the physical domain with the geometric map being simultaneously preserved. A couple of new schemes are also proposed to empower the shape functions with the interpolation property that in turn enables a precise imposition of Dirichlet boundary conditions. We illustrate the parametric mesh-free method in the context of strong/weak solutions of a few linear and non-linear boundary and initial value problems of engineering interest.