| dc.description.abstract | A family of new mesh-free and mesh-based methods is developed and explored in the context of linear and nonlinear boundary and initial value problems of typical interest in engineering mechanics. A common feature in these methods is the use of non-uniform rational B-splines (NURBS) to construct the basis functions, an aspect that potentially helps bring in local support, convex approximation and variation diminishing properties in the functional approximation.
First, a NURBS-based error reproducing kernel method (ERKM) is proposed. The ERKM is based on an initial approximation of the target function by NURBS basis functions. The errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. An accurate computation of derivatives of the ERKM basis function is also proposed. Denoting ? to be a multi-index, the scheme is based on the principle that the ?-th derivative of the ERKM basis functions will exactly reproduce the derivative of space of polynomial of degree p ? |?|. This derivative reproduction scheme does not require differentiations of moment matrices and window functions. We also propose a couple of additional modifications, viz. a point inverse scheme and another via bridging to seamlessly empower the ERKM shape functions with the interpolation (Kronecker delta) property.
Since NURBS in higher dimensions (>1) is usually constructed through tensor products of its one-dimensional component functions, higher dimensional NURBS basis is generally defined over rectangular (cuboid) grid structures. Thus, in many problems of practical interest, geometric complexity of the domain would prevent making use of NURBS in the ERKM. Keeping these issues in mind, a NURBS-based parametric mesh-free method is proposed. This reformulation allows the method to be applicable to non-rectangular (or non-cuboid) physical domains in higher dimensions. The first step in the parametric formulation is to define a parametric space ? = [0,1]^n such that there is a local bijection ? : ? ? Q. The transformation ? is constructed through NURBS. Once the geometric map ? is available, any subsequent refinement is done using ?. Then the shape function and its derivatives are constructed over the parametric space ? so that polynomial reproduction and interpolation properties get satisfied over ? and the geometric map ? is simultaneously preserved. Given that NURBS is constructed only over ?, the issue of domain complexity is effectively tackled. Moreover, since the parametric mesh-free method uses only NURBS in the derivation of shape functions, the issue of choosing a support size for the window function is rendered irrelevant.
For still more complex domain geometries, a unique inverse of the geometric map may not exist and this puts a restriction on the use of the parametric mesh-free method. In order to remove this restriction, a "NURBS-based smooth finite element method (NSFEM)" is next proposed. The first key step in the NSFEM is to decompose a given domain Q into a finite number of sub-domains such that there is a bijection ? ? [1,N] where ?_i = [0,1]^n is the so-called parametric space. Then a set of NURBS-based basis functions are constructed over ? so that polynomial reproduction properties are satisfied over each ?_i. Moreover, the geometric map and the inter-domain continuity (to any desired order) of the approximated field variables are also simultaneously preserved.
We re-emphasize below a few noteworthy outcomes of the studies undertaken in this thesis. First, the issue of support size of the window function is rendered either irrelevant or not-so-relevant. Second, computing derivatives, especially the higher order ones, is faster and less prone to numerical errors. Third, no complicated meshing algorithm is required to discretize the geometry - discretization of the geometry or its refinement is possible in run time as part of the solver (without actually having repeated interactions with the CAD). This is possible through an exploitation of the ability of NURBS in representing a highly irregular geometric object precisely. Moreover, given that an alignment of the support domain with the background mesh is readily enforceable, these methods may be made conforming with very little effort. Finally, by adjusting the number and location of knots in the NURBS functions, one achieves a close C^0 approximation to even a discontinuous target function (such as a step function) without appreciable numerical instability (arising out of the so-called Gibbs or Runge phenomena). In order to bring out these features and the associated computational advantages, several numerical examples are presented, including a new formulation for wrinkled or slack membranes, space-time and semi-discrete formulation of Burger's equation and several others. | |
