Mathematical Problems in Engineering

Volume 2016, Article ID 2549213, 11 pages

http://dx.doi.org/10.1155/2016/2549213

## A Numerical Method Based on Daubechies Wavelet Basis and B-Spline Patches for Elasticity Problems

China Special Equipment Inspection and Research Institute, Beijing 100029, China

Received 11 May 2016; Revised 6 July 2016; Accepted 25 July 2016

Academic Editor: Lihua Wang

Copyright © 2016 Yanan Liu and Keqin Din. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Daubechies (DB) wavelets are used for solving 2D plane elasticity problems. In order to improve the accuracy and stability in computation, the DB wavelet scaling functions in comprising boundary scaling functions are chosen as basis functions for approximation. The B-spline patches used in isogeometry analysis method are constructed to describe the problem domain. Through the isoparametric analysis approach, the function approximation and relevant computation based on DB wavelet functions are implemented on B-spline patches. This work makes an attempt to break the limitation that problems only can be discretized on uniform grids in the traditional wavelet numerical method. Numerical examples of 2D elasticity problems illustrate that this kind of analysis method is effective and stable.

#### 1. Introduction

Wavelet is a powerful mathematical tool in solving many problems in science and engineering. In recent years, there has been an increasing interest in wavelet-based methods due to their successes in some applications, such as mathematical analysis and signal processing. The wavelet-based numerical methods have been developed by many researchers. At present, there are mainly three kinds of wavelet-based numerical methods: wavelet finite element method, wavelet collocation method, and wavelet-Galerkin method. The wavelet finite element method [1–4] is based on traditional FEM. In this method, the scaling functions and wavelet functions in wavelet analysis are used as basis functions to construct the so-called shape functions on elements. In wavelet collocation method [5], the scaling functions and wavelet functions are directly used as basis functions to approximate the unknown functions instead of constructing shape functions as done in the finite element method, and the collocation approach is used for discretization. Based on Daubechies wavelet, some numerical examples for 1- and 2-dimensional model problems show that the wavelet collocation methods are stable and effective for PDE.

The wavelet-Galerkin method [6–13] is the most popular wavelet-based numerical method. Unlike the wavelet collocation method, the Galerkin approach is used for discretization. Although the computational efficiency of Galerkin method is lower than that of collocation method, the accuracy and stability are improved remarkably. The desirable advantage of wavelet is the multiresolution property. Based on the property, the wavelet-based multiscale analysis is easy to be realized [13–19].

Although wavelets have demonstrated potential in numerical simulation, there are still some works to do for engineering application. The traditional wavelets, such as some orthogonality or biorthogonality wavelets, can describe details of problems and perform well in resolving high gradients. However, such wavelets are best suitable for problems that are discretized on uniform grids, a constraint that can be rather restrictive when it comes to modeling problems with complex geometry. By contrast, the finite elements method is very well suited for complex meshes. Some works have been made to use this kind of wavelets to solve problems with general domains [12, 13] and the wavelet-based multiscale analysis method can be used to conduct local analysis [13]. But the computational efficiency of this kind of methods still needs to be improved. The second generation wavelets can be constructed on nonuniform grids for numerical simulation [20, 21]. However, these kinds of wavelets are either complex in construction or too simple to simulate complicated deformation.

The isogeometry analysis method [22, 23] developed in recent years presented some new ideas in numerical simulation. In this method, the B-spline functions or nonuniform rational B-spline functions are used to describe the problem geometry and the total solution domain can be divided into many B-spline patches which are similar to the elements in finite element method. Function approximation and relative computation can be implemented on B-spline patches through isoparametric analysis approach. It can be found that the traditional wavelet basis functions are similar to B-spline basis functions in framework that they must be constructed on structure grids. So it is reasonable to introduce the B-spline patches into the problems in which the traditional wavelet basis functions are used. This is an attempt to break the limitation that the traditional wavelet-based numerical methods are only restricted on uniform grids.

In this paper, the Daubechies (DB) wavelet which has orthogonality and compact support is chosen for analysis because of its good performance in numerical simulation. In order to improve the accuracy and stability in computation, the DB wavelet scaling functions in which comprise boundary scaling functions are used as basis functions for approximation. The B-spline patches constructed by the B-spline basis functions are used to describe the problem geometry. The function approximation based on DB wavelet basis functions and relevant computations are implemented on B-spline patches through the isoparametric analysis approach. Numerical examples for 2D elasticity problems are given to illustrate the effectiveness of the present method.

#### 2. The Function Approximation by DB Wavelet Scaling Function

##### 2.1. The Basic Properties of DB Wavelet

According to the theory of DB wavelet, the so-called scaling function and wavelet function of DB wavelet both satisfy two-scaling relation:Here, the index denotes the ordinal number of DB wavelet series (DB wavelet with ordinal number is abbreviated as “” in the following), and denotes the place. are called filter coefficients. The supports of scaling function and wavelet function of DB wavelet are, respectively,DB wavelet function has consecutive moments equal to zero. That is vanishing moment :The ordinal number of DB wavelet series is equal to the number of its vanishing moments. Furthermore, the smoothness of DB wavelet scaling functions and wavelet functions will be improved with the increase of number of vanishing moments.

From translation and dilation of a basic scaling function , we haveIn the above equation, and denote, respectively, the scale and the place in wavelet space. It is obvious that the support of the function is

According to the principle of multiresolution, the scaling function can be used to build the wavelet space :

In addition, the scaling function can be used to exactly represent polynomial to some degrees. For , we can write are defined bywhere

##### 2.2. The DB Wavelet Basis Functions on

For , the boundary scaling functions are defined byFrom the boundary scaling functions , we have the formula for all in :The boundary scaling functions , , are linearly independent and are orthogonal to the functions for (called the ordinary scaling functions on ). Figure 1 shows the boundary scaling functions and one of the ordinary scaling functions of DB6 wavelet.