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This paper examined and discussed a Meshless Wavelet Galerkin Method (MWGM) formulation for a first-order shear deformable beam, the properties of the MWGM, the differences between the MWGM and EFG, and programming methods for the MWGM. The first-order shear deformable beam (FSDB) consists of a pair of second-order elliptic differential equations. The weak forms of two differential equations are deduced using Hat wavelet series. The exact integration and reduced integration were used to analyze the problems. Some indeterminate beam problems are considered. Condition numbers of the stiffness matrix were analyzed with exact integration and reduced integration for two cases of these problems. Consequently, the results were converged on the analytic solutions. The shear-locking phenomenon also occurred in the MWGM as it occurs in the conventional FEM. The stiffness matrix calculated from the reduced integration causes a similar numerical error to the stiffness matrix calculated from the exact integration in the MWGM. The MWGM showed desirable results in the examples.

The aim of this study was to formulate a Meshless Wavelet Galerkin Method (MWGM) to solve a pair of 2nd-order elliptic differential equations, in other words, the Timoshenko beam differential equation using Hat wavelets. Usually the formulation of a MWGM is similar to the conventional formulation of FEM. However, some differences exist and this paper is focused on these differences. Additionally, the property of the 1st-order shear deformable theory, which is the locking phenomenon, is discussed.

Meanwhile, the wavelet theory has evolved over the last few decades. Similar to the Fourier series expansion, wavelets are used to express arbitrary signals. Even though wavelets so far have been used to process signals, they can be used to solve differential equations. Moreover, wavelet numerical methods have been investigated by many researchers in numerical analysis as well as structural analysis fields. For instance, the B-spline wavelet finite element method was developed to analyze vibrations in structures [

This study used the Hat wavelet series for expandability and for the ability of inosculation to other numerical methods. The possibility of the inosculation of numerical methods has become important recently because existing methods have already come into wide use such as the standard h-FEM, BEM, and Meshless Method. It is better if expansion and inosculation as well as realization are easy.

Usually, the first derivative of the shape functions of FEM must be square integrable function in the Lévesque sense. In most FEMs, approximations of variational boundary-value problems are concerned with the square integral of the first derivative of functions. As shown in Figures

A scaling function

Derivatives of a scaling function and wavelet functions.

The definition of Sobolev classes may be summarized as follows. A function

With the above considerations, the following properties should be satisfied to use these functions as basic functions for MWGM.

To summarize,

The basic scaling functions and wavelet functions should be combined linearly to construct the basis functions for the wavelet series analysis as in (

In the expression for

The subspace of

The next equation is used for approximating signals or functions over the span

Wavelet basis functions consist of scaling functions and wavelet functions as mentioned above. The generalized expression of extension is shown in (

The following expressions represent examples of the relationship for each level,

The extension of the spaces shows that the spaces are hierarchical, and the refinement can be done by using a suitable value for

There are many wavelet series including the Daubechies, Trigonometric, Hermite Cubic, Haar, and Hat. Because of its suitable properties, this study used the Hat wavelet series as basis functions for the numerical analysis of the weak form of the differential equation, in other words, test functions and trial functions.

Hat wavelet functions have a continuity property but not an orthonormality. After differentiation, the Hat wavelet functions are discontinuous and have a jump property. However the wavelet functions can be integrable after differentiation. The first derivative of the Hat wavelet functions is square integrable. This property is suitable for basis functions for the weak form of the 2nd-order elliptic differential equation. Equations (

Basic scaling functions (in

Wavelet functions (in

The first-order shear deformable beam (FSDB) consists of a pair of second-order elliptic differential equations (

The weak forms of two differential equations are deduced by multiplying each differential equation with the test functions

It is clear that (

If the integrands are integrable on the problem domain

As in (

Equation (

Equation (

Equation (

Integration should be performed with (

Some sets of functions are introduced to code the above equations.

At this point, the similarities and difference of the EFG and MWGM are discussed. As for the similarities, first, the EFG like the MWGM also uses basis functions that exist over the cell which is introduced to integrate the basis functions in the problem domain. Thus, integration and assembly of the stiffness matrix for these two methods can be performed very similarly. As previously stated, sections are introduced to integrate the problem domain in the MWGM. These sections have an effect on the procedures of the MWGM as cells affect the procedures of the EFG. Second, EFG cannot obtain the solutions of the nodes directly; thus, it has to perform additional calculations with the parameters of the basis functions. MWGM also follows the same procedures using the parameters of the basis functions.

Finally, sections have also a property similar to that of the standard FEM elements, which is a discontinuous attribute in the boundary of the elements. This property stems from the folded shape of the wavelet basis functions. If the folded wavelet basis functions exist in one section, sections as many as wavelet basis functions that are folded should be inserted into a preexisting section to integrate wavelet basis functions. This is equivalent to the mesh refinement of standard FEM and to extending subspaces, which consist of piecewise polynomials.

As for the differences, the EFG performs a Gauss elimination to calculate the basis functions value of the integration points. However, the MWGM does not perform the gauss elimination to obtain the basis functions value of the integration points. The MWGM only follows the method of the standard FEM, which uses simplified procedures for constructing stiffness matrices.

A similar case of sections that are used in this paper is found in [

Problem domain is

Governing Equations are

Boundary Conditions are

We considered the indeterminate beam problem shown in Figure

The fixed end supported beam applied by a Dirac delta function.

7 functions and 8 sections (

15 functions and 16 sections (

As shown in Figure

Deflection with the exact and reduced integration (

Deflection with the exact and reduced integration (

Deflection with the exact and reduced integration (

Deflection with the exact and reduced integration (

Deflection with the exact and reduced integration (

As shown in Figure

Rotations with the exact and reduced integration (

Rotations with the exact and reduced integration (

Rotations with the exact and reduced integration (

Rotations with the exact and reduced integration (

Rotations with the exact and reduced integration (

Additionally, condition numbers of the stiffness matrix were analyzed with exact integration and reduced integration for two cases (

Condition numbers of

Condition numbers of

Problem domain and properties of the cross sections are as follows:

Governing equations are as follows:

Boundary conditions are as follows:

Next, we considered the indeterminate beam problem shown in Figure

The fixed end supported beam with a nonuniform section applied by a Dirac delta function.

As shown in the MWGM and analytic solution, maximum deflection occurs on the left side of the domain. Figure

Deflection with the exact and reduced integration (

Both solutions with the exact integration and reduced integration agree well with each other in the previous problem. However, both solutions do not agree with each other in this problem. More specifically, the solutions with the exact integration correspond to the analytic solutions in domain

Rotations with the exact and reduced integration (

This paper examined and discussed the MWGM formulation for a first-order shear deformable beam, the properties of the MWGM, the differences between the MWGM and EFG, and programming methods for the MWGM. The MWGM has also the shear-locking phenomenon. Circumventing assembling the global stiffness matrix K enables the program to be written easily. Multiresolution and localization analysis, which is similar to adaptive analysis of the finite element method, can be performed with the MWGM. It is carefully predicted that all analysis methods could be used to solve one problem such as structure, fluid, soil, or multiphysics problems in the future. The MWGM could be used as part of the analyses. It is easy to perform multiresolution analysis, in other words, adaptive analysis using the MWGM, because integration and interpolation are separated, and linear functions are used as the bases.

This study used the Hat wavelet series for expandability as well as for the ability of inosculation to other numerical methods. As mentioned in the introduction section of this paper, the MWGM is a good alternative and complementary method with traditional and recently developed methods.

The authors declare that they have no conflicts of interest.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2016R1A2B4016632).