1. Introduction
The classical finite element method (FEM), which is based on the weighted residual technique, is a versatile and well-developed numerical approach in the field of modern computation mechanics [
1]. Many mature commercial software packages (such as ANSYS, ABAQUS and NASTRAN) based on the FE approach have been developed and used in various engineering applications. Though the standard FEM has achieved great success in practical engineering computation, the FEM still suffers from several inherent shortcomings compared to other advanced numerical techniques [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12]. Among them, one important issue is that the FEM is essentially a mesh-based method and the involved problem domain should be firstly discretized into a series of elements that are connected by nodes for FE analysis. Therefore, the additional burdensome tasks for meshing operations cannot always be circumvented. Additionally, the solution accuracy of the FEM is usually sensitive to mesh qualities and the solutions from low-quality meshes are always not sufficiently accurate. To obtain sufficiently fine solutions, more attention should be given to obtain high-quality meshes. These issues will be greater when the standard FEM is employed to manage problems related to dynamic cracks and large deformation of complicated geometric shapes.
To alleviate the dependence of the conventional FE approach on predefined meshes, a series of smoothed FEMs [
13,
14,
15,
16,
17] and meshfree techniques [
18,
19,
20,
21,
22,
23,
24] have been proposed, such as the element-free Galerkin method (EFGM) [
25,
26], the meshless local Petrov–Galerkin method (MLPG) [
27], the reproducing kernel particle method (RKPM) [
28], the radial point interpolation method (RPIM) [
29,
30], the general finite difference method GFDM [
31,
32,
33,
34,
35], and the boundary-based numerical methods [
36,
37,
38,
39,
40], to name a few. Actually, the meshfree methods can be classified into different types according to different formulation procedures [
18]. Among them, several meshless methods are based on the weak form of the governing equation [
41,
42,
43,
44], while some others are based on the strong weak form [
45,
46,
47,
48,
49,
50,
51]. In this work, we mainly focus on discussing the meshfree methods based on the well-known Galerkin weighted residual technique (such as the EFEM and RPIM), which are the typical weak-form-based numerical techniques. Compared to the standard FEM, one outstanding advantage of these meshfree methods is that the required nodal shape functions can be built entirely by using a set of scattered nodes, rather than as elements in the conventional FEM. In consequence, the field function approximation also can be constructed by the scattered nodes. This property enables the meshfree methods to have distinct advantages over the conventional FEM in managing the dynamic crack problem and larger deformation problem. In addition, adaptive analysis also can be implemented much more easily in the meshfree framework than in the standard FEM framework. More importantly, the meshfree methods usually possess other excellent features that the standard FEM does not have. A very good comparison and overview on the meshfree methods and the FEM can be found in a published monograph [
18].
Although the meshfree methods have achieved considerable success both in theory and practical engineering applications, they still cannot match the classical FEM in terms of universality; further, there still exists several crucial issues that should be addressed very carefully. For example, the radial point interpolation method (RPIM), which is a typical meshfree numerical method, has been employed for solving many engineering problems owing to several excellent features, such as relatively high computation accuracy, good numerical stability and the possession of the Kronecker-delta function property. However, the compatibility of the standard RPIM cannot be automatically ensured, which may lead to numerical integration error. The main reason is that in the standard RPIM the local support domains of nodal interpolation functions are not always in accord with the constructed integration cells for numerical integration. The related issues have been investigated in [
52,
53] and in the so-called bounding box technique that has been proposed by proposed by Dolbow and Belytschko [
52]. The related numerical results show that this scheme is indeed quite effective in addressing the issues mentioned; however, the implementation of this scheme is quite complicated and, hence, it is not very practical in engineering computation.
In this work, a simple and elegant scheme is developed to make the local support domains of the nodal shape functions in RPIM entirely align with the constructed quadrature cells for numerical integration; hence, the possible integration error can be markedly decreased. The main idea of this scheme is to design a new node selection scheme for the field function approximation. In this scheme, a fixed support domain (not a moving support domain in the original RPIM), which is determined by the geometrical center of the quadrature cell, is used for any quadrature point in the integration cell. For the convenience of notation, the proposed scheme in this work is called the modified RPIM (M-RPIM). We have further employed the present M-RPIM to analyze the free vibration of two-dimensional solids. It can be found that the M-RPIM behaves much better than the original RPIM for free vibration analysis, and many more numerical solutions can be provided with the totally identical node distributions.
2. Formulation of the Original RPIM and the Present M-RPIM
Consider a problem domain
with boundary
, and a field function
is defined on it. A series of scattered field nodes are employed to totally discretize the problem domain and its boundary. For a sampling point in the problem domain, the corresponding field function approximation
can be expressed in the following form by using the radial basis function (RBF) and polynomial basis function (PBF) [
18]:
in which
stands for the RBF used and
represents the PBF used;
n denotes the number of RBF used for interpolation, namely, there are
n field nodes in the support domain of the sampling point
,
m denotes the number of PBF used for interpolation, and the complete linear polynomial ([1
x y]) is used in this work, namely,
m = 3;
and
are the unknown interpolation coefficients.
There are many different types of RBF that can be used to formulate the RPIM, and different RBFs have different features [
18]. In this work, the well-known multiquadrics (MQ) function is used to construct the required field function approximation owing to its several excellent characteristics. The expression of the MQ function is as follows [
18,
21]:
in which
denotes the distance from the field node to the sampling point,
is the average nodal interval of the field nodes used, and
and
q denote two undetermined parameters that are closely related to the computation accuracy of the RPIM;
q = 1.03 and
are used in this work because very good numerical results can always be obtained for solid mechanics with these parameters.
With the aim to determine the coefficients
and
, Equation (1) should satisfy a series of reasonable constraint conditions. Firstly, it is usually assumed that the constructed field function approximation can exactly pass through the function values of all the nodes located in the support domain of the sampling point
; these constraints can be expressed by:
in which
and
are the so-called moment matrices corresponding to the RBF and PBF, respectively.
To uniquely determine the unknown interpolation coefficients
and
, the following additional constraints should also be satisfied:
The combination of all the constraining conditions shown in Equations (2) and (6) can result in the following matrix equation:
Then, the undetermined interpolation coefficients can be calculated by
Substituting the interpolation coefficients obtained into Equation (1) and following the standard formulation of the conventional RPIM, the required nodal interpolation shape function can be obtained by
In the standard RPIM, the field nodes participating in building the field function approximation for the sampling point, which are usually quadrature points, are determined by a support domain. The shape of the support domain can be a square or a circle. The sampling point is usually the center of the defined support domain, while the background cells for numerical integration are always constructed independently of the support domain. As a result, the different sampling points (or quadrature points) in one integration cell may have different support domains, namely, the required field nodes to construct the field function approximation are different. In summary, since the moving support domain is used in the traditional RPIM, the support domain of the nodal shape functions always do not align with the background integration cells, which then leads to considerable numerical integration error and degrades the quality of the numerical solutions obtained.
To effectively overcome the abovementioned misalignment between the nodal shape function supports and the background integration cells, in this work a modified RPIM (M-RPIM) is employed to analyze the free vibration of two-dimensional solids. In this M-RPIM, a fixed support domain (as shown in
Figure 1) rather than the moving support domain in the standard RPIM is used to select the required field nodes for the construction of the field function approximation. In other words, the identical field nodes are used for interpolation for any quadrature points in one background integration cell. The fixed support domain used can still be a square or a circle (the square support domain is used in this work); however, this fixed support domain is always centered by the geometrical center of the integration cell, not centered by the sampling points (which are usually the quadrature points) as in the conventional RPIM. The difference between the original RPIM and the present M-RPIM in constructing the field function approximation can be shown as follows:
in which
stands for the moving support domains, which are centered by the quadrature points in one background cell,
represents the fixed support domains that are directly centered by the centroids of the background integration cells.
3. Formulation of the Elastodynamics of Two-Dimensional Solids
Based on the small displacement assumption, the partial differential equation (PDE) of the boundary-value problem for the elastodynamics of solids can be written by
in which
denotes the problem domain considered,
stands for the body force,
represents the stress tensor,
is the mass density,
is the displacement vector and
signifies second derivatives of
.
As usual, the following two kinds of boundary conditions are always considered for the two-dimensional elastodynamics of solids:
in which
and
denote the essential boundary condition and the natural boundary condition, respectively;
and
are the imposed displacement vector and traction vector on the corresponding boundary conditions.
Using the boundary conditions shown in Equation (12) and following the virtual displacement principle, the weak form of Equation (11) for the elastodynamics of two-dimensional solids can be obtained by
in which
and
stand for the virtual displacement and strain, respectively.
Using the Galerkin weighted residual techniques and the field function approximation shown in Equation (1), the matrix equation for the weak form shown in Equation (13) can be obtained [
1,
18] by the following relationship:
in which
is the usual mass matrix,
is the usual stiffness matrix,
is the applied force vector and
is the matrix containing the damping effects.
Without considering the damping effects and the external force, Equation (14) reduces to
Equation (15) is the governing matrix equation obtained for the free vibration analysis of two-dimensional solids.
Assuming that the displacement solution to Equation (15) is time harmonic, namely,
in which
,
denotes the angular frequency, and
is the amplitude of the displacement distribution.
Substituting Equation (16) into Equation (15), then Equation (15) can be rewritten as
From Equation (17), we can observe that the typical eigenvalue problem should be solved to perform the analysis of free vibration problems.
5. Conclusions
In this work, a modified radial point interpolation method (M-RPIM) is proposed to enhance the capacities of the original RPIM for the free vibration analysis of two-dimensional solids. In the present M-RPIM, the numerical approximation established in integration cells is continuously differentiable while the corresponding numerical approximation in the original RPIM is always not continuously differentiable. Therefore, the possible numerical integration error in the original RPIM can be markedly reduced by the present M-RPIM.
Several supporting numerical examples are employed to investigate fully and in detail the performance of the proposed M-RPIM in solving free vibration problems. It is demonstrated that the proposed M-RPIM not only is able to surpass the original RPIM and the standard FEM-Q4 in terms of computation accuracy and convergence properties when the identical node arrangement scheme is employed, but the proposed method also has higher computation efficiency. This is because the fixed support domain is employed for any quadrature points in the integration cells; hence, the additional operations to determine the support domain for each quadrature point are not required. Owing to these excellent features, the present M-RPIM has great potential for solving more complex problems in practical engineering application.