4.1.1 Overview of Mesh-Free Methods

Mesh-free methods can be traced back to the 1970s, but they did not receive much attention from researchers until recently. In 1977, Lucy [35] and Gingold and Monaghan [36] proposed the smooth particle hydrodynamics (SPH) method and successfully applied it in the astrophysical field. Subsequently, Johnson and Beissel [37] improved the accuracy of the SPH method using a normalized smoothing function algorithm which can calculate accurately the normal strain rates for conditions of constant strain rates. Swegle et al. [38], Dyka et al. [39], and Chen et al. [40] pointed out the possible reasons for instability of the SPH method and gave an improvement. Vignjevic et al. [41] presented an approach for the treatment of zero-energy modes in the SPH method. Monaghan [42–44] concluded the main characteristic of the SPH method. Although the SPH originated for the purpose of studying astrophysical problems, it has been widely applied in high velocity impact, penetration, underwater explosion, shock wave, solids, and fluids [45–55].

Nayroles et al. [56–58] proposed a diffuse element method (DEM) by introducing moving least square (MLS) [59] into the Galerkin method in 1992. Afterwards, Belytschko et al. [60] modified the DEM by preserving the omitted terms in the derivatives of the interpolations in the DEM and thus proposed an element-free Galerkin (EFG) method. They introduced the Lagrangian multiplier method for the imposition of essential boundary conditions since the trial function lacks Kronecker delta property. This proposed EFG method costs much more computational resource than SPH but has a better computational stability. Chung and Belytschko et al. [61–64] gave an error estimate for the EFG method and conducted thorough research on the trial function and numerical integration. The EFG method has overcome the shortcoming of FEM which needs remeshing when simulating the moving discontinuity problems. In addition, the use of MLS can easily construct a shape function with C¹-continuity. The above superior properties allowed the EFG method to be successfully applied in thin plates and shells [65,66], dynamic and static fracture and crack propagation problems [67–77], water flows and impact [78–80], and phase interface [81].

Inspired by the theory of wavelets, Liu et al. [82,83] proposed the reproducing kernel particle method, by introducing a correction function into the kernel function, to overcome the problems of the lack of consistency in the SPH method in 1995. Chen et al. [84] proposed a strain smoothing stabilization for nodal integration to eliminate spatial instability. This method has been widely employed in numerical analysis such as rubber hyperelasticity [85,86], acoustics [87], structural dynamics [83], fluid dynamics [88,89], stress concentration [90], and large deformation problems [85,91–94].

In 1996, Duarte and Oden [95] constructed a partition of unity using the MLS functions to develop an meshless method, termed h-p clouds. Garcia et al. [96] and Mendonca et al. [97] applied this h-p clouds method to investigate the bending for thick plate and Timoshenko beam problems. Later, Zhu et al. [98–100] and Atluri et al. [101–103] proposed the local boundary integral equation method and meshless local Petrov-Galerkin approach based on the MLS approximation. The main advantage of these methods was the simplification of integration process.

In addition, there have been also many other mesh-free methods in the literature, i.e., finite point method (FPM) [104], finite spheres method (FSM) [105–107], free-mesh method (FMM) [108], material point method (MPM) [109], particle-in-cell (PIC) method [110,111], and point interpolation method (PIM) [112,113].

Note that the imposition of essential boundary conditions is a crucial point in mesh-free methods since most of the mesh-free methods lack the Kronecker delta property. It needs a special treatment in the imposition of essential boundary conditions in contrast to mesh-based methods. Some appropriate techniques have been introduced to impose the essential boundary conditions, such as penalty function method [114], Lagrange multiplier method [69], and modified variational principle [115], etc. In recent years, a novel interpolation, Kriging interpolation, has attracted much attention as it inherently possesses the delta property. This superior property makes it capable of enforcing the essential boundary conditions directly as FEM.

4.1.2 Advantages and Disadvantages of Mesh-Free Methods

Mesh-free methods have been proved to have many particular advantages. One distinct advantage is that they are well-suited to the treatment of discontinuities that do not coincide with the original mesh lines, e.g., the growth of cracks with arbitrary and complex paths and the simulation of phase transformation. Mesh-free methods have been successfully applied to the study of crack propagation [67,69,116] and the treatment of discontinuity [81]. In the FEM, the most viable technique for handling these complications is to remesh the domain of the structure at every computational step during the simulation. This remeshing process avoids severe mesh distortion and keeps the mesh line coincident with any discontinuity. However, the remeshing technique requires the projection of field variables between the meshes in successive stages of the problem, which leads to projection error and reduces the accuracy of the numerical simulation. In mesh-free methods, the shape functions are developed using the sophisticated interpolating strategy of field variables at a global level, the approximate solution can be constructed entirely in terms of a set of nodes, and the discrete equations can be subsequently developed without the need for meshes. The burdensome work of meshing and remeshing that is associated with the FEM can thus be avoided.

Most of the mesh-free methods are constructed by embedding a highly smooth window function with a certain range of support sizes and reproducing the polynomials exactly. This methodology leads to higher-order polynomial approximation or a highly smooth shape function with higher-order continuity, which alleviates many of the numerical ill-conditions encountered in the FEM. Mesh-free methods have been successfully applied to the simulation of material with the effects of the strain gradient and higher-order deformation gradient [26].

Moreover, mesh-free methods can be viewed as nonlocal approximations, as every evaluated point in the domain is actually covered by the multiple shape functions of the node, as schematically demonstrated in Fig. 4.1. This property may smooth the discontinuous fields and significantly delay the mesh distortion in large deformation computations. It has been found that mesh-free methods can achieve much higher precision levels than the FEM for large deformation problems [85,86,117].

There are several key issues that need to be pointed out here. An important point is the imposition of essential boundary conditions in mesh-free methods. As the mesh-free shape functions generally lack the Kronecker delta property, the approximation generated by the MLS method does not pass through the nodal parameter values. Thus, the essential boundary conditions cannot be enforced as in the FEM. An appropriate technique is needed to impose the essential boundary conditions. Moreover, in the FEM, the integrals can be carried out on a series of discretized elements. However, the concept of elements disappears in mesh-free methods, and the evaluation of the integrals requires the use of certain techniques. The integration method is always an important factor that affects the stability and precision of numerical results. Furthermore, irregular geometrical shapes and irregular boundaries often have an effect on the performance of mesh-free methods and the accuracy of computational results.

4.4.1 Transformation of SWCNTs

An undeformed SWCNT can be considered as the rolling up of an initial equilibrium graphite sheet into a hollow cylindrical structure. It is noteworthy that the atomic structure of an initial equilibrium graphite sheet is different from that of an initial equilibrium SWCNT. All of the C–C bonds are the same in the initial equilibrium graphite sheet, while there are slight differences in C–C bonds in the counterpart SWCNT. Therefore, three geometrical parameters, λ₁, λ₂, and θ, corresponding to the uniform longitudinal, circumferential stretches, and the skew of any surface longitudinal line, respectively, are introduced to describe this shifting process. The three geometrical parameters can be determined by minimizing the strain energy of the representative cell to obtain an initial equilibrium configuration. The initial equilibrium graphite sheet and the optimized equilibrium SWCNT are selected as the reference configuration and the current configuration, respectively. The deformation process mapped from Fig. 4.6A–C can then be written:

$x_1=X_1{\lambda }_1\cos \theta$ (4.38)

(4.38)

$x_2=\frac{\mathrm{\Gamma }{\lambda }_2}{2\pi }\sin \left[2\pi \left(\frac{X_2}{\mathrm{\Gamma }}+\frac{X_1{\lambda }_1\sin \theta }{\Gamma {\lambda }_2}\right)\right]$ (4.39)

(4.39)

$x_3=\frac{\mathrm{\Gamma }{\lambda }_2}{2\pi }-\frac{\mathrm{\Gamma }{\lambda }_2}{2\pi }\cos \left[2\pi \left(\frac{X_2}{\mathrm{\Gamma }}+\frac{X_1{\lambda }_1\sin \theta }{\mathrm{\Gamma }{\lambda }_2}\right)\right]$ (4.40)

(4.40)

where $\mathbf{x}=\mathbf{x}({\mathit{x}}_1,x_2,x_3)$ and $\mathbf{X}=\mathbf{X}(X_1,X_2)$ are the current configuration and reference configuration, respectively. The first- and second-order deformation gradients F and G can be exactly calculated from the deformation process of SWCNTs in Eqs. (4.38)– (4.40).

When the three parameters λ₁, λ₂, and θ are introduced, the first- and second deformation gradients are determined by the three parameters and Eq. (4.14) can be further rewritten as:

$v_I[\mathbf{F},\mathbf{G},\mathbf{\eta }]=v_I({\lambda }_1,{\lambda }_2,\theta ,{\eta }_1,{\eta }_2)$ (4.41)

(4.41)

Denote the unknown vector as:

$\mathbf{\lambda }={({\lambda }_1,{\lambda }_2,\theta ,{\eta }_1,{\eta }_2)}^T$ (4.42)

(4.42)

Structural parameters can be determined by minimizing the strain energy of the representative cell:

$\frac{\partial {\nu }_I}{\partial \mathbf{\lambda }}=\mathbf{0}$ (4.43)

(4.43)

The Taylor expansion of ${\nu }_I$ around an initial guess of ${\mathbf{\lambda }}_0$ for the equilibrium state is

$v_I(\mathbf{\lambda })=v_I({\mathbf{\lambda }}_0)+\frac{\partial v_I(\mathbf{\lambda })}{\partial \mathbf{\lambda }}{|}_{\mathbf{\lambda }={\mathbf{\lambda }}_0}\cdot \mathrm{\Delta }\mathbf{\lambda }+\frac{1}{2}\mathrm{\Delta }\mathbf{\lambda }\cdot \frac{{\partial }^2{v}_I(\mathbf{\lambda })}{\partial {\mathbf{\lambda }}^2}{|}_{\mathbf{\lambda }={\mathbf{\lambda }}_0}\cdot \mathrm{\Delta }\mathbf{\lambda }+o(\mathrm{\Delta }{\mathbf{\lambda }}^2)$ (4.44)

(4.44)

Substitution of Eq. (4.44) into (4.43) yields the following governing equation

$\frac{{\partial }^2{\nu }_I}{\partial {\mathbf{\lambda }}^2}\cdot \mathrm{\Delta }\mathbf{\lambda }=-\frac{\partial {\nu }_I}{\partial \mathbf{\lambda }}$ (4.45)

(4.45)

The increment of structural vector $\mathrm{\Delta }\mathbf{\lambda }$ can be determined by iteratively solving Eq. (4.45) until the right term reaches zero.

The aforementioned strain energy density ${\omega }_0$ in Eq. (4.14) is defined as the area strain energy density, while the volume strain energy density is required in order to calculate the elastic and structural properties. Nevertheless, there is no agreed thickness concept for CNTs. Scholars have suggested an interlayer distance of approximately 0.33~0.34 nm for CNTs. In many publications [118,119], the interlayer distance of graphite sheet is chosen as 0.334 nm, hence we choose it as the thickness. Consequently, the volume strain energy density is calculated by:

${\hat{\omega }}_0={\omega }_0/t$ (4.46)

(4.46)

where t is the thickness.

Since geometrical parameters λ₁ and λ₂ correspond to the uniform axial and circumferential stretches, respectively, the axial and circumferential strains are calculated by:

${\epsilon }_1={\lambda }_1-1,{\epsilon }_2={\lambda }_2-1.$ (4.47)

(4.47)

Thus, axial and circumferential stresses are determined by:

${\sigma }_1=\frac{\partial {\hat{\omega }}_0}{\partial {\epsilon }_1}=\frac{\partial {\hat{\omega }}_0}{\partial {\lambda }_1},{\sigma }_2=\frac{\partial {\hat{\omega }}_0}{\partial {\epsilon }_2}=\frac{\partial {\hat{\omega }}_0}{\partial {\lambda }_2}.$ (4.48)

(4.48)

In case of hydrostatic pressure, CNTs deform uniformly along the circumference. The radial strains of the CNTs equal the circumferential strains, i.e., ${\epsilon }_r={\epsilon }_2$. Moreover, the hydrostatic pressure p has a relationship with ${\sigma }_2$ as

$p2R={\sigma }_{2}2t$ (4.49)

(4.49)

Young’s moduli along the axial and circumferential directions of CNTs E₁ and E₂ can be obtained by:

$E_1=\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_1^2}-\frac{{\left(\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_1\partial {\lambda }_2}\right)}^2}{\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_2^2}},E_2=\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_2^2}-\frac{{\left(\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_2\partial {\lambda }_1}\right)}^2}{\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_1^2}}$ (4.50)

(4.50)

Using Eq. (4.49), the radial Young’s modulus can be calculated as

$E_r=E_{2}t/R$ (4.51)

(4.51)

The angle θ is the shear strain at the surface and thus the shear stress at the cross-section is:

$\tau =\frac{\partial {\hat{\omega }}_0}{\partial \theta }$ (4.52)

(4.52)

The shear modulus is calculated by:

$G=\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\theta }^2}$ (4.53)

(4.53)

The Poisson’s ratio is calculated by:

$v_{12}=\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_2\partial {\lambda }_1}/\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_1^2},{\nu }_{21}=\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_1\partial {\lambda }_2}/\frac{{\partial }^2{\hat{\omega }}_0}{\partial {\lambda }_2^2}$ (4.54)

(4.54)

Moreover, we have the relationship

$\frac{v_{12}}{E_1}=\frac{v_{21}}{E_2}$ (4.55)

(4.55)

Similar to the calculation of tangential modulus, it is noteworthy that the contribution of the inner shift vector should not be neglected when calculating elastic constants. Structural parameters of CNTs can be obtained by iteratively solving the incremental governing equation Eq. (4.45), and elastic properties are the values of Eqs. (4.50)– (4.54) at the initial equilibrium state of CNTs.

4.4.2 Structural Parameters

With the solution of Eq. (4.45), the structural parameters of the initial undeformed SWCNT can be obtained. Here, the second set of parameters [120] for the Brenner potential is employed. In the calculation, we need to present a number as the initial bond length for the representative cell in the reference graphite sheet (Fig. 4.7A). Essentially, we can choose any number as the initial optimization value, and the choice of this value has no effect on the final optimized results. Moreover, by setting a very large tube radius, we can obtain the bond length in the initial equilibrium graphite sheet. For the second set of parameters of the Brenner potential, our computation indicates that the bond length in the initial equilibrium graphite sheet is 0.14507 nm which is used as the initial value of the bond lengths in the reference configuration. With this choice, the evaluated values of λ₁, λ₂, and θ are, respectively, the axial and circumferential stretches and twisting angle of the tube relative to the initial equilibrium graphite sheet.

Table 4.1 shows the evaluated values of λ₁, λ₂, and θ for chiral (9, 6), armchair (5, 5), and zigzag (10, 0) SWCNTs. It is noted that θ=0 for the armchair and zigzag SWCNTs, whereas a nonzero twisting angle θ occurs for the chiral CNT. This is because the representative cells in the armchair and zigzag CNTs are axisymmetric with respect to the axial and circumferential directions of the tube. The evaluated bond lengths and tube radii are listed in Table 4.2, along with the calculated atomic energy. Table 4.3 gives the results obtained by Chandraseker and Mukherjee [121]. The present results for bond lengths and atomic energy display excellent agreement with those of Chandraseker and Mukherjee [121]. However, the tube radius shows a slight discrepancy, and this may be because of a difference in the radius definitions in the two methods. (9, 6), (5, 5), and (10, 0) SWCNTs belong to tubes of small radius. The present computations show that the approximation with the first- and second-order deformation gradient is precise enough even for tubes of small radius.

4.4.3 Elastic Properties

Computations have been carried out for different types of CNTs. In the results analysis, we use (3n, 2n) CNTs as an example to illustrate the properties of chiral CNTs. In all of the calculations, the first- and second-order derivatives are exactly computed. Fig. 4.8 shows the variation of the axial and circumferential Young’s moduli with the tube radius. With an increasing tube radius, the axial and circumferential Young’s moduli trend to the same constant that corresponds to the Young’s modulus of the graphite sheet. The present estimated value is 0.7049 TPa, which is very close to that predicted by Arroyo and Belytschko [18]. When the tube radius is larger than 0.5 nm, the moduli are very close to those of the graphite sheet, and the maximum difference is less than 2%. Moreover, we also notice a special phenomenon: the axial and circumferential Young’s moduli of the CNTs are generally less than those of the graphite sheet, but the circumferential Young’s moduli of the zigzag CNTs are slightly larger than those of the graphite sheet. This indicates that the chirality of CNTs has an effect on their properties, but this effect is very slight when the tube radius is larger than 0.5 nm. Plotted in Fig. 4.9 is the radial Young’s modulus versus the tube radius. The radial Young’s modulus has a large dependence on the tube radius, because it is equal to the circumferential Young’s modulus divided by the tube radius. Fig. 4.10 shows the variation of Poisson’s ratio $v_{21}$ with the tube radius. Similar to the axial and circumferential Young’s moduli, $v_{21}$ trends to the Poisson’s ratio of the graphite sheet (the present estimated value is 0.4123). When the tube radius is larger than 0.5 nm, the Poisson’s ratio is very close to 0.4123. Fig. 4.11 shows the dependence of the shear modulus on the tube radius. When the tube radius is larger than 0.5 nm, the shear modulus is very close to that of the graphite sheet. The present estimated shear modulus of the graphite sheet is approximately 0.25 TPa. In conclusion, the elastic properties of CNTs have a dependence on the chirality and the tube radius, but when the tube radius is larger than 0.5 nm, the axial and circumferential Young’s moduli, Poisson’s ratio, and shear modulus are pretty close to those of the graphite sheet. Therefore, CNTs can be viewed as a curved surface on which the elastic properties at the tangent direct are almost the same with the graphite sheet.

4.4.4 Pressure–Radial Strain Curve

The above method can be viewed as an analytical-numerical method, in which the first- and second-order derivatives of the strain energy density are calculated exactly, and Newton’s method is used only to determine the minimum point of energy. Moreover, by varying the values of λ₁, λ₂, and θ, and then minimizing the energy density, we can calculate the mechanical response of CNTs under axial tension or compression (varying λ₁), hydrostatic pressure (varying λ₂), and twisting (varying θ). This method can also be used to study the interaction of different loading conditions, e.g., the twisting behavior of different chiralities of CNTs under axial or radial compression. Here, the hydrostatic pressure–radial strain and axial strain–stress curves are presented as two examples.

The axial strain–stress curves of SWCNTs can be obtained by varying the value of λ₁. For a given λ₁ that corresponds to an axial strain, the energy minimization results in the values of λ₂ and θ and the axial stress can be calculated using Eq. (4.48). By varying λ₁, the axial strain–stress can be obtained. Plotted in Fig. 4.12 is the axial strain–stress curve of a (15, 0) SWCNT.

The pressure–radial strain curve of SWCNTs under hydrostatic pressure can be obtained by varying the value of λ₂. For a given λ₂ that corresponds to a radial compressive strain, energy minimization gives the values of λ₁ and θ, and the hydrostatic pressure can be calculated as $p=\frac{\partial W_0}{R\partial {\lambda }_2}$. By varying λ₂, the curve of the hydrostatic pressure versus the radial strain (λ₂−1) can be obtained. Plotted in Fig. 4.13 is the pressure–radial strain curve of a (10, 10) SWCNT.

4.5.1 Moving least-squares approximation

In the MLS approximation, any variable $\psi (\mathbf{X})$ that is defined in a domain $\mathrm{\Omega }$ can be approximated by ${\psi }^h(\mathbf{X})$ in the subdomain ${\mathrm{\Omega }}_{\mathbf{X}}$, as follows.

${\psi }^h(\mathbf{X})=\sum _{i=1}^m{g}_i(\mathbf{X})a_i(\mathbf{X})={\mathbf{g}}^T(\mathbf{X})\mathbf{a}(\mathbf{X})$ (4.58)

(4.58)

where $g_i(\mathbf{X})$ is the monomial basis function, m is the number of terms in the basis, and $a_i(\mathbf{X})$ are the coefficients of the basis function. The commonly used basis is the linear basis

${\mathbf{g}}^T=(1,X_1),m=2,\mathrm{in}1\mathrm{D}$ (4.59)

(4.59)

${\mathbf{g}}^T=(1,X_1,X_2),m=3,\mathrm{in}2\mathrm{D}$ (4.60)

(4.60)

and the quadratic basis is

${\mathbf{g}}^T=(1,X_1,X_1^2),m=3,\mathrm{in}1\mathrm{D}$ (4.61)

(4.61)

${\mathbf{g}}^T=(1,X_1,X_2,X_1^2,X_1{X}_2,X_2^2),m=6,\mathrm{in}2\mathrm{D}$ (4.62)

(4.62)

Certain basis functions can be used to satisfy the special properties of the variable. For example, the following basis function can capture stress singularity in linear-elastic fracture mechanics [116].

${\mathbf{g}}^T=(1,X_1,X_2,\sqrt{r}\cos \frac{\theta }{2},\sqrt{r}\sin \frac{\theta }{2},\sqrt{r}\sin \frac{\theta }{2}\cos \theta ,\sqrt{r}\cos \frac{\theta }{2}\sin \theta ),m=7$ (4.63)

(4.63)

where r and θ are polar coordinates with the crack tip as the origin.

The unknown coefficients $a_i(\mathbf{X})$ in Eq. (4.58) can be determined by the minimization of the weighted discrete L₂ norm as

$J=\sum _{I=1}^{NP}w(\mathbf{X}-{\mathbf{X}}_I){[{\mathbf{g}}^T({\mathbf{X}}_I)\mathbf{a}({\mathbf{X}}_I)-{\psi }_I]}^2$ (4.64)

(4.64)

where $w(\mathbf{X}-{\mathbf{X}}_I)$ is the weight function with compact support, and NP is the number of nodes with $w(\mathbf{X}-{\mathbf{X}}_I)>0$. The minimum of J in Eq. (4.64) with respect to $\mathbf{a}(\mathbf{X})$ leads to a set of linear equations:

$\mathbf{A}\mathbf{(}\mathbf{X}\mathbf{)}\mathbf{a}\mathbf{(}\mathbf{X}\mathbf{)}=\mathbf{B}\mathbf{(}\mathbf{X}\mathbf{)}\mathbf{\psi }$ (4.65)

(4.65)

where $\mathbf{\psi }={({\psi }_1,{\psi }_2,\dots ,{\psi }_N)}^T$, and

$\mathbf{A}\mathbf{(}\mathbf{X}\mathbf{)}=\sum _{I=1}^{NP}w(\mathbf{X}-{\mathbf{X}}_I)\mathbf{g}\mathbf{(}{\mathbf{X}}_I\mathbf{)}{\mathbf{g}}^T\mathbf{(}{\mathbf{X}}_I\mathbf{)}$ (4.66)

(4.66)

and

$\mathbf{B}\mathbf{(}\mathbf{X}\mathbf{)}=[w(\mathbf{X}-{\mathbf{X}}_1)\mathbf{g}\mathbf{(}{\mathbf{X}}_1\mathbf{)},w(\mathbf{X}-{\mathbf{X}}_2)\mathbf{g}\mathbf{(}{\mathbf{X}}_2\mathbf{)},\dots ,w(\mathbf{X}-{\mathbf{X}}_N)\mathbf{g}\mathbf{(}{\mathbf{X}}_N\mathbf{)}]$ (4.67)

(4.67)

Thus, the unknown coefficients $\mathbf{a}(\mathbf{X})$ can be obtained from Eq. (4.64) as

$\mathbf{a}\mathbf{(}\mathbf{X}\mathbf{)}={\mathbf{A}}^{-1}\mathbf{(}\mathbf{X}\mathbf{)}\mathbf{B}\mathbf{(}\mathbf{X}\mathbf{)}\mathbf{\psi }$ (4.68)

(4.68)

Substituting Eq. (4.68) into Eq. (4.58), the MLS approximation can be written in standard form as

${\psi }^h(\mathbf{X})=\sum _{I=1}^{NP}{\phi }_I\mathbf{(}\mathbf{X}\mathbf{)}{\psi }_I\mathbf{(}\mathbf{X}\mathbf{)}$ (4.69)

(4.69)

where the MLS shape function ${\varphi }_I\mathbf{(}\mathbf{X}\mathbf{)}$ is defined as

${\phi }_I\mathbf{(}\mathbf{X}\mathbf{)}=\sum _{j=1}^m{g}_j(\mathbf{X}){[{\mathbf{A}}^{-1}\mathbf{(}\mathbf{X}\mathbf{)}\mathbf{B}\mathbf{(}\mathbf{X}\mathbf{)}]}_{jI}$ (4.70)

(4.70)

The partial derivatives of ${\varphi }_I\mathbf{(}\mathbf{X}\mathbf{)}$ with respect to X_K can be obtained as

${\phi }_{I,K}=\sum _{j=1}^m[g_{j,K}{({\mathbf{A}}^{-1}\mathbf{B})}_{jI}+g_j{({\mathbf{A}}^{-1}{\mathbf{B}}_K+{\mathbf{A}}_K^{-1}\mathbf{B})}_{jI}]$ (4.71)

(4.71)

in which

${\mathbf{B}}_K(\mathbf{X})=[w_{1,K}(\mathbf{X})\mathbf{g}({\mathbf{X}}_1),w_{2,K}(\mathbf{X})\mathbf{g}({\mathbf{X}}_2),\dots ,w_{N,K}(\mathbf{X})\mathbf{g}({\mathbf{X}}_N)]$ (4.72)

(4.72)

${\mathbf{A}}_K^{-1}=-{\mathbf{A}}^{-1}{\mathbf{A}}_K{\mathbf{A}}^{-1}$ (4.73)

(4.73)

$A_K(\mathbf{X})=\sum _{I=1}^N{w}_{I,k}(\mathbf{X})\mathbf{g}({\mathbf{X}}_I){\mathbf{g}}^T({\mathbf{X}}_I)$ (4.74)

(4.74)

The second partial derivatives of ${\varphi }_I\mathbf{(}\mathbf{X}\mathbf{)}$ are obtained as

$\begin{array}{l}{\phi }_{I,KL}=\sum _{j=1}^m[g_{j,KL}{({\mathbf{A}}^{-1}\mathbf{B})}_{jI}+g_{j,K}{({\mathbf{A}}_L^{-1}\mathbf{B}+{\mathbf{A}}^{-1}{\mathbf{B}}_L)}_{jI}+g_{j,L}{({\mathbf{A}}^{-1}{\mathbf{B}}_K+{\mathbf{A}}_K^{-1}\mathbf{B})}_{jI}\hfill \\ +g_j{({\mathbf{A}}_L^{-1}{\mathbf{B}}_K+{\mathbf{A}}^{-1}{\mathbf{B}}_{KL}+{\mathbf{A}}_{KL}^{-1}\mathbf{B}+{\mathbf{A}}_K^{-1}{\mathbf{B}}_L)}_{jI}]\hfill \end{array}$ (4.75)

(4.75)

in which

${\mathbf{B}}_{KL}(\mathbf{X})=[w_{1,KL}(\mathbf{X})\mathbf{g}({\mathbf{X}}_1),w_{2,KL}(\mathbf{X})\mathbf{g}({\mathbf{X}}_2),\dots ,w_{N,KL}(\mathbf{X})\mathbf{g}({\mathbf{X}}_N)]$ (4.76)

(4.76)

${\mathbf{A}}_{KL}^{-1}={\mathbf{A}}^{-1}{\mathbf{A}}_L{\mathbf{A}}^{-1}{\mathbf{A}}_K{\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}{\mathbf{A}}_{KL}{\mathbf{A}}^{-1}+{\mathbf{A}}^{-1}{\mathbf{A}}_K{\mathbf{A}}^{-1}{\mathbf{A}}_L{\mathbf{A}}^{-1}$ (4.77)

(4.77)

In the present research, the quadratic basis (Eq. 4.62) is used, and the weight function is chosen as the cubic spline function (Eq. 4.80). Shown in Fig. 4.16 is a plot of the MLS shape function. Fig. 4.16A presents the 2-D view, and Fig. 4.16B the variation of the shape function along the X₁ direction (X₂=0), which corresponds to the 1-D MLS shape function. Fig. 4.17 shows the first-order derivative of the shape function ${\varphi }_1$, and plotted in Figs. 4.18 and 4.19 are the second-order derivatives ${\varphi }_{11}$ and ${\varphi }_{12}$.

4.5.2 Weight Functions

The choice of weight functions is an important ingredient in mesh-free methods, as these functions affect the continuity and accuracy of the resulting mesh-free approximation. Generally, the weight functions can result in C² or higher-order continuity of the shape functions. To generate a set of sparse discrete equations, the weight functions are defined as having a compact support, which means that they should be nonzero only over a small neighborhood of node X_I.

The commonly used weight functions are in the following forms.

$\mathrm{Exponential}:w(d)=\{\begin{array}{ll}{e}^{-{(d/\alpha )}^2,}\hfill & d\le 1\hfill \\ 0,\hfill & d>1\hfill \end{array}$ (4.78)

(4.78)

$\mathrm{Quadratic}\mathrm{spline}:w(d)=\{\begin{array}{ll}1-2d^2,\hfill & d\le \frac{1}{2}\hfill \\ 2{(1-d)}^2,\hfill & \frac{1}{2}<d\le 1\hfill \\ 0,\hfill & d>1\hfill \end{array}$ (4.79)

(4.79)

$\mathrm{Cubic}\mathrm{spline}:w(d)=\{\begin{array}{ll}\frac{2}{3}-4d^2+4d^3,\hfill & d\le \frac{1}{2}\hfill \\ \frac{4}{3}+4d+4d^2-\frac{4}{3}{d}^3,\hfill & \frac{1}{2}<d\le 1\hfill \\ 0,\hfill & d>1\hfill \end{array}$ (4.80)

(4.80)

$\mathrm{Quartic}\mathrm{spline}:w(d)=\{\begin{array}{ll}1-6d^2+8d^3-3d^4,\hfill & d\le 1\hfill \\ 0,\hfill & d>1\hfill \end{array}$ (4.81)

(4.81)

where d is the normalized distance, and α is a dilation parameter of the weight function.

In the 1-D case, the weight function is written as

$w(X_1-X_{1I})=w\left(\frac{d_I}{c_I}\right)$ (4.82)

(4.82)

where $d_I=|X_1-X_{1I}|$, and c_I is the size of the support of node X_I.

In the 2-D case, there are two commonly used supports. For the circular compact support shown in Fig. 4.1A, the weight function can be written as

$w(\mathbf{X}-{\mathbf{X}}_I)=w\left(\frac{d_I}{R_I}\right)$ (4.83)

(4.83)

where $d_I=\left|\right|\mathbf{X}-{\mathbf{X}}_I\left|\right|=\sqrt{{(X_1-X_{1I})}^2+{(X_2-X_{2I})}^2}$, and R_I is the radius of the support of node X_I.

For the rectangular compact support shown in Fig. 4.1B, the product weight function is given by

$w(\mathbf{X}-{\mathbf{X}}_I)=w\left(\frac{d_1}{a_{1I}}\right)w\left(\frac{d_2}{a_{2I}}\right)$ (4.84)

(4.84)

where $d_1=|X_1-X_{1I}|$,$d_2=|X_2-X_{2I}|$, and $(a_{1I},a_{2I})$ is the size of the rectangular support of node X_I.

The MLS approximation is closely related to the weight functions and the basis functions. Dolbow and Belytschko [122] stated that if the continuity of the basis function is greater than the continuity of the weight function, then the resulting approximation would inherit the continuity of the weight function. The exponential weight can resemble a weight with C¹ or higher-order continuity. The cubic and quartic weights possess C² continuity, which is ideal for constructing the mesh-free shape function with C² continuity.

4.5.3 Domain of Influence of Nodes

The weight functions are defined as having a compact support, i.e., the subdomain over it is nonzero. Each subdomain ${\mathrm{\Omega }}_I$ is associated with a node ${\mathbf{X}}_I$, which can be termed the node’s domain of influence (DOI). The way in which the DOI for each node is determined has a tremendous impact on the total performance of a mesh-free method. The node’s DOI is inherent in the weight function’s compact support, which makes the interpolation limited on the several nodes whose supports cover the evaluated point. The most commonly used supports for 2-D problems are circular and rectangular supports, as shown in Fig. 4.1. The shape of the support has almost no effect on the results.

In the 1-D case, the size of the support can be defined as

$c_I=D_{\max }{c}_{bI},I=1,\dots ,n,$ (4.85)

(4.85)

where $D_{\max }$ is a scaling factor, and $c_{bI}$ is the basic support of node $X_I$. $c_{bI}$ can be determined by the maximum distance to its nearest neighboring node.

The radius of the circular support can be defined as

$R_I=D_{\max }{r}_I,I=1,\dots ,n,$ (4.86)