(4.86)

where $r_I$ is the radius of a basic circular support. The radius $r_I$ is determined by searching for sufficient adjacent nodes to satisfy the basis. For example, we can designate the support to cover just the three nearest neighboring nodes.

The size of a rectangular support can be defined by

$\{\begin{array}{c}{a}_{1I}=D_{\max }{c}_{1I}\\ a_{2I}=D_{\max }{c}_{2I}\end{array},I=1,\dots ,n,$ (4.87)

(4.87)

where $(c_{1I},c_{2I})$ defines a basic support that is determined by searching for sufficient neighboring nodes to satisfy the basis in both directions. For the regular collocation of nodes, $c_{1I}$ and $c_{2I}$ can be determined by the distance to the nearest neighboring nodes in two directions.

The size of the compact support (or DOI) determines the number of nodes N in the construction of matrix $\mathbf{A}(\mathbf{X})$. The number of nodes N should be large enough to ensure that $\mathbf{A}(\mathbf{X})$ is invertible. The size of each node’s DOI can be different, and it depends on the spacing density of the nodes in the local context. Small support sizes may affect the convergence of the mesh-free solutions. Large support sizes ensure convergence, but increase the computational time. Furthermore, if the support of a node extends far outside the boundary of the problem domain, then the computation size of the domain that is decided by all of the supports will be very different from the actual, which will inevitably affect the computational results. Therefore, for a given $D_{\max }$, increasing the number of nodes will decrease the support size and make the computational size of the domain coincide with its actual size. According to the literature, $D_{\max }$ is typically in the range of 2.0–4.0.

In the later numerical simulations, the mesh-free nodes are uniformly collocated along the axial and circumferential directions (see Fig. 4.15). Two simple examples are tested to discuss the choice of the parameter $D_{\max }$ for the present problems.

4.5.4 MK Interpolation

Kriging can be traced to the first application in geostatistics [123] for spatial interpolation and was used for construction of the shape function and its derivatives. Similar to the MLS approximation, Kriging interpolation can also be extended to any subdomain $\mathrm{\Omega }(x_i)$—termed MK interpolation. We consider a distribution function $u(x_i)$ in a subdomain $\mathrm{\Omega }(x_i)$. The subdomain is represented by a series of dispersive nodes x_i (i=1, 2, …, n), where n is the total number of nodes covered in the subdomain. With n given field values u(x₁), u(x₂), … u(x_n), the approximate field function $\hat{u}(x)$ can be interpolated by a weighted linear combination:

$\hat{u}(x)=\sum _{i=1}^n{\lambda }_i(x)u(x_i)$ (4.88)

(4.88)

where ${\lambda }_i(x)$ is the weight function of the ith node. There are two conditions which should be satisfied for the selection of coefficients ${\lambda }_i(x)$, i.e., unbiasedness and minimum variance.

The first term is unbiasedness, which implies that the expected value of approximate function $\hat{u}(x)$ must be in accordance with that of the given nodal values u(x):

$E[u(x)]=E[\hat{u}(x)]$ (4.89)

(4.89)

Substitution of Eq. (4.88) into Eq. (4.89) yields:

$E[u(x)]=E\left[\sum _{i=1}^n{\lambda }_i(x)u(x_i)\right]$ (4.90)

(4.90)

A polynomial drift model $p_l(x)$ is summarized by:

$\sum _{i=1}^n{\lambda }_i(x)p_l(x_i)=p_l(x),1\le l\le m,p_1(x)=1$ (4.91)

(4.91)

A commonly used linear basis in one-, two-, and three-dimensional space is given as:

${\mathbf{p}}^T(\mathit{x})=(1,x_1),m=2$ (4.92)

(4.92)

${\mathbf{p}}^T(\mathit{x})=(1,x_1,x_2),m=3$ (4.93)

(4.93)

and a quadratic basis is:

${\mathbf{p}}^T(\mathit{x})=(1,x_1,x_1^2),m=3$ (4.94)

(4.94)

${\mathbf{p}}^T(\mathit{x})=(1,x_1,x_2,x_1^2,x_1{x}_2,x_2^2),m=6$ (4.95)

(4.95)

Secondly, the condition of minimum variance requires a minimum estimation error in the mean square sense:

$\prod =E\left[{(u(x)-\hat{u}(x))}^2\right]$ (4.96)

(4.96)

Regarding Eq. (4.91) as a constraint by using a Lagrange multiplier ${\mu }_l$ in Eq. (4.11) yields:

$\begin{array}{ll}\prod \hfill & =E\left[{(u(x)-\hat{u}(x))}^2\right]+2\sum _{l=1}^m{\mu }_l\left[\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)\right]\hfill \\ \hfill & =Var[{\hat{u}}_{(x)}-u_{(x)}]+2\sum _{l=1}^m{\mu }_l\left[\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)\right]\hfill \\ \hfill & =E{\left[\sum _{i=1}^n{\lambda }_i{u}_{(x_i)}-u_{(x)}\right]}^2-E^2[{\hat{u}}_{(x)}-u_{(x)}]+2\sum _{l=1}^m{\mu }_l\left[\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)\right]\hfill \\ \hfill & =E{\left[\sum _{i=1}^n{\lambda }_i{u}_{(x_i)}-u_{(x)}\right]}^2+2\sum _{l=1}^m{\mu }_l\left[\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)\right]\hfill \\ \hfill & =\sum _{i=1}^n\sum _{j=1}^n{\lambda }_i{\lambda }_{j}E[(u_{(x_i)}-u_{(x)})(u_{(x_j)}-u_{(x)})]-\sum _{i=1}^n\sum _{j=1}^n{\lambda }_i{\lambda }_{j}E[(u_{(x_i)}-u_{(x)})]E[(u_{(x_j)}-u_{(x)})]\hfill \\ \hfill & +2\sum _{l=1}^m{\mu }_l\left[\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)\right]\hfill \\ \hfill & =\sum _{i=1}^n\sum _{j=1}^n{\lambda }_i{\lambda }_{j}Cov(u_{(x_i)}-u_{(x)},u_{(x_j)}-u_{(x)})+2\sum _{l=1}^m{\mu }_l\left[\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)\right]\hfill \end{array}$ (4.97)

(4.97)

By introducing a semivariogram model, covariance E[u(x)u(x_i)] can be substituted by semivariogram $\gamma (h)$, defined as:

$\gamma (h)=\gamma (x,x_i)=E[u(x)u(x_i)]=\frac{1}{2}E\left[{\{u(x)-u(x_i)\}}^2\right]$ (4.98)

(4.98)

Many theoretical models are available for semivariograms. Due to the simplicity and wide application of the Gaussian function, we select this as the correlation function in the present numerical calculation:

$\gamma (x,x_i)=\gamma (h)=2\left(1-e^{-\theta {\left(\frac{h}{a_0}\right)}^2}\right)(h\le a_0)$ (4.99)

(4.99)

where $h=\left|\right|\mathit{x}-{\mathit{x}}_i\left|\right|$ is the vectorial distance between x and x_i; θ is a correlation parameter and $a_0$ is the range of the compact support domain. It can be seen from the foregoing formulations that the selection of parameters for the correlation function plays an important role in the accuracy of MK interpolation and the numerical solution. Since trigonometric functions possess the higher-order continuity, the optimal value for correlation parameter θ is determined by fitting the trigonometric functions as well as their first- and second-order derivatives for both one- and two-dimensional problems. Finally, we concluded that a value of one for the correlation parameter in the numerical calculation would give a good prediction with the analytical solution. We select a circle with radius r=a₀, as the compact support domain. In the same way, E[u(x)u(x_j)] and E[u(x_i)u(x_j)] can be replaced by $\gamma (x,x_j)$ and $\gamma (x_i,x_j)$, respectively. Thus, we obtain:

$\begin{array}{ll}2\gamma (x_i,x_j)\hfill & =E{[u(x_i)-u(x_j)]}^2\hfill \\ \hfill & =E{[(u(x_i)-u(x))-(u(x_j)-u(x))]}^2\hfill \\ \hfill & =E{[(u(x_i)-u(x))]}^2+E{[(u(x_j)-u(x))]}^2\hfill \\ \hfill & -2E[(u(x_i)-u(x))(u(x_j)-u(x))]\hfill \\ \hfill & =2\gamma (x,x_i)+2\gamma (x,x_j)-2\mathrm{cov}[u(x_i)-u(x),u(x_j)-u(x)]\hfill \end{array}$ (4.100)

(4.100)

Eq. (4.100) can be rewritten as:

$Cov(u_{(x)}-u_{(x_j)},u_{(x_i)}-u_{(x_j)})={\gamma }_{(x,x_j)}+{\gamma }_{(x_i,x_j)}-{\gamma }_{(x,x_i)}$ (4.101)

(4.101)

Substituting Eq. (4.101) into Eq. (4.97), we obtain the following equation:

$\begin{array}{ll}\prod \hfill & =\sum _{i=1}^n\sum _{j=1}^n{\lambda }_i{\lambda }_j({\gamma }_{(x_i,x)}+{\gamma }_{(x_j,x)}-{\gamma }_{(x_i,x_j)})+2\sum _{l=1}^m{\mu }_l\left[\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)\right]\hfill \\ \hfill & =2\sum _{i=1}^n{\lambda }_i{\gamma }_{(x_i,x)}-\sum _{i=1}^n\sum _{j=1}^n{\lambda }_i{\lambda }_j{\gamma }_{(x_i,x_j)}+2\sum _{l=1}^m{\mu }_l\left[\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)\right]\hfill \end{array}$ (4.102)

(4.102)

The minimum mean square error is determined by equating all of the first derivatives with respect to ${\lambda }_i(x)$ and ${\mu }_i(x)$ to zero, which results in the following series of linear algebraic equations:

$\begin{array}{l}\frac{\partial \mathrm{\Pi }}{\partial {\lambda }_i}=2{\gamma }_{(x,x_i)}-2\sum _{j=1}^n{\lambda }_j{\gamma }_{(x_i,x_j)}+2\sum _{l=1}^m{\mu }_l{p}_l(x_i)=01\le i\le n\hfill \\ \frac{\partial \mathrm{\Pi }}{\partial {\mu }_l}=\sum _{i=1}^n{\lambda }_i{p}_l(x_i)-p_l(x)=01\le l\le m\hfill \end{array}$ (4.103)

(4.103)

For convenience, Eq. (4.103) is rewritten in matrix form as follows:

$\mathbf{G}\mathbf{W}=\mathbf{g}$ (4.104)

(4.104)

where

$\mathbf{G}\mathit{=}\left[\begin{array}{cc}\mathbf{R}& \mathbf{P}\\ {\mathbf{P}}^{\mathbf{T}}& \mathbf{0}\end{array}\right]$ (4.105)

(4.105)

$\mathbf{R}\mathbf{=}\left[\begin{array}{lll}\gamma (x_1,x_1)\hfill & \cdots \hfill & \gamma (x_1,x_n)\hfill \\ \cdots \hfill & \cdots \hfill & \cdots \hfill \\ \gamma (x_n,x_1)\hfill & \cdots \hfill & \gamma (x_n,x_n)\hfill \end{array}\right]$ (4.106)

(4.106)

$\mathbf{P}=\left[\begin{array}{lll}{p}_1(x_1)\hfill & \cdots \hfill & p_m(x_1)\hfill \\ \cdots \hfill & \cdots \hfill & \cdots \hfill \\ p_1(x_n)\hfill & \cdots \hfill & p_m(x_n)\hfill \end{array}\right]$ (4.107)

(4.107)

$\mathbf{W}\mathit{=}\left[\begin{array}{llllllll}{\lambda }_1\hfill & {\lambda }_2\hfill & \cdots \hfill & {\lambda }_n\hfill & {\mu }_1\hfill & {\mu }_2\hfill & \cdots \hfill & {\mu }_m\hfill \end{array}\right]$ (4.108)

(4.108)

${\mathbf{g}}^T=\left[\begin{array}{cc}{\mathbf{r}}^T(x)& {\mathbf{p}}^T(x)\end{array}\right]$ (4.109)

(4.109)

${\mathbf{r}}^T(x)=\left[\begin{array}{ccc}\gamma (x,x_1)& \cdots & \gamma (x,x_n)\end{array}\right]$ (4.110)

(4.110)

${\mathbf{p}}^T(x)=\left[\begin{array}{ccc}{p}_1(x)& \cdots & p_m(x)\end{array}\right]$ (4.111)

(4.111)

Based on Eqs. (4.88) and (4.104), the estimated value can be calculated numerically as:

$\hat{u}(x)={\mathbf{p}}^T(x)\stackrel{\mathbf{ˆ}}{\mathbf{\beta }}+{\mathbf{r}}^T(x){\mathbf{R}}^{-1}(\mathbf{u}-\mathbf{P}\stackrel{\mathbf{ˆ}}{\mathbf{\beta }})$ (4.112)

(4.112)

where

$\stackrel{\mathbf{ˆ}}{\mathbf{\beta }}={({\mathbf{P}}^T{\mathbf{R}}^{-1}\mathbf{P}\mathbf{)}}^{-1}{\mathbf{P}}^T{\mathbf{R}}^{-1}\mathbf{u}$ (4.113)

(4.113)

The approximation function $\hat{u}(x)$ can be expressed in the form of u(x):

$\begin{array}{ll}\hat{u}(x)\hfill & ={\mathbf{p}}^T(x_i)\stackrel{\mathbf{ˆ}}{\mathbf{\beta }}+{\mathbf{r}}^T(x_i){\mathbf{R}}^{-1}(\mathbf{u}(x_i)-\mathbf{P}\stackrel{\mathbf{ˆ}}{\mathbf{\beta }})\hfill \\ \hfill & =[{\mathbf{p}}^T(x_i)\mathbf{A}+{\mathbf{r}}^T(x_i)\mathbf{B}]\mathbf{u}(x_i)\hfill \\ \hfill & =\sum _{i=1}^n{\varphi }_i(x_i)u(x_i)=\mathbf{\Phi }(x)\mathbf{u}(x)\hfill \end{array}$ (4.114)

(4.114)

where $\mathbf{\Phi }(x)$ is the interpolation function, which can be expressed by:

$\mathbf{\Phi }(x)={\mathbf{p}}^T(x)\mathbf{A}+{\mathbf{r}}^T(x)\mathbf{B}$ (4.115)

(4.115)

$\mathbf{u}(x)=\left[\begin{array}{cccc}u(x_1)& u(x_2)& \cdots & u(x_n)\end{array}\right]$ (4.116)

(4.116)

$\mathbf{A}={({\mathbf{P}}^T\mathbf{R}\mathbf{P})}^{-1}{\mathbf{P}}^T{\mathbf{R}}^{-1}$ (4.117)

(4.117)

$\mathbf{B}\mathit{=}{\mathbf{R}}^{-1}(\mathbf{I}-\mathbf{P}\mathbf{A})$ (4.118)

(4.118)

where I is a n×n unit matrix. The first- and second-order derivatives of the shape function can be derived from Eq. (4.115):

${\mathbf{\Phi }}_i(x)={\mathbf{p}}_i^T(x)\mathbf{A}+{\mathbf{r}}_i^T(x)\mathbf{B}$ (4.119)

(4.119)

${\mathbf{\Phi }}_{ij}(x)={\mathbf{p}}_{ij}^T(x)\mathbf{A}+{\mathbf{r}}_{ij}^T(x)\mathbf{B}$ (4.120)

(4.120)

where ${\mathbf{\Phi }}_i(x)$ and ${\mathbf{\Phi }}_{ij}(x)$ denote the first- and second-order spatial derivatives, respectively. Compared with MLS approximation, the shape function $\mathbf{\Phi }(x)$ constructed using the MK interpolation in Eq. (4.115) has a distinct property:

${\varphi }_i(x_j)={\delta }_{ij}$ (4.121)

(4.121)

Fig. 4.20A illustrates the one-dimensional shape functions which demonstrate that the MK interpolation has the delta function property; first- and second-order derivatives of the shape function are plotted in Fig. 4.20A and C, respectively.

4.5.5 The Mesh-Free Computational Framework

The displacement of a deformed SWCNT is divided into two parts: the displacement of an undeformed SWCNT relative to the reference configuration and the displacement of the deformed SWCNT relative to the undeformed SWCNT. The first part can be exactly calculated (Fig. 4.21). The second part is approximated using the MLS approximation, i.e.,

$\mathbf{u}=\left(\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right)=\sum _{I=1}^{NP}{\varphi }_I{\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_I$ (4.122)

(4.122)

where ${\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_I={({\hat{u}}_{I1},{\hat{u}}_{I2},{\hat{u}}_{I3})}^T$ is the nodal parameter, ${\varphi }_I$ is the mesh-free shape function, and NP is the number of nodes whose supports cover the evaluating point. It is noted that any node has 3 degrees of freedom, but the interpolation shape function is built on the 2-D domain (Fig. 4.22).

The first- and second-order gradients are computed using the following formulas.

$\mathbf{F}={\mathbf{F}}_{\mathrm{ini}}+{\mathbf{F}}_{\mathrm{apr}},\mathbf{G}={\mathbf{G}}_{\mathrm{ini}}+{\mathbf{G}}_{\mathrm{apr}}$ (4.123)

(4.123)

where ${\mathbf{F}}_{\mathrm{ini}}$ and ${\mathbf{G}}_{\mathrm{ini}}$ are the gradients of the evaluated point when no loading is applied, and can be analytically calculated using the mapping equation (4.56), and ${\mathbf{F}}_{\mathrm{apr}}$ and ${\mathbf{G}}_{\mathrm{apr}}$ are approximated as

$\{\begin{array}{l}{\mathbf{F}}_{\mathrm{apr}}=\sum _{I=1}^{NP}{\nabla }_X({\phi }_I{\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_I)=\sum _{I=1}^{NP}\left[\begin{array}{c}{\phi }_{I,1}\\ {\phi }_{I,2}\end{array}\right]{\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_I\hfill \\ {\mathbf{G}}_{\mathrm{apr}}=\sum _{I=1}^{NP}{\nabla }_{XX}({\phi }_I{\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_I)=\sum _{I=1}^{NP}\left[\begin{array}{cc}{\phi }_{I,11}& {\phi }_{I,12}\\ {\phi }_{I,21}& {\phi }_{I,22}\end{array}\right]{\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_I\hfill \end{array}$ (4.124)

(4.124)

with ${\varphi }_{I,K}$ and ${\varphi }_{I,KL}(K,L=1,2)$, respectively, being the first- and second-order derivatives of the mesh-free shape function ${\varphi }_I$.

Newton’s method is used to solve the nonlinear governing equations in which the loading is decomposed into a series of time-steps, and at each iterative step the equilibrium state is determined by iteratively solving the incremental equation.

With the definition of the deformation gradient vector grad and the stress vector $\mathbf{\sigma }$, the weak form of the equilibrium equation can be written as

${\int }_{\mathrm{\Omega }}\delta \mathbf{g}\mathbf{r}\mathbf{a}{\mathbf{d}}^T\cdot \mathbf{\sigma }dV-{\int }_{\partial \mathrm{\Omega }}\delta {\mathbf{u}}^T\cdot {\mathbf{t}}_0^{P}dS-{\int }_{\partial \mathrm{\Omega }}{({\nabla }_N\delta \mathbf{u})}^T\cdot {\mathbf{t}}_0^{Q}dS=0$ (4.125)

(4.125)

Newton’s method is adopted to solve the nonlinear governing equations. The stress at the n+1th iterative step can be obtained by:

${\mathbf{\sigma }}_{n+1}={\mathbf{\sigma }}_n+\mathrm{\Delta }{\mathbf{\sigma }}_{n+1}$ (4.126)

(4.126)

Thus, Eq. (4.125) can be rewritten as:

${\int }_{\mathrm{\Omega }}\delta \mathbf{g}\mathbf{r}\mathbf{a}{\mathbf{d}}_{n+1}^T\cdot \mathrm{\Delta }{\mathbf{\sigma }}_{n+1}dV=-{\int }_{\mathrm{\Omega }}\delta \mathbf{g}\mathbf{r}\mathbf{a}{\mathbf{d}}_{n+1}^T\cdot {\mathbf{\sigma }}_n+{\int }_{\partial \mathrm{\Omega }}\delta {\mathbf{u}}_{n+1}^T\cdot {\mathbf{t}}_0^{P}dS+{\int }_{\partial \mathrm{\Omega }}{({\nabla }_N\delta {\mathbf{u}}_{n+1})}^T\cdot {\mathbf{t}}_0^{Q}dS$ (4.127)

(4.127)

At the n+1th iterative step,

$\delta {\mathbf{u}}_{n+1}=\delta ({\mathbf{u}}_n+\mathrm{\Delta }{\mathbf{u}}_{n+1})=\delta \mathrm{\Delta }{\mathbf{u}}_{n+1}$ (4.128)

(4.128)

$\delta \mathbf{gra}{\mathbf{d}}_{n+1}=\delta (\mathbf{gra}{\mathbf{d}}_n+\mathrm{\Delta }\mathbf{gra}{\mathbf{d}}_{n+1})=\delta \mathrm{\Delta }\mathbf{gra}{\mathbf{d}}_{n+1}$ (4.129)

(4.129)

Based on Eqs. (4.128) and (4.129), Eq. (4.127) can be rewritten as:

${\int }_{\mathrm{\Omega }}\delta \mathrm{\Delta }\mathbf{gra}{\mathbf{d}}_{n+1}^T\cdot \mathrm{\Delta }{\mathbf{\sigma }}_{n+1}dV=-{\int }_{\mathrm{\Omega }}\delta \mathrm{\Delta }\mathbf{gra}{\mathbf{d}}_{n+1}^T\cdot {\mathbf{\sigma }}_n+{\int }_{\partial \mathrm{\Omega }}\delta \mathrm{\Delta }{\mathbf{u}}_{n+1}^T\cdot {\mathbf{t}}_0^{P}dS+{\int }_{\partial \mathrm{\Omega }}{({\nabla }_N\delta \mathrm{\Delta }{\mathbf{u}}_{n+1})}^T\cdot {\mathbf{t}}_0^{Q}dS$ (4.130)

(4.130)

where $\mathrm{\Delta }\mathbf{\sigma }\mathbf{=}\mathbf{M}\cdot \mathrm{\Delta }\mathbf{g}\mathbf{r}\mathbf{a}\mathbf{d}$ and M is a combination of the tangential moduli which can be expressed as:

$\mathbf{M}\mathbf{=}\left[\begin{array}{cc}{\mathbf{M}}_{\mathbf{F}\mathbf{F}}& {\mathbf{M}}_{\mathbf{F}\mathbf{G}}\\ {\mathbf{M}}_{\mathbf{G}\mathbf{F}}& {\mathbf{M}}_{\mathbf{G}\mathbf{G}}\end{array}\right]$ (4.131)

(4.131)

Consequently, Eq. (4.130) can be expressed as:

${\int }_{\mathrm{\Omega }}\delta \mathrm{\Delta }\mathbf{gra}{\mathbf{d}}_{n+1}^T\cdot \mathbf{M}\cdot \mathrm{\Delta }\mathbf{gra}{\mathbf{d}}_{n+1}dV=-{\int }_{\mathrm{\Omega }}\delta \mathrm{\Delta }\mathbf{gra}{\mathbf{d}}_{n+1}^T\cdot {\mathbf{\sigma }}_n+{\int }_{\partial \mathrm{\Omega }}\delta \mathrm{\Delta }{\mathbf{u}}_{n+1}^T\cdot {\mathbf{t}}_0^{P}dS+{\int }_{\partial \mathrm{\Omega }}{({\nabla }_N\delta \mathrm{\Delta }{\mathbf{u}}_{n+1})}^T\cdot {\mathbf{t}}_0^{Q}dS$ (4.132)

(4.132)

The displacement increment of the evaluating point can be approximated as

$\mathrm{\Delta }{\mathbf{u}}_{n+1}=\left(\begin{array}{c}\mathrm{\Delta }{u}_1\\ \mathrm{\Delta }{u}_2\\ \mathrm{\Delta }{u}_3\end{array}\right)=\mathbf{\Phi }\cdot \mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}^e$ (4.133)

(4.133)

$\mathbf{\Phi }=\left[\begin{array}{llllllllllll}{\phi }_1\hfill & 0\hfill & 0\hfill & {\phi }_2\hfill & 0\hfill & 0\hfill & \ddots \hfill & \hfill & \hfill & {\phi }_{NP}\hfill & 0\hfill & 0\hfill \\ 0\hfill & {\phi }_1\hfill & 0\hfill & 0\hfill & {\phi }_2\hfill & 0\hfill & \hfill & \ddots \hfill & \hfill & 0\hfill & {\phi }_{NP}\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\phi }_1\hfill & 0\hfill & 0\hfill & {\phi }_2\hfill & \hfill & \hfill & \ddots \hfill & 0\hfill & 0\hfill & {\phi }_{NP}\hfill \end{array}\right]$ (4.134)

(4.134)

$\mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}^e=\left[\begin{array}{l}\mathrm{\Delta }{\hat{u}}_{11}\hfill \\ \mathrm{\Delta }{\hat{u}}_{12}\hfill \\ \mathrm{\Delta }{\hat{u}}_{13}\hfill \\ \mathrm{\Delta }{\hat{u}}_{21}\hfill \\ \mathrm{\Delta }{\hat{u}}_{22}\hfill \\ \mathrm{\Delta }{\hat{u}}_{23}\hfill \\ ⋮\hfill \\ ⋮\hfill \\ ⋮\hfill \\ \mathrm{\Delta }{\hat{u}}_{NP1}\hfill \\ \mathrm{\Delta }{\hat{u}}_{NP2}\hfill \\ \mathrm{\Delta }{\hat{u}}_{NP3}\hfill \end{array}\right]$ (4.135)

(4.135)

The gradient increment can be calculated as

$\mathrm{\Delta }\mathbf{g}\mathbf{r}\mathbf{a}{\mathbf{d}}_{n+1}=\mathbf{L}\cdot \mathrm{\Delta }\mathbf{u}=\mathbf{L}\cdot \mathbf{\Phi }\cdot \mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}^e$ (4.136)

(4.136)

$\mathbf{L}=\left[\begin{array}{lll}\partial /\partial X_1\hfill & 0\hfill & 0\hfill \\ \partial /\partial X_2\hfill & 0\hfill & 0\hfill \\ 0\hfill & \partial /\partial X_1\hfill & 0\hfill \\ 0\hfill & \partial /\partial X_2\hfill & 0\hfill \\ 0\hfill & 0\hfill & \partial /\partial X_1\hfill \\ 0\hfill & 0\hfill & \partial /\partial X_2\hfill \\ {\partial }^2/\partial {X_1}^2\hfill & 0\hfill & 0\hfill \\ {\partial }^2/\partial X_1\partial X_2\hfill & 0\hfill & 0\hfill \\ {\partial }^2/\partial {X_2}^2\hfill & 0\hfill & 0\hfill \\ 0\hfill & {\partial }^2/\partial {X_1}^2\hfill & 0\hfill \\ 0\hfill & {\partial }^2/\partial X_1\partial X_2\hfill & 0\hfill \\ 0\hfill & {\partial }^2/\partial {X_2}^2\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\partial }^2/\partial {X_1}^2\hfill \\ 0\hfill & 0\hfill & {\partial }^2/\partial X_1\partial X_2\hfill \\ 0\hfill & 0\hfill & {\partial }^2/\partial {X_2}^2\hfill \end{array}\right]$ (4.137)

(4.137)

$\mathbf{L}\cdot \mathbf{\Phi }=\left[\begin{array}{llllllllllll}{\phi }_{1,1}\hfill & 0\hfill & 0\hfill & {\phi }_{2,1}\hfill & 0\hfill & 0\hfill & \ddots \hfill & \hfill & \hfill & {\phi }_{NP,1}\hfill & 0\hfill & 0\hfill \\ {\phi }_{1,2}\hfill & 0\hfill & 0\hfill & {\phi }_{2,2}\hfill & 0\hfill & 0\hfill & \hfill & \ddots \hfill & \hfill & {\phi }_{NP,2}\hfill & 0\hfill & 0\hfill \\ 0\hfill & {\phi }_{1,1}\hfill & 0\hfill & 0\hfill & {\phi }_{2,1}\hfill & 0\hfill & \hfill & \hfill & \ddots \hfill & 0\hfill & {\phi }_{NP,1}\hfill & 0\hfill \\ 0\hfill & {\phi }_{1,2}\hfill & 0\hfill & 0\hfill & {\phi }_{2,2}\hfill & 0\hfill & \ddots \hfill & \hfill & \hfill & 0\hfill & {\phi }_{NP,2}\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\phi }_{1,1}\hfill & 0\hfill & 0\hfill & {\phi }_{2,1}\hfill & \hfill & \ddots \hfill & \hfill & 0\hfill & 0\hfill & {\phi }_{NP,1}\hfill \\ 0\hfill & 0\hfill & {\phi }_{1,2}\hfill & 0\hfill & 0\hfill & {\phi }_{2,2}\hfill & \hfill & \hfill & \ddots \hfill & 0\hfill & 0\hfill & {\phi }_{NP,2}\hfill \\ {\phi }_{1,11}\hfill & 0\hfill & 0\hfill & {\phi }_{2,11}\hfill & 0\hfill & 0\hfill & \ddots \hfill & \hfill & \hfill & {\phi }_{NP,11}\hfill & 0\hfill & 0\hfill \\ {\phi }_{1,12}\hfill & 0\hfill & 0\hfill & {\phi }_{2,12}\hfill & 0\hfill & 0\hfill & \hfill & \ddots \hfill & \hfill & {\phi }_{NP,12}\hfill & 0\hfill & 0\hfill \\ {\phi }_{1,22}\hfill & 0\hfill & 0\hfill & {\phi }_{2,22}\hfill & 0\hfill & 0\hfill & \hfill & \hfill & \ddots \hfill & {\phi }_{NP,22}\hfill & 0\hfill & 0\hfill \\ 0\hfill & {\phi }_{1,11}\hfill & 0\hfill & 0\hfill & {\phi }_{2,11}\hfill & 0\hfill & \ddots \hfill & \hfill & \hfill & 0\hfill & {\phi }_{NP,11}\hfill & 0\hfill \\ 0\hfill & {\phi }_{1,12}\hfill & 0\hfill & 0\hfill & {\phi }_{2,12}\hfill & 0\hfill & \hfill & \ddots \hfill & \hfill & 0\hfill & {\phi }_{NP,12}\hfill & 0\hfill \\ 0\hfill & {\phi }_{1,22}\hfill & 0\hfill & 0\hfill & {\phi }_{2,22}\hfill & 0\hfill & \hfill & \hfill & \ddots \hfill & 0\hfill & {\phi }_{NP,22}\hfill & 0\hfill \\ 0\hfill & \hfill & {\phi }_{1,11}\hfill & 0\hfill & 0\hfill & {\phi }_{2,11}\hfill & \ddots \hfill & \hfill & \hfill & 0\hfill & 0\hfill & {\phi }_{NP,11}\hfill \\ 0\hfill & \hfill & {\phi }_{1,12}\hfill & 0\hfill & 0\hfill & {\phi }_{2,12}\hfill & \hfill & \ddots \hfill & \hfill & 0\hfill & 0\hfill & {\phi }_{NP,12}\hfill \\ 0\hfill & \hfill & {\phi }_{1,22}\hfill & 0\hfill & 0\hfill & {\phi }_{2,22}\hfill & \hfill & \hfill & \ddots \hfill & 0\hfill & 0\hfill & {\phi }_{NP,22}\hfill \end{array}\right]$ (4.138)

(4.138)

The outward normal gradient is calculated as

${\nabla }_N\mathrm{\Delta }{\mathbf{u}}_{n+1}=N_1\frac{\partial \mathrm{\Delta }\mathbf{u}}{\partial X_1}+N_2\frac{\partial \mathrm{\Delta }\mathbf{u}}{\partial X_2}=(N_1{\mathbf{\Phi }}_1+N_2{\mathbf{\Phi }}_2)\cdot \mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}^e$ (4.139)

(4.139)

Substituting Eqs. (4.133), (4.136), and (4.139) into (4.132), we have

$\begin{array}{l}{\int }_{\mathrm{\Omega }}[{(\mathbf{L}\cdot \mathbf{\Phi })}^T\cdot \mathbf{M}\cdot (\mathbf{L}\cdot \mathbf{\Phi })]\mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}^{e}dV={\int }_{\partial \mathrm{\Omega }}{(\mathbf{\Phi })}^T\cdot {\mathbf{t}}_0^{P}dS+{\int }_{\partial \mathrm{\Omega }}{((N_1{\mathbf{\Phi }}_1+N_2{\mathbf{\Phi }}_2))}^T\cdot {\mathbf{t}}_0^{Q}dS-{\int }_{\mathrm{\Omega }}{(\mathbf{L}\cdot \mathbf{\Phi })}^T\cdot {\mathbf{\sigma }}_{n}dV\hfill \end{array}$ (4.140)

(4.140)

Thus, we can obtain the incremental system equation

${\mathbf{K}}_{n+1}\mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}={\mathbf{f}}_{n+1}$ (4.141)

(4.141)

where $\mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}$ contains the nodal parameters of all of the mesh-free nodes. The global stiffness matrix is

${\mathbf{K}}_{n+1}={\int }_{\mathrm{\Omega }}[{(\mathbf{L}\cdot \mathbf{\Phi })}^T\cdot \mathbf{M}\cdot (\mathbf{L}\cdot \mathbf{\Phi })]dV$ (4.142)

(4.142)

The nonequilibrium force vector is

${\mathbf{f}}_{n+1}={\mathbf{f}}_{n+1}^{\mathrm{ext}}-{\mathbf{f}}_n^{\mathrm{int}}$ (4.143)

(4.143)

${\mathbf{f}}_{n+1}^{\mathrm{ext}}={\int }_{\partial \mathrm{\Omega }}{(\mathbf{\Phi })}^T\cdot {\mathbf{t}}_0^{P}dS+{\int }_{\partial \mathrm{\Omega }}{((N_1{\mathbf{\Phi }}_1+N_2{\mathbf{\Phi }}_2))}^T\cdot {\mathbf{t}}_0^{Q}dS$ (4.144)

(4.144)

${\mathbf{f}}_n^{\mathrm{int}}={\int }_{\mathrm{\Omega }}{(\mathbf{L}\cdot \mathbf{\Phi })}^T\cdot {\mathbf{\sigma }}_{n}dV$ (4.145)

(4.145)

Eq. (4.144) expresses the external forces. The stable state is obtained by iteratively solving Eq. (4.141) until f reaches zero.

4.5.6 Integration Scheme for a Discrete Equation

The evaluation of integrals in Eq. (4.142) requires an appropriate integral technique. In mesh-free methods, the evaluation of these integrations is a key point. Several integrations have been proposed in the literature.

One approach is to use the background mesh-free or background cell for quadrature integration, which is used in most mesh-free methods. In this approach, the domain is divided into a series of background cells, and the quadrature rule is used in each. Different from the FEM, the background cells are used only to implement the integration, and they need not be compatible with the mesh-free nodes. However, this approach is not very stable or accurate. The setup of the background cells and the distribution of the quadrature points at the global level do not ensure sufficient quadrature points in the compact support, which is crucial to the accuracy of the integration. It has been found that if quadrature integration is performed using background cells whose corners coincide with the nodes, then significant integration errors may occur in irregular cells. At the same time, if the background cells do not match the compact support of the mesh-free interpolations, then considerable integration errors may also arise. Nevertheless, this method is still widely used due to its simplicity both in theory and implementation. To alleviate integration error, one can use the integration cells that coincide with the nodal arrangements and apply a higher-order Gaussian quadrature, although this may result in significantly increased computational time. For example, 2×2 quadrature points can achieve a high level of accuracy, but the mesh-free method may require 4×4 quadrature points.

To avoid the “ghost” of the background cell, a nodal integration procedure was developed by Beissel and Belytschko [125]. This approach added a stabilization term consisting of the square of the residual of the equilibrium equation in the potential energy functional, and the resulting equations were integrated only at the nodes. However, introducing this additional term in the potential energy sacrifices variational consistency, and hence compromises the accuracy of the formulation.

Another development is the local support integration that was proposed by Li and Liu [126]. Due to the compact support of the mesh-free shape functions, effective integration should be performed for the common parts of the supports for the associated nodes. If each node takes on the same support shape, then we can fix the same quadrature rule for each support and integrate the weak form locally from one compact to another. By carrying out integration on the compact support, we are free from the background cell. One difficulty associated with local support integration is the integration of the compact that is cut by the boundary. A particular technique is needed for integration on these cut compacts because they are not regular. Moreover, in this approach, we still need to set up the quadrature points in the compact support.

To avoid the quadrature points completely, Chen et al. [84] put forward a stabilized conforming nodal integration method for the mesh-free Galerkin method. A smoothing stabilization technique was used to overcome the numerical instabilities occurring in the nodal integration.

From the above review, we can see that there are several integration strategies that can be used in the mesh-free method. However, in the present research, background quadrature meshes are used for integration due to their simplicity and easy implementation. The mesh-free nodes are collocated uniformly along the axial and circumferential directions, and the background cells are chosen to coincide with the nodal arrangements, i.e., each integral cell is a rectangle. The Gaussian quadrature rule is used for each cell. In the practical calculation, the Gaussian points can be globally marked. To avoid repeating the calculation for each Gaussian point, the information that includes the nodes whose supports cover the Gaussian points and the corresponding shape functions and their first- and second-order derivatives is calculated and deposited in advance. Assuming that the total number of Gaussian points for the entire problem domain is NQ, we can evaluate the stiffness matrix in Eq. (4.142) as follows.

${\mathbf{K}}_{n+1}={\int }_{\mathrm{\Omega }}[{(\mathbf{L}\cdot \mathbf{\Phi })}^T\cdot \mathbf{M}\cdot (\mathbf{L}\cdot \mathbf{\Phi })]dV=\sum _{q=1}^{NQ}{w}_q{[{(\mathbf{L}\cdot \mathbf{\Phi })}^T\cdot \mathbf{M}\cdot (\mathbf{L}\cdot \mathbf{\Phi })]}_q$ (4.146)

(4.146)

where $w_q$ is the Gaussian weight. For each Gaussian point, we can obtain the substiffness matrix

${\mathbf{K}}_{n+1}^q=w_q{[{(\mathbf{L}\cdot \mathbf{\Phi })}^T\cdot \mathbf{M}\cdot (\mathbf{L}\cdot \mathbf{\Phi })]}_q$ (4.147)

(4.147)

Global stiffness can be obtained by assembling all of these substiffness matrices.

4.5.7 Enforcement of Essential Boundary Conditions

In comparison with the FEM, the mesh-free method has an obvious difficulty: the imposition of essential boundary conditions. Similar to most of the mesh-free shape functions, the MLS shape functions lack the Kronecker delta property, i.e., ${\varphi }_I(X_J)\ne {\delta }_{IJ}$. Thus, the approximation that is generated by the MLS method does not pass through the nodal parameter values:

${\psi }^h({\mathbf{X}}_I)=\sum _{iJ}{\phi }_J({\mathbf{X}}_I){\psi }_J\ne {\psi }_I$ (4.148)

(4.148)

where I and J denote the I-th and J-th mesh-free nodes, respectively. Therefore, the essential boundary conditions cannot be treated directly in mesh-free methods as they can in the FEM.

Some techniques have been proposed to impose the essential boundary conditions in mesh-free computations. Belytschko et al. [69] proposed the Lagrange multiplier method, in which not only the discrete field variables, but also the Lagrange multipliers, must be solved. Therefore, a separate set of interpolations for the Lagrange multipliers is required, which results in an increasing problem size. The introduction of the Lagrange multipliers to the method also leads to the singularity of the stiffness matrix, and thus a special solver is needed. Krongauz [127] introduced the EFG method coupled with the FEM method, in which finite elements are placed along the essential boundary so that the combined shape functions exactly satisfy the Kronecker delta property along the boundary. Ren and Liew [128] proposed the nodal interpolation method (NIM) and the direct imposition method (DIM). In the NIM, the polynomial interpolations are constructed near the essential boundary, and the combination of these interpolants with the mesh-free shape function allows the essential boundary conditions to be satisfied along the boundary. In the DIM, after a mathematical treatment is introduced to the resulting system matrix equations, the essential boundary conditions can be imposed directly as in the FEM. Chen et al. [85,91] developed a full transformation method, in which a global transformation matrix is introduced to transform the unknown variable vector and the stiffness matrix into those in the real deformation space. This method can achieve good efficiency and is very suitable to vibration problems. However, it is quite expensive. For nonlinear problems in particular, the repeated operations of the matrix multiplication requires a large amount of computational time. Zhu and Atluri [114] employ the penalty function method to impose the essential boundary conditions. The process in this method is similar to that in the FEM except that the mesh-free approximation at the boundary points is computed in advance. Other treatments of the essential boundary conditions include the Gram-Schmidt orthogonalization method [129], the singular weight function [130], and the development of the modified variational principle [115].

The full transformation method is found to be very accurate to impose the essential boundary conditions. However, the frequent matrix transformation often requires a large amount of computational resources for the present nonlinear problem. Therefore, the penalty function method was mainly used in preparing this thesis.

Assuming that the i-th displacement is prescribed at a boundary node ${\Gamma }_K$, this given displacement can be interpolated with the MLS approximation as follows.

$\mathrm{\Delta }{\overline{u}}_{iK}=\sum _{I=1,NP}{\phi }_I({\mathbf{X}}_{{\mathrm{\Gamma }}_K})\mathrm{\Delta }{\hat{u}}_{iI},at{\mathrm{\Gamma }}_K$ (4.149)

(4.149)

where ${\mathrm{\Gamma }}_K$ denotes a boundary node at which the i-th displacement is given, $\mathrm{\Delta }{\overline{u}}_{iK}$ is the prescribed displacement increment, and $\mathrm{\Delta }{\hat{u}}_{iI}$ is the nodal parameter.

Eq. (4.149) is performed at all of the boundary nodes at which the essential boundary conditions are prescribed, and all of these equations can be assembled as

$\mathrm{\Delta }\overline{\mathbf{u}}=\mathbf{\Psi }\mathrm{\Delta }\stackrel{\mathbf{ˆ}}{\mathbf{U}}$ (4.150)

(4.150)

$\mathbf{\Psi }=\left[\begin{array}{llll}{\mathbf{\phi }}_1({\mathbf{X}}_{{\mathrm{\Gamma }}_1})\hfill & {\mathbf{\phi }}_2({\mathbf{X}}_{{\mathrm{\Gamma }}_1})\hfill & \cdots \hfill & {\mathbf{\phi }}_{NP}({\mathbf{X}}_{{\mathrm{\Gamma }}_1})\hfill \\ {\mathbf{\phi }}_1({\mathbf{X}}_{{\mathrm{\Gamma }}_2})\hfill & {\mathbf{\phi }}_2({\mathbf{X}}_{{\mathrm{\Gamma }}_2})\hfill & \cdots \hfill & {\mathbf{\phi }}_{NP}({\mathbf{X}}_{{\mathrm{\Gamma }}_2})\hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ {\mathbf{\phi }}_1({\mathbf{X}}_{{\mathrm{\Gamma }}_K})\hfill & {\mathbf{\phi }}_2({\mathbf{X}}_{{\mathrm{\Gamma }}_K})\hfill & \cdots \hfill & {\mathbf{\phi }}_{NP}({\mathbf{X}}_{{\mathrm{\Gamma }}_K})\hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \end{array}\right]$ (4.151)

(4.151)

where ${\mathbf{\phi }}_I({\mathbf{X}}_{{\mathrm{\Gamma }}_K})$ is determined by the index i of the prescribed displacement $\mathrm{\Delta }{\overline{u}}_{iK}$ at the boundary ${\mathrm{\Gamma }}_K$, and it takes $({\varphi }_I({\mathbf{X}}_{{\mathrm{\Gamma }}_K}),0,0)$ (for i=1), $(0,{\varphi }_I({\mathbf{X}}_{{\mathrm{\Gamma }}_K}),0)$ (for i=2) or $(0,0,{\varphi }_I({\mathbf{X}}_{{\mathrm{\Gamma }}_K}))$ (for i=3).

The weak equilibrium equation is

$\begin{array}{l}{\int }_{\mathrm{\Omega }}{(\mathrm{\delta }\mathrm{\Delta }\mathbf{g}\mathbf{r}\mathbf{a}{\mathbf{d}}_{n+1})}^T\cdot \mathbf{M}\cdot \mathrm{\Delta }\mathbf{g}\mathbf{r}\mathbf{a}{\mathbf{d}}_{n+1}dV={\int }_{\partial \mathrm{\Omega }}{(\delta \mathrm{\Delta }{\mathbf{u}}_{n+1})}^T\cdot {\mathbf{t}}_0^{P}dS+{\int }_{\partial \mathrm{\Omega }}{({\nabla }_N\delta \mathrm{\Delta }{\mathbf{u}}_{n+1})}^T\cdot {\mathbf{t}}_0^{Q}dS\hfill \\ -{\int }_{\mathrm{\Omega }}{(\mathrm{\delta }\mathrm{\Delta }\mathbf{g}\mathbf{r}\mathbf{a}{\mathbf{d}}_{n+1})}^T\cdot {\mathbf{\sigma }}_{n}dV+\sum _{{\mathrm{\Gamma }}_K}\alpha {(\delta \mathrm{\Delta }\mathbf{u})}^T(\mathrm{\Delta }\mathbf{u}-\mathrm{\Delta }\overline{\mathbf{u}})\hfill \end{array}$ (4.152)

(4.152)

where α is the penalty number.

Using Eq. (4.150), the discretized equation can be obtained as

$\{{\mathbf{K}}_{n+1}+\sum _{{\Gamma }_K}\left[{\mathbf{\Psi }}^T\alpha \mathbf{\Psi }\right]\}\mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}={\mathbf{f}}_{n+1}+\sum _{{\Gamma }_K}\left[{\mathbf{\Psi }}^T\alpha \mathrm{\Delta }\overline{\mathbf{u}}\right]$ (4.153)

(4.153)

The displacement in the first step can be precisely calculated, while that in the second step is accomplished by using the MK interpolation:

$\stackrel{\mathbf{ˆ}}{\mathbf{u}}=\stackrel{\mathbf{ˆ}}{\mathbf{u}}{({\hat{u}}_1,{\hat{u}}_2,{\hat{u}}_3)}^T=\sum _{i=1}^n{\mathit{\varphi }}_i{\mathbf{u}}_i=\mathbf{\Phi }\mathbf{U}$ (4.154)

(4.154)

where the matrix ${\mathbf{K}}_{n+1}$ and the vectors $\mathrm{\Delta }{\stackrel{\mathbf{ˆ}}{\mathbf{U}}}_{n+1}$ and f are of the same form as Eq. (4.141).

4.5.8 Stability of the Algorithm

The stable state of the structures is determined by iteratively solving the incremental equation (4.141) with the nonequilibrium force and the stiffness matrix at each iterative step. The present algorithm is actually a standard Newton–Raphson method [131], which is faster than gradient-based methods, such as the conjugate gradient method [132] in which only the first-order derivative of the potential energy is required. However, the stiffness matrix ${\mathbf{K}}_{n+1}$ may not be positive definite in some cases, such as when buckling and material softening occur; thus, the iterative solution does not converge to the minimum point of the potential energy. A simple way to remedy this problem is to replace ${\mathbf{K}}_{n+1}$ with ${\mathbf{K}}_{n+1}+\beta \mathbf{I}$, where I is the identity matrix, and $\beta$ is a positive number. The repeated replacement can ensure that the solution converges to the real one. The stiffness matrix ${\mathbf{K}}_{n+1}$ generally becomes positive definite after a few cycles of replacement, and the standard Newton–Raphson method can then be resumed. This modification has been used in the atomic-scale finite element [133,134].

The choice of $\beta$ should ensure that ${\mathbf{K}}_{n+1}+\beta \mathbf{I}$ is positive definite. The ideal value of $\beta$ is a positive number that is slightly larger than the magnitude of the most negative eigenvalue of ${\mathbf{K}}_{n+1}$ [131] because the larger the value of $\beta$, the slower the convergence of the solutions. Therefore, to achieve a good convergence rate, we can first calculate the eigenvalue of ${\mathbf{K}}_{n+1}$ at each iterative step. However, the calculation of this eigenvalue consumes additional computational time. Therefore, for small-sized structures, the value of $\beta$ can be chosen by calculating the eigenvalue of ${\mathbf{K}}_{n+1}$. For large-scale structures, it can be determined by frequent attempts.

4.5.9 Procedures for Equilibrium Solution

Based on the above illustration, the solution procedure for a loading step can be summarized as follows (the MLS shape functions and their first- and second-order derivatives are calculated and deposited for each Gaussian point prior to the first loading step).