## 2 An Improved RPIM for Stress Concentration Evaluation of Welded Component
### Formulation of RPIM Shape Functions  
In terms of RPIM, the approximation function \(u(\mathbf{x})\) at a point of interest \(x\) can be written as  
\[u(\mathbf{x})=\sum_{i=1}^{n}R_{i}(\mathbf{x})a_{i}+\sum_{j=1}^{m}p_{j}(\mathbf{x})b_{j}=R^ {T}(\mathbf{x})\mathbf{a+p}^{T}(\mathbf{x})\mathbf{b} \tag{1}\]  
where \(R_{i}(\mathbf{x})\) is a radial basis function, \(p_{j}(\mathbf{x})\) is monomial in the space coordinates \(\mathbf{x}_{T}=[x,y]\), \(n\) represents the number of radial basis function, \(m\) represents the number of polynomial basis function. \(a_{i}\) and \(b_{j}\) are unknown constants. There exists variable \(r\) in radial basis function \(R_{i}(\mathbf{x})\):  
\[r=\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}} \tag{2}\]  
The geometric significance of \(r\) is the distance between the point at \(x\) and node at \(x_{i}\). With the variable \(r\), the multi-quadrics (MQ) function can be used as the radial basis function (RBF):  
\[R_{i}(x,y)=\left(r_{i}^{2}+\left(a_{c}d_{c}\right)^{2}\right)^{q} \tag{3}\]  
where \(a_{c}\) and \(q\) are shape parameters, \(d_{c}\) is the characteristic length relating to the nodal spacing. Applying Equation (1) to all the \(n\) nodes in the support domain of point \(\mathbf{x}\), we can obtain \(n\) linear equations which can be express in the matrix form:  
\[\mathbf{U_{s}}\mathbf{=}\mathbf{R_{0}}\mathbf{a+}\mathbf{P_{m}}\mathbf{b} \tag{4}\]  
where  
\(\mathbf{U_{s}}\) is the vector of nodal function values:  
\[\mathbf{U_{s}}=\left\{ \begin{array}{cccc}u_{1}&u_{2}&\cdots&u_{n}\end{array} \right\} \tag{5}\]  
\(\mathbf{a}\) is the vector of unknown coefficients:  
\[\mathbf{a}=\left[ \begin{array}{cccc}a_{1}&a_{2}&\cdots&a_{n}\end{array} \right]^{T} \tag{6}\]  
\(\mathbf{b}\) is the vector of unknown coefficients:  
\[\mathbf{b}=\left[ \begin{array}{cccc}b_{1}&b_{2}&\cdots&b_{m}\end{array} \right]^{T} \tag{7}\]  
\(\mathbf{R_{0}}\) is the moment matrix of RBFs:  
\[\mathbf{R_{0}}=\left[ \begin{array}{cccc}R_{1}(r_{1})&R_{2}(r_{1})&\cdots&R_{n}( r_{1})\\ R_{1}(r_{2})&R_{2}(r_{2})&\cdots&R_{n}(r_{2})\\ \vdots&\vdots&\cdots&\vdots\\ R_{1}(r_{n})&R_{2}(r_{n})&\cdots&R_{n}(r_{n})\end{array} \right]_{(n\cap n)} \tag{8}\]\(\mathbf{P_{m}}\) is the polynomial moment matrix:  
\[\mathbf{P_{m}}=\left[ \begin{array}{ccccc}1&x_{1}&y_{1}&\cdots&p_{m}(x_{1})\\ 1&x_{2}&y_{2}&\cdots&p_{m}(x_{2})\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&x_{n}&y_{n}&\cdots&p_{m}(x_{n})\\ \end{array} \right]_{(n\times m)} \tag{9}\]  
Additional \(m\) equations are employed as constraint conditions:  
\[\sum_{i=1}^{n}p_{j}(x_{i})a_{i}=\mathbf{P}_{m}^{T}\mathbf{a}=0,j=1,2\cdots,m \tag{10}\]  
Combing Equations (4) and (10) we get:  
\[\overline{\mathbf{U}}_{s}=\left[ \begin{array}{c}\mathbf{U}_{s}\\ 0\\ \end{array} \right]=\left[ \begin{array}{cc}R_{0}&\mathbf{P}_{m}\\ \mathbf{P}_{m}^{T}&0\\ \end{array} \right]\left\{ \begin{array}{c}\mathbf{a}\\ \mathbf{b}\\ \end{array} \right\}=Ga_{0} \tag{11}\]  
where  
\[\mathbf{G}=\left[ \begin{array}{cc}\mathbf{R}_{0}&\mathbf{P}_{m}\\ \mathbf{P}_{m}^{T}&0\\ \end{array} \right] \tag{12}\]  
\[\mathbf{a}_{0}=\left\{ \begin{array}{c}\mathbf{a}\\ \mathbf{b}\\ \end{array} \right\} \tag{13}\]  
A unique solution can be obtained by solving Equation (11):  
\[\mathbf{a}_{0}=\left\{ \begin{array}{c}\mathbf{a}\\ \mathbf{b}\\ \end{array} \right\}=\mathbf{G}^{-1}\overline{\mathbf{U}}_{s} \tag{14}\]  
Substituting Equation (14) back into Equation (1), we obtain:  
\[u(\mathbf{x})=\left\{ \begin{array}{cc}\mathbf{R}^{T}(\mathbf{x})&\mathbf{p}^{T}(\mathbf{x})\\ \end{array} \right\}\left\{ \begin{array}{c}\mathbf{a}\\ \mathbf{b}\\ \end{array} \right\}=\left\{ \begin{array}{cc}\mathbf{R}^{T}(\mathbf{x})&\mathbf{p}^{T}( \mathbf{x})\\ \end{array} \right\}\mathbf{G}^{-1}\overline{\mathbf{U}}_{s}=\overline{\mathbf{\Phi}}^{T}( \mathbf{x})\overline{\mathbf{U}}_{s} \tag{15}\]  
Then, the approximation function \(u(\mathbf{x})\) can be expressed as:  
\[u(\mathbf{x})=\overline{\mathbf{\Phi}}^{T}(\mathbf{x})\overline{\mathbf{U}}_{s}=\left\{