# Improved Homotopy Perturbation Method for Geometrically Nonlinear Analysis of Space Trusses
## 3 Nonlinear Analysis Algorithms
### Improved Perturbation Method for the Nonlinear Problem  
The perturbation method solution could be described as [27]  
\[D(u(x))=f(r(x)) \tag{7}\]  
With boundary conditions shown in Equation (8). In Equation (7), \(u(x)\) denotes the approximation function of the solution.  
\[B(u(x),\partial u(x)/\partial n)=0 \tag{8}\]  
Figure 1: Incremental–iterative solution method [24].  
where \(D\) is the differential operator and B indicates the boundary condition, and \(f(r(x))\) is a general analytic function. If the Operator \(D\) is divided into two parts of linear LL and NL functions, the analytic function could be reestablished as shown in Equation (9).  
\[LL(u(x))+NL(u(x))=f(r(x)) \tag{9}\]  
Using the homotopy procedure over the constructed homotopy would result in  
\[H(u(x),p)=(1-p)[LL(U(x))-u_{0}(x)]+p[D(U(x))-f(r(x)]=0 \tag{10}\]  
In which \(p\) is a small value between 0 and 1. \(u_{0}(x)\) is the primary approximation response. Equation (10) could be described for specific \(p\)-values. Shown in Equations (11) and (12).  
\[H(u(x),0)=LL(U(x))-u_{0}(x)=0 \tag{11}\]  
\[H(u(x),1)=D(U(x))-f(r(x))=0 \tag{12}\]  
From the above-mentioned equations, it is concluded that varying p from zero to one is equivalent to varying \(H(U(x),\)\(p\) from \(LL(U(x))-u_{0}(x)\) to \(D(U(x))-f(r(x))\). By considering the perturbation technique, it could be assumed that the solution for Equations (11) and (12) are given as powers series of p.  
\[U(x)=U_{0}(x)+pU_{1}(x)+p^{2}U_{2}(x)+\ldots \tag{13}\]  
From Equation (13) it is concluded that \(u(x)=\underset{p\to 1}{\lim}U(x)=U_{0}(x)+U_{1}(x)+U_{2}(x)+\ldots\), which indicates the approximate solution. The equivalence convex homotopy is considered to illuminate NHPM solution procedure as it is indicated in Equation (14).  
\[H(U(x),p)=LL(U(x))-u_{0}(x)+pu_{0}(x)+p[NL(U(x))-f(r(x))]=0 \tag{14}\]  
Equation (14) could be simplified and rewritten as shown in Equation (15)  
\[LL(U(x))=u_{0}(x)+p[f(r(x))-u_{0}(x)-NL(U(x))] \tag{15}\]  
Considering the \(S\) as the second derivative differential operator with respect to \(x\) (or _d2/dx2_), it is concluded that S\({}^{-1}\) is the two-fold integration from 0 to \(x\). Therefore, Equation (15) could be reestablished as shown in Equation (16).  
\[U(x)=C(x)+S^{-1}(u_{0}(x))+p[S^{-1}(f(r(x))-S^{-1}(u_{0}(x))-S^{-1}(N(U(x))) \tag{16}\]  
It is noted that C is the constant incorporated in the integration; therefore, CS = 0. To apply the nonlinear perturbation method, the initial approximation is considered as shown in Equation (17).  
\[u_{0}(x)=\sum_{n=0}^{\infty}a_{n}F_{n}(x) \tag{17}\]where \(a_{0}\), \(a_{1}\), \(a_{2}\),... are the unknown factors and \(F_{0}\)(\(x\)), \(F_{1}\)(\(x\)), \(F_{2}\)(\(x\)),... are functions. In addition, if the factor \(p\) is considered as insignificant varying from the range 0 to 1, the solution for Equations (13) and (17) could be written as  
\[\begin{array}{c}\sum\limits_{n=0}^{\infty}p^{n}U_{n}(x)=C(x)+\\ S^{-1}(\sum\limits_{n=0}^{\infty}a_{n}F_{n}(x))+\\ p[S^{-1}(f(r(x)))-S^{-1}(\sum\limits_{n=0}^{\infty}a_{n}F_{n}(x))\\ -S^{-1}NL(\sum\limits_{n=0}^{\infty}p^{n}U_{n}(x))]\end{array} \tag{18}\]  
If the coefficients of terms from the left-hand side and right-hand side of the Equation (18) are compared; therefore, Equation (19) could be obtained.  
\[\begin{array}{c}p^{0}:U_{0}(x)=C(x)+S^{-1}(\sum\limits_{n=0}^{\infty}a_{n}F_ {n}(x))\\ p^{1}:U_{1}(x)=S^{-1}(f(r(x))+S^{-1}[\sum\limits_{n=0}^{\infty}a_{n}F_{n}(x)-S ^{-1}NL(\sum\limits_{n=0}^{\infty}p^{n}U_{n}(x))]\\ \vdots\\ p^{i}:U_{i}(x)=-S^{-1}NL(U_{0}(x),U_{1}(x),U_{2}(x),\ldots,U_{i-1}(x))\end{array} \tag{19}\]  
Therefore, the equations could be solved assuming that \(U_{1}(x)=0\). Then from Equation (19), \(U_{2}\)(\(x\)) = \(U_{3}\)(\(x\)) = \(\ldots\) = 0, Equation (20) could be obtained  
\[u(x)=U_{0}(x)=C(x)+S^{-1}(\sum\limits_{n=0}^{\infty}a_{n}F_{n}(x)) \tag{20}\]