## 2 Introduction to a Strong Fluid-Structure Coupling Framework--Optimal Transport Meshless Method
### Form Function  
The primary distinction among different meshless methods lies in the construction of the interpolation function for the displacement field within the continuous medium domain. In the OTM method, this interpolation function takes the form of a local maximum entropy (LME) shape function. LME shape functions possess the desirable property of a weak Kronecker delta at boundaries, thereby overcoming limitations associated with interpolation functions in SPH methods. This article aims to introduce and highlight the theoretical advantages of using LME shape functions within the OTM method for simulating fluid-structure coupling problems. Given a discrete set of nodes \(x_{i}\) within a continuous medium, the displacement field of that medium can be expressed as a combination of node sets:  
\[u(x)=\sum_{i=1}^{n}p_{i}(x_{i})u(x_{i}) \tag{1}\]  
where \(p_{i}(x_{i})\) is the form coefficient and \(u(x_{i})\) is the displacement at node \(x_{i}\). According to [32], the value of \(p_{i}(x_{i})\) should satisfy the following equation:  
Figure 2: OTM discretizes the continuous medium into two distinct nodes [31]. \[ \begin{array}{l}min\ \mathbf{f}_{g}(x,p)=\\ \beta\sum\limits_{i=0}^{n}p_{i}|x-x_{i}|^{2}+\sum\limits_{i=0}^{n}p_{i}log(p_{i} )\end{array}  \tag{2}\]  
where \(\sum p_{i}\log(p_{i})\) describes the information entropy of the node-set, and \(\sum p_{i}|x-x_{i}|^{2}\) represents the effect of other nodes in the near neighborhood of the node at \(x\). The constraint ensures the continuity of the shape function up to zero and first order, thereby satisfying the continuity requirements of the displacement field. The solution of \(p_{i}\) can be obtained by  
\[ \begin{array}{l}p_{i}=\frac{1}{Z(x,\lambda^{*}(x))}*\\ exp\left[-\beta|x-x_{i}|^{2}+\lambda^{*}(x)*(x-x_{i})\right]\end{array}  \tag{3}\]  
where parameter \(\lambda^{*}(x)\) and \(Z(x,\lambda^{*}(x))\) satisfy the following equation:  
\[\lambda^{*}(x)=arg\ \text{min}\ log(Z(x,\lambda^{*}(x))) \tag{4}\]  
Equation (4) can be easily solved by Newton-Rapson iteration. \(\beta\in R^{+}\) is a Pareto optimization parameter that measures the maximum entropy and the effects of other nodes in the near neighborhood. In practice, to account for the effect of node distribution, [PERSON] [33] et al. adopted a size parameter \"\(\gamma\)\" to control the shape function of the LME, and \(\gamma\) satisfies the following equation  
\[\gamma=\beta h^{2} \tag{5}\]  
where \(h\) is the characteristic scale of the node set and can generally be set to compute the shortest distance of the neighborhood. When \"\(\gamma\)\" varies from 0.8 to 6.8 (see Figure 3), the search range of the neighborhood of the LME shape functions decreases gradually from its large initial value. The derivation above reveals that the LME must encompass all nodes of the boundary, adhere to the convex approximation, and satisfy the weak Kronecker delta attribute. Figure 3: The neighborhood of LME follows changes in “\(\gamma\)”. Framework for the Solution of Fatigue Life of Pipelines under Combined Excitation Based on the OTM Method  
The pipeline system in aviation hydraulic systems plays a crucial role in connecting the pump source and actuator for the efficient transmission of hydraulic fluid. In aircraft, the aviation hydraulic pipeline system is typically routed from the engine accessory casing transmission device to the landing-gear wheel compartment and rear accessory compartment. Due to spatial limitations, numerous pipelines intersect each other, resulting in minimal gaps between adjacent accessories (as depicted in Figure 4). This configuration represents a part of the hydraulic system of Garrett GTCP85-98D gas turbine engines. Aero-engine pipelines operate under prolonged high-frequency vibration conditions, leading to significant fatigue in vulnerable sections of the lines and subsequently resulting in failure modes such as fractures and oil leaks, particularly at welded and bent areas.