本文提出了一種基於牽引力的深度能量法(traction-based deep energy method, tDEM)，考慮了牽引力的邊界效應。此方法是由混合深度能量法(mixed Deep Energy Method, mDEM)擴展出來的方法，結合了物理資訊神經網路(Physics-Informed Neural Network, PINN)和深度能量法(Deep Energy Method, DEM)。由於mDEM在訓練的過程中多施加了本構行為(constitutive behavier)，造成很高的計算成本。因此，本文提出tDEM只考慮牽引力邊界條件。而tDEM又可以細分成兩種類型，分別為t1DEM和t2DEM。其中，t1DEM考慮了整個牽引力邊界條件，而t2DEM只考慮了自由牽引力邊界條件。此外，對於幾何形狀較複雜的問題，本文結合基於參數化深度能量法(Parametric Deep Energy Method, PDEM)的映射方法，提出了參數化牽引力深度能量法(Parametric traction-based deep energy method, PtDEM)，該方法透過NURBS基函數將訓練座標點由物理域(physical domain)映射到參數域(parametric domain)。損失函數是在參數域中使用高斯求積法計算而得。在參數域中訓練完成後，再將結果映射回物理域。本文以三個彈性體問題為例，展示了tDEM和PtDEM的性能，並與解析解進行比較。證明本文提出方法t2DEM和Pt2DEM提供了更準確的結果。

In this study, we propose a traction-based deep energy method (tDEM) that accounts for the boundary effects of traction. This method is an extension of the mixed Deep Energy Method (mDEM), which combines the Physics-Informed Neural Network (PINN) and the Deep Energy Method (DEM). Since mDEM imposes constitutive behavier in the training process, it causes high computational cost. Therefore, in this study, we propose tDEM to consider only the traction boundary conditions. The tDEM can be subdivided into two types, t1DEM and t2DEM, where t1DEM considers the whole traction boundary condition and t2DEM considers only the free traction boundary condition. In addition, for geometrically complex problems, this study combines the Parametric Deep Energy Method (PDEM)-based mapping method and proposes the Parametric traction-based deep energy method (PtDEM), which maps the training coordinates from the physical domain to the parametric domain by means of the NURBS basis functions. The loss function is calculated in the parametric domain using Gaussian quadrature. After the training in the parametric domain, the result is mapped back to the physical domain. In this study, the performance of tDEM and PtDEM is demonstrated and compared with the analytical solution for three elastic problems. It is shown that the proposed methods t2DEM and Pt2DEM provide more accurate results.

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