在本研究中針對半無限域問題提出了一種創新的分析方法，為利用無網格法與動態無限點法的耦合方式來解決問題。大自然中土壤為一無窮遠的範圍，欲直接分析將十分的耗時，因此在分析時會以一個人工的邊界劃分出有限的範圍，在邊界內部稱之為近域，邊界外部稱為遠域。研究中將使用無網格法中的再生核質點法(Reproducing Kernel Particle Method, RKPM)針對近域進行分析，而動態無限點法(Dynamic Infinite Meshfree Method, DIMM)則針對遠域進行分析。再生核質點法為直接在定義域建構再生核形狀函數，並採用穩定一致節點積分法(Stabilized Conforming Nodal Integration, SCNI)進行數值積分，若結果產生誤差，則使用自然穩定節點積分法(Naturally Stabilized Nodal Integration, NSNI)進行修正。動態無限點法為利用邊界上所建構的一維再生核形狀函數與波傳函數相互結合，以模擬波源消散至無窮遠的狀況，並採用Newton-Cotes積分法進行無窮遠處的積分，此外在邊界處也會使用邊界奇異核法(Boundary Singular Kernel Method, BSKM)來滿足無網格法中所缺乏的克羅內克函數性質。
在動態和靜態的半無限域問題上，使用RK-DIMM耦合方法分析時調整近域離散點間距以及針對問題調整無限點長度，都可以達到穩定且精準的結果。為了使分析更為自動化，也針對問題採非均勻離散化下的分析，並透過自然穩定節點積分法進行修正，以達到精準的分析結果。本研究成功只以無網格離散點分析的方式運用在半無限域的問題中，避免傳統使用有限元素法及無限元素法的方式模擬導致計算耗時且網格易失真的缺點，為後續相關領域提供了一種新的數值解決方法。

In this study, an innovative analytical approach is proposed for addressing the semi-infinite domain problems by a new coupled method of RK-DIMM. In nature, soil extends indefinitely, making direct analysis extremely time-consuming. Therefore, in the analysis, a finite domain is artificially delineated by a boundary, termed as the near-field within the boundary and the far-field outside it. The reproducing kernel particle method (RKPM) from the meshfree method will be employed to analyze the near-field, while the method of dynamic infinite meshfree method (DIMM) will be used for analyzing the far-field. RKPM involves constructing reproducing kernel shape functions directly in the domain of interest and employing stabilized conforming nodal integration (SCNI) for numerical integration. In case of errors, naturally stabilized nodal integration (NSNI) will be utilized for correction. DIMM involves the integration of one-dimensional reproducing kernel shape function and wave propagation function constructed on the boundary to simulate the phenomenon of waves propagating to infinity. Newton-Cotes integration method is used for integration at infinity. Additionally, at the boundary, the boundary singular kernel method (BSK) is employed to fulfill the lacking properties of the Kronecker delta function in the meshfree method.
By utilizing RK-DIMM, adjusting the spacing between discrete points, the wave number and decay factor for the problem both lead to stable and accurate results. In order to make the analysis more automated, the problem is also analyzed under non-uniform discretization and corrected through the NSNI to achieve accurate analysis results. This study successfully employs a meshfree discretization approach solely for analyzing problems in semi- infinite domain, thus avoiding the drawbacks of traditional finite and infinite element methods, which often lead to time-consuming computations and grid distortions. This provides a new numerical solution method for subsequent related fields.

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