顧承宇特聘教授之研究方向與成果綜整如下：

數值分析在大地工程之應用

無網格法 (Meshless Methods)

- 研究重點：無網格方法常被用於求解各類偏微分方程 (PDEs)，針對數值解法的發展，特別是在處理空間與時間維度問題方面，如熱傳導、地表下流動、和擴散問題等。論文探討了多種無網格方法（Meshless Methods），如Trefftz方法、徑向基函數（RBF）法與基本解法（Method of Fundamental Solutions, MFS），並將其應用於熱傳導、地下水流動及自由表面問題。例如：無網格方法應用於熱傳導問題（論文45、53、55）和逆向熱傳導問題（論文42、43）。MFS方法用於解決三維非線性自由表面流（論文40）和層狀土壤中的自由表面問題（論文51）。
- 應用領域：土壤流動、地下水流、擴散過程中的逆問題、以及異質多層材料中的流動模擬。例如：Collocation Trefftz方法應用於地下水潮汐效應分析（論文21）、使用徑向基函數的無網格方法解逆熱傳導問題（論文24）。

非線性與移動邊界問題

- 非線性問題和移動邊界問題在顧教授多篇研究中也被探討，尤其是對於異質性材料中的流動進行建模（論文41、44）。這些研究採用無網格方法進行模擬，強調方法的高效性及對異質性複雜問題的適應性。

Trefftz方法與混合型無網格法

- Trefftz方法（Trefftz Method）是一種重要的數值方法，許多研究將其與其他方法結合使用，例如MFS（論文44）和多尺度Trefftz方法（論文55、58）。這些混合方法有效應用於熱傳導、地下水流動等複雜物理問題。

徑向基函數法（RBF）

- 無網格方法（Meshless Methods）與徑向基函數技術的應用，如解決橢圓邊界值問題、傳輸和擴散過程的逆問題、以及多層土壤中的地下水流動問題。例如，Ku 和 Liu（2023）的研究使用無網格徑向基函數方法來解決地下水問題，並且Liu 等人（2024）提出使用徑向基函數來解決未飽和流問題。

同倫法與數值求解技術

- 研究方向：動態牛頓法和同倫方法的改進與應用，解決數值分析中的病態系統及非線性問題，特別是具邊界值問題的應用。研究包括同倫方法的最佳化和步長自適應的數值解法。Ku et al., 2012-2011 — 提出動態牛頓法，並應用於解決病態系統。Ku et al., 2010 — 使用牛頓-同倫延續法解決非線性代數方程。無網格法應用於地下水流問題

反算問題 (Inverse Problems)

- 研究重點：研究如何反算技術來解決擴散和熱傳導等逆問題。主要應用於傳熱、擴散過程、以及地層壓縮等現象的模型構建和數據重建。許多研究專注於逆問題的解決方法，特別是在傳輸與擴散過程中，使用如徑向基函數、空間時間多項式以及多源無網格方法等技術進行模擬與求解，為地下水流、熱傳等問題提供了創新的數值方法。例如，Xiao 等人（2022）的研究使用徑向基函數與多項式來解決穩態對流擴散方程的逆問題。討論了逆向問題（Inverse Problems）和邊界值問題（Boundary Value Problems）。逆向問題常見於需要從有限的觀測數據中重建未知的物理量，例如：逆向熱傳導問題（論文42、43）利用時空無網格方法進行解決。逆向Cauchy問題（論文56）使用基本解法和指數收斂的標量同倫演算法進行求解。

離岸風電基礎數值分析

- 研究重點：對離岸風電基礎的承載特性和土壤有效應力進行數值模擬。
- 應用領域：臺灣離岸風電的結構安全與穩定性評估。

人工智慧在大地工程之應用

機器學習在災害防救之應用

- 研究重點：應用隨機森林技術和神經網絡來進行土壤分類、地層下陷、火災預測、和淹水潛勢評估。
- 應用領域：土壤分類、地層下陷、火災預測、淹水和邊坡穩定的風險評估。

機器學習整合GIS應用於災害分析

- 研究重點：應用數值方法和機器學習技術來模擬因降雨、邊坡、火災等自然災害的觸發條件。研究討論如何將深度神經網絡（DNN）、隨機森林（Random Forest）等機器學習技術應用於解決各種複雜的工程與環境問題，如土壤分類、火災預測以及地層下陷等。這些研究展示了機器學習技術在處理大量資料與非線性問題時的效能。例如，Liu 等人（2024）的研究應用隨機森林技術進行土壤分類，並且Ku 等人（2024）利用深度學習預測火災的發生。
- 應用領域：特別是在台灣地區進行的邊坡和火災風險預測研究，以及相關的空間分析和地理資訊系統（GIS）技術的應用。研究探索了地層下陷與水文過程之間的關係，尤其是使用數學模型和人工神經網絡來模擬和預測地層下陷地層下陷以及地下水提取所引發的問題。此類研究應用於台灣的雲林縣與濁水溪三角洲等區域。例如，Ku 等人（2022）研究濁水溪三角洲的地層下陷與地下水抽取的空間變異性，並應用GIS技術進行模擬。

岩石力學與區域地質災害

- 研究方向：地質災害的原因分析與模擬，特別關注於中臺灣地區的地質風險（如落石風險與土石流）。這些研究重點在於對地區性地質災害進行深入的原因探討，並開發模擬技術，以提高預測和風險管理能力。Ku et al., 2014 — 探討中台灣的地質災害成因。Ku, 2014 — 3D數值模型用於落石風險評估。Hsu et al., 2010 — 應用模擬技術對土石流危險區域進行劃定。
- 研究方向：岩石裂縫的力學行為與裂縫擴展的數值模擬，特別針對異向性岩石材料。這些研究有助於提升對岩石裂縫擴展的理解，並對工程結構設計提供依據。相關文獻：Ke et al., 2009 — 使用邊界元方法模擬異向性岩石中的裂縫擴展路徑。開發等效連續體和離散模型方法，應用於裂隙岩體的力學模擬。該類研究為解決多尺度岩體力學行為提供了新的模型框架。Lin & Ku, 2017 — 針對裂隙岩體的多尺度建模。

研究論文亦可詳下連結

Researchgate

https://www.researchgate.net/profile/Cheng-Yu-Ku

The research focus and achievements of Distinguished Professor Cheng-Yu Ku are summarized as follows:

Application of Numerical Analysis in Geotechnical Engineering

Meshless Methods

Research Focus: Meshless methods are commonly used to solve various partial differential equations (PDEs), particularly for developing numerical solutions for space and time-related problems, such as heat conduction, subsurface flow, and diffusion problems. The papers explore various meshless methods, such as the Trefftz Method, Radial Basis Function (RBF) method, and Method of Fundamental Solutions (MFS), applied to heat conduction, groundwater flow, and free surface problems. For instance, meshless methods were applied to heat conduction problems (Papers 45, 53, 55) and inverse heat conduction problems (Papers 42, 43). The MFS method was used to solve 3D nonlinear free surface flows (Paper 40) and free surface problems in layered soils (Paper 51). Applications: Soil flow, groundwater flow, inverse problems in diffusion processes, and flow simulations in heterogeneous multilayer materials. For example, the Collocation Trefftz Method was applied to groundwater tidal effect analysis (Paper 21), and the meshless RBF method was used to solve inverse heat conduction problems (Paper 24).

Nonlinear and Moving Boundary Problems

Nonlinear and moving boundary problems are also explored in several of Professor Ku’s studies, particularly in modeling flows in heterogeneous materials (Papers 41, 44). These studies use meshless methods, highlighting their efficiency and adaptability for solving complex problems in heterogeneous systems.

Trefftz Method and Hybrid Meshless Methods

The Trefftz Method is a key numerical method explored in many studies, often combined with other techniques such as MFS (Paper 44) and multi-scale Trefftz methods (Papers 55, 58). These hybrid methods are effectively applied to complex physical problems, including heat conduction and groundwater flow.

Radial Basis Function (RBF) Method

Meshless methods and RBF techniques are applied to solve problems such as elliptic boundary value problems, inverse problems in transport and diffusion processes, and groundwater flow in multilayered soils. For example, Ku and Liu (2023) applied the meshless RBF method to solve groundwater problems, and Liu et al. (2024) proposed using RBF to solve unsaturated flow problems.

Homotopy Method and Numerical Solving Techniques

Research Focus: Improvements and applications of the dynamic Newton method and homotopy methods to solve ill-posed systems and nonlinear problems in numerical analysis, especially for boundary value problems. The studies include the optimization of homotopy methods and adaptive step-size numerical solvers. Ku et al. (2012-2011) proposed a dynamic Newton method applied to solve ill-posed systems. Ku et al. (2010) used the Newton-homotopy continuation method to solve nonlinear algebraic equations. Applications: Meshless methods applied to groundwater flow problems.

Inverse Problems

Research Focus: Investigating inverse techniques to solve problems in diffusion and heat conduction. These methods are primarily applied to model and reconstruct data for phenomena such as heat transfer, diffusion processes, and geological compression. Many studies focus on solving inverse problems, especially in transport and diffusion processes, using techniques such as RBF, space-time polynomials, and multi-source meshless methods. These provide innovative numerical methods for groundwater flow and heat transfer problems. For example, Xiao et al. (2022) used RBF and polynomials to solve the inverse problem for steady-state convection-diffusion equations. Inverse problems (Inverse Problems) and boundary value problems (Boundary Value Problems) are frequently discussed. Inverse problems often involve reconstructing unknown physical quantities from limited observational data, such as inverse heat conduction problems (Papers 42, 43) solved using space-time meshless methods. The inverse Cauchy problem (Paper 56) was solved using MFS and a scalar homotopy algorithm with exponential convergence.

Offshore Wind Turbine Foundation Numerical Analysis

Research Focus: Numerical simulations of the bearing characteristics and effective soil stress of offshore wind turbine foundations. Applications: Structural safety and stability assessment of offshore wind turbines in Taiwan.

Artificial Intelligence in Geotechnical Engineering

Application of Machine Learning in Disaster Prevention and Mitigation

Research Focus: Applying techniques such as random forests and neural networks for soil classification, ground subsidence, fire prediction, and flood susceptibility assessment. Applications: Soil classification, ground subsidence, fire prediction, and risk assessment of flooding and slope stability.

Machine Learning Integrated with GIS for Disaster Analysis

Research Focus: Applying numerical methods and machine learning techniques to simulate triggering conditions for natural disasters such as rainfall, slope instability, and fires. Studies discuss how deep neural networks (DNN), random forests, and other machine learning techniques are used to address complex engineering and environmental problems like soil classification, fire prediction, and ground subsidence. These studies demonstrate the efficiency of machine learning in handling large datasets and nonlinear problems. For example, Liu et al. (2024) applied random forest techniques for soil classification, while Ku et al. (2024) used deep learning to predict fire occurrence.
Applications: Fire and slope risk prediction studies in Taiwan, along with spatial analysis and GIS technology applications. Research explores the relationship between ground subsidence and hydrological processes, particularly using mathematical models and artificial neural networks to simulate and predict ground subsidence issues caused by groundwater extraction. These studies are applied to areas such as Yunlin County and the Zhuoshui River Delta in Taiwan. For instance, Ku et al. (2022) studied the spatial variability of ground subsidence and groundwater extraction in the Zhuoshui River Delta, using GIS for simulation.

Rock Mechanics and Regional Geological Disasters

Research Focus: Investigating the causes and simulation of geological disasters, with a particular focus on geological risks (e.g., rockfall and debris flow) in Central Taiwan. These studies aim to develop simulation techniques to enhance prediction and risk management. Ku et al. (2014) explored the causes of geological disasters in Central Taiwan, while Ku (2014) used 3D numerical modeling for rockfall risk assessment. Hsu et al. (2010) applied simulation techniques to delineate debris flow hazard zones. Research Focus: The mechanical behavior of rock fractures and numerical simulation of crack propagation, particularly in anisotropic rock materials. These studies improve understanding of crack propagation in rocks and provide a basis for engineering design. Relevant literature: Ke et al. (2009) used the boundary element method to simulate crack propagation paths in anisotropic rocks. The development of equivalent continuum and discrete models was applied to the mechanical simulation of fractured rock masses. This research provides a new framework for addressing multi-scale mechanical behavior in rock masses. Lin & Ku (2017) focused on multi-scale modeling of fractured rock masses.