本論文提出一創新的拓樸設計方法—固體體積函數法(Volume-of-Solid Method, VOS method)，並將固體體積函數法應用於熱傳導路徑之最佳化設計。本研究利用本法設計一最佳路徑可在使用最少材料的情況下，將最大的熱傳量由高溫物件傳到低溫物件。本方法第一步先同時求解固體體積函數及熱傳導方程式，進而得到設計領域的溫度分佈及固體體積函數分佈。本方法第二步以取不同固體體積函數之截斷值(Cut-off value)，來得到熱傳路徑的形狀，再以選擇具備最大單位固體材料質量之熱傳率者為最佳之熱傳導路徑形狀。
透過不同的案例分析來探討固體體積函數法之有效性及可行性，並且結合快速共軛梯度法(Simplified Conjugate-Gradient Method, SCGM)及格點影像法以縮短其求解時間。本法首先應用於求解不同配置的熱傳邊界條件之二維熱傳導問題；其結果顯示本法可預測得到一熱傳導路徑之最佳化形狀。由於固體體積函數法在計算不同熱傳路徑的單位固體質量之熱傳率時，需不斷地執行費時的格點產生程序，本研究結合固體體積函數法、快速共軛梯度法及格點影像取代格點產生程序，可有效地縮短其目標函數之收斂所需時間。
此外，本研究將固體體積函數法用來最佳化多層金屬複合材料之熱傳導路徑。本研究以銅、鋁、鐵及不鏽鋼等組合的三層金屬複合材料為例，討論其不同組成及不同排列方向之熱傳行為。研究發現本法可彈性應用於不同的目標函數最小化，且可有效地進行提高單位固體質量之熱傳率(Q/m)、單位固體體積之熱傳率(Q/V)、或單位成本之熱傳率(Q/USD)等不同設計目標之設計。

The aim of this thesis is to develop a novel computational approach, which is based on a non-constrained formulation with a volume-of-solid (VOS) function equation, for topology design of heat conductive solid paths between constant-temperature objects. In the first step of the approach, the distributions of the VOS function and the temperature in the original design domain are carried out by simultaneously solving the VOS function equation and the heat conduction equation. Secondly, the shape outline of the heat conduction path leading to a maximum heat transfer rate per unit solid mass is determined by selecting a cut-off value of the VOS function.
Validity and capability of the VOS method are investigated and tested by several different cases. However, it is found that with this method one needs to carry out the grid generation repeatedly during the evolution of the shape of the heat conduction paths. Thus, the design process is rather time-consuming. In this regard, the VOS method is improved by introducing the simplified conjugate-gradient method (SCGM) and grid image visualization into the topology design process. This improved approach, VOS+SCGM, is applied to the two-dimensional test cases with various thermal boundary configurations. Results show that the optimal shapes of the heat conduction paths can be more efficiently predicted by using the VOS+SCGM method. In the VOS+SCGM method, the grid generation is no longer required, and instead the grid image visualization is used to portray the shape of the objects.
Furthermore, the VOS method is applied to optimize the conduction path in laminated metallic materials between unequal isothermal surfaces. Three-layer laminated metallic composite materials which are made of copper, aluminum, stainless steel or iron are considered in this study. Two possible orientations of the composite materials, vertical and horizontal, are investigated. Optimal shapes of the thermal conduction path between different temperature objects are determined toward minimization of the objective function. This approach is able to reach optimal shape corresponding to different objective functions. It implies that the selection of the objective function becomes more flexible. By using this approach, optimal thermal conduction paths leading to maximum heat transfer rate per unit mass (Q/m) , maximum heat transfer rate per unit volume (Q/V) , or maximum heat transfer rate per unit cost (Q/USD) can be readily yielded.

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