新型態數值模擬之研發-可適性計算

Abstract

隨著資訊科技的快速發展,數值模擬模式之計算能力大幅提昇,因此已被廣泛應用在各工程領域中,且其重要性日趨顯著。數值模式之開發一般包含四大步驟,分別為「概念模式描述」、「數學模式定義」、「數值離散推導」與「電腦程式開發」四個階段,任一數值模式之開發均需經歷上述四個階段,因此使得更新或擴充一個既存數值模式之問題範疇,則需重頭至尾歷經上述四個步驟,使得修改工作變得極為複雜且耗時,限制了既存數值模式模擬範疇擴充與更新的彈性。有鑑於此,本研究提出全新的數值建模方法 − 「可適性計算架構」,突破傳統的數值建模方式的限制,使得應用「可適性計算架構」開發的數值模式,具有容易擴充與更新模擬功能之特點。

「可適性計算架構」並非只是一般的數值方法 (如有限元素法或有限差分法等) ,而是一種數值建模方法。與傳統建模方法作比較,可適性計算架構從「數學模式定義」開始著手,直接從分散之多條基礎的數學方程式(組)直接進行離散,而不需額外以數學推導與假設進行整合。此外,為了可以處理多條基礎方程式之計算,且檢驗基礎方程組之定義完備性,本研究提出「一致性分析」檢驗多條方程式之相依關係,並決定彼此間之求解順序。在「數值離散推導」方面,本研究採簡單差分法處理微分運算子,且相較於傳統計算方式,本研究毋須建立矩陣方程式,以各節點直接運算。此外,本研究以「Voronoi Diagram」作為空間切割法,網格形狀極具彈性,可適應不同的空間型態。在「電腦程式開發」階段,相較於傳統之矩陣解法,本研究提出「內、外迭代」流程負責求得符合邊界條件與初始條件之解,惟仍維持各格點計算上的獨立性。本計算架構相較於與傳統方式,雖然整體開發方式不同,惟若在「數學模式定義」階段之初,選取相同之數學方程組,則本計算架構與傳統方法所解的為相同之問題,且本計算架構毋須額外之數學推導與假設,除了可以節省開發模式之心力外,在概念上更貼近原始定義之問題。

在案例驗證上,本研究以可適性計算架構建立「地下水流」、「熱流傳輸」與「溶質傳輸」三個子問題之模擬,並建立五個模擬案例進行驗證,證實本計算架構之正確性與彈性。藉由案例實作上證實,應用「可適性計算架構」建立之模式,如欲擴張模式模擬能力,新增其他運動機制,僅需撰寫替換或增加之方程式,證實「可適性計算架構」的擴充能力。因此,應用「可適性計算架構」開發數值模式可以大幅減輕模式開發的負擔,使得工程師或研究人員可以更加專注於問題本質上,而非工具或模式開發上。

This study proposed a innovative methodology for developing numerical simulation models that overwhelm conventional developing process and greatly increase the efficiency of model development. The advancement of information technology (IT) have significantly improved the computational capa- bility of numerical model, thus increased the importance of numerical simulation in various engineering analysis. The conventional process of numerical model development consists four steps that includes “conceptual model description”, “mathematical model definition”, “numerical model derivation” and “computer program development”. Once a numerical model has developed, one still has to repeat the four steps to modify the code even if only part of the original problem was modified with the conven- tional model developing process. The modification process is always complicated and time consuming. Hence, the traditional development process is lack of flexibility and difficult to update the computing functionalities of an existed numerical model. Therefore, to resolve these model developing issues, the Adaptive Computation Framework (ACF), a novel methodology to develop numerical simulation method, is proposed in this study. By using the proposed ACF method, a new computing function is easy to add into a existing model, i.e. a numerical model can grow with new computing functions.

The ACF is much more than just a new numerical scheme such as the finite element (FEM) or finite difference method (FDM). At the “mathematical model definition” step, the ACF define a problem by the set of originally fundamental equations without further artificial combination and simplification to get a more compact set of PDEs. An ”equation consistence analysis” is proposed in this step to ensure the consistence of these fundamental equations and variables, and also determine the sequence to solve the equations. In the “numerical model derivation” step, instead of applying complicated numerical scheme such as FEM or FDM, only simple difference method is needed to discretize the equations and the “Voronoi Diagram” is proposed as the griding method for spatial discretization. In the “computation program development” step, instead of solving a matrix equation, a general iteration method consists of inner and outer iteration is proposed to compute the solutions at each grids.

To demo the effectivity of the proposed methodology, three different groundwater numerical mod- els, “groundwater flow only”, “groundwater flow with heat transport” and “groundwater flow with head and solute transport”, are developed by using ACF. Five different cases are examined to ver- ify the correctness and the flexibility of ACF. The cases studies demonstrated that, using the ACF method, a model computing functions can be extended by only adding the required equations and thus increase the model computing capability with minimum coding effort. By using the ACF, engineers or scientists can get relief from the time consuming model redeveloping process, thus can focus more on the problem analysis instead of tool (model) development.