沉浸有限元素法於彈性界面與流構耦合問題的計算與誤差估計

Abstract

在國科會計畫 96-2115-M-009-014 和 97-2119-M-009-006, 我們使用 streamline upwinding Petrov-Galerkin (SUPG) finite element 的方法來模擬二度空間中黏性不可壓縮流與非線性尤拉彈性桿的相互作用其中流體與結構的互動則經由區域移動網格來達成。我們也研究拉普拉斯方程的奇異解並以奇異元素與自適性網格加密的多重網格法得到精確的數值解我們希望進一步研究彈性方程與bi-harmonic方程的奇異解 (這些問題也對應到平板形變與彎曲的裂縫問題)。 我們也希望能運用新的沈浸有限元素法來解決彈性方程與bi-harmonic方程中的界面問題並籍此進一步克服我們在流構互動模擬中所面臨的網格糾纏的問題。 在本計劃中， 我們將先推導沈浸有限元素法於拉普拉斯方程, 彈性方程 與bi-harmonic方程的前驗與後驗誤差估計並將之運用於流構耦合的界面。經由適當的修正流體與彈性桿數值解的有限元素的基底函數，介面的連續條件與跳躍條件可以自動被滿足。經由後驗誤差估計所提供自動網格加密，介面區域解的精確度應可以有效的提高。加上沈浸有限元素法允許介面與網格交叉通過對流構互動問題也自然沒有網格糾纏的問題。 我們將進一步經由計算更多 benchmark problems (包含主動式與被動式) 的流構互動問題來確認沈浸有限元素法於流構互動模擬中的精確度與可靠度。最後如果時間與研究人力上許可我們會進一步運用知名的BCIZ元素來模擬彈性平板的型變與彎曲並進一步對相關的裂縫問題做研究。

Under the support of grants 96-2115-M-009-014 and 97-2119-M-009-006, we have implemented the streamline upwinding Petrov-Galerkin (SUPG) finite element for the incompressible flow solver and the nonlinear beam finite element on a local moving mesh for simulating the interactions between an incompressible fluid and a non-linear Bernulli-Euler beam. We also initiated our studies in the singular solutions of Laplace equation using singular element and multigrid method. It is our interests to further investigate the singular solutions of elasticity partial differential equation and bi-harmonic equation in 2D since they are related to the crack problems in a plate and shell structure. Moreover, we shall employee the newly developed immersed finite element method (IFEM) to prevent the mesh tangling problem that we encountered in the fluid-structure interaction (FSI) simulation and to solve interface problems rising in plate and shell structures. In the current proposal, we shall first develop a priori and a posteriori error estimations for the IFEM method for both the Laplace equation, elastic system equations and bi-harmonic equation to ensure the accuracy of the IFEM solutions. Next, the IFEM approximation will be applied to the FSI problem in which the fluid-structure velocity continuity condition and the interface flux-jump conditions are automatically satisfied by properly modifying the finite element basis functions both in fluid equations and structure equations. Since the elastic bar immersed in fluid can be considered as an interface and the IFEM method allow interface across elements, we anticipate the IFEM method can naturally prevent the mesh tangling problem in our FSI simulation. Both active and passive benchmark problems (such as the flapping-flagproblem) concerning fluid-structure interactions (FSI) will be computed to confirm the accuracy and stability of our simulators. Finally, we shall study the singular solutions in the in-plane deformation and bending of an elastic plate. For the plate bending problem, the well-known BCIZ element (a modified Hermite element) will be employed. We are interested in deriving an a posteriori error estimator and apply the adaptive mesh refinement and local mesh moving scheme to obtain a better approximation to the singular solution.