流構耦合的多尺度數值模擬與特徵值問題的探討

Abstract

在國科會計畫 96-2115-M-009-014, 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。在這個工作裡我們已完成不可壓縮流數值模擬器與尤拉彈性桿模擬器的精確度測試。 而流體與結構的互動則經由區域移動網格來達成。 在本計劃中， 我們將延續先前的工作。 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我們的流構互動模擬器的精確度與可靠度。並希望引進多尺度的方法來增加模擬器的精確度與穩定度(R. A. Khurram and A. Masud, 2006). 另一方面如何提高微幫浦的效率一直是生物晶片設計中的難題， 因此，流構耦合的自然頻率的特徵值問題是非常重要的課題。 由於這類的特徵值問題有其特殊結構， 我們也將應用保結構算法來提高特徵值的精確度。同時我們知道當外力與自然頻率發生共振時，大型變極有可能進而造成結構產生 bulking 與 failure 的現象。本計畫也將對這個問題做進一步的探討。 最後，我們也將應用我們的研究成果來計算二維的微幫浦流構耦合問題。

In the project of 96-2115-M-009-014, we used the streamline upwinding Petrov-Galerkin (SUPG) finite element to simulate the interactions between an incompressible fluid and a non-linear Bernulli-Euler elastic beam. So far, we have finished the accuracy testing of the simulators for both objects. The interactions between the fluid and the structure were simulated using local moving meshes. The work will be continued in the current proposal. Both active and passive benchmark problems concerning fluid-structure interactions (FSI) will be computed to confirm the accuracy and stability of our simulators. Recently, Khurram and Masud have successfully applied the multiscale method in solving the flapping-flag problem. We will also employee the multiscale method proposed by Hughes to enhance the accuracy and reliability of our FSI simulators. The development of the FSI simulators has its potential to help solve an issue that have long posed an obstacle in the design of biological chips –how to improve the efficiency of micropumps. This problem directly pertains to the eigenvalues of natural frequencies of fluid-structure devices, which will consequently be dealt with in the proposal. Because the associated problems are large and sparse, efficient algorithm that takes advantage the special structure of the eigenvalue problems and its multiscale nature from discretization is needed. We are also concerned about the bulking effect and failure in the structure resulted from rapidly large deformation when external forces resonate with the natural frequency. The proposed project will address these issues in finer details, and the developed simulators will be applied to compute the 2D FSI problems in micro-pump.