多重網格法與保結構二次特徵值問題計算對預測壓電材料裂縫成長的重要性

Abstract

在很多化學製品和生物醫學的應用內，微幫浦在運送一些精密測量的微量液體中扮演重 要的角色。 特別是，在化學方面的應用上，微幫浦經常是生物晶片設備中的零件。 對 於設計微幫浦而言, 如何增加幫浦對抽入與送出微流體的效能, 往往建立在工程師對流 體與彈性薄膜間的交互作用有充分理解。微幫浦一般是由像矽氮化物那樣的彈性材料所 構造而開關閥基本上可視為一些彈性薄膜。我們所考慮的微幫浦的構造基本如下︰ 其中頂部薄膜的週期震動是由一般所謂的壓電材料(PZT)受週期性電壓變化產生的形變 所造成的。 在國科會計畫 96-2115-M-009-014, 我們使用 immersed finite element 的方法來模擬二 度空間中黏性不可壓縮流與彈性薄膜的相互作用。 在這個工作裡我們已完成解二維 isotropic 彈性薄膜與三維彈性體方程的程式。 然而當以電壓極化 PZT 時壓電材料通常 是 anisotropic。 而且由於壓電材料的晶體特性， 在不斷的伸縮與彎曲的過程中，裂縫 非常容易產生。 而電場於裂縫下的奇異性更會加速裂縫的成長。 如此加成的作用，會 迅速導致微幫浦的故障。因此對微幫浦的設計而言，應用數值模擬來嘹解並預估壓電材 料的裂縫成長是非常重要的。 在本計劃中， 我們將應用多重網格法求電位與 anisotropic 彈性體於裂縫尖點的奇異解。 我們將用一般含 crack tip element 的有限元素法或 Dimitrov, Andräand 與Schnack 提出 的 Galerkin-Petrov 有限元素法來作方程的離散化。所得的二次固有值問題， 因有 SHH (Skew Hamiltonian Hamiltonian) 的特殊結構， 我們也將用保結構的數值方法對期 求解。以增加裂縫尖端應力強度估計的精確度。

In many chemical and biomedical applications, micro pumps play an important role in transporting a small, accurately measured liquid. Especially, when utilized in chemical applications, micro pumps are often a component of a lab-on-a-chip device. A micro pump is generally fabricated by some elastic material such as silicon nitride, and consists of a pump chamber, passive flap valves on the flow inlets and outlets and an active flexible diaphragm on one side of the pump where a periodic force is provided by some piezoelectric component. With the supports from NSC grant 96-2115-M-009-014, we have studied the interaction between the fluid and elastic membranes using immerse finite element method. Computer codes for solving two dimensional and three dimensional isotropic elastic equations have been implemented. However, the piezoelectric ceramics are usually anisotropic due to electrical poling. Moreover, the crystal nature of this kind of materials is subjected to develop cracks under frequent constriction, dilation and deflection. The singularity of the electric fields near crack tips can further accelerate the growth of cracks which usually results in the failure of micro pumps. Therefore, it is very important to be able to predict the crack development of piezoelectric ceramics numerically for better design of a micro pump. In this proposal, we would like to use multigrid methods to compute the singular solutions of electrical potential and anisotropic elasticity near cracks. The general finite element method with crack tip element (singular element) or the Galerkin-Petrov method proposed by Dimitrov, Andräand and Schnack shall be used for discretization. The associated structure quadratic eigenvalue problems shall be solved by using structure-preserving scheme to ensure the accuracy of the stress intensity at the crack tips. 表