This paper presents a novel Petrov-Galerkin free element method (PGPZ-FREM) based on a combination of the strong form free element method (FREM), sub-domain mapping technique, and Petrov-Galerkin method for analyzing piezoelectric structures. This is a brand new numerical method that combines the ideas of isogeometric method and meshless method. Similar to the isogeometric method, the computational domain is divided into a lot of patches or subdomains firstly. In each subdomain, local collocation Lagrangian elements are generated according to the location of the nodes. Additionally, the Heaviside step function is selected as the weight function to simplify the calculations. By constructing equations point by point, a set of linear algebraic equations is established to solve the piezoelectric problem. Finally, the accuracy and stability of the piezoelectric zonal Petrov-Galerkin free element method are verified by numerical examples, including a symmetric piezoelectric block, a piezoelectric tuning fork, a dual-material MFC sensor, and the wing skin pressure sensing system.

中文翻译：

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用于压电结构的 Petrov-Galerkin 区域自由元方法


本文提出了一种新颖的 Petrov-Galerkin 自由元方法 （PGPZ-FREM），该方法基于强形式自由元法 （FREM）、子域映射技术和 Petrov-Galerkin 方法的组合来分析压电结构。这是一种全新的数值方法，结合了等几何方法和无网格方法的思想。与等几何方法类似，计算域首先分为许多 patches 或子域。在每个子域中，根据节点的位置生成局部搭配拉格朗日元素。此外，选择 Heaviside 阶跃函数作为权重函数以简化计算。通过逐点构建方程，建立了一组线性代数方程来解决压电问题。最后，通过对称压电块、压电音叉、双材料MFC传感器和翼皮压力传感系统等数值算例验证了压电带状Petrov-Galerkin自由元法的准确性和稳定性。