การประดิษฐ์และการประเมินเอลิเมนต์เมทริกซ์รูปแบบปิดของเอลิเมนต์สี่เหลี่ยมสี่จุดต่อสำหรับปัญหาความเค้นเนื่องจากความร้อน - เบญจาภา ยนต์สกุล (จุฬาลงกรณ์มหาวิทยาลัย, 2017)
In the past, the derivation of finite element equation using quadrilateral element is very difficult because there is no closed-form solution as triangular element. Therefore, the widely used numerical integration, Gauss-Legendre method, is applied to obtain the approximated numerical result. The accuracy of the numerical integration can be improved by increasing the number of Gauss’s point but also increase the computational time. In this research, the closed-form four-node quadrilateral element matrices for thermal stress problem are derived by using the symbolic algebra software, Mathematica, along with hand manipulation. The proposed closed-form expression can be classified into four cases based on the element’s shape to increase the performance of the computational process. The finite element matrices are also presented in detail and can be used to create the computer program directly. The accuracy and computational time of the derived closed-form expression are evaluated by the single element test problem and the problems with and without exact solution for heat transfer and thermal stress problems, respectively. The computational results show that the derived closed-form expression can provide the high solution accuracy comparing up to 8x8 Gauss’s point. Moreover, the presented method also spends less CPU time comparing with the conventional method.