Barycentric Finite Element Method For Solving Torsional Problems

Torsional problems are usually appeared in the elasticity. The key for solving torsional problems is to make out the second order partial differential equations, which the prandtl function is satisfied to. For generally cross-section, we can use the analytical method to obtain the prandtl function, and then calculate the shear stress, torsional stiffness, unit twist and warp function depending on the equation of elasticity. But for complicated or complex cross-section, it is difficult to analyze the torsional problems using analytical method. The Finite Element Method plays an important part in solving partial differential equations. So we can use it to solve torsional problems with complex cross-section. Traditional Finite Element Method usually divided the whole regional into triangular or quadrilateral elements, so it is difficult to fit the coordinate requirement of the interpolation function, In this dissertation, depending on mean value interpolation, the Barycentric Finite Element Method is used to analyze the torsional problems of different conditions.In this dissertation, torsional problem and Barycentric Finite Element are used for the main clue, and then the following work and subjects were investigated.First, the paper summarized the main method for solving torsional problems, including analytical methods and numerical methods. Analytical methods are divided into displacement method, conjugate function method, warping function method and so on. Numerical methods are divided into Finite Element Method, Finite Difference Method, Boundary Element Method and so on.Second, the paper derived the mathematical expression of mean interpolation function on polygon element. And the paper also uses Matlab language to solve the mean value interpolation function. The mean value interpolation function can be used on polygon element. It can fit the requirement of coordination, and don't have the determined parameters. In addition, it doesn't require isoparametric transformation. Compared the traditional Wachspress interpolation function with the Laplace interpolation function, the mean value interpolation function have more advantages.Third, the paper use Taylor expansion and the nature of Wachspress to derive the error theorem of polygonal mean value interpolation function. The paper observe this following rules, the error of polygonal mean value interpolation is reduced with the decreases of cell size. With the decreases of the polygonal cell size, the numerical solution of the Barycentric Finite Element Method will gradually converge to the exact solution. Forth, this dissertation takes elliptical cross-section and circle regional cross-section with semi-circular groove for example, at the premise of a known torque, demonstrate the impact of elastic modulus's changes to the results.The Barycentric Finite Element Method uses polygons to divide the regional, and breaking the limitations of traditional finite element method. It makes the dividing of polygons regional grid more casual and more flexible. What's more, the Barycentric Finite Element Method uses center triangle numerical integration program, which simplifies the process of preparing the computer program. In addition, the polygonal element reduced the total number of units, thus reducing the computational cost and improving the computational efficiency. Numerical results show that, the Barycentric Finite Element Method has a better calculation accuracy. Comparing with traditional Finite Element Method, the Barycentric Finite Element Method has more advantages.