The Application Of Space Curve Element With Finite Element Method Of Line In The Field Of Heat Conduction

Finite Element Method of Lines (FEMOL) is a semi-analytical technique supported by solver of ordinary differential equations. Compared with complete nodal discretization feature of FEM, FEMOL semi-discretize the structure with lines, which makes the elements needed in the element discretization decrease significantly; For FEMOL, the ordinary differential equations system (ODEs) of nodal line functions that could be gotten by the variation principle would be solved by written program with the ODE solver COLSYS, which means the computational accuracy of FEMOL is higher than other traditional numerical methods. Based on FEMOL, the following works have been done:(1)Theory establishment:space curve element was developed by cubic B-spline, which successfully applied FEMOL to solve the thermal problems of irregular space area. First, map space curve element to the regular element in the interval of [-1,1], including nodal line mapping and element mapping. For nodal line mapping, Lagrange interpolation is used for the simple curve mapping; while the space curve is complex, B-spline interpolation would be adopted. For element mapping, the linear interpolation of nodal lines is employed. Second, introduce matrix operation into thermal functional variation, which obtains the unified form of ordinary differential equations system under different boundary conditions.(2)Program writing:In the paper, heat conduction program TFEMOL of space curve element by FEMOL was written by FORTRAN. Modular writing of program and uniform of ordinary differential equation system makes programming work significantly decrease. The first cubic B-spline interpolation is used for the complex curve mapping in this paper, and the first-order derivative of the endpoint is given approximately by the first-order derivative of Lagrange interpolation function of the three nodes near the endpoint, which simplify the data input and makes the accuracy stable. Coefficient matrix element is calculated by Gaussian integrity, and in this paper Gaussian integrity points from2to6are given for the user’s choice. The result is output in unified.TXT format, and MATLAB is used to draw temperature field nephogram.(3)Examples making and computation:according to the principle of semi-inverse method of elastic mechanics, some examples are made to testify the program. Meanwhile, the program is used to compute the calculation models simplified based on the engineering project, like screw conveyor and coil spring. Results comparison between FEMOL and exact answer indicate that space curve element of FEMOL has a remarkable performance of less element discretization and higher computation accuracy for the solution of heat conduction problems.