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Extra resources for Finite Element Analysis Applications in Mechanical Engineering (2012)
2345 % Table 1. Maximum density error for three different meshes Figure 2. 5 s. □ line: Analytical solution, ◊ line: Mesh 1, 500 nodes, x line: Mesh 2, 1000 nodes and + line: Mesh 3, 5000 nodes Electromagnetic and Fluid Analysis of Collisional Plasmas 49 Figure 3. 5 s. 2. 002 m, with the simulation being run for 5000 steps at a constant time step increments of 3x10-8 s. The contour plot of the momentum in the x direction is shown in Figure 4, where at the exterior of the wave front, the wave is moving outwards in the x direction with maximum momentum.
1. Advective predictor step The advective predictor step is formulated using the Taylor series expansion for the unknown variable Q at a first order approximation, thereby resulting in the half time n 1/ 2 values of the independent variable Q : Electromagnetic and Fluid Analysis of Collisional Plasmas 47 n 1/2 Q Ve 4 4 n Qi i 1 n n n n 4 t 4 t 4 t 4 t K i bi K i ci K i di Ve M i 12 i 1 12 i 1 12 i 1 8 i 1 (85) where Ve is the volume of a tetrahedral element, bi, ci and di are the linearly interpolated piecewise shape functions.
Corrector step The corrector step utilizes the Taylor series expansion for the full time step to the second order approximation to get: n r e Dc Q t r e t r e n n 1/ 2 bie Kr 2 A cie L z t r e e M 2 3 t r e n 1/ 2 t r e n 1/ 2 cie Kz 2 n 1/ 2 (K n Ae The above discretization results in the following equations: t r e n n bie Lr 2 Ln )N ie dA (80) 46 Finite Element Analysis – Applications in Mechanical Engineering n ' 'n Dc Q B (81) where: ' Dc r Dc (82) and 'n B r Bn (83) ' where r denoting a matrix of r e entries everywhere, Dc is the consistent mass matrix and 'n B is the vector of added element contributions to the nodes in the cylindrical axisymmetric case.