tailieunhanh - Fundamentals of Finite Element Analysis phần 4

Lưu ý mối quan hệ giữa các chức năng nội suy được xác định trong phương trình 5,20 và các chức năng thử nghiệm trong phương trình 5,11. Các chức năng nội suy tương ứng với phần chồng chéo chức năng thử nghiệm áp dụng trong một lĩnh vực yếu tố duy nhất. Cũng lưu ý rằng các chức năng nội suy đáp ứng các điều kiện | I Text Hutton Fundamentals of 5. Method of Weighted Finite Element Analysis Residuals The McGraw-Hill Companies 2004 The Galerkin Finite Element Method 143 Note the relation between the interpolation functions defined in Equation and the trial functions in Equation . The interpolation functions correspond to the overlapping portions of the trial functions applicable in a single element domain. Also note that the interpolation functions satisfy the conditions N1 x Xj 1 N1 x Xj 1 0 N X Xj 0 N2 X Xj 1 1 such that the element boundary nodal conditions Equation are identically satisfied. Substitution of the assumed solution into Equation gives the residual as R e X yj yj 1 d2 y e d2 f x yjN1 X yj 1 N2 x f X 0 dX2 dX2 where the superscript is again used to indicate that the residual is for the element. Applying the Galerkin weighted residual criterion results in j 1 y Ni x R e X yj yj 1 dX Xj 1 _ d2y e . Ni X hr f X J L dX2 dX 0 xj xj or xix1 . j- d2y e I Ni x dX 2 dx J Ni X f x dx 0 i 1 2 Xj Xj as the element residual equations. Applying integration by parts to the first integral results in i 1 2 x dy e Ni X dX Xj 1 Xj j . 6A dNi dy e dx dX dX j 1 y Ni X f X dx 0 Xj Xj i 1 2 which after evaluation of the nonintegral term and rearranging is equivalent to the two equations is j 1. d N1 d y e dX Xj dX 1 dX dy e N1 X f x dx dX Xj Xj J r xj 1 dN2 dy e dy e dX N2 x f x dX - Note that in arriving at the form of Equation explicit use has been made of Equation in evaluation of the interpolation functions at the element nodes. I Text Hutton Fundamentals of 5. Method of Weighted Finite Element Analysis Residuals The McGraw-Hill Companies 2004 144 CHAPTER 5 Method of Weighted Residuals Integration of Equation by parts results in three benefits 2 1. The highest order of the derivatives appearing in the element equations has been reduced by one. 2. As will be observed explicitly the stiffness matrix was made .