Variational and Weighted Residual Methods 1 The Weighted

Variational and Weighted Residual Methods 1

The Weighted Residual Method The governing equation for 1 -D heat conduction A solution to this equation for specific boundary conditions was sought in terms of extremising a functional A solution can be found by making use of a trial function which contains a number of parameters to be determined 2

The Weighted Residual Method In general, the trial function will not satisfy at all points in the region in which the solution is sought and that where r(x) is the residual at the point x in the region 3

The Weighted Residual Method If : Φ produces the exact solution as the number of parameters α in Φ is increased indefinitely. Wi are linearly independent functions of x (weighting functions) It can be shown that Φ → u, the exact solution, if for all Wi 4

The Weighted Residual Method Noting that the values of Wi are linearly independent if none of them can be expressed as a linear combination of the other. No set of numbers bj exists such that 5

The Weighted Residual Method In the FE version of the weighted method, the trial function is expressed in terms of its nodal values Φ i=1. . M , where M is the total number of nodes where are the global shape functions 6

The Weighted Residual Method For the series of linear elements. . e-1, e, e+1, e+2. . . 7

The Weighted Residual Method is as shown in elements e and e+1, but is zero in all other elements. The parameters to be determined are the nodal values, Φi The weighting functions are chosen the same as the shape functions (Galerkin method) 8

The Weighted Residual Method The weighted residual statement is for i= 1. . M The method fails with piecewise linear shape functions, since is everywhere zero (except at nodes where is discontinuous) 9

The Weighted Residual Method Integrating the above equation by parts The right-hand side second term is zero unless both and belong to the same finite element. The first term on the right-hand side is also zero, apart from the two elements at the extremities of Ω. 10

The Weighted Residual Method For the first element Since Then 11

The Weighted Residual Method This can be written as Again , since This becomes 12

The Weighted Residual Method Similarly for the last element Hence gives the component Kiα of the global stiffness matrix [K]g and 13

The Weighted Residual Method where 14

The Weighted Residual Method In the j-th row of the matrix product [K]g {Φ}, the only non-zero terms are 15

The Weighted Residual Method Reverting to the shape functions within the individual elements, these three non-zero terms can be expressed as This is an integral over element e, since is zero for elements numbers less than e and is zero for element numbers greater than e 16

The Weighted Residual Method Similarly the second term can be expressed as and the third as 17

The Weighted Residual Method Hence we can express the global stiffness matrix by assembling the element stiffness matrices [K]e, where α and β could each take on the values j-1 and j for element e and the superscript has been removed from the element shape functions. 18

Review of results The result obtained by Galerkin weighted residual method is exactly the same as that obtained by the piecewise application of the Variational approach The stiffness matrices are symmetrical (advantageous in reducing the computation in solving the system simultaneous equations) The Variational methods are extremely powerful in engineering mathematics 19

Review of results If a variational formulation of a certain problem is possible the same results can be obtained by the Galerkin weighted residual method. The Galerkin weighted residual method can be used when no variational formulation is available. The Galerkin weighted residual method is the most commonly used of the weighted residual methods. 20