Dear Amir

You can find the answer for your question in:

Crandall (1956) noticed that the weighted residual methods can catch most approximation procedures. He gave definitions to different choices of weight functions and shape functions. He assigned the name Galerkin method to the procedure in which the weight functions and the shape functions are identical and other names to the procedures in which the weight functions differ from the shape functions, such as the collocation method, the subdomain method and the least squares method.

" Finite Elements Based on the Piece-Wise Linear Weight Functions in Contact Problems "

DOCTORAL THESIS

By: Chouping Luo

Luleå University of Technology Department of Civil and Environmental Engineering, Division of Structural Mechanics

2004

Best wishes

Dear Mohannad,

Thanks very much for guidance and references.

Amir

I guess that this is a matter of presentation.

Each independent _test_ function leads to an independent equation in the set of equation to solve. So (unless in the case of MPC - multi-point constraint - where they are not independent), the equation number i corresponds to the test function number i only, say Yi, and this function is Yi = di*Ni (Galerkin Finite Element method). Since all the equation is multiplied by di 0 (non zero...), this is equivalent to take Yi = Ni.

But now, consider the unknowns and not the equations: unknown number j usually corresponds to the searched value at node j, so is Uj = uj*Nj. In the case where the system to solve is linear, then the coefficient multipling the unknown uj in equation i is obtained by using as test function Yi = Ni (line i in the system of equation), and computed for Nj (column j in the matrix multiplying the vector of unknowns u).

Hope this helps.

Hi David,

The point I was looking for; thanks very much.

When weighting function and shape functions are considered same then they yield symmetric stiffness matrix. This basically reduces the computational time to solve the linear simultaneous equations.