In Galerkin's Method, the weighting functions are chosen to be equal to the corresponding what?

In Galerkin's Method, the weighting functions are chosen to be equal to the corresponding what?

• A. Shape functions Nb
• B. Displacement field u
• C. Forces f
• D. Material properties K

Answer: (A)

In Galerkin's method, you approximate the unknown field (for example, displacement) with a sum of interpolation shape functions N_i multiplied by nodal values. The residual of the governing equation is then projected onto a finite set of weighting functions. The key idea is to select those weighting functions from the same space as the trial functions, so each weight function is equal to the corresponding shape function (w_i = N_i). This choice makes the projection consistent with the variational (energy) form of the problem, so the resulting discretized equations inherit the structure and stability of the continuous problem. It also leads to a symmetric stiffness matrix when material properties are symmetric, provided the problem is posed accordingly. In practice, the discrete system takes the familiar form where the stiffness matrix comes from integrals of derivatives of the shape functions, and the nodal loads come from the external terms weighted by the same shape functions. This is what makes the Galerkin approach natural and effective for finite element analysis.

In Galerkin's method, you approximate the unknown field (for example, displacement) with a sum of interpolation shape functions N_i multiplied by nodal values. The residual of the governing equation is then projected onto a finite set of weighting functions. The key idea is to select those weighting functions from the same space as the trial functions, so each weight function is equal to the corresponding shape function (w_i = N_i). This choice makes the projection consistent with the variational (energy) form of the problem, so the resulting discretized equations inherit the structure and stability of the continuous problem. It also leads to a symmetric stiffness matrix when material properties are symmetric, provided the problem is posed accordingly. In practice, the discrete system takes the familiar form where the stiffness matrix comes from integrals of derivatives of the shape functions, and the nodal loads come from the external terms weighted by the same shape functions. This is what makes the Galerkin approach natural and effective for finite element analysis.