What substitutions are made for the method of weighted residuals?

What substitutions are made for the method of weighted residuals?

• A. u = Σ N_a u_a and v = Σ w_b δ u_b
• B. u = Σ N_b u_b and v = Σ w_a δ u_a
• C. u = Σ N_a δ u_a and v = Σ w_b u_b
• D. u = Σ N_a u_a and v = Σ w_b δ v_b

Answer: (A)

In the method of weighted residuals, you turn the differential problem into a weighted integral by representing the unknown field with a finite basis and choosing weighting functions from the same variation space. The field is approximated as a sum of shape functions times nodal values, u ≈ Σ_a N_a u_a. The corresponding weighting (test) function is built from the variations, so you form v as a linear combination of these variations, v = Σ_b w_b δu_b, where the w_b are chosen weights. This setup makes the residual orthogonal to the chosen weight space and leads directly to a set of discrete equations for the nodal values. Other formulations would mix in variations where the field itself is being represented in the approximation or would use inconsistent or incorrectly indexed terms, which breaks the intended projection.

In the method of weighted residuals, you turn the differential problem into a weighted integral by representing the unknown field with a finite basis and choosing weighting functions from the same variation space. The field is approximated as a sum of shape functions times nodal values, u ≈ Σ_a N_a u_a. The corresponding weighting (test) function is built from the variations, so you form v as a linear combination of these variations, v = Σ_b w_b δu_b, where the w_b are chosen weights. This setup makes the residual orthogonal to the chosen weight space and leads directly to a set of discrete equations for the nodal values. Other formulations would mix in variations where the field itself is being represented in the approximation or would use inconsistent or incorrectly indexed terms, which breaks the intended projection.