Which approach is used to derive the equation Ku+f=0 by projecting residuals onto test functions?

Which approach is used to derive the equation Ku+f=0 by projecting residuals onto test functions?

• A. Finite Element Method
• B. Boundary Element Method
• C. Galerkin Method
• D. Newton-Raphson Method

Answer: (C)

This question tests the Galerkin method, a weighted-residual approach. You take the residual from the governing equation, R = Ku + f, and require its projection onto a set of test functions to vanish: ∫ w^T (Ku + f) dΩ = 0 for every test function w in the chosen space. If you select test functions from the same space used to approximate the solution (w = φ_i), you obtain the Galerkin weak form. When you discretize the displacement u as u ≈ ∑ φ_i u_i and assemble over the domain, this weak form becomes the algebraic system Ku + f = 0, with K as the stiffness matrix and f as the force vector. The finite element method often uses this Galerkin formulation; boundary element methods rely on boundary-integral formulations, and Newton-Raphson is a nonlinear solver, not a residual-projection approach.

This question tests the Galerkin method, a weighted-residual approach. You take the residual from the governing equation, R = Ku + f, and require its projection onto a set of test functions to vanish: ∫ w^T (Ku + f) dΩ = 0 for every test function w in the chosen space. If you select test functions from the same space used to approximate the solution (w = φ_i), you obtain the Galerkin weak form. When you discretize the displacement u as u ≈ ∑ φ_i u_i and assemble over the domain, this weak form becomes the algebraic system Ku + f = 0, with K as the stiffness matrix and f as the force vector. The finite element method often uses this Galerkin formulation; boundary element methods rely on boundary-integral formulations, and Newton-Raphson is a nonlinear solver, not a residual-projection approach.