Name two methods to obtain Ku+f=0 from an approximate solution.

• A. Collocation Method
• B. Rayleigh-Ritz Method
• C. Galerkin Method
• D. Galerkin, Variational Principles (Raleigh-Ritz)

Answer: (D)

The idea is to turn the continuum equation Ku + f = 0 into a solvable algebraic system using an approximate displacement u_h expressed in a chosen finite-dimensional space. Two standard, complementary ways to do this are Galerkin projection and Ritz (variational) principles.

In Galerkin, you form the residual R = K u_h + f and require it to be orthogonal to every test function in the same space as the trial displacement. Practically, if you write u_h = Φ c with basis Φ and coefficients c, you enforce Φ^T (K Φ c + f) = 0, giving the reduced system (Φ^T K Φ) c = - Φ^T f. This is how you ensure the approximate solution satisfies equilibrium in a weighted-average sense across the chosen space.

With Ritz (variational) methods, you start from the energy principle. For linear elasticity, the potential energy is Pi(u) = 1/2 u^T K u − u^T f. Minimizing Pi with respect to the coefficients c in u_h = Φ c yields the same reduced system (Φ^T K Φ) c = - Φ^T f. So the Ritz approach derives the same equations by seeking the stationary point of the energy, while Galerkin does so by minimizing the residual in a weighted sense.

Thus, the two methods to obtain Ku + f = 0 from an approximate solution are Galerkin projection and Ritz (variational) principles.