Problems -

1. Consider the one-dimensional boundary value problem (BVP):

u_xx + x = 0, x ∈ (0, 1)

-ux(0) = ku(0) + q (mixed BC also know as Robin's BC)

and u(1) = g,

where q, k, and g are given constants.

(i) Solve the BVP exactly.

(ii) Obtain the weak formulation of this problem.

(iii) Using the method of Lagrange multipliers, convert the Dirichlet boundary condition at x = l into a natural boundary condition and obtain the corresponding weak formulation.

(iv) Divide the domain into two uniform elements. Using linear elements and the weak formulation of (iii), calculate by hand the stiffness matrix and solve for the values of u at the various nodes.

(v) Compare the calculated values with the exact solution. Use d₂(u_exact, u_FE) = √(₀∫¹[u_exact(x)-u_FE(x)]²dx) as the measure of the error.

(vi) Calculate the flux u_x(1) at x = 1 from the finite element approximation and from the Lagrange multiplier. Compare both values to the exact one. Which one is a better approximation? Why?

(vii) Repeat the calculation (with 2 elements) within COMSOL. Use g = 1, k = 2, and q = -1. Compare the COMSOL results with the results of your hand calculations.

(viii) Repeat the calculation in COMSOL using 10 uniform, linear elements. Compare the COMSOL solution with the exact solution. Use d₂(u_exact, u_FE) = √(₀∫¹[u_exact(x)-u_FE(x)]²dx) as the measure of the error.

(ix) Examine the structure of the stiffness matrix. Compare the COMSOL generated stiffness matrix with the stiffness matrix that you calculated by hand in (iv).

(x) Implement the weak formulation directly in COMSOL and solve.

2. Consider the set of second order differential equations, defined over the domain 0 ≤ x ≤ L,

-(d/dx)(S((du₁/dx) + u₂)) + Cu₁ = q

-(d/dx)(D(du₂/dx)) + S ((du₁/dx) + u₂) = 0

with the boundary conditions

u₁(0) = u₂(0) = 0,

S((du₁/dx) + u₂)_x=L - F = (D(du₂/dx))_x=L - M = 0

Determine the equivalent weak formulation for the Timoshenko beam equations. Specify clearly the boundary conditions to be satisfied by the test functions.

[The following is provided as background information and is not needed for the problem's solution. The above equations describe the bending of a Timoshenko beam. S is the shear stiffness, S=K_sGA, where K_s is a geometry-dependent correction coefficient; G is the shear modulus; A is the cross-sectional area. D=EI is the bending stiffness. u₁ is the transverse deflection. u₂ is the rotation. C is the foundation modulus. q is distributed transverse load. In contrast to the Euler-Bernoulli theory, of beams, the angular deflection is an independent variable and is not approximated by the slope of the deflection.]

Hint: use two (independent) test functions w₁ and w₂. Multiply the first equation by w₁ and the second equation by w₂ and integrate each product over the domain.