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Politecnico di Milano Computational Biomechanics Laboratoy Final Exam (Call: 22/2/2018) Problem 1 (12pts) Consider a the linear elastic one-dimensional bar subjected to an axial load, F (see Figure 1). The bar has a cross sectional area A = 1000 mm 2, and Young’s modulus E = 210 GPa. If the bar is discretized with one two-node linear element as shown, determine: 100 mm A B F 100 mm 1 2 F y x y x Figure 1: Problem 1. 1. Shape functions for the element and the expression for the displacement field within the element. (2pts) 2. Value of the strain, ϵ, and stress, σ within the element. (3pts) 3. Write the expression for the principle of virtual work of the problem and calculate the stiffness matrix of the element. (5pts) 4. Determine the displacement of point B if F = 10 kN (2pts) Problem 2 (3pts) Which are the properties of the stiffness matrix for the element calculated in Problem 1. Problem 3 (12pts) Consider a property φ transported by means of convection and diffusion through a one-dimension domain, with cross section area A, in absence of body forces shown in Figure 2 1 m A B u 1 2 y x y x 3 dx Figure 2: Problem 2. 1. Write the governing equation for steady state transport. (2pts) 2. The domain is discretized with three finite volumes as shown in Fig. 2. Write the discretized system using first-order upwind for the convective term and central difference for the diffusive term. (8pts) 3. If A = 0.01 m2, dx = 1/3 m, u = 0.01 m/s, the density ρ = 1 Kg/cm3, the diffusion coefficient Γ = 0.1 Kg/m·s, and boundary conditions φ(0) = 10 and φ(1) = 0, determine the value of φ along the domain for the previous discretization. (2pts) Problem 4 (3pts) What type of discretization schemes are the forward-Euler and the Crank Nicolson methods?. Which are the differences between these two…