Biomedical Engineering - Computational Biomechanics Laboratory

Full exam

1 Politecnico di Milano – Master Degree in Biomedical Engineering Computational Biomechanics Laborator y Prof. D. Gastaldi A.A. 20 20 - 2 1 – Dat e 2 0 / 0 1 / 20 2 1 Surname...................................................................... ........ ... .. ......Name................................................................................. .. .......... Person Code ............................................ MARK …………… …. T he written examination will last 120 minutes. FINITE ELEMENT MODELING AND STRUCTURAL MECHANICS Consider the FEM structure in the Figure composed by 2D plane str ess isoparametric 4 - node elements with full integration . 1) Define the basics of the isoparametric element formulation and d etermine the shape functions. [ 4 pts] 2) D etermine the Jacobian matrix for all the elements . [ 4 pts] 3) Determine the nodal force vector equivalent to reaction forces and the external loads: distr ibuted traction ( q1, q2 ) and concentrated force F =q 1 L . [ 6 pts] 4) Make a graphical representation of the nodal forces acting on the structure . [ 2 pts] 2 FINITE VOLUME MODELING AND COMPUTATIONAL FLUID DYNAMICS 5 ) Write the general transport equation with comments about the different terms. Then, apply the general transport equation to the energy conservation concept in order to obtain the equation of energy conservation . [ 2 pts] 6) Consider a generic property Φ tra n sported in steady state condition by means of convection and diffusion in two - dimensio ns without body forces with the direction of flow as indicated in the figure . Adopting the finite volume approach, consider the central differenc e scheme for the diffusive term and the 1 st order upwind scheme for the advective term . 6a) Write the contribution of each face of cell P to the advective term . [ 4 pts] 6b) Assuming !" !# = 0 in the whole domain write the diffusive term. [ 4 pts] 7 ) W rite the general transport equation of generic property Φ in its general transient formulation and make a discussion about time integration strateg ies . [ 3 pts ] 8) Focusing on thermo - flui d dynamics problems, make a schematic discussion about the difference between segregated and coupled methods. [ 4 pts]