Final exam PHƯƠNG PHÁP SỐ

Calculate analytically the first derivative and the second derivative of function (t) at xo 2l Estimate numerically the flrst derivative and the second derivative of function (1) at xo using centered finitedivided difference formula Q) e (3). Compare the numerical results with analytical ones to determine the errors

Ouestion (L.o.1.3; L.O.2.2) (2 marks) Consider the following function: f{x}: -0.6xa- 0.8rB -Ex? -x + L.g "rp: S.$,Ii: {1} p.t$ 1/ Calculate analytically the first derivative and the second derivative of function (t) at xo 2l Estimate numerically the flrst derivative and the second derivative of function (1) at xo using centered finite-divided difference formula Q) e (3) Compare the numerical results with analytical ones to determine the errors F,{ j.,, : -.f(*o*r} + fi,ftxr+r} - I f{xr-r} +gqx,-r} _ -i- \- f "(;r1) -"i'{x;+r} - * 16 f{x6ar} - sn f{x,} + 16 f{ni:r.} L4,J - f,(xc-a} La hr Ouestion (L.O.1.3; L.O.2.1) i/ t3) (2 marks) Compute numerically the following integral using Newton- C6tes method: +l J (*' +9x2 * 7x-rrs ) dx -1 The abscissae and factors of the points of integration are given as follows xr:-1, @1 : *t *r:o, cDr:!rx3:1, -r:1 2/ Estimate the error (compare with the exact solution given by the analytical integration NewtonLeibnitz) Ouestion (L.O.4.3 ; L.o.4.4; L.O.6 ) (2 marks) Consider a 3-node triangular (1- 2- 3) with node 1(x: 0; y: 0), node 2(x: 1; y: 0); node 3(x: 0; y: 1) 1/ Determine the three interpolation functions Nr(x, y), Nz(x, y), Nl(x, y) 2l Given 800; t2:830i t3: 810 are temperatures at node 1, node 2, node3, respectively Interpolate temperature at point M(x: 0.2;y:0.2) tr: Student's ID: Student's full name: Page ll2 .-t -a Ouestion (L.O.4.3L.O 4.4; L.O.6.2) (2 marks) Consider the plane truss in Fig 4-l with the given: Et: E2: 70,000 MPa (Young's modulus of aluminium); Ar: A2: 64 mm2 (area of the cross section of the 2bars);Lr: Lz:800 mm (length of two bars); P: 50,000 N; element l(l- 2); element 2(2- 3) v 81 l, t &,.I0 L ffi s Figure 4- I Use the finite element method to 1/ Determine: 2l a/ Stiffness matrices of element I [Kr] and of b/ Stiffrress matrix of structure [K] elemett2lKzT CaLcrlate the nodal displacements uz (horizontal component component at node 2) Question (L.O.1.3; L.O.2.2) at node 2) afi vz (vertical (2 marks) Consider a cantilevered beam (fixed at one end and free at the other end) with the given parameters E (Young's modulus of beam material (concentrated load) as shown in fig 5-1 ,I (inertia moment of cross section), L(length of beam) ,P Fig.5-1 With the finite element method, use one-element model: 1/ Determine the stiffness matrix of this element [Kr], the stiffness matrix of the whole structure [K] Notice that this beam structure consists of only one element 2l Calculate nodal displacements: linear displacment vz and angular displacement #3 - END Student's ID Student's full name: Page2l2