Exam paper: Numerical Analysis of Continua I

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1 Exam paper: Numerical Analysis of Continua I Tuesday April th: Code: 8W3, BMT 3. Biomedical Technology Eindhoven University of Technology This is an open book exam. It comprises questions. The questions until 8 yield a maximum of points each. Question yields 2 points. The use of a laptop computer and MATLAB is recommended. Question ( points) A tendon with length l is stretched by fixing it at x = and by hanging a mass M at the other end of the tendon (see figure). The tendon has a density ρ, the gravitational constant is g and the cross section of the tendon is A, the Young s modulus of the tendon is E. The deformation of the tendon is described by the following equation: d dx ( EA du dx ) + ρga = The numerical value of the parameters in the equation are given in the table below: l [mm] E [N/mm 2 ] ρ [kg/mm 3 ] A [mm 2 ] g [mm/s 2 ] M [kg] Adapt the file demo_femd in the directory oned to calculate the displacement field in the tendon with the Finite Element program mlfem_nac. Describe on your exam paper which changes you have made to the script demo_femd. Calculate the displacement u of the point x = l.

2 Question 2 ( points) The following partial differential equation for u=u(x,t): u t + 2 u x u u x 2 = is solved on the domain x 6. Derive the weak form of this equation, using the weighted residual method. Question 3 ( points) Consider the two-dimensional element mesh as given in the figure below, consisting of 4 linear triangular elements and 6 global nodes. Every node has two degrees of freedom, the displacements u and v in x- and y-direction respectively. Let the global solution array (sol) be arranged as follows : sol = [u u 2 u 3 v v 2 v 3 u 4 u 5 u 6 v 4 v 5 v 6 ] Determine the array dest. Question 4 ( points) We want to solve the following ordinary differential equation: du dt + 3u = 2 with u = u(t) and initial condition u() = u at time t =. Show that the following, recursive scheme can be derived if we use a θ-scheme with θ = 2 (Crank-Nicholson): u i+ = ( 3 2 t)u i + 2 t t with: u i = u(t i ) and t a finite timestep. 2

3 Question 5 ( points) A one-dimensional element has 4 nodes at the positions with respect to the global coordinate system as depicted in the figure below. Derive the 4 shape functions N i ( i =, 2, 3, 4) that belong to this element. Question 6 ( points) The following integral is given: I = 3 x sin(x) + dx x + 4 Determine an approximate solution of this integral I by means of 2-point Gauss integration. 3

4 Question 7 ( points) A finite element mesh for a two-dimensional elasticity problem with two triangular elements is given in the figure below. There is no distributed load working. The associated dest matrix (in MATLAB format) is given by: dest=[7 8 ; 3 4 ; 2 ; 5 6] The displacements in node 4 are suppressed in both directions. The displacement of node is suppressed in y-direction. The displacement of node 3 is suppressed in x-direction. On the nodes and 2 forces with a magnitude F work in x-direction. After all the steps to solve the problem with the Finite Element Method the equilibrium equations are written as a set of linear equations of the form: K u = f. Depict which entries in the right-hand-side vector f are unknown, which entries have a value of zero and which entries are known and unequal to zero. Question 8 ( points) For an element the following shape functions are given: N (x, y) = (x 2)(y 3) N 2 (x, y) = (x + )(y 3) N 3 (x, y) = (x + )y N 4 (x, y) = (x 2)y In the nodes of the element the following solution is found: u T = [, 5, 6, ] Determine u/ x in the point (x, y) = (,.5) 4

5 Question (2 points) We want to solve a linear elastic problem. Consider the rectangular domain as given in the figure below. The domain is [mm] long and 5 [mm] high. We consider a plane stress situation. The domain is divided in two subareas, with equal sizes, as given in the figure. The material can be described with Hooke s law and has a Young s modulus E = [MPa] and a Poisson s ratio ν =.3. We are going to adjust the file demo_bar in directory twode to solve the problem with mlfem_nac. Define userpoints, usercurves and subareas as given in the figure. Create a mesh with 5 5 elements in each of the subareas (so total mesh is 5 elements). Use linear quadrilateral elements. For the curves and 2 the displacements in y-direction are suppressed. For curve 5 the displacement is suppressed in x-direction. Curve 3 is given a displacement v = in y-direction (dotted line shows the y-position of the curve after displacement). Adjust the file demo_bar to solve this problem and after that calculate the total vertical force that had to be applied to curve 3 to achieve this displacement (if you do not succeed in letting the program run, give the command lines on your exam paper that you would use to solve the problem and to get the total force out of the solution vector) 5

6 Answers Question u =.2 Question 2 6 w u t dx + 6 ( ) u 2w x u dx 6 5 w u u dx + 5w x x x 6 = Question 3 dest = Question 5 N = x(x )(x 3) 8 N2 = (x + )(x )(x 3) 3 N3 = x(x + )(x 3) 4 N4 = (x + )x(x ) 24 Question 6 I =.6 6

7 Question 7 f T = [? F?? F? ] Question 8 y=.5; u=[ ; 5 ; 6 ; ] dndx=[(/)*(y-3) ; -(/)*(y-3) (/)*y -(/)*y]; dudx=dndx *u u x =.5 Question 8 The code to calculate the reaction force on curve 3: inodes=nonzeros(usercurves(3,:)); idof=dest(inodes,2); freac=q*sol; force=freac(idof) xx=coord(inodes,) plot(xx,force) totforce=sum(force) F =.238 7

Analysis of ANSI W W 6x9-118,

Page 1 of 8 Analysis of ANSI W W 6x9-118,110236220472 Author: Analysis Created: Analysis Last Modified: Report Created: Introduction Administrator, 08:29:09, 08:29:09 09:26:02 Database: Z:\ENGENHARIA\ESTUDOS