MecE 390 Final examination, Winter 2014

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1 MecE 390 Final examination, Winter 2014 Directions: (i) a double-sided formula sheet is permitted, (ii) no calculators are permitted, (iii) the exam is 80 minutes in duration; please turn your paper in promptly when asked, (iv) any communication with other students is prohibited, (v) please sign and date below. Good luck. Please do not turn the page until instructed to do so Name: ID number: Question Grade Max. score Total 85 I have not received help, hints, etc. from other exam takers, nor have/will I offer such assistance Signature, date: 1

2 Problem 1 [15 marks] The multidimensional Newton algorithm provides a means of solving systems of nonlinear equations of the form f 1 (x 1, x 2,... x n ) = 0, f 2 (x 1, x 2,... x n ) = 0,. f n (x 1, x 2,... x n ) = 0. but this algorithm should work equally well if f 1, f 2,... f n are linear functions of x 1, x 2,... x n. consider the following 2 2 system of linearly-independent equations: Let s where a, b, c and α, β, γ are constants. (i) [5 marks] Show that the solution of (1) and (2) is given by x 1 = b ( ) γa αc c x 2 = a βa αb a f 1 = ax 1 + bx 2 + c, (1) f 2 = αx 1 + βx 2 + γ, (2) αc γa βa αb. (ii) [5 marks] How is the Jacobian matrix defined given (1) and (2)? (iii) [5 marks] Suppose that our initial guesses for the roots of (1) and (2) are x 1 = x 2 = 0. Show that Newton s algorithm will converge to the solution identified in (i) in a single iteration. (You can assume that the matrix equation is solved using, say, Gaussian elimination). 2

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4 Problem 2 [20 points] Boundary value problems naturally arise in the study of horizontal beams. Consider an elastic beam for which the non-dimensional vertical deflection, y, is Y L when x = 0. Conversely when x = 1, y = 0 where y = dy/dx. Here x is the non-dimensional horizontal coordinate. Supposing the beam is subjected to a uniform transverse load, y is given by the solution of y α 2 y = βx(x 1), 0 x 1, (3) where y = d 2 y/dx 2 and α and β > 0 are additional fixed constants that incorporate the load, the modulus of elasticity, the moment of inertia, etc. Using a finite-difference approach similar to those studied in class, outline the procedure by which you would obtain a numerical solution to the above problem. Your solution must clearly identify the manner in which the left- and right-hand side boundary conditions are incorporated. 4

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6 Problem 3 [20 points] The general second-order explicit Runge-Kutta equations read k 1 = f(t n, y n ), (4) k 2 = f(t n + αh, y n + αhk 1 ), (5) [( y n+1 = y n + h 1 1 ) k ] 2α 2α k 2, (6) where 0 < α 1. Consider the application of the above equations to the following autonomous IVP: dy dt = λy, y(t = 0) = y 0, (7) in which λ < 0. Show that the time-step restriction required for numerical stability is the same as for the forward Euler equation, i.e. h < 2/ λ. In spite of this result, why might it be preferable to use (4) through (6) in place of forward Euler assuming a fixed value for h? 6

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8 Problem 4 [10 points] According to Newton s algorithm, the slope of the tangent line is used in finding an updated estimate for the root, r, of the function f. Equivalently, and with reference to the following figure, x 2 is computing from the slope of the tangent line at x 1. Using geometric arguments, demonstrate that x 2 = x 1 f(x 1) f (x 1 ), where f df/dx. f f x 1 x 2 x f(x 1) 8

9 Problem 5 [20 points] Multiple choice questions, each worth 5 points. Please circle the correct answer. Incorrect answers receive 0 points. (I) In class, we spent a good deal of time discussing the heat equation and its application in finding the temperature distribution in a cooling fin. Suppose that we consider a fin of uniform area so that the non-dimensional governing equation reads d 2 θ = Rθ, (8) dξ2 where 0 ξ 1. At ξ = 1, we have considered both Dirichlet and Neumann boundary conditions on θ. Saving the best for last, let s now examine a Robin boundary condition of the form [ ] dθ dξ Λθ = 0, (9) where Λ is a positive, non-dimensional constant. In solving the associated problem Aθ = b, which of the following discretized equations must be embodied in the last row of A? (a) 1 h 2 θ n 1 ( 2 h 2 + R)θ n + 1 h 2 θ n+1 = 0 (b) ( 2 h 2 + R)θ n + Λ h 2 θ n+1 = 0 (c) θ n 2hΛθ n+1 + θ n+2 = 0 (d) θ n + 2hΛθ n+1 θ n+2 = 0 (II) The Thomas algorithm is (a) An efficient way of computing the inverse of a square matrix. (b) An iterative method akin to, though more efficient than, the successive-over-relaxation (SOR) scheme. (c) An efficient way of solving the matrix equation Ax = b, where A is a tridiagonal matrix. (d) (a) and (b). (III) Which of the following statements is incorrect? (a) Cramer s Rule is not considered to be an efficient method for solving the matrix equation Ax = b because of the large number of matrix determinants that must be evaluated. (b) A fourth-order Runge Kutta scheme is typically more accurate, but also more computationally expensive, than a second-order scheme. (c) The determinant of a lower triangular matrix, L, is given by the product of the diagonal elements of L. (d) When connecting discrete data points using cubic splines, the composite curve has a continuous first derivative, but a discontinuous second derivative. (IV) Which of the following statements is correct? (a) Newton s method for solving the scalar equation f(x) = 0 is a special case of the fixed point iteration (FPI) technique in which g(x) = x f(x)/f (x). (b) As a consequence of truncation errors, there are only a finite number of values that can be represented within a given range e.g. between -1 and 1. (c) Shooting methods can be used to solve boundary value problems, but not when coupled to a bisection-type root finding scheme. (d) The truncation errors associated with trapezoidal integration and Simpson s rule integration are O(h 2 ) and O(h 3 ), respectively. ξ=1 9

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11 SCRATCH PAPER (Material on this page will not be graded unless you specifically request otherwise) 11

5. Hand in the entire exam booklet and your computer score sheet.

5. Hand in the entire exam booklet and your computer score sheet. WINTER 2016 MATH*2130 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie 19 April, 2016 INSTRUCTIONS: 1. This is a closed book examination, but a calculator is allowed. The test