Biomedical Engineering - Mathematical and Numerical Methods in Engineering

Multiple choice test

*Questo modulo registrerà il tuo nome, inserire il nome. 21/1/2025 - MATHEMATICAL AND NUMERICAL METHODS IN ENGINEERING - NUMERICAL PART Prof. C. Vergara TIME : 1h 45' Rispondere direttamente su Forms o su un foglio di carta in base alle istruzioni indicate punto per punto. Per le risposte della parte 3 (scritte su un foglio di carta), consegnarle direttamente al docente --------------------------------------------------------------------------------------------------------------------------------- Respond directly on Forms or on a sheet of paper according to the step-by-step instructions. For answers of part 3 (written on a sheet of paper), please give them to the teacher Part I Quiz Answer the followin g questions: U^{n+1} = U^n + beta*dt/(2h)*B*U^n - dt*F^n U^{n+1} = U^n - beta*dt/(2h)*B*U^n + dt*F^n B*U^{n+1} = U^n - beta*dt/(2h)*U^n + dt*F^n (I+beta*dt/2h)*B)*U^{n+1} = U^n dt*F^n Given the matrix B with 0 on the main diagonal, +1 on the up main diagonal (elements (j,j+1)) and -1 on the low main diagonal (elemens (j,j-1)), and vectors F collecting approximation of the forcing terms/boundary conditions, which is the vectorial expression for the Forward Euler centered method applied to the unsteady advection problem with parameter beta?(1 punto) 1 It is not affected neither by numerical dissipation nor by numerical dispersion It is affected by a relevant numerical dissipation It is affected by a relevant numerical dispersion (i.e. a phase error) It is affected by relevant numerical dissipation and dispersion Which is true about the numerical solution obtained by means of the Lax-Wendroff scheme? (1 punto) 2 A diffusion/reaction partial differential equation A linear system with matrix A A system of ordinary differential equations with matrix A An ordinary differential equation Consider the Heat equation and the matrix A: A(i,i)=2, A(i,i-1)=A(i,i+1)=-1. The semi-discrete approximation arising after space discretization by means of the centered/II order method, leads to (1 punto) 3 21/1/2025 - MATHEMATICAL AND NUMERICAL METHODS I... https://forms.office.com/Pages/DesignPageV2.aspx?origin=NeoPorta.. . 1 di 627/01/2025, 17:03 It is unconditionally (absolute) stable for theta>=1/2 It leads to a linear system with matrix I+mu*dt*theta/h^2 * A It is of order at least one in time It leads to a linear system with matrix I-mu*dt*theta/h^2 * A Regarding the Theta method with parameter theta for the Heat equation, which of the following is false?(1 punto) 4 U^1_3 = 0.12 U^1_3 = 0.01 U^1_3 = 0.31 U^1_3 = 0.57 Given the unsteady advection problem in [0,1] with beta=-3 and the Upwind method, with f=0, u0=x(1-x), dt=h=0.1, which of the following is true? U^n_j means solution at discrete time n and node j (node 0 is the left boundary one). (1 punto) 5 The local truncation error is always zero, for any h It satisfies the Galerkin ortogonality, i.e. a(u-uh,vh)=0 for any vh in Vh The subspace Vh should well approximate V, i.e. for any v in V, inf_{vh in Vh} ||v-vh||_H^1->0 for h->0 It is stable with constant independent of the data Which of the following condition is false related to the Galkerkin approximationobtain with space Vh?(1 punto) 6 If u in H^2, it is a second order method The solution is a parabolic function The Lagrangian basis functions are defined as continuous functions which are parabolae on each interval The Lagrangian basis functions are all convex Regarding P2 Finite Elements, which of the following is true(1 punto) 7 They are piecewise linear functions They assume always value 1 in correspondence of all the mesh nodes They are global polynomials of degree N They are not overall continuous functions Regarding the Lagrangian basis functions for linear Finite Elements in 1D over N intervals, which sentence is true ?(1 punto) 8 21/1/2025 - MATHEMATICAL AND NUMERICAL METHODS I... https://forms.office.com/Pages/DesignPageV2.aspx?origin=NeoPorta.. . 2 di 627/01/2025, 17:03 PA RT 2 Consider the following function and text, and answer the questions below: function [x,u] = esame(a, b, h, ua, ub, f, mu, beta, phi) M = floor((b-a)/h);x = linspace(a, b, M+1)'; u = zeros(M+1, 1); Pe = abs(beta)*h/(2*mu);muh = mu*(1+phi(Pe)); e = ones(M-1, 1); Adiff = spdiags([-e, 2*e, -e], [-1, 0, 1] , M-1, M-1)/(h^2); Aadve = spdiags([-e e], [-1 1], M-1, M-1)/(2*h);A = muh*Adiff + beta*Aadve; F = f(x(2:end-1));F(1) = F(1)+ (muh/h^2 + beta/(2*h)) * ua;F(end) = F(end) + (muh/h^2 - beta/(2*h)) * ub;u(1) = ua; u (end) = ub;u (2:end-1 ) = A\F; Answer to point 2a. (2 punti) 9 Answer to point 2b reporting all the Matlab instructions introduced (1 punto) 10 Answer to point 2c reporting all the Matlab instructions introduced (1 punto) 11 Carica file Limite del numero di file: 1 Limite di dimensioni del file singolo: 10MB Tipi di file consentiti: Word, Excel, PPT, PDF, Immagine Upload a plot of the solution as requested at point 2d. Please name the file as "codice-persona_2d.png" (ex 10169517-2d.png)(2 punti) 12 Answer to point 2e(2 punti) 13 21/1/2025 - MATHEMATICAL AND NUMERICAL METHODS I... https://forms.office.com/Pages/DesignPageV2.aspx?origin=NeoPorta.. . 3 di 627/01/2025, 17:03 Carica file Limite del numero di file: 1 Limite di dimensioni del file singolo: 10MB Tipi di file consentiti: Word, Excel, PPT, PDF, Immagine Upload a plot of the solution as requested at point 2f. Please name the file as "codice-persona_2f.png" (ex 10169517-2f.png)(2 punti) 14 Answer to point 2g(2 punti) 15 21/1/2025 - MATHEMATICAL AND NUMERICAL METHODS I... https://forms.office.com/Pages/DesignPageV2.aspx?origin=NeoPorta.. . 4 di 627/01/2025, 17:03 Part 3 Carica file Limite del numero di file: 1 Limite di dimensioni del file singolo: 10MB Tipi di file consentiti: Word, PDF, Immagine Points 3a-b-c) : write the answers on paper and then give them to the teacher. Upload after photo only for online students (Erasmus,...) (12 punti) 16 21/1/2025 - MATHEMATICAL AND NUMERICAL METHODS I... https://forms.office.com/Pages/DesignPageV2.aspx?origin=NeoPorta.. . 5 di 627/01/2025, 17:03 Questo contenuto non è stato creato né approvato da Microsoft. I dati che invii verranno recapitati al proprietario del modulo. Microsoft Forms Option for withdrawin g the exam and writin g notes Please use the text below for any relevant communication to the instructor 17 Ye s No Do you want to withdraw your exam (si vuole ritirare)? 18 21/1/2025 - MATHEMATICAL AND NUMERICAL METHODS I... https://forms.office.com/Pages/DesignPageV2.aspx?origin=NeoPorta.. . 6 di 627/01/2025, 17:03