From 6e748d46b2600a80ec978fe3b95c44baedb0f6d1 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Sun, 15 Oct 2023 16:12:48 +0100 Subject: [PATCH] add exerices sheets, worked example matlab scripts --- .../nm-week1-practice-sheets-solutions.docx | Bin 0 -> 980465 bytes .../nm-week1-practice-sheets.docx | Bin 0 -> 58504 bytes .../worked_example_matlab_scripts/l1_wx1.m | 65 +++++++++++++++++ .../worked_example_matlab_scripts/l2_wx2.m | 68 ++++++++++++++++++ 4 files changed, 133 insertions(+) create mode 100644 uni/mmme/3086_computer_modelling_techniques/exercise_sheets/nm-week1-practice-sheets-solutions.docx create mode 100644 uni/mmme/3086_computer_modelling_techniques/exercise_sheets/nm-week1-practice-sheets.docx create mode 100644 uni/mmme/3086_computer_modelling_techniques/worked_example_matlab_scripts/l1_wx1.m create mode 100644 uni/mmme/3086_computer_modelling_techniques/worked_example_matlab_scripts/l2_wx2.m diff --git 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Dx/2; + +% create matrix +A = zeros(n, n); +B = zeros(n, 1); + +A(1, 1) = 1; % Dirichlet boundary +B(1) = Ta; + +A(n, n) = 1; % Dirichlet boundary +B(n) = Tb; + +for i=2:n-1 + A(i, i-1) = -conductivity/dx; + A(i, i) = 2*conductivity/dx; + A(i, i+1) = -conductivity/dx; + + B(i) = 0; +end + +T = A\B; % matrix backslash + +figure('color', 'w'); +plot(x0, T, 'rs'); +grid on; +xlabel('x [m]'); ylabel('T [K]'); +hold on; + +% comparison with theoretical solution +Tteo = Ta + (Tb-Ta)*x0/L; +plot(x0, Tteo, 'k-'); +legend('backslash', 'Exact'); +error = mean(abs(T-Tteo')) % Tteo transposed by ' operator + +% check of residuals +residual = sum(abs(B-A*T))/sum(abs(diag(A).*T)) % defined as |B-A*T|/|diag(A).T| + +% check of energy balance +num=0; den = 0; + +for i=2:n-1 + num = num+abs(-conductivity*(T(i)-T(i-1))/dx+conductivity*(T(i+1)-T(i))/dx); % sum(|q_w-q_e|) + den = den+abs(conductivity*(T(i)-T(i-1))/dx); % sum(|q_w|) +end + +unbalance = num/den*100 \ No newline at end of file diff --git a/uni/mmme/3086_computer_modelling_techniques/worked_example_matlab_scripts/l2_wx2.m b/uni/mmme/3086_computer_modelling_techniques/worked_example_matlab_scripts/l2_wx2.m new file mode 100644 index 0000000..77d7d58 --- /dev/null +++ b/uni/mmme/3086_computer_modelling_techniques/worked_example_matlab_scripts/l2_wx2.m @@ -0,0 +1,68 @@ +% author: Akbar Alvi Rahman +% Lecture 1 Worked Example 1 (Section 10.1) + +clear all; close all; clc + +% define physical constants +L = 1; +S = 0; +conductivity = 400; +Ta = 300; +Tb = 320; + +Sp = -100; +Sc = 5000; + +% define numerical parameters +n = 21; + +% grid generation +x0 = linspace(0, L, 21); +dx = L/(n-1); +Dx = dx; +DxB = Dx/2; + +% create matrix +A = zeros(n, n); +B = zeros(n, 1); + +A(1, 1) = 1; % Dirichlet boundary +B(1) = Ta; + +A(n, n) = 1; % Dirichlet boundary +B(n) = Tb; + +for i=2:n-1 + A(i, i-1) = -conductivity/dx; + A(i, i) = 2*conductivity/dx-Sp*Dx; + A(i, i+1) = -conductivity/dx; + + B(i) = Sc*Dx; +end + +T = A\B; % matrix backslash + +figure('color', 'w'); +plot(x0, T, 'rs'); +grid on; +xlabel('x [m]'); ylabel('T [K]'); +hold on; + +% comparison with theoretical solution +Tteo = Ta + (Tb-Ta)*x0/L; +plot(x0, Tteo, 'k-'); +legend('backslash', 'Exact'); +error = mean(abs(T-Tteo')) % Tteo transposed by ' operator + +% check of residuals +residual = sum(abs(B-A*T))/sum(abs(diag(A).*T)) % defined as |B-A*T|/|diag(A).T| + +% check of energy balance +num=0; den = 0; + +for i=2:n-1 + num = num+abs(-conductivity*(T(i)-T(i-1))/dx+conductivity*(T(i+1)-T(i))/dx); % sum(|q_w-q_e|) + den = den+abs(conductivity*(T(i)-T(i-1))/dx); % sum(|q_w|) +end + +unbalance = num/den*100 \ No newline at end of file