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Project/Main.m
298
Project/Main.m
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% Axisymmetric mug
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% r–z plane, rotation around r = 0
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clear; clc; close all;
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%% General values that we use in the entire script
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maxIter = 5000;
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tol = 1e-6;
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%Task 3) Thermal conductivity
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lambda_wall = 2.0; % ceramic RANDOM NUMBER
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lambda_fluid = 0.6; % water RANDOM NUMBER
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lambda_air = 0.025; % air RANDOM NUMBER
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%Task 4)
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alpha = 10; % heat transfer coefficient
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u_out = 18; % ambient air temperature (°C)
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%Task 6) Heat capacities (example values)
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c_wall = 900; % ceramic RANDOM NUMBER
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c_fluid = 4180; % water RANDOM NUMBER
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c_air = 1005; % air RANDOM NUMBER
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% This numbers don't simulate real life because in real life we have
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% volumetric heat capacities. this means we are having a reduced factore of
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% approx 10^4.
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%% Task 1: Mesh definition
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% Create PDE model
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model = createpde();
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% Points (meters)
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A = [0, 0];
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B = [0.055, 0];
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C = [0.083, 0.105];
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H = [0.078, 0.105];
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F = [0.050, 0.005];
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E = [0, 0.005];
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G = [0.067, 0.066];
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I = [0, 0.066];
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D = [0, 0.105];
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% Geometry matrix (edges)
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% Axis (split by material)
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g1 = [2; A(1); E(1); A(2); E(2); 1; 0]; % ceramic
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g2 = [2; E(1); I(1); E(2); I(2); 2; 0]; % fluid
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g3 = [2; I(1); D(1); I(2); D(2); 3; 0]; % air
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% Outer ceramic
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g4 = [2; A(1); B(1); A(2); B(2); 1; 0];
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g5 = [2; B(1); C(1); B(2); C(2); 1; 0];
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% Top rim: C -> H is ceramic-air (ONLY)
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g6 = [2; C(1); H(1); C(2); H(2); 1; 3];
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% Inner ceramic wall (H -> F) ceramic-air
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g7 = [2; H(1); F(1); H(2); F(2); 1; 3];
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% Inner ceramic bottom (F -> E) ceramic-fluid
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g8 = [2; F(1); E(1); F(2); E(2); 1; 2];
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% Fluid surface (F -> G) fluid-air
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g9 = [2; F(1); G(1); F(2); G(2); 2; 3];
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% Fluid surface (G -> I) fluid-air
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g10 = [2; G(1); I(1); G(2); I(2); 2; 3];
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% Air top boundary: D -> H (air-outside)
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g11 = [2; D(1); H(1); D(2); H(2); 3; 0];
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% Assemble geometry
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g = [g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11];
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% Load geometry into PDE model
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geometryFromEdges(model, g);
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% Plot geometry with labels
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% figure(1);
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% pdegplot(model, 'EdgeLabels','on', 'FaceLabels','on');
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% axis equal;
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% title('Geometry with edge and face labels');
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% Generate mesh
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mesh = generateMesh(model, 'Hmax', 0.006, 'GeometricOrder','linear');
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% Plot mesh
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% figure(2);
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% pdemesh(model);
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% axis equal;
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% title('Generated mesh');
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%% Task 2: Jacobi solver with constant lambda
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nodes = mesh.Nodes; % coordinates of all mesh nodes
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elements = mesh.Elements; % which nodes make up each triangle element
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Nnodes = size(nodes,2);
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Nelems = size(elements,2);
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% Define material properties (for simplicity, lambda = 1 everywhere)
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lambda = ones(Nelems,1); % thermal conductivity
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% Initialize global stiffness matrix and RHS
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K = sparse(Nnodes, Nnodes);
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F = zeros(Nnodes,1);
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% Assemble K and F
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for e = 1:Nelems %Loop over each triangle element
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vert = elements(:,e); %nodes of element
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x = nodes(1,vert);
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y = nodes(2,vert);
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Ae = polyarea(x,y); % Compute area of the triangle
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% Linear triangle gradients
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b = [y(2)-y(3); y(3)-y(1); y(1)-y(2)]; % derivative with respect to x
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c = [x(3)-x(2); x(1)-x(3); x(2)-x(1)]; % derivative with respect to y
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% Element stiffness matrix
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Ke = (lambda(e)/(4*Ae)) * (b*b.' + c*c.');
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% Assemble
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K(vert,vert) = K(vert,vert) + Ke;
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% Element load vector (f=0)
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F(vert) = F(vert) + zeros(3,1);
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end
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% Find boundary nodes (couldn't find a better way)
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edgesAll = [elements([1 2],:), elements([2 3],:), elements([3 1],:)]; % all edges -> 2x(3*Nelems) awway since each triangle has 3 edges
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edgesSorted = sort(edgesAll,1); % sort nodes of each edge, ensure that [i,j] and [j,i] are the same
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[~,~,ic] = unique(edgesSorted','rows'); % identifies unique edges and assignes indices ic
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counts = accumarray(ic,1); % counts how many times each edge appears in the mesh
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boundaryEdges = find(counts==1); %edges belonging to only 1 element: hence on the boundary
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boundaryNodes = unique(edgesSorted(:,boundaryEdges)); % nodes belonging to these boundary edges
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% Jacobi iteration
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D = diag(K);
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R = K - diag(D);
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u = zeros(Nnodes,1);
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for iter = 1:maxIter
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u_new = (F - R*u) ./ D;
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u_new(boundaryNodes) = 0; % Enforce Dirichlet BC
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if norm(u_new - u, inf) < tol % Convergence check
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fprintf('Task 2) Jacobi converged in %d iterations.\n', iter);
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break;
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end
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u = u_new;
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end
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% % Plot solution
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% figure(3)
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% pdeplot(model, 'XYData', u, 'Mesh','on');
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% axis equal;
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% title('Stationary Dirichlet solution via Jacobi');
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% colorbar;
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%% Task 3: Laplace with multiple lambdas
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[K, F] = CalculateLaplace_mult(model, lambda_wall, lambda_fluid, lambda_air);
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D = diag(K);
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R = K - diag(D);
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u = zeros(Nnodes,1);
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for iter = 1:maxIter
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u_new = (F - R*u) ./ D;
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u_new(boundaryNodes) = 0; % Dirichlet BC
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if norm(u_new - u, inf) < tol
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fprintf('Task 3) Jacobi converged in %d iterations.\n', iter);
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break;
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end
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u = u_new;
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end
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% figure(4)
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% pdeplot(model, 'XYData', u, 'Mesh','on');
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% axis equal
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% title('Task 3: Stationary solution with multiple conductivities');
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% colorbar
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%% Task 4: Robin boundary condition
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[K, F] = ApplyRobinBC_mult(model, K, F, alpha, u_out);
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D = diag(K);
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R = K - diag(D);
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u = zeros(Nnodes,1);
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for iter = 1:maxIter
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u_new = (F - R*u) ./ D;
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if norm(u_new - u, inf) < tol
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fprintf('Task 4) Jacobi (Robin BC) converged in %d iterations.\n', iter);
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break;
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end
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u = u_new;
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end
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figure(5)
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pdeplot(model, 'XYData', u, 'Mesh','on');
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axis equal
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title('Task 4: Stationary solution with Robin BC');
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colorbar
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%% Task 5: Axisymmetric Laplace + Robin BC
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[K, F] = CalculateLaplace_mult_rot(model, lambda_wall, lambda_fluid, lambda_air);
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[K, F] = ApplyRobinBC_mult_rot(model, K, F, alpha, u_out);
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D = diag(K);
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R = K - diag(D);
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u = zeros(Nnodes,1);
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for iter = 1:maxIter
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u_new = (F - R*u) ./ D;
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if norm(u_new - u, inf) < tol
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fprintf('Task 5) Jacobi (axisymmetric Robin) converged in %d iterations.\n', iter);
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break;
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end
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u = u_new;
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end
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% Plot solution
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% figure(6)
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% pdeplot(model, 'XYData', u, 'Mesh','on');
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% axis equal
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% title('Task 5: Axisymmetric stationary solution with Robin BC');
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% colorbar
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%To see it in 3D paste here the code in "AdditionalPlotCodes.txt".
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%% Task 6: Axisymmetric mass matrix
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M = sparse(Nnodes, Nnodes);
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M = AddMass_mult_rot(model, M, c_wall, c_fluid, c_air);
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%% Task 7: Initial solution
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u0 = Init_Solution_mult(model, 18, 80, 18);
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% figure(7)
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% pdeplot(model, 'XYData', u0, 'Mesh','on');
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% axis equal
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% title('Initial temperature distribution');
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% colorbar
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%% Task 8: Time-dependent simulation (explicit scheme)
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tau = 0.5; % time step in seconds
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T_end = 400; % total simulation time (seconds)
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Nt = ceil(T_end/tau); % number of time steps
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A = (1/tau)*M+K; % Left-hand side matrix
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[L,U,P,Q] = lu(A); % A is constant -> factorize it once: PAQ=LU
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% Initialize solution
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u = u0;
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for k = 1:Nt
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b = (1/tau)*M*u + F; % F is the load vector, F=0
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% Solve for next time step
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u_next = Q*(U\(L\(P*b))); % efficient solution using LU factors
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u = u_next; % Update
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if mod(k,20) == 0
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% figure(8)
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% pdeplot(model, 'XYData', u, 'Mesh','on');
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% axis equal
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% title(['Temperature at t = ', num2str(k*tau), ' s']);
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% colorbar
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% drawnow
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end
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end
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%To see the 9 snapshots paste here the codes in "AdditionalPlotCodes.txt"
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%% Task 9 (i): Heating time using inner ceramic wall temperature
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T_target = 67; % [°C]
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innerWallNodes = findNodes(model.Mesh,'region','Edge',8); % Edge 8 = ceramic–fluid interface
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u = u0;
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% Storage
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timeVec = (0:Nt-1)' * tau;
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innerWallTemp = zeros(Nt,1);
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Twarm = NaN;
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for k = 1:Nt
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b = (1/tau)*M*u + F;
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u = Q*(U\(L\(P*b)));
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% Average inner wall temperature
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innerWallTemp(k) = mean(u(innerWallNodes));
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% Check heating criterion
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if innerWallTemp(k) >= T_target
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Twarm = k * tau;
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fprintf('Task 9 (i): Inner wall reaches %.1f°C at T = %.1f s\n', ...
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T_target, Twarm);
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break
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end
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end
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%% --- Plot inner wall temperature evolution
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figure(9)
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plot(timeVec(1:k), innerWallTemp(1:k), 'LineWidth', 2)
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hold on
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yline(T_target,'r--','67°C','LineWidth',1.5)
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xlabel('Time [s]')
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ylabel('Average inner wall temperature [°C]')
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title('Heating of the inner ceramic wall')
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grid on
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