ME 582 Handout 11 1D Unsteady Code and Sample Input File

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1 METU Mechanical Engineering Department ME 582 Finite Element Analysis in Thermofluids Spring 2018 (Dr. Sert) Handout 11 1D Unsteady Code and a Sample Input File Download the complete code and the sample input files from the course web site. Explanations given below focus on the parts that are different compared to the steady code. % unsteady1d.m % % Middle East Technical University % % Department of Mechanical Engineering % % ME 582 Finite Element Analysis in Thermofluids % % % % Author : Dr. Cuneyt Sert PDE that we solve % is % csert@metu.edu.tr % % u % % Last Update : 17/04/2018 t u u %(a ) + b + cu = f x x x Important: a, b, c and f are assumed to be % functions of x only, % not time. % FEATURES and LIMITATIONS % Warning: There is already a minus sign in % This code % % - is written for educational purposes. front of the first % term. So be careful in proving % - solves the following model ODE in a 1D domain. the function a. % % du/dt - d/dx [a(x) du/dx] + b(x) du/dx + c(x) * u = f(x) % % with a(x), b(x), c(x) and f(x) being known functions and u(x) being % % the unknown. Note that none of the functions dep on time. % % - uses Galerkin Finite Element Method (GFEM). % % - uses linear or quadratic Lagrange type elements. % % - uses theta-family schemes (1st order forward Euler, 1st order % % backward Euler or Crank-Nicolson % % - uses uniform mesh, i.e. all elements have the same length. % % - does not support time depent boundary condtions. % % - uses 3 point Gauss Quadrature integration. % % - stores [M] and [K] as full matrices. % % HOW TO USE? % % Change the initialcond function. % % Prepare an input file with the extension.inp and provide the name of % % the file (without the extension) when asked for. % % VARIABLES % % xmin : Minimum x value of the problem domain % % xmax : Maximum x value of the problem domain % % NE : Number of elements % % NN : Number of nodes % % NEN : Number of element nodes (2 or 3) % % coord : Coordinates of mesh nodes. Array of length NN % % LtoG : Local to global node mapping matrix NExNEN % % NGQP : Number of GQ integration points (= 3 in this code) % 11-1

2 % GQksi : Coordinates of GQ points. Array of size NGQP % % GQW : Weights of GQ points. Array of size NGQP % % prob : Structure that holds a(x), b(x), c(x) and d(x) functions % % of the DE. % % S : Shape functions evaluated at GQ points. Matrix of size % % NENxNGQP % % ds : Derivatives of shape functions wrt to ksi evaluated at % % GQ points. Matrix of size NENxNGP % % BC : Structure for boundary conditions % % - leftbc : BC type at the left boundary. 1:EBC, 2:NBC, 3:MBC % % - leftpv : PV at the left boundary, if known % % - leftsv : SV at the left boundary, if known % % - leftbeta : Beta value at the left boundary, if known % % - leftgamma : Gamma value at the left boundary, if known % % - rightbc : BC type at the right boundary. 1:EBC, 2:NBC, 3:MBC % % - rightpv : PV at the right boundary, if known % % - rightsv : SV at the right boundary, if known % % - rightbeta : Beta value at the right boundary, if known % % - rightgamma: Gamma value at the right boundary, if known % % elem : Structure for elements % % - M : Elemental mass matrix of size NENxNEN % % - K : Elemental stiffness matrix of size NENxNEN % % - F : Elemental force vector of size NENx1 % % - h : Element's length % % - J : Element's Jacobian = h/2 % % time : Structure for time integration % A new structure called time, % - s : Current time level % % - t : Current time holds % time related variables. % - dt : Time step (constant) % % - initial : Initial time % % - final : Final time % % - theta : Parameter used to select a time integration scheme % % 0.0: 1st order forward Euler % % 0.5: Crank Nicolson % % 1.0: 1st order backward Euler % % - nupdate : Solution plot will be refreshed at every nupdate steps % % - pause : Amount in seconds that the animation pauses between % % updates % % M : Global mass matrix of size NNxNN % % K : Global stiffness matrix of size NNxNN % % F : Global force vector of size NNx1 % % U : Global unknown vector of size NNx1 % % FUNCTIONS % % unsteady1d : Main function % % readinputfile : Reads the input file % % generatemesh : Generates the 1D mesh % % setupgq : Generates variables for GQ points and weights % % calcshape : Evaluates shape functions and their derivatives at % % GQ points % % calcglobalsys : Calculates global M, K and F by the assembly of % % elemental ones % % calcelemsys : Calculates elemental M, K and F % % applybc : Applies BCs to the global system of equations % % initialcond : Initializes the solution with the specified initial initialcond % is a new function. % condition. This part is problem depent % % solve : Solves the global system in a time loop % 11-2

3 function [] = unsteady1d clc; % Clear MATLAB's command window clear all; % Clear variables and functions in the memory close all; % Close figure windows disp('**********************************************************'); disp('** Middle East Technical University **'); disp('** Department of Mechanical Engineering **'); disp('** ME 582 Finite Element Analysis in Thermofluids **'); disp('** **'); disp('** **'); disp('** 1D, Unsteady Solver **'); disp('**********************************************************'); readinputfile(); generatemesh(); setupgq(); calcshape(); calcglobalsys(); solve(); fprintf('\nprogram finished successfully.\n'); % End of function unsteady1d() Warning: [M], [K] and {F} are assumed to be time indepent and they are calculated once outside the time loop. solve function has the main time loop in it. It calls the initialcond and applybc functions. function readinputfile % Asks for the problem name and reads the input file. % Variables defined as global can be used in other functions too. time.dt = fscanf(inputfile, 'dt :%f'); fgets(inputfile); time.initial = fscanf(inputfile, 't0 :%f'); fgets(inputfile); time.final = fscanf(inputfile, 'tfinal :%f'); fgets(inputfile); time.theta = fscanf(inputfile, 'theta :%f'); fgets(inputfile); time.nupdate = fscanf(inputfile, 'nupdate :%d'); fgets(inputfile); time.pause = fscanf(inputfile, 'pause :%f'); fgets(inputfile); Time related new inputs. Other lines are the same as the steady code. % End of function readinputfile() function generatemesh % Coordinates of mesh nodes are calculated. LtoG matrix is formed. Exactly the same as the steady code. % End of function generatemesh() 11-3

4 function setupgq % Sets up coordinates and weights of GQ points Exactly the same as the steady code. % End of function setupgq() function calcshape() % Calculates the values of the shape functions and their derivatives with % respect to ksi coordinate at GQ points. Exactly the same as the steady code. % End of function calcshape() function calcglobalsys() % Calculates the global mass matrix [K], stiffness matrix [K] and the % force vector {F}. All these are assumed to be time indepent. global NE NN M K F LtoG elem; M = zeros(nn,nn); K = zeros(nn,nn); F = zeros(nn,1); for e = 1:NE calcelemsys(e); % M and K are NNxNN square matrices. In this 1D code they % are stored as a full matrices without memory concerns. % F is a NNx1 vector % Calculate elemental M, K and F New mass matrix % Assemble Me into M, Ke into K and Fe into F. M(LtoG(e,:), LtoG(e,:)) = M(LtoG(e,:), LtoG(e,:)) + elem(e).m(:,:); K(LtoG(e,:), LtoG(e,:)) = K(LtoG(e,:), LtoG(e,:)) + elem(e).k(:,:); F(LtoG(e,:)) = F(LtoG(e,:)) + elem(e).f(:); Assembly of the mass matrix % End of function calcglobalsys() function calcelemsys(e) % Calculates the element level mass matrix, stiffness matrix and force vector. global prob elem NEN LtoG NGQP GQksi GQW coord S ds; elem(e).f = zeros(nen,1); elem(e).m = zeros(nen,nen); elem(e).k = zeros(nen,nen); Elemental mass matrix 11-4

5 for i = 1:NEN for j = 1:NEN elem(e).m(i,j) = elem(e).m(i,j) + S(i,k) * S(j,k) * elem(e).j * GQW(k); elem(e).k(i,j) = elem(e).k(i,j) + (avalue * ds(i,k)/elem(e).j * ds(j,k)/elem(e).j + bvalue * S(i,k) * ds(j,k)/elem(e).j + cvalue * S(i,k) * S(j,k) ) * elem(e).j * GQW(k); Calculate the elemental mass matrix. for i = 1:NEN elem(e).f(i) = elem(e).f(i) + fvalue * S(i,k) * elem(e).j * GQW(k); % End of GQ loop % End of function calcelemsys() function [] = solve() % Main time loop of the solution. global M K F U Kcap Fcap NN time; New function % Initialize the solution using the specified initial condition. time.t = time.initial; time.s = 0; initialcond(); updateplot(); %hold on; Uncomment this to see the results of intermediate time steps. Set the initial condition. Important: For each problem the correct I.C. needs to be defined inside this Kcap = zeros(nn,nn); Mbar = zeros(nn,nn); Fcap = zeros(nn,1); Fbar = zeros(nn,1); Warning: Excessive memory allocation. Kcap = M + time.theta * time.dt * K; Mbar = M - (1 - time.theta) * time.dt * K; Fbar = F; At each time step we solve the following system where [K] = [M] + θ t[k]s+1 [K] {φ} s+1 = {F}, {F} = [M] {φ} s + t{f} [M] = [M] (1 θ) t[k]s while (time.final - time.t > 1e-10) time.s = time.s + 1; time.t = time.t + time.dt; Fcap = Mbar * U + time.dt * Fbar; Time loop {F} = θ{f}s+1 + (1 θ){f} s Warning: In this code we assumed that [K] and {F} are time indepent. applybc(); % Calculate the solution at the new time step. U = Kcap \ Fcap; {F} deps on {φ} s, therefore it is updated inside the time loop. if mod(time.s, time.nupdate) == 0 updateplot(); time.t % End of function timeloop() Update the solution plotat every nupdate time steps. Boundary conditions need to be applied inside the time loop at each time step. 11-5

6 function applybc() % K and F are modified according to the given BCs global NN Kcap Fcap BC; Important: Very similar to the steady code, but instead of modifying [K] and {F}, we modify [K] and {F} % End of function applybc() function [] = updateplot() global NN coord U time; plot(transpose(coord(1:nn)), U, 'black', 'LineWidth', 2); axis([coord(1) coord(nn) ]); Warning: Change the axis limits properly if these default values are not suitable for your problem. grid on; title(['s = ' num2str(time.s), ', pause(time.pause); drawnow; % End of function updateplot() t = ' num2str(time.t)]); Show the current time and time step on the title of the function [] = initialcond() % Apply initial condition. This function is problem depent. global NN coord U; U = zeros(nn,1); Important: Change this I.C. function for each problem. for i = 1:NN x = coord(i); U(i) = 0; %U(i) = 1-x^2; % Heated bar example % Cooling bar example I.C. of Example 6 of Handout 9. I.C. of Example 1 of Handout 9. %if (x < 0.4) % U(i) = 0.5 * (1 + cos(5*pi*(x-0.2))); % Cosine hill example %else % U(i) = 0.0; % This inactive part is the initial condition of Example 7 of Handout 9. % End of function initialcond() This initial condition is for Example 1 of Handout 9. function [f] = heating(x) if x >= -0.1 && x <= 0.1 f = 4; else f = 0; % End of function heating() This function provides the special f(x) of Example 6 of Handout

7 Sample input file for the problem solved in Example 1 of Handout 9 Warning: The format of input files is very strict. Only change the entries after the colons. Do not change the variables names that appear before the colons. Do not change the places of the colons. Do not remove the dummy lines that are used for clarity. Example 1 of Handout 9 (Spring 2018) ================================================ NE : 5 NEN : 2 xmin : -1 xmax : 1 dt : 0.01 t0 : 0 tfinal : 1 theta : 1.0 nupdate : 1 pause : 0 afunc : 1 bfunc : 0 cfunc : 0 ffunc : 0 New time related inputs. ================================================ leftbctype : 1 leftpv : 0.0 leftsv : 0.0 leftbeta : 0.0 leftgamma : 0.0 ================================================ rightbctype : 1 rightpv : 0.0 rightsv : 0.0 rightbeta : 0.0 rightgamma : 0.0 When you download unsteady1d.m code it comes with two more sample input files corresponding to problems of Handout

Download the complete code and the sample input files from the course web site. DE that we solve is

METU Mechanical Engineering Department ME 582 Finite Element Analysis in Thermofluids Spring 2018 (Dr. Sert) Handout 4 1D FEM Code and a Sample Input File Download the complete code and the sample input