Your Question:

Write a program (use MATLAB) that accepts user input for the following parameters: the total num- ber of elements, the endpoints of each element, the degree of the polynomial, and information regarding boundary conditions. This program's main tasks are to construct the matrix K and the load vector F. Here is an overview of the program's structure:

function [K.F]-buildkf(nelem,h,deg.alpha,beta,gamma)

% alpha,beta.gamma-vectors of length 2 which determine the

% boundary conditions: alpha(1) u'(0)+beta(1) u(0)=gamma(1) % alpha (2) u'(1)+beta(2)*u(1)=gamma(2)

correct K and F for boundary conditions

You should create separate external functions for the coefficients k, b, c, and f, and include a Gauss- quadrature routine to handle the integration process. Additionally, consider developing functions that generate the shape functions and their derivatives on the master element.

Step By Step Answers with Explanation

function k = coefficient_k(x)

r = 0; % You can change r as needed

end

function c = coefficient_c(x)

end

Define the Gauss Quadrature Routine:

% a, b - integration limits

% n - number of quadrature points

x_i = 0.5 * (b - a) * x(i) + 0.5 * (a + b);

w_i = 0.5 * (b - a) * w(i);

Create functions to generate the shape functions and their derivatives on the master element.

function [N, dNdx] = shape_functions(x, deg)

if deg == 1

N = [0.5 * (1 - x); 0.5 * (1 + x)];

else

error('Degree not supported.');

function [K, F] = buildkf(nelem, h, deg, alpha, beta, gamma)

% Initialize global stiffness matrix K and load vector F

% Get the element endpoints

a = (e - 1) * h(e);

for i = 1:(deg + 1)

for j = 1:(deg + 1)

fe(i) = fe(i) + gauss_quadrature(@(x) source_term(x) * shape_functions(x, deg)(i), a, b, 5);

end

end

end

nelem = 8;

h = ones(1, nelem) * (1 / nelem);

[K, F] = buildkf(nelem, h, deg, alpha, beta, gamma);

% Display the matrix K and the load vector F

This program will construct the stiffness matrix K and the load vector F as per the given problem, including shape functions, integration using Gauss quadrature, and handling boundary conditions. You can modify the coefficients and boundary conditions as needed for different problems.