%
% EFG1D - ONE DIMENSIONAL Galerkin-based MESHLESS PROGRAM FOR SOLVING A 1D BAR
% SUBJECTED TO A LINEAR BODY FORCE OF MAGNITUDE X WHOSE EXACT SOLUTION IS GIVEN BY
% u = \frac{50}{3} (x - x^3)/E, \sigma = \frac{50}{3}*(1-3x^2)
%
% BACKGROUND CELL QUADRATURE IS EMPLOYED TO EVALUATE INTEGRALS
% - CELLS ARE COINCIDE WITH THE INTERVALS BETWEEN THE NODES
%
% LAGRANGIAN MULTIPLIER METHOD IS EMPLOYER TO IMPOSE THE ESSENTIAL BOUNDARY CONDITIONS
clear all
% SET UP GLOBAL CONTROL PARAMETERS
scale = 2.4; % Scale used to determine the radius of support for nodes
nint = 3; % Order of Gauss quadrature
dx = 0.1; % Distance between adjacent nodes
base = 2; % Basis type - 1: Constant basis; 2: Linear basis; 3: Quadratic basis
type = 2; % Type of quadrature
% 1: Gauss quadrature; 2: Nodal quadrature; 3: Particle quadrature
WeightType = 'SPLIN'; % Type of weight function ('GAUSS', 'QUART', 'SPLIN','CSRBF')
% SET UP NODAL COORDINATES ALONG BAR, DETERMINE NUMBER OF CELLS
L = 1.0; % Length of the bar
xi = [0.0 : dx : L]; % Nodal coordinates
nnodes = length(xi);
ncells = nnodes-1;
% SET MATERIAL PROPERITES
E = 1.0; % Elastic modulus
area = 1.0; % Area of cross section
% DETERMINE RADIUS OF SUPPORTS FOR EACH NODE
dm = scale*dx*ones(1,nnodes);
% INITIALIZE MATRICES
K = zeros(nnodes);
P = zeros(nnodes,1);
G = zeros(nnodes,2);
% ----------------------------- Quadrature ----------------------------------------
if type == 1 % Gauss quadrature
% LOOP OVER CELLS
for i = 1:ncells
x1 = xi(i); % Left point of cell i
x2 = xi(i+1); % Right point of cell i
x0 = (x1+x2)/2; % Coordinate of the mid-point of cell i
h = x2-x1; % Length of cell i
jac = h/2; % Jacobian for cell i
[r,w] = Gauss(nint); % Natural coordinates of Gauss quadrature points and weights
% LOOP OVER GAUSS POINTS
for j = 1:nint
xq = x0 + h*r(j)/2; % Coordinates of Gauss quadrature points
% EVALUATE SHAPE FUNCTIONS AND THEIR DERIVATIVES AT GAUSS POINT xg
