efg1d.m

一篇优秀的无网格伽辽金方法入门文章。这篇文章最好的地方在于——提供了matlab源码。方便大家了解和学习无网格的编程实现

% ONE DIMENSIONAL EFG PROGRAM
% SET UP NODAL COORDINATES ALONG BAR, DETERMINE NUMBER OF CELLS
x = [0.0:.1:1.0];
nnodes = length(x);
ncells = nnodes-1;

% SET PARAMETERS FOR WEIGHT FUNCTION, MATERIAL PROPERITES
dmax = 2.0;
E=1.0; area=1.0;

% DETERMINE DMI FOR EACH NODE
dm = dmax*(x(2)-x(1))*ones(1,nnodes);

%SET UP GAUSS POINTS, WEIGHTS, AND JACOBIAN FOR EACH CELL
gg = zeros(1,ncells);
jac = (x(2)-x(1))/2;
weight = 2;
gg = -.05:.1:0.95; gg(1) = 0.0;

% INITIALIZE MATRICES
k = zeros(nnodes);
f = zeros(nnodes,1);
GG = zeros(nnodes,1);

% LOOP OVER GAUSS POINTS
for j = 1:length(gg)
    xg = gg(j);

    % DETERMINE DISTANCE BETWEEN NODES AND GAUSS POINT
    dif = xg*ones(1,nnodes)-x;;

    % SET UP WEIGHTS W AND DW FOR EACH NODE
    clear w dw
    for i=1:nnodes
        drdx = sign(dif(i))/dm(i);
        r = abs(dif(i))/dm(i);
        if r<=0.5
            w(i) = (2/3) - 4*r*r + 4*r^3;
            dw(i) = (-8*r + 12*r^2)*drdx;
        elseif r<=1.0
            w(i) = (4/3)-4*r+4*r*r -(4/3)*r^3;
            dw(i) = (-4 + 8*r-4*r^2)*drdx;
        elseif r>1.0
            w(i) = 0.0;
            dw(i) = 0.0;
        end
    end

    %SET UP SHAPE FUNCTIONS AND DERIVATIVES
    won = ones(1,nnodes);
    p = [won;x];
    B = p.*[w;w];
    pp = zeros(2);
    A = zeros(2);
    dA = zeros(2);
    for i=1:nnodes
        pp = p(1:2,i)*p(1:2,i)';
        A = A+w(1,i)*pp;
        dA = dA+dw(1,i)*pp;
    end
    Ainv = inv(A);
    pg = [1 xg];
    phi = pg*Ainv*B;
    db = p.*[dw;dw];
    da = -Ainv*(dA*Ainv);
    dphi = [0 1]*Ainv*B+pg*(da*B+Ainv*db);

    %ASSEMBLE DISCRETE EQUATIONS
    if j == 1
        GG(1:3,1) = -phi(1:3)';
    else if j>1
            k = k+(weight*E*area*jac)*(dphi'*dphi);
            fbody = area*xg;
            f = f+(weight*fbody*jac)*phi';
        end
    end
end

% ENFORCE BOUNDARY CONDITIONS USING LAGRANGE MULTIPLIERS
q = [0];
m = [k GG;GG' zeros(1)];
% SOLVE FOR NODAL PARAMETERS
d = m\[f' q]';
u = d(1:nnodes);