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Main_ArcLength.m
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% C0 Finite element Formulation
% Geometrically Nonlinear Analysis of Laminated Composite Plate
%
% Problem : To find the Central(Maximum) bending deflection and
% stresses of composite plate when sinusoidally distributed
% transverse load sin((pi/a)*x)*sin((pi/b)*y)
%
% Origin is assumed to be at centre of the plate
% (-L/2,B/2) (0,B/2) (L/2,B/2)
% (-L/2,0) (0,0) (L/2,0)
% (-L/2,-B/2) (0,-B/2) (L/2,-B/2)
% By doing this only we need to change the force component
% ( replace 'sin' term with 'cos' term)
%
% y_new
% y ^
% ^ |
% | |
% | L/2 |
% |<-----> o--------> x_new
% | ^
% | |B/2
% | |
% | v
% o---------------------------->x
% x = L/2 + x_new , y = B/2 + new
% sin((pi/L)*x) = cos((pi/L)*x_new)
% sin((pi/B)*y) = cos((pi/B)*y_new)
%
% Layer layout
% Upper layer (3) (z axis positive)
% Middle layer (2) (z = 0)
% Lower layer (1) (z axis negative)
%%
folder = 'plots';
if(~exist(folder,'dir'))
mkdir(folder)
end
clc; % clear command windows
close all; % close all pop up windows
clear all; % clear all variables in Workspace
format shortG;
fprintf('Please wait programme is under run...\n');
%%
%------------------------
% Problem_L_1 : 0/90/90/0
%------------------------
% Parameter.Load = 'UDL';
% Parameter.Load = 'SSL';
% Parameter.gamma = 1e7; % Penalty Parameter
% Parameter.NL = 0; % Linear : 0; NonLinear : 1
% Parameter.Type = 'Linear';
% Parameter.Type = 'vonKarman';
% Parameter.Type = 'Green';
% Parameter.Theory = 'Ferreira';
% Parameter.Theory = 'IHSDT';
% Parameter.Theory = 'TSDT';
% Parameter.typeBC = 'SSSS1';
% Parameter.typeBC = 'SSSS2';
% E2 = 1;
% E1 = 25*E2;
% NU12 = 0.25;
% G12 = 0.5*E2;
% G13 = 0.5*E2;
% G23 = 0.2*E2;
% NU21 = (E2/E1)*NU12;
% rho = 1;
% E2 = 1;
% E1 = 40*E2;
% NU12 = 0.25;
% G12 = 0.6*E2;
% G13 = 0.6*E2;
% G23 = 0.5*E2;
% NU21 = (E2/E1)*NU12;
% rho = 1;
% E2 = 1;
% E1 = E2;
% NU12 = 0.25;
% G12 = 0.5*E2/(1+NU12);
% G13 = G12;
% G23 = G12;
% NU21 = (E2/E1)*NU12;
% rho = 1;
% Lx = 1;
% Ly = Lx;
% h = Lx/4;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
% theta = [0;90;90;0;];
% theta = [0;];
% theta = [0;90;];
% theta = [0;90;0;90;0;90];
% thetaAngle = 5;
% theta = [thetaAngle; -thetaAngle;];
% theta = [thetaAngle; -thetaAngle; thetaAngle; -thetaAngle; thetaAngle; -thetaAngle];
% LoadFactor = 1;
% TotalLoad = 1/LoadFactor;
% % LoadStep = 1;
% DiscreateLoad = unique(1)/LoadFactor;
% LoadStep = length(DiscreateLoad);
%%
%---------------------------
% Problem_NL_1 : 0 Isotropic (Backup of Problem : 3)
%---------------------------
% % Parameter.Load = 'UDL';
% Parameter.Load = 'SSL';
% Parameter.gamma = 1e7; % Penalty Parameter
% % Parameter.NL = 1; % Nonlinearity Parameter
% % Parameter.Type = 'Linear';
% % Parameter.Type = 'vonKarman';
% Parameter.Type = 'Green';
% Parameter.Theory = 'IHSDT';
% % Parameter.Theory = 'TSDT';
% Parameter.typeBC = 'CCCC'; % Boundary Condition
%
% Material Properties
% E2 = 206.84e9;
% E1 = E2;
% NU12 = 0.316;
% G12 = 206.84e9/(2+2*NU12);
% G13 = G12;
% G23 = G12;
% NU21 = (E2/E1)*NU12;
% rho = 1;
%
% Plate Geometry
% Lx = 300;
% Ly = Lx;
% h = 3;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
%
% Ply Orientation
% theta = 0;
%
% LoadFactor = (Lx^4)/(E2*h^4);
% % TotalLoad = 400/LoadFactor;
% % LoadStep = 16;
% DiscreateLoad = unique([20:20:100,100:50:400])/LoadFactor;
% LoadStep = length(DiscreateLoad);
%
% file = sprintf('Poblem_NL_1_%s_%s_h_a_%g_%g_%s_%s',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC)
% save(fullfile(pwd,'plots',[file,'.mat']),'Data_W','Data_xx');
%%
%---------------------------
% Problem_NL_3
%---------------------------
% Parameter.Load = 'UDL';
% % Parameter.Load = 'SSL';
% Parameter.gamma = 1e7; % Penalty Parameter
% % Parameter.NL = 1; % Nonlinearity Parameter
% % Parameter.Type = 'Linear';
% % Parameter.Type = 'vonKarman';
% Parameter.Type = 'Green';
% Parameter.Theory = 'IHSDT';
% % Parameter.Theory = 'TSDT';
% % Parameter.typeBC = 'SSSS1';
% Parameter.typeBC = 'SSSS4';
% % Parameter.typeBC = 'CCCC';
%
% % Material Properties
% E2 = 7.8e6;
% E1 = E2;
% NU12 = 0.3;
% G12 = E2/(2+2*NU12);
% G13 = G12;
% G23 = G12;
% NU21 = (E2/E1)*NU12;
% rho = 1;
%
% % Plate Geometry
% Lx = 10;
% Ly = Lx;
% h = 1;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
%
% % Ply Orientation
% theta = 0;
%
% LoadFactor = (Lx^4)/(E2*h^4);
% TotalLoad = 200/LoadFactor;
% % LoadStep = 16;
% DiscreateLoad = unique([6.25,12.5,25:25:200])/LoadFactor;
% LoadStep = length(DiscreateLoad);
%
% file = sprintf('Poblem_NL_3_%s_%s_h_a_%g_%g_%s_%s',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC)
% save(fullfile(pwd,'plots',[file,'.mat']),'Data_W','Data_xx');
%%
%---------------------------
% Problem_NL_5 : Different Boundary
%---------------------------
% Parameter.Load = 'UDL';
% % Parameter.Load = 'SSL';
% Parameter.gamma = 1e7; % Penalty Parameter
% % Parameter.NL = 1; % Nonlinearity Parameter
% % Parameter.Type = 'Linear';
% % Parameter.Type = 'vonKarman';
% Parameter.Type = 'Green';
% Parameter.Theory = 'IHSDT';
% % Parameter.Theory = 'TSDT';
% % Parameter.typeBC = 'CCCC1';
% % Parameter.typeBC = 'SSSS1';
% % Parameter.typeBC = 'SSSS2';
% % Parameter.typeBC = 'SSSS3';
% % Parameter.typeBC = 'SSSS4';
% Parameter.typeBC = 'SSFS1';
% % Parameter.typeBC = 'SSFS2';
% % Parameter.typeBC = 'SSFS3';
% % Parameter.typeBC = 'SSFS4';
% % Parameter.typeBC = 'CCFC4';
% % Parameter.typeBC = 'CFFF4';
%
% % Material Properties
% E2 = 7.8e6;
% E1 = E2;
% NU12 = 0.3;
% G12 = E2/(2+2*NU12);
% G13 = G12;
% G23 = G12;
% NU21 = (E2/E1)*NU12;
% rho = 1;
%
% % Plate Geometry
% Lx = 10;
% Ly = Lx;
% h = 1;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
%
% % Ply Orientation
% theta = 0;
%
% LoadFactor = (Lx^4)/(E2*h^4);
% % TotalLoad = 400/LoadFactor;
% % LoadStep = 16;
% DiscreateLoad = unique([6.25,12.5,25:25:400])/LoadFactor;
% LoadStep = length(DiscreateLoad);
%
% file = sprintf('Poblem_NL_5_%s_%s_h_a_%g_%g_%s_%s',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC)
% % save(fullfile(pwd,'plots',[file,'.mat']),'Data_W','Data_xx');
%%
% %---------------------------
% % Problem_NL_6 : Orthotropic Plate
% %---------------------------
% Parameter.Load = 'UDL';
% % Parameter.Load = 'SSL';
% Parameter.gamma = 1e7; % Penalty Parameter
% % Parameter.NL = 1; % Nonlinearity Parameter
% % Parameter.Type = 'Linear';
% % Parameter.Type = 'vonKarman';
% Parameter.Type = 'Green';
% Parameter.Theory = 'IHSDT';
% % Parameter.Theory = 'TSDT';
% % Parameter.typeBC = 'CCCC';
% Parameter.typeBC = 'SSSS1';
%
% %---------------
% % Problem_NL_6_1
% %---------------
% % Material Properties : Orthrotropic
% E2 = 1.28e6;
% E1 = 3e6;
% NU12 = 0.32;
% G12 = 0.37e6;
% G13 = G12;
% G23 = G12;
% NU21 = (E2/E1)*NU12;
% rho = 1;
%
% % Plate Geometry : : Orthrotropic
% Lx = 12;
% Ly = Lx;
% h = 0.138;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
%
% % Ply Orientation
% theta = 0;
%
% % file = sprintf('Poblem_NL_6_1_%s_%s_0_%s_%s_Penalty_%g',...
% % Parameter.Type,Parameter.Load,Parameter.Theory,Parameter.typeBC,log10(Parameter.gamma))
%
% %---------------
% % Problem_NL_6_2
% %---------------
% % Material Properties : 0/90/90/0
% E2 = 1.8315e6;
% E1 = 1.8282e6;
% NU12 = 0.2395;
% G12 = 3.125e5;
% G13 = G12;
% G23 = G12;
% NU21 = (E2/E1)*NU12;
% rho = 1;
%
% % Plate Geometry : : 0/90/90/0
% Lx = 12;
% Ly = Lx;
% h = 0.096;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
%
% % Ply Orientation
% theta = [0;90;90;0];
%
% file = sprintf('Poblem_NL_6_2_%s_%s_0_90_90_0_%s_%s_Penalty_%g',...
% Parameter.Type,Parameter.Load,Parameter.Theory,Parameter.typeBC,log10(Parameter.gamma))
%
% LoadFactor = 1;
% DiscreateLoad = unique([0.2:0.2:2])/LoadFactor;
% LoadStep = length(DiscreateLoad);
% %
% % save(fullfile(pwd,'plots',[file,'.mat']),'Data_W');
%%
%---------------------------
% Problem_NL_7 : Hinged 0/90/90/0
%---------------------------
% Parameter.Load = 'UDL';
% % Parameter.Load = 'SSL';
% Parameter.gamma = 1e7; % Penalty Parameter
% % Parameter.NL = 1; % Nonlinearity Parameter
% % Parameter.Type = 'Linear';
% Parameter.Type = 'vonKarman';
% % Parameter.Type = 'Green';
% % Parameter.Theory = 'IHSDT';
% Parameter.Theory = 'TSDT';
% % Parameter.typeBC = 'CCCC';
% % Parameter.typeBC = 'SSSS1';
% % Parameter.typeBC = 'SSSS2';
% % Parameter.typeBC = 'SSSS3';
% % Parameter.typeBC = 'SSSS4';
% % Parameter.typeBC = 'SSFS1';
% % Parameter.typeBC = 'SSFS2';
% Parameter.typeBC = 'SSFS3';
% % Parameter.typeBC = 'SSFS4';
% % Parameter.typeBC = 'CCFC4';
% % Parameter.typeBC = 'CFFF4';
%
% % Material Properties
% E2 = 1;
% E1 = 25*E2;
% NU12 = 0.25;
% G12 = 0.5*E2;
% G13 = G12;
% G23 = 0.2*E2;
% NU21 = (E2/E1)*NU12;
% rho = 1;
%
% % Plate Geometry
% Lx = 1;
% Ly = Lx;
% h = Lx/4;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
%
% % Ply Orientation
% theta = [0;90;90;0];
%
% LoadFactor = (Lx^4)/(E2*h^4);
% DiscreateLoad = unique([50:50:250])/LoadFactor;
% LoadStep = length(DiscreateLoad);
%
% file = sprintf('Poblem_NL_7_%s_%s_h_a_%g_%g_%s_%s',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC)
% % save(fullfile(pwd,'plots',[file,'.mat']),'Data_W','Data_xx');
%%
%---------------------------
% Problem_NL_8 : SSSS1 0/90/0 and 0/90/90/0
%---------------------------
% % Parameter.Load = 'UDL';
% Parameter.Load = 'SSL';
% Parameter.gamma = 0*1e7; % Penalty Parameter
% % Parameter.NL = 1; % Nonlinearity Parameter
% % Parameter.Type = 'Linear';
% % Parameter.Type = 'vonKarman';
% Parameter.Type = 'Green';
% Parameter.Theory = 'IHSDT';
% % Parameter.Theory = 'TSDT';
% % Parameter.typeBC = 'CCCC';
% Parameter.typeBC = 'SSSS1';
% % Parameter.typeBC = 'SSSS2';
% % Parameter.typeBC = 'SSSS3';
% % Parameter.typeBC = 'SSSS4';
% % Parameter.typeBC = 'SSFS1';
% % Parameter.typeBC = 'SSFS2';
% % Parameter.typeBC = 'SSFS3';
% % Parameter.typeBC = 'SSFS4';
% % Parameter.typeBC = 'CCFC4';
% % Parameter.typeBC = 'CFFF4';
%
% % Material Properties
% % E2 = 1;
% % E1 = 25*E2;
% % NU12 = 0.25;
% % G12 = 0.5*E2;
% % G13 = G12;
% % G23 = 0.2*E2;
% % NU21 = (E2/E1)*NU12;
% % rho = 1;
%
% % Plate Geometry
% Lx = 1;
% Ly = Lx;
% h = Lx/10;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
%
% % Ply Orientation
% theta = [0;90;90;0];
% % theta = [0;90;0];
%
% LoadFactor = (Lx^4)/(E2*h^4);
% TotalLoad = 300/LoadFactor;
% DiscreateLoad = unique([50,100:100:300])/LoadFactor;
% LoadStep = length(DiscreateLoad);
% file = sprintf('Poblem_NL_8_%s_%s_h_a_%g_%g_%s_%s_0_90_0',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC)
% file = sprintf('Poblem_NL_8_%s_%s_h_a_%g_%g_%s_%s_0_90_90_0',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC)
% save(fullfile(pwd,'plots',[file,'.mat']),...
% 'Data_W','IndexZ','temp_xx','temp_yy','temp_xy','temp_yz','temp_xz');
%%
%---------------------------
% Problem_NL_9 : SSSS1 (0/90)n and -theta/theta/-theta/theta and -theta/theta/theta/-theta
%---------------------------
% Parameter.Load = 'UDL';
% % Parameter.Load = 'SSL';
% Parameter.gamma = 1e7; % Penalty Parameter
% % Parameter.NL = 1; % Nonlinearity Parameter
% % Parameter.Type = 'Linear';
% % Parameter.Type = 'vonKarman';
% Parameter.Type = 'Green';
% Parameter.Theory = 'IHSDT';
% % Parameter.Theory = 'TSDT';
% Parameter.typeBC = 'SSSS1';
% % Parameter.typeBC = 'SSSS2';
%
% % E2 = 1;
% % E1 = 25*E2;
% % NU12 = 0.25;
% % G12 = 0.5*E2;
% % G13 = G12;
% % G23 = 0.2*E2;
% % NU21 = (E2/E1)*NU12;
% % rho = 1;
%
% % Old Data for Problem 9 _1 _2 _3 _4
% E2 = 1;
% E1 = 40*E2;
% NU12 = 0.25;
% G12 = 0.6*E2;
% G13 = G12;
% G23 = 0.5*E2;
% NU21 = (E2/E1)*NU12;
% rho = 1;
%
% % Plate Geometry
% Lx = 1;
% Ly = Lx;
% h = Lx/4;
% [ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
%
% % Problem_NL_9_1
% % theta = [0;90;];
% % theta = [0;90;0;90;];
% % theta = [0;90;0;90;0;90];
% % theta = [0;90;0;90;0;90;0;90];
% % theta = [0;90;0;90;0;90;0;90;0;90;];
%
% % % Problem_NL_9_2
% theta_angle = 45;
% theta = [-theta_angle;theta_angle;-theta_angle;theta_angle;];
%
% % % Problem_NL_9_3
% % theta_angle = 0;
% % % theta = [-theta_angle;theta_angle;theta_angle;-theta_angle;];
%
% LoadFactor = (Lx^4)/(E2*h^4);
% DiscreateLoad = unique([25:25:300])/LoadFactor;
% % DiscreateLoad = unique([50,100,200,300])/LoadFactor;
% LoadStep = length(DiscreateLoad);
%
% % file = sprintf('Poblem_NL_9_1_%s_%s_h_a_%g_%g_%s_%s_0_90_%g',...
% % Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC,length(theta)/2)
% file = sprintf('Poblem_NL_9_2_%s_%s_h_a_%g_%g_%s_%s_AS_%g',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC,abs(theta(1)))
% % file = sprintf('Poblem_NL_9_3_%s_%s_h_a_%g_%g_%s_%s_AS_%g',...
% % Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC,abs(theta(1)))
% % file = sprintf('Poblem_NL_9_3_%s_%s_h_a_%g_%g_%s_%s_S_%g',...
% % Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC,abs(theta(1)))
% % save(fullfile(pwd,'plots',[file,'.mat']),'Data_W');
%%
% %---------------------------
% % Problem_NL_10 : Laminated different boundary condition
% %---------------------------
Parameter.Load = 'UDL';
% Parameter.Load = 'SSL';
Parameter.gamma = 1e7; % Penalty Parameter
% Parameter.NL = 1; % Nonlinearity Parameter
% Parameter.Type = 'Linear';
% Parameter.Type = 'vonKarman';
Parameter.Type = 'Green';
Parameter.Theory = 'IHSDT';
% Parameter.Theory = 'TSDT';
% Parameter.typeBC = 'CCCC4';
% Parameter.typeBC = 'SSSS1';
% Parameter.typeBC = 'SSSS2';
% Parameter.typeBC = 'SSSS3';
% Parameter.typeBC = 'SSSS4';
% Parameter.typeBC = 'CCFC4';
Parameter.typeBC = 'SSFS1';
% Parameter.typeBC = 'SSFS2';
% Parameter.typeBC = 'SSFS3';
% Parameter.typeBC = 'SSFS4';
E2 = 1;
E1 = 25*E2;
NU12 = 0.25;
G12 = 0.5*E2;
G13 = G12;
G23 = 0.2*E2;
NU21 = (E2/E1)*NU12;
rho = 1;
% E2 = 7.8e6;
% E1 = E2;
% NU12 = 0.3;
% G12 = E2/(2+2*NU12);
% G13 = G12;
% G23 = G12;
% NU21 = (E2/E1)*NU12;
% rho = 1;
% Plate Geometry
Lx = 10;
Ly = Lx;
h = Lx/10; %<====
[ar_N,ar_D] = rat(h/Lx); % Aspect Ratio
% Problem_NL_10_1
theta = [0;90;90;0;];
% Problem_NL_10_2
% theta = [0;90;0;90;];
% theta = [0;90;0;90;0;90;0;90;];
% Problem_NL_10_3
% theta = [0;90;0;90;];
% theta = [45;-45;45;-45;];
DeflectionFactor = 1/h;
LoadFactor = (Lx^4)/(E2*h^4);
DiscreateLoad = 1000/LoadFactor;
% DiscreateLoad = unique([25,50,100:100:1000])/LoadFactor;
% DiscreateLoad = unique([25,50,100:25:1000])/LoadFactor;
LoadStep = length(DiscreateLoad);
file = sprintf('Poblem_NL_10_1_%s_%s_h_a_%g_%g_%s_%s',...
Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC)
% file = sprintf('Poblem_NL_10_2_%s_%s_h_a_%g_%g_%s_%s_0_90_%g',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC,length(theta)/2)
% file = sprintf('Poblem_NL_10_3_%s_%s_h_a_%g_%g_%s_%s_AS_CrossPly_%g',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC,length(theta)/2)
% file = sprintf('Poblem_NL_10_3_%s_%s_h_a_%g_%g_%s_%s_AS_AnglePly_%g',...
% Parameter.Type,Parameter.Load,ar_N,ar_D,Parameter.Theory,Parameter.typeBC,length(theta)/2)
% save(fullfile(pwd,'plots',[file,'.mat']),'Data_W');
%%
if strcmp(Parameter.Type,'Linear')
fprintf('Linear analysis\n');
elseif strcmp(Parameter.Type,'vonKarman')
fprintf('Nonlinear analysis : von Karman\n');
elseif strcmp(Parameter.Type,'Green')
fprintf('Nonlinear analysis : Green-Lagrange\n');
else
error('Analysis Type\n');
end
fprintf('%s Load\n',Parameter.Load);
fprintf('%s\n',Parameter.typeBC);
fprintf('%s\n',Parameter.Theory);
%%
%--------------------------------------------------------------------------
% Material properties
%--------------------------------------------------------------------------
%%
%--------------------------------------------------------------------------
% Geometrical properties
%--------------------------------------------------------------------------
nol = length(theta); % number of layers
disp(table(theta));
fprintf('Number of Layers : %d\n',nol);
%%
%--------------------------------------------------------------------------
% Transverse uniform pressure on plate
%--------------------------------------------------------------------------
%%
%--------------------------------------------------------------------------
% Discretization
%--------------------------------------------------------------------------
noe = 12;
fprintf('Number of element : %d X %d\n',noe,noe);
nx = noe;
ny = noe;
nel = nx*ny; % Total number of elements nel=nx*ny;
nnel = 9; % number of control points per element
ndof = 7; % number of dofs per node
mx = 2*nx + 1; % Number of basis functions/control points in xi direction
my = 2*ny + 1; % Number of basis functions/control points in eta direction
nnode = mx*my; % Total number of basis functions/control points
edof = nnel*ndof; % degrees of freedom per element
sdof = nnode*ndof; % Global degrees of freedom= total CP's*DOF assumed for each CP i.e. total system dofs
%--------------------------------------------------------------------------
% ELEMENT TOPOLOGY
%--------------------------------------------------------------------------
[Coordinate,IEN] = Mesh(Lx,Ly,mx,my,nnode);
%%
%--------------------------------------------------------------------------
% Material Matrix
%--------------------------------------------------------------------------
if strcmp(Parameter.Theory,'TSDT')
% Third order theory
funF = @(z) z - (4/(3*h^2))*z.^3;
fundF = @(z) 1 - (4/h^2)*z.^2;
end
if strcmp(Parameter.Theory,'IHSDT')
r = 3;
funG = @(z) asinh(r*z/h);
fundG = @(z) (r)./sqrt(r^2*z.^2 + h^2);
funO = -2*r/(h*sqrt(r^2+4));
funF = @(z) funO*z + funG(z);
fundF = @(z) funO + fundG(z);
end
if strcmp(Parameter.Theory,'Ferreira')
% Ferreira
funF = @(z) sin(pi/h*z);
fundF = @(z) (pi/h)*cos(pi/h*z);
end
[Qbar,B1Matrix,B2Matrix,B3Matrix,B4Matrix,...
S1Matrix,S2Matrix,S3Matrix,S4Matrix] = MaterialProperty(...
E1,E2,NU12,NU21,G12,G13,G23,h,theta,nol,funF,fundF);
%--------------------------------------------------------------------------
% Boundary conditions
%--------------------------------------------------------------------------
typeBC = Parameter.typeBC;
[prescribedbdof,activedof] = BoundaryCondition(my,mx,nnode,ndof,sdof,Coordinate,typeBC);
%--------------------------------------------------------------------------
% Computation of element matrices and vectors and their assembly
%--------------------------------------------------------------------------
LambdaDisp = [];
% Point 1
Disp = zeros(sdof,1);
Lambda = 0;
DispPrev2 = Disp;
LambdaPrev2 = Lambda;
uTangentPrev2 = zeros(size([LambdaPrev2;DispPrev2]));
converged2 = false;
IterPrev2 = 1; % Default
LambdaDisp = [LambdaDisp,[Lambda;Disp]];
[NonLoad,NonCen_W] = PostProcess(DiscreateLoad,DeflectionFactor,LoadFactor,nx,ny,mx,my,nnode,ndof,sdof,...
Lambda,Disp);
plot(NonLoad,NonCen_W,':ob','LineWidth',3);
drawnow;
hold on;
% Point 2
Disp = zeros(sdof,1);
Lambda = 0;
DispPrev = Disp;
LambdaPrev = Lambda;
uTangentPrev = zeros(size([LambdaPrev;DispPrev])); % Default
converged1 = false;
IterPrev = 1; % Default
LambdaDisp = [LambdaDisp,[Lambda;Disp]];
[NonLoad,NonCen_W] = PostProcess(DiscreateLoad,DeflectionFactor,LoadFactor,nx,ny,mx,my,nnode,ndof,sdof,...
Lambda,Disp);
plot(NonLoad,NonCen_W,':ob','LineWidth',3);
drawnow;
hold on;
%% Arc-Length Method
iStep_Max = 500;
iNR_Max = 30;
tolerance = 1e-2;
LambdaIncr = 0.025; % + for increase/ - for decrease
% Flow direction
Lambda_S = 0; Lambda_E = 1;
LambdaDir = sign(Lambda_E-Lambda_S); % or simply dive +1 for loading; -1 for unloading
BreakFlag = false; % To stop the simulation from going outside range
for iStep = 1:iStep_Max
fprintf('\n');
converged = false;
%%
% Prector Step
if iStep == 1
LambdaScale = 1;
Ds = LambdaIncr; % Assumed
uTangentPrev = zeros(size([LambdaPrev;DispPrev]));
uTangentPrev(1) = LambdaDir; % Assumed load increment only
else
% Adaptive Step scale
if converged1
if converged2
% Adapative Method
% LambdaScale = IterPrev2/IterPrev; % for cylindrical
% LambdaScale = 5/IterPrev; % for cylindrical
% LambdaScaleTemp = LambdaScale;
% LambdaScale = 4/IterPrev;
LambdaScale = LambdaScale*1.1;
% Restrict Lambda Scale Factor
LambdaScale = min(LambdaScale,2); % maximum scaling; 2 times for spherical; 4 for cylndrical
fprintf('Stepsize increased\n');
else
fprintf('##Stepsize kept same\n');
end
else
LambdaScale = LambdaScale/2;
LambdaScale = max(LambdaScale,1/2^20);
fprintf('**Stepsize decreased\n');
if LambdaScale < 1/2^20
fprintf('Cant further reduce the stepsize\n');
break;
end
end
% Tangent stiffness, Secant stiffness and Force
[tangent,stiffness,force] = TSDT_5dof(Parameter,DiscreateLoad(end),Lx,Ly,h,...
nel,nnel,ndof,nnode,edof,sdof,Coordinate,IEN,...
B1Matrix,B2Matrix,B3Matrix,B4Matrix,...
S1Matrix,S2Matrix,S3Matrix,S4Matrix,Disp);
Stiffness_Reduced = stiffness(activedof,activedof);
Tangent_Reduced = tangent(activedof,activedof);
Force_Reduced = force(activedof,1);
Disp_Reduced = Disp(activedof,1);
% Calculation of Unit tangent for (prediction step) or (initial incnrement)
dR = Stiffness_Reduced*Disp_Reduced - Lambda*Force_Reduced;
dRdu = Tangent_Reduced;
dRdl = -Force_Reduced;
LambdaDot = LambdaDir/sqrt(norm(dRdu\dRdl)+1);
DispDot = -(dRdu\dRdl)*LambdaDot;
DispDot_Full = zeros(sdof,1);
DispDot_Full(activedof,1) = DispDot;
uTangentPrev = [LambdaDot;DispDot_Full];
uTangentPrev = uTangentPrev/norm(uTangentPrev); % just for assurance
% if uTangentPrev2'*uTangentPrev < 0
% if uTangentPrev2(1)'*uTangentPrev(1) <= 0 || uTangentPrev2(2:end)'*uTangentPrev(2:end) <= 0
% if uTangentPrev2(2:end)'*uTangentPrev(2:end) <= 0
if uTangentPrev2(2:end)'*uTangentPrev(2:end) <= 0 || (DispPrev-DispPrev2)'*uTangentPrev(2:end) <=0
% [uTangentPrev2(1)'*uTangentPrev(1),uTangentPrev2(2:end)'*uTangentPrev(2:end),(DispPrev-DispPrev2)'*uTangentPrev(2:end)]
uTangentPrev = -uTangentPrev;
end
end
% Increasing StepSize is constraint
if LambdaScale > 1
LambdaScale = 1; % for spherical
end
fprintf('Step : %d, StepScale : %g\n',iStep,LambdaScale);
Dul = uTangentPrev*LambdaScale*Ds;
Dl = Dul(1);
Du = Dul(2:end);
Lambda = LambdaPrev + Dl;
Disp = DispPrev + Du;
[NonLoad,NonCen_W] = PostProcess(DiscreateLoad,DeflectionFactor,LoadFactor,nx,ny,mx,my,nnode,ndof,sdof,...
Lambda,Disp);
plot(NonLoad,NonCen_W,':ok','LineWidth',3);
drawnow;
hold on;
%%
% Corrector Step
for iNR = 1:iNR_Max
% Tangent stiffness, Secant stiffness and Force
[tangent,stiffness,force] = TSDT_5dof(Parameter,DiscreateLoad(end),Lx,Ly,h,...
nel,nnel,ndof,nnode,edof,sdof,Coordinate,IEN,...
B1Matrix,B2Matrix,B3Matrix,B4Matrix,...
S1Matrix,S2Matrix,S3Matrix,S4Matrix,Disp);
Stiffness_Reduced = stiffness(activedof,activedof);
Tangent_Reduced = tangent(activedof,activedof);
Force_Reduced = force(activedof,1);
Disp_Reduced = Disp(activedof,1);
dR = Stiffness_Reduced*Disp_Reduced - Lambda*Force_Reduced;
dRdu = Tangent_Reduced;
dRdl = -Force_Reduced;
% Solving Approach : 1
[L, U, P] = lu(sparse(dRdu));
%
duu = L\(P*dRdl);
duCap = U\duu;
duCap = -duCap;
%
duu = L\(P*dR);
duBar = U\duu;
duBar = -duBar;
% % Solving Approach : 2
% duBar = -dRdu\dR;
% duCap = -dRdu\dRdl;
% phi = 0; % Original; % for cylindrical
phi = 1; % for spherical
FtF = 1;
% FtF = dRdl'*dRdl;
phi = phi*FtF; % Modified
% Constraint Approach : 1
% Spherical constraint
% Method : 1
% if (iStep > 1)
% Du_Reduced = Du(Activedof);
% dF = Du_Reduced'*Du_Reduced + phi*Dl*Dl - LambdaScale^2*Ds^2;
% dFdu = 2.0*Du_Reduced';
% dFdl = 2.0*phi*Dl;
% else
% Du_Reduced = Du(Activedof);
% dF = 0.0;
% dFdu = 0.0*Du_Reduced';
% dFdl = 1.0;
% end
% dl = (-dFdu*duBar - dF)/(dFdl+dFdu*duCap);
% Constraint Approach : 1
% Spherical constraint
% Method : 2
if (iStep > 1)
Du_Reduced = Du(activedof);
a1 = duCap'*duCap + phi;
a2 = (Du_Reduced+duBar)'*duCap + phi*Dl;
a3 = (duBar+Du_Reduced)'*(duBar+Du_Reduced) + phi*Dl*Dl - LambdaScale^2*Ds^2;
dF = Du_Reduced'*Du_Reduced + phi*Dl*Dl - LambdaScale^2*Ds^2;
else
a1 = 1;
a2 = 0;
a3 = 0;
dF = 0;
end
% solution possibility
if (4*a2*a2-4*a1*a3)<0
converged = false;
break;
end
%
dl_1=(-2*a2+sqrt(4*a2*a2-4*a1*a3))/(2*a1);
dl_2=(-2*a2-sqrt(4*a2*a2-4*a1*a3))/(2*a1);
%
du_1 = duBar + dl_1*duCap;
du_2 = duBar + dl_2*duCap;
%
du_1_Full = zeros(sdof,1); du_2_Full = zeros(sdof,1);
du_1_Full(activedof,1) = du_1; du_2_Full(activedof,1) = du_2;
%
Du_1 = Du + du_1_Full;
Du_2 = Du + du_2_Full;
%
if Du'*Du_1 > 0 && Du'*Du_2 < 0
dl = dl_1;
elseif Du'*Du_1 < 0 && Du'*Du_2 > 0
dl = dl_2;
else %deltaDis'*deltaDis1 > 0 && deltaDis'*deltaDis2 > 0
if (dl_1+(0.5*a3/a2))^2 < (dl_2+(0.5*a3/a2))^2
dl = dl_1;
else
dl = dl_2;
end
end
% checking the limit of Lambda and according modified the sub-increment of Lambda
if LambdaDir == 1 % Loading : Initial increase in load
if Lambda + dl < Lambda_S
dl = Lambda_S - Lambda;
fprintf('Forward Start\n');
BreakFlag = true;
keyboard
end
if Lambda + dl > Lambda_E
dl = Lambda_E - Lambda;
fprintf('Forward End\n');
BreakFlag = true;
% keyboard
end
else % UnLoading : Initial decrease in load
if Lambda + dl < Lambda_E
dl = Lambda_E - Lambda;
fprintf('Backward End\n');
BreakFlag = true;
keyboard
end
if Lambda + dl > Lambda_S
dl = Lambda_S - Lambda;
fprintf('Backward Start\n')
BreakFlag = true;
keyboard
end
end
du = duBar + dl*duCap;
du_Full = zeros(sdof,1);
du_Full(activedof,1) = du;
Du = Du + du_Full;
Dl = Dl + dl;
Disp = Disp + du_Full;
Lambda = Lambda + dl;
du_Norm = sqrt(du_Full'*du_Full + dl*dl)/sqrt(Disp'*Disp + Lambda*Lambda);
dR_Norm = sqrt(dR'*dR + 1*dF*dF);
if BreakFlag
dR_Norm = sqrt(dR'*dR);
end
fprintf('Iter : %d, residual : %g\n',iNR,dR_Norm);
if du_Norm < tolerance && dR_Norm < 1
converged = true;
break;
end
[NonLoad,NonCen_W] = PostProcess(DiscreateLoad,DeflectionFactor,LoadFactor,nx,ny,mx,my,nnode,ndof,sdof,...
Lambda,Disp);
plot(NonLoad,NonCen_W,':or','LineWidth',3);
drawnow;
hold on;
end
%%
% After Corrector Step
if converged
if iStep == 1
Du_Reduced = Du(activedof);
Ds = sqrt(Du_Reduced'*Du_Reduced + phi*Dl*Dl); % Set for default step size
% Ds = sqrt(Du'*Du + phi*Dl*Dl); % Set for default step size
% Tangent stiffness, Secant stiffness and Force
[tangent,stiffness,force] = TSDT_5dof(Parameter,DiscreateLoad(end),Lx,Ly,h,...
nel,nnel,ndof,nnode,edof,sdof,Coordinate,IEN,...
B1Matrix,B2Matrix,B3Matrix,B4Matrix,...
S1Matrix,S2Matrix,S3Matrix,S4Matrix,Disp);