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% Copyright (C) 2009-2016 Luis Saavedra, Yves Renard.
%
% This file is a part of GetFEM++
%
% GetFEM++ is free software; you can redistribute it and/or modify it
% under the terms of the GNU Lesser General Public License as published
% by the Free Software Foundation; either version 3 of the License, or
% (at your option) any later version along with the GCC Runtime Library
% Exception either version 3.1 or (at your option) any later version.
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
% License and GCC Runtime Library Exception for more details.
% You should have received a copy of the GNU Lesser General Public License
% along with this program; if not, write to the Free Software Foundation,
% Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
%
%%% Linear Elastostatic problem with a crack. Use of XFem.
%
% This program is used to check that matlab-getfem is working. This is
% also a good example of use of GetFEM++.
%
%%%
variant = 4;
% variant : 1 : a single crack with cutoff enrichement
% 2 : a single crack with a fixed size area Xfem enrichment
% 3 : a supplementary crossing crack with a fixed size area
% Xfem enrichment
% 4 : variant 3 with the second crack closed by a penalisation of
% the jump (example of use of xfem_plus and xfem_minus).
% 5 : variant 3 with the first crack closed by a penalisation of
% the jump (example of use of xfem_plus and xfem_minus).
% 6 : variant 3 with the two cracks closed by a penalisation of
% the jump (example of use of xfem_plus and xfem_minus).
gf_workspace('clear all');
% Parameters
nx = 20;
DIRICHLET = 101;
Lambda = 1.25e10; % Lame coefficient
Mu = 1.875e10; % Lame coefficient
% Mesh definition:
m = gf_mesh('regular_simplices', -0.5:1.0/nx:0.5+1.0/nx, -0.5:1.0/nx:0.5+1.0/nx);
% m = gf_mesh('import','gmsh','quad.msh')
% Boundary set:
gf_mesh_set(m, 'region', DIRICHLET, gf_mesh_get(m,'outer_faces'));
% Global functions for asymptotic enrichment:
ck0 = gf_global_function('crack',0);
ck1 = gf_global_function('crack',1);
ck2 = gf_global_function('crack',2);
ck3 = gf_global_function('crack',3);
if (variant == 1) % Cutoff enrichement
coff = gf_global_function('cutoff',2,0.4,0.01,0.4);
ckoff0 = gf_global_function('product', ck0, coff);
ckoff1 = gf_global_function('product', ck1, coff);
ckoff2 = gf_global_function('product', ck2, coff);
ckoff3 = gf_global_function('product', ck3, coff);
end
% Levelset(s) definition
ls = gf_levelset(m,1,'y','x');
mls = gf_mesh_levelset(m);
gf_mesh_levelset_set(mls,'add',ls);
if (variant > 2)
ls2 = gf_levelset(m,1,'x+0.125','abs(y)-0.375');
gf_mesh_levelset_set(mls,'add',ls2);
end
gf_mesh_levelset_set(mls,'adapt');
% Basic mesh_fem without enrichment
mf_pre_u = gf_mesh_fem(m);
gf_mesh_fem_set(mf_pre_u,'fem',gf_fem('FEM_PK(2,1)'));
% Definition of the enriched finite element method
mfls_u = gf_mesh_fem('levelset',mls,mf_pre_u);
if (variant == 1) % Cutoff enrichement
mf_sing_u = gf_mesh_fem('global function',m,ls, {ckoff0,ckoff1,ckoff2,ckoff3},1);
mf_u = gf_mesh_fem('sum',mf_sing_u,mfls_u);
else
mf_part_unity = gf_mesh_fem(m);
gf_mesh_fem_set(mf_part_unity, 'classical fem', 1);
DOFpts = gf_mesh_fem_get(mf_part_unity, 'basic dof nodes');
% Search the dofs to be enriched with the asymptotic displacement.
Idofs_center = find((DOFpts(1,:)).^2 + (DOFpts(2,:)).^2 <= (0.1)^2);
mf_sing_u = gf_mesh_fem('global function',m,ls, {ck0,ck1,ck2,ck3}, 1);
mf_xfem_sing = gf_mesh_fem('product', mf_part_unity, mf_sing_u);
gf_mesh_fem_set( mf_xfem_sing, 'set enriched dofs', Idofs_center);
if (variant > 2)
Idofs_up = find((DOFpts(1,:)+0.125).^2 + (DOFpts(2,:)-0.375).^2 <= (0.1)^2);
Idofs_down = find((DOFpts(1,:)+0.125).^2 + (DOFpts(2,:)+0.375).^2 <= (0.1)^2);
mf_sing_u2 = gf_mesh_fem('global function',m,ls2, {ck0,ck1,ck2,ck3}, 1);
mf_xfem_sing2 = gf_mesh_fem('product', mf_part_unity, mf_sing_u2);
gf_mesh_fem_set(mf_xfem_sing2, 'set enriched dofs', [Idofs_up Idofs_down]);
end
if (variant == 2)
mf_u = gf_mesh_fem('sum', mf_xfem_sing, mfls_u);
else
mf_u = gf_mesh_fem('sum', mf_xfem_sing, mf_xfem_sing2, mfls_u);
end
end
gf_mesh_fem_set(mf_u,'qdim',2);
% MeshIm definition:
mim = gf_mesh_im('levelset', mls, 'all', ...
gf_integ('IM_STRUCTURED_COMPOSITE(IM_TRIANGLE(6),3)'), ...
gf_integ('IM_STRUCTURED_COMPOSITE(IM_GAUSS_PARALLELEPIPED(2,6),9)'), ...
gf_integ('IM_STRUCTURED_COMPOSITE(IM_TRIANGLE(6),5)'));
% Exact solution for a single crack:
mf_ue = gf_mesh_fem('global function',m,ls,{ck0,ck1,ck2,ck3});
A = 2+2*Mu/(Lambda+2*Mu);
B = -2*(Lambda+Mu)/(Lambda+2*Mu);
Ue = zeros(2,4);
Ue(1,1) = 0; Ue(2,1) = A-B; % sin(theta/2)
Ue(1,2) = A+B; Ue(2,2) = 0; % cos(theta/2)
Ue(1,3) = -B; Ue(2,3) = 0; % sin(theta/2)*sin(theta)
Ue(1,4) = 0; Ue(2,4) = B; % cos(theta/2)*cos(theta)
Ue = Ue / 2*pi;
Ue=reshape(Ue,1,8);
% Model definition:
md = gf_model('real');
gf_model_set(md,'add_fem_variable', 'u', mf_u);
% data
gf_model_set(md,'add_initialized_data','lambda', [Lambda]);
gf_model_set(md,'add_initialized_data','mu', [Mu]);
gf_model_set(md,'add_isotropic_linearized_elasticity_brick',mim,'u','lambda','mu');
gf_model_set(md,'add_initialized_fem_data','DirichletData', mf_ue, Ue);
gf_model_set(md,'add_Dirichlet_condition_with_penalization',mim,'u', 1e12, DIRICHLET, 'DirichletData');
if (variant == 5 || variant == 6) % Penalisation of the jump over the first crack
mim_bound1 = gf_mesh_im('levelset', mls, 'boundary(a)', gf_integ('IM_STRUCTURED_COMPOSITE(IM_TRIANGLE(6),3)'));
% gf_asm('generic', mim_bound, 0, '1', -1) % length of the crack
gf_model_set(md, 'add linear generic assembly brick', mim_bound1, '1e17*(Xfem_plus(u)-Xfem_minus(u)).(Xfem_plus(Test_u)-Xfem_minus(Test_u))');
end
if (variant == 4 || variant == 6) % Penalisation of the jump over the second crack
mim_bound2 = gf_mesh_im('levelset', mls, 'boundary(b)', gf_integ('IM_STRUCTURED_COMPOSITE(IM_TRIANGLE(6),3)'));
gf_model_set(md, 'add linear generic assembly brick', mim_bound2, '1e17*(Xfem_plus(u)-Xfem_minus(u)).(Xfem_plus(Test_u)-Xfem_minus(Test_u))');
end
% Assembly of the linear system and solve:
gf_model_get(md,'solve');
U = gf_model_get(md,'variable','u');
% Interpolation of the solution on a cut mesh for the drawing purpose
cut_mesh = gf_mesh_levelset_get(mls,'cut_mesh');
mfv = gf_mesh_fem(cut_mesh,2);
gf_mesh_fem_set(mfv,'classical_discontinuous_fem',2,0.001);
gf_mesh_fem_set(mf_ue,'qdim',2);
V = gf_compute(mf_u,U,'interpolate_on',mfv);
Ve = gf_compute(mf_ue,Ue,'interpolate_on',mfv);
% Computation of the Von Mises stress
mfvm = gf_mesh_fem(cut_mesh);
gf_mesh_fem_set(mfvm,'classical_discontinuous_fem',2,0.001);
gf_model_set(md,'add initialized fem data', 'u_cut', mfv, V);
VM = gf_model_get(md,'compute_isotropic_linearized_Von_Mises_or_Tresca', 'u_cut', 'lambda', 'mu', mfvm);
% export to pos
gf_mesh_fem_get(mfv,'export_to_pos','crack.pos',V,'V',Ve,'Ve', mfvm, VM,'Von Mises');
disp('You can view the solution with (for example): gmsh crack.pos\n');
% drawing the solution
gf_plot(mfvm, VM, 'deformed_mesh', 'on', 'deformation_mf', mfv, 'deformation', V, 'deformation_scale', 0.10);
colorbar;
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