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Functions/Subroutines | |
| program | elasticity_dirichlet_2d |
| SOLVES THE LINEAR ELASTICITY PROBLEM with a Dirichlet boundary conditions More... | |
| real(rp) function | lambda (x) |
| Lame coefficient 'lambda'. More... | |
| real(rp) function | mu (x) |
| Lame coefficient 'mu'. More... | |
| real(rp) function | u_1 (x) |
| exact solution u=(u_1, u_2) More... | |
| real(rp) function | u_2 (x) |
| real(rp) function, dimension(3) | grad_u_1 (x) |
| gradient of the exact solution More... | |
| real(rp) function, dimension(3) | grad_u_2 (x) |
| real(rp) function | f_1 (x) |
| right hand side 'f'= =(f_1, f_2) More... | |
| real(rp) function | f_2 (x) |
| real(rp) function | gam (x) |
| To characterise . More... | |
| real(rp) function | cc (x) |
| real(rp) function | ss (x) |
| real(rp) function | cs (x) |
| real(rp) function | sc (x) |
| real(rp) function | cc4 (x) |
| real(rp) function | ss4 (x) |
| real(rp) function | cs4 (x) |
| real(rp) function | sc4 (x) |
Function/Subroutine Documentation
◆ cc()
| real(rp) function elasticity_dirichlet_2d::cc | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 407 of file elasticity_Dirichlet_2D.f90.
◆ cc4()
| real(rp) function elasticity_dirichlet_2d::cc4 | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 437 of file elasticity_Dirichlet_2D.f90.
◆ cs()
| real(rp) function elasticity_dirichlet_2d::cs | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 421 of file elasticity_Dirichlet_2D.f90.
◆ cs4()
| real(rp) function elasticity_dirichlet_2d::cs4 | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 451 of file elasticity_Dirichlet_2D.f90.
◆ elasticity_dirichlet_2d()
| program elasticity_dirichlet_2d | ( | ) |
SOLVES THE LINEAR ELASTICITY PROBLEM with a Dirichlet boundary conditions
Search for \(u:~\Omega \mapsto \R^2\) with \( \Omega= [0,1]^2 \) that satisfies
\(~~~~~~~~~ -\dv(A(x) e(u)) = f ~~~\) on \(~~~ \Omega \)
\(~~~~~~~~~~ u = g ~~~\) on \(~~~\Gamma = \partial \Omega ~~~\) with
- \( f:~\Omega\mapsto \R^2\)
- \( g:~\Gamma\mapsto \R^2\)
PROBLEM DATA See choral/maple/example_elasticity_dirichlet.mw
for the definition of \( f(x),~u(x),~\lambda(x),~\mu(x)\).
HOOKE TENSOR \( A(x)\xi = \lambda(x) Tr(\xi) Id + 2 \mu(x)\xi \) with \( \lambda,~\mu~: \Omega \mapsto \R \)
SYMMETRISED GRADIENT \( e(u) = (\nabla u + ^T\nabla u)/2 \)
VARIATIONAL FORMULATION
- The finite element space is \( X_h\subset \Hu\).
- Let \( X_{h,0} \subset X_h\) the subspace of all functions vanishing on \( \Gamma \).
NUMERICAL PROBLEM find \( u\in [X_h]^2 \) such that \( ~~\forall ~v \in [X_{h,0}]^2, ~~~~ \)
\[ \int_\Omega A(x) e(u):e(v) \,\dx ~=~ \int_\Omega f \cdot v \,\dx \]
with \( \xi:\zeta = \sum_{1\le i,j \le 2} \xi_{ij}\zeta_{ij}\) for the 2 matrices \( \xi~, \zeta \in \R^{2\times 2}\),
and such that for all finite element node \( x\in \Gamma \)
\[ u(x) = g(x). \]
NUMERICAL RESOLUTION
- The finite element space \( X_h \) has a finite element bassi,
- it provides a basis \( (v_i)_{1\le i\le N} \) for \( Y = [X_h]^2 \).
- Computation of the stiffness matrix \( S =[s_{i,\,j}]_{1\le i,\,j\le N}\)
\[ s_{i,\,j} = \int_\Omega A(x) e(v_i):e(v_j)\,\dx \]
- Computation of the right hand side \( F = (f_i)_{1\le i\le N} \)
\[ f_i = \int_\Omega f \cdot v_i \,\dx \]
For the basis functions \( u_i \notin [ X_{h,0}]^2 \) (i.e. \( u_i \) is associated to a finite element node \( x_i \in\Gamma\) the domain boundary),
- Modification of the stiffness matrix: \( s_{i,\,j} = \delta_{ij} \) (Kronecker symbol)
- Modification of the right hand side \( F_{i} = g(x_i) \)
- Resolution of the (non-symmetric) system
\[ S U_h = F + G \]
POST TREATMENT
- Computation of the numerical error between the exact solution \(u\) and the numerical solution \(u_h\),
\[ \int_\Omega |u-u_h|^2 \dx~,\quad\quad \int_\Omega |\nabla u-\nabla u_h|^2 \dx \]
- Graphical display of the numerical solution with gmsh
Charles PIERRE, December 2019
Definition at line 88 of file elasticity_Dirichlet_2D.f90.
◆ f_1()
| real(rp) function elasticity_dirichlet_2d::f_1 | ( | real(rp), dimension(3), intent(in) | x | ) |
right hand side 'f'= =(f_1, f_2)
Definition at line 361 of file elasticity_Dirichlet_2D.f90.
◆ f_2()
| real(rp) function elasticity_dirichlet_2d::f_2 | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 378 of file elasticity_Dirichlet_2D.f90.
◆ gam()
| real(rp) function elasticity_dirichlet_2d::gam | ( | real(rp), dimension(3), intent(in) | x | ) |
To characterise .
Definition at line 398 of file elasticity_Dirichlet_2D.f90.
◆ grad_u_1()
| real(rp) function, dimension(3) elasticity_dirichlet_2d::grad_u_1 | ( | real(rp), dimension(3), intent(in) | x | ) |
gradient of the exact solution
Definition at line 339 of file elasticity_Dirichlet_2D.f90.
◆ grad_u_2()
| real(rp) function, dimension(3) elasticity_dirichlet_2d::grad_u_2 | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 348 of file elasticity_Dirichlet_2D.f90.
◆ lambda()
| real(rp) function elasticity_dirichlet_2d::lambda | ( | real(rp), dimension(3), intent(in) | x | ) |
Lame coefficient 'lambda'.
Definition at line 300 of file elasticity_Dirichlet_2D.f90.
◆ mu()
| real(rp) function elasticity_dirichlet_2d::mu | ( | real(rp), dimension(3), intent(in) | x | ) |
Lame coefficient 'mu'.
Definition at line 310 of file elasticity_Dirichlet_2D.f90.
◆ sc()
| real(rp) function elasticity_dirichlet_2d::sc | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 428 of file elasticity_Dirichlet_2D.f90.
◆ sc4()
| real(rp) function elasticity_dirichlet_2d::sc4 | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 458 of file elasticity_Dirichlet_2D.f90.
◆ ss()
| real(rp) function elasticity_dirichlet_2d::ss | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 414 of file elasticity_Dirichlet_2D.f90.
◆ ss4()
| real(rp) function elasticity_dirichlet_2d::ss4 | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 444 of file elasticity_Dirichlet_2D.f90.
◆ u_1()
| real(rp) function elasticity_dirichlet_2d::u_1 | ( | real(rp), dimension(3), intent(in) | x | ) |
exact solution u=(u_1, u_2)
Definition at line 320 of file elasticity_Dirichlet_2D.f90.
◆ u_2()
| real(rp) function elasticity_dirichlet_2d::u_2 | ( | real(rp), dimension(3), intent(in) | x | ) |
Definition at line 327 of file elasticity_Dirichlet_2D.f90.
