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Functions/Subroutines | |
| program | diffusion_neumann |
| SOLVES THE DIFFUSION PROBLEM with Neumann boundary conditions More... | |
| real(rp) function | u (x) |
| exact solution 'u' More... | |
| real(rp) function, dimension(3) | grad_u (x) |
| gradient of the exact solution More... | |
| real(rp) function | f (x) |
| right hand side 'f' More... | |
| real(rp) function | rho (x) |
| density '(x)' More... | |
| real(rp) function | bl_form_b (x, w1, w2) |
| real(rp) function | theta (x) |
| Angle of the principal direction with e_x. More... | |
| real(rp) function, dimension(2, 2) | b (x) |
| Diffusion matrix B(x) at point x. More... | |
| real(rp) function | x_le_0 (x) |
| To characterise {x=0}. More... | |
| real(rp) function | x_ge_1 (x) |
| To characterise {x=1}. More... | |
| real(rp) function | y_le_0 (x) |
| To characterise {x=0}. More... | |
| real(rp) function | y_ge_1 (x) |
| To characterise {x=1}. More... | |
| real(rp) function | g_x_0 (x) |
| g(x) on {x=0} More... | |
| real(rp) function | g_x_1 (x) |
| g(x) on {x=1} More... | |
| real(rp) function | g_y_0 (x) |
| g(x) on {y=0} More... | |
| real(rp) function | g_y_1 (x) |
| g(x) on {y=1} More... | |
Function/Subroutine Documentation
◆ b()
| real(rp) function, dimension(2,2) diffusion_neumann::b | ( | real(rp), dimension(3), intent(in) | x | ) |
Diffusion matrix B(x) at point x.
Definition at line 432 of file diffusion_Neumann.f90.
◆ bl_form_b()
| real(rp) function diffusion_neumann::bl_form_b | ( | real(rp), dimension(3), intent(in) | x, |
| real(rp), dimension(3), intent(in) | w1, | ||
| real(rp), dimension(3), intent(in) | w2 | ||
| ) |
◆ diffusion_neumann()
| program diffusion_neumann | ( | ) |
SOLVES THE DIFFUSION PROBLEM with Neumann boundary conditions
\(~~~~~~~~~ -\dv(B(x) \nabla u) + \rho(x) u = f ~~~\) on \(~~~ \Omega= [0,1]^2 \)
\(~~~~~~~~~~ B(x)\nabla u.n = g ~~~\) on \(~~~\Gamma = \partial \Omega ~~~\) with
- \( g \) = g_x_0 on \(\Gamma \cap \{x=0\} \)
- \( g \) = g_x_1 on \(\Gamma \cap \{x=1\} \)
- \( g \) = g_y_0 on \(\Gamma \cap \{y=0\} \)
- \( g \) = g_y_1 on \(\Gamma \cap \{y=1\} \)
DIFFUSIVITY TENSOR \( B(x) = P(x) D ^TP(x) \) with \( D = \left(\begin{array}{cc} L_1 & 0 \\ 0 & L_2 \end{array}\right)\)
and \( P(x) \) is the rotation matrix with angle \( \theta = \theta(x)\).
See choral/maple/example_diffusion_Neumann.mw for the computation of the problem data.
VARIATIONAL FORMULATION
- The finite element space is \( X_h\subset \Hu\).
Numerical probelm: find \(u\in X_h\) such that \( ~~~~\forall ~v \in X_h, ~~~~ \)
\[ \int_\Omega B(x) \nabla u\cdot\nabla v \,\dx ~+~ \int_\Omega \rho(x)\, u\, v \,\dx ~=~ \int_\Omega f \, v \,\dx ~+~ \int_\Gamma g \, v \,\dx \]
NUMERICAL RESOLUTION
The basis functions are \( (v_i)_{1\le i\le N} \) (basis of \(X_h\)).
- Computation of the stiffness matrix
\[ S =[s_{i,\,j}]_{1\le i,\,j\le N}, \quad \quad s_{i,\,j} = \int_\Omega B(x) \nabla v_i\cdot\nabla v_j \,\dx \]
- Computation of the mass matrix
\[ M =[s_{i,\,j}]_{1\le i,\,j\le N}, \quad \quad m_{i,\,j} = \int_\Omega \rho(x)\, v_i\, v_j \,\dx \]
- Computation of the right hand side for the volumoc source term \( f \)
\[ F = (f_i)_{1\le i\le N}, \quad \quad f_i = \int_\Omega f \, v_i \,\dx \]
- Computation of the right hand side for the surfacic source term \( g \)
\[ G = (g_i)_{1\le i\le N}, \quad \quad g_i = \int_\Gamma g \, v_i \,\dx \]
- Resolution of the (symmetric positive definite) system
\[ (M+S) U_h = F + G \]
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, October 2019
Definition at line 84 of file diffusion_Neumann.f90.
◆ f()
| real(rp) function diffusion_neumann::f | ( | real(rp), dimension(3), intent(in) | x | ) |
right hand side 'f'
Definition at line 356 of file diffusion_Neumann.f90.
◆ g_x_0()
| real(rp) function diffusion_neumann::g_x_0 | ( | real(rp), dimension(3), intent(in) | x | ) |
g(x) on {x=0}
Definition at line 499 of file diffusion_Neumann.f90.
◆ g_x_1()
| real(rp) function diffusion_neumann::g_x_1 | ( | real(rp), dimension(3), intent(in) | x | ) |
g(x) on {x=1}
Definition at line 516 of file diffusion_Neumann.f90.
◆ g_y_0()
| real(rp) function diffusion_neumann::g_y_0 | ( | real(rp), dimension(3), intent(in) | x | ) |
g(x) on {y=0}
Definition at line 527 of file diffusion_Neumann.f90.
◆ g_y_1()
| real(rp) function diffusion_neumann::g_y_1 | ( | real(rp), dimension(3), intent(in) | x | ) |
g(x) on {y=1}
Definition at line 544 of file diffusion_Neumann.f90.
◆ grad_u()
| real(rp) function, dimension(3) diffusion_neumann::grad_u | ( | real(rp), dimension(3), intent(in) | x | ) |
gradient of the exact solution
Definition at line 343 of file diffusion_Neumann.f90.
◆ rho()
| real(rp) function diffusion_neumann::rho | ( | real(rp), dimension(3), intent(in) | x | ) |
density '(x)'
Definition at line 392 of file diffusion_Neumann.f90.
◆ theta()
| real(rp) function diffusion_neumann::theta | ( | real(rp), dimension(3), intent(in) | x | ) |
Angle of the principal direction with e_x.
Definition at line 422 of file diffusion_Neumann.f90.
◆ u()
| real(rp) function diffusion_neumann::u | ( | real(rp), dimension(3), intent(in) | x | ) |
exact solution 'u'
Definition at line 332 of file diffusion_Neumann.f90.
◆ x_ge_1()
| real(rp) function diffusion_neumann::x_ge_1 | ( | real(rp), dimension(3), intent(in) | x | ) |
To characterise {x=1}.
Definition at line 467 of file diffusion_Neumann.f90.
◆ x_le_0()
| real(rp) function diffusion_neumann::x_le_0 | ( | real(rp), dimension(3), intent(in) | x | ) |
To characterise {x=0}.
Definition at line 456 of file diffusion_Neumann.f90.
◆ y_ge_1()
| real(rp) function diffusion_neumann::y_ge_1 | ( | real(rp), dimension(3), intent(in) | x | ) |
To characterise {x=1}.
Definition at line 488 of file diffusion_Neumann.f90.
◆ y_le_0()
| real(rp) function diffusion_neumann::y_le_0 | ( | real(rp), dimension(3), intent(in) | x | ) |
To characterise {x=0}.
Definition at line 477 of file diffusion_Neumann.f90.
