Functions/Subroutines
program	elasticity_neumann_2d
	SOLVES THE LINEAR ELASTICITY PROBLEM with Neumann 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_u1 (x)
	gradient of the exact solution More...

real(rp) function, dimension(3)	grad_u2 (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	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_1_x_0 (x)
	g(x) on {x=0}, first component More...

real(rp) function	g_2_x_0 (x)
	g(x) on {x=0}, second component More...

real(rp) function	g_1_x_1 (x)
	g(x) on {x=1}, first component More...

real(rp) function	g_2_x_1 (x)
	g(x) on {x=1}, second component More...

real(rp) function	g_1_y_0 (x)
	g(x) on {y=0}, first component More...

real(rp) function	g_2_y_0 (x)
	g(x) on {y=0}, second component More...

real(rp) function	g_1_y_1 (x)
	g(x) on {y=1}, first component More...

real(rp) function	g_2_y_1 (x)
	g(x) on {y=1}, second component More...

Function/Subroutine Documentation

◆ elasticity_neumann_2d()

program elasticity_neumann_2d ( )

SOLVES THE LINEAR ELASTICITY PROBLEM with Neumann boundary conditions

Search for \(u:~\Omega \mapsto \R^2\) with \( \Omega= [0,1]^2 \) that satisfies

\(~~~~~~~~~ -\dv(A(x) e(u)) = f ~~~\) on \(~~~ \Omega \)

\(~~~~~~~~~~ A(x) e(u) n = g ~~~\) on \(~~~\Gamma = \partial \Omega ~~~\) with

\( f:~\Omega\mapsto \R^2\)
\( g:~\Gamma\mapsto \R^2\) given on each side of \(\Omega\): \( \left\{\begin{array}{c} g = \text{ g_x_0 on } \Gamma \cap \{x=0\} \\ g = \text{ g_x_1 on } \Gamma \cap \{x=1\} \\ g = \text{ g_y_0 on } \Gamma \cap \{y=0\} \\ g = \text{ g_y_1 on } \Gamma \cap \{y=1\} \end{array}\right.\)

HOOKE TENSOR \( A(x)\xi = \lambda(x) Tr(\xi) Id + 2 \mu(x)\xi \) with \( \lambda,~\mu~: \Omega \mapsto \R \)

SYMMETRISED GRADIENT \( e(u) = (\nabla u + ^T\nabla u)/2 \)

PROBLEM DATA See choral/maple/example_elasticity_Neumann.mw
for the computation of the problem data.

VARIATIONAL FORMULATION find \(u\in \Hu^2\) such that \( ~~~~\forall ~v \in \Hu^2, ~~~~ \)

\[ \int_\Omega A(x) e(u):e(v) \,\dx ~=~ \int_\Omega f \cdot v \,\dx ~+~ \int_\Gamma g \cdot v \,\dx \]

with \( \xi:\zeta = \sum_{1\le i,j \le 2} \xi_{ij}\zeta_{ij}\) for the 2 matrices \( \xi~, \zeta \in \R^{2\times 2}\).

NUMERICAL RESOLUTION

Definition of a finite element space \( X_h \subset \Hu \)
Definition of the finite element space \( Y = [X_h]^2 \)
and of the associated basis functions \( (v_i)_{1\le i\le N} \)

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 \( F = (f_i)_{1\le i\le N} \)
\[ f_i = \int_\Omega f \cdot v_i \,\dx \]

Computation of the right hand side \( G = (g_i)_{1\le i\le N} \)
\[ g_i = \int_\Gamma g \cdot v_i \,\dx \]

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, November 2019

Definition at line 81 of file elasticity_Neumann_2D.f90.

Here is the call graph for this function:

◆ f_1()

real(rp) function elasticity_neumann_2d::f_1 ( real(rp), dimension(3), intent(in) x )

right hand side 'f'= =(f_1, f_2)

Definition at line 373 of file elasticity_Neumann_2D.f90.

Here is the call graph for this function:

◆ f_2()

real(rp) function elasticity_neumann_2d::f_2 ( real(rp), dimension(3), intent(in) x )

Definition at line 380 of file elasticity_Neumann_2D.f90.

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◆ g_1_x_0()

real(rp) function elasticity_neumann_2d::g_1_x_0 ( real(rp), dimension(3), intent(in) x )

g(x) on {x=0}, first component

Definition at line 434 of file elasticity_Neumann_2D.f90.

◆ g_1_x_1()

real(rp) function elasticity_neumann_2d::g_1_x_1 ( real(rp), dimension(3), intent(in) x )

g(x) on {x=1}, first component

Definition at line 454 of file elasticity_Neumann_2D.f90.

◆ g_1_y_0()

real(rp) function elasticity_neumann_2d::g_1_y_0 ( real(rp), dimension(3), intent(in) x )

g(x) on {y=0}, first component

Definition at line 474 of file elasticity_Neumann_2D.f90.

◆ g_1_y_1()

real(rp) function elasticity_neumann_2d::g_1_y_1 ( real(rp), dimension(3), intent(in) x )

g(x) on {y=1}, first component

Definition at line 494 of file elasticity_Neumann_2D.f90.

◆ g_2_x_0()

real(rp) function elasticity_neumann_2d::g_2_x_0 ( real(rp), dimension(3), intent(in) x )

g(x) on {x=0}, second component

Definition at line 444 of file elasticity_Neumann_2D.f90.

◆ g_2_x_1()

real(rp) function elasticity_neumann_2d::g_2_x_1 ( real(rp), dimension(3), intent(in) x )

g(x) on {x=1}, second component

Definition at line 464 of file elasticity_Neumann_2D.f90.

◆ g_2_y_0()

real(rp) function elasticity_neumann_2d::g_2_y_0 ( real(rp), dimension(3), intent(in) x )

g(x) on {y=0}, second component

Definition at line 484 of file elasticity_Neumann_2D.f90.

◆ g_2_y_1()

real(rp) function elasticity_neumann_2d::g_2_y_1 ( real(rp), dimension(3), intent(in) x )

g(x) on {y=1}, second component

Definition at line 504 of file elasticity_Neumann_2D.f90.

◆ grad_u1()

real(rp) function, dimension(3) elasticity_neumann_2d::grad_u1 ( real(rp), dimension(3), intent(in) x )

gradient of the exact solution

Definition at line 351 of file elasticity_Neumann_2D.f90.

◆ grad_u2()

real(rp) function, dimension(3) elasticity_neumann_2d::grad_u2 ( real(rp), dimension(3), intent(in) x )

Definition at line 360 of file elasticity_Neumann_2D.f90.

◆ lambda()

real(rp) function elasticity_neumann_2d::lambda ( real(rp), dimension(3), intent(in) x )

Lame coefficient 'lambda'.

Definition at line 312 of file elasticity_Neumann_2D.f90.

◆ mu()

real(rp) function elasticity_neumann_2d::mu ( real(rp), dimension(3), intent(in) x )

Lame coefficient 'mu'.

Definition at line 322 of file elasticity_Neumann_2D.f90.

◆ u_1()

real(rp) function elasticity_neumann_2d::u_1 ( real(rp), dimension(3), intent(in) x )

exact solution u=(u_1, u_2)

Definition at line 332 of file elasticity_Neumann_2D.f90.

◆ u_2()

real(rp) function elasticity_neumann_2d::u_2 ( real(rp), dimension(3), intent(in) x )

Definition at line 339 of file elasticity_Neumann_2D.f90.

◆ x_ge_1()

real(rp) function elasticity_neumann_2d::x_ge_1 ( real(rp), dimension(3), intent(in) x )

To characterise {x=1}.

Definition at line 402 of file elasticity_Neumann_2D.f90.

◆ x_le_0()

real(rp) function elasticity_neumann_2d::x_le_0 ( real(rp), dimension(3), intent(in) x )

To characterise {x=0}.

Definition at line 391 of file elasticity_Neumann_2D.f90.

◆ y_ge_1()

real(rp) function elasticity_neumann_2d::y_ge_1 ( real(rp), dimension(3), intent(in) x )

To characterise {x=1}.

Definition at line 423 of file elasticity_Neumann_2D.f90.

◆ y_le_0()

real(rp) function elasticity_neumann_2d::y_le_0 ( real(rp), dimension(3), intent(in) x )

To characterise {x=0}.

Definition at line 412 of file elasticity_Neumann_2D.f90.

Functions/Subroutines

Function/Subroutine Documentation

◆ elasticity_neumann_2d()

◆ f_1()

◆ f_2()

◆ g_1_x_0()

◆ g_1_x_1()

◆ g_1_y_0()

◆ g_1_y_1()

◆ g_2_x_0()

◆ g_2_x_1()

◆ g_2_y_0()

◆ g_2_y_1()

◆ grad_u1()

◆ grad_u2()

◆ lambda()

◆ mu()

◆ u_1()

◆ u_2()

◆ x_ge_1()

◆ x_le_0()

◆ y_ge_1()

◆ y_le_0()