diff --git a/docs/examples/ex02.py b/docs/examples/ex02.py index 0d218b22..cae75251 100644 --- a/docs/examples/ex02.py +++ b/docs/examples/ex02.py @@ -1,8 +1,6 @@ -r""" +r"""Kirchhoff plate problem. -This example demonstrates the solution of a slightly more complicated problem -with multiple boundary conditions and a fourth-order differential operator. We -consider the `Kirchhoff plate bending problem +This example demonstrates the solution a fourth order `Kirchhoff plate bending problem `_ which finds its applications in solid mechanics. For a stationary plate of constant thickness :math:`d`, the governing equation reads: find the deflection :math:`u @@ -16,20 +14,6 @@ The Young's modulus of steel is :math:`E = 200 \cdot 10^9\,\text{Pa}` and Poisson ratio :math:`\nu = 0.3`. -In reality, the operator - -.. math:: - \frac{Ed^3}{12(1-\nu^2)} \Delta^2 -is a combination of multiple first-order operators: - -.. math:: - \boldsymbol{K}(u) = - \boldsymbol{\varepsilon}(\nabla u), \quad \boldsymbol{\varepsilon}(\boldsymbol{w}) = \frac12(\nabla \boldsymbol{w} + \nabla \boldsymbol{w}^T), -.. math:: - \boldsymbol{M}(u) = \frac{d^3}{12} \mathbb{C} \boldsymbol{K}(u), \quad \mathbb{C} \boldsymbol{T} = \frac{E}{1+\nu}\left( \boldsymbol{T} + \frac{\nu}{1-\nu}(\text{tr}\,\boldsymbol{T})\boldsymbol{I}\right), -where :math:`\boldsymbol{I}` is the identity matrix. In particular, - -.. math:: - \frac{Ed^3}{12(1-\nu^2)} \Delta^2 u = - \text{div}\,\textbf{div}\,\boldsymbol{M}(u). There are several boundary conditions that the problem can take. The *fully clamped* boundary condition reads @@ -61,10 +45,7 @@ `_ which is a piecewise quadratic :math:`C^0`-continuous element for biharmonic problems. -The full source code of the example reads as follows: - -.. literalinclude:: examples/ex02.py - :start-after: EOF""" +""" from skfem import * from skfem.models.poisson import unit_load from skfem.helpers import dd, ddot, trace, eye