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update docstring in second example
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kinnala committed Jun 15, 2024
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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
<https://en.wikipedia.org/wiki/Kirchhoff%E2%80%93Love_plate_theory>`_ 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
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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
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<https://users.aalto.fi/~jakke74/WebFiles/Slides-Niiranen-ADMOS-09.pdf>`_ 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
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