Commit 5682d82b by Jeremy BLEYER

parent 4c8e7114
abstract.pdf 0 → 100644
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 ... @@ -9,7 +9,7 @@ ... @@ -9,7 +9,7 @@ # # # We will investigate here the case of a hollow hemisphere of inner (resp. outer) radius $R_i$ (resp. $R_e$). Due to the revolution symmetry, the 2D cross-section corresponds to a quarter of a hollow cylinder. # We will investigate here the case of a hollow hemisphere of inner (resp. outer) radius $R_i$ (resp. $R_e$). Due to the revolution symmetry, the 2D cross-section corresponds to a quarter of a hollow cylinder. # In[50]: # In[51]: from __future__ import print_function from __future__ import print_function ... @@ -65,7 +65,7 @@ ds = Measure("ds", subdomain_data=facets) ... @@ -65,7 +65,7 @@ ds = Measure("ds", subdomain_data=facets) # # # > **Note**: we could also express the strain components in the form of a vector of size 4 in alternative of the 3D tensor representation implemented below. # > **Note**: we could also express the strain components in the form of a vector of size 4 in alternative of the 3D tensor representation implemented below. # In[43]: # In[52]: x = SpatialCoordinate(mesh) x = SpatialCoordinate(mesh) ... @@ -104,7 +104,7 @@ def sigma(v): ... @@ -104,7 +104,7 @@ def sigma(v): # # # The final formulation is therefore pretty straightforward. Since a uniform pressure loading is applied on the outer boundary, we will also need the exterior normal vector to define the work of external forces form. # The final formulation is therefore pretty straightforward. Since a uniform pressure loading is applied on the outer boundary, we will also need the exterior normal vector to define the work of external forces form. # In[44]: # In[53]: n = FacetNormal(mesh) n = FacetNormal(mesh) ... @@ -125,7 +125,7 @@ u = Function(V, name="Displacement") ... @@ -125,7 +125,7 @@ u = Function(V, name="Displacement") # \quad u_z=0 # \quad u_z=0 # # # In[45]: # In[54]: bcs = [DirichletBC(V.sub(1), Constant(0), facets, 1), bcs = [DirichletBC(V.sub(1), Constant(0), facets, 1), ... @@ -135,7 +135,7 @@ print("Inwards radial displacement at (r=Re, theta=0): {:1.7f} (FE) {:1.7f} (Exa ... @@ -135,7 +135,7 @@ print("Inwards radial displacement at (r=Re, theta=0): {:1.7f} (FE) {:1.7f} (Exa print("Inwards radial displacement at (r=Ri, theta=0): {:1.7f} (FE) {:1.7f} (Exact)".format(-u(Ri, 0.)[0], float(Re**3/(Re**3-Ri**3)*((1-2*nu)*Ri+(1+nu)*Ri/2)*p/E))) print("Inwards radial displacement at (r=Ri, theta=0): {:1.7f} (FE) {:1.7f} (Exact)".format(-u(Ri, 0.)[0], float(Re**3/(Re**3-Ri**3)*((1-2*nu)*Ri+(1+nu)*Ri/2)*p/E))) # In[46]: # In[55]: plt.figure() plt.figure() ... @@ -145,7 +145,7 @@ plt.show() ... @@ -145,7 +145,7 @@ plt.show() # The second loading case corresponds to a fully clamped condition on $z=0$, the vertical boundary remaining in smooth contact. # The second loading case corresponds to a fully clamped condition on $z=0$, the vertical boundary remaining in smooth contact. # In[48]: # In[56]: bcs = [DirichletBC(V, Constant((0., 0.)), facets, 1), bcs = [DirichletBC(V, Constant((0., 0.)), facets, 1), ... ...
 ... @@ -22,6 +22,18 @@ illustrating the versatility of FEniCS. ... @@ -22,6 +22,18 @@ illustrating the versatility of FEniCS. The full set of demos can be obtained from the *COmputational MEchanics Toolbox* (COMET) available at The full set of demos can be obtained from the *COmputational MEchanics Toolbox* (COMET) available at https://gitlab.enpc.fr/jeremy.bleyer/comet-fenics. https://gitlab.enpc.fr/jeremy.bleyer/comet-fenics. If you find these demos useful for your research work, please consider citing them using the following Zenodo DOI https://doi.org/10.5281/zenodo.1287832 .. code-block:: none @article{bleyer2018numericaltours, title={Numerical Tours of Computational Mechanics with FEniCS}, DOI={10.5281/zenodo.1287832}, publisher={Zenodo}, author={Jeremy Bleyer}, year={2018}} ----------------------- ----------------------- How do I get started ? How do I get started ? ... ...
 ... @@ -243,7 +243,7 @@ symmetry, the 2D cross-section corresponds to a quarter of a hollow ... @@ -243,7 +243,7 @@ symmetry, the 2D cross-section corresponds to a quarter of a hollow cylinder.

cylinder.

In [50]:                                                                                        In [51]:

...  @@ -1090,7 +1090,7 @@ a vector of size 4 in alternative of the 3D tensor representation                 ...  @@ -1090,7 +1090,7 @@ a vector of size 4 in alternative of the 3D tensor representation
implemented below.
implemented below.

In [43]:                                                                                        In [52]:

...  @@ -1148,7 +1148,7 @@ need the exterior normal vector to define the work of external forces             ...  @@ -1148,7 +1148,7 @@ need the exterior normal vector to define the work of external forces
form.

form.

In [44]:                                                                                        In [53]:

...  @@ -1175,7 +1175,7 @@ u_r(r) = -\dfrac{R_e^3}{R_e^3-R_i^3}\left((1 − 2\nu)r + (1 + \nu)\dfrac{R_i^3}    ...  @@ -1175,7 +1175,7 @@ u_r(r) = -\dfrac{R_e^3}{R_e^3-R_i^3}\left((1 − 2\nu)r + (1 + \nu)\dfrac{R_i^3}

In [45]:                                                                                        In [54]:

...  @@ -1200,7 +1200,7 @@ Inwards radial displacement at (r=Ri, theta=0): 0.0020879 (FE) 0.0020894 (Exact)  ...  @@ -1200,7 +1200,7 @@ Inwards radial displacement at (r=Ri, theta=0): 0.0020879 (FE) 0.0020894 (Exact)

In [46]:                                                                                        In [55]:

...  @@ -1996,7 +1996,7 @@ if (IPython.notebook.kernel != null) {                                            ...  @@ -1996,7 +1996,7 @@ if (IPython.notebook.kernel != null) {
$$z=0$$, the vertical boundary remaining in smooth contact.

$$z=0$$, the vertical boundary remaining in smooth contact.

In [48]:                                                                                        In [56]:

...                                                                                                          ...
 ... @@ -64,6 +64,16 @@ getting started with FEniCS using solid mechanics examples.

... @@ -64,6 +64,16 @@ getting started with FEniCS using solid mechanics examples.

illustrating the versatility of FEniCS.

illustrating the versatility of FEniCS.

The full set of demos can be obtained from the COmputational MEchanics Toolbox (COMET) available at

The full set of demos can be obtained from the COmputational MEchanics Toolbox (COMET) available at https://gitlab.enpc.fr/jeremy.bleyer/comet-fenics.

https://gitlab.enpc.fr/jeremy.bleyer/comet-fenics.

If you find these demos useful for your research work, please consider citing them using the following Zenodo DOI https://doi.org/10.5281/zenodo.1287832

@article{bleyer2018numericaltours,
title={Numerical Tours of Computational Mechanics with FEniCS},
DOI={10.5281/zenodo.1287832},
publisher={Zenodo},
author={Jeremy Bleyer},
year={2018}}

This diff is collapsed.
 ... @@ -22,6 +22,18 @@ illustrating the versatility of FEniCS. ... @@ -22,6 +22,18 @@ illustrating the versatility of FEniCS. The full set of demos can be obtained from the *COmputational MEchanics Toolbox* (COMET) available at The full set of demos can be obtained from the *COmputational MEchanics Toolbox* (COMET) available at https://gitlab.enpc.fr/jeremy.bleyer/comet-fenics. https://gitlab.enpc.fr/jeremy.bleyer/comet-fenics. If you find these demos useful for your research work, please consider citing them using the following Zenodo DOI https://doi.org/10.5281/zenodo.1287832 .. code-block:: none @article{bleyer2018numericaltours, title={Numerical Tours of Computational Mechanics with FEniCS}, DOI={10.5281/zenodo.1287832}, publisher={Zenodo}, author={Jeremy Bleyer}, year={2018}} ----------------------- ----------------------- How do I get started ? How do I get started ? ... ...
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