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‎_tutorials/structural_mechanics/Nonlinear_Elasticity.md‎

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---
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title: Non-linear Elasticity
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permalink: /tutorials/Nonlinear_Elasticity/
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---
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| Written by | for Version | Revised by | Revision date | Revised version |
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| --- | --- | --- | --- | --- |
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| [@rsanfer](https://github.com/rsanfer) | 7.0.0 | [@rsanfer](https://github.com/rsanfer) | Jan 28, 2020 | 7.0.1 |
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###### **At one glance**
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###### Solver: ```ELASTICITY```
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###### Uses: ```SU2_CFD```
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###### Complexity: <span style="color: orange;">**Intermediate**</span>
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###### Prerequisites: [**Linear Elasticity**](../Linear_Elasticity/)
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### Goals
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Once completed the tutorial on [Linear Elasticity](../Linear_Elasticity/), we can move on to non-linear problems. This document will guide you throught the following capabilities of SU2:
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- Setting up a non-linear structural problem with large deformations
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- Non-linear (hyperelastic) material law
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- Follower boundary conditions
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- Incremental loads
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In this tutorial, we use the same problem definition as for the linear elastic case: a vertical, slender cantilever, clamped in its base, and subject to a horizontal, follower load $$P$$ on its left boundary. This is shown in Fig. 1.
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![ProblemSetup](../structural_mechanics/images/lin1.png)
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### Resources
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You can find the resources for this tutorial in the same [structural_mechanics/cantilever](https://github.com/rsanfer/Tutorials/blob/master/structural_mechanics/cantilever) folder in the [Tutorials repository](https://github.com/rsanfer/Tutorials). You can reuse the mesh file [mesh_cantilever.su2](https://github.com/rsanfer/Tutorials/blob/master/structural_mechanics/cantilever/mesh_cantilever.su2)
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from the Linear Elasticity tutorial, but you will need a new config file, [config_nonlinear.cfg](https://github.com/rsanfer/Tutorials/blob/master/structural_mechanics/cantilever/config_nonlinear.cfg).
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### Background
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SU2 has been designed using a finite-deformation framework$$^1$$ to account for geometrical and material non-linearities.
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We can write the non-linear structural problem via the residual equation
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$$\mathscr{S}(\mathbf{u}) = \mathbf{T}(\mathbf{u}) - \mathbf{F}_b - \mathbf{F}_{\Gamma}(\mathbf{u})$$
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which has been obtained from the weak formulation of the structural problem defined using the principle of virtual work and discretized using FEM. The state variable $$\mathbf{u}$$ corresponds to the displacements of the structure with respect to its reference configuration $$\mathbf{R}$$. $$\mathbf{T}(\mathbf{u})$$ are the internal equivalent forces, $$\mathbf{F}_b$$ are the body forces and $$\mathbf{F}_{\Gamma}(\mathbf{u})$$ are the surface forces applied over the external boundaries of the structure.
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We can linearize the structural equation as
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$$\mathscr{S}(\mathbf{u}) + \frac{\partial \mathscr{S}(\mathbf{u})}{\partial \mathbf{u}} \Delta \mathbf{u} = \mathbf{0}$$
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where the tangent matrix is $$\mathbf{K} = \partial \mathscr{S}(\mathbf{u})/\partial \mathbf{u}$$. The linearized problem is solved using a Newton method until convergence. Further description of the approach adopted in SU2 can be found in a [paper](https://doi.org/10.1002/nme.5700) by Sanchez _et al_$$^2$$.
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#### Mesh Description
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The cantilever is discretized using 1000 4-node quad elements with linear interpolation. The boundaries are defined as follows:
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![Mesh](../structural_mechanics/images/lin2.png)
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#### Configuration File Options
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We start the tutorial by definining the structural properties. In this case, we will solve a geometrically non-linear problem, with a Neo-Hookean material model.
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```
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GEOMETRIC_CONDITIONS= LARGE_DEFORMATIONS
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MATERIAL_MODEL= NEO_HOOKEAN
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```
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We adopt a **plane strain** formulation for the 2D problem, and reduce the stiffness of the cantilever by 2 orders of magnitude as compared to the [Linear Elasticity](../Linear_Elasticity/) tutorial, so that the beam undergoes large deformations:
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```
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ELASTICITY_MODULUS=5E7
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POISSON_RATIO=0.35
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FORMULATION_ELASTICITY_2D = PLANE_STRAIN
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```
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Now, we define the Newton-Raphson strategy to solve the non-linear problem. We set the solution method and maximum number of subiterations using
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```
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NONLINEAR_FEM_SOLUTION_METHOD = NEWTON_RAPHSON
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INNER_ITER = 15
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```
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Then, the convergence criteria is set to evaluate the norm of the displacement vector (```RMS_UTOL```), the residual vector (```RMS_RTOL```) and the energy (```RMS_ETOL```). We set the convergence values to -8.0, which is the log<sub>10</sub> of the previously defined magnitudes.
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```
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CONV_FIELD= RMS_UTOL, RMS_RTOL, RMS_ETOL
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CONV_RESIDUAL_MINVAL= -8,
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```
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Finally, we define the boundary conditions. In this case, the load applied to the ```left``` boundary will be an uniform, _follower_ pressure, that is, its direction will change with the changes in the normals of the surface. This is done as
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```
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MARKER_CLAMPED = ( clamped )
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MARKER_PRESSURE = ( left, 1E3, upper, 0, right, 0)
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```
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### Running SU2
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Follow the links provided to download the [config](https://github.com/rsanfer/Tutorials/blob/master/structural_mechanics/cantilever/config_nonlinear.cfg) and [mesh](https://github.com/rsanfer/Tutorials/blob/master/structural_mechanics/cantilever/mesh_cantilever.su2) files. Execute the code with the standard command
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```
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SU2_CFD config_nonlinear.cfg
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```
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which will show the following convergence history:
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```
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+----------------------------------------------------------------+
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| Inner_Iter| rms[U]| rms[R]| rms[E]| VonMises|
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+----------------------------------------------------------------+
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| 0| -1.454599| -0.001635| -2.082200| 1.1259e+06|
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| 1| -2.406133| 3.664361| -0.038068| 1.1224e+06|
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| 2| -3.623182| 2.288350| -2.827362| 1.1053e+06|
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| 3| -3.096627| -0.433125| -5.369087| 1.1229e+06|
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| 4| -4.547761| 0.493533| -6.401479| 1.1222e+06|
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| 5| -5.306858| -2.420278| -9.740703| 1.1221e+06|
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| 6| -6.878854| -3.854770| -12.877010| 1.1221e+06|
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| 7| -8.478641| -6.065417| -16.078409| 1.1221e+06|
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| 8| -10.081222| -7.667567| -19.283559| 1.1221e+06|
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| 9| -11.683669| -8.697378| -22.488189| 1.1221e+06|
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```
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The code is stopped as soon as the values of ```rms[U]```, ```rms[R]``` and ```rms[E]``` are below the convergence criteria set in the config file. The displacement field obtained in _nonlinear.vtk_ is shown below:
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![Nonlinear Results](../structural_mechanics/images/nlin1.png)
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### Increasing the load
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The magnitude of the residual can limit the convergence of the solver, particularly for those cases in which the problem is very non-linear. Say, for example, that we multiply the load in the ```left``` boundary by 4, using
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```
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MARKER_PRESSURE = ( left, 4E3, upper, 0, right, 0)
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```
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Running SU2 now would result in the divergence of the solver,
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```
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+----------------------------------------------------------------+
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| Inner_Iter| rms[U]| rms[R]| rms[E]| VonMises|
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+----------------------------------------------------------------+
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| 0| -0.852539| 0.600425| -0.878080| 4.5753e+06|
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| 1| -1.130461| 4.628370| 2.207826| 4.4618e+06|
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| 2| -2.198592| 4.658260| 1.539054| 2.8742e+08|
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| 3| -nan| -nan| -nan| 0.0000e+00|
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| 4| -nan| -nan| -nan| 0.0000e+00|
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```
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due to a large imbalance between the internal and external loads. It can be observed that the order of magnitude of the residual vector grows above 10<sup>4</sup>. To solve this problem, we have incorporated an incremental approach to SU2, that applies the load in linear steps,
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$$\mathbf{F}_i = \frac{i}{N}\mathbf{F}$$
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where $$i$$ is the step and $$N$$ the maximum number of increments allowed. To apply the load in increments, we need to use the following commands
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```
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INCREMENTAL_LOAD = YES
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NUMBER_INCREMENTS = 25
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INCREMENTAL_CRITERIA = (2.0, 2.0, 2.0)
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```
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where $$N$$ = 25, and the criteria to apply the increments is that either one of ```rms[U] > 2.0```, ```rms[R] > 2.0``` or ```rms[E] > 2.0``` is met. We add an option to print out the load increment to the screen output:
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```
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SCREEN_OUTPUT = (INNER_ITER, LOAD_INCREMENT, RMS_UTOL, RMS_RTOL, RMS_ETOL, VMS)
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```
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If we now run the code,
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```
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+-----------------------------------------------------------------------------+
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| Inner_Iter| Load[%]| rms[U]| rms[R]| rms[E]| VonMises|
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+-----------------------------------------------------------------------------+
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| 0| 100.00%| -0.852539| 0.600425| -0.878080| 4.5753e+06|
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| 1| 100.00%| -1.130461| 4.628370| 2.207826| 4.4618e+06|
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Incremental load: increment 1
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+-----------------------------------------------------------------------------+
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| Inner_Iter| Load[%]| rms[U]| rms[R]| rms[E]| VonMises|
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+-----------------------------------------------------------------------------+
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| 0| 4.00%| -2.250479| -0.797515| -3.673960| 1.7944e+05|
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| 1| 4.00%| -4.005792| 2.096002| -3.207727| 1.7948e+05|
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| 2| 4.00%| -5.555428| -0.881184| -9.113457| 1.7949e+05|
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| 3| 4.00%| -7.611025| -4.228316| -14.254076| 1.7949e+05|
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| 4| 4.00%| -10.343696| -7.532184| -19.528442| 1.7949e+05|
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| 5| 4.00%| -13.138042| -8.791438| -24.949847| 1.7949e+05|
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```
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given that ```4.628370 > 2.0``` and ```2.207826 > 2.0```, the load is applied in $$\Delta \mathbf{F} = 1/25 \mathbf{F}$$, completing the simulation in the 25th step,
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```
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Incremental load: increment 25
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+-----------------------------------------------------------------------------+
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| Inner_Iter| Load[%]| rms[U]| rms[R]| rms[E]| VonMises|
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+-----------------------------------------------------------------------------+
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| 0| 100.00%| -2.188244| -0.786306| -3.616397| 4.2701e+06|
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| 1| 100.00%| -3.417010| 2.241142| -2.911125| 4.2325e+06|
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| 2| 100.00%| -2.802795| -0.383540| -4.853108| 4.1931e+06|
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| 3| 100.00%| -3.461966| 1.059169| -5.203155| 4.2102e+06|
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| 4| 100.00%| -4.053497| -0.412173| -7.024221| 4.2103e+06|
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| 5| 100.00%| -4.505028| -1.211625| -8.188042| 4.2090e+06|
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| 6| 100.00%| -5.044900| -2.434771| -9.288963| 4.2093e+06|
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| 7| 100.00%| -5.572267| -3.418852| -10.345812| 4.2092e+06|
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| 8| 100.00%| -6.099258| -4.140827| -11.400141| 4.2093e+06|
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| 9| 100.00%| -6.625915| -4.693663| -12.453409| 4.2093e+06|
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| 10| 100.00%| -7.152645| -5.224056| -13.506885| 4.2093e+06|
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| 11| 100.00%| -7.679350| -5.750731| -14.560291| 4.2093e+06|
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| 12| 100.00%| -8.206063| -6.277523| -15.613718| 4.2093e+06|
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| 13| 100.00%| -8.732773| -6.804154| -16.667138| 4.2093e+06|
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| 14| 100.00%| -9.259484| -7.329682| -17.720560| 4.2093e+06|
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| 15| 100.00%| -9.786195| -7.846390| -18.773983| 4.2093e+06|
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| 16| 100.00%| -10.312907| -8.293905| -19.827405| 4.2093e+06|
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```
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The displacement field is now
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![Nonlinear Incremental Results](../structural_mechanics/images/nlin2.png)

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