<h1>Results</h1>
-Firstly, we present a comparison of a series of results with those
+Firstly, we present a comparison of a series of results with those
in the literature (see Reese et al (2000)) to demonstrate that the program works as expected.
We begin with a comparison of the convergence with mesh refinement for the $Q_1-DGPM_0-DGPM_0$ and
$Q_2-DGPM_1-DGPM_1$ formulations, as summarised in the figure below.
The vertical displacement of the midpoint of the upper surface of the block is used to assess convergence.
-Both schemes demonstrate good convergence properties for varying values of the load parameter $p/p_0$.
+Both schemes demonstrate good convergence properties for varying values of the load parameter $p/p_0$.
The results agree with those in the literature.
-The lower-order formulation typically overestimates the displacement for low levels of refinement,
+The lower-order formulation typically overestimates the displacement for low levels of refinement,
while the higher-order interpolation scheme underestimates it, but be a lesser degree.
This benchmark, and a series of others not shown here, give us confidence that the code is working
as it should.
</table>
-A typical screen output generated by running the problem is shown below.
+A typical screen output generated by running the problem is shown below.
The particular case demonstrated is that of the $Q_2-DGPM_1-DGPM_1$ formulation.
-It is clear that, using the Newton-Raphson method, quadratic convergence of the solution is obtained.
-Solution convergence is achieved within 5 Newton increments for all time-steps.
+It is clear that, using the Newton-Raphson method, quadratic convergence of the solution is obtained.
+Solution convergence is achieved within 5 Newton increments for all time-steps.
The converged displacement's $L_2$-norm is several orders of magnitude less than the geometry scale.
@code
Timestep 1 @ 0.1s
___________________________________________________________________________________________________________________________________________________________
- SOLVER STEP | LIN_IT LIN_RES RES_NORM RES_U RES_P RES_J NU_NORM NU_U NU_P NU_J
+ SOLVER STEP | LIN_IT LIN_RES RES_NORM RES_U RES_P RES_J NU_NORM NU_U NU_P NU_J
___________________________________________________________________________________________________________________________________________________________
- 0 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 786 2.118e-06 1.000e+00 1.000e+00 0.000e+00 0.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
- 1 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 552 1.031e-03 8.563e-02 8.563e-02 9.200e-13 3.929e-08 1.060e-01 3.816e-02 1.060e-01 1.060e-01
- 2 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 667 5.602e-06 2.482e-03 2.482e-03 3.373e-15 2.982e-10 2.936e-03 2.053e-04 2.936e-03 2.936e-03
- 3 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 856 6.469e-10 2.129e-06 2.129e-06 2.245e-19 1.244e-13 1.887e-06 7.289e-07 1.887e-06 1.887e-06
- 4 ASM_R CONVERGED!
+ 0 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 786 2.118e-06 1.000e+00 1.000e+00 0.000e+00 0.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
+ 1 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 552 1.031e-03 8.563e-02 8.563e-02 9.200e-13 3.929e-08 1.060e-01 3.816e-02 1.060e-01 1.060e-01
+ 2 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 667 5.602e-06 2.482e-03 2.482e-03 3.373e-15 2.982e-10 2.936e-03 2.053e-04 2.936e-03 2.936e-03
+ 3 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 856 6.469e-10 2.129e-06 2.129e-06 2.245e-19 1.244e-13 1.887e-06 7.289e-07 1.887e-06 1.887e-06
+ 4 ASM_R CONVERGED!
___________________________________________________________________________________________________________________________________________________________
Relative errors:
Displacement: 7.289e-07
Timestep 10 @ 1.000e+00s
___________________________________________________________________________________________________________________________________________________________
- SOLVER STEP | LIN_IT LIN_RES RES_NORM RES_U RES_P RES_J NU_NORM NU_U NU_P NU_J
+ SOLVER STEP | LIN_IT LIN_RES RES_NORM RES_U RES_P RES_J NU_NORM NU_U NU_P NU_J
___________________________________________________________________________________________________________________________________________________________
- 0 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 874 2.358e-06 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
- 1 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 658 2.942e-04 1.544e-01 1.544e-01 1.208e+13 1.855e+06 6.014e-02 7.398e-02 6.014e-02 6.014e-02
- 2 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 790 2.206e-06 2.908e-03 2.908e-03 7.302e+10 2.067e+03 2.716e-03 1.433e-03 2.716e-03 2.717e-03
- 3 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 893 2.374e-09 1.919e-06 1.919e-06 4.527e+07 4.100e+00 1.672e-06 6.842e-07 1.672e-06 1.672e-06
- 4 ASM_R CONVERGED!
+ 0 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 874 2.358e-06 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
+ 1 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 658 2.942e-04 1.544e-01 1.544e-01 1.208e+13 1.855e+06 6.014e-02 7.398e-02 6.014e-02 6.014e-02
+ 2 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 790 2.206e-06 2.908e-03 2.908e-03 7.302e+10 2.067e+03 2.716e-03 1.433e-03 2.716e-03 2.717e-03
+ 3 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 893 2.374e-09 1.919e-06 1.919e-06 4.527e+07 4.100e+00 1.672e-06 6.842e-07 1.672e-06 1.672e-06
+ 4 ASM_R CONVERGED!
___________________________________________________________________________________________________________________________________________________________
Relative errors:
Displacement: 6.842e-07
-Using the Timer class, we can discern which parts of the code require the highest computational expense.
+Using the Timer class, we can discern which parts of the code require the highest computational expense.
For a case with a large number of degrees-of-freedom (i.e. a high level of refinement), a typical output of the Timer is given below.
-Much of the code in the tutorial has been developed based on the optimisations described,
-discussed and demonstrated in Step-18 and others.
-With over 93% of the time being spent in the linear solver, it is obvious that it may be necessary
-to invest in a better solver for large three-dimensional problems.
-The SSOR preconditioner is not multi-threaded but is effective for this class of solid problems.
+Much of the code in the tutorial has been developed based on the optimisations described,
+discussed and demonstrated in Step-18 and others.
+With over 93% of the time being spent in the linear solver, it is obvious that it may be necessary
+to invest in a better solver for large three-dimensional problems.
+The SSOR preconditioner is not multi-threaded but is effective for this class of solid problems.
It may be beneficial to investigate the use of another solver such as those available through the Trilinos library.
The first was for the coarsest grid and the lowest-order interpolation method: $Q_1-DGPM_0-DGPM_0$.
The second was on a refined grid using a $Q_2-DGPM_1-DGPM_1$ formulation.
The vertical component of the displacement, the pressure $\widetilde{p}$ and the dilatation $\widetilde{J}$ fields
-are shown below.
+are shown below.
For the first case it is clear that the coarse spatial discretisation coupled with large displacements leads to a low quality solution
(the loading ratio is $p/p_0=80$).
-Additionally, the pressure difference between elements is very large.
-The constant pressure field on the element means that the large pressure gradient is not captured.
+Additionally, the pressure difference between elements is very large.
+The constant pressure field on the element means that the large pressure gradient is not captured.
However, it should be noted that locking, which would be present in a standard $Q_1$ displacement formulation does not arise
-even in this poorly discretised case.
-The final vertical displacement of the tracked node on the top surface of the block is still within 12.5% of the converged solution.
-The pressure solution is very coarse and has large jumps between adjacent cells.
-It is clear that the volume nearest to the applied traction undergoes compression while the outer extents
-of the domain are in a state of expansion.
-The dilatation solution field and pressure field are clearly linked,
+even in this poorly discretised case.
+The final vertical displacement of the tracked node on the top surface of the block is still within 12.5% of the converged solution.
+The pressure solution is very coarse and has large jumps between adjacent cells.
+It is clear that the volume nearest to the applied traction undergoes compression while the outer extents
+of the domain are in a state of expansion.
+The dilatation solution field and pressure field are clearly linked,
with positive dilatation indicating regions of positive pressure and negative showing regions placed in compression.
-As discussed in the Introduction, a compressive pressure has a negative sign
-while an expansive pressure takes a positive sign.
-This stems from the definition of the volumetric strain energy function
+As discussed in the Introduction, a compressive pressure has a negative sign
+while an expansive pressure takes a positive sign.
+This stems from the definition of the volumetric strain energy function
and is opposite to the physically realistic interpretation of pressure.
</tr>
</table>
-Combining spatial refinement and a higher-order interpolation scheme results in a high-quality solution.
-Three grid refinements coupled with a $Q_2-DGPM_1-DGPM_1$ formulation produces
-a result that clearly captures the mechanics of the problem.
-The deformation of the traction surface is well resolved.
+Combining spatial refinement and a higher-order interpolation scheme results in a high-quality solution.
+Three grid refinements coupled with a $Q_2-DGPM_1-DGPM_1$ formulation produces
+a result that clearly captures the mechanics of the problem.
+The deformation of the traction surface is well resolved.
We can now observe the actual extent of the applied traction, with the maximum force being applied
-at the central point of the surface causing the largest compression.
-Even though very high strains are experienced in the domain,
-especially at the boundary of the region of applied traction,
-the solution remains accurate.
-The pressure field is captured in far greater detail than before.
-There is a clear distinction and transition between regions of compression and expansion,
-and the linear approximation of the pressure field allows a refined visualisation
-of the pressure at the sub-element scale.
+at the central point of the surface causing the largest compression.
+Even though very high strains are experienced in the domain,
+especially at the boundary of the region of applied traction,
+the solution remains accurate.
+The pressure field is captured in far greater detail than before.
+There is a clear distinction and transition between regions of compression and expansion,
+and the linear approximation of the pressure field allows a refined visualisation
+of the pressure at the sub-element scale.
It should however be noted that the pressure field remains discontinuous
and could be smoothed on a continuous grid for the post-processing purposes.
</tr>
</table>
-This brief analysis of the results demonstrates that the three-field formulation is effective
-in circumventing volumetric locking for highly-incompressible media.
-The mixed formulation is able to accurately simulate the displacement of a
-near-incompressible block under compression.
-The command-line output indicates that the volumetric change under extreme compression resulted in
+This brief analysis of the results demonstrates that the three-field formulation is effective
+in circumventing volumetric locking for highly-incompressible media.
+The mixed formulation is able to accurately simulate the displacement of a
+near-incompressible block under compression.
+The command-line output indicates that the volumetric change under extreme compression resulted in
less than 0.01% volume change for a Poisson's ratio of 0.4999.
-In terms of run-time, the $Q_2-DGPM_1-DGPM_1$ formulation tends to be more computationally expensive
-than the $Q_1-DGPM_0-DGPM_0$ for a similar number of degrees-of-freedom
-(produced by adding an extra grid refinement level for the lower-order interpolation).
+In terms of run-time, the $Q_2-DGPM_1-DGPM_1$ formulation tends to be more computationally expensive
+than the $Q_1-DGPM_0-DGPM_0$ for a similar number of degrees-of-freedom
+(produced by adding an extra grid refinement level for the lower-order interpolation).
This is shown in the graph below for a batch of tests run consecutively on a single 4-core (8-thread) machine.
-The increase in computational time for the higher-order method is likely due to
-the increased band-width required for the higher-order elements.
-As previously mentioned, the use of a better solver and precondtioner may mitigate the
-expense of using a higher-order formulation.
-It was observed that for the given problem using the multithreaded Jacobi preconditioner can reduce the
+The increase in computational time for the higher-order method is likely due to
+the increased band-width required for the higher-order elements.
+As previously mentioned, the use of a better solver and precondtioner may mitigate the
+expense of using a higher-order formulation.
+It was observed that for the given problem using the multithreaded Jacobi preconditioner can reduce the
computational runtime by up to 72% (for the worst case being a higher-order formulation with a large number
-of degrees-of-freedom) in comparison to the single-thread SSOR preconditioner.
-However, it is the author's experience that the Jacobi method of preconditioning may not be suitable for
+of degrees-of-freedom) in comparison to the single-thread SSOR preconditioner.
+However, it is the author's experience that the Jacobi method of preconditioning may not be suitable for
some finite-strain problems involving alternative constitutive models.
(detection and stress calculations) itself. An alternative to
additional penalty terms in the free-energy functional would be to
use active set methods such as the one used in step-41.
-- Finally, adaptive mesh refinement, as demonstrated in step-18, could
- provide additional solution accuracy.
+- Finally, adaptive mesh refinement, as demonstrated in step-6 and
+ step-18, could provide additional solution accuracy.
projects / dealii-svn.git / commitdiff