From 7c392a045a29083ed6bd5b908c894c088258daba Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Tue, 24 Apr 2012 15:02:52 +0000 Subject: [PATCH] Fix formatting issues. git-svn-id: https://svn.dealii.org/trunk@25439 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-44/doc/results.dox | 130 +++++++++++------------ 1 file changed, 65 insertions(+), 65 deletions(-) diff --git a/deal.II/examples/step-44/doc/results.dox b/deal.II/examples/step-44/doc/results.dox index 8e0e243b97..c598762c7a 100644 --- a/deal.II/examples/step-44/doc/results.dox +++ b/deal.II/examples/step-44/doc/results.dox @@ -1,14 +1,14 @@

Results

-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. @@ -31,10 +31,10 @@ as it should. -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 @@ -47,13 +47,13 @@ Triangulation: 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 @@ -66,13 +66,13 @@ v / V_0: 1.000e-09 / 1.000e-09 = 1.000e+00 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 @@ -83,13 +83,13 @@ v / V_0: 1.000e-09 / 1.000e-09 = 1.000e+00 -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. @@ -114,24 +114,24 @@ We then used ParaView to visualise the results for two cases. 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. @@ -158,19 +158,19 @@ and is opposite to the physically realistic interpretation of pressure. -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. @@ -200,25 +200,25 @@ It should however be noted that the pressure field remains discontinuous -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. @@ -275,5 +275,5 @@ There are a number of obvious extensions for this work: (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. -- 2.39.5