From: mcbride Date: Fri, 24 Feb 2012 10:55:27 +0000 (+0000) Subject: step:44 rewording and formatting of results.dox X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=61c44c5ce3f7602b61633fc7199b0994fd0e6f81;p=dealii-svn.git step:44 rewording and formatting of results.dox git-svn-id: https://svn.dealii.org/trunk@25161 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-44/doc/results.dox b/deal.II/examples/step-44/doc/results.dox index ebff2dcf9c..f3f67c7f5b 100644 --- a/deal.II/examples/step-44/doc/results.dox +++ b/deal.II/examples/step-44/doc/results.dox @@ -1,26 +1,41 @@

Results

-Firstly, a comparison of a batch of results with that in the literature demonstrates that the program works as expected. +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 $p/p_0$. +The results agree with those in the literature. +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 conidence that the code is working +as it should +
@image html "step-44.Q1-P0_convergence.png"

- Convergence for the Q1-P0-P0 formulation. + Convergence of the $Q_1-DGPM_0-DGPM_0$ formulation.

@image html "step-44.Q2-P1_convergence.png"

- Convergence for the Q2-P1-P1 formulation. + Convergence of the $Q_2-DGPM_1-DGPM_1$ formulation.

-Using the appropriate material and loading parameters, a set of results matches that presented in Reese (2000) is produced. Both schemes demonstrate good convergence properties upon grid refinement, with an accurate measure of the centre-point vertical displacement attained within a few grid refinements. The lower order formulation typically overestimates the displacement solution at low levels of refinement, while the higher order interpolation scheme underestimates it, but be a lesser degree. This result gives confidence that the program produces the correct output. -A typical output generated by running the problem looks is shown below. The particular case -demonstrated is that of the Q2-P1-P1 element. + +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. +Other than in the first timestep, solution convergence is achieved within 5 Newton increments. +The converged displacement's $L_2$-norm is several orders of magnitude less than the geometry scale. @code Grid: @@ -69,9 +84,17 @@ Dilatation: 1.525e-06 v / V_0: 9.999e-10 / 1.000e-09 = 9.999e-01 @endcode -It is clear that, using the Newton-Raphson method, quadratic convergence of the solution is obtained. Other than the for the first timestep, solution convergence is achieved within 5 Newton increments. The end displacement L2-norm is several orders of magnitude less than the geometry scale. -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, a typical output of the Timer may look like + +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. +It may be beneficial to investigate the use of another solver such as those available through the Trilinos library. + @code +---------------------------------------------+------------+------------+ @@ -89,11 +112,31 @@ Using the Timer class, we can discern which parts of the code require the highes +---------------------------------+-----------+------------+------------+ @endcode -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 Trilinos. -Using ParaView, we visualise the results for the vertical displacement of the centre-point of the traction surface and pressure field from two of the cases. +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 seond 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. + + +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. +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, +with positive dilatation indicating regions of positive hyprodstatic 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 +and is opposite to the physically realistic interpretation of pressure. -The first case shown is that using the coarsest grid using the lowest order interpolation method, namely the Q1-P0-P0 element. It is clear that the coarse spatial discretisation coupled with large displacements leads to a low quality solution. Additionally, the pressure difference between elements is very large. The constant pressure field on the element ensures that the large pressure gradient is not captured. However, it should be noted that locking that would be present in a standard Q1 displacement formulation still does not arise in this poorly discretised case. The images below show the results for the tri-linear displacement formulation with p/p0=80. @@ -117,9 +160,24 @@ The first case shown is that using the coarsest grid using the lowest order inte
-Although the displacement solution is coarse, the final vertical displacement of the node at the centre of the top surface of the block is within 12.5% of the converged solution. The pressure solution is very course and has large jumps between adjacent cells. It is clear that the volume nearest the applied traction undergoes compression while the extents of the domain are in a state of dilatation. The dilatation solution field and pressure field are clearly linked, with positive dilatation indicating regions of positive hyprodstatic pressure and negative showing regions placed in compression. It should be noted that in these, a compressive pressure generated in each element has a negative sign while 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. -Both spatial refinement and the use of a higher-order interpolation scheme results in a high quality displacement solution. Three grid refinements coupled with a Q2-P1-P1 formulation produce a result that clearly captures the mechanics of the problem. The deformation of the traction surface is well resolved. The images below show the result for tri-quadratic displacement solution with p/p0=80. +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. +It should however be noted that the pressure field remains discontinuous + and could be smoothed on a continuous grid for the post-processing purposes. + + @@ -144,11 +202,28 @@ Both spatial refinement and the use of a higher-order interpolation scheme resul
-Here we observe the true 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 junction between the regions of the upper surface when the traction is and isn't applied, 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 tension, and the linear approximation of the pressure field allows a refined visualisation of the pressure at a 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. -A brief analysis of the results for an selection of input parameters for the described problem demonstrates that the three-field formulation is effective in circumventing volumetric-locking of 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. +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 a bit more computationally expensive +than the $Q_1-DGPM_0-DGPM_0$ for a similar number of degree-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 serially 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 using the multithreaded Jacobi preconditioner does reduce the +computational runtime by up to 25% 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. + -In terms of run-time, the Q2-P1-P1 formulation tends to be a bit more computationally expensive than the Q1-P0-P0 for a similar number of degree-of-freedom (produced by adding an extra grid refinement level for the lower order interpolation method). This is demonstrated in the graph below for a batch of tests run serially on a single 4-core (8 thread) machine.
@@ -159,7 +234,6 @@ In terms of run-time, the Q2-P1-P1 formulation tends to be a bit more computatio
-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 using the multithreaded Jacobi preconditioner does reduce computational runtime by up to 25% 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 different constitutive models.

Possibilities for extensions

@@ -173,7 +247,7 @@ There are a number of obvious extensions for this work: augented Lagrange multipliers. This is demonstrated in Simo and Taylor (1991) . - The constitutive relationship used in this - model is very basic. It may be beneficial to split the material + model is relatively basic. It may be beneficial to split the material class into two separate classes, one dealing with the volumetric response and the other the isochoric response, and produce a generic materials class (i.e. having abstract virtual functions that derived @@ -194,7 +268,7 @@ There are a number of obvious extensions for this work: limit the step size within a Newton increment to ensure optimum convergence. It may also be necessary to use a load limiting method, such as the Riks method, to solve unstable problems involving - geometric nonlinearity such as buckling and snap-through. + geometric instability such as buckling and snap-through. - Many physical problems involve contact. It is possible to include the effect of frictional or frictionless contact between objects