Prolongation extends naturally to two-dimensional problems by weighting the coarse grid data, as shown in Fig. Nodes identified by squares belong to the Ω 2 h grid while nodes corresponding to the Ω h grid are shown as circles. In analogy to the one-dimensional case, values at nodes on the fine grid that coincide with nodes on the coarse grid are directly transferred. For fine grid nodes lying along coarse grid lines, values are computed by averaging of the two neighbors along the grid line. Finally, for fine grid nodes located at the center of a coarse grid cell, values from all four nodes corners of the cell are used in the averaging process. Thus, for i = 1, 2, N x 2 − 1 and j = 1, 2, N x 2 − 1, the prolongation algorithm proceeds as follows. (9.17) W → 2 h ( 0 ) = I ^ h 2 h W → h n + 1,where the subscript 2 h denotes the coarse grid 1 and I ^ h 2 h is the interpolation operator.
![]()
NVIDIA® Virtual GPU (vGPU) Software Documentation. Learn how to use NVIDIA® Quadro® Virtual Data Center Workstation (Quadro vDWS), NVIDIA vComputeServer, NVIDIA GRID™ Virtual PC, and GRID Virtual Applications. To connect to a older version of OpenSimulator, you may need to use a older version of your favority viewer. Consult your grid team, or the viewer team This may also limit Hypergrid capability. Alchemy Viewer - v3 based TPV. Cool VL Viewer - the oldest of all actively maintained Third Party Viewers (former name: Cool SL Viewer).
The residuals have to be transferred to the coarse grid as well, so that their low-frequency error components can be smoothed. A conservative transfer operator is employed for this purpose. This means that when the control volume size increases, the value of the residual must increase by the same amount. The residuals of the fine grid are also required in order to retain the solution accuracy of the fine grid on the coarse grid. For this purpose, a source term, the so-called forcing function 19, 22, is formed as the difference between the residual transferred from the fine grid and the residual computed using the initial solution W → 2 h ( 0 ) ( Eq.
(9.17)) on the coarse grid, i.e. In, 2000 5.5 Cycle PerformanceIn the previous section the coarse grid equations have been derived. In this section the performance of the constructed coarse grid correction cycle will be evaluated as a function of the coefficients ω 1 and ω 2 used in the relaxation of the pressure and the relaxation of the force balance condition. The coarse grid correction cycle will be used with a coarsest grid of (4 + 1) × (4 + 1) points and results will be presented as a function of the finest grid used in the cycle.
The domain used in the calculations extended over −2.0 ≤ X ≤ 2.0 and −2.0 ≤ Y ≤ 2.0. As an initial approximation P h = 0.999 p h is used where p h is Hertzian pressure solution given by Equation (5.11).
Indeed, this implies that a first approximation is used that, on fine grids, is already very close to the exact solution to the problem. On coarse grids the choice of the first approximation is not critical. However, on fine grids an accurate first approximation is needed for the process to converge as will be explained in the following section. Notice that this is by no means a restriction of the solver because an excellent tool is available to obtain a good first approximation: the Full Multigrid algorithm. However, in this section the aim is to illustrate cycle performance only, and therefore an accurate first approximation had to be chosen. First the influence of the value of ω 1 is studied, while freezing the rigid body motion ( H 0 = −1, ω 2 = 0). Tables 5.1–5.3 show the residual norm of the contact equation as a function of the gridlevel and the number of V (2,1) cycles performed for ω 1 = 1.0, ω 1 = 0.8 and ω 1 = 0.6 respectively.
From these three tables it can be concluded that the value of ω 1 is not critical. Using a moderate underrelaxation coefficient like ω 1 = 0.6 a very stable convergence is obtained, independent of the level and the number of V-cycles. Note that, based on the contact region alone (the linear problem), the Fourier analysis predicted an optimal value of ω 1 = 0.9 with a potential error reduction of a factor of 20 per cycle. The fact that this factor is not reached is attributed to the effects of the intergrid transfer, and the free boundary. rn number of V-cycles level 1 2 3 4 5 6 7 24.7 10 −21.4 10 −24.1 10 −31.2 10 −33.6 10 −41.1 10 −43.3 10 −539.7 10 −32.3 10 −36.6 10 −41.7 10 −14.6 10 −51.2 10 −53.2 10 −642.9 10 −31.1 10 −32.8 10 −48.0 10 −52.3 10 −56.6 10 −61.9 10 −653.0 10 −46.2 10 −51.6 10 −54.4 10 −61.2 10 −63.1 10 −78.2 10 −864.6 10 −41.7 10 −44.8 10 −51.8 10 −56.3 10 −62.2 lO −67.9 10 −176.1 10 −42.7 10 −42.8 10 −58.0 10 −79.0 10 −12.2 10 −73.8 10 −8. Choosing ω 1 = 0.6 the influence of the value of ω 2 on the convergence behaviour of the cycle is studied.
Again V (2,1) cycles are used. Table 5.4 shows the residual of the contact equation and of the force balance equation as a function of the grid level, the number of V-cycles, and the value of ω 2.
From this table it can be concluded that slow convergence of the residual norm can be caused in two different ways. For a value of ω 2 which is too small, the force balance equation will be undercorrected. This causes the convergence of the force balance equation to lag behind, and generates continuous changes of the H 0 value. These continued changes will eventually cause a reduction of the overall convergence speed.
This type of convergence can be called creeping convergence. rn number of V-cycles level 1 2 3 4 5 6 7 21.3 10 −25.6 10 −42.7 10 −51.3 10 −66.4 10 −83.1 10 −91.5 10 −1034.8 10 −31.8 10 −46.1 10 −62.1 10 −77.5 10 −92.8 10 −101.1 10 −1148.9 10 −43.5 10 −51.4 10 −65.8 10 −83.0 10 −91.3 10 −107.5 10 −1252.9 10 −41.7 10 −51.2 10 −67.9 10 −85.5 10 −93.7 10 −102.5 10 −1161.6 10 −47.0 10 −61.0 10 −67.8 10 −88.9 10 −97.8 10 −108.0 10 −1171.9 10 −41.9 10 −53.4 10 −64.7 10 −77.2 10 −81.1 10 −81.6 10 −9. rn number of V-cycles level 1 2 3 4 5 6 7 24.8 10 −32.4 10 −41.6 10 −51.1 10 −67.2 10 −84.7 10 −93.1 10 −1032.0 10 −31.5 10 −41.3 10 −51.2 10 −61.3 10 −71.7 10 −82.3 10 −944.3 10 −44.1 10 −54.3 10 −64.7 10 −75.3 10 −86.4 10 −98.2 10 −1051.1 10 −49.9 10 −61.1 10 −61.3 10 −71.5 10 −81.9 10 −92.4 10 −1068.5 10 −58.4 10 −61.1 10 −61.5 10 −72.2 10 −83.2 10 −94.7 10 −1071.0 10 −46.5 10 −67.4 10 −78.5 10 −81.1 10 −81.3 10 −91.8 10 −10When ω 2 is too large, the force balance equation tends to be overcorrected causing the H 0 value to oscillate. Small oscillations are not very harmful. But larger ones tend to hinder the overall convergence as much as undercorrection.
This type of convergence can be called oscillating convergence. An extreme example is seen in the table for ω 2 = 0.6. The H 0 value is overcorrected such that the residual of the force balance equation changes sign, but its value hardly reduces. Even larger ω 2 will cause the solution process to diverge. A value of ω 2 = 0.3 for the smooth circular contact problem was found to be a good compromise, causing something close to critical convergence.
The optimal value of ω 2 can also be obtained explicitly by studying the residual of the force balance equation wr h as a function of H 0, as is done in Figure 5.7. The derivative∂ W r h / ∂ H 0 corresponds to the contact stiffness and a value close to 3.0 was found. The exact value can be obtained using a Hertzian analysis, as the load is proportional to the rigid body displacement to the power 3/2. For the residual of the dimensionless force balance equation one finds w r = 2π/3(− H 0) 3/2 yielding a dimensionless stiffness of π around H 0 = −1.0. The ω 2 value is the inverse of this stiffness, and should thus have an optimum value of 0.318 However, for problems involving the contact between extremely rough surfaces, a value of ω 2 = 0.2 yields a very robust convergence behaviour. (3.16) r I H = ( 1 × r 2 I - 1 h + 2 × r 2 I h + 1 × r 2 I + 1 h ) / 4The Equations (3.11) and (3.14) generally need not be used in this matrix form in practice. Assuming that two arrays are available, one containing the values r h i for all fine grid points, and one to store r H I for all coarse grid points all one needs to do is to scan the coarse grid array point by point computing r H I according to (3.13) or (3.16).
For the boundary points (3.16) can not be used asr 2 I - 1 h orr 2 I + 1 h are not defined. For these points a modified stencil should be used. If boundary values are really needed one could use injection for these points, or, as is done in (3.14) use the stencil without the contribution of the points which lie outside of the domain,r 2 I - 1 h andr 2 I + 1 h respectively for the left and right boundary.
This implies that it is assumed that r h is zero outside the domain. In any case, the main point is that one generally does not need to store the matrix, as the coefficients represented in the stencil are all that is needed. A formal analysis of the restriction process can be carried out using Local Mode Analysis, see Section 3.12.3. For the present section it is sufficient to note that full weighting, because of its averaging, acts as a filter removing high frequency components. The injection operator on the other hand takes such components along to the coarser grid but, because this grid can not resolve them, they alias with low frequency components. This difference between injection and full weighting is illustrated in the Figures 3.5 and 3.6.
Restriction of a fine grid vectorr h to the coarse grid with injection and full weighting for a function with high frequency components that can not be seen by the coarse grid. Fine grid: h = I/64, coarse grid: H = 2h = 1/32.Figure 3.5 shows the restriction of a relatively smooth function from a fine grid to a coarser grid with twice the mesh size. Because the function hardly contains frequencies that the coarse grid can not represent, both injection and full weighting represent it accurately on the coarse grid.Figure 3.6 shows the restriction of a more oscillatory function from a fine to a coarse grid for the same choice of grids.
In this case the function does contain high frequencies that the coarse grid can not represent. Figure 3.6 shows that the coarse grid representation obtained with full weighting indeed accurately approximates the behaviour of the fine grid function with respect to the components that the coarse grid can represent.
However, if injection is used this is not the case. Aliasing causes high frequency components from the fine grid that can not be represented accurately on the coarse grid to appear on the coarse grid as low frequency components. Consequently the coarse grid representation obtained by injection forms a poor approximation of the smooth part of the fine grid function.In the coarse grid correction process the coarse grid is introduced to solve the smooth error components.
These components are associated with the smooth components of the residual. As a result the restriction should thus accurately represent all components of the fine grid residual that can also be represented on the coarse grid. If a large error is made in the representation of these components of the residuals it will lead to a poor coarse grid correction which adversely affects the performance of the cycle. Full weighting is therefore commonly preferred for residual transfers.
It ensures that high frequency components still present in the residual on the fine grid after relaxation. Even though they may have a small amplitude, are not transferred to the coarser grid.Restriction operators are ranked according to their order where order refers to the magnitude of the power of h of the error resulting in the coarse grid representation. One even distinguishes a secondary order distinguishing the contribution from high frequency components on the fine grid from the contribution of low frequency components to this error. Injection has order 0 and full weighting has order 2. Higher order restrictions can also be derived. However, for the transfer of residuals they are of little use.
For the case of the transfer of a function they may be needed and in that case they are generally derived from a higher order interpolation as will be shown in the following section. Domenico Borello. Franco Rispoli, in, 2003 3.1 Multigrid techniqueIn the following the main issues of the MG implementation are stated.
The coarse grid operator is carried out through a direct discretization method. On each nested grid level the same discretization has been used. For the adopted standard grid coarsening technique the restriction phase is straightforward, while the prolongation is carried out by using the standard finite element shape functions. The prolongation depends on the finite element space discretization order, that is linear for equal order Q1-Q1 spaces. On mixed order Q2-Q1 elements it becomes quadratic for primary-turbulent variables and linear for constraint ones.The LMG, LFMG and HLFMG solvers have been applied to the fully coupled turbulent Navier-Stokes problem.
The Gauss Siedel Preconditioned GMRes (PGMRes) 2 has been used as a solver for the coarse mesh and as MG smoother. Although many authors discourage the use of non-linear solvers for the smoothing phase due to their poor attitude in eliminating selectively the high frequency error components, however it has been demonstrated that the LMG solver with a non-linear PGMRes based smoother is actually competitive in terms of computational time, if compared to PGMRes iterative solver for single grid problems. 1 a comparison of the results for single and MG simulations have been reported with reference to the solution of a 2D turbulent 90° elbow with synthetic treatment of wall layer.
The Reynolds number is set equal to 40.000. A Q2-Q1 241×81 discretisation in streamwise and crosswise directions has been adopted. The MG schemes feature one V cycle for each linearized iteration. The PGMRes(20) has been used as a solver for the single grid case such as for the MG smoothing. The coarse grid solution has been carried out by using the same PGMRes(20) with 20 restarts. SolverPGMRes(20)LMGLFMGHLFMGCoarseFineTotalCoarseFineTotalIterations205433275Time ratio1.003.445.425.80The convergence threshold parameters here adopted are r / r 0.
In this paragraph the problem without the complementarity condition is considered. For generality the coarse grid correction is described using the Full Approximation Scheme, see Section 3.8.2, although strictly FAS is only needed if cavitation is taken into account as the complementarity condition causes the problem to become non-linear. For simplicity only two grids are assumed.
A fine grid with mesh size h θ and h y and a coarse grid with H θ and H y. The fine grid discretised dimensionless Reynolds Equation (4.30) reads.
(4.43) ξ I + 1 / 2, J H P I + 1, J H - ( ξ I + 1 / 2, J H + ξ I - 1 / 2, J H ) P I, J H + ξ I - 1 / 2, J H P I - 1, J H H θ 2 + 1 k 2 ξ I, J + 1 / 2 H P I, J + 1 H - ( ξ I, J + 1 / 2 H + ξ I, J - 1 / 2 H ) P I, J H + ξ I, J - 1 / 2 H P I, J - 1 H H Y 2 - H I + 1, J H - H I - 1, J H 2 H θ = P f ˆ I, J Hwhere the initial value of P H is defined byP ˜ I, J H = I h H P ˜ h I, J. Note that H θ and H y denote the coarse grid mesh size, H θ = 2 h θ and H Y = 2 h y. The right hand sideP f i, j h = 0 on the finest grid. On coarser grids the right hand side isP f ˆ I, J H and defined using (3.74) as. (4.44) f ˆ I, J H = I h H r h I, J + ξ I + 1 / 2, J H P ˜ I + 1, J H - ( ξ I + 1 / 2, J H + ξ I - 1 / 2, J H ) P ˜ I, J H + ξ I - 1 / 2, J H P ˜ I - 1, J H H θ 2 + 1 k 2 ξ I, J + 1 / 2 H P ˜ I, J + 1 H - ( ξ I, J + 1 / 2 H + ξ I, J - 1 / 2 H ) P ˜ I, J H + ξ I, J - 1 / 2 H P ˜ I, J - 1 H H Y 2 - H I + 1, J H - H I - 1, J H 2 H θAfter an approximationP ¯ H to P H has been obtained from Equation (4.43), the fine grid approximationP ¯ h is corrected according to Equation (3.75) which gives. The next table studies the convergence as a function of the number of V(2,1) cycles and as a function of the level, using the initial approximation P h = 0.
All results in this section have been obtained for ∈ = 0.2, k = L/ R = 1 and 24 × 4 points on the coarsest grid. As can be seen from Table 4.1, the residual reduction per V-cycle is O (10) and is not affected by either the number of preceding V-cycles, nor by the number of points on a level. On the whole, the convergence of the hydrodynamic lubrication solver is as fast as the convergence of the Poisson (2d) program.
![]()
This is exactly what one could expect. On a sufficiently fine grid the coefficient ξ will vary smoothly over the grid and the problem for k = L/ R = 1 locally approximates the Poisson 2d problem.
For that problem the worst amplification factor for high frequency components wasμ ¯ = 0.5 which predicts an error reduction of a factor of 8 per V(2,1) cycle. At this point it is noted that this convergence speed is valid as long as the coefficienth θ 2 / ( k 2 h y 2 ) = O ( 1 ), see Section 4.10.1. Having shown good convergence in terms of the reduction of residuals next convergence of the solution with decreasing mesh size can be investigated. For this purpose the load capacity of the bearing is taken.
In accordance with the Sommerfeld approximation only the positive pressures are taken. This implies that Equation (4.32) is used with max(0,P i, j h) replacingP i, j h. The value of W h thus obtained can be compared on different grids. As such it allows us to estimate the discretization and the numerical error in the solution. Table 4.2 gives the computed load capacity as a function of the gridlevel and the number of V (2,1) cycles that is used. V-cyclelevel 5level 6level 7384 × 64768 × 1281536 × 7 10 −22.39728 10 −22.39733 10 −222.55129 10 −22.55185 10 −22.55199 10 −232.56097 10 −22.56157 10 −22.56172 10 −242.56156 10 −22.56216 10 −22.56232 10 −252.56159 10 −22.56220 10 −22.56235 10 −262.56159 10 −22.56220 10 −22.56235 10 −2202.56159 10 −22.56220 10 −22.56235 10 −2The table shows that after three to five cycles the solution has converged to an error that is small compared to the discretization error. This can be seen from the fact that the difference between the solution on a grid after some cycles compared to the value on the same grid after many cycles is small compared to the difference between the solution on that grid after many cycles and the one on the next finer grid after many cycles.
With respect to the discretization error the table shows that the difference between the computed load capacity on level 6 and 7 is four times smaller than the difference between the computed load capacity on level 5 and 6. This is in accordance with the second order accuracy of the discretization that is used. The subject of accuracy will be addressed in more detail in Section 4.7.The fact that with decreasing mesh size an increasing number of cycles is required to attain this precision is caused by the bad initial approximation ( P = 0), and implies that using an FMG algorithm one would reduce the amount of work needed to attain this precision.
The parallel efficiency was also analysed, using 1 to 16 processors on a CRAY-T3E. The two grids were used: a coarse grid consisting of 6 × 24 points and a fine grid with 48 × 196 points. On the coarse grid no MG was applied. Four grid levels were used in the MG algorithm on the fine grid. The results are shown in Table 1.
![]()
The left number shows the number of MG-cycles which were needed for convergence (residual less then 10 −16). The right number is the time used per MG-cycle.
For the coarse grid the number of cycles increases dramatically when the number of processors increases. This is due to the implicit solver which was employed. This is not the case for the MG algorithm on the fine grid, and the is main reason for using the algorithm. The CPU-time consumed per iteration is nearly constant for the coarse grid and decreases for the fine grid. The small problem size is the main reason for the bad speed-up for the coarse grid. Also the MG algorithm on the finer grid scales poorly with the number of processors.
This is explained by the fact that some iterations are still needed on the coarse grid. These iterations do not scale with the number of processors, as already mentioned. The solution and residuals are transferred onto successive independent coarser grids until the coarsest grid is reached.
And after calculating the flow-field variables on the coarsest grid, the correction is evaluated and then has to be interpolated back to the fine grid. The correction is the difference between the newly computed value on the coarse grid, W h + 1 +, and the initial value that was transferred from the finer grid, W h + 1 ( 0 ), and this correction is added to the values on the fine grid as follows.
(5.4) W h + = W h + I h + 1 h ( W h + 1 + − W h + 1 ( 0 ) )where I h + 1 h is an prolongation operator from the coarse grid to the fine grid, which will be defined in Section 5.6. W h + is the updated variables of the solution and W h is the solution from the fine grid.To speed up the calculations at the coarser grids, the viscous terms are only evaluated on the fine grid and not on the coarser grids. Since the coarser grids are used to cancel the dominating low-frequency errors on the fine grid and the solution on the coarser grids are driven by the residual from the fine grid; hence, this treatment does not affect the accuracy of the solution. The upwind-biased discretization scheme is set to first-order in the coarser levels where the left-right states of Eqs. (3.26a) and (3.26b) are taken as the values of the two nodes of the edge to calculate the flux associated with the edge. Rong Li, in, 1996 C ADAPTIVE GRID (AG) SCHEMEA third means of obtaining the supporting digraphs is to make adaptive the grid of the parameters that characterize the possible modes. In this scheme, a coarse grid is set up initially and then the grid is adjusted recursively according to an adaptation scheme based possibly on the current estimate, mode probabilities, and measurement residual.
This approach is particularly advantageous in the case where the set of possible system modes is large.Consider the problem of state estimation with an uncertain parameter over a two-dimensional continuous region. If ten quantization levels are used for each dimension, a fixed structure MM estimator would consist of 10 × 10 = 100 filters. An adaptive grid MM estimator with only 3 × 3 = 9 filters can yield equivalent or even better performance if the parameter is constant and the state estimation is sensitive to the parameter.An adaptive grid technique was implemented in 67 for the adaptive multiple-model PDA filter presented there. The moving-bank MM estimators of 107, 71 follows an essentially same idea. The reader is referred to these publications for their simulation results.
Graph Paper Template 10 by 10 Grid Where these photos came from and how you can use themWe are just like you, persons who really honor original idea from every one, with no exception. Because of that we always keep the original pictures without single change including the copyright mark. And we ensure to include website or blog link where it belongs to be, below each pictures.
Many message came to us about the proper right in relation with the photos on our gallery. In case you want to know what you can do, please contact the website on each photos, the reason is we are not able to determine your true right. We notice you, no watermark does not mean the images is able to freely used without permission.
![]() Comments are closed.
|
AuthorWrite something about yourself. No need to be fancy, just an overview. Archives
January 2023
Categories |