3 Numerical Methods of Lubrication Calculation
3.4 Multi-Grid Method for Solving EHL Problems
3.4.5 Multi-Grid Integration Method
The computing time of the multi-grid integration method is nearly proportional to the node number. Therefore, for an EHL problem with many nodes, the advantage of multi-grid integra- tion method is obvious [9].
We introduce integration for a line contact problem between two meshes first. If we under- stand this, the multi-grid integration will be similar.
The multi-grid integration method is first used to transfer the variable from the finer mesh to the coarser mesh. Then, integrate the variable on the coarser mesh and transfer the integrated result back to the finer mesh. After modification, a result meeting the accuracy requirement is obtained.
Suppose there are two meshes. The finer one is indicated by a superscripthand the node number with subscriptsiandj; for the coarser one withH,IandJ. IfIorJis equal to 0, 1…, or N, andiorjare equal to 0, 1,…,n, we haven=2N−1 because we always assume that the node numbers of the finer mesh is twice that of the coarser one.
k k For a line contact EHL problem, the integral formula of elastic deformation is
w(x) =
∫
xb
xa
ln|x−x′|p(x′)dx′. (3.126)
The numerical integral calculation formula on the finer mesh is equal to whi =
∑n
j=0
Ki,jhhphj, (3.127)
whereKi,jhhis the integration coefficient. It uses two superscripth’s. The firsthindicates the finer mesh and node numberi, while the secondhcorresponds to the finer mesh and node number j;phj is the dimensionless pressure and its superscript and subscript are similar toKi,jhh.
We know that the amount of calculation on the finer mesh is very heavy. If the numerical integration is carried out on the coarser mesh, it will be
wHI =
∑N
J=0
KI,JHHpHJ . (3.128)
Although the computational times of Equation 3.128 are much less than those of Equation 3.127, because the result of the coarser grid is not as accurate as that of the finer, it needs to be modified in order to get the same accuracy. The modification includes the following steps.
3.4.5.1 Transfer Pressure Downwards
Although we can directly transfer the pressure from the finer mesh to the coarser mesh node to node, in order to consider the variation of pressure, it is better to transfer pressure with an interpolation formula:
pHI = 1
32(−ph2I−3+9ph2I−2+16ph2I−1+9ph2I−ph2I+1) I=2,3,…,N−1. (3.129) In this equation, the pressure on the left-hand side is on the coarser mesh, while the pressures on the right-hand side are on the finer mesh. ForI=1 andN, the formulas become
pHI = 1
32(16ph1+18ph3−2ph5), (3.130)
pHN = 1
32(−2phn−2+18phn−1−16phn). (3.131)
3.4.5.2 Transfer Integral Coefficients Downwards
A mapper operator is used to transfer the integral coefficients:
KI,JHH =K2I−1,hh 2J−1, (3.132)
where the superscripts and subscripts are the same as previously mentioned.
3.4.5.3 Integration on the Coarser Mesh
The integration on the coarser mesh is of the same form as Equation 3.128. However, the inte- gral coefficient and the pressure are transferred downwards from the finer mesh rather than generated in the mesh itself:
k k wHI =
∑N
J=0
KI,HHJ pHJ . (3.133)
3.4.5.4 Transfer Back Integration Results
Because the integral value is not calculated on the finer mesh, the known value should be inter- polated back. First, map the results on the coarser nodes to the finer nodes:
̃
wh2I−1=wHI . (3.134)
For the node for which the coarser mesh has no related node to the finer mesh, interpolation is
̃ wh2I = 1
16(−wHI−1+9wHI +9wHI+1−wHI+2). (3.135) Fori=2 ori=n−1, the finer mesh nodes can be calculated as the average of two adjacent nodes.
3.4.5.5 Modification on the Finer Mesh
Modifications on the finer mesh include three parts: the integrated coefficient modification, the mapped value modification and the interpolated value modification.
1. Integral coefficient modification
First, calculate the interpolated coefficients. Then, subtract the interpolated values from the integrated coefficients to obtain a difference. Because a mapped value of the nodes is usu- ally not equal to an interpolated value, we must modify them. The interpolated value of the integral coefficient is determined by using
K̃2I−1,hh 2J−1= 1
16(9KI+1,JHH +9KI−1,JHH −KI+3,JHH −KI−3,HHJ). (3.136) Because the adjacent integral nodes are not suitable for high-order interpolation, the follow- ing interpolation formulas can be used instead:
K̃I,hh2J−1= 1 8
(9K2,JHH−K4,JHH)
(3.137) K̃2,hh2J−1= 1
16
(9K1,HHJ +9K3,JHH−K3,HHJ −K5,JHH)
(3.138) K̃3,hh2J−1= 1
16
(9K2,HHJ +9K4,JHH−K2,HHJ −K6,JHH)
(3.139) For a mapped node, the difference between the calculated integral coefficient and the inter- polated integral coefficient is equal to
ΔK̃i,jhh=Ki,hhj −K̃i,jhh. (3.140)
For an interpolated node, the difference between the calculated integral coefficient and the interpolated integral coefficient is equal to
ΔK̃i,jhh=
{0 mapped node
Ki,hhj −K̃i,jhh interpolated node. (3.141)
k k 2. Mapped value modification
Use the difference between the integral coefficient to modify the integral value of a mapped node:
wh2I−1=w̃H2I−1+
∑M
j=1
ΔK̃i,jhhpjΔx. (3.142)
3. Interpolated value modification
Use the difference between the integral coefficient to modify the integral value of an inter- polated node:
wh2I =w̃h2I+
∑M
j=1
ΔK̂i,jhhpjΔx. (3.143)
In the above equations,M≥3+2ln(n), M and it should be rounded;nis the node number.
The above steps are only for the two meshes, for an entire multi-grid mesh, the multi-grid integration method follows the following steps.
1. According toM≥3+2ln (n) calculateM.
2. According to Equations 3.129–3.132 transfer node parameters (pressure, integral coefficient, etc.) downwards to the coarsest mesh.
3. According to Equation 3.133 numerically calculate integration on the coarsest mesh.
4. According to Equation 3.132 calculate the integral coefficient of the corresponding node on the upper mesh.
5. According to Equations 3.136–3.140 interpolate the integral coefficients of the upper mesh.
6. According to Equation 3.134 map the corresponding value to the upper mesh node.
7. According to Equation 3.142 modify the mapped integral values.
8. According to Equation 3.135 interpolate the non-corresponding values of the upper mesh nodes.
9. According to Equation 3.143 modify the interpolated integral values.
10. Return to the step (4) for other calculations until all the nodes integral values have been obtained and calculation is completed.
It should be noted that by using the multi-grid integration method, we must correctly under- stand the importance of all the modifications. The main purpose of the modifications is through a less coarse grid calculation and modifications to obtain the result with the same accuracy of the fine grid.
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