3 Numerical Methods of Lubrication Calculation
3.1.2 Finite Element Method and Boundary Element Method
The following gives a brief introduction to the use of the finite element method and the bound- ary element method in solving lubrication problems.
3.1.2.1 Finite Element Method (FEM)
The finite element method was first developed in terms of elasticity theory and was applied to hydrodynamic lubrication in the 1990s. Compared with the finite difference method the main advantages of the finite element method are its adaptability and convenience in complicated geometric shapes. Its element size and node number can be arbitrarily selected using accurate calculation. However, its calculating procedures are more complex.
The finite element method obeys the variation principles for solving functional equations.
The general form of Reynolds equation used for incompressible hydrodynamic lubrication is
𝜕
𝜕x ( h3
12𝜂
𝜕p
𝜕x )
+ 𝜕
𝜕y ( h3
12𝜂
𝜕p
𝜕y )
= 1 2
𝜕(hU)
𝜕x +1 2
𝜕(hV)
𝜕y +𝜕h
𝜕t. (3.25)
The vector form of Equation 3.25 is
∇⋅ ( h3
12𝜂∇p )
−1
2∇⋅(hU) +h,̇ (3.26)
where∇ =i𝜕∕𝜕x+j𝜕∕𝜕y;Uis the velocity vector;ḣ =𝜕h∕𝜕t.
As shown in Figure 3.5, the lubrication region is divided into a number of triangular elements.
On the border there are two boundary conditions. The pressure onspis known asp=p0; and the flow onsqis known asq=q0.
Supposeeis an element with pressurepe, the functional equation of the element can be writ- ten as
k k Je= −
∫ ∫ [
− h3
12𝜂∇pe⋅∇pe+hU⋅∇pe−2hpe ]
dA+2
∫sqq0peds, (3.27) whereAis the solution zone;sis the border.
If the lubrication region is divided into a total ofnelements, the total function will be equal to the sum of functions of all elements.
J=
∑n
e=1
Je. (3.28)
According to the variation principle, the extreme of the total function exists while 𝛿J=
∑n
e=1
𝛿Je=0. (3.29)
By using the Euler–Lagrange equation it can be proven that the solutionp(x,y) of Equation 3.26 satisfies Equation 3.29 and the given boundary conditions. Or, p(x, y) obtained from Equation 3.29 must be the solution of the Reynolds equation (3.26) and satisfy the given boundary conditions. Therefore, the finite element method need not directly solve the two integral equations, but it transforms the Reynolds equation into a functional equation and, by solving Equation 3.29, we can also obtain the solution.
The solution process of the finite element method can be generally summarized as follows:
1. Divide the solution region into a number of triangular or quadrilateral elements.
2. Write out the functional equation according to the variation principles.
3. Establish interpolation functions to express the variables by the node values of each element.
4. Based on the boundary conditions algebraic equations are established in terms of the unknown variables of each node.
5. Use an iteration or elimination method to solve the algebraic equations.
3.1.2.2 Boundary Element Method
The basic feature of the boundary element method is to solve the unknown parameters in the region by the known borders. First, divide the border into a number of elements. Then, solve the other unknown border variables by the known and then the unknown variables in the solu- tion region. Therefore, the main advantage of the boundary element method is that it has a very limited number of algebraic equations so as to significantly reduce the amount of data. In addition, the boundary element method has a higher accuracy than the other methods and can be easily used in a mixed problem. However, the establishment of equations of the boundary element method is not so easy.
At present, the boundary element method is mainly used in the analysis of theory of elasticity and heat transfer. The author took the Rayleigh step slider lubrication as an example to calculate its lubrication properties by the boundary element method [2].
The slider is as shown in Figure 3.6. It can be divided into two different partsΩ1,Ω2. The pressurepin each part depends on the following Reynolds equation.
∇2p= 𝜕2p
𝜕x2 + 𝜕2p
𝜕y2 =0. (3.30)
k k Figure 3.6 Rayleigh step slider.
Because the problem is symmetric to thexaxis, only half of the slider needs to be consid- ered in the analysis, for example,OBCE. If the total boundary iss, it is divided intos1ands2, ors=s1+s2. The known boundary conditions are:p|s1=p0=0 and q|s2 =𝜕p∕𝜕y|s2 =q0=0. Now, let us introduce a weighting functionf to meet the basic Equation 3.30. The equation of the boundary element method by the weighted residual method is
∫Ω(∇2p)fdΩ =∫s2(q−q0)fds−∫s
1
(p−p0)Qds, (3.31)
whereQ=𝜕f∕𝜕y.
The weight functionf can be obtained through mathematical analysis f = −In r
2𝜋, (3.32)
whereris the distance from pointito other points.
The unknown variablespiin the region has a relationship with the boundary variables as follows:
pi+∫spQds=∫sfds. (3.33)
Similarly, the unknown variablespion the border can be solved by the known border variables from the following integral equations:
1 2pi+
∫spQds=
∫sqfds. (3.34)
With Equation 3.34, the unknown variables along the border can be obtained. Then, by using Equation 3.33, the unknown variables can be calculated.
k k Figure 3.7 Boundary element division.
Simply use straight lines to divide the boundary inton elements, as shown in Figure 3.7.
Then, use the n straight line segments instead of the actual curve border. If the midpoint of each segment is taken as the node, the unknown variables on each element will vary linearly.
Applying Equation 3.34 to the equivalent element boundaries, we have 1
2pi+
∑n
j=1
pj∫sjQds=
∑n
j=1
qj∫sjfds. (3.35)
Because each node has two variables,pandq, the total variables are 2n. Here,n=n1+n2, whilen1is the number of the knownpiandn2of the knownqj. Therefore, there arenunknown variables. With Equation 3.35, we havenalgebraic equations, which are equal to the number of the unknown variables. Therefore, the total equations have definite solutions to obtain the unknownpandqof each node on the border. Then, use the border variables to calculate the inner unknown variables with Equation 3.33. The discrete form of Equation 3.33 is
pi=
∑n
i=1
qj∫sjfds−
∑n
j=1
pj∫sjQds. (3.36)