Fast Solution of Pressure and Stress of Dry Contact Using Multigrid Techniques

Abstract

Fast computation of the elastic deformation integrals of dry contact is accomplished using multigrid techniques. The method is called multi-level multi-integration, which enables the efficient numerical solution of contact pressure for large dry-contact problems. The computing effort can be reduced to O (NlogN) operations, compared to that of classical solution methods, O (N²). The fast integration technique can be straightforwardly applied to computing sub-surface stresses. As an example, the pressure and stress distribution of the contact of wire rope and friction lining, a large calculation due to the wire rope’s complicated construction, were fast solved. Subsequently, the friction lining’s transient temperature, when the wire rope was sliding, was solved using multigrid techniques.

1. Introduction

Knowledge of pressure and stress distributions may contain valuable information concerning the failure of dry contacts. This requires the calculation of elastic deformation integrals which depend on the contact surface topographies. For precise modeling of the surfaces, many points are needed, which increases the difficulties in calculation. It is impossible to use analytical methods. Numerical calculations can no longer make use of symmetry and have to be performed on fine grids. It needs a very long calculation time and requires much effort, when using traditional numerical methods (like the finite element method) [1].

However, multigrid techniques have been applied successfully to large calculations, demanding a great number of points, for dry contacts [2,3,4], mechanical behavior of heterogeneous materials [5,6,7,8], hydrodynamic lubrication (HL) [9,10,11,12,13], and elastohydrodynamic lubrication (EHL) [14,15,16,17,18,19]. The application of multi-level techniques, compared to other numerical methods, reduces the calculation time by several orders of magnitude. The solution time is dominated by computing multi-integrals [1]. Calculating the multi-integrals by a multi-level technique, known as multi-level multi-integration (MLMI), is outlined by Brandt [20,21]. The technique is worked out in detail in [22].

This paper applied MLMI to calculate elastic deformation integrals, so as to obtain the pressure of dry contact. This involved little work, O (NlogN), approximately proportional to N, which was the number of the points for discretizing the contact surfaces. The integration method was then ingeniously applied to solving stresses, which was similar to computation of elastic deformation integrals if one considers solving the stress field, depth by depth [1]. The proposed method was competent, compared to other numerical methods, for solving large dry contact problems (for instance, the dry contact of two surfaces with realistic roughness), in a reasonable amount of time and demanding little computing effort.

In subsequent sections, MLMI and the fast solutions of pressure and subsurface stresses of dry contact are addressed. Then, as a validation, the pressure and subsurface stress distributions were fast solved for a wire rope-friction lining contact which involved large calculations. Then, the transient temperature of the friction lining was solved using multi-level techniques.

2. Theory

2.1. Outline of the Fast Integration Method, MLMI

For calculating the elastic deformation integrals, pressure times a coefficient is integrated over the whole calculation domain. Using a general notation, such an integral is written as:

$$w\left(x\right)={{\displaystyle \int }}_{\Omega }K\left(x,x^{\prime }\right)u\left(x^{\prime }\right)\mathrm{d}{x}^{\prime },x\in \Omega \subseteq {\Re }^n.$$

(1)

The coefficient $K\left(x,x^{\prime }\right)$, called the kernel, and the function $u\left(x^{\prime }\right)$ are given. For the 2-dimensional (2-D) contact pressure, there is $x=\left(x_1, x_2\right)=\left(x, y\right)$. For the 3-dimensional (3-D) sub-surface stress, $x=\left(x_1, x_2, x_3\right)=\left(x, y, z\right)$.

Following is a short mathematical description of rapidly calculating Equation (1) by MLMI. The discretized integrals are defined as:

$$w^h\left(x_i^h\right)=w_i^h=h^n{\displaystyle \sum }_{i^{\prime }}{K}_{i,i^{\prime }}^{hh}{u}_{i^{\prime }}^h.$$

(2)

The domain, $\Omega$, is discretized with an equidistant grid and the mesh size is h; i is the index of the point. The integration can be accelerated by means of conducting it on the coarser grids, whenever K is smooth. This is about multigrid techniques. For convenience, only one coarser grid is introduced here. On the coarser grid, the grid distance is H, H = 2 h, and the related indices are denoted by capitals.

Information is transferred between the two grids by two operators: $I{I}_H^h$ (interpolation, from the coarse grid to the fine one) and ${(I{I}_H^h)}^T$ (restriction, from the fine grid to the coarse grid). The value $w_i^h$ is obtained by $I{I}_H^h$ from $w_I^H$:

$$w_i^h\simeq {[I{I}_H^h{w}_{\cdot }^H]}_i.$$

(3)

The subscript dot (.) denotes the index of the point where one transferring operator works. The newly obtained index is placed outside the square bracket. However, such an approach is only possible when K is smooth, so as to be certain to extend over the entire region.

The kernels of potential types, such as $K={\left|x-x^{\prime }\right|}^{-1}$, are singular (non-smooth) close to Point i ($\left|i-i^{\prime }\right|\le m$). Fortunately, the smoothness increases fast with an increase of $r=\left|x-x^{\prime }\right|$. Since $K$ is smooth in most of the domain, one can possibly maintain the approach outlined previously by adding only a local correction to Equation (3) so as to reduce the integration error on the coarser grid [23].

The resulting algorithms, for rapidly calculating integrals with a smooth or singular kernel, given in [23], are as follows (when the kernel is singular smooth, please include the two steps (b’) and (c’); otherwise, neglect them):

(a)

Anterpolation

Calculate $u_{I^{\prime }}^H$, for every point I′, in accordance with

$$u_{I^{\prime }}^H\stackrel{\scriptscriptstyle\mathrm{def}}{=}{2}^{-n}{\left[{(I{I}_H^h)}^T{u}_{\cdot }^h\right]}_{I^{\prime }}.$$

(4)

(b)

Coarse grid summation

Calculate $w_I^H$, for every point I, in accordance with

$$w_I^H=H^n{{\displaystyle \sum }}_{I^{\prime }}{K}_{I,I^{\prime }}^{HH}{u}_{I^{\prime }}^H,$$

(5)

where $K_{I,I^{\prime }}^{HH}=K_{2I,2I^{\prime }}^{hh}$

(b’)

Coarse grid correction

$$w_I^H=w_I^H+h^n{{\displaystyle \sum }}_{\left|2I-i^{\prime }\right|\le m}\left(K_{2I,i^{\prime }}^{hh}-{\widehat{K}}_{2I,i^{\prime }}^{hh}\right)u_{i^{\prime }}^h.$$

(6)

(c)

Interpolation

For each point I, compute $w_i^h$ by using Equation (3).

(c’)

Fine grid correction

$$w_i^h={[I{I}_H^h{w}_{\cdot }^H]}_i+h^n{{\displaystyle \sum }}_{\left|i-i^{\prime }\right|\le m}\left(K_{i,i^{\prime }}^{hh}-{\widehat{K}}_{i,i^{\prime }}^{hh}\right)u_{i^{\prime }}^h.$$

(7)

This approximate kernel is defined as ${\widehat{K}}_{i,i^{\prime }}^{hh}={[I{I}_H^h{K}_{\cdot ,i^{\prime }}^{Hh}]}_i$, where ${\widehat{K}}_{I,i^{\prime }}^{Hh}=K_{2I,i^{\prime }}^{hh}$. The correction term $\left(K_{i,i^{\prime }}^{hh}-{\widehat{K}}_{i,i^{\prime }}^{hh}\right)\to 0$ as $\left(i-i^{\prime }\right)\to \infty$, indicating that the kernel $K^{hh}$ is becoming smoother with the increase of $\left|i-i^{\prime }\right|$.

In the following sections, we no longer use the general notation but are restricted to the dry-contact equations.

2.2. Fast Solution of Dry-Contact Pressure

The dry-contact problem can be expressed mathematically as:

$$\{\begin{array}{c}f\left(x,y\right)=0, p\left(x,y\right)>0, \mathrm{contact} \\ f\left(x,y\right)>0,p\left(x,y\right)=0, \mathrm{no} \mathrm{contact} \end{array}$$

(8)

where p is the pressure and f is the film thickness.

The film thickness equation is given by:

$$f\left(x,y\right)=f_0+g\left(x,y\right)+\frac{2}{\pi E^{\prime }}{{\displaystyle \int }}_{-\infty }^{+\infty }{{\displaystyle \int }}_{-\infty }^{+\infty }\frac{p\left(x^{\prime },y^{\prime }\right)}{\sqrt{{(x-x\prime )}^2+{(y-y\prime )}^2}}\mathrm{d}{x}^{\prime }\mathrm{d}{y}^{\prime }.$$

(9)

$E^{\prime }$ is the reduced modulus of elasticity; g is the gap between the undeformed geometries with a minimum distance of 0. The last term is the elastic deformation.

The correct value of f₀ satisfies the following condition, called the force balance equation:

$$w={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }p\left(x,y\right)\mathrm{d}x\mathrm{d}y,$$

(10)

where w is the load applied.

To solve the problem numerically, it is convenient to make Equations (9) and (10) dimensionless by introducing the variables: P = p/p_m, Y = y/b, X = x/b and F = f /$\delta$. Then, discretizing dimensionless equations with an equidistant grid in the X and Y directions, one gets:

$${\displaystyle \sum }_{i\prime }{\displaystyle \sum }_{j\prime }{P}_{i\prime ,j\prime }^h=\frac{w}{h^2{p}_m}$$

(11)

and

$$F_{i,j}^h=F_0+G_{i,j}^h+D_{i,j}^h.$$

(12)

$D_{i,j}$ is the discretized elastic deformation, expressed:

$$D_{i,j}^h={\sum }_{i\prime }{\sum }_{j\prime }{K}_{i,i\prime ,j,j\prime }^{hh}{P}_{i\prime ,j\prime }^h.$$

(13)

The coefficients $K_{i,i\prime ,j,j\prime }^{hh}$ are defined as:

$$K_{i,i\prime ,j,j\prime }^{hh}=K_{\left|i-i\prime \right|,\left|j-j\prime \right|}^{hh}=\frac{2p_{h}b}{\pi E^{\prime }\delta }{{\displaystyle \int }}_{Y_j-h/2}^{Y_j+h/2}{{\displaystyle \int }}_{X_i-h/2}^{X_i+h/2}\frac{\mathrm{d}X\prime \mathrm{d}Y\prime }{\sqrt{{(X_i-X\prime )}^2+{(Y_j-Y\prime )}^2}}$$

(14)

with $X_i=X_0+i\cdot h/b$ and $Y_j=Y_0+j\cdot h/b$. One can see that $K_{i,i\prime ,j,j\prime }^{hh}$ is only dependent of $\left|i-i^{\prime }\right|$ and $\left|j-j^{\prime }\right|$.

${\delta }_{i,j}$ can be obtained as:

$${\delta }_{i,j}^h=\frac{r_{i,j}^h}{K_{0,0}^{hh}-K_{1,0}^{hh}}$$

(15)

where $K_{0,0}^{hh}=K_{i=i\prime ,j=j\prime }^{hh}$ and $K_{1,0}^{hh}=K_{i=i\prime \pm 1,j=j\prime }^{hh}=K_{0,1}^{hh}=K_{i=i\prime ,j=j\prime \pm 1}^{hh}$.

Let ${\stackrel{-}{P}}_{i,j}^h$ denote the current approximation of $P_{i,j}$. The distributed relaxation can be implemented in a Jacobi-type process. The new approximation ${\stackrel{-}{P}}_{i,j}^h$ is obtained which satisfies:

$${\bar{P}}_{i,j}^h={\tilde{P}}_{i,j}^h+{\delta }_{i,j}^h-\frac{1}{4}\left({\delta }_{i-1,j}^h+{\delta }_{i+1,j}^h+{\delta }_{i,j-1}^h+{\delta }_{i,j+1}^h\right).$$

(16)

The influence of the distributed changes decays fast like the second derivative of K: $r^{-3}$ ($r=\sqrt{{\left|X-X^{\prime }\right|}^2+{\left|Y-Y^{\prime }\right|}^2}$). The detailed description of the distributed relaxation and its effect on the magnitude of ${\delta }_{i,j}$ are given in [23]. The full approximation scheme (FAS) (proposed by Venner and Lubrecht [23]) are used, since the problem is non-linear, to accelerate the relaxation’s convergence. The computation of the elastic deformation on a grid point already needs O (N) operations. Consequently, the computation in all the grid points will require O (N²) operations. Incorporating the MLMI algorithm into a full multigrid (FMG) solver can obtain optimal efficiency O (Nln(N)). The details for the fast solver can be seen in [23]. The fast solver has been validated (see the earlier work of one of the authors [3]), acquiring a high accuracy on 6-level multi-grids and taking 100 times less time than the computation on the single finest grid. The accuracy increases with increasing mesh size (the level of the multi-grids).

2.3. Fast Solution of Sub-Surface Stress

Solving the sub-surface stresses of an elastic half space, with the pressure being given, is a 3-D problem. Calculating the stresses on a plane (Z constant) resembles the fast integration of the elastic deformation outlined in Section 2.1. The integrals (the six stress components) can be written as:

$$S_l\left(X,Y,Z\right)={\int }_{\Omega }{K^{\prime }}_l\left(X,X^{\prime },Y,Y^{\prime },Z\right)P\left(X\prime ,Y\prime \right) \mathrm{d}{X}^{\prime }\mathrm{d}{Y}^{\prime }, l=1, 2,\dots ,6.$$

(17)

It becomes a 2-D problem in each Z plane. The equation can be discretized, similar to Equation (13), as:

$$S_{l,i,j}^h\left(Z\right)={\sum }_{i\prime }{\sum }_{j\prime }K{\prime }_{l,i,i\prime ,j,j\prime }^{hh}\left(Z\right)P_{i\prime ,j\prime }^h.$$

(18)

The discrete kernels were obtained according to the work of Johnson [24] and Kalker [25]. One can find in [25] the detailed expressions for the kernels K_l which, therefore, are not listed here. These K_l resemble the kernels for solving elastic deformation, so the same techniques for accelerating the calculating as in Section 2.2 can be maintained, which can reduce a great deal of programming effort. Calculating the stress tensors plane by plane can avoid storage problems, because the number of points dealt with at a time is much smaller [1].

The fast solver has been validated by Lubrecht et al. [1], by comparing the numerical solutions of the stress component S_yz with the analytical solutions given in [26,27] for Hertzian contact. Using 6th order transfer operators, sufficient accuracy was acquired for all Z planes.

3. Results

As an example, the contact pressure, stress and sliding temperature of the dry contact of wire rope-friction lining and wire rope were calculated, to test the ability of the solvers (codes) to deal with large numbers of points.

The stranded wire rope in Figure 1 was constructed with a fiber core and 6 twisted strands, each strand formed of 6 twisted wires and a core, resulting in 36 double-helical outermost wires. The envelop of the wire rope surface complemented the friction lining groove (the nominal diameter of the wire rope was approximately equal to the groove’s diameter). Therefore, the contact produced multiple discrete contact areas. That the wire rope slid along the groove could be seen as many protuberances moving on a smooth flat surface.

The surface of the groove was cylindrical, so it was easily mathematically described. The expression for the wire rope surface can be referred to the earlier work of one of the authors [3], omitted here. Then, one could easily compose the film thickness equation, Equation (9), for this dry contact.

3.1. Pressure of Friction Lining-Wire Rope Contact

The contact of the wire rope with the friction lining’s groove could be simplified into a line contact, by seeing it as a cylinder. Therefore, we chose the Hertzian line contact solutions for the three reference parameters b, $p_m$ and $\delta$, which were, respectively, the contact half width, maximum Hertzian pressure and maximum deformation [23]. They were expressed as:

$$b=\sqrt{\frac{8w_{1}R\prime }{\pi E\prime }},$$

(19)

$$p_m=\frac{2w_1}{\pi b}$$

(20)

and

$$\delta =(\frac{1}{4}+\frac{1}{2}\ln 2)\frac{b^2}{R^{\prime }}.$$

(21)

The value w₁ was the load of unit length, $w_1=w/L_l$; $R^{\prime }\mathrm{and} E^{\prime }$ were the reduced radius of curve and reduced elastic modulus, $1/R^{\prime }=2/D_w-1/R_l$.

The parameters involved in solving the contact pressure and sub-surface stresses are listed in

Table 1. Parameters for solving the friction lining-wire rope dry contact.

Rl (mm)	Ll (mm)	Dw (mm)	dwc (mm)	dwo (mm)	Rw (mm)	Rs (mm)	${\mathsf{\beta }}_s$ (rad)	${\mathit{E}}^{\prime }$ (MPa)	Poisson Ratio (−)
14.1	70	28	3.11	3.11	3.11	9.33	1.18	657.23	0.3

. The values of $b$,$p_m$ and $\delta$ obtained with load w = 4900 N were 23.1 mm, 1.926 MPa and 0.16 mm. The dimensional computational domain was $x\in \left[-L_l/2, L_l/2\right]$ and $y\in \left[-R_l, R_l\right]$, so the dimensionless one was $X\in \left[-1.515, 1.515\right]$ and $Y\in \left[-0.610, 0.610\right]$. The code involved 8-level multi-grids with the coarsest grid being of 10 × 4 points. The mesh size on a finer grid was half that on the adjacent coarse grid.

Figure 2 shows the distribution of contact pressure on the friction lining surface for load w = 1960 N. One can see that the contact was of the multi-point and elliptical type. From Equations (7)–(9) in [3], it can be seen that with an increase in w, the local maximum pressure and the area of each contact spot increased.

3.2. Sub-Surface Stress of the Wire Rope-Friction Lining Contact

Figure 3 presents xy and xz slices of the von-Mises stresses and those of the z-component stress distributions under different load conditions. One can see that on the surface, the pattern of the von-Mises stress, S_VM, was similar to that of the z-direction stress component, S_z. One should notice that S_z on the surface was actually the contact pressure. The larger the load, the larger the local contact areas, and with an increase in the load, S_VM and S_z at a point increased.

Figure 4 shows the variations of S_VM and S_z at Point (0, 0, z) with depth z. One can see that the maximum S_VM occurred at some place below the surface. z = 0.33, 0.43, 0.44 (mm) for w = 1960, 3920, 5880 (N), and the maximum S_z occurred on the surface.

3.3. Sliding Temperature of the Wire Rope-Friction Lining Contact

Here we conducted an extended study on the temperature in the friction lining with the wire rope sliding. The calculation could be approximately transformed into solving the transient temperature in a semi-infinite space with heat sources moving on the surface [28].

Regarding the solution, at any timestep one first obtained the pressure distribution and then calculated the transient sliding temperature with the pressure solution as the load. Details of the solver using the multi-level techniques (proposed by one of the authors) can be seen in [4], but are omitted here.

Table 2. Physical parameters for sliding temperature solutions of the friction lining-wire rope contact.

sw (mm)	${\mathit{\alpha }}_{\mathit{w}}$ (m2/s)	${\mathit{k}}_{\mathit{w}}$ (W/(m $\cdot$ K))	${\mathit{\alpha }}_{\mathit{l}}$ (m2/s)	${\mathit{k}}_{\mathit{l}}$ (W/(m $\cdot$ K))	μ (−)	v (m/s)
100	1.37 × 10−5	44.9	1.7 × 10−5	0.405	0.385	5 m/s

lists the parameters for solving the sliding temperature. These values were kept constant for all the cases unless statements were made locally. Together with the values of $\mathrm{b}$ and $p_m$ (23.1 mm and 1.926 MPa), it was possible to obtain the temperature results.

Figure 5 shows the transient temperature distributions on the xy and xz planes with different loads. At the initial time (t = 0), the wire rope was still and generated many elliptical contact spots on the friction lining’s groove and, then, with the wire rope sliding (t > 0), the moving contact spots formed the contact bands. One can see from Figure 5 that the larger the load w, the larger the width of the contact spots (bands) and the larger the temperature rise.

Further, in Figure 6 we present the temperature rise at some depths below the surface point (x = 0, y = 0). One could find that with time, the temperature near the surface showed a saw-tooth pattern. With time, the surface and sub-surface temperatures increased. With increasing depth z, the temperature decreased and temperature fluctuation reduced.

4. Conclusions

A fast solver for dry-contact problems was addressed. Using MLMI, fast calculation of the elastic-deformation integrals for obtaining the contact pressure was accomplished. The fast integration method could be applied directly to the computation of stresses if one considers solving the stress field depth by depth.

The complexity of the efficient solver was O (NlogN), which reduced computing time a great deal, compared to many traditional solution methods with O (N²).

As an example, the fast solution schemes were successfully applied to a complicated dry-contact problem: the contact pressure and the stress distribution of the friction lining-wire rope contact were obtained with little effort. Finally, the transient temperature of the friction lining while the wire rope was sliding was numerically solved by using multigrid techniques. It was found that the contact was multi-point and the contact spots were elliptical. The larger the load, the larger the contact areas and the larger the local maximum pressure of each contact spot. With an increase in the load, both the von-Mises stress and the z-component stress increased. The temperature near the surface showed a saw-tooth pattern. With time, both the surface and sub-surface temperatures increased and with increasing depth z, the temperature decreased and temperature fluctuation reduced.