Micro-texture design and optimization in hydrodynamic lubrication via two-scale analysis

Abstract

A novel computational surface engineering framework is developed to design micro-textures which can optimize the macroscopic response of hydrodynamically lubricated interfaces. All macroscopic objectives are formulated and analyzed within a homogenization-based two-scale setting and the micro-texture design is achieved through topology optimization schemes. Two non-standard aspects of this multiscale optimization problem, namely the temporal and spatial variations in the homogenized response of the micro-texture, are individually addressed. Extensive numerical investigations demonstrate the ability of the framework to deliver optimal micro-texture designs as well as the influence of major problem parameters.

Appendices

Appendix: A: Filter formulations

Topology optimization based on an element-wise constant design variable distribution is prone to checkerboard patterns (Diaz and Sigmund 1995; Sigmund and Petersson 1998). This instability may be avoided by employing filters. The morphology filter $\mathcal {F}$ of Section 2.2 operates on the design variable distribution s ∈ [0, 1] within a neighborhood $\mathcal {D}^{K}$ in order to deliver the morphology variable degree of freedom $\rho ^{K} = \mathcal {F}(\mathcal {D}^{K},s)$. Different filter formulations and their performances have recently been discussed in Svanberg and Svärd (2013) — see also Bourdin (2001) and Sigmund (2007). For their formulation, conic weights w ^KI ∈ [0, 1] with the property ${\sum }_{I} w^{KI} = 1$ are defined in order to ensure that ρ ^K remain within [0, 1]:

$$ w^{KI} = \left\{ \begin{array}{cc} \frac{R - d(K,I)}{{\sum}_{J\in \mathcal{D}^{J}} R - d(K,J)} & I \in \mathcal{D}^{K}\\ 0 & I \ne \mathcal{D}^{K} \end{array} \right. \quad. $$

(A.1)

The neighborhood $\mathcal {D}^{K}$ is defined as the set of elements I whose distance d(K,I) to element K are less than or equal to a radius R. The distance of element K to I is measured by $d(K,I) = \sqrt {(X_{K} - X_{I})^{2}+(Y_{K} - Y_{I})^{2}}$ where (X,Y ) denote the element index coordinates, each coordinate lying in the interval [1,N _m ] where N _m = 40 is the number of elements per edge (Section 2.3). With respect to this measure of distance, the default value of the radius is chosen as R = 4 in all numerical investigations, unless otherwise noted. It is remarked that checkerboard patterns do not appear if the design and morphology variables are interpolated (Matsui and Terada 2004; Guest et al. 2004). Even in this case, a filter is useful for assigning a length scale to the micro-texture pattern — see Waseem et al. (2016) for a study of filter radius influence on the micro-texture. In this work, a linear (LN) filter will be employed for 0-to-1 type micro-texture design and an exponential erode (EE) filter for 0-or-1. For a given s-distribution, they deliver ρ ^K which respectively satisfy the following formulations:

$$ \rho^{K}_{LN} = \sum\limits_{I} w^{KI} s^{I} \quad,\quad e^{\alpha(1 - \rho^{K}_{EE})} = \sum\limits_{I} w^{KI} e^{\alpha \left( 1 - s^{I} \right)} \quad. $$

(A.2)

For the exponential erode formulation, α = 100 is employed.

Appendix: B: Sensitivity expressions

In Section 2.4, the sensitivity analysis of the governing macroscopic equation (2.3) had been outlined. Note that the right-hand side of (2.3) applies only to the time-dependent problem but its sensitivity vanishes in that case as well, because the mean film h ₀ is independent of the micro-texture by construction. Moreover, the rate of the primary solution field p ₀ does not enter the governing macroscopic equation, thus rendering the present analysis different with respect to other transient optimization problems (Michelaris et al. 1994). Hence, the expression (2.13) is a generic form that applies to both temporal and spatial variations of h ₀, with the term involving C further vanishing in the former case due to the special structure of the squeeze-film problem. In that case, in order to eliminate the influence of a particular choice of the time instant for optimization (see also the discussion in Section 3.3), the macroscopic objective function (3.4) was formulated in terms of a time integral. Upon solving for $\pi _{I} = \frac {\partial p_{0}}{\partial s^{I}}$ from (2.13) at each discrete time, its sensitivity may be evaluated as

$$ \frac{\partial \varphi}{\partial s^{I}} = - \frac{{\int}_{T} {\int}_{\Omega} 2 \pi_{I} p_{0}(s) \ \ \mathrm{d}{\Omega} \ \ \mathrm{d}{t}}{{\int}_{T} {\int}_{\Omega} p^{2}_{\text{ref}} \ \ \mathrm{d}{\Omega} \ \ \mathrm{d}{t}} \quad. $$

(B.1)

For the wedge problem, the relevant expression is simplified due to the absence of a time-integral in (4.2). However, because the sensitivity of both A and C must be evaluated in order to determine π _I from (2.13) and because these terms are spatially varying, the overall cost is significantly higher.

The microscopic analysis dominates the overall computation time, both in obtaining the solution p ₀ as well as in the sensitivity analysis (Table 1). Consequently, as noted in Section 2.4, gains due to an adjoint sensitivity approach for the macroscopic sensitivity analysis are negligible and the more straightforward direct approach has been preferred instead for calculating π _I . However, it should be highlighted that the gains due to an efficient microscopic sensitivity analysis are significant and an appropriate formulation for determining $\{\frac {\partial \boldsymbol {A}}{\partial s^{I}},\frac {\partial \boldsymbol {C}}{\partial s^{I}}\}$ is crucial to an effective optimization algorithm construction. In this context, it is noted that the variational approach presented in Waseem et al. (2016) for this purpose does not follow a standard adjoint sensitivity approach but the results are consistent with known adjoint sensitivity results that apply to one part of the analysis, namely $\frac {\partial \boldsymbol {A}}{\partial s^{I}}$, in view of its similarity to sensitivity analysis for heat conduction problems (Bendsøe and Sigmund 2004). The term $\frac {\partial \boldsymbol {C}}{\partial s^{I}}$ has no direct counterpart in material design problems.

The sensitivity expressions (2.12) for $\{\frac {\partial \boldsymbol {A}}{\partial s^{I}},\frac {\partial \boldsymbol {C}}{\partial s^{I}}\}$ have been verified in Waseem et al. (2016) by comparing the analytical (A) expressions with numerical (N) differentiation results. Here, a similar study is carried out for verifying the analytical sensitivity expressions for $\frac {\partial \varphi }{\partial s^{I}}$ which were discussed above. For this purpose, representative 0-or-1 and 0-to-1 type designs are selected from both types of macroscopic problems. For numerical differentiation, following the earlier study for $\{\frac {\partial \boldsymbol {A}}{\partial s^{I}},\frac {\partial \boldsymbol {C}}{\partial s^{I}}\}$, second-order finite difference is employed where each s ^I is sequentially perturbed by 10⁻⁴. The aim is to observe, at each point of the micro-texture, a factor of less than 10⁻³ between $\left | (\partial {\varphi }/\partial {s^{I}})_{A}-(\partial {\varphi }/\partial {s^{I}})_{N} \right |$ and the maximum value of $\left | (\partial {\varphi }/\partial {s^{I}})_{A} \right |$ throughout the micro-texture. The results summarized in Fig. 19 demonstrate this, thereby verifying the derived sensitivity expressions. Employing a perturbation value smaller than 10⁻⁴ is generally not desirable due to possible numerical ill-conditioning errors but a fourth-order scheme would clearly further decrease the relative difference. However, the evaluation of the second-order scheme is already significantly more costly than the analytical expressions, which also highlights the efficiency gains due to use of the latter.

Fig. 19

The analytical (A) sensitivity distributions (∂ φ/∂ s ^I) _A are provided and compared with the numerical (N) sensitivities (∂ φ/∂ s ^I) _N through the difference $\left | (\partial {\varphi }/\partial {s^{I}})_{A}-(\partial {\varphi }/\partial {s^{I}})_{N} \right |$ on converged 0-or-1 and 0-to-1 type textures: for (a)–(b) from Fig. 8 of the squeeze-film problem (S) for the sinusoidal velocity profile, and for (c)-(d) from Fig. 11 of the wedge problem (W) for 𝜃 = 0. Note that the ratio between the difference and the analytical sensitivity is less than 10⁻³ for all cases

Figure 20 depicts the change of the objective function value with the number of iterations in attaining the micro-textures employed in the analysis of Fig. 19. For completeness, the results for the square-wave velocity profile have also been included. Note that the initial guess employed does not necessarily satisfy the optimization constraint, which can lead to initial rapid changes in the objective. Once the micro-texture starts to develop, the objective function decreases almost monotonically to convergence. The rate of convergence can be strongly influenced by the MMA parameters. In view of the large number of test cases considered, no attempt was made to tune these parameters for faster or smoother convergence on a case-by-case basis and, instead, the default choices which were previously outlined in Waseem et al. (2016) were preserved.

Fig. 20

The variation of the objective function with the number of iterations is summarized for the squeeze-film problem results of Fig. 8 and for the wedge problem results of Fig. 11 (for 𝜃 = 0)