Additive manufacturing technologies are a key point of the current era of Industry 4.0, promoting the production of mechanical components via the addition of subsequent layers of material. Then, they may be also used to produce surfaces tailored to achieve a desired mechanical contact response. In this work, we develop a method to prototype profiles optimizing a suitable trade-off between two different target mechanical responses. The mechanical design problem is solved relying on both physical assumptions and optimization methods. An algorithm is proposed, exploiting an analogy between genetics and the multiscale characterization of roughness, where various length-scales are described in terms of rough profiles, named chromosomes. Finally, the proposed algorithm is tested on a representative example, and the topological and spectral features of roughness of the optimized profiles are discussed.
The role of the interface between different material constituents/phases is a main research topic in this current era of Industry 4.0, which is radically changing perspectives of industry about the technological manufacturing of components and/or materials [
Nowadays, most approaches to design roughness are based either on mimicking natural surfaces [
The article is structured as follows. In Section
This study refers to the modeling and optimization of the mechanical behavior of two contacting rough surfaces. First, the contact problem and the roughness model are presented. Then, the mechanical interaction of different roughness length-scales is used to identify two categories of roughness. Finally, an algorithm is proposed to optimize roughness, expressing the objective function of the optimization problem as a suitable trade-off between two different target responses. To model contact, the frictionless elastic normal case is considered, taking as targets a desired stiffness-load curve,
Following the seminal work of Johnson [
As an example, in Figure
(a) Topography of a numerically generated rough surface. (b) Its deformed configuration during contact with a half-plane for the frictionless elastic normal problem.
In the work, roughness is described in terms of various parameters (see Section
Referring now for simplicity to one-dimensional rough profiles, their topography
A rough profile
In the absence of any subscript, the base 10 is used to compute the logarithm. In (
The indices
The particular combination of genes associated with each fixed value of the index
The nonlinear interaction of multiple length-scales of roughness provides the collective mechanical response of a rough profile. For example, the thermal/electric contact conductance is intimately related to the stiffness-load curve
In the following, the parameters that identify the stiffness-load curve
The profile
The profile
This distinction is shown in Figure
Topography of the rough profiles
(a) Stiffness-load curve
Regarding the
From the operative stand point, chromosomes associated with macro-roughness are selected as follows. First, the
In the following, we consider the problem of designing a rough profile in such a way that its stiffness-load curve and contact area-load curve are close, respectively, to a given target stiffness-load curve
Before going into the details of the proposed M-CCO algorithm, we introduce some notations and concepts. First, we define the similarity score
To compute consistently the similarity scores in (
The steps of the M-CCO algorithm are reported in Algorithm
(10) (11) (12) (13) (14) (15) (16) (17) (18) (19)
Then, the macroscale roughness is firstly identified, finding all the genomes with a similarity score
Similarly, the genomes with a similarity score
The resulting sets
For each genome
Now, the three genomes in
The square of the mixed similarly score is maximized by applying the Globally Convergent Method of Moving Asymptotes (GCMMA) algorithm [
Finally, the new genome
The Globally Convergent Method of Moving Asymptotes (GCMMA) [
The term
In Steps (16) and (17) of Algorithm
To apply the GCMMA algorithm inside Algorithm
The GCMMA algorithm is applied with several different initializations, in order to increase the probability of obtaining a good constrained local maximizer of the original objective function. Finally, the best solution, i.e., the one associated with the greatest (square of the) mixed similarity score obtained during the various optimizations, is produced as output, for both Steps (16) and (17) of Algorithm
To save computational time, the number of variables in each optimization problem is reduced as described in the following. The parameters (genes)
The M-CCO algorithm is now tested, designing roughness in a realization length
Target mechanical responses imposed to validate the algorithm, taking a realization length
For all the profiles in the database, the amplitude gene is fixed to
Each profile
The three best genomes obtained from the M-CCO algorithm are summarized in Table
For the three best genomes obtained from the M-CCO algorithm: mixed similarity score
genome | | | | GCMMA initialization | GCMMA initialization |
for Step (16) of Algorithm | for Step (17) of Algorithm | ||||
| |||||
| | | | 5-1 | 2-2 |
| | | | 11-1 | 2-2 |
| | | | 5-1 | 2-1 |
Genome database: (a) description of all the pairs (
The best approximation of the target contact responses is provided by the genome
Mechanical responses of the best rough profiles obtained from the M-CCO algorithm to achieve the two targets
Results of the M-CCO algorithm. For each of the three obtained genomes: (a) shows the macro-roughness; (b) shows the micro-roughness; (c) shows the complete profile.
Macro-roughness
Micro-roughness
Roughness
As far as the stiffness-load curve is concerned (see Figure
The
For both genomes
The three rough profiles obtained from the M-CCO algorithm are visualized in Figure
In Figure
As observed before, these solutions are originated from the same set of chromosomes as regards the macro-roughness level. Moreover, they have also quite close values of similarity scores (see Table
The micro-roughness of genomes
The topography of genome
Finally, the spectral decomposition of the three genomes is shown in Figure
Power spectral densities of the three genomes obtained from the M-CCO algorithm, whose associated profiles are shown in Figure
All the results of Fast Fourier Transform (FFT) filtering show a power spectral density of the complete profile containing several peaks. For all the three genomes, the first part of the PSD does not present any evident peak. For the genome
Looking at the solutions
In this paper, the Mixed Chromosomes Cross-Over (M-CCO) algorithm has been proposed to optimize a roughness topography to match two different target mechanical contact responses, namely the normal contact stiffness
Still, the results show that there is room for possible improvement, which could be obtained, e.g., by future variations of the M-CCO algorithm. Particularly, the algorithm could be improved in the selection of the chromosomes to combine. Future work will also concern the introduction of additional interactions between the rough surfaces in contact, such as friction, adhesion, wear, and so on. Such mechanical interactions are difficult to predict using BEM, and it is probably necessary to move to the Finite Element Method (FEM) to tackle nonlinear multi-field coupled contact and fracture problems with complex and realistic geometries. For example, the case of elasto-plastic contact is certainly a fundamental aspect to be investigated, and plastic deformation can be taken into account by using ad hoc computational models.
This work is a first step in the generation of surface topographies achieving desired target mechanical responses. The problem investigated in the article and the M-CCO algorithm are potentially useful for the production of surface pressure sensors in sealing applications. In this case, the contact has to be assured for a contact pressure larger than some threshold. At the same time, the electrical conductivity is associated with the
Another important potential application refers to the problem of surface morphing. In this case, roughness may be modified in time based on external stimuli. Morphing is a quite novel concept, which has been recently applied to smart structures and devices. However, its application to surfaces is a very new research area.
The problem considered in the work is important also in view of the fact that, nowadays, additive manufacturing technologies simplify the realization of topographies, enabling the production of surface prototypes. 3D printing is one of the most fascinating and robust technology, due to its very fast progress and impact on several industrial sectors, which promotes technology transfer and patent applications. Applications of the proposed methodology to surface printing could open new research areas.
The MATLAB data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
M. Paggi acknowledges the support of MIUR-DAAD to the Joint Mobility Program 2017 “Multi-Scale Modeling of Friction for Large Scale Engineering Problems”. G. Gnecco acknowledges the support of FFABR (Fondo per il Finanziamento delle Attività Base di Ricerca) from the Italian Ministry of Education, University and Research (MIUR).