Advances in Production Technology pp 69-84 | Cite as

# Meta-Modelling Techniques Towards Virtual Production Intelligence

## Abstract

Decision making for competitive production in high-wage countries is a daily challenge where rational and irrational methods are used. The design of decision making processes is an intriguing, discipline spanning science. However, there are gaps in understanding the impact of the known mathematical and procedural methods on the usage of rational choice theory. Following Benjamin Franklin’s rule for decision making formulated in London 1772, he called “Prudential Algebra” with the meaning of prudential reasons, one of the major ingredients of Meta-Modelling can be identified finally leading to one algebraic value labelling the results (criteria settings) of alternative decisions (parameter settings). This work describes the advances in Meta-Modelling techniques applied to multi-dimensional and multi-criterial optimization in laser processing, e.g. sheet metal cutting, including the generation of fast and frugal Meta-Models with controlled error based on model reduction in mathematical physical or numerical model reduction. Reduced Models are derived to avoid any unnecessary complexity. The advances of the Meta-Modelling technique are based on three main concepts: (i) classification methods that decomposes the space of process parameters into feasible and non-feasible regions facilitating optimization, or monotone regions (ii) smart sampling methods for faster generation of a Meta-Model, and (iii) a method for multi-dimensional interpolation using a radial basis function network continuously mapping the discrete, multi-dimensional sampling set that contains the process parameters as well as the quality criteria. Both, model reduction and optimization on a multi-dimensional parameter space are improved by exploring the data mapping within an advancing “Cockpit” for Virtual Production Intelligence.

## Keywords

Sheet Metal Radial Basis Function Network Training Point Latin Hypercube Design Ripple Structure## 6.1 Introduction

### Routes of Application

At least two routes of direct application are enabled actually by Meta-Modelling, namely, decision making and evaluation of description models. While calculating multi-objective weighted criteria resulting in one algebraic value applies for decision making, multi-parameter exploration for the values of one selected criterion is used for evaluation of the mathematical model which was used to generate the Meta-Model.

### Visual exploration and Dimensionality Reduction

More sophisticated usage of Meta-Modelling deals with visual exploration and data manipulation like dimensionality reduction. Tools for viewing multidimensional data (Asimov 2011). are well known from iterature. Visual Exploration of High Dimensional Scalar Functions (Gerber 2010) today focusses on steepest-gradient representation on a global support, also called Morse-Smale Complex. The scalar function represents the value of the criterion as function of the different parameters. As result, at least one trace of steepest gradient is visualized connecting an optimum with a minimum of the scalar function. Typically, the global optimum performance of the system, which is represented by a specific point in the parameter space, can be traced back on different traces corresponding to the different minima. These trace can be followed visually through the high-dimensional parameter space revealing the technical parameters or physical reasons for any deviation from the optimum performance.

Analytical methods for dimensionality reduction, e.g. the well-known Buckingham Π-Theorem (Buckingham 1914), are applied since 100 years for determination of the dimensionality as well as the possible dimensionless groups of parameters. Buckingham’s ideas can be transferred to data models. As result, methods for estimating the dimension of data models (Schulz 1978), dimensionality reduction of data models as well as identification of suitable data representations (Belkin 2003) are developed.

### Value chain and discrete to continuous support

The value chain of Meta-Modelling related to decision making enables the benefit rating of alternative decisions based on improvements as result of iterative design optimization including model prediction and experimental trial. One essential contribution of Meta-Modelling is to overcome the drawback of experimental trials generating sparse data in high-dimensional parameter space. Models from mathematical physics are intended to give criterion data for parameter values dense enough for successful Meta-Modeling. Interpolation in Meta-Modelling changes the discrete support of parameters (sampling data) to a continuous support. As result of the continuous support, rigorous mathematical methods for data manipulation are applicable generating virtual propositions

### Resolve the dichotomy of cybernetic and deterministic approaches

Meta-Modelling can be seen as a route to resolve the separation between cybernetic/empirical and deterministic/rigorous approaches and to bring them together as well as making use of the advantages of both. The rigorous methods involved in Meta-Modelling may introduce heuristic elements into empirical approaches and the analysis of data, e.g. sensitivity measures, may reveal the sound basis of empirical findings and give hints to reduce the dimension of Meta-Models or at least to partially estimate the structure of the solution not obvious from the underlying experimental/numerical data or mathematical equations.

## 6.2 Meta-Modelling Methods

In order to gain a better insight and improve the quality of the process, the procedure of conceptual design is applied. The conceptual design is defined as creating new innovative concept from simulation data (Currie 2005). It allows creating and extracting specific rules that potentially explain complex processes depending on industrial needs.

Before applying this concept, the developers are validating their model by performing one single simulation run (s. Application: sheet metal drilling) fitting one model parameter to experimental evidence. This approach requires sound phenomenological insight. Instead of fitting the model parameter to experimental evidence multi-physics, complex numerical calculations can be used to fit the empirical parameters of the reduced model. This requires a lot of scientific modelling effort in order to achieve good results that could be comparable to real life experimental investigation.

Once the model is validated and good results are achieved, the conceptual design analysis is then possible either to understand the complexity of the process, optimize it, or detect dependencies. The conceptual design analysis is based on simulations that are performed on different parameter settings within the full design space. This allows for a complete overview of the solution properties that contribute well to the design optimization processes (Auerbach et al. 2011). However the challenge rises when either the number of parameters increases or the time required for each single simulation grows.

Theses drawbacks can be overcome by the development of fast approximation models which are called metamodels. These metamodels mimic the real behavior of the simulation model by considering only the input output relationship in a simpler manner than the full simulation (Reinhard 2014). Although the metamodel is not perfectly accurate like the simulation model, yet it is still possible to analyze the process with decreased time constraints since the developer is looking for tendencies or patterns rather than values. This allows analyzing the simulation model much faster with controlled accuracy.

- 1.
Sampling

- 2.
Interpolation

- 3.
Exploration.

### 6.2.1 Sampling

Sampling is concerned with the selection of discrete data sets that contain both input and output of a process in order to estimate or extract characteristics or dependencies. The procedure to efficiently sampling the parameter space is addressed by many Design of Experiments (DOE) techniques. A survey on DOE methods focusing on likelihood methods can be found in the contribution of Ferrari et al. (Ferrari 2013). The basic form is the Factorial Designs (FD) where data is collected for all possible combinations of different predefined sampling levels of the full the parameter space (Box and Hunter 1978).

However for high dimensional parameter space, the size of FD data set increases exponentially with the number of parameters considered. This leads to the well-known term “Curse of dimensionality” that was defined by Bellman (Bellman 1957) and unmanageable number of runs should be conducted to sample the parameter space adequately. When the simulation runs are time consuming, these FD design could be inefficient or even inappropriate to be applied on simulation models (Kleijnen 1957).

The suitable techniques used in simulation DOE are those whose sampling points are spread over the entire design space. They are known as space filling design (Box and Hunter 1978), the two well-known methods are the Orthogonal arrays, and the Latin Hypercube design.

The appropriate sample size depends not only on the number of the parameter space but also on the computational time for a simulation run. This is due to the fact that a complex nonlinear function requires more sampling points. A proper way to use those DOE techniques in simulation is to maximize the minimum Euclidean distance between the sampling points so that the developer guarantees that the sampling points are spread along the complete regions in the parameter space (Jurecka 2007).

### 6.2.2 Interpolation

### 6.2.3 Exploration

Visualization is very important for analyzing huge sets of data. This allows an efficient decision making. Therefore, multi-dimensional exploration or visualization tools are needed. 2D Contour plots or 3D cube plots can be easily generated by any conventional mathematical software. However nowadays, visualization of high dimensional simulation data remains a core field of interest. An innovative method was developed by Gebhardt (2013). In the second phase of the Cluster of Excellence “Integrative Production Technology for High-Wage Countries” the Virtual Production Intelligence (VPI). It relies on a hyperslice-based visualization approach that uses hyperslices in combination with direct volume rendering. The tool not only allows to visualize the metamodel with the training points and the gradient trajectory, but also assures a fast navigation that helps in extracting rules from the metamodel; hence, offering an user-interface. The tool was developed in a virtual reality platform of RWTH Aachen that is known as the aixCAVE. Another interesting method called the Morse-Smale complex can also be used. It captures the behavior of the gradient of a scalar function on a high dimensional manifold (Gerber 2010) and thus can give a quick overview of high dimensional relationships.

## 6.3 Applications

In this section, the metamodeling techniques are applied to different laser manufacturing processes. The first two applications (Laser metal sheet cutting and Laser epoxy cut) where considered a data driven metamodeling process where models where considered as a black box and a learning process was applied directly on the data. The last two applications (Drilling and Glass Ablation) a model driven metamodeiling process was applied.

The goal of this section is to highlight the importance of using the proper metamodeling technique in order to generate a specific metamodel for every process. The developer should realize that generating a metamodel is a user demanding procedure that involves compromises between many criteria and the metamodel with the greatest accuracy is not necessarily the best choice for a metamodel. The proper metamodel is the one which fits perfectly to the developer needs. The needs have to be prioritized according to some characteristics or criteria which was defined by Franke (1982). The major criteria are accuracy, speed, storage, visual aspects, sensitivity to parameters and ease of implementation.

### 6.3.1 Sheet Metal Cutting with Laser Radiation

Process design domain

Beam Parameters | Minimum | Maximum | Sampling Points |
---|---|---|---|

Beam Quality M2 | 7 | 13 | 7 |

Astigmatism [mm] | 15 | 25 | 9 |

Focal position [mm] | −8 | 20 | 11 |

Beam Radius x-direction [µm] | 80 | 200 | 6 |

Beam Radius y-direction [µm] | 80 | 200 | 6 |

Data of the metamodels

Metamodel | MAE/[µm] | R |
---|---|---|

A | 7 | 13 |

B | 15 | 25 |

C | −8 | 20 |

D | 80 | 200 |

E | 80 | 200 |

The results show that the quality of the metamodel is dependent on the number of sampling points; the quality is improved when the number of training points is increased. As visualization technique contour plots were used, which in their entirety form the process map. The star-shaped marker, denoting the seed point of the investigation, represents the current cutting parameter settings and the arrow trajectory shows how an improvement in the cut quality is achieved. The results show that in order to minimize the cutting surface roughness in the vicinity of the seed point, the beam radius in the feed direction x should be decreased and the focal position should be increased Eppelt and Al Khawli (2014) In the special application case studied here the minimum number of sampling points with an RBFN model is already a good choice for giving an optimized working point for the laser cutting process. These metamodels have different accuracy values, but having an overview of the generated tendency can support the developer with his decision making step.

### 6.3.2 Laser Epoxy Cut

The metamodel takes the discrete training data set as an input, and provides the operator with a continuous relationship of the pulse duration and laser power (parameters) and cutting width (quality) as an output.

In order to address the first goal which is to minimize the lower cutting width, the metamodel above allows a general overview of the the 2D process model.

It can be clearly seen that one should either decrease the pulse duration or the laser power so that the cutting limits (the blue region determines the no cut region) are not achieved. The second goal was to include the cutting limits in the Meta-modelling generation. When performing a global interpolation techniques, the mathematical value that represents the no cutting regions (the user set it to 0 in this case) affects the global interpolation techniques.

From the results in Fig. 6.5, the developer can totally realize that process that contains discontinuities, or feasible and non-feasible points (in this case cut and no cut) should be classified first into feasible metamodel and a dichotomy metamodel.

The feasible metamodel improves the prediction accuracy in the cut region and the dichotomy metamodel states whether the prediction lies in a feasible or a non-feasible domain. This is one of the development fields that the authors of this paper are focusing on.

### 6.3.3 Sheet Metal Drilling

_{Th}.The reduced model assumes that there exists an ablation threshold characterized by the threshold fluence F

_{th}which is material specific and has to be determined to apply the model.

_{Th}where the width of the drill at the bottom is fitted. As consequence the whole asymptotic shape of the drilled hole is calculated and is illustrated in Fig. 6.7.

### 6.3.4 Ablation of Glass

As an example for classification of a parameter space we consider laser ablation of glass with ultrashort pulses as a promising solution for cutting thin glass sheets in display industries. A numerical model describes laser ablation and laser damage in glass based on beam propagation and nonlinear absorption as well as generation of free electrons (Sun 2013). Free-electron density increases until reaching the critical electron density \( \uprho_{\text{crit}} = (\omega^{2} m_{e} \varepsilon_{0} )/e = 3.95 \times 1021\,{\text{cm}}^{ - 3} , \) which yields the ablation threshold.

_{damage}for laser damage is a material dependent quantity, that typically has the value ρ

_{damage}= 0.025 ρ

_{crit}and is used as the damage criterion in the model. Classification of the parameter region where damage and ablation takes place reveal the threshold in terms of intensity, as shown in Fig. 6.8 below, change from an intensity threshold at ns-pulses to a fluence threshold at ps-pulses.

## 6.4 Conclusion and Outlook

This contribution is focused on the application of the Meta-Modeling techniques towards Virtual Production Intelligence. The concept of Meta-modelling is applied to laser processing, e.g. sheet metal cutting, sheet metal drilling, glass cutting, and cutting glass fiber reinforced plastic. The goal is to convince the simulation ana-lysts to use the metamodeling techniques in order to generate such process maps that support their decision making. The techniques can be applied to almost any economical, ecological, or technical process, where the process itself is described by a reduced model. Such a reduced model is the object of Meta-Modeling and can be seen as a data generating black box which operates fast and frugal. Once an initial reduced model is set then data manipulation is used to evaluate and to im-prove the reduced model until the desired model quality is achieved by iteration. Hence, one aim of Meta-Modeling is to provide a concept and tools which guide and facilitate the design of a reduced model with the desired model quality. Evalu-ation of the reduced model is carried out by comparison with rare and expensive data from more comprehensive numerical simulation and experimental evidence. Finally, a Meta-Model serves as a user friendly look-up table for the criteria with a large extent of a continuous support in parameter space enabling fast exploration and optimization.

The concept of Meta-Modeling plays an important role in improving the quality of the process since: (i) it allows a fast prediction tool of new parameter settings, providing mathematical methods to carry out involved tasks like global optimiza-tion, sensitivity analysis, parameter reduction, etc.; (ii) allows a fast user interface exploration where the tendencies or patterns are visualized supporting intuition; (iii) replaces the discrete data of current conventional technology tables or catalogues which are delivered almost with all the manufacturing machines for a good operation by continuous maps.

It turns out that a reduced model or even a Meta-Model with the greatest accuracy is not necessarily the “best” Meta-Model, since choosing a Meta-Model is a decision making procedure that involves compromises between many criteria (speed, accuracy, visualization, complexity, storage, etc.) of the Meta-Model quality. In the special application case studied here the minimum number of sampling points with a linear regression model is already a good choice for giving an optimized working point for sheet metal cutting, if speed, storage and fast visualization are of dominant interest. On the other hand when dealing with high accuracy goals especially when detecting physical limits, smart sampling techniques, nonlinear interpolation models and more complex metamodels (e.g. with classification techniques) are suitable.

Further progress will focus on improving the performance of generating the met-amodel especially developing the smart sampling algorithm, and verifying it on other industrial applications. Additional progress will focus on allowing the crea-tion of a metamodel that handles distributed quantities and not only scalar quanti-ties. Last but not least, these metamodels will be interfaced to global sensitivity analysis technique that helps to extract knowledge or rules from data.

## Notes

### Acknowledgments

The investigations are partly supported by the German Research Association (DFG) within the Cluster of Excellence “Integrative Production Technology for High-Wage Countries” at RWTH Aachen University as well as by the European Commission within the EU-FP7-Framework (project HALO, see http://www.halo-project.eu/).

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