Design of experiments are used in various fields such as the chemical formulation of new cosmetic products [MAT00], the creation of recipes in the food industry [SAB06] but also the optimisation of industrial processes such as brazing or welding [BID16].
One might wonder how experimental designs can be used for such varied applications. In fact, in all the examples cited, there is always a process-related objective to be achieved, for example: a cream that nourishes the skin without causing allergies, a pizza dough that crumbles and cooks in 10 minutes, a soldered connection that looks good and has the mechanical strength required for the application.
In all cases there are also variables, so-called factors, for example: the composition of the materials to be joined (copper, steel, aluminium, etc.) and their properties (melting temperature, thermal conductivity, etc.), the welding intensity, the pressure, etc.
The goals to be achieved can be as varied as our imagination. The factors may depend, among other things, on the process chosen to achieve the objective.
It should be noted that there are different methodologies for experimental design depending on the objective to be achieved [KAR04], examples are given below:
- I want to identify the influential factors of a new multi-parameter process → I will conduct a factor screening design of experiments
- I seek to quantify the factors and predict their influence → I will conduct a modeling design of experiments
- I seek to obtain the best operating conditions for my process → I will conduct an optimisation design of experiments
- I seek to take into account environmental factors, equipment wear etc. → I will conduct a robustness design of experiments
In this chapter, the experimental design approach is described for the specific case of resistance brazing process optimisation. It is therefore a question of modeling and optimizing my process.
At this stage, it is agreed that the operative and metallurgical weldability is demonstrated. The choice of the joining technique and the materials to be joined are not questioned during the experimental design.
The process of optimising my process is carried out in 7 steps described in figure 1.
This diagram immediately reminds us of the steps of a DMAIC:
The DOE is therefore not only an integral part of the DMAIC but also follows such a process.
1. Definition of the objectives
The definition of the objectives to be achieved is done by several people, for example: the project manager, the technical manager, the quality department etc...
It is a question of putting the approach into context by asking yourself several questions detailed below (the answers in blue correspond to the answers given for the concrete case).
- What is the context? industrialisation of a new interconnector
- Which product(s) are concerned? The interconnector and the stator
- How long does it take to complete the process optimisation? Expected completion before JPC for JPC with optimised processes
- Where is the design of experiments carried out? On the production line → provide slots on the line for testing
- Why carry out a design of experiments? To ensure that the solder joint between the interconnector and the stator is maintained during the life of the product (customer request)
2. Determining the study area
The first step is to make an exhaustive list of factors that can influence the quality of the process. It is recommended to classify them according to the Ishikawa diagram (Figure 2).
Next, it is necessary to determine which factors will become experimental factors by classifying them according to whether they are
- studied: variables in the experimental design
- controlled: variables fixed in the process parameters
- uncontrolled: variables that can change from one trial to another without control
- blocks: variables that can be sorted as differences from one day to the next
Table 1 shows the approach for the case of the resistance brazing process optimisation example.
Tableau 1 : Classification of factors
2.2. Area of study
The factors studied thus determine the space of the design of the experiment (Figure 3), but there are still elements missing to obtain the study domain (Figure 4).
In order to determine the study domain, the so-called high and low configurations of the design of the experiment must be known (Figure 5).
Sometimes experience is sufficient to know the high and low configurations of the experimental design domain, but it is usually necessary to carry out some tests to determine these bounds. In the case of the inter-connector and stator brazing, the configurations obtained are detailed in Table 2.
Table 2: Details of the high and low configurations of the design of the experiment
2.3. Answers to be optimized
At this stage, the answers have not yet been described, but they will determine the outcome of the process optimisation.
In the process, the question must be asked: What are we trying to achieve?
In the case of the example, the mechanical strength of the brazed assembly is one answer, but it is not the only one. In addition, there are visual criteria related to the indentation of the electrode, the wetting of the filler metal, or the non-degradation of the plastic overmoulding, for example.
For each answer, a success criterion must be determined. Table 3 shows the optimisation criteria for the soldering of the inter-connector.
Table 3: Optimisation criteria for resistance brazing
2.4. Choice of the experimental design model
In the context of the optimisation design of experiments, the choice of a test matrix based on the analysis of response surfaces is appropriate.
The name of response surface denotes the concern to visualise by a geometrical representation what is the response of a physical process to stimuli [DUP12]. The response studied, Y, results from the application of a transfer function that characterises the system with the input parameters (Figure 6).
Polynomial response surfaces are mainly used in mechanics [DUP12]. The polynomial quadratic model is a simple polynomial that describes the relationship between the factors under study and the responses. For a response Y (e.g. mechanical strength), the relationship with the factors (e.g. time: A, and intensity B) can be written according to [1.1].
With βx the coefficients of the response surface are determined by the least squares method from the experimental design. An illustration of the response surfaces is shown in Figure 7.
This type of model is suitable when the domain is well defined and allows the number of tests to be reduced.
3. Building the design of the experiment
The use of Box-Behnken designs has the advantage of reducing the number of test points compared to centred composite designs [MIN18]. Other constructions are possible, described in [FRA08], the objective always being to obtain a reliable response at a lower cost.
For each factor, the Box-Behnken design takes into account only three levels of the design domain defined upstream. It never includes points at the extremes of each factor. These matrices are then completed with a point at the centre of the domain.
An illustration of the points to be made is shown in Figure 8. The black dots are the test points to be carried out in the study domain, itself represented in blue and determined in §2.2.
Software such as Design Expert® or Minitab® can help to determine the experimental design matrix. For the purposes of this example, the test matrix is detailed in Table 4.
Table 4: Matrix of tests to be performed
4. Conducting the tests
The experimental design can now be carried out.
Carrying out the tests in accordance with the defined design is the most difficult part of the process, as it requires rigour and organisation and is time-consuming.
The tests must be carried out in order and duplicated. Duplicating the tests allows the reliability of the results to be adjusted. It is recommended that each configuration be repeated three times.
Each response should be measured and the non-destructive tests repeated to improve the reliability of the response.
After the tests have been carried out, the experimental design matrix table is completed with the responses obtained (Table 5).
Table 5: Design of experiment completed with responses
5. Analysis of the results
As described above, the analysis of the results is based on a quadratic model.
For each response :
- visual appearance
- mechanical behaviour
- fracture appearancecorresponds to a law meeting the quadratic polynomial model [eq 1.1].
The complexity of the model for each response can be reduced according to the coefficients obtained, in the case where the coefficients governing the curvature terms are almost zero, the model is simplified. The use of DesignExpert® or Minitab® allows the results to be processed in order to obtain the behaviour laws.
The analysis of the model is then based on the analysis of the rij residuals. The residual is the difference between the measured response rij (for replication J of test number i) and the value of the response predicted by the model Yi.
rij=Yij-Yi [eq 5.1]
The objective is to have for each answer [VIV16]:
- The normal residual plot shows that the residuals follow a normal distribution that describes the random variation of the process.
- The residual value with respect to the predicted values shows a random dispersion, which means that the same level of accuracy is expected for small and large values of the response.
- The residuals with respect to the order of execution are also random, which means that the study environment did not change during the tests.
- The predicted versus actual value is distributed along the 45° axis and indicates a better prediction for the higher mechanical strength.
In this case the models are accepted and the optimisation process can start.
6. Optimisation of the process
Process optimisation is achieved by means of a desirability function
For each response Yk a desirability function dk(Yk) assigns a value between 0 and 1 to the possible value of the response; 0 being the least desirable response and 1 the most desirable response meeting all the process optimisation criteria.
In the example, the objective of the experimental design is to have :
- a maximised mechanical strength of more than 400N
- a visual aspect after welding at 2 (slight visible indentation)
- a visual appearance of the fracture surface at 3 (fracture outside the brazed area)
The individual desirabilities are then combined using a geometric mean
With Wk, the relative importance given to the indexed criterion k, with
D=0 if one of the dk is zero and D=1 if all dk=1. The optimum configuration corresponds to the configuration giving the highest D. The result obtained in the case of resistance brazing is given below.
The optimum configuration is often a configuration outside the design of experiment matrix. Indeed, the optimum configuration (red dot in figure 9) is not a point of the design of experiment matrix. Therefore, this optimum must be confirmed and validated.
The practice, in the case of welding, is to repeat the welds 30 times in the optimum configuration and within the process cycle times and to confirm that the mechanical strengths are still above 400N.
The practice of design of experiments in the era of big data remains a key method in the search for industrial robustness. Lean 6 sigma tools provide rapid efficiency in the field. After several months of production, traceability data in the context of Big Data can provide additional optimisation leads with the implementation of machine learning algorithms, in addition to feedback from the field.
[SAB06] | R. SABRE, planification expérimentale en agroalimentaire, Techniques de l’ingénieur, 2006, F1005
[BID16] L. BIDI et al., Etude de l’influence des paramètres opératoires sur la morphologie des cordons de soudure - cas de soudage hybride (laser/mig), ICEMAEP2016, 2016
[MAT00] D. MATHIEU, R. PHAN-TAN-LUU, Planification d’éprience en formulation : criblage, Techniques de l’ingénieur 2000, J2240
[KAR04] S. KARAM, Application de la méthodologie des plans d’expériences et de l’analyse de données à l’optimisation des processus de dépôt, thèse de doctorat soutenue le 26 novembre 2004 à Limoges
[DUP12] F. DUPRAT et al.,Surfaces de réponse physiques et polynomiales, Fiabilité des ouvrages Sûretés, variabilité, maintenance, sécurité, pp. 165-193, 2012
[VIV16] L. Vivet (GSI-RD-H01-0000-424 Process optimization Standard)
Claire SCHAYES has a PhD in materials science. She started at Valeo in R&D. She then defined a method for predicting the fatigue behaviour of a steel used in alternator rotors. The aim was to be able to design the rotor as accurately as possible with regard to its mechanical reliability. Today, she uses her theoretical knowledge of materials on the choice of technologies, the choice of materials and the production of resistance brazing and LASER welding processes. Using DESIGN EXPERT, and permanently in the field, she testifies here about her practice and her conclusions between prediction and field reality.