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Numerical 3D simulation of wire deposition process to predict distortion of parts
Mechanics of Advanced Materials and Modern Processes volume 5, Article number: 3 (2019)
Abstract
In the work, on the basis of the analysis of publications, existing approaches to the numerical modeling of additive processes of product formation are shown. The work itself is devoted to the study of the influence of parameters of the process of surfacing wire materials on the formation of residual deformations in parts. A mathematical formulation of the nonstationary thermomechanical problem is presented, and algorithms for solving the problem using the technology of birth and death of finite elements in the ANSYS package are described. Verification of the model created by the finite element was carried out on the basis of data from the experiment on surfacing a multilayer sample. The effect on the level of residual warping of the following process parameters was studied: the exposure time before the next layer was deposited; the motion path of the burner; ambient temperature. It is shown that the change in ambient temperature is the most effective way to reduce the residual distortions of the form.
Introduction
Additive manufacturing (AM)  the process of building threedimensional (3D) parts based on a digital model by gradually adding thin layers of material. This feature allows you to produce complex or unique parts directly from a digital model without the need to develop expensive tooling or molds, reducing the need for many conventional processing steps (Petrick and Simpson 2013; Wray 2014). Complex details can be performed in one step without the limitations inherent in traditional processing methods.
When creating products by using methods of AM, large temperature gradients and technological residual stresses arise in the volume of the material, resulting in disruption of the product shape, changes in the mechanical and operational characteristics of the object, and its destruction during manufacturing (King et al. 2015; Li et al. 2010; Ibiye et al. 2011; Parry et al. 2015; Wu et al. 2014; Baufeld et al. 2010; Riedlbauer et al. 2012). In order to work out the regimes and optimize the technological process, it is advisable to carry out a preliminary modeling of the process of layerbylayer formation of the product, which boils down to a multivariate solution of thermoconversion problems.
The numerical simulation of the process of reflow wire material is considered in the works (Agelet de Saracibar et al. 2014; Lundbäck 2011).
Sequential coupled analysis of nonstationary thermal conductivity with subsequent analysis of elastoplastic deformations is a frequently used approach to the numerical solution of temperature deformations and residual stresses (Labudovic et al. 2003). A fullfeatured analysis that solves the equations of heat conduction and the mechanics of a deformable solid at the same time was used by some researchers (Smith et al. 2016), however, conducting fullfeatured analysis requires much more computing resources. The calculation of residual stresses and buckling for the process of layerbylayer surfacing of wire remains the most difficult aspect in numerical simulation. As discussed earlier, adding material is modeled by adding and / or activating new preplaced items. There are three most commonly used methods for simulating material deposition: «element birth», «quiet element» and «hybrid activation» (Martukanitz et al. 2014; Michaleris 2014).
For the AM with a relatively small number of weld passes, it is permissible to have detailed modeling of each pass when constructing a part (Denlinger and Michaleris 2016). With this method of modeling, the heat input from the beam energy is usually applied as a volumetric heat flow, the center of which moves along the deposition path, thus representing a moving heat source. However, the AM process usually has a large number of layers, which makes it impractical to model each individual pass when creating a part. To ensure greater efficiency of calculations, the principle is used in which successive steps of melting and even layers are grouped together for subsequent simultaneous activation (Korner 2016; Prabhakar et al. 2015).
Also known are authoring methods for modeling stress and strain fields in an additive process. Thus, in order to increase the efficiency of calculations, Li et al. (2016) proposed a multiscale modeling method for quickly predicting distortion of a part. Another approach for effective modeling of warping in the AM is to use the inherentstrain method (developed by Yuan and Ueda 1996). Finally, Mukherjee et al. (2016, 2017) constructed an analytical expression for a special deformation parameter to estimate maximum residual deformations as a function of linear heat generation, substrate stiffness, maximum temperature, thermal expansion coefficient, which expresses the relationship between the velocity changes in thermal conditions in the environment and the rate of restructuring of the temperature field inside the system under consideration.
Thus, to study the complex thermodynamic processes occurring during the formation of the product in an additive way, various methods and approaches are used, described in the modern scientific literature. The main methods of studying are mathematical modeling, including computer modeling based on numerical algorithms, digital models and a multivariate analysis of the process with various implementations of technological parameters of the AM, taken into account using a different set of boundary conditions. There is a rather large amount of publications in terms of modeling the processes of formation of residual stress fields and heat shrinkable deformations in products obtained using additive technologies, including the method of wire welding. Often this uses wellknown software, such as ANSYS.
Methods
Mathematical model of the surfacing process
In a paper Smetannikov et al. (2017), a mathematical formulation of a similar fusion problem was described, adapted to the material Inconel 718 (Chiumenti et al. 2010). Taking into account the smallness of the deformations and the negligibly small dissipative heat release, we can separate the boundaryvalue problem of unsteady heat conduction and the boundaryvalue problem of thermomechanics on the stressstrain state, which in this formulation are unrelated. To solve them, as before, the technology of birth and death of elements in ANSYS is used. The results obtained by the Chiumenti et al. (2010) in comparing the measured and calculated temperatures at the points on the upper surface of the base plate showed that there are no experimentally observed sharp peaks in the graphs of calculated temperatures corresponding to the moments of closest approach to the heat source. In addition, the level of experimental temperatures on a given surface is noticeably higher than those determined in the calculation. On this basis, the radial transfer of heat from the heat source to the part of the plate surface visible from it is included in the updated formulation of the problem of unsteady heat conduction. S_{f}.
Another difference of the presented model from the previously published article (Chiumenti et al. 2010) is the hypothesis of maintaining a constant temperature of the deposited material. It is assumed that the power of the heat source is regulated in order to comply with this condition by feedback with the measuring element.
At the kth substep of the solution, the formulation of the boundaryvalue problem of unsteady heat conduction for determining temperature fields T(x, t) in the region V_{k} with the boundary S, taking into account the accepted hypotheses, includes:
Heat equation
\( \rho \left(\mathbf{x}\right)c\left(\mathbf{x},T\right)\frac{\partial T}{\partial t}=\operatorname{div}\left(\lambda \left(\mathbf{x},T\right)\operatorname{grad}(T)\right) \), x ∈ V_{k}, (1).
where c(x, T), λ(x, T), ρ(x)  respectively, heat capacity, thermal conductivity and density of heterogeneous doped material, \( {V}_k={V}_k^{old}+{V}_k^{new} \)  estimated volume, \( {V}_k^{old}={V}_{k1} \)  structure volume created in the previous substep, \( {V}_k^{new} \)  the volume of the structure builtup on the current step. On the reweldable part of the structure during the time of exposure to the beam t_{a} (см. раздел 4) the temperature is applied.
T(x, t) = T_{fl}, \( \mathbf{x}\in {V}_k^{new} \), t ∈ [t_{k − 1}, t_{k − 1} + t_{a}], (2).
where t_{k − 1}  end time of previous substep. The condition (2) does not act on the time interval t ∈ [t_{k − 1} + t_{a}, t_{k}] remaining until the end of the substep.
Boundary conditions
where the first component of the righthand side describes convective heat transfer, the second is radiation (the StefanBoltzmann law), and the third is radiation heating from a heat source (welding arc); ε_{f} – surface blackness coefficient, σ_{0}– StefanBoltzmann constant, h(T) – heat transfer coefficient, T_{c}(t)– ambient temperature, n – external unit normal to the boundary S of the cooled body, q_{rf}  power of heat flow from radiation heating of the surface by the arc. Assuming that the arc is a point source of thermal energy, the expression for the calculation q_{rf} takes the form
where \( F{\hbox{'}}_{ij}=\frac{F_{ij}}{F_{ij}\left(1{\varepsilon}_f\right)+{\varepsilon}_f} \), F_{ij}  integral angular radiation coefficient, \( {F}_{ij}={A}_e\frac{\cos {\theta}_j}{\pi {r}^2} \)_{,} r – distance between surface point and source, θ – the angle that the normal to the surface forms with the direction to the source.
Initial conditions
T(x, t_{0, k}) = T_{k − 1}(x), x ∈ V_{k}, (3).
where T(x, t_{0, k})  initial temperature distribution for kth substage, T_{k − 1}(x)  temperature determined at the end of the previous substage.
These ratios take into account that the research area \( {V}_k={V}_k^{liv}\cup {V}_k^{kil} \) remains unchanged during the substage. Here, the zones occupied by the “live” and “killed” elements, respectively, are denoted by \( {V}_k^{liv} \) и \( {V}_k^{kil} \). At the same time, the thermophysical properties of the material in the zone of “killed” elements are subject to degradation:
\( c\left(\mathbf{x}\right),\mathbf{x}\in {V}_k^{kil}<<c\left(\mathbf{x},T\right),\mathbf{x}\in {V}_k^{liv} \), \( \rho \left(\mathbf{x}\right),\mathbf{x}\in {V}_k^{kil}<<\rho \left(\mathbf{x},T\right),\mathbf{x}\in {V}_k^{liv} \), \( \lambda \left(\mathbf{x}\right),\mathbf{x}\in {V}_k^{kil}>>\lambda \left(\mathbf{x},T\right),\mathbf{x}\in {V}_k^{liv} \).
The unbound quasistatic boundary value problem of mechanics, taking into account the insignificance of the contribution of mass forces in the kth substage, includes:
Equilibrium equations
\( \operatorname{div}\ \hat{\sigma}=0 \), x ∈ V_{k}, (4).
where \( \hat{\sigma}\left(\mathbf{x},t\right) \)  stress tensor;
Geometric relations Cauchy
\( \hat{\varepsilon}=\frac{1}{2}\left(\nabla \mathbf{u}+{\left(\nabla \mathbf{u}\right)}^T\right) \), x ∈ V_{k}. (5).
where u(x, t)  displacement vector, \( \hat{\varepsilon}\left(\mathbf{x},t\right) \)  total strain tensor.
Boundary conditions
u = U, x ∈ S_{u, k}, (6).
\( \hat{\sigma}\cdot \mathbf{n}=\mathbf{P} \), x ∈ S_{σ, k}, (7).
where S_{u,} S_{σ}  parts of the boundary with given displacements and loads, respectively.
Thermomechanical parameters of the material in the killed elements zone exclude physical nonlinearity, are ideally elastic with degraded values.:
where \( {}^4\hat{C} \)  fourth rank tensor of elastic material constants.
Viscoelasticplastic behavior of the alloy from which the product is made, taking into account the temperature range, including the phase transition, suggests the possibility of using several different physical models of plasticity. In particular, the Anand model was used in [28]. In this work, the model of plasticity with bilinear isotropic hardening (BISO) and temperature dependence of parameters is used. The behavior of the material in the coordinates of equivalent stress equivalent strain is shown in Fig. 1.
In Fig. 1, the value σ_{y} corresponds to the yield strength.
The model has the following form:
where \( {}^4\hat{C} \)  fourth rank tensor of elastic material constants; \( \hat{\varepsilon}\left(\mathbf{x},t\right) \)  total strain tensor; \( {\hat{\varepsilon}}_e\left(\mathbf{x},t\right)=\hat{\varepsilon}\left(\mathbf{x},t\right){\hat{\varepsilon}}_P\left(\mathbf{x},t\right){\hat{\varepsilon}}_T\left(\mathbf{x},t\right) \)  elastic strain tensor; \( {\hat{\varepsilon}}_P\left(\mathbf{x},t\right) \)  plastic strain tensor; \( {\hat{\varepsilon}}_T\left(\mathbf{x},t\right) \)  thermal strain tensor; α(x, T)  coefficient of thermal expansion of the material; T_{0}  reference temperature; \( \hat{E} \)  rank two unit tensor.
The criterion of plasticity, which determines the level of stresses at which plasticity begins, has the form
where σ_{e} equivalent stress, σ_{y} yield strength,
If the equivalent stresses σ_{e} are less than the yield strength σ_{y}, then the deformation is described by Hooke’s law
where \( \hat{\sigma} \) stress tensor; \( {}^4\hat{C}={}^4\hat{C}\left(x,T\right) \)  fourthrank tensor of elastic material constants in the volume of “live” elements.
If σ_{e} ≥ σ_{y}, material undergoes plastic deformation. In the numerical implementation of the BISO model, the von Mises plasticity criterion (12) and the associative flow law are used; the work of plastic deformation is chosen as a measure of hardening. The law of hardening is determined by the ratio (13):
where \( \hat{s}\left(\mathbf{x},t\right)=\hat{\sigma}\left(\mathbf{x},t\right)\sigma \left(\mathbf{x},t\right)\hat{E} \)  stress tensor deviator;
where σ_{k}  plastic work function. For the BISO model, a σ_{k} is determined directly from the equivalent plastic deformations and the specified deformation diagram (Fig. 1).
The problem is solved using an iterative stepbystep procedure in terms of the increments of deformations and stresses. Plastic deformation is described by the associative flow law:
where \( d{\hat{\varepsilon}}_P \)  plastic strain increment; λ  Lagrange multiplier, in the numerical implementation of the BISO model it is determined in the NewtonRaphson iterative procedure; Q – plastic potential, which is already used as a function of fluidity.
The increment of stresses in the stepbystep procedure is calculated by the formula:
Taking into account the peculiarities of the behavior of elements are birthed by the technology used in ANSYS, relations (11) are converted to
where \( {\hat{\varepsilon}}_{k1} \)  total deformation, calculated by the end of the k1th substep (the actual state at the time of the “birthing” of the element is natural for him), \( {\hat{\varepsilon}}_B \)– strain of birthed elements.
Numerical model and verification
The mathematical formulation described above is implemented as software modules in the APDL language in the ANSYS Mechanical.
The algorithm used for the calculation of temperature fields in the ANSYS finiteelement package in the numerical simulation of the process of electron beam fusion involves the following calculation procedures.

1.
Creation of a finite element model, including the volumes divided into separate horizontal layers occupied by the deposited material, the plate  the base and part of the platform on which the sample is mounted, with corresponding thermophysical properties.

2.
“Killing” (EKILL command) parts of the elements of the product that are missing in the actual process of building up to its beginning.

3.
In a loop on incremental layers, starting from the bottom:

3.1.
Setting the conditions of convective heat exchange by the formula (3) on the upper boundary of the layer

3.2.
In a cycle by deposition zones of the current layer:

3.2.1.
Removal at the lower boundary of the part of the layer under the kth zone of the previously specified conditions of convective heat transfer

3.2.2.
The “birthing” (EALIVE command) of all elements of the kth zone

3.2.3.
Heating elements of the zone to a temperature T_{fl} (eq. 2) and exposure during the time of exposure to the beam t_{a}

3.2.4.
Removing temperature T_{fl} from nodes of elements and waiting for a time interval equal to the waiting time t_{wi}.

3.2.1.

3.3.
Waiting until next layer over time t_{wl}

3.1.
Waiting to complete (partial) cooling system
The exposure time of the beam t_{a}, taking into account the possible overlap of the burner movement trajectories, is calculated by the equation
where S_{s} = πD_{s}^{2}/4  spot square, v_{s}  burner speed, ds  distance between adjacent tracks. The waiting time t_{w, k} for the kth weld zone is determined by the expression
where S_{k}  zone square.
The described algorithm is used to calculate nonstationary temperature fields in experimental samples made of aluminum alloy D16 (Table 1).
In Fig. 2 shows photographs of fullscale specimens representing the bars welded to the base plate, which are used to verify the finite element model.
The following are presented the main parameters of the experiments. The size of the base (plate)  5х200х60 mm. The diameter of the wire  1,2 mm. The parameters are seted in the slicer: the height of the first layer  3.0 mm; roller width  4 mm; nozzle diameter  1 mm; filling strategy  longitudinal filling. Surfacing mode: slicer speed 400 mm/min. The trajectory of the burner corresponds to the scheme shown in Fig. 7,a. The geometrical parameters of the weld bars and the characteristics of the surfacing cycles are given in Tables 2, 3, respectively.
The sample fixing scheme, close to the cantilever, is chosen as the most free in terms of the formation of technological and residual deformations. In the finiteelement analogue of the temperature problem, the plate and the assembly table have a common border. During the solution 20node isoparametric elements Solid279 are used. The temperature was measured using a Mastech MS 6550B contactless pyrometer with a range of −32  + 1650 ° C and an error of not more than 1.5% of the readings. The distance from the pyrometer to the measuring point is 180 mm. The center of the measuring point was located 78 mm from the clamped end of the plate at the midline of its lower face. (KT on the Fig. 2,a).
Graphs of temperature evolution in KT are shown in Figs. 3 and 4. As can be seen from the figures, the data obtained qualitatively and quantitatively agree well with the experiment, the relative discrepancy does not exceed 5%. For comparison and evaluation of the convergence of the calculation algorithm in Figs. 3b and 4b presents the results of the calculation with doubled in comparison with the initial (see Fig. 2) number of fusion zones along the length of the weld bar. The relative discrepancy does not exceed 2%.
When solving the problem of thermomechanics (4)–(16), the discrete analog is identical to that shown in Fig. 2. Before starting the calculation, the Solid279 temperature element is replaced with 20node isoparametric element Solid186, which uses displacements as degrees of freedom and adds thermomechanical properties. The calculation algorithm includes:
1. The “killing” of the buiding up part of the elements of the product and elements of the assembly table. Setting the boundary conditions in displacements (symmetric fastening along the cut plane, cantilever fastening on the clamped faceplate).
2. In a loop on incremental layers, starting from the bottom:
3.2. In a cycle by deposition zones of the current layer:
3.2.1. The “birthing” of all elements of the kth zone.
3.2.2. Consideration of previously calculated for this point in time temperature to the nodes of the model during the time of exposure to the beam t_{a}.
3.4.2. Consideration of temperatures previously calculated for the moment of the end of the waiting time and exposure t_{wi} seconds.
4. Reading the temperature field for the time of partial cooling of the system, the calculation of stress and strain fields.
5. Cooling to ambient temperature, the calculation of stress and strain fields.
In Figs. 5 and 6 shows the results of solving the problem of thermomechanics. Figure 5 illustrates the dynamics of the longitudinal deflection of the plate. From the graphs, in particular, it can be seen that each transit of burner is accompanied by an initial downward bend, due to a higher temperature at the top of the plate. Further equalization of the temperature over the volume during the waiting time until the next pass causes, ultimately, the plate deflection by approximately equal to 0.42 mm. The Fig. 7 shows the calculated and experimental residual warping curves of the released samples placed on a horizontal surface. Measurements were made with a caliper with an accuracy of 50 μm. The maximum relative discrepancy does not exceed 5%, which confirms the adequacy of the proposed model.
Results and discussion
The study of the influence of technological parameters of the process on residual distortion
As an object of multivariant analysis of residual deformations, a plate made of aluminum alloy D16 with a weldable 10layer bar was considered. The dimensions of the plate, the thickness of the layers and the method of attachment are identical to the sample S4.
The following process characteristics were varied:

the exposure time before the start of surfacing the next layer;

burner trajectory;

the ambient temperature.
The effect of exposure time between the deposition of layers
In a fullscale experiment with sample S4, as follows from its description, a exposure time of 90 s was used. This is due to the need to prevent overheating of the structure in order to avoid uncontrolled spreading of the metal. In the calculation, we assume that the sample temperature control system provides this condition automatically and there is the possibility of a quick burner transition to the next layer. The burner path follows the pattern shown in Fig. 7, a: a layer is created in one pass, the starting points for all layers are the same. Further, this strategy is designated as tr_{1}.
The results of the study are presented in Table 4.
Analysis of the data presented in the table shows that by reducing the waiting time between the overlaying of the layers, it is possible to reduce the residual deflection by about 30%. This effect is explained by the fact that the minimum exposure time provides the smallest difference between the heating temperature of the material being deposited and the average temperature of the sample that does not have time to cool down after the previous approach, and, as a result, the lowest values of the temperature gradient. This should take into account the risk of melt spreading due to overheating.
The effect of surfacing strategy (burner trajectories)
The studied kinematic modes are shown schematically in Fig. 7. The trajectory of “continuous” surfacing, in which the beginning of the growth of the next layer coincides with the end of the current, was not considered due to possible overheating at the transition point. The strategies shown in Fig. 7, b, c (tr_{2} and tr_{3} respectively) imply the formation of each layer in 2 passes. In the first variant (Fig. 7, b), half of the layers pass from the periphery to the center of the bar, in the second (Fig. 7, c) from the center to the periphery. The exposure time between adjacent layers is 20 s. The calculation results are shown in Fig. 8.
From Fig. 8a, it can be seen that the buildup along the trajectories tr_{2} and tr_{3} is accompanied by the number of local extremes 2 times higher than that for the trajectory tr_{1}, which can be explained by the thermal wave reached by the CT from the slicer passage at each of the halflayers. The extension process ends for all the variants at about the same time, the subsequent cooling takes place almost synchronously. As Fig. 8b shows, the deflection of the free end of the sample decreases as the height of the bar increases, reaching 20 mm by the end of the process, and, as a result, its flexural rigidity. In the process of final cooling, the deflection grows slightly, by 0.2 mm. Its total value is maximum for the trajectory, and differs from the other two by no more than 1 mm (5%). A similar difference is also observed in the size of the deflection bowl (Fig. 8, c). As follows from the analysis of the results, a change in the deposition path, other things being equal, does not make it possible to significantly reduce residual warpage.
The effect of temperature
As follows from section 4.2, one of the ways to reduce residual deformations can be the approximation of the average sample temperature to the melt temperature. This, in particular, can be achieved by raising the ambient temperature. For the study selected 3 levels of temperature: = 25, 150 and 250 °C. At the same time, the deposition strategy corresponds to that shown in Fig. 7,a. Exposure between adjacent layers is 2 s. The calculation results are presented in Fig. 9.
From the figure, in particular, it follows that as a result of a decrease in temperature gradients in the sample, when the ambient temperature is increased to 250 °C, the level of residual deflections can be reduced more than 4 times (Fig. 9, b, c). In addition, at a given temperature, as can be seen from Fig. 9b, a slight decrease in the deflection is observed, starting from the 4th layer of surfacing from 4.8 to 3.7 mm at the last, followed by a slight increase to 4 mm by the end of cooling. Also observed is the asymmetry of the deflection bowl, which is exacerbated with increasing T_{s}. The latter is due to increased heat transfer on the side of the plate attached to the table. Thus, this method of reducing residual warpage is the most effective. The disadvantages include the need to work in a heat chamber. This places high demands on the heat resistance of the equipment and its compactness. In addition, the probability of melt spreading due to overheating increases, therefore, the accuracy of temperature control in the zone of the heating spot should be high.
Conclusion
1. The algorithm for calculating nonstationary temperature fields and the stressstrain state of the structure in the process of its creation by 3D arc welding of wire materials, taking into account the radiant transfer of heat energy of the welding arc to the surface of the product, was developed and implemented as a program on the APDL ANSYS.
2. Verification experiments on surfacing aluminum alloy bars on a plate were performed. The acceptable accuracy of the calculation results according to the developed numerical method is shown..
3. The effect on the level of residual warping of the following process parameters was studied: the exposure time before the next layer was deposited; the motion path of the burner; ambient temperature. It is shown that the increasing of T_{s} is the most effective way to reduce the residual distortions of the form, but it places increased demands on the thermal resistance of equipment and the accuracy of arc energy control.
Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 AM:

Additive manufacturing
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Acknowledgements
The authors thank the Ministry of Science and Higher Education of Russian Federation for creating conditions for interaction with colleagues from China and India in the framework of the implementation of the BRICS international project.
Funding
This work was carried out with financial supported by the Ministry of Science and Higher Education of the Russian Federation (RFMEFI58317X0062) within the framework of the BRICS project.
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OS: development of a modeling method, conducting numerical experiments, analysis of results. PM: model verification, simulation algorithm testing. DT: problem statement, organization of research, analysis of results. GP: conducting fullscale experiments. VB: organization of experimental studies, synthesis of results. AF: processing the results of fullscale experiments.
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Correspondence to Petr V. Maksimov.
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Smetannikov, O.Y., Maksimov, P.V., Trushnikov, D.N. et al. Numerical 3D simulation of wire deposition process to predict distortion of parts. Mech Adv Mater Mod Process 5, 3 (2019). https://doi.org/10.1186/s4075901900432
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Keywords
 Additive manufacturing
 Additive technology
 Deposition
 Wire deposition
 Finiteelement modeling
 Multivariate analysis