Development of a calibration technique of a finite element model for calculating compensation of deformations from the action of residual stresses in additive manufacturing

. In additive manufacturing, it is necessary to take into account and compensate for the warping of the part due to the action of residual stresses. One of the effective fast methods for calculating the magnitude of deformations in CAE-systems is mechanical finite element analysis, which does not require numerous iterations. To ensure the accuracy of calculations in CAE-systems, they must be calibrated on special samples. The paper proposes a method of such calibration on annular samples for the process of direct metal deposition (DMD), the result of which are the values of internal deformations necessary to calculate the deformations of parts during DMD. With the help of the Simufact Additive CAE system, a digital model of the DMD process was designed and the effect of residual stresses was simulated. Based on the obtained results of comparison with full-scale samples, the calculated parameters have been adjusted, which can be applied to optimize the geometry of the workpiece parts, taking into account the features of the DMD process.


Introduction
In recent years, additive manufacturing (AM), also known as 3D printing, has been widely recognized and has become one of the fastest growing technologies in the field of manufacturing.To obtain large-sized blanks of complex geometric shape from heat-resistant nickel alloys, an additive technological process of direct supply of energy and material is used, known as direct metal deposition (DMD) [1,2].Currently, the use of nickel materials in the field of additive technologies is limited by the features of ultrafast crystallization processes, which causes the accumulation of significant internal stresses, which leads to the formation of micro-and macro-defects [3].
In general, the residual stresses acting on the part during welding are the result of the action of residual deformations: thermal, mechanical, shrinkage, creep, phase transition.These residual deformations are the result of the action of the heat source [4].
One of the most frequently used approaches in predicting residual stresses and deformations for AM is thermomechanical analysis [5,6], in which the fields of deformations and stresses are calculated from the thermal load.There are various software products, usually based on the finite element method -CAE systems, which can be used in modeling the heat-stressed state of a part.In addition to the traditional thermomechanical analysis consisting of two cycles of thermal and mechanical, there is a simplified mechanical analysis based on the method of internal deformations [7].
Internal deformation is defined as the ratio of the change in the distance between points in the stress relief state to the initial state.To apply this theory in practice, it is assumed that elastic deformation is not as important as plastic deformation [8].
With acceptable accuracy, simplified mechanical analysis is significantly more productive than conjugate thermomechanical analysis, since it is not an iterative process requiring recalculation of the stress-strain state as new layers of material are formed in the case of conjugate analysis.To ensure sufficient accuracy of calculations in CAE systems, they must be calibrated on special samples.Typically, samples of the "console" type [9] (Fig. 1 (a)) or the "bridge" type (Fig. 1 (b)) are used [10].As already noted, in additive manufacturing, it is necessary to take into account and compensate for the warping of the part due to the action of residual stresses.One of the effective fast methods for calculating the magnitude of warping in CAE systems is mechanical finite element analysis, which does not require numerous iterations.The accuracy of this method depends on the calibration of the CAE system for specific growing conditions on special samples.If the calibration technology has been developed for the selective laser melting process, then there is an urgent need for the development of such a method for DMD.This is especially true for the cultivation of ring parts, including size of their warping when separating from the platform. in a computer simulation system, for example Simufact Additive, a virtual experiment is conducted using a mechanical calculation method using various values of the main components of the internal residual strain tensor in accordance with the developed experimental plan and calculated deviations from roundness are determined;  regression analysis is carried out and regression models of deviations from roundness are obtained according to the data of the virtual experiment;  the components of internal deformations that minimize the differences between the data of virtual and full-scale experiments are determined by the method of least squares, i.e. calibrated values of the tensor of internal residual deformations are obtained;  the parameters of the workpiece correction are calculated using the calibrated values of the internal residual strain tensor.
In order to test this technique, full-scale annular calibration samples were grown (Fig. 3).Deposition was carried out at the ILIST-L installation (SPbGMTU, St. Petersburg, Russia) using a four-jet coaxial nozzle with a 45° solution angle on pre-optimized technological modes (Table 1), from a metal-powder composition of a heat-resistant alloy EP648 (HN50VMTUB), Table 2, substrate material -steel 3. The outer diameter of the rings was 50.0 mm, wall thickness 6.0 mm, height 15.0 mm.The reference values characterizing internal residual deformations at a given optimal growing regime (Table 1) were the diameters of the sample in four sections according to the scheme shown in Fig. 2. Samples were measured before and after separation from the substrate using electrical discharge machining to exclude the introduction of additional stresses.The height of the cut samples was 10 mm.

Results of experiments
Table 3 shows the measurement results of calibration ring samples before and after separation from the substrate for construction.In finite element modeling by mechanical calculation method in the Simufact Additive CAE system, the initial values of the internal strain tensor (Table 4) were taken for this alloy according to data for selective laser fusion.Based on these data, the estimated coefficients of the tensor of internal residual deformations are calculated (Table 4), and a plan of virtual experiments is prepared (Table 5).According to the fractional plan of the experiment (Table 5), a series of calculations was carried out in order to obtain and further compare the control values of the diameters in four sections.The simulation results of one of the virtual experiments in the series are described below.
During the deposition process, stresses are concentrated in the lower layers and on the outer diameter, while reaching a maximum value of 1056 MPa.In the area of the lower inner diameter, the stresses are minimal and amount to 100 MPa (Fig. 7).After the deposition of all layers, the stresses are distributed over all layers of the manufactured ring sample, but the stresses in the lower layers are preserved (Fig. 8).

Fig. 8. Final stresses on one of the virtual experiments
Despite the concentration of stresses in the lower layers, the deformation is limited to the base plate on which the cultivation takes place.The simulation results show that the maximum value of deformation from the action of residual stresses is 0.03 mm (Fig. 9).
We introduce into consideration a matrix of factors, which are internal deformations (3), as well as a matrix of responses, which are deviations in thickness from the average value obtained from the results of a full-scale experiment on growing rings (4).
  In this case, the true values of internal deformations are (5), which are the solution of a system of nonlinear equations (6).
The quadratic form of F is essentially an implementation of the least squares method (7).
  The solution of the system of equations F=0 (8) was found in MATLAB using the lsqnonlin solver, which implements the algorithm (2). where The vector of initial approximations was used in the solution (10)  -for the upper bound.
The results of the calculation of internal deformations for rings 1, 2, 3 are given below ( 13

Discussion of experimental results
The solution of a system of nonlinear equations F = 0 (6) for calculating the values of internal deformations using the lsqnonlin algorithm in MATLAB has good convergence -no more than 10 iterations.A characteristic graph of the convergence of solutions depending on the number of iterations is shown in Fig. 10.As a result, without taking into account outliers according to the calculated values of internal deformations for 3 rings, we obtain an average vector of internal deformations, which is taken as the result of calibration (14).er = (ex ey ez)r = (-0,00677 -0,003 -0,01521) It should be noted that elastic deformation cannot be neglected in the additive process, as in the model of the traditional welding process.Let's consider this difference in more detail.Each passage forming the layer consists of a heating and cooling cycle.Due to this heat cycle, deformation occurs in the grown workpiece.The generated mechanical deformation is strongly localized at this point.These two states are associated with the formation of deformation at localized points due to the melting and solidification process [11,12].In addition, stresses from previously deposited layers arise in each newly added layer.After adding several new layers, the previous layer goes into a stable solid state and is exposed to elastic deformations from the upper layers.As the workpiece grows, this effect changes with a change in the heat-stressed state.In general, the deposition of the material can be divided into three stages: initial deposition, intermediate deposition, and stationary state.The initial deposition is the deposition of the first layer of material on the substrate.The intermediate state is the state when the heat source moves to the next point to weld a new layer.Stationary state is a state in which the entire workpiece is cooled to ambient temperature.The intermediate state is a direct result of the solidification process.Thus, the traditional theory of internal deformation developed for welding processes, applied to the additive manufacturing process due to the multilayer effect in the additive manufacturing process, can give a significant error [13].In order to get away from this error, it is necessary, according to the modified method of internal deformations [13,14], to replace the deformation caused by the thermal cycle with a tensor of internal deformations for each layer, which balances the stresses that arise.
When modeling an additive stress process, the temperature and displacement of a certain point in the layer depend on the number of layers applied earlier.In order to reduce the time for predicting deformations in finite element analysis, the mechanical formulation of the problem uses the technique of layer-by-layer nucleation and destruction of finite elements.The process of layer-by-layer nucleation and destruction of elements is shown in Fig. 11.When using this method at the initial stage, all elements are deactivated.The elements are activated layer by layer in the direction of growing the part.Their values of internal deformations are superimposed on each activated layer or several layers [13,14].To determine the internal deformations of a group of layers within the framework of the developed calibration method, it is enough to grow rings of different heights.

Conclusions
The simulation results show that the internal deformation method is a viable alternative modeling method to replace expensive thermomechanical modeling that requires significant computational resources.The method of modeling internal deformation significantly reduces the calculation time from several days to several hours.
The thermal boundary condition has a significant impact on the accuracy of results, since modeling with a realistic boundary condition increases the accuracy of calibration results.
The simulation results show that the built-in methodology minimizes the need for expensive computational thermomechanical modeling and reduces the calculation time.The internal deformation method can be used to ensure the manufacturability and quality of the product at an earlier stage of product development.A faster and more accurate way to predict deformations and stresses shortens the product development cycle for additive manufacturing.
1.A new method of calibration of the CAE system has been developed for calculating compensation for distortion of the geometry of the grown workpiece from the action of residual stresses, combining a full-scale and virtual experiment.A distinctive feature of the method is the use of ring samples structurally similar to axisymmetric parts, the cultivation of which uses the technology of forming tracks by equidistant.
2. Approbation of the DMD calibration method using annular samples from MPC EP648 allowed to obtain values of internal deformations, which can be further applied in the calculation of distortions of the geometry of products in CAE systems by the accelerated method.
The work was carried out with the financial support of the Ministry of Education and Science of Russia as part of the implementation of a comprehensive project to create a high-tech production on the topic: "Organization of high-tech production of industrial gas turbine engines with an intelligent system of design and technological training to improve functional characteristics" (Grant Agreement No. 075-11-2021-042 dated 06/24/2021.).

E3SFig. 7 .
Fig. 7. Operating stresses during the deposition process in one of the virtual experiments

Fig. 9 .
Fig. 9.The total displacement during the deposition process in one of the virtual experiments

Fig. 10 .
Fig. 10.Convergence of solutions of nonlinear equations for calculating the values of internal deformations

Fig. 11 .
Fig. 11.Illustration of the method of layer-by-layer assignment of internal deformation

Table 4 .
Estimated coefficients of the tensor of internal residual deformations and levels of its variation

Table 5 .
Fractional factor plan of virtual experiments ):