Today, the
development of powder metallurgy and laser optics allows not only to conduct
high-precision research [1], but also to develop new production methods. The
technology of layer-by-layer selective laser melting SLM (selective laser
melting) is a new technology of additive manufacturing, which allows to obtain
products of complex shapes, which can not be created with the help of classical
production methods. It is most in demand in the aerospace industry [2], for
example, when creating parts from various types of steels, aluminum and
titanium alloys [3], and the biomedical industry [4]. It is worth noting that
this technology is implemented on Russian equipment, for example, in the ML6-1
unit produced by the group of companies "Lasers and equipment", fig.
1.
Fig.
1. Experimental equipment ML-6-1.
Fig.
2. Detail made on equipment ML-6-1.
Fig.
3. Schematic diagram of SLM process. [9]
The narrowness of
the technological parameters is caused by the complexity of the geometry of
manufactured parts, fig. 2, and the complexity of the manufacturing process
itself, fig. 3. These aspects lead to the need for numerical simulation [5-7].
In the first approximation, the problem of modeling the SLM process is reduced
to solving the boundary value problem of thermal conductivity taking into
account the phase transitions of the first kind (melting and solidification).
The paper [8]
describes in more detail the numerical simulation of heat transfer in the
process of SLM and comparison with the calculations of other authors. On the
basis of the obtained results, a software package was created, including:
1. calculation of heat transfer in
SLM process, 2. visualization of temperature field simulation results. An
important part of the software package are the visual components responsible
for displaying the results. The program provides two modes of visualization:
two-dimensional and three-dimensional.
In the first
approximation, a mathematical model of the process of heating a metal object
with a mobile heat source is created for modeling the SLM process. The heat
source is a unimodal Gaussian beam of laser radiation. Some of the incident
radiation is reflected. The heated material undergoes melting and hardening
processes during the calculations.
To set the
Gaussian beam, the surface of the heated body was divided into several regions.
At each region, a boundary condition of the second kind was set, where the
incoming heat flow was calculated through the intensity of the radiation incident
on this region at a certain point in time. Thus, the Gaussian function of the
transverse spatial distribution of the laser beam power density is approximated
by a piecewise linear function. From the ratios (1-2)
(1)
,
(2)
have
,
(3)
where ω - the characteristic radius of the beam (the distance from the
center of the beam at which the intensity decreases e2
times), I0 - maximum intensity, P0 - laser power, I - intensity, r - distance to the center of the spot. Thus
we come to the dependence of the intensity at a particular point in space.
For the problem to
be solved it was necessary to consider the significant influence of the
reflection of the radiation from the metal surface. The formula for radiation
intensity, which is used by us in further calculations, is as follows
(4)
where R – the reflection
coefficient of the metal surface.
The basis of the
mathematical model, as already mentioned, is the equation of thermal
conductivity, which describes the distribution of temperature in a given region
of space and its change in time.
(5)
where p - density, C(T)
- a function of the dependence of heat capacity on temperature, K - the
coefficient of thermal conductivity, T - temperature, x,y,z -
spatial variables, t - time, q - the bulk density of heat sources
at time t at point x.
With initial conditions
and mixed boundary
conditions
The heat flux at
the boundary is calculated by using the formula (4).
This differential
boundary value problem using numerical differentiation formulas:
it is transformed
into a finite difference boundary value problem of the form
(6)
The model of phase
transitions is changed in comparison with the work [10] by means of
replacement:
Thus, have
where H - the enthalpy, K
- the coefficient of thermal conductivity, h - the spatial step, T
- the temperature, Δt
- the time step, i, j, m - the spatial coordinate indices, n -
the time index.
And the
temperature dependence on enthalpy (H) has the form [10].
,
where 
-
melting point, 
-
heat capacity before the phase transition, 
- heat capacity after the phase transition.
This explicit
finite difference scheme (6) has the first order of accuracy in time and the second
in space.
The heat capacity
of the material varies with temperature, this dependence is taken into account
by changing the model of temperature recalculation through enthalpy. The method
of temperature recalculation by means of enthalpy is somewhat modified by us,
in comparison with the work [10].
If we have data on
the dependence of the bulk heat capacity of the material on the temperature Ñ1
- Ò1,
Ñ2
-Ò2
, …, Ñn
- Tn, then, determining recursively the
enthalpy value corresponding to the specified temperatures
…
and,
having previously calculated these values, we come to a new model of
temperature recalculation from enthalpy
To simulate heat
transfer in the SLM process, SLMT1 programs were created to calculate the
temperature distribution in the melt bath region and SLMT2 to find the
temperature field in the entire part.
The SLMT1 program
served as the basis for the thermal simulation module (hereinafter
SimulationLib) for the LaserAprBuildProcessor software complex-a part of the
software of the laser 3D printing machine using the selective laser fusion
ML6-1 technology Produced by the Lasers and equipment group of companies.
LaserAprBuildProcessor
is a software complex, those tasks are: 1) providing a comprehensive
assistance to the process engineer and the operator of the machine, 2) the
division of a part model into layers, which are the input data for the
operation of the machine, 3) filling the layers with a plurality of
segments-laser trajectories, 4) building supports, 5) generation of laser modes
SimulationLib
(SLMT2) is a thermal simulation module, a companion program that makes it
easier for a process engineer to perform his or her duties.
Unlike SLMT1,
SimulationLib calculates the temperature distribution in the entire part rather
than in the melt bath area. In the new program SLMT2 became possible to
consider the trajectory of the laser beam, the geometry of the part, sprinkling
layers were taken into account the dependence of thermophysical parameters of
material temperature.
Ray paths are
segments defined by two points (the start point and the end point of the
segment). SimulationLib gets them, along with the geometry of the part from the
external environment of LaserAprBuildProcessor.
The program code
is written in C# using .NET platform technologies (WPF). The installation
program was created using NSIS (Nullsoft Scriptable Install System). The
program is designed to run on Microsoft Windows and requires the .NET Framework
4.7. For visualization, the capabilities of the open library SharpGL for C# are
actively used, which allows a user to easily use the capabilities of OpenGL.
Application of MVVM (Model-View-ViewModel) architectural design pattern allowed
to separate application logic from the visual part (representation) and to
achieve a clearer and more meaningful structure. The input data are CLI (Common
Layer Interface) files, a universal format for entering geometric data into
model fabrication systems based on layer fabrication technologies (LMT). It is
suitable for systems using layer-by-layer photo-curing of resin, sintering or
binding of powder, cutting of sheet material, solidification of molten
material, and any other systems that create models in layers. This format is
actively used in the additive manufacturing industry. Data of this format is
obtained as a result of the work of various programs for the preparation and
editing of data in the STL format (from the English. stereolithography) for
additive manufacturing. A striking example of such a program is Materialize
Magics.
In 2–D mode, the
part is displayed in layers, the projections of the supports necessary to
compensate for the thermal stresses formed during the process and prevent
deformation. If necessary, the program:
1.
Creates one or more equidistants outlining
the contour of the part, which allows a user to bypass the part along the
contour of the laser in different modes. This allows a user to achieve a higher
surface quality later.
2. Splits the interior space into areas
of varying area and shading them. Each resulting bar-line is the trajectory of
the laser. The procedure of such "shading" is a stage of preparation
of the technical specification of the machine.
Fig.
4. The circuit details.
In consequence, on each layer, fig. 4,
after the simulation, the temperature field is superimposed, fig. 5, fig. 6.
The calculation takes into account the trajectory of the laser, its power,
speed, form of radiation distribution, thermophysical parameters of the
material, powder deposition. Information about each step of the numerical
calculation is stored in a buffer, which allows a user to determine the
temperature value at any time of the upcoming fusion process for each layer. To
simplify the analysis of the results, the function of creating custom palettes
is provided, with which the technologist can determine the number of color intervals,
their length, color, as well as which fill option (gradient or solid) for each
interval to use. It is possible to dynamically change the monitored interval
using a two-finger slider, which allows a user to focus on a specific area of
interest to the technologist. Since even in layers with a high grouping of
objects and their dense shading, the number of displayed elements is relatively
low, performance problems do not arise even on weak systems. On the contrary,
this problem is acute in three-dimensional mapping.
Fig.
5. Visualization of the heating of the part in the manufacturing.
Fig.
6. The selection of the critical zone of the greatest heating is shown in red.
In
3-D mode, all layers are displayed together, allowing a user to see the part in
its original form, Fig. 7. It is possible to view temperatures, Fig. 8, Fig. 9,
but unlike the 2-D mode, the temperature field can be viewed in three
projections. The colors displayed in this mode are superimposed on three
perpendicular secant planes. When each of these planes moves, parts that
interfere with the view are removed from the display. Thus, in 3–D mode, the
operator is able to see the temperatures not only on the X0Y plane, but also in
the Y0Z and X0Z planes. Since the number of displayed graphical primitives
based on the layer map exceeds half a million for large parts in some cases,
the displayed geometry is simplified before rendering, so that the difference
is barely noticeable to the operator. Simplification is
implemented according to three principles:
1.
Cutting a three-dimensional model of the manufactured part occurs in
the data preparation programs, as indicated earlier. Because the geometry of a
layer is determined by looking for the intersection of the triangular
primitives that make up the object model, in the resulting file, very often
even straight lines represent more than one object. Such segments, without the
risk of losing accuracy, can be re-combined into one, thereby reducing the
number of primitives displayed in the aftermath. The computational complexity
of this simplification is 
,
where
N is the number of elements in the layer, and is applied only once.
2.
Because
of the high demands placed on the parts produced in the industry, the lines of
varying degrees of curvature are approximated by straight lines of small
dimensions, which allow maintaining high surface quality. When visualizing,
this accuracy is often unnecessary. Therefore, it is not the original layer map
that is used for the display, but its simplified version. If for two segments
the beginning of one segment coincides with the end of the second, and one
segment lies at a slight angle to the other (if this angle is less than a
predetermined value, depending on the distance between the point of view and
the segments in question), then such segments can be replaced by one in the
simplified version of the layer map. Since the segments in the layer are in
order and only the adjacent segments need to be compared, the computational
complexity is also , where N
is the number of elements in the layer.
3.
Most often, any two adjacent layers do not
differ much. This allows, with a small approximation, to exclude entire layers
from the simplified map. Layers are considered dissimilar if there are segments
in one layer that are at a great distance from all segments of another layer.
The computational complexity of comparing layers to determine
"similarity" is 
,
where N and M are the number of elements in the two layers being
compared. It is worth noting that a special numerical mark equal to the
approximation value is put in accordance with each layer, after which the
differences in the layers are invisible and this layer potentially should not
be drawn – this allows to further reduce the number of side calculations.
The second and
third simplification methods are applied when a user change the display scale.
Thus, on zoom out, the number of displayed primitives in most cases decreases,
which reduces the load on the system and speeds up the program. On zoom out,
the detail of the displayed part increases, which allows a user to avoid losses
in information content. The result is to increase the frame rate for large and
complex parts from an average of 7 to 15 FPS (frames per second). Despite the
fact that 15 FPS is quite a small number, this is quite enough, due to the fact
that purely stationary objects are displayed and the movement is represented
only in the form of rotation and movement of the camera, movement of secant
planes, changes in the number of displayed layers.
Fig.
7. Three-dimensional model of the created part.
Fig.
8. Part model with superimposed point-to-point temperature display model.
Fig.
9. A model of a part with a cutting plane on which the calculated temperature
field is superimposed. Intermediate temperature values are obtained by linear
interpolation.
Most
often, thermal calculations and similar numerical studies are carried out in
special programs, such as products of the Ansys and Comsol family [11].
Creating custom user models of the SLM process in these systems is very
difficult, setting all the geometry in manual mode requires an incredible
amount of time, and automation of this process requires a high knowledge of
specific languages, such as the parametric APDL language specially developed
for use in Ansys, and a high level of skills programming. Therefore, such
systems have their own solutions to this problem; we wrote in more detail about
the differences between these models from ours in [8].
Here it is worth
to dwell on the differences in visualization. The Ansys solution for SLM
process the received temperature is displayed on the surface of the created
items in the appropriate color, to reach values in depth details are not so
simple, this can be achieved by pre-split items apart in the graphical editor
or by using APDL, here we come again to the necessity knowledge of special of
the language. There are problems with setting the color palette, by default the
colors in the palette are set, and the interval to which the colors correspond
is tied to the maximum and minimum temperature. For all time of use of this
product, it is convenient to adjust a palette at us and it did not turn out. In
our work, the user can easily pick up a color palette, bind colors to certain temperatures
or temperature intervals, highlight areas of interest, use piecewise linear or
gradient view of painting, using secant temperature sensors at any point at any
time. All these features are available to the technologist through the user
interface, which significantly reduces the requirements for the level of
training of the user. Also, our program is included in the software for
installation of Russian production ML-6-1. The process of calculation and
display of its results works asynchronously, which allows at this time to
continue to use other functions of the software of the machine or to conduct
research on a third-party personal computer (PC).
Among the main disadvantages of the
presented implementation of the solution at this stage of work, is that the
calculation is mainly single-threaded, which does not allow full use of the
computational capabilities of multi-core processors. In more professional
programs Ansys and Comsol much attention is paid to the appearance of the work
and visualization of the calculation, texturing, working with lighting, our
solution visually looks more rough. Improving the texture quality is possible,
but it makes the program code more cumbersome, and the work in General more
time-consuming, which at this stage of the work we consider impractical.
As a result, the
creation of two display modes previously developed model was able to simplify
the process of research and analysis of the results. Extensive, but not yet
all-encompassing functionality allows technologists to see the fullness of the
information obtained during the simulation of the SLM process, to see the
dynamics of what is happening, to track the areas of interest. The additional
ability to use these two modes simultaneously (with the output of the picture
on one or more screens) allows a user to simultaneously see the picture as a
whole and concentrate on individual, local areas.
The authors
Express their gratitude to the management of the group of companies
"Lasers and Apparatus" for assistance in the material and technical
support of experimental studies and modeling of the process under
consideration.
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