=Paper= {{Paper |id=Vol-2744/paper53 |storemode=property |title=Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics of Objects |pdfUrl=https://ceur-ws.org/Vol-2744/paper53.pdf |volume=Vol-2744 |authors=Aleksandr Plaksin,Alexey Tolok }} ==Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics of Objects== https://ceur-ws.org/Vol-2744/paper53.pdf 
     Functional-Voxel Method in Problems of Geometric
      Modeling of Thermal Characteristics of Objects*

          Aleksandr Plaksin 1[0000-0001-9390-8322], Alexey Tolok 1[0000-0002-7257-9029],
1 Laboratory of Computer Graphics, V.A. Trapeznikov Institute of Control Science of Russian

           Academy of Sciences, 65 Profsoyuznaya street, Moscow, 117997, Russia
                 a.m.plaksin@gmail.com, tolok61@mail.ru



        Abstract. The paper presents an approach developed on the basis of the func-
        tional voxel method to the geometric representation of the thermal expansion of
        objects and temperature stresses in a material when exposed to a surface of a heat
        source. A discrete geometric law of a single temperature stress in an isotropic
        heat-conducting body is derived, applicable in the concept of functional voxel
        modeling. Based on this law, functional-voxel models of thermal stress are de-
        veloped for a single and distributed application of a heat source. Algorithms of
        functional-voxel modeling of temperature stress and expansion in the case of dis-
        tributed thermal loading are presented, which make it possible to construct a load-
        ing region of a complex configuration, uniformly form a contour (surface) after
        material expansion and obtain information about changes in the length (volume)
        of products. The advantages of the proposed functional-voxel approach to mod-
        eling thermal expansion and stress over approaches based on the FEM are sub-
        stantiated.

        Keywords: Discrete Geometric Model, Finite Element Method (FEM), Func-
        tional Voxel Method (FVM), Temperature Stress, Thermal Expansion.


1       Introduction

   Modern automated solutions to the problem of calculating thermal characteristics
are based on solving differential equations using numerical methods using the finite
element method, which is a complex computational problem [7]. To apply this approach
in the case of dynamic calculations of contact loads, it is necessary to constantly real-
locate the position of the nodes of the finite element grid, which requires powerful com-
puting resources. Moreover, it is a difficult task to include additional criteria in the
calculation scheme that affect the change in the geometric shape of the modeled object.
The solution of the presented problems is reduced to simplifying the problem statement,



Copyright c 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).

*   Supported by V.A. Trapeznikov Institute of Control Science of Russian
2 A.Plaksin, A. Tolok


which entails a loss of accuracy of calculations as a result of neglecting various condi-
tions.
   Solving the problem of finding the circumference contour of the processing object,
taking into account the influence of differential properties of physical processes, is an
urgent task of automated calculations. The method of functional-voxel modeling is one
of the possible approaches to the complex solution of this problem [2–10], due to the
possibility of linking the analytical formulations of physical laws applicable at a given
point in a single context of the entire discrete space under consideration. The use of the
functional voxel method simplifies the calculations by reducing the calculated expres-
sion to a linear polynomial that describes the local functional dependence at a specific
point in the simulated space.
   Functional-voxel method has high calculated applicability in solving analytical mod-
eling problems. This is achieved by the possibility of quick access to the differential
and integral characteristics of the modeled function at each point by means of local
geometric characteristics stored in multidimensional graphic images [2 - 10].
   Thus, the application of the functional-voxel method to the solution of the problem
posed will make it possible to have information on the stressed and expanded state of
the body at each of its points in computer modeling of the geometric characteristics of
thermal stress and thermal expansion.


2       Functional-Voxel Model of Temperature Stress

Consider the point application of the thermal load πœƒ to an isotropic heat-conducting
body. The heat applied to the body spreads evenly in the depths of the material in all
directions [11], which is geometrically expressed in the representation of the heat dis-
tribution region as a sphere in the three-dimensional case and as a circle in the two-
dimensional [1, 12]. Thus, the value of the function of the applied heat load changes as
the applied heat is distributed over the surface area of the formed sphere:
                                               𝛩
                                      πœŽπ›©π‘“ =         ,                               (1)
                                             4πœ‹π‘…2
   The presented expression will describe the distribution of heat when heat is applied
inside the material, but from a practical point of view, it is necessary to consider the
application of heat to the surface of a solid. In the case of an isotropic heat-conducting
body under the surface influence of heat, the formed heat distribution sphere will be cut
off, thus forming a hemisphere, the surface area of which is two times smaller than the
surface area of the sphere:
                                                𝛩
                                      πœŽπ›©π‘“ =                                            (2)
                                              2πœ‹π‘…2
    From the general equation of a sphere (2), it is possible to express its radius R:
                                        2
                                  𝑅 = √π‘₯ 2 + 𝑦 2 + 𝑧 2 .                                 (3)
 Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics… 3


   However, it is necessary to take into account the situation in which the radius of the
sphere in the denominator of the expression is zero. There is also a situation when the
                                                                          1
calculated value πœŽπ›©π‘“ exceeds the value of the applied heat load πœƒ at 1       > 𝑅 β‰₯ 0 in
                                                                              √2πœ‹
the case of considering a continuous representation of this propagation law. It is neces-
sary to make a transition to its discrete representation.
    The heat source is applied pointwise, then a sphere with a surface area of one is the
abstract β€œunit” application area:
                                       𝛩
                               πœŽπ›©π‘“ =      = 𝛩, Π³Π΄Π΅ 𝑆 1 = 1.                            (4)
                                       𝑆1
                                          1
   From 𝑆 1 = 4πœ‹π‘…2 we obtain 𝑅 = 2 πœ‹. To describe a discrete law on an entire geo-
                                          √
metric body, it is necessary to take a discrete parameter that describes the distance from
a certain point of application of heat to the studied point of the body. We denote it as
βˆ†π‘…π‘–                                          for                                        the
i-th application point (π‘₯𝑖 , 𝑦𝑖 , 𝑧𝑖 ). Then, the calculated area is represented as
                              1           2
𝑆𝑖 = 4πœ‹(𝑅 + βˆ†π‘…π‘– )2 = 4πœ‹ (2βˆšΟ€ + βˆ†π‘…π‘– ) = 1 + 4βˆšπœ‹βˆ†π‘…π‘– + 4πœ‹βˆ†π‘…π‘– 2 . Thus, the discrete
geometric law of temperature stress at some i-th point will have the form:
                                                        𝛩
                     πœŽπ‘– = 𝑓(π‘₯𝑖 , 𝑦𝑖 , 𝑧𝑖 ) =                              ,            (5)
                                               1 + 4βˆšπœ‹βˆ†π‘…π‘– + 4πœ‹βˆ†π‘…π‘– 2

    where βˆ†π‘…π‘– = √π‘₯𝑖2 + 𝑦𝑖2 + 𝑧𝑖2 provided that the origin of the coordinate system is
located at the point of application of heat.
    The presented expression geometrically partially corresponds to a quadratic hyper-
bole. The calculated value of πœŽπ‘– will coincide with the applied heat load πœƒ if the discrete
parameter βˆ†π‘…π‘– is equal to zero. With a single applied heat load, the described law takes
the form illustrated in Figure 1, which also shows the subfunctional space that is related
in volume to the heat load applied to the body.
    In the future, this temperature volume can be used in modeling problems in the
calculations, which circumvents the need to use the physical concept of temperature.




                       Fig. 1. The geometric law of temperature stress.

  The ratio between the applied heat load and the distributed heat will be:
4 A.Plaksin, A. Tolok


                                     πœŽπ›©π‘“ = π‘˜πœŽπ‘‰π‘“                                  (6)
  To determine the indicated equivalent volume of the temperature stress field, it is
necessary to calculate the quadruple integral of the form:
                             𝛩 𝑙 𝑀 β„Ž
                                                   𝛩
                   πœŽπ‘‰π‘– = ∫ ∫ ∫ ∫                                  𝑑π‘₯𝑑𝑦𝑑𝑧𝑑𝑒 ,                    (7)
                                       1 + 4βˆšπœ‹βˆ†π‘…π‘– + 4πœ‹βˆ†π‘…π‘– 2
                             0 0 0 0

   where 𝑙, 𝑀 and β„Ž are the size of the studied area.
   The presented expression is applied in the case of research in three-dimensional
space. To consider the two-dimensional case, it suffices to consider a cube, in the in-
creased space of which the value of the function corresponds, i.e., temperature.
   The necessary calculations are carried out by means of the functional-voxel method,
according to the technique proposed in [13]:
                          𝑛                                   𝑛
                         βˆ’            βˆ’ 𝑒𝑛        βˆ’                      2 βˆ’ 𝑒𝑛
                 1 + 4βˆšπœ‹βˆ†π‘…π‘– + 4πœ‹βˆ†π‘…π‘– 2               1 + 4 βˆšπœ‹βˆ†π‘… 𝑖 + 4πœ‹βˆ†π‘…π‘–
                                             + ||                                 |
                                                                                  |
                         𝑛                                   𝑛
       𝛩 𝑙 𝑀 β„Ž |                     βˆ’ 𝑒 𝑛 |      |                      βˆ’  𝑒 𝑛 |
                1 + 4βˆšπœ‹βˆ†π‘…π‘– + 4πœ‹βˆ†π‘…π‘– 2               1 + 4βˆšπœ‹βˆ†π‘…π‘– + 4πœ‹βˆ†π‘…π‘– 2
 πœŽπ‘‰π‘” = βˆ‘ βˆ‘ βˆ‘ βˆ‘                                                                      𝑆π‘₯ 𝑆𝑦 𝑆𝑧 𝑆𝑛 , (8)
                                              2
       𝑛=0 𝑖=0 𝑗=0 π‘˜=0


   where 𝑆π‘₯ , 𝑆𝑦 , 𝑆𝑧 ΠΈ 𝑆𝑛        are the discretization of the voxel space, and
                             2
βˆ†π‘… = √(𝑖𝑆π‘₯ )2 + (𝑗𝑆𝑦 ) + (π‘˜π‘†π‘§ )2 .

  The solution of both equations leads to similar results, allowing us to conclude their
equivalence. In the case of a single thermal effect in a space of 100 Γ— 100 Γ— 100, the
coefficient is:
                                                  1
                                            π‘˜=       .                                          (9)
                                                  30
    As a result, we obtain the equilibrium equation of volume distribution with which it
is possible to supplement the developed model of unit temperature stress. When simu-
lating heat propagation, this equation will allow one to determine the heating of the
application point k times less than the applied thermal loading.
    The obtained discrete geometric law can be applied to carry out functional-voxel
modeling of temperature stress in a solid isotropic heat-conducting body with a single
and distributed application of a heat source [6, 7].
    Figures 2 and 3 show a comparison of the visualization of the functional-voxel mod-
els (FVM) with a similar study conducted by the finite element method (FEM). As can
be seen from the figures, the proposed approach allows us to obtain a geometrically
adequate law of heat distribution.
 Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics… 5




                     FEM [12, 13]                                    FVM
Fig. 2. Comparison of the temperature stress distribution of the FEM and the FVM when apply-
                                     ing a load at a point




                     FEM [12, 13]                                       FVM
Fig. 3. Comparison of the temperature stress distribution of the FEM and the FVM when apply-
                                    ing a distributed load.




3      Functional-Voxel Model of Thermal Expansion

The interpreted law of the classical theory of thermal expansion for a discrete model
will be: βˆ†V𝑖 = 𝛼𝑇𝑖 𝑉𝑖 , where 𝛼 is the coefficient of thermal expansion, 𝑇𝑖 is the temper-
ature stress at the i-th point, and 𝑉𝑖 is the volume of the sphere formed by an increment
of the radius βˆ†π‘…π‘– .
    Then, to obtain a discrete geometric model of thermal expansion of the surface, it is
necessary to calculate the increment of coordinates at the i-th point (for example βˆ†π‘§π‘– ).
To do this, it is necessary to divide the obtained volume βˆ†V𝑖 into a discrete model for
determining the current area of the circle 𝐾𝑖 as a platform that is incremented by the
radius βˆ†π‘…π‘– :
   𝑆 1 = πœ‹π‘…2 = 1, and therefore 𝑅 = 1/βˆšπœ‹;
                                 1 2         1
  𝑆𝑖 = πœ‹(𝑅 + βˆ†π‘…π‘– )2 = πœ‹ (( ) + 2               βˆ†π‘… + βˆ†π‘…2 ) = 1 + 2βˆšπœ‹βˆ†π‘…π‘– + πœ‹βˆ†π‘…π‘–2 ; (10)
                                βˆšπœ‹          βˆšπœ‹
                                              βˆ†V𝑖
                              βˆ†π‘§π‘– =                         ,                         (11)
                                      1 + 2βˆšπœ‹βˆ†π‘…π‘– + πœ‹βˆ†π‘…π‘–2
6 A.Plaksin, A. Tolok


   where βˆ†V𝑖 = 𝛼𝑇𝑖 𝑉𝑖 .
   The volume βˆ†V𝑖 is determined similarly to the considered 𝑆𝑖 with the expression of
the accumulated volume 𝑉𝑖 :
                                4                                3   3
                          𝑉 1 = 3 πœ‹π‘…3 = 1 and therefore 𝑅 = √4πœ‹;

                    4πœ‹(𝑅 + βˆ†π‘…π‘– )3 4πœ‹ 3
             𝑉𝑖 =                =   (𝑅 + 3𝑅2 βˆ†π‘…π‘– + 3π‘…βˆ†π‘…π‘–2 + βˆ†π‘…π‘–3 ) =
                         3         3
                              3 4        3 4 2       4
                        1 + 3 √ πœ‹βˆ†π‘…π‘– + 3 √( πœ‹) βˆ†π‘…π‘–2 + πœ‹βˆ†π‘…π‘–3 ,                           (12)
                                3          3         3

where βˆ†π‘…π‘– = √π‘₯𝑖2 + 𝑦𝑖2 + 𝑧𝑖2 .
   With an increase in the radius of the generatrix of the sphere, the temperature value
decreases according to the hyperbolic law, which means that the obtained temperature
volume at the greatest distance minimally affects the geometry of the object. From this
we can conclude that the thermal expansion of the body by the volume Δ𝑉𝑖 at a specific
iteration will be carried out over the entire area of the circle with a radius βˆ†π‘…π‘– , the value
of which is the maximum remote distance, which is shown in Figure 4.




                Fig. 4. The change in body volume by βˆ†π‘‰π‘– for the current 𝑅.

   The presented model makes it possible to exert a point thermal effect on the surface,
using the concept of an abstract β€œunit” site, in contrast to the finite element method,
where it is necessary to use the region that makes up the element of the computational
grid as the loading point. Comparison of thermal expansion results by the presented
method and finite element method is shown in Figure 5, where geometric similarity of
simulation results can be observed [1, 12].
 Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics… 7




                   FEM [12, 13]                                                     FVM
Fig. 5. Comparison of simulation results with the point impact of a heat source to the body sur-
                                             face.

   The obtained model of heat distribution from a point source can be applied to obtain
a model of heat distribution from a distributed application of the source to the surface
of the body. In this case, the law of thermal expansion for a unit load must be applied
to each point of the contact area between the surface and the heat source. Thus, we
obtain a change in the volume of the current iteration βˆ†π‘‰π‘– for a point with the distribu-
tion over the area of a sphere with radius R distributed along a certain straight line L
(see Figure 6).




   Fig. 6. Change in body volume by βˆ†π‘‰π‘– for the current R during load distribution along L.

   Figure 7 presents the simulation results of the proposed method and the finite ele-
ment method, which indicate the similarity of the results of both modeling methods
[1, 7].




                      FEM [12, 33]                                            FVM
Fig. 7. Comparison of simulation results with the distributed effect of the heat source to the sur-
                                       face of the body.

  Due to the modeling of the application of both point and distributed application of
heat to the surface of a complex geometric shape [8-10], it is possible to create a full-
8 A.Plaksin, A. Tolok


fledged device for accounting for thermal characteristics during the passage of a cutting
tool. At the same time, the construction of complex geometric contours is realized
through set-theoretic operations, and specifically, with the R-functional modeling ap-
paratus [2, 4, 8]. Figures 8 and 9 show examples of the implementation of modeling the
application of thermal influence to the contour of a part of a complex configuration
with the occurring thermal expansion.




            Fig. 8. Example of applying a heat load at a point in a contour section.




    Fig. 9. Example of applying a distributed heat load to a specified section of the contour.


4      Refine Workpiece Bypass Contour

The developed approach can be applied to clarify the contour of the bypass of the pro-
cessing tool during the machining of parts. Consider this by the example of milling on
a universal milling machine of a beam workpiece. The measurement results of the ac-
tual contour obtained during processing are presented in Figure 10. In this case, the
measurements were carried out after the workpiece reached thermal equilibrium.
 Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics… 9




 Fig. 10. Comparison of the actual and required contour of a workpiece of the beam-type after the
                                        milling operation.

    To form a contour based on the functional-voxel method, it is necessary to use tem-
perature values that affect the distortion of the shape of the workpiece. Figure 11 shows
a functional image that visualizes the shape of the resulting thermal expansion of the
workpiece as a result of its processing and the tool bypass circuit. Using functional
voxel modeling tools, the resulting image was scaled along the abscissa for clarity.




         Fig. 11. The formation of the contour for the passage of the processing tool.

    The circuit obtained as a result of functional voxel modeling was used to compile
the control program code for a CNC milling machine for processing a workpiece of the
"beam" type. Figure 12 shows the results of measurements taken at the end of pro-
cessing and after the workpiece reaches thermal equilibrium, in comparison with the
results of processing on a universal machine. According to the above data, we observe
a more rectilinear contour of the part due to a decrease in thermal effects.
10 A.Plaksin, A. Tolok




       Fig. 12. Comparison of results before and after refinement of the circuit using the FVM
                                            method.


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