The paper presents an approach developed on the basis of the functional 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 developed for a single and distributed application of a heat source. Algorithms of functional-voxel modeling of temperature stress and expansion in the case of distributed thermal loading are presented, which make it possible to construct a loading 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 modeling thermal expansion and stress over approaches based on the FEM are substantiated.
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 [
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 [
Functional-voxel method has high calculated applicability in solving analytical
modeling 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 [
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 to an isotropic heat-conducting
body. The heat applied to the body spreads evenly in the depths of the material in all
directions [
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:
. To describe a discrete law on an entire geometric 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 ∆ i-th application point ( , , ). Then, the calculated area is represented as = 4 ( + ∆ )2 = 4 ( 1 2√π
+ ∆ ) = 1 + 4√ ∆ + 4 ∆ 2. Thus, the discrete geometric law of temperature stress at some i-th point will have the form: 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 calculated value exceeds the value of the applied heat load at 1 1 >
≥ 0 in √2 the case of considering a continuous representation of this propagation law. It is necessary 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 = ( , , ) =
, where ∆ = √ 2 + 2 + 2 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 hyperbole. The calculated value of will coincide with the applied heat load if the discrete parameter ∆ 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.
The ratio between the applied heat load and the distributed heat will be: (4) the (5)
To determine the indicated equivalent volume of the temperature stress field, it is necessary to calculate the quadruple integral of the form:
ℎ = ∫ ∫ ∫ ∫ where , and ℎ 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 increased 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 [
= = ∑ ∑ ∑
∑ =0 =0 =0 =0 ℎ |1 + 4√ ∆ + 4 ∆ 2 − | − 1 + 4√ ∆ + 4 ∆ 2 − + ||− 1 + 4√ ∆ + 4 ∆ 2 − | | |1 + 4√ ∆ + 4 ∆ 2 − | 2
, (8) (6) (7) (9) where , , и
are the discretization of the voxel space, and ∆ = √( )2 + ( ) + ( )2.
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 × 100 × 100, the coefficient is: = 1 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 simulating 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 [
Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics… 5 will be: ∆V =
, where is the coefficient of thermal expansion, is the temperature stress at the i-th point, and is the volume of the sphere formed by an increment of the radius ∆ .
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 ∆ ). To do this, it is necessary to divide the obtained volume ∆V into a discrete model for determining the current area of the circle as a platform that is incremented by the radius ∆ : 1 = 2 = 1, and therefore = 1/√ ; = ( + ∆ )2 = ((
∆ + ∆ 2) = 1 + 2√ ∆ + ∆ 2; 1 √ ∆ =
2 ) + 2 1 √
∆V 1 + 2√ ∆ + ∆ 2 (11) where ∆V =
. the accumulated volume :
The volume ∆V is determined similarly to the considered with the expression of 1 = 4 3 = 1 and therefore = √ 1 + 3√ ∆ + 3√( ) ∆ 2 + ∆ 3, (12) where ∆ = √ 2 + 2 + 2.
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 Δ at a specific
iteration will be carried out over the entire area of the circle with a radius ∆ , the value
of which is the maximum remote distance, which is shown in Figure 4.
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 [
Functional-Voxel Method in Problems of Geometric Modeling of Thermal Characteristics… 7
FEM [
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 ∆ for a point with the distribution over the area of a sphere with radius R distributed along a certain straight line L (see Figure 6).
Figure 7 presents the simulation results of the proposed method and the finite
element method, which indicate the similarity of the results of both modeling methods
[
FEM [
Due to the modeling of the application of both point and distributed application of
heat to the surface of a complex geometric shape [
Refine Workpiece Bypass Contour The developed approach can be applied to clarify the contour of the bypass of the processing 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 actual 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 temperature 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.
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 processing 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.