Despite the
increased interest in the task of visualization of simulated physical
quantities, to date, no hardware or software tools has been proposed that can
provide an adequate result.Even obtaining the value of the stress at a specific
point of an isotropic body entails calculations with relative to the values
which can be measured by instruments [1,2] or are calculated by complex
differential equations (FEM). For example, the resulting deformation of the
body helps to simulate the intense field acting in this case. The fact that
stress is a structural quantity that influences the design process is shown by
developing new directions for designing optimal geometry of structures
relative to premodeled stresses. Thus, there is a need to obtain local stress (stress
from a point load) as a structural unit for a minimum site. The selection of
such a stress element allows you to proceed to the design of stress fields of
any configuration. Existing CAD design modules such as SolidWorks, PTC Creo, ANSYS,
etc. [3] do not allow effective work with local stress and are focused on
modeling grid regions. At the same time, any attempts to model the point load
lead to a complex problem of singularity of the simulated mesh, which
negatively affects the stability of the FEM calculation (Fig.1). The fact that
stress is a structural quantity that influences the design process is evidenced
by developing new directions in designing the optimal geometry of structures
relative to previously modeled stresses. Thus, it becomes necessary to obtain
local stress (stress from a point load) as a structural unit for a minimum
site. The selection of such a stress element allows us to proceed to the design
of stress fields of any configuration. Existing CAD design modules such as
SolidWorks, PTC Creo, ANSYS, etc. [3] do not allow effective work with local stress
and focus on modeling grid regions. Moreover, any attempts to simulate a point
load lead to the difficult problem of the singularity of the simulated mesh,
which negatively affects the stability of the FEM calculation (Fig. 1). Uniform
thickening of the grid leads to an increase in the estimated time, and even to
a lack of RAM in the computer.
|
|
|
|
Figure 1:
Example of the result of calculating the stress
after applying the grid singularity at the point of force application
|
Figure 2:
Distribution of the force flux in a area near of
point of application of force
|
The approach to the construction a local stress model considered in
the article is one of the directions of a whole series of studies conducted by
prof. Tolok A.V.It is devoted to the application of the functional voxel
modeling method (VFM) [4] in solving engineering problems of the life
cycle.Prior to this, such problems as finding the path with obstacles [5],
problems of geometric modeling [6], problems of calculating the integral values
of objects of complex geometry [7], solving mathematical programming problems
[8], etc. were considered. The
basis of the PCM method is the principle of organizing symbolic and graphical
information on a computer that combines the analytical form for describing the
space of a function
with a multidimensional model of a voxel
representation of its local geometric characteristics. That is, any continuous
function of the form
in a given space m can be represented as a
computer model containing a linear polynomial
(local
function) at each point in this space, describing it tangent.
In this case the normal components
are aligned with the gradation of the color palette, and each
such component forms its own m-dimensional voxel.
As a result, the space of the original function of
m-dimension can be represented on the computer by the number of voxel images
equal to
. Such an approach allows one to obtain
differential and integral characteristics for research at points of a functional
multidimensional domain in a computer representation [9].
The
proposed approach to computer modeling of local stresses is based on the theory
of strength of materials [10], which relates to the study of tensor elements.
With the transition to grid methods of calculation (FE, MGE, etc.), the
theoretical aspects of this subject are undeservedly reduced to student study
and are rarely used in computer practice.This is due to the fact that the
approaches of strength of materials describe the principle of modeling stress
at a single point of application of force and writes the law for the selected
slope of a given cross section. Having settled on this, the theory of sopromat
does not allow simulating the transition from the vector of the applied force
to the volumetric stress vector, as a geometric object that occurs in a solid
isotropic medium and allows modeling of local stress.
We
introduce the geometric concept of a volume vector as a unit of the volume distribution
of a force vector in a solid isotropic medium.
Definition.
A
volume vector
should be understood as a
geometric object, defined by analogy with a conventional vector (as a directed
segment from the starting point, having an angle of direction
α
and a distant
of
),
however, the direction function
and the function of the value p(p)
are defined for the starting point.
Figure 3:
Directional Flow force F
For
example, we simulate the volume vector of local stress arising in a solid
isotropic body as a result of the action of a vertically applied force vector.
In this case, the force application point is considered the starting point of
the volume vector.
Let's start
by building the function
. To do this, it is necessary to determine certain
conventions, without which it is impossible to ensure the transition from a
continuous law, characterized by infinite approximations, to its discrete
model. We localize the point of force application by certain unit neighborhood,
i.e. sphere with a unit surface are
,
ãäå
. We generalizes the law by adding the
variable parameter
as an increment of the distribution
radius of the force vector
. Opening
the brackets and transform the right part, we get
. Considering
that the increase in the area under the applied force acts inversely with the
applied force
, the desired law
can be written as
. Further,
it should be noted that in the case of the application of the force
to the surface of a solid body, the
considered neighborhood of the point turns into a hemisphere, which means that
the law changes to
respectively.
Now
let's turn to the construction of the law
. Figure 2 demonstrates the principle of
projecting the force
on a perpendicular to the main site
of normal stress. The perpendicular to such a site is determined by the
direction of the straight line passing between the point of the body under
consideration A and the application’s point of force
(the starting point of the volume vector).
The projection of the force
. Next, we should return to physical
conventions and understand that the applied force must have some of a planar
neighborhood’s radius of the application, in our case, the radius of the
neighborhood is taken as
.
Figure 3
shows the cross section of the application of force
in the form of a directed flow to a
flat area bounded by radius
. Taking the body as an infinite beam of
bounded planes intersected at point
À
you can imagine an infinite number of rotatable minimal
neighborhoods with the unidirectional flow of force
applied to them (Fig. .4a). Figure
4b demonstrates a separate case of such a turn of the neighborhood relative to
the flow
, where there is a decrease in the number
of flow elements (in the form of arrows) falling on the site of the
neighborhood area when turning by the angle
In Fig. 4b, the rotation is shown by the
arrow. Given the obtained property, the projection
takes the following form:
.
|
|
|
|
Figure 4:
Changing
in load flow when a single
neighborhood is rotated
|
Combining
the functional laws
and
by multiplication, we obtain
the general functional law for constructing the volume stress vector
:
|
|
(1)
|
If the
origin of the coordinate system is set at the application’s point of force,
then
In this case, it is also easy to calculate the value
In [3], the
process of constructing a functional voxel model for the domain of analytic
function is described in sufficient detail. It will ensure the transition of
the analytical continuous representation of the volume vector function
to a discrete functional-voxel computer
representation. In this case, a computer organization of voxel images is
created that provide an area‘s description of the domain’s volume vector
function at the presentation’s level by its local geometric characteristics. For
illustration purposes, Figure 5 illustrates an example of a two-dimensional
representation of the central section in the
plane of a volume vector in the form of
four model images (M-images). The first two M-images store information about
the normal components
and
on the
and
,
ò
the third M-image represents the values of
the
for the σ value axis , and the fourth
M-image is the n4 component and provides the necessary information to determine
the position of the normal on the function area. Moreover, the function
is replaced by a local function of the form
|
where
,
l=
1…4, P=256 – halftone palette.
|
(2)
|
|
|
|
|
|
|
|
|
|
|
|
Figure 5:
M-images depicting local geometric characteristics
of the unit for a single normal stress
|
Calculating
σ
using the formula (2) allows us to calculate the palette for the
image of the unit stress values:
|
|
(3)
|
Figure 6a shows an
image of normal values for a unit pressure distributed over a tone gradation.
|
|
|
|
|
a
|
b
|
c
|
|
Figure 6:
Visualization of the normal stress during
localization of the power load: a) Image of the unit stress by the FVM
method, b) Image of the pressure when the load localization by the FEM
method, c) distribution of the unit vector over the local area of the force
application by the PCM method.
|
Figure 6b is
presented for comparison. It was obtained on the FEM grid when trying to set
the minimum area of force application. It is clear that it is not possible to
bring such a localization to a single point as it is done in the FVM, however,
it will not be difficult to solve the inverse problem of reducing the unit
stress at the point (Fig.6a) obtained by the FVM to a distribution over a
certain area of the minimum areas’section values of unit stresses, taking
into account their spatial location. Figure 6c shows the distribution of unit
stresses for the local loading section. It should be noted that the results are
similar to Figure 6b, but the image (6c) obtained by the sum of unit stresses
is more attractive, since it looks continuously smooth in accordance with the
physical law itself and its image is stable to changes in the space
discretization step and does not depend on the shape mesh element simulated
for FEM.
Since the tangent stress is orthogonal to the normal one, the module
of the volume vector model
takes
the form
|
|
(4)
|
At the same time, M-images of the created FV-model display
the corresponding local geometric characteristics on the considered function
area (Fig. 7).
|
|
|
|
|
|
|
|
|
|
|
Figure 7:
M-images displaying local geometric characteristics
for a unit tangent stress
|
In Figure 8, by
analogy with the sixth figure, the results of modeling images of tangent stresses
applied to the minimum area inside the body are shown sequentially. Figure 8a
is derived from M-images and a local function
|
,
l=
1…4, P=256 – halftone palette
|
(5)
|
The image in
Figure 8b depicts the result of displaying the image of tangent stresses for a
local load, modeled by the FEM method. The red region is the region of maximum
positive values of the function
, and the blue region is negative - the minimum.
A similar simulation using the FVM allows you to get an image similar in
content, but only with a more pronounced smooth shape of the formed stress
surface.
|
|
|
|
a
|
b
|
|
Figure 8:
Visualization of the tangential stress during
localization of the power load: a) Image of a single voltage using the FVM
method, b) Image of the stress during localization of the load using the FEM
method
|
To build an image
of a unit of temperature stress, you can use the rules for constructing a
volume vector, the principles of which are described in the first paragraph. In
this case, the direction’s absence of the applied temperature should be taken
into account, which means that the function
, and the value of the temperature applied
to the surface of the body allows us to use the function of the value
to model the stress distribution in the
body. An analytically volume vector for a unit of temperature stress can be
written:
where
is the temperature
applied at a point on the boundary of the body.
Functionalvoxel
model is determined by similarity with the voltage arising from the force
effects and leads to a similar local function
where
,
l=
1…4, P=256 – halftone palette.
Figure 9 presents
M-images characterizing the local geometric characteristics in the considered area.
|
|
|
|
|
|
|
|
|
|
|
Figure 9:
M-images showing local geometric characteristics for
a single temperature stress
|
Figure 10a shows
the result of modeling images of temperature stress applied at a surface point
on the surface of an isotropic body by the proposed approach. For visual comparison
the obtained model calculation results by analogy with the previous cases, the
results of the calculating the same problem using the FEM method are presented
(Fig. 10b). The distribution of temperature load is carried out by ordinary
summation of point loads uniformly distributed along the selected direction. Figures
10b, d show the result of heat load distribution along the
axis for both approaches,
respectively. The results show the shape of the resulting temperature
distribution is identical in both cases.
|
|
|
a
|
b
|
|
|
|
c
|
d
|
|
Figure 10:
Examples of visualization of thermal stress
simulation results: a) Calculation of thermal stress for the temperature
applied at a point by the FEM method, b) FEM method for the same task, c)
Calculation of thermal stress for the distributed temperature by the FEM
method, d) the FEM method for that same task.
|
The thermal
expansion of the material is an important parameter for any mechanical process.
Its accounting is necessary in various technologies related to the accuracy of
material processing, etc. Consider the problem of modeling the shape of thermal
expansion for the possibility of visual diagnostics. Based on the obtained
functional-voxel model, which allows one to obtain
values for each point of the
considered area without any difficulties, the shape of the additional relative
volume is simulated with the expected expansion of the material depending on
temperature values:
,
where
– coefficient of thermal expansion.
Figure 11a, b
gives an example of the result of modeling the shape of thermal expansion by
the proposed method and the FE method on a triangulated grid for visual
comparison. In both cases, the result was obtained in a comparable time period
of calculation.There is an obvious difference in the visibility of presenting
the same result for the case of using the voxel (Fig. 11a) and polygonal (Fig.
11b) models.
|
|
|
|
a
|
b
|
|
Figure 11:
Visualization’s examples of the thermal expansion
results of modeling : a) calculation of the cross section shape of the
thermal expansion of the material for the temperature, applied at the point
by the FWM method, b) calculation of the cross section shape of the thermal
expansion of the material for the temperature, localized small neighborhood
by the FEM method.
|
Figure 12 shows an
example of spatial modeling of the form of thermal expansion, constructed by
the functional-voxel method as one of the means of visual diagnostics of the
thermal process.
|
|
|
|
a
|
b
|
|
Figure 12:
Examples of constructing the form of thermal
expansion: a) For uniformly distributed thermal loading, b) For point thermal
loading.
|
In this paper, we
considered the means of local geometry in computer representation for modeling
physical quantities. At the present stage, geometric modeling accompanies
engineering tasks, limited to the procedure for constructing a finite element
mesh and its adjustment. Further, methods related to mathematical physics are
based on the construction of complex differential equations enter into the
calculation. At the same time, the local (differential) geometry, which is
intended for the subsequent stage of modeling physical quantities on the minimal
sections of the object, has not yet received its computer representation. The
main advantage of the proposed approach is the absence in the differential
equations’ calculations of the grid decomposition used in the FEM. The discretization
of the function space proposed in the FVM basically contains the derivation of
differential characteristics that make it possible to model complex physical
processes with simple algebraic expressions of laws. Additional advantages of
the proposed approach can also be considered the possibility of significantly
simplifying the calculation at the investigated point in space, as well as the
possibility of parallelizing the computational process with the existing
simplest means. The prospect of further research is to develop tools modeling
more complex problem statements for engineering calculations based on the
developed approach.
[1]
Device
for measuring mechanical stresses in metal products // RF Patent No. 2079825.
1997.
Zhukov S. V., Zhukov V. S.
[2]
Method
for determining mechanical stresses and a device for its implementation // RF
Patent No. 2195636.
2001.
Zhukov S. V., Zhukov V. S., Kapitsa N. N.
[3]
A.B.
Kaplun, E.M. Morozov, M.A. Olfepyeva
ANSYS in the hands of an engineer: Practical guide/
Ì
.
Book house «LIBROKOM», 2015. – 272 p.
[4]
Tolok A.V.
Functional-voxel
method in computer modeling. – Moscow.:Phismatlit, 2016. 105 p.
[5]
Grigoryev
S.N., Tolok N.B., Tolok A.V.
Local Search Gradient
Algorithm Based on Functional Voxel Modeling // Programming and Computer
Software. 2017. Vol. 43, No. 5.
Ñ
.
300–306.
[6]
Maksimenko-Sheyko K. V., Sheyko T. I.,
Tolok A. V.
R-function in the analytical design
using a system of "RANOK"] / / Vestnik MSTU "Stankin".
Scientific
peer-reviewed journal. M.: MSTU "Stankin". ¹4(12). 2010. P. 139-151
[7]
Tolok, A.V., Silantyev D.
A., Lotarevich E. A., Pushkarev S. A. Voxel-mathematical modelling at the
solution of problems of determining the area of surfaces // Information
technologies in designing and manufacturing No. 3, 2013, 29-33.
[8]
Tolok A.V., Tolok
N.B.
Mathematical
Programming Problems Solving by Functional Voxel Method // Automation and
Remote Control. 2018. Volume 79, ¹ 9. pp 1703–1712.
[9]
Tolok N.B., Tolok, A.V.,
Functional voxel method, computing / Scientific visualization. M.:
MEPhI. 2017. Vol. 2, vol. 9. P. 1-12
http://sv-journal.org/2017-2/01.php?lang=ru
[10]
Starovoitov
E. I.
Strength of materials. — M.: Phismatlit,
2008. — P. 384. — 1000 copies. — ISBN 978-5-9221-0883-6
