=Paper=
{{Paper
|id=Vol-2744/paper13
|storemode=property
|title=Visualization of the Stress-Strain State of Shell Structures Using Virtual and Augmented Reality Technologies
|pdfUrl=https://ceur-ws.org/Vol-2744/paper13.pdf
|volume=Vol-2744
|authors=Alexey Semenov,Iurii Zgoda
}}
==Visualization of the Stress-Strain State of Shell Structures Using Virtual and Augmented Reality Technologies ==
https://ceur-ws.org/Vol-2744/paper13.pdf
Visualization of the Stress-Strain State of Shell Structures
Using Virtual and Augmented Reality Technologies *
Alexey Semenov1[0000-0001-9490-7364],
Iurii Zgoda 2[0000-0001-6714-500X]
1 Saint Petersburg State University of Architecture and Civil Engineering, Saint-Petersburg,
190005, Russia.
sw.semenov@gmail.com
2 Saint Petersburg State University of Architecture and Civil Engineering, Saint-Petersburg,
190005, Russia.
yurii.zgoda@mail.ru
Abstract. The paper describes a mathematical model of changes in the geometry
of thin-shell structures for visualization of the analysis data on their stress-strain
state (SSS). Based on this mathematical model, a visualization module for shell
SSS visualization using VR and AR technologies was developed. The interactive
visualization environment Unity 2019.3 and C# programming language were
used. The interactive visualization module makes a 3D image of a shell structure
and visualizes the SSS either through heat maps over the shell or through the
changes in the shell geometry on the basis of the shell type, its geometric charac-
teristics, and SSS analysis data (transferred to the visualization module by means
of a JSON file). While working on the visualization module, the authors devel-
oped a system of components that makes it possible to visualize any 3D surface
with coordinate axes (including numbers with a pitch determined automatically),
visualize heat maps with a graduated scale, visualize a mesh over the graph to
improve the perception of the surface deformations. The middle surface can also
be deformed on the basis of SSS analysis data. This solution increases the effi-
ciency of the work of specialists in civil engineering and architecture and can be
used when training specialists in courses on thin-shell structures and procedural
geometry.
Keywords: Shells, Stress-Strain State, Virtual Reality, Augmented Realty,
Unity.
1 Introduction
Thin-shell structures actively studied due to their unique properties are highly strong
and stable while using the minimum amount of materials [1]. Thin-shell structures are
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
* The research was supported by RSF project № 18-19-00474.
2 A. Semenov, I. Zgoda
often used in shipbuilding, aircraft and spacecraft construction, mechanical and civil
engineering. Great attention is paid to studies on plates and shells since such structures
are highly strong and stiff and have a variety of design shapes [2].
However, the issues of shell visualization have not been solved in full. The majority
of studies on shells cover only the stress-strain state (hereinafter — the SSS) of a shell
structure relative to the middle surface (Fig. 1), while deformations in the global coor-
dinates are not considered. While such software packages as ANSYS and LIRA-SAPR
enable graphic visualization of deformations, there is no standard technique or algo-
rithm of visualization of deformed shells for variational analysis methods. In the mean-
time, the use of variational methods such as the Ritz method can significantly improve
the accuracy of the analysis and reduce its time [3].
Informative visualization of the SSS of shells is important for their detailed study.
In many cases, a researcher will examine the analysis data more efficiently if they are
presented as a 3D animation rather than static contoured maps or 2D graphs.
Fig. 1. Visualization of vertical displacements relative to the middle surface
VR and AR technologies are particularly important in this context [4]. Virtual reality
gives a 3D representation of structural deformations. Augmented reality makes it pos-
sible to see the structure in the real world, which is also useful for studies on shell
structures.
The purpose of this study is to develop a software package (hereinafter — the SP)
for the analysis of the SSS and visualization of shell structures using VR and AR tech-
nologies
2 SP Architecture
The software package consists of two modules: the SSS analysis module and the visu-
alization module. The SP architecture is shown in Fig. 2.
Visualization of the Stress-Strain State of Shell Structures Using Virtual and Augmented… 3
Software package
Analysis module Visualization module
SSS analysis Analysis results export Analysis results import SSS visualization
Fig. 2. Software package for shell analysis
The visualization module deserializes the file with the SSS analysis data, performs pro-
cedural generation of the geometry of the shell structure, and, then, visualizes the SSS
through either heat maps presented over the shell or through changing the geometry of
the shell structure.
The main feature of the described solution is the ability to render shell SSS using
virtual and augmented reality technologies. At the moment, there are no solutions that
allow visualizing the calculation results stored in standard formats (such as VTK) on
mobile augmented or virtual reality platforms. For this reason, existing renderers (such
as ParaView) couldn’t be used and a custom rendering solution was developed. Visu-
alization module imports the analysis results in custom data format used for SSS data
storage in the analysis module because using other data formats doesn’t provide bene-
fits within the investigated problem.
3 SSS of a Shell
Analysis of the SSS of a shell structure means minimization of the functional of the
total potential strain energy (which is a sum of the work of internal and external forces)
on the basis of the Timoshenko (Reissner–Mindlin) model. The Ritz method is used for
numerical search for the functional minimum. It reduces the variational problem to a
problem of unconstrained optimization of the function of several variables. For this
U (x, y ), V (x, y ), W (x, y )
purpose, the required displacements functions as well as func-
x (x, y ), y (x, y )
tions of normal segment turning angles to the middle surface are
replaced with the following approximations:
N N N N
U = U (x, y ) = U kl X1kY1l , V = V (x, y ) = Vkl X 2k Y2l ,
k =1 l =1 k =1 l =1
N N
W = W (x, y ) = Wkl X 3k Y3l ,
k =1 l =1 (1)
N N N N
x = x (x, y ) = PSkl X 4k Y4l , y = y (x, y ) = PN kl X 5k Y5l ,
k =1 l =1 k =1 l =1
4 A. Semenov, I. Zgoda
where approximation functions X 1 − X 5 , Y1 − Y5 are known and predetermined by the
k k l l
U − PN kl
conditions of shell fixing, and parameters kl are unknown numeric coeffi-
cients; N – is the quantity of expansion terms.
Thus, the functional s
(
E = Es U , V , W , x , y )
is approximated by the function of sev-
eral variables, and it is sufficient to use the approximation functions and the numeric
coefficients’ values ensuring the minimum of the functional to recover the SSS analysis
data.
4 Geometry of a Shell
When using variational principles for making a mathematical model, the geometry of a
shell structure is found through Lame parameters and principal curvatures. However, it
does not seem too comfortable to make a curvilinear coordinate system to generate the
geometry of a shell structure on the basis of these parameters only. Papers dealing with
shell visualization [5] suggest using a parametric notation for shell structures instead.
Many shells can be described in the parametric form, which relates each of the points
of the middle surface in a 2D space to a point in a 3D coordinate system. Therefore, the
question of how deformations are applied to the shell middle surface should be solved.
In this paper, the local basis in each point of the middle surface is used to solve this
problem, which makes it possible to use displacements in the global coordinates instead
of displacements U (x, y ), V (x, y ), W (x, y ) relative to the middle surface. To build the
geometry of a shell with a certain thickness h, ratios for displacements in an arbitrary
layer of the shell that follow from the Timoshenko model are used on the basis of the
analysis for the middle surface.
Let us describe a parametric shell in a generalized form below. Each point of such a
shell is determined through the following ratios:
X = X (x, y ),
Y = Y (x, y ), (2)
Z = Z (x, y ).
To apply deformations, i.e. displacement of the points in a horizontal, vertical and nor-
X Y Z
NU = , ,
mal directions, to such a geometry, we need to find vectors x x x ,
X Y Z
NV = , ,
y y y for each point of the shell, after which they should be normal-
ized. These vectors determine the horizontal and vertical directions of displacement of
the points, respectively. The vector product of these normalized vectors is the normal
N ( x, y )
to the surface of the shell in the point W . Analytical expressions of the basis of
these vectors are derived for various types of shell structures in this paper.
Visualization of the Stress-Strain State of Shell Structures Using Virtual and Augmented… 5
4.1 Doubly Curved Shallow Shell
The input parameters of a doubly curved shallow shell are linear dimensions a, b,
R1 , R2
and radii of circular arcs . Let us introduce additional parameters
R = max R1 , R2 − min R1 , R2 r = min R1 , R2
and . In this case, the parametric form
for this shell will be as follows:
min a, b min a, b
X = (R + r cos x )sin y, x − , ,
2r 2r
Y = (R + r cos x )cos y,
Z = r sin x. y − max a, b , max a, b,
2(R + r ) 2(R + r )
(3)
where x is the turning angle of a small radius; y is the turning angle of a large radius.
The expression for the basis in each point of the shell is as follows:
NU = (− sin x sin y, − sin x cos y, cos x ),
NV = (cos y, − sin y, 0),
N = (cos x sin y, cos x cos y, sin x ).
W (4)
The middle surface of a shallow shell is given in Fig.3.
Fig. 3. Middle surface of a doubly curved shallow shell
4.2 Spherical Shell
The input parameters of a spherical shell are linear parameters a, a1, b and radius R.
The parametric form of a spherical shell is as follows:
6 A. Semenov, I. Zgoda
X = R sin x sin y, x a1 , a ,
Y = R cos x, b b
Z = − R sin x cos y. y − 2 , 2 ,
(5)
where x and y coincide with the latitude and longitude, respectively. The basis in the
point of the middle surface is determined as follows:
NU = (cos x sin y, − sin x, − cos x cos y ),
NV = (cos y, 0, sin y ),
N = (sin x sin y, cos x, − sin x cos y ).
W (6)
The appearance of the middle surface of a spherical shell is given in Fig. 4.
Fig. 4. Middle surface of a spherical shell
4.3 Toroid-Shape Shell
The model of a toroid-shape shell coincides with the model of a spherical shell but
includes displacement d1 from the vertical axis of rotation of the sphere. The parametric
form of a toroid-shape shell is as follows:
X = R sin x sin y + d1 sin y, x a1 , a ,
Y = R cos x, b b (7)
Z = − R sin x cos y − d cos y. y − 2 , 2 .
1
The basis in the point of the middle surface is determined as follows:
NU = (cos x sin y, − sin x, − cos x cos y ),
NV = (cos y, 0, sin y ), (8)
N = (sin x sin y, cos x, − sin x cos y ).
W
The appearance of the middle surface of a toroid-shape shell is given in Fig.5.
Visualization of the Stress-Strain State of Shell Structures Using Virtual and Augmented… 7
Fig. 5. Middle surface of a toroid-shape shell
4.4 Cylindrical Shell
The input parameters of a cylindrical shell are linear dimensions a, a1, b and radius R.
The parametric form of a cylindrical shell is as follows:
a − a1
X = x − 2 , x a1 , a ,
Y = R cos y − R, b b (9)
Z = − R sin y. y − 2 , 2 .
Curvilinear coordinate x is along the element, while curvilinear coordinate y is along
the circle made by the cross-section of the cylinder with a plane parallel to its base.
The basis in an arbitrary point of the middle surface is as follows:
NU = (1, 0, 0),
NV = (0, − sin y, − cos x ), (10)
N = (0, cos y, − sin y ).
W
The appearance of the middle surface of a cylindrical shell is given in Fig.6.
Fig. 6. Middle surface of a cylindrical shell
8 A. Semenov, I. Zgoda
The input parameters of a catenoid shell are linear dimensions a, a1, b and parameter
c. The parametric form of a catenoid shell is as follows:
a − a1
X = x − ,
x a1 , a ,
2
a − a1
Y = c cosh x − cos y − c, b b (11)
2 y − 2 , 2 .
a − a1
Z = −c cosh x − sin y.
2
The basis for such a shell takes the following form:
a − a1 a − a1
1, c sinh − x cos y, c sinh − x sin y
N = 2 2 ,
U a −a
c 2 sinh 2 x − 1 +1
2
NV = (0, − sin y, − cos y ), (12)
a − a1
c sinh − x , cos y, − sin y
2 .
NW =
a − a
c 2 sinh 2 1 − x +1
2
The appearance of the middle surface of a catenoid shell is given in Fig.7.
Fig. 7. Middle surface of a catenoid shell
Visualization of the Stress-Strain State of Shell Structures Using Virtual and Augmented… 9
4.5 Conical Shell
The input parameters of a conical shell are linear dimensions a, a1, b. The parametric
form of a conical shell is as follows:
a1 + a
X = x cos − 2 , x a1 , a ,
Y = x sin cos y, b b (13)
Z = − x sin sin y. y − 2 , 2 .
Curvilinear coordinate x is along the element, while curvilinear coordinate y is along
the circle made by the cross-section of the cone with a plane parallel to its base.
The basis for a conical shell is as follows:
NU = (cos , sin cos y, − sin sin y ),
NV = (0, − sin y, − cos y ), (14)
N = (− sin(), cos y cos , − cos sin y ).
W
The appearance of the middle surface of a conical shell is given in Fig.8.
Fig. 8. Middle surface of a conical shell
5 Shell Visualization
To implement the shell visualization module that uses VR and AR technologies, the
interactive visualization environment Unity 2019.3 and C# programming language
were used. The interactive visualization module makes a 3D image of a shell structure
and visualizes the SSS either through heat maps over the shell or through the changes
10 A. Semenov, I. Zgoda
in the shell geometry on the basis of the shell type, its geometric characteristics, and
SSS analysis data (transferred to the visualization module by means of a JSON file).
There is an option of using the proposed module without SSS visualization. This
visualization mode can be useful to architects when they examine various forms of
structures and to students studying shell structures. This allows for a better understand-
ing of a relationship between the parameters of a shell structure and its final appearance.
While working on the visualization module, the authors developed a system of com-
ponents that makes it possible to visualize any 3D surface with coordinate axes (includ-
ing numbers with a pitch determined automatically), visualize heat maps with a gradu-
ated scale, visualize a mesh over the graph to improve the perception of the surface
deformations. The middle surface can also be deformed on the basis of SSS analysis
data.
6 Results and Discussion
The SP for visualization of shell structures was tested for shells of various shapes and
using various visualization options. Fig.9 provides a demonstration of the SSS of a shell
structure using heat maps and geometry changes.
Fig. 9. Deformation of a shallow shell when exposed to a load
As follows from Fig. 9, visualization has great information capacity and describes data
on the shell deformation both visually and numerically. The suggested visualization
module helps to study the shell SSS in a real-world scale and with the initial proportions
preserved. The clarity of such visualization in comparison to traditional visualization
of the deflection relative to the middle surface (Fig. 1) is obvious.
As stated above, the suggested visualization module not only makes it possible to
visualize the shell SSS but also can be used by students studying procedural modeling
in architecture allowing them to clearly depict shell structures depending on their geo-
metric parameters. Fig. 10 presents a shallow shell with lesser curvature radii and larger
linear dimensions than those in Fig. 9. As you can see, a change in geometric parame-
ters of shells of the same type can lead to significant changes in the final shape.
Visualization of the Stress-Strain State of Shell Structures Using Virtual and Augmented… 11
Fig. 10. Visualization of a shell with a small radius of rotation
The visualization module can be implemented using VR and AR technologies. AR vis-
ualization is demonstrated in Fig. 11.
Fig. 11. AR visualization of a shell
7 Conclusion
Thus, the visualization module for the SSS of shell structures, enabling VR and AR
visualization, was developed.
The solution can be used as a tool for informative and clear visualization of the shell
SSS or when training students majoring in architecture and civil engineering in courses
on thin-shell structures.
12 A. Semenov, I. Zgoda
8 Acknowledgments
The research was supported by RSF (project No. 18-19-00474).
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