This paper is an extended version of a contribution presented
at the Graphiñon 2025 conference.
In
today's world, where digitalization and intellectualization are becoming key
factors of competitiveness in manufacturing industries, digital twin technology
(Digital Twins are becoming increasingly important. The use of digital twins is
particularly relevant during the development of complex and expensive technical
systems, allowing for design optimization, risk reduction, and faster
time-to-market for innovative products.
This
article explores the use of digital twin technology to optimize the design of a
wind turbine blade assembly currently under development. It presents the
experience of a team from the Keldysh Institute of Applied Mathematics of the
Russian Academy of Sciences in creating a digital twin designed for
high-fidelity virtual testing and modeling.
This
study, presented in a series of papers [1-7], makes a valuable contribution to
the fields of mathematical modeling, numerical methods, and high-performance
computing, enabling the creation of digital twins that accurately reflect the
physical processes occurring in real systems. The developed technology enables
parametric optimization of blade assembly design, taking into account various
design constraints and maximizing wind turbine efficiency.
The
aim of this paper is to demonstrate the capabilities of a digital twin, created
by a team from the Keldysh Institute of Applied Mathematics (RAS), for
optimizing the design of a wind turbine blade assembly during the development
phase. The paper describes the workflow and reveals how the application of
modern mathematical modeling and visualization methods enables the
identification of optimal geometric parameters that ensure maximum wind turbine
efficiency and mitigate the risks associated with implementing new technology.
Using the Navier-Stokes equations, the open object-oriented OpenFOAM library,
and modern computing resources, such as the K-100 hybrid computing cluster at
the Keldysh Institute of Applied Mathematics (RAS), it is possible to create an
accurate model simulating airflow around the blade assembly. Particular
attention is paid to the visualization of parametric analysis results, which
allows for the determination of the optimal blade assembly geometry, taking
into account mass and moment of inertia constraints.
The
article will present the results of numerical experiments demonstrating the
effectiveness of the developed approach and the prospects for its application
to the design of other complex technical systems.
A
digital twin is a virtual copy of a physical object or system that is
constantly updated with data obtained from the real prototype. This allows not
only to monitor the current state of the object but also to predict its
behavior under various conditions and optimize its performance.
It's
important to distinguish between a digital twin and a digital shadow. A digital
shadow can predict the behavior of a real object only under the conditions
under which the data was collected, but cannot simulate situations in which the
real object was not used. A digital twin, on the other hand, allows for the
simulation of a wide range of scenarios and virtual testing, which is
especially important in developing new products and optimizing existing ones.
In
the context of developing complex and expensive installations, such as wind
turbines, creating a digital twin becomes a powerful tool for optimizing the
design process and reducing the risks associated with implementing new
technology. During the development phase, when a physical prototype does not
yet exist or is in the early stages of creation, a digital twin enables virtual
testing and simulation of various design solutions, identifying potential
problems, and optimizing system parameters before costly physical production
begins.
Wind
turbines, even at the design stage, are complex systems that require
consideration of numerous factors, including aerodynamics, material strength,
load dynamics, and energy conversion efficiency. Traditional design methods,
based on sequentially completing individual stages and physically testing
prototypes, can be time-consuming, expensive, and do not always allow for the
early identification of all potential problems.
A
digital twin enables an iterative design process, in which virtual testing and
simulation are conducted at each stage of development, enabling rapid
evaluation of various design solutions and identification of optimal system
parameters. Through the continuous exchange of data between the digital model
and the results of physical experiments (if any), the digital twin becomes an
increasingly accurate and reliable tool for predicting the behavior of the
actual system. This reduces the risks associated with implementing new
technology, reduces development time and costs, and improves the efficiency and
reliability of the final product.
The
economic feasibility of creating a digital twin during the development of wind
turbines is driven by several factors. First, it reduces the costs of physical
testing and prototyping through virtual experiments. Second, it reduces
development time by accelerating decision-making and identifying potential issues
early. Third, it improves the efficiency and reliability of the final product
by optimizing system parameters based on virtual testing results. Fourth, it
reduces the risks associated with implementing new technology through virtual
testing and modeling various scenarios.
The
concept of digital twins is actively developing in the Russian economy, as
evidenced by works [8-11]. A separate paradigm for their application on a
global economic scale has been developed – Smart Digital Twin [( Simulation
& Optimization )- Based Smart Big Data Driven Advanced ( Design
& Manufacturing ), the driver of which is a “smart” digital twin, formed as
a result of multidisciplinary multi-scale numerical modeling and the
application of many optimization technologies [8].
In
the context of complex and expensive installations such as wind turbines,
creating a digital twin is becoming not just desirable but practically
essential to ensure their efficient and safe operation. Wind turbines operate
under highly uncertain conditions, exposed to variable wind loads, temperature
fluctuations, corrosion, and other factors that can lead to performance
degradation and accidents. Traditional monitoring and diagnostic methods based
on periodic inspections and historical data analysis are often insufficient for
the timely detection and prevention of potential problems.
Creating
a digital twin during the development of wind turbines is a strategically
important step, allowing for the optimization of the design process, mitigation
of risks, increased efficiency and reliability of the final product, and
reduction of development time and costs. This technology opens new horizons for
the development of wind energy and contributes to the creation of more
efficient and reliable energy systems.
To
effectively create a digital twin of a wind turbine blade assembly, it is
necessary to combine the efforts of specialists from various fields: design
engineers, technologists, materials scientists, calculators, and others. This
leads to the emergence of a new type of engineer – a systems engineer –
possessing competencies in various fields of knowledge and capable of
effectively interacting with a variety of modeling and analysis tools.
The process
of creating a digital twin involves several stages, each of which plays a vital
role in ensuring the adequacy and accuracy of the virtual model. The key stages
are listed below:
1. Construction of a CAD model.
2. Construction of the computational grid.
3., the problem of flow simulation around the blade assembly is
solved based on the Navier-Stokes equations using specialized software such as
OpenFOAM [12-16].
4. Visualization of results.
The
model developed for simulating the operation and variation of blade assembly
shape is a unified process chain of algorithms. This chain includes the
construction of a CAD model to describe the complex blade assembly geometry,
the generation of a computational mesh based on the resulting geometry, the
solution of a flow simulation problem based on the full system of Navier-Stokes
equations, and the visualization and animation of the results in mono- and
stereo modes. Based on this unified process chain of algorithms, technologies
for varying blade assembly shape based on key parameters have been developed to
determine the optimal blade assembly shape in terms of performance
characteristics.
The
objective of this study is to find the optimal shape for a power plant blade
assembly in terms of power load. It is necessary to find a shape that provides
maximum torque while varying three key geometric parameters of the assembly—two
blade pitch angles and blade width. The potential increase in the assembly's
mass and moment of inertia should not significantly exceed those of the base
assembly shape.
The
calculations were performed using the open-source object-oriented OpenFOAM
library, written in C ++, which supports massive parallelization mechanisms and
is intended for numerical modeling of continuum mechanics problems [12-14]. The
library is based on finite-volume approximations written in operator form.
OpenFOAM elements are actively used in industry, academia, and the expert
community, in particular, in the numerical analysis of the energy
characteristics of horizontal-type plants [14-16]. All calculations for solving
the problems of mathematical modeling of the flow around a power plant were
performed on the K-100 hybrid computing cluster at the Keldysh Institute of
Problems of Materials Science, Russian Academy of Sciences [17] in parallel
computing mode. The complete system of Navier-Stokes equations, describing the
motion of a viscous, heat-conducting, compressible gas, was used as a
mathematical model [18]. To analyze the results, a method for animated visual
representation of the operation of the blade assembly in mono- and stereo-modes
[3] was developed, based on modern methods and concepts of visual display of
the results of numerical calculations.
As a
result of previous work, a computer technology was developed that enables
mathematical modeling of a power unit's blade assembly with specific geometric
parameters and determines the force load acting on the unit in the airflow.
This created the basis for further research to determine the optimal blade
assembly shape for wind load, taking into account technological constraints.
Building
a robust and effective numerical technology involves several necessary steps,
including inputting geometric data, processing it, constructing a mesh model,
selecting specialized flow calculation algorithms, analyzing current results,
calculating functionals (forces and moments), visually representing the
results, and processing and analyzing the results of multivariate calculations.
The foundation of this technology is the process of mathematical modeling of a
wind turbine blade assembly with specific geometric parameters under a given
wind load. The mathematical modeling process itself involves creating an
experimental computational system for calculating the flow around the complete
wind turbine assembly based on solving the Navier-Stokes equations and
calculating the corresponding integral flow characteristics.
The
next important stage is the creation of a methodology for conducting
optimization calculations to select the optimal shape of a wind turbine based
on the optimization of the values of the selected integral characteristics.
The
mesh generation process begins with the description of the surface bounding the
three-dimensional body being considered. Typically, the surface of a real three-dimensional
object with a complex shape is fully or partially imported from a CAD package
and can be edited if necessary. Surface meshes have fairly "obvious"
requirements—no self-intersections, closedness, and a few others. When working
with real industrial objects, the resulting CAD surfaces rarely meet these
criteria, and additional surface preparation is required to obtain a volumetric
computational mesh of the required quality.
To
achieve this goal, a number of automatic tools with a wide range of quality
control tools are used, such as tools for creating and editing surface meshes
as " surface wrapping " and " remeshing ", which allows to
reduce the time spent by orders of magnitude and almost completely eliminate
the need for manual mesh preparation. " Surface Wrapper corrects CAD
geometry defects (closes holes, eliminates self-intersections, etc.), resulting
in a closed surface with the desired level of detail. To create a high-quality
initial triangular surface mesh, use the " surface " tool. remesher
", which allows, using a fairly wide range of settings, to obtain a
surface with the required parameters (the degree of smoothness in areas with
high curvature, the degree of resolution of thin areas, the rate of growth of
the characteristic size of surface cells when moving away from areas with high
detail, the preservation of topological features, local mesh refinement, etc.).
Fig.
1 already presents an “error-free mesh” – a triangular surface mesh with the
necessary parameters (degree of smoothness in areas of high curvature, level of
resolution of fine points, rate of growth of the characteristic size of surface
cells with distance from areas of high detail, preservation of topological
features, local mesh refinement, etc.).
Fig. 1. Corrected
surface triangular mesh constructed for the original CAD surface
The
resulting surface mesh forms the basis for constructing a volumetric
computational mesh. The specifics of numerical simulation of liquid and gas
flows dictate certain rules for constructing volumetric meshes. When solving
such problems, the volumetric mesh typically consists of two main parts: a
prismatic mesh near the flowed surfaces and an arbitrary polyhedral mesh at a
sufficient distance from the surfaces.
Currently,
flows of a continuous gas medium (i.e., a gas medium under the assumption that
it can be considered without taking into account individual particles) are
calculated on the basis of the Navier-Stokes equations. This system of
equations is as follows, see, for example, [18].
Here
is the average
flow velocity vector with components
- molecular and
turbulent (obtained by averaging various functionals from small-scale
pulsations) components of the viscous stress tensor;
- specific total
energy of the gas,
- specific
internal energy of the gas
- total enthalpy;
- molecular and
turbulent components of the heat flux density vectors.
The
type of the remaining turbulent components is no longer universal; their
selection involves so -called turbulence models. Turbulence models must be
selected taking into account the properties of real physical flows within the
selected parameter range. Note that the influence of turbulence on the physical
characteristics of the entire process was not considered in the calculations
performed.
To
determine the optimal wind turbine geometry for load bearing, a series of blade
assembly models were constructed with varying blade geometry. We selected three
key parameters, varying which allows us to describe a wide variety of possible
geometric shapes.
Fig.
2 schematically shows these variable parameters:
is the angle
between the direction of the main blade and the vertical,
is the angular
size of the main blade in the direction of the axis of rotation,
=L is the width of
the main blade. Note that throughout the following we will present the results
for the three parameters
(
,
,
L), where
.
Fig. 2. Variable
geometric parameters
(
,
,
L) for determining
the optimal shape of the blade assembly geometry.
To
determine the main parameters and dimensions of the product (spacing between
supports, average blade cross-sectional shape, etc.), the wind turbine geometry
was used, obtained by scanning a prototype test sample. Surface construction
and modification were performed using the SolidWorks CAD package. All
constructed models were exported in SLDPRT format.
The values
(,
,
L)
change in a certain range, with the base value being
,
,
the result of
laser scanning of the prototype was selected:
= 55 degrees,
=120 degrees,
~ 20 cm.
takes on values of 45, 50, 55, 60 and 65 degrees,
— values of 100, 110, 120 and 130 degrees,
L
is selected from the set of values
(
),
(
),
1.2
(
).
For
clarity, let us supplement the schematic picture presented in Fig. 2 with
three-dimensional images, see Figs. 3-4. The drawings represent the change in
the shape of the blade assembly with variations in the geometric parameters
described above.
Fig. 3. Changing the angular parameter
:
45 (green), 50 (gray), 55 (basic version, red), 60 (gray) and 65 (blue) degrees.
,
L
.
Fig. 4. Changing the angular parameter
:
100, 110, 120 (basic version, red) and 130 degrees. In this case, the parameters
,
L
.
The
formal general statement of the optimization problem is as follows: find among
the elements
that form the set
such an element
on which the
given function
reaches a
minimum (or maximum) value, i.e.
(or
,
respectively).
Therefore, in order to formulate an optimization problem, it is necessary to
specify: the feasible set
,
the objective function
, and the search
criterion – what we are looking for – ( max or min ).
Solving
such a problem means either finding the desired extremum
or showing that
no solution exists. If, when specifying a feasible set of
constraints,
are absent, then
we are dealing with an unconstrained optimization problem. If the conditions
exist, then
such a problem is called a constrained optimization problem.
Assuming
a number of design constraints inherent to the problem under consideration, we
formally deal with a conditional multiparameter optimization problem, which is
typical of design optimization problems in general. Our case is a
multiparameter optimization problem, since variations of three key geometric
parameters are considered. The variations of these parameters are limited by
ranges and, therefore, impose constraints on the varied parameters. In general,
the exploratory parametric search problem can be formulated as follows: find
the values of the key geometric parameters of the blade assembly
that ensure the
maximum value of the objective function M* = maxM
(
).
The main
aerodynamic characteristics were chosen as the objective function: the total
aerodynamic force
and torque
.
To
solve the optimization problem, we propose using a grid method, which is
appropriate for an initial assessment, as multidimensional problems are
significantly more complex and time-consuming than one-dimensional ones. The
essence of the proposed method for finding the smallest value is to determine
the values of the objective function at a discrete set of nodes that do not
exceed the feasible set
.
In other words,
the ranges of variation for each key geometric parameter are partitioned at a
specific step.
Thus,
the spatial region defined by the parameter ranges is covered by a grid. The
objective function is calculated at each grid node. The largest of the set of
objective function values on a given grid is taken as the maximum. Previously,
this method was traditionally considered practically unsuitable for problems of
higher dimension due to the long calculation time required. However, the
development of parallel computing allows for calculations to be accelerated by
orders of magnitude. This makes the most unpretentious and simple methods truly
applicable to practical problems. Moreover, their simplicity and reliability
give them significant advantages in this context.
The
conducted numerical simulations allowed for exploratory research and a rough
optimization estimate of the optimal set of key geometric parameters for maximizing
blade assembly torque. To achieve this, each key parameter
was sequentially
varied, while the two remaining parameters were held constant.
An
analysis of the calculated values for the blade assembly volume and its moment
of inertia for various assembly geometry variants reveals that variations in
angular parameters and blade width have different effects on the geometric
characteristics—the assembly volume and moment of inertia. Specifically, width
variations lead to a more significant increase in volume and moment of inertia,
making it pointless to consider the constraints associated with combined
variations in angular parameters and blade width. Therefore, a decision was
made to consider design constraints for angular parameter variations separately
for each data layer corresponding to a given blade width.
Below
are the results of taking into account design constraints according to a
previously developed methodology [7] for varying angular parameters
and
for a given blade
width L+20%.
In
Figure 5, the calculated volumes are represented as a three-dimensional surface
depending on the
variations in the main angles. The plane bounded by red corresponds to the
volume value for the base case. Accordingly, the plane bounded by blue
corresponds to the volume value for the base case increased by 10%. The
intersection lines of both planes with the surface of volume values limit the
variation of the product shape. In Figure 6, the range of acceptable values,
taking into account the volume constraints, is shown on the plane of angular
parameter variations and is enclosed in the area bounded by the thick red and
blue lines.
Fig. 5. Organization of accounting for volume restrictions for the L+20% data layer.
Fig. 6. Volume limitation area for varying angles for the L+20% data layer.
Limits
for the moment of inertia are determined in a similar manner. They are shown in
Figures 7 and 8. In Figure 7, the calculated moments of inertia are represented
as a three-dimensional surface
depending on the
variations in the main angles. The plane bounded by red corresponds to the
moment of inertia value for the base case. Accordingly, the plane bounded by
blue corresponds to the moment of inertia value for the base case increased by
10%. The intersection lines of both planes with the surface of the moments of
inertia values limit the variation in the product shape. In Figure 8, the range
of permissible values is represented on the plane of angular parameter
variations as the area between the thick red and blue lines.
Fig. 7.
Organization of accounting for restrictions on the moment of inertia for the
L+20% data layer.
Fig. 8.
Limitation region for the moment of inertia when varying angles for the L+20%
data layer.
Thus,
for a given data layer corresponding to a blade width of L+20%, we have
obtained the volume and moment of inertia constraints shown in Figures 6 and 8.
Now we need to organize their combined consideration. To do this, they should
be combined in a single image and the range of angle variations selected that
corresponds to the most stringent constraint.
For a
more visual representation, let us consider Figure 9. This representation
allows us to more accurately determine the range of variation of the angular
parameters that ensures the maximization of the torque, taking into account the
imposed restrictions on the change in volume and the moment of inertia of the
blade assembly.
Fig. 9.
Two-dimensional representation of the distribution of the moment of inertia
taking into account the limitations when varying the angular parameters.
Figure 9 provides
a clear picture of the desired range of angular parameter variation. It can be
stated that, given the selected volume and moment of inertia constraints, the
desired range lies within the angle variation range
of 55° to 60°, and the angle variation range
of 120° to 125°.
The digital twin
technology developed and tested by a team from the Keldysh Institute of Applied
Mathematics of the Russian Academy of Sciences using a wind turbine blade
assembly as an example represents not only an effective solution for optimizing
a specific design but also a potentially transformative paradigm in digital
engineering. The achieved results demonstrate the feasibility of creating a
scalable and adaptive platform capable of integrating various methods of
mathematical modeling, numerical analysis, and data visualization to solve a
wide range of problems in various industries.
Further
development of the technology involves in-depth research and expansion of the
platform's functionality in the following areas:
1. Developing
adaptive modeling methods: Adaptive modeling methods need to be developed that
automatically adjust the level of model detail depending on the problem being
solved and available computing resources. This will optimize the modeling and
analysis process, ensuring the required accuracy of results at minimal cost.
2. Research
into methods for verifying and validating models: Rigorous methods for
verifying and validating models, based on comparison of modeling results with
experimental data and the results of other independent calculations, will
increase confidence in the modeling results and ensure the reliability of
decisions made.
3. Developing
tools for automatic model generation: Tools for automatically generating models
based on data about the geometry, materials, and operating conditions of a
system offer potential for reducing the time and cost of creating digital twins
and making them accessible to a wider range of users.
The developed
technology can be applied to solve a wide range of problems in various
industries, including:
• Design
and optimization of complex technical systems: Developing digital twins for
aircraft, automobiles, power plants, and other complex technical systems will
optimize their design, improve efficiency and reliability, and reduce
development and operating costs.
• Monitoring
and diagnostics of technical condition: Developing digital twins for monitoring
and diagnostics of equipment's technical condition will enable timely detection
of defects and prevention of accidents, as well as optimization of maintenance
and repair schedules.
• Complex
Process Management: Developing digital twins for complex process management
will optimize equipment operation, improve production efficiency, and reduce
energy and material costs.
• Development
of new materials and technologies: Developing digital twins to model the
properties and behavior of new materials and technologies will accelerate their
development and implementation into production.
Implementing
these promising development areas will require the consolidated efforts of the
scientific community, industrial enterprises, and government agencies, as well
as significant investment in research and development. However, the results of
this work will create a powerful platform for digital engineering, which will
contribute to the competitiveness of Russian industry and strengthen Russia's
position in the global high-tech market.
This
paper presents the experience of a team from the Keldysh Institute of Applied
Mathematics of the Russian Academy of Sciences in developing and applying
digital twin technology to optimize the design of a wind turbine blade
assembly. It is demonstrated that creating a digital twin enables highly
accurate virtual testing and modeling, reducing development and testing time
and costs, and enabling the identification of optimal geometric parameters for
maximum turbine efficiency. The developed technology represents an effective
solution, readily scalable and applicable to a wide range of engineering
problems. It can be used for the design and optimization of complex technical
systems in various industries, as well as for monitoring and diagnostics of
their technical condition.
Prospects
for further development of the technology include expanding the platform's
functionality, developing specialized model libraries, integrating with
computer-aided design systems, and applying machine learning and artificial
intelligence methods. Implementing these promising areas will create a powerful
digital engineering platform that will enhance the competitiveness of Russian
industry.
In
conclusion, digital twin technology is a key area of development in modern
science and technology. Its application allows us to solve complex engineering
problems, create innovative products, and improve production efficiency.
Further research and development in this area will contribute to the
development of the Russian economy and improve the quality of life.
The calculations
were performed using the K100 hybrid supercomputer installed at the Keldysh
Institute of Applied Mathematics and Mathematics Supercomputer Center of the
Russian Academy of Sciences.
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