Open AccessFeature PaperArticle

Frequency-Based Finite Element Updating Method for Physics-Based Digital Twin

Youngjae Jeon

¹,

Geomji Choi

²,

Kwanghyun Ahn

³,

Kang-Heon Lee

^4,* and

Seongmin Chang

^2,*

Department of Automotive Engineering (Automotive-Computer Convergence), Hanyang University, Seoul 04763, Republic of Korea

School of Mechanical Engineering, Chungnam National University, Daejeon 34134, Republic of Korea

SMART System Development Division, Korea Atomic Energy Research Institute, Daejeon 34057, Republic of Korea

⁴

MSILABS Inc., Daejeon 34127, Republic of Korea

Authors to whom correspondence should be addressed.

Mathematics 2025, 13(5), 738; https://doi.org/10.3390/math13050738

Submission received: 21 January 2025 / Revised: 13 February 2025 / Accepted: 18 February 2025 / Published: 24 February 2025

(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)

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Figure 1
Comparison of conventional method and proposed method. "> Figure 2
Effect of damping on the accuracy of the result: (a) magnitude of updating results including and excluding damping; (b) absolute error. "> Figure 3
Strategy for frequency-based finite element updating method. "> Figure 4
Numerical model of CRDM. "> Figure 5
Input acceleration time history applied to x, y, z axes for dynamic analysis. "> Figure 6
Input acceleration data transformed into frequency domain for dynamic analysis. "> Figure 7
Verification of accuracy using frequency-based method and time-based method: (a) magnitude of updating results including and excluding damping; (b) absolute error. "> Figure 8
Comparison of calculation time and number of iterations using frequency-based method and time-based method. ">

Versions Notes

Abstract

This study proposes a frequency-based finite element updating method for an effective physics-based digital twin (DT). One approach to constructing a physics-based DT is to develop a mechanics-based mathematical model that accurately simulates the behavior of an actual structure. The proposed method utilizes finite element updating, adjusting model parameters to improve model accuracy. Unlike simple modal analysis, which focuses on vibration characteristics, this method recognizes that accurate dynamic transient-based vibration analysis requires considering the damping effect, as well as mass and stiffness, during the updating process. Moreover, a frequency-based analysis is employed instead of the computationally expensive time-based analysis for more efficient dynamic modeling. By transforming data into the frequency domain, the method efficiently represents dynamic behavior within relevant frequency ranges. We further enhance the computational efficiency using the model reduction technique. To validate the method’s accuracy and efficiency, we compare the analysis results and computation time using a numerical example of the control rod drive mechanism. The proposed method shows significantly reduced computation time, by a factor of 8.9 compared to conventional time-based methods, while preserving high accuracy. Therefore, the proposed method can effectively support the development of physics-based DTs.

Keywords:

digital twin; frequency-based analysis; finite element updating method; vibration analysis; control rod drive mechanism

MSC:

74H45; 74S05

1. Introduction

Digital twin (DT) has gained significant attention recently and is defined differently depending on the field of application. One definition by Glaessgen and Stargel describes DT as “an integrated multi-physics, multi-scale, probabilistic simulation of a vehicle or system that uses the best available physical models, sensor updates, fleet history, etc., to mirror the life of its flying twin” [1,2]. Another definition by Grieves refers to it as “a virtual, digital equivalent to a physical product” [3]. In essence, a DT is a virtual representation that mirrors an actual physical structure. The creation of such a replica involves developing a mathematical model that resembles the real structure. In mechanical engineering, methods based on mechanics are often used for structures with physical behavior. Computational techniques such as CAE can simulate the physical behavior under external loads or excitation, which is a key step in constructing a physics-based DT [4,5,6]. Constructing a physics-based DT involves an inverse analysis to determine the properties of a physical system from its response. In dynamic analysis, the inverse process aims to determine the system’s dynamic properties, improving the simulation’s accuracy in reflecting the actual structure.

To match the vibration behavior of the analysis model with that of the actual structure, the finite element updating method is widely used [7,8,9,10,11,12,13]. This method adjusts the model’s properties to improve correlation with the real structure. In dynamic analysis, model behavior includes displacement, velocity, and acceleration, while the dynamic properties include mass, damping, and stiffness [14,15]. Typically, for materials or structures with negligible damping, dynamic analysis considers only mass and stiffness, as this effectively predicts vibration behavior. However, research on constructing physics-based DTs for mechanical structures has shown that damping must be considered for accurate dynamic DT modeling. In this study, a dynamic analysis and updating were performed, and the vibration behavior was compared with experimental data. When mass and stiffness were considered alone, the model did not accurately reflect the experimental results. However, incorporating damping led to effective updating. Thus, to construct a physics-based DT with behavior that matches actual performance, mass, stiffness, and damping must be considered in the updating process.

Dynamic analysis is typically classified into modal and transient response analysis. Modal analysis focuses on identifying vibrational characteristics such as mode shapes and natural frequencies. For applications such as resonance avoidance, vibration behavior is calculated at specific lower-order modes or frequencies. When acceleration time histories or time-varying loads are involved, transient analysis is more effective than modal analysis, as it captures the response of the system to time-dependent loads. For a transient-based vibration analysis of physics-based DTs, it is essential to match the model’s vibration behavior to the actual structure at each time step. However, analyzing every time step is computationally expensive. As the frequency range of interest in dynamic analyses typically extends up to approximately 50 Hz, focusing on this frequency range via frequency-based analysis can be more efficient. Thus, this study proposes a frequency-based finite element updating method to efficiently perform a dynamic analysis and updating.

The proposed method incorporates a frequency-based analysis and damping-inclusive updating, achieving efficient analysis within the frequency range of interest through fast Fourier transform (FFT) and model order reduction. FFT transforms data from the time domain to the frequency domain, while model order reduction reduces the matrix order required for analysis. Effective updating is achieved by considering mass, stiffness, and damping. The impact of damping on accuracy is validated by comparing the updating results with and without damping. The accuracy and efficiency of the method are demonstrated through the dynamic analysis of a numerical example using the control rod drive mechanism (CRDM), a key component of nuclear power plants.

The main contributions of this study are as follows:

While previous updating strategies have mainly focused on matching mass and stiffness parameters, our method extends this by also incorporating damping. By doing so, the model more accurately captures the true dynamic behavior of the physical system. This is particularly important when constructing a physics-based DT for dynamic mechanical structures.
Rather than performing a computationally expensive time domain analysis, our approach leverages FFT to transform the time domain data into the frequency domain. This allows us to concentrate the analysis on the frequency range of interest. Model order reduction techniques are used to simplify the mathematical model without compromising accuracy.
The proposed method integrates damping into the frequency-based updating process. The accuracy and computational efficiency of the proposed method are evaluated using numerical examples based on CRDM experimental data. The results demonstrate that the proposed method reduces computational costs by approximately 90% while maintaining accuracy compared to conventional time-based methods.

The remainder of this paper is organized as follows: Section 2 presents the formulas for a dynamic analysis using the reduction method. Section 3 introduces the proposed frequency-based finite element updating method. The numerical results are discussed in Section 4. Finally, conclusions are provided in Section 5.

2. Applying the Reduction Method to Dynamic Analysis

This section describes the formulation for a dynamic analysis. The equation of motion for a multi-degree-of-freedom damped system is described, and the application of reduction methods to the dynamic equation for computational efficiency is discussed. The reduction method decreases the size of the stiffness, mass, damping, and force matrices [16,17,18,19]. In the dynamic analysis, we use the mode-based reduction method that is known as generalized coordinate reduction. The reduction method can reduce the degree of freedom of a matrix by using the eigenvector of the matrix. The dynamic analysis uses the reduction method on the stiffness, mass, damping, and force matrices. As the order of the full model is large, reduced matrices enable efficient response calculation.

In dynamic analysis, the response of the structure is obtained through the dynamic equation of a damped system:

M \ddot{u} + C \dot{u} + K u = F,

(1)

where

M

C

, and

K

are mass, damping, and stiffness matrices, respectively.

\ddot{u}

\dot{u,}

and

u

are acceleration, velocity, and displacement vectors, respectively, and

F

is the force vector. Vector

u

can also be referred to as the physical space.

u

can be transformed into the modal space

z

using the reduction method.

u

can be expressed as

u = φ z,

(2)

where

φ

is the eigenvector. Damping matrix

C

is expressed using Rayleigh damping as

C = α K + β M,

(3)

where

α

and

β

are stiffness and mass proportional damping coefficients, respectively. Rayleigh damping approximates the damping in systems with known mass and stiffness matrices through a linear combination of these matrices. Rayleigh damping is one of the most general methods for calculating damping. By substituting Equation (2), the dynamic equation can be expressed as

M φ \ddot{z} + C φ \dot{z} + K φ z = F .

(4)

The acceleration vector

\ddot{u}

and velocity vector

\dot{u}

are expressed as

φ \ddot{z}

and

φ \dot{z}

, respectively. Multiplying both sides of Equation (4) by

φ^{T}

yields

φ^{T} M φ \ddot{z} + φ^{T} C φ \dot{z} + φ^{T} K φ z = φ^{T} F,

(5)

where superscript

T

denotes the matrix transpose. The reduced dynamic equation can be formulated as

M_{R} \ddot{z} + C_{R} \dot{z} + K_{R} z = F_{R},

(6)

where

M_{R}

C_{R}

K_{R}

, and

F_{R}

are reduced mass, damping, stiffness, and force matrices, respectively. Assuming

z

Z e^{j ω t}

, the reduced equation can be expressed as

- ω^{2} M_{R} z + j ω C_{R} z + K_{R} z = F_{R},

(7)

where

Z,

ω,

and

j

denote amplitude, frequency, and the imaginary number, respectively.

z

can be obtained as follows

z = {(- ω^{2} M_{R} + j ω C_{R} + K_{R})}^{- 1} F_{R},

(8)

z = K_{e}^{- 1} F_{R},

(9)

where

K_{e}

is the effective stiffness matrix. The calculated

z

is substituted into Equation (2) to determine

u

. The relationship between

\ddot{u}

and

u

is expressed as

\ddot{u} = - ω^{2} u .

(10)

By substituting

u

into Equation (10), the acceleration vector

\ddot{u}

can be calculated.

3. Frequency-Based Finite Element Updating Method

This section presents a frequency-based finite element updating method to efficiently obtain accurate dynamic responses of physics-based DT. It compares time- and frequency-based analysis methods, analyzes the influence of damping on the accuracy of results, and presents details on the proposed method.

3.1. Comparison of Time-Based Analysis and Frequency-Based Analysis

To perform a dynamic analysis of a physics-based DT, it is crucial to accurately match the model’s vibration behavior with the actual behavior. However, accounting for every time step in the analysis is computationally expensive. The proposed method addresses this challenge by employing a frequency-based approach that considers only the frequency range of interest.

Figure 1 compares the time-based and frequency-based analysis methods. The key distinction lies in the domain of analysis: the time-based method analyzes data at every time step in the time domain, resulting in high computational costs. By contrast, the proposed method utilizes FFT to convert input data into the frequency domain, allowing the analysis to be confined to the relevant frequency range. By excluding high-frequency regions that are not of interest, the frequency-based approach significantly reduces computational time [20,21]. The frequency-based method involves an additional transformation process, resulting in higher computational complexity compared to the time-based methods. Nevertheless, the computational efficiency of the frequency-based analysis has increased because the computational advantages from frequency range restriction are greater. The accuracy and efficiency of the proposed method are validated by comparing its analysis results to those obtained using the conventional time-based method and experimental data.

Both methods commonly evaluate the response to dynamic loads through the response spectrum. The frequency-based analysis transforms the input data in the frequency domain data using FFT. However, it does not represent the response of a structure to dynamic loads and may not effectively capture the response characteristics. To address this issue, the response spectrum method is employed to adequately represent the maximum response in the frequency domain. The response spectrum represents how a structure reacts to a specific earthquake or dynamic load. Since it shows how a structure responds across various frequencies, it is generally used for structural dynamic analysis.

3.2. Effect of Damping on Accuracy of Result

A finite element updating method was applied to improve the accuracy of the analysis results. Damping is a parameter used to obtain a model representing vibration behavior that matches well with the actual behavior. Figure 2 compares the updating results, including and excluding damping with the experimental results, to show the effect of damping on the accuracy of the results. Figure 2 shows the magnitude and the absolute error of the calculated vibration behavior in the frequency domain. As shown in Figure 2a, the peak of the response excluding damping is inaccurate, but the result that includes damping aligns with the experiment. The absolute error decreases through damping, as shown in Figure 2b. Thus, damping should be selected as a parameter along with stiffness and mass when updating.

Stiffness, mass, and damping matrices after updating are expressed as

K^{'} = \sum_{i = 1}^{n} p_{i}^{(K)} K_{i}, i = 1, 2, \dots, n,

(11)

M^{'} = \sum_{i = 1}^{n} p_{i}^{(M)} M_{i}, i = 1, 2, \dots, n,

(12)

C^{'} = α K^{'} + β M^{'},

(13)

where

p_{i}^{(K)}

and

p_{i}^{(M)}

are updating parameters of the

i

-th part stiffness and mass matrices, respectively.

K_{i}

and

M_{i}

are the

i

-th part stiffness and mass matrices before updating. The parameters are updated until the response accurately matches the experimental results. The update process becomes an optimization problem of minimizing an objective function. Iterative optimization can find the optimal parameter p by minimizing the difference between the analytical behavior and the actual experimental behavior. The objective function is a mean squared error between the experimental and analytical values. The optimization code was implemented in MATLAB R2022b and employs the Levenberg–Marquardt algorithm for optimization. The Levenberg–Marquardt algorithm is widely used in nonlinear least-squares problems for fast and robust optimization and is typically applied to minimize the sum of squared errors between observed data and model predictions.

3.3. Process of Frequency-Based Finite Element Updating Method

Figure 3 shows the process of the proposed method. It involves using FFT to transform excitation acceleration data for dynamic analysis from the time domain to the frequency domain. A reduction method is applied for efficient dynamic analysis. As the size of the stiffness and mass matrices affect the computation time, the reduction method can improve the computational efficiency. An accurate response is obtained through a frequency-based dynamic analysis using the reduction and updating methods considering damping. To adequately reflect the response of a structure to dynamic loads, the dynamic response is represented by a response spectrum. As the response spectrum takes the time domain data as input, the outputted data are converted into the time domain through inverse FFT before using the response spectrum.

4. Verification of Proposed Method Using Numerical Model

Dynamic analysis was performed using a numerical example to verify the accuracy and efficiency of the proposed method. To validate the accuracy, the analysis result of the proposed method was compared to the results of the conventional time-based method and experiment. To verify the efficiency, the computation time of the proposed method was compared to that of the conventional method.

4.1. Information on the Numerical Model

The numerical model is CRDM, which is one of the components installed inside a reactor vessel. The power of the reactor can be controlled by modifying the position of the control rod. Figure 4 shows the finite element model of CRDM for dynamic analysis. The CRDM model was divided into 11 parts. The maximum height of the model is 4487 mm. The model is constructed using a three-dimensional eight-node hexahedron. The number of elements, nodes, and total degrees of freedom are 9584, 14,013, and 42,039, respectively.

4.1.1. Input Data

A dynamic experiment using a CRDM prototype was conducted by the Korea Atomic Energy Research Institute [22,23,24]. The experimental data were used for the dynamic analysis. To implement the proposed method, the acceleration data obtained from the experiment were used as input. Figure 5 presents the input acceleration data in the x-, y-, and z-axis directions in the time domain, which were used for both the experiment and analysis.

For the frequency-based analysis, the acceleration data in the time domain were transformed into the frequency domain using FFT, which decomposes the input signal into a sum of sinusoidal components across a range of frequencies. The seismic design criteria for nuclear reactors typically consider frequencies up to 50 Hz [22,23,24]. For conservative consideration of the frequency range of interest, the range was set to 150 Hz. However, the transformed acceleration data initially spanned frequencies up to 800 Hz, resulting in excessive computational requirements. By excluding unnecessary frequency components (150–800 Hz), the analysis frequency range was optimized, and computation time was significantly reduced. Figure 6 illustrates the acceleration data for the x-, y-, and z-axis directions transformed into the frequency domain.

4.1.2. Updating Result

The dynamic analysis of the CRDM used the frequency domain input data. Initially, the dynamic response obtained from the analysis exhibited low accuracy. To enhance accuracy, stiffness, mass, and damping were updated as key parameters. For detailed parameter refinement, the CRDM model was divided into 11 parts. By updating the parameters for each segment, accuracy was significantly improved. Stiffness and mass were updated individually for each part, while damping was adjusted uniformly for the entire model. Table 1 and Table 2 present the parameter values before and after the updates, respectively.

4.2. Verification of Proposed Method

4.2.1. Verification of Accuracy

To validate the accuracy and efficiency of the proposed method, its analysis results were compared using a numerical example. Accuracy was assessed by comparing the analysis results of the proposed method to the experimental data and the results of the conventional method. Figure 7 shows the magnitude of the dynamic response using both methods before and after updating: dashed line for before updating; solid line for after updating; blue line for the proposed method; and red line for the conventional method. In the dynamic analysis of CRDM, the focus should be on the peak of the response. It is crucial that the peak of the analysis aligns with that of the experiment. Figure 7a shows that the peak of both methods closely aligns with the experimental data following the parameter update. Both methods demonstrated a reduction in absolute error after updating the parameters to include damping effects, as shown in Figure 7b.

4.2.2. Verification of Efficiency

To validate the efficiency of the proposed method, its computation time and the number of updating iterations were compared to those of the conventional method. As illustrated in Figure 8, the two methods exhibited significant differences in both metrics. The conventional method required 18,525 s (5.15 h) for updating and 18,533 s (5.15 h) for total computation, while the proposed method required only 2013 s (0.56 h) and 2083 s (0.58 h), respectively. The updating time and computation time were reduced by factors of 9.20 and 8.90, respectively.

Similarly, the number of updating iterations was significantly reduced, from 250 iterations with the conventional method to 69 iterations with the proposed method—a 3.6-fold reduction. These comparisons demonstrate that the proposed method achieved substantially higher efficiency than the conventional approach. Furthermore, the validation confirms that the proposed method enables more accurate and computationally efficient analysis compared to the conventional method.

5. Conclusions

We proposed a frequency-based finite element updating method to construct an accurate and efficient physics-based DT. The method integrates frequency-based analysis with a finite element updating process that accounts for damping effects. By incorporating damping with stiffness and mass, the proposed method achieves accurate dynamic updates. Efficiency is further enhanced through the use of a frequency-based analysis and reduction techniques. A comparison of analyses including and excluding damping confirmed that considering damping is essential for precise updates of the dynamic physics-based DT. The method’s accuracy and efficiency were validated using a numerical example of CRDM. It demonstrated reduced computational time and fewer iterations compared with the conventional time-based analysis, without compromising accuracy.

The proposed method has broad applicability in fields requiring frequency-based dynamic analyses and computational efficiency. For example, it can effectively predict dynamic responses in critical frequency ranges for structures vulnerable to seismic activity, such as buildings and bridges. It is particularly useful for design and safety assessments by determining dynamic behavior in relevant frequency domains. By focusing on frequency-based analysis and updating, this method significantly improves the computational efficiency of dynamic analyses. Additionally, this approach can accurately capture the vibrational behavior of metal components such as robotic arms, thereby broadening its utility in industrial automation. Therefore, this method offers a foundation for advancing physics-based DT development.

However, the proposed method has several limitations. Since our approach focuses on a reduced design frequency range of the physics-based DT, it may not efficiently capture high-frequency dynamic responses. And, efficiency should be validated in cases where the frequency range is excessively wide, the transformation process is computationally intensive, or the system size and number of modes are large. Additionally, while incorporating damping improves accuracy, complementary strategies are necessary to fully account for the interplay between damping and nonlinear behavior in dynamic environments. The proposed approach struggles with nonlinear effects, especially at connection points when combining non-metal components like rubber. These issues require further investigation. Further study aimed at addressing the limitations inherent in modal-based model order reduction, high frequencies, and nonlinear elements are currently in progress. Moreover, integrating structural, thermal, and dynamic analyses will be essential for developing more comprehensive and effective physics-based digital twin models.

Author Contributions

Conceptualization, K.A. and S.C.; methodology, Y.J., G.C., K.A., K.-H.L. and S.C.; software, Y.J., G.C. and S.C.; validation, Y.J. and S.C.; formal analysis, Y.J., G.C. and S.C.; investigation, Y.J., G.C., K.A., K.-H.L. and S.C.; resources, G.C., K.A., K.-H.L. and S.C.; data curation, Y.J., K.A., K.-H.L. and S.C.; writing—original draft, Y.J.; writing—review and editing, K.A., K.-H.L. and S.C.; visualization, Y.J. and G.C.; supervision, K.A., K.-H.L. and S.C.; project administration, S.C.; funding acquisition, S.C. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the Innovative Small Modular Reactor Development Agency grant funded by the Korea Government (MOTIE) (No. RS-2024-00405419).

Data Availability Statement

The data presented in this study are available from the corresponding author on request due to privacy and confidentiality concerns.

Conflicts of Interest

The authors declare no conflicts of interest. One of the authors is affiliated with MSILABS Inc., but no financial or personal interests influenced the research.

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Figure 1. Comparison of conventional method and proposed method.

Figure 2. Effect of damping on the accuracy of the result: (a) magnitude of updating results including and excluding damping; (b) absolute error.

Figure 3. Strategy for frequency-based finite element updating method.

Figure 4. Numerical model of CRDM.

Figure 5. Input acceleration time history applied to x, y, z axes for dynamic analysis.

Figure 6. Input acceleration data transformed into frequency domain for dynamic analysis.

Figure 7. Verification of accuracy using frequency-based method and time-based method: (a) magnitude of updating results including and excluding damping; (b) absolute error.

Figure 8. Comparison of calculation time and number of iterations using frequency-based method and time-based method.

Table 1. Material properties and damping coefficients of numerical model before updating.

Part	Elastic Modulus [Pa]	Density [kg/m³]	Poisson’s Ratio	Damping Coefficients
Part	Elastic Modulus [Pa]	Density [kg/m³]	Poisson’s Ratio	α	β
1–8	$1.20 \times 10^{11}$	7720	0.3	$5 \times 10^{- 3}$	$1 \times 10^{- 4}$
9–10	$1.41 \times 10^{11}$	8110
11	$2.10 \times 10^{17}$	7850

Table 2. Material properties and damping coefficients of numerical model after updating.

Part	Elastic Modulus [Pa]	Density [kg/m³]	Poisson’s Ratio	Damping Coefficients
Part	Elastic Modulus [Pa]	Density [kg/m³]	Poisson’s Ratio	α	β
1	$1.5088 \times 10^{11}$	8888	0.3	$1.009 \times 10^{- 3}$	$1 \times 10^{- 4}$
2	$1.2677 \times 10^{11}$	12,064
3	$1.1587 \times 10^{11}$	8526
4	$9.7180 \times 10^{10}$	6957
5	$9.9129 \times 10^{10}$	7321
6	$1.2832 \times 10^{11}$	7657
7	$1.0452 \times 10^{11}$	7188
8	$1.3447 \times 10^{11}$	8280
9	$9.3077 \times 10^{10}$	7880
10	$1.3288 \times 10^{11}$	8199
11	$2.2968 \times 10^{17}$	7598

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Jeon, Y.; Choi, G.; Ahn, K.; Lee, K.-H.; Chang, S. Frequency-Based Finite Element Updating Method for Physics-Based Digital Twin. Mathematics 2025, 13, 738. https://doi.org/10.3390/math13050738

AMA Style

Jeon Y, Choi G, Ahn K, Lee K-H, Chang S. Frequency-Based Finite Element Updating Method for Physics-Based Digital Twin. Mathematics. 2025; 13(5):738. https://doi.org/10.3390/math13050738

Chicago/Turabian Style

Jeon, Youngjae, Geomji Choi, Kwanghyun Ahn, Kang-Heon Lee, and Seongmin Chang. 2025. "Frequency-Based Finite Element Updating Method for Physics-Based Digital Twin" Mathematics 13, no. 5: 738. https://doi.org/10.3390/math13050738

APA Style

Jeon, Y., Choi, G., Ahn, K., Lee, K.-H., & Chang, S. (2025). Frequency-Based Finite Element Updating Method for Physics-Based Digital Twin. Mathematics, 13(5), 738. https://doi.org/10.3390/math13050738

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Frequency-Based Finite Element Updating Method for Physics-Based Digital Twin

Abstract

1. Introduction

2. Applying the Reduction Method to Dynamic Analysis

3. Frequency-Based Finite Element Updating Method

3.1. Comparison of Time-Based Analysis and Frequency-Based Analysis

3.2. Effect of Damping on Accuracy of Result

3.3. Process of Frequency-Based Finite Element Updating Method

4. Verification of Proposed Method Using Numerical Model

4.1. Information on the Numerical Model

4.1.1. Input Data

4.1.2. Updating Result

4.2. Verification of Proposed Method

4.2.1. Verification of Accuracy

4.2.2. Verification of Efficiency

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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