M-LFM: a multi-level fusion modeling method for shape−performance integrated digital twin of complex structure

Abstract

As a virtual representation of a specific physical asset, the digital twin has great potential for realizing the life cycle maintenance management of a dynamic system. Nevertheless, the dynamic stress concentration is generated since the state of the dynamic system changes over time. This generation of dynamic stress concentration has hindered the exploitation of the digital twin to reflect the dynamic behaviors of systems in practical engineering applications. In this context, this paper is interested in achieving real-time performance prediction of dynamic systems by developing a new digital twin framework that includes simulation data, measuring data, multi-level fusion modeling (M-LFM), visualization techniques, and fatigue analysis. To leverage its capacity, the M-LFM method combines the advantages of different surrogate models and integrates simulation and measured data, which can improve the prediction accuracy of dynamic stress concentration. A telescopic boom crane is used as an example to verify the proposed framework for stress prediction and fatigue analysis of the complex dynamic system. The results show that the M-LFM method has better performance in the computational efficiency and calculation accuracy of the stress prediction compared with the polynomial response surface method and the kriging method. In other words, the proposed framework can leverage the advantages of digital twins in a dynamic system: damage monitoring, safety assessment, and other aspects and then promote the development of digital twins in industrial fields.

Key words： shape−performance integrated digital twin (SPI-DT); multi-level fusion modeling (M-LFM); surrogate model; telescopic boom crane; data fusion

Nomenclature

Abbreviations
3D	Three-dimensional
DT	Digital twin
FEM	Finite element method
KRG	Kriging
M-LFM	Multi-level fusion modeling
NMAE	Normalized maximum absolute error
NRMSE	Normalized root mean square error
PRS	Polynomial response surface
RUL	Remaining useful life
SNR	Signal-to-noise ratios
SPI-DT	Shape–performance integrated digital twin
Variables
a	Unknown coefficient used to change the result size of the predicted model
A	Amplitude of the signal
Anoise	Amplitude of the noise
b	Scale factor
Df	Remaining useful life
e	Error between the training value and the predicted value at training point
E	Elastic modulus,
fi(∙) (i = 1,2,…,m)	Unknown approximate function
Fi (i = 1,2,…,5)	Thrust produced by the hydraulic cylinder and the pulling force produced by the transmission chain
Ffi (i = 2,3,4)	Fraction between segments
F	Force vector
g	gth cycle
H	Vector whose elements are ones
k	Number of measured data
Kt	Elastic stress concentration factor
L	Cost function
m	Number of input variables
Mg	Total cycle number
M, C, K	Mass matrix, damping matrix, and stiffness matrix, respectively
ng, Ng	Cycle number in practical and cycles to failure, respectively
Nf	Cycle number to failure
Nt	Test point number
P	Signal power
Pnoise	Noise power
\mathit{Q},\dot{\mathit{Q}},\ddot{\mathit{Q}}	Displacement, velocity, and acceleration vector, respectively
r	Related vector between the unknown predicted point and the trained point
R	Relationship function
R2	Coefficient of determination
R	Relationship matrix whose elements are the value of relationship function
t	Time
Ttest	Measured data
ui, vi, wi	Space coordinates in the ith point
uC, vC, wC	Space coordinates in the compensation point
xi (i = 1,2,…,m)	ith input variable
X	Input vector
\overline{y}	Mean value of the true output at all test points
\hat{y}	Predicted value of output variable
y_i^{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}	Response value in the ith training point
{\hat{\mathit{y}}}^{\mathrm{P}\mathrm{R}\mathrm{S}}	Predicted data of PRS model
{\hat{y}}_C^{\mathrm{P}\mathrm{R}\mathrm{S}},{\hat{y}}_C^{\mathrm{K}\mathrm{R}\mathrm{G}}	Predicted values of the PRS model and KRG model at the compensation point, respectively
{\mathit{y}}_{\mathrm{s}}	True response value at samples.
\mathit{Y}	Output vector
{\mathit{Y}}_{\mathrm{P}\mathrm{R}\mathrm{S}},{\mathit{Y}}_{\mathrm{K}\mathrm{R}\mathrm{G}}	Predicted vectors of the PRS model and the KRG model, respectively
\mathit{\text{z}}(\cdot )	Error function of KRG model
C{f}_i (i = 1,2,…,n)	Compensation factor of the influence domain
C{f}_{min}, C{f}_{max}	Minimum and maximum values of the compensation factor set, respectively
C{f}_i^{\mathrm{n}\mathrm{o}\mathrm{r}}	Normalized compensation factor
\mathit{C}{\mathit{f}}_{\mathrm{n}\mathrm{o}\mathrm{r}}	Compensation factor set
{\alpha }_i (i = 1,2,…,m)	Unknown weight coefficient
{\alpha }_i^{\delta },{\alpha }_i^u,{\alpha }_i^a	Unknown weight coefficient of different parameters model
\mathit{\alpha }	Vector of unknown coefficients
\hat{\beta }	Maximum likelihood estimation value of hyper-parameter
{\beta }_i	Unknown weighted coefficients of the KRG model
\delta	Parameter of the influence domain
\hat{\delta },\hat{\mu },\hat{a}	Predicted values of the model parameters under different sensor states
{\delta }_{\mathrm{l}},{\delta }_{\mathrm{u}}	Upper and lower bounds of the parameter \delta, respectively
δopt, {\mathit{\mu }}_{\mathrm{o}\mathrm{p}\mathrm{t}},{\mathit{a}}_{\mathrm{o}\mathrm{p}\mathrm{t}}	Optimal parameters of \delta, \mu, and a, respectively
\mathit{\epsilon }	Vector of error function
\mu	Deviation coefficient of the compensation point
{\mu }_{\mathrm{l}},{\mu }_{\mathrm{u}}	Upper and lower bounds of the parameter \mu, respectively
{\rho }_i (i = 1,2,…,n)	Euclidean distance between the compensation point and the surrounding points
{\rho }_{min}, {\rho }_{max}	Minimum and maximum values of the Euclidean distance set, respectively
{\rho }_i^{\mathrm{n}\mathrm{o}\mathrm{r}}	Normalized Euclidean distance
σ2	Variance of error function
σa	Stress amplitude
σm	Mean stress
σy	Tensile yield stress
σe(Soder)	Modified mean stress
σ	Performance vector
σs, σh	Predicted stresses in the extension and retraction processes, respectively
∆σnom	Nominal stress range
ψi (x)	Function correlated with the physical properties of problem itself
{\psi }_i^{\delta }(x), {\psi }_i^u(x), {\psi }_i^a(x)	Functions correlated with the physical properties of different PRS model

Acknowledgements

This work was supported by the National Key R&D Program of China (Grant No. 2018YFB1700704) and the National Natural Science Foundation of China (Grant No. 52075068).

Electronic Supplementary Materials

The supplementary materials can be found in the online version of this article at https://doi.org/10.1007/s11465-022-0708-0 and are accessible to authorized users.

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