Toward a shape-performance integrated digital twin based on hybrid reduced-order modeling for engineering structures

Abstract

With the increasing demand for structural visualization, health monitoring, and fault diagnosis, the requirements for real-time and highly accurate digital twins (DTs) to realize interactions between the physical and digital spaces have increased. However, the real-time requirements of DT have become an obstacle to their applications in practical engineering. In this work, a hybrid reduced-order modeling (HROM) method was proposed to improve calculation efficiency and ensure the accuracy of the shape-performance integrated DT (SPI-DT). The proposed method is driven by dynamic sensor data and consisted of the reduced-order model (ROM), the Kriging (KRG) model, and the isoparametric element formulation. It also introduces an iterative infilling strategy to select master degrees of freedom (MDOFs) in this method. In addition, the main beam of a gantry crane was used as a numerical example to verify the accuracy and efficiency of the HROM method. The results show that the online computationally cost online of the HROM method is lower than the KRG and ROM methods, which can improve the computational efficiency of the SPI-DT for complex-large structures by significantly reducing the delay time. Moreover, the HROM method had higher prediction accuracy (R-squared = 0.8986) compared with the KRG method (R-squared = 0.8599) and closer to the ROM method (R-squared = 0.9052) since the HROM method based on the ROM data and the data-driven method. This research provides a feasible method for realizing the applications of DT in major engineering equipment.

Appendix

This section introduces the derived formula of the strain transformation matrix. As shown in Fig. 14, combining adjacent nodes can degenerate into a 4 nodes 3D tetrahedral element for a linear 3D hexahedron element with 8 nodes. The processing of equation deducing as follows:

Fig. 14

The 8-node spatial elements degenerate into tetrahedral elements

First, the shape functions of linear 3D hexahedron elements as an example can be expressed as follows:

$$N_{K} \left( {\xi ,\eta ,\zeta } \right) = \frac{1}{8}\left( {1 + \xi_{K} \xi } \right)\left( {1 + \eta_{K} \eta } \right)\left( {1 + \zeta_{K} \zeta } \right),$$

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where $\xi_{1} = - 1,\eta_{1} = - 1,\zeta_{1} = - 1, \ldots ,\xi_{8} = - 1,\eta_{8} = 1,\zeta_{8} = 1$, $\xi_{K} ,\eta_{K} ,\zeta_{K}$ are the element coordinates system of the node [46]. The coordinates of the node in the linear 3D tetrahedral element can be expressed by the shape function:

$$\begin{aligned} u & = \sum\limits_{i = 1}^{8} {N_{i} u_{i} } = \left( {N_{1} + N_{2} + N_{3} + N_{4} } \right)u_{3} + \left( {N_{5} + N_{6} } \right)u_{6} + N_{7} u_{7} + N_{8} u_{8} \\ v & = \sum\limits_{i = 1}^{8} {N_{i} v_{i} } = \left( {N_{1} + N_{2} + N_{3} + N_{4} } \right)v_{3} + \left( {N_{5} + N_{6} } \right)v_{6} + N_{7} v_{7} + N_{8} v_{8} \\ w & = \sum\limits_{i = 1}^{8} {N_{i} w_{i} } = \left( {N_{1} + N_{2} + N_{3} + N_{4} } \right)w_{3} + \left( {N_{5} + N_{6} } \right)w_{6} + N_{7} w_{7} + N_{8} w_{8} . \\ \end{aligned}$$

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The shape function is given in the local coordinate system and according to differentiation:

$$\frac{{\partial N_{i} }}{\partial \xi } = \frac{{\partial N_{i} }}{\partial u}\frac{\partial u}{{\partial \xi }} + \frac{{\partial N_{i} }}{\partial v}\frac{\partial v}{{\partial \xi }} + \frac{{\partial N_{i} }}{\partial w}\frac{\partial w}{{\partial \xi }}.$$

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Similarly, $\frac{{\partial N_{i} }}{\partial \eta }$ and $\frac{{\partial N_{i} }}{\partial \zeta }$ can be combined as follows:

$$\left\{ {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial \xi }} \\ {\frac{{\partial N_{i} }}{\partial \eta }} \\ {\frac{{\partial N_{i} }}{\partial \zeta }} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\frac{\partial u}{{\partial \xi }}} & {\frac{\partial v}{{\partial \xi }}} & {\frac{\partial w}{{\partial \xi }}} \\ {\frac{\partial u}{{\partial \eta }}} & {\frac{\partial v}{{\partial \eta }}} & {\frac{\partial w}{{\partial \eta }}} \\ {\frac{\partial u}{{\partial \zeta }}} & {\frac{\partial v}{{\partial \zeta }}} & {\frac{\partial w}{{\partial \zeta }}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial u}} \\ {\frac{{\partial N_{i} }}{\partial v}} \\ {\frac{{\partial N_{i} }}{\partial w}} \\ \end{array} } \right\} = [{\mathbf{J}}]\left\{ {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial u}} \\ {\frac{{\partial N_{i} }}{\partial v}} \\ {\frac{{\partial N_{i} }}{\partial w}} \\ \end{array} } \right\},$$

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where $[{\mathbf{J}}]$ denotes the Jacobian matrix,

$$[{\mathbf{J}}] = \left[ {\begin{array}{*{20}c} {\frac{\partial u}{{\partial \xi }}} & {\frac{\partial v}{{\partial \xi }}} & {\frac{\partial w}{{\partial \xi }}} \\ {\frac{\partial u}{{\partial \eta }}} & {\frac{\partial v}{{\partial \eta }}} & {\frac{\partial w}{{\partial \eta }}} \\ {\frac{\partial u}{{\partial \zeta }}} & {\frac{\partial v}{{\partial \zeta }}} & {\frac{\partial w}{{\partial \zeta }}} \\ \end{array} } \right].$$

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The strain transformation matrix can be expressed as:

$$[{\mathbf{B}}_{i} ] = \left[ {{\mathbf{J}}^{ - 1} } \right]\left[ \begin{gathered} \begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial \xi }} & 0 & 0 \\ 0 & {\frac{{\partial N_{i} }}{\partial \eta }} & 0 \\ 0 & 0 & {\frac{{\partial N_{i} }}{\partial \zeta }} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial \eta }} & {\frac{{\partial N_{i} }}{\partial \xi }} & 0 \\ 0 & {\frac{{\partial N_{i} }}{\partial \zeta }} & {\frac{{\partial N_{i} }}{\partial \eta }} \\ {\frac{{\partial N_{i} }}{\partial \zeta }} & 0 & {\frac{{\partial N_{i} }}{\partial \xi }} \\ \end{array} \hfill \\ \end{gathered} \right].$$

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