# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
## 3 Evaluation Criteria  
It is essential to validate the accuracy of the dynamic reduction approaches outlined in the preceding section in the context of off-road vehicles. In general, the quality of the reduced finite element models obtained by these dynamic reduction approaches characterizes their accuracy. The frequencies and mode shapes of a reduced model are typically compared to those of the full-order finite element model or reference model to determine its accuracy. An important metric for evaluating the precision of dynamic reduction techniques is the deviation of the natural frequencies from the full-order model for the reduced structure. To represent the frequency deviations between the reduced-order model and the full-order model in mathematical terms, we can use  
\[\epsilon_{i}=\left|\frac{\omega_{*i}-\omega_{i}}{\omega_{i}}\right|\times 1 00\% \tag{41}\]  
where \(\epsilon_{i}\) represents the percentage error between the \(i\)th reduced-order and full-order mode, \(i\in\{Q\}\), and where \(\{Q\}=\{1,\dots,R\}\cap\{1,\dots,N\}\). For the most part, the client defines the maximum frequency deviations permitted. The cross-orthogonality check, also known as the modal vector orthogonality check, is often used to assess the quality of reduced-order finite element models in conjunction with the frequency deviation check. In the cross-orthogonality check [74], the mass normalized modal matrix (\([\phi]\)) obtained by the modal analysis is primarily employed in combination with the mass matrix (\([M_{r}]\)) of the reduced-order model to assess the orthogonality of the modal vectors. Because reduced-order models have linearly independent degrees of freedom, each modal vector in the modal matrix should be orthogonal to the other vectors. As a result, the orthogonality relations may be expressed as  
\[\{\phi_{i}\}^{\intercal}[M_{r}]\{\phi_{j}\}= \begin{cases}0:i\
eq j\\ 1:\text{otherwise}\end{cases}  \tag{42}\]  
Dynamic reduction techniques are based on the notion of transforming a substructure's physical coordinates into a set of generalized coordinates and then truncating modal coordinates to obtain a reduced set of equations for the system. As a consequence, exact system solutions with reduced computational load are obtained over a limited frequency range. Modal truncation was essential in the past for structural finite element model analysis when the number of degrees of freedom exceeded the computing capability. In the last two decades, the exponential increase in processor performance, the introduction of general-purpose graphics processing units, and the availability of efficient and accurate numerical algorithms have essentially mitigated the computational constraints in the design and analysis processes of earlier times. Nonetheless, in the context of Industry 4.0, dynamic reduction methods are significant for the development of physics-based digital models for off-road vehicles. In essence, virtual models consist of the modal characteristics of physical off-road vehicles. In addition, digital twins are intended to emulate the simultaneous activities of their physical counterparts operating in an industrial environment. Consequently, solving the equation of motion in real-time that encompasses the complete set of modes constituting the virtual system is impractical. As a result, dynamic reduction-based reduced-order models are appropriate for accomplishing the requirements. In practice, for well-formulated structural analysis problems, the lower order modes derived by dynamic reduction methods are extremely accurate, but substantial inaccuracies may occur in the higher frequency range. Furthermore, the general principle for dynamic reduction is to incorporate subcomponent modes up to 1.5-2.0 times the frequency range of interest [75], which is a value that the client specifies. However, certain subcomponents for off-road vehicles may have all the modal components exceeding the frequency range of interest.