# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
###### Abstract  
Digital models are the foundation of digital twins, which form the basis of autonomous off-road vehicles. Developing virtual models of off-road vehicles using dynamic reduction techniques is one of several approaches. The article commences with a comprehensive overview of the most widely used dynamic reduction methods and then introduces performance metrics for assessing their efficacies in the context of digital twins. The paper additionally includes a detailed mathematical derivation of the state-space representation for reduced-order finite element models. The state-space representation of the reduced finite element models facilitates their export to problem-solving environments for dynamic analysis. The state-space models are eventually solved utilizing the built-in libraries of numerical solvers in textual and graphical programming platforms. In addition, the article identifies the set of solvers that best suit the simulation of virtual models for off-road vehicles. This article also includes an evaluation of the simulation results for digital models with modes ranging from 0 to 30 Hz. In addition, the article demonstrates the lower bound of the frequency range necessary and sufficient to be retained in off-road vehicle virtual models. Finally, the paper presents the simulation outcomes for digital models of commercial off-road vehicles with custom-built virtual modules of powertrain, electrical, and control systems in a problem-solving environment. digital twin; industry 4.0; MATLAB; ANSYS; simulation; crane; modal analysis; dynamic substructuring: dynamic reduction; component modal synthesis +
Footnote †: journal: Applied  
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interdependent vehicle systems\" [51]. As stated in the definition, the primary objective of a digital twin is to replicate the life of an aircraft in cyberspace using the \"best\" physics-based digital model that consists of the integration of sub-components that represent the overall structure. While the aerospace industry and space agencies, in particular, have reaped immense benefits, the adoption of digital twins for heavy off-road vehicles is rare. There are potential benefits of digital twin models for the latter sector, including but not limited to enhanced operational efficiency, improved maintenance regimes that can prevent expensive faults, optimized asset utilization, reduced operating expenses, and remote commissioning. A virtual model that encompasses the entire set of coordinates of a full-order model for a structure accurately and consistently represents the deformed shape, but it requires solving large matrices. Consequently, simulation of virtual models with the entire set of principal coordinates is computationally expensive and has a prolonged runtime, preventing real-time synchronization with the activities of their physical counterparts operating in an industrial environment. This article proposes dynamic reduction-based virtual models as a solution to these challenges. Dynamic reduction techniques came into existence from 1960 onwards and are referred to as \"component modal synthesis\" in the field of structural dynamics [52; 53], and \"substructuring\" in statics [54], with the original objectives of (i) minimizing the computational cost of analysis of structures with a large number of degrees of freedom and (ii) analyzing constituent substructures of a structure independently in such a manner that it would be feasible to amalgamate the results. Nonetheless, dynamic model reduction schemes have the potential to make a substantial contribution to the development of physics-based virtual models. Substructure coupling approaches may be primarily categorized based on the types of modes employed for modeling the reduced finite element representation of a component as  
1. Fixed-Interface methods [55; 56; 57];
2. Free-Interface methods [58; 59; 60; 61];
3. Loaded-Interface [62];
4. Hybrid methods [63; 64].