###### Abstract  
The computation of the electric field in composite materials at the microscopic scale results in an immense number of degrees of freedom. Consequently, this often leads to prohibitively long computation times and extensive memory requirements, making direct computation impractical. In this study, one employs an innovative approach that integrates domain decomposition and model order reduction to retain local information while significantly reducing computation time. Domain decomposition allows for the division of the computational domain into smaller, more manageable subdomains, enabling parallel processing and reducing the overall complexity of the problem. Model order reduction further enhances this by approximating the solution in a lower-dimensional subspace, thereby minimising the number of unknown variables that need to be computed. Comparative analysis between the results obtained from the reduced model and those from direct resolution demonstrates that our method not only reduces computation time but also maintains accuracy. The new method effectively captures the essential characteristics of the electric field distribution in composite materials, ensuring that the local phenomena are accurately represented. This study provides a contribution to the field of computational electromagnetics by presenting a feasible solution to the challenges posed by the high computational demands of simulating composite materials at the microscopic scale. The proposed methodology offers a promising direction for future research and practical applications, enabling more efficient and accurate simulations of complex material systems. model order reduction (MOR); domain decomposition (DD); multi-scale simulation 20242024146013390/app1446013390/20242024146013390/20242024146013390/2024202414601390/2024202414 to the future mechanical solicitations of the pile up. Composites are highly anisotropic and heterogeneous, especially from an electrical point of view. Flaws may occur within these materials with complex structures and are difficult to detect, including voids, delamination, fibre misalignments, and matrix cracks [3,4]. These flaws can significantly impact the material's performance and integrity. For example, voids can affect the level of strength and elasticity; delamination can lead to a drastic reduction in load-bearing capacity; fibre misalignments can weaken the composite structure; and matrix cracks can initiate and propagate under stress, leading to a gradual deterioration of mechanical properties. In order to carry out the inspection on composites and guarantee their integrity, Non-Destructive Testing (NDT) is commonly used. Among them, induction thermography (Figure 1) is a testing method where the eddy currents in the material to inspect, induced by an external variable magnetic source, are disturbed by the presence of an internal flaw. The resulting heat flow is then also disturbed, which causes an abnormal temperature variation at the surface of the material. The latter can be detected by an infrared camera and correlated to the internal defect [5, 6, 7]. Numerical simulation of such a material is useful to understand and predict its behaviour but can be complicated. To deal with composite materials, there are three scales to study (Figure 2): the microscopic scale (fibre scale, micrometres), the mesoscopic scale (layers scale, hundreds of micrometres), and the macroscopic scale (structure scale, several metres). Due to the huge ratio between microscopic and macroscopic scales, direct simulation at both scales at the same time is impossible because it would generate numerous unknowns, resulting in a long computation time and high computational resources. In previous work, the equivalent circuit approach is used to model the electrical behaviour of CFRP at a microscopic scale. The equivalent circuit approach has the advantage of eliminating the 3D mesh of the fibres, in particular by considering electrical resistance to represent  
Figure 1: General operating principles of induction thermography [8].