Constructing a holistic digital twin of a system composed of multiple physical domains is crucial for various tasks. In particular, when the simulation is extended with faults, it becomes a very important resource to achieve robust functional safety analysis. This article proposes a new methodology to build non-electrical fault models for the thermal domain. Such thermal faults are defined through an electrical circuit representing the thermal behavior of the system, known as the Cauer network, based on the physical analogies between the two domains. Including this thermal representation in a multi-domain system allows to simulate the interconnections between different physical domains, thus achieving a more realistic system behavior and evaluating the mutual impact of different domains (e.g., mechanical, electrical and thermal). The entire methodology is applied to a complex case of study implemented by using Verilog-AMS as a proof of concept.