## 1 Introduction  
Microwave energy has been used in a variety of applications, including radar, the food industry and other fields [1; 2; 3; 4]. Microwave heating is a new heating method due to its high efficiency, environmental protection and selective heating [5; 6; 7]. However, the nonuniformity of microwave heating limits the application of microwave energy in the food industry [8]. Moving elements, such as turntables and mode stirrers, are usually introduced into the microwave cavities to solve the inhomogeneity of microwave heating [9; 10; 11; 12]. Unfortunately, due to the dynamic movement of the elements in the heating process, the mathematical modeling is complex, so it is still a challenge to accurately compute the microwave heating process in the cavity with moving elements [13]. There is a rotation model designed by [PERSON] et al. [14], and the microwave heating process of liquid rotating with a cylindrical cavity was studied based on [PERSON]'s law. This brings about the fact that this method is challenging for computing a small penetration depth inside of the sample, and the distribution of electromagnetic field in the microwave oven without load cannot be estimated [15; 16; 17]. For instance, [PERSON] et al. [15; 16] proposed a two-dimensional microwave cavity with a laminate mode stirrer, and the finite-element method (FEM) was used to compute the effects of the mode stirrer on the electromagnetic field distribution and heating pattern of the sample inside the cavity. [PERSON] et al. [17] computed the turntable's rotation inside a microwave cavity by rotating each angular degree of \(\pi/12\) using FEM. However, this method does not consider that the sample's dielectric properties will vary with the temperature during the heating process. Subsequently, [PERSON] et al. [18] updated the sample's dielectric properties after each temperature computation step to overcome this defect. This also means the mesh needs to be redivided with each rotation angle, which leads to the mesh not being able to be matched well. Recently, [PERSON] et al. [19] proposed a more accurate method based on the implicit function and level set methods to compute the microwave heating process of a heating cavity with a turntable. According to this method, the moving region's properties are a function of space coordinates and time, and the change in parameters represents the elements' movement. Compared with the traditional method, this method reduces the grid density by 71% and the computing time by 81%. A method based on transformation optics was proposed by [PERSON] et al. [20], which utilized both a 2D and 3D model with turntables that were computed by FEM. In this method, the moving region is surrounded by a layer of a time-varying anisotropic medium, which can control the electric field. The anisotropic region's variations represent the rotating motion of elements. However, when computing the 3D rotating turntable model, influenced by the mesh density, they divided a circle into four discrete positions, which means that the mesh needs to be redivided after each \(\pi/4\) rotation, and deviation might occur. This paper proposed a method based on transformation optics, implicit function and level set methods to compute the continuous microwave heating process with spiral motion. In the proposed method, the properties of the lifting region vary with the movement equations of space and time. Meanwhile, the rotating region is surrounded by two cylindrical regions of a time-varying anisotropic medium. Compared with one region [20], the two-region transformation medium can solve the problem of continuous rotation computation well and avoid the need for re-meshing with several discrete positions. Namely, this method completely avoids the division of the mesh. Moreover, by using the coordinate transformation method, the electromagnetic field and temperature field can be coupled. In Section 2, a 3D model with a turntable and a support rod is presented, and the theoretical basis is also studied.