## 1 Introduction  
Experiments are expensive and challenging with increasing complexity. As an alternative research method, numerical simulations have attracted much attention in the past twenty years [1]. As an alternative method of computational fluid dynamics (CFD), the lattice Boltzmann method (LBM) has received remarkable attention since its origin [2, 3]. After the proposal of the Bhatnagar-Gross-Krook (BGK) collision operator, the efficiency and flexibility of the LBM have been enhanced significantly. Consequently, the BGK-based LBM has been successfully applied to various fluid dynamics problems, such as multiphase flow [4, 5, 6, 7], complex flow in porous media [8], and non-Newtonian flow [9, 10]. To address the numerical instability in single-relaxation-time LBM due to low viscosity, entropic [11, 12], two-relaxation-time (TRT), and multi-relaxation-time (MRT) LBM methods have been developed [4]. The major advantages of the LBM are its simplicity, easy implementation, explicit calculations, and intrinsic parallel nature. However, the uniform mesh used in the standard LBM limits its applications in engineering [13]. Many efforts have been made to remove this drawback, including differential type and interpolation type LBM. Grid refinement is another way to retain the simplicity of the LBM and enhance computational efficiency. The mesh is locally refined where high resolution is needed. The multi-block technique proposed by [PERSON] et al. [14] divides the fluid field into several blocks with different mesh sizes. As the mesh is initialized before the computation in this method, the mesh refinement is not involved with the fluid field. [PERSON] and [PERSON] proposed a solution-adaptive refinement scheme, where the refinement procedure is decided by the fluid field [13]. By using the refinement technique, the computational efficiency can be highly enhanced. Fluid-structure interaction (FSI) problems extensively exist in engineering. Thanks to the representation of the forcing term introduced by [PERSON] et al. [15], the involved external or internal force can be considered approximately, which also enables the successful combination of the LBM and the immersed boundary method (IBM) (know as IB-LBM) [16; 17; 18]. The IB-LBM proved to be an efficient method for FSI and has been applied to various problems successfully [19; 20; 21; 22; 23; 24; 25; 26]. Non-Newtonian fluid is extensively involved in many industrial substances, i.e., blood flow, polymer, and melt solutions. The high viscosity levels in conjunction with the complexity in the behaviour of the stress terms of momentum equations for non-Newtonian fluids generally give rise to the limitation of fluid processing under laminar flow conditions [27; 28]. Due to the potential of rheological characteristics (shear-thinning, shear-thickening, etc.) in the applications of engineering, remarkable efforts have been made to develop efficient numerical methods to handle non-Newtonian flows and heat transfer over the last few decades [9; 27; 29]. As the LBM has proven to be an efficient method for CFD [3], it is thought to be a good idea to employ the LBM to handle non-Newtonian flow and heat transfer. In terms of the simulation of thermal fluid systems, most of the previous thermal LBM models adopt the multi-speed approach [30; 31; 32], in which additional particle speeds are needed to obtain the energy equation at the macroscopic level [33]. The additional particle speeds decrease the computational efficiency; the stability is also limited. In order to keep the simplicity property of the LBM, [PERSON] et al. proposed a simplified thermal LBM, neglecting the compression work done by the pressure and viscous heat dissipation [34]. In this study, an alternative method to solve the heat transfer in non-Newtonian fluid flow is presented, conserving the simplicity and efficiency of the LBM.