New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method

Zhang et al. Chin. J. Mech. Eng. New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method XuJ‑iu Zhang 0 Yong‑Sheng Zhu 0 Ke Yan 0 YouY‑un Zhang 0 0 Key Laboratory of Education Ministry for Modern Design & Rotor‐Bearing System, Xi'an Jiaotong University , Xi'an 710049 , China Currently, many studies on the local discontinuous Galerkin method focus on the Cartesian grid with low computational efficiency and poor adaptability to complex shapes. A new immersed boundary method is presented, and this method employs the adaptive Cartesian grid to improve the adaptability to complex shapes and the immersed boundary to increase computational efficiency. The new immersed boundary method employs different boundary cells (the physical cell and ghost cell) to impose the boundary condition and the reconstruction algorithm of the ghost cell is the key for this method. The classical model elliptic equation is used to test the method. This method is tested and analyzed from the viewpoints of boundary cell type, error distribution and accuracy. The numerical result shows that the presented method has low error and a good rate of the convergence and works well in complex geometries. The method has good prospect for practical application research of the numerical calculation research. Immersed boundary method; Adaptive Cartesian grid; Local discontinuous Galerkin method; Reconstruction; Heat transfer equation 1 Introduction The immersed boundary method (IBM) is an effective method for studying complex boundary. Imposing a boundary condition is not straightforward. To solve this problem, different IBMs have been proposed in literature. Generally, IBMs can be classified into two categories, i.e., the continuous force approach and discrete force approach [1]. The continuous force approach is not suitable for computing the high Reynolds number flows. Therefore, many researchers focus on the discrete force approach. Fadlun et al. [2] implemented the discrete-time forcing approach on a standard marker- and -cell (MAC) staggered grid. Tseng et al. [3] extended the idea of Verzicco et al. [4] and proposed the ghost-cell IBM (GCIBM) for simulating turbulent flows in complex geometries. Mittal et al. [5] used a sharp interface IBM to simulate incompressible viscous flows past three-dimensional immersed bodies. Using the ghost point treatment as a starting point, Gao et  al. [6] improved the method of Tseng et  al. [3]. The method effectively eliminates numerical instabilities caused by matrix inversion and flexibly. To improve the accuracy at the boundaries, Shinn et al. [7] implemented the immersed boundary method using the ghost cell approach, whereby the incompressible flows are solved on a staggered grid. To control the spurious force oscillations, Lee et  al. [8] proposed a fully-implicit ghost-cell IBM for simulating flows over complex moving bodies on a Cartesian grid. The method is well capable of controlling the generation of spurious force oscillations on the surface of a moving body, thereby producing an accurate and stable solution. To simulate high-Reynolds number compressible viscous flows on adaptive Cartesian grids, Hu et  al. [9] present a new ghost-cell turbulent wall boundary condition. In the frame of adaptive Cartesian grids, a cell-centered, secondorder accurate finite volume solver has been developed for predicting turbulent flow fields. The robustness and accuracy of the methodology have been validated against welldocumented turbulent flow test problems. Now, the IBM method is applied to many fluid dynamic problems such as heat transfer problems [10–12], fluid-solid interaction Let us multiply Eqs.  ( 4 ) and ( 5 ) by arbitrary smooth test function v and r, respectively, integrate them over an arbitrary element Ω, and apply Green’s theorem to write qˆ := {{q}} − C11[[u]] − C12 · [[q]], uˆ := {{u}} + C12 · [[u]]. Γ Γ d d Γ Γ ( 8 ) ( 9 ) ( 10 ) ( 11 ) ( 12 ) ( 13 ) ( 14 ) ( 16 ) ( 17 ) q · rdxdy = ur · nds − u∇ · rdxdy, q · ∇vdxdy = vq · nds + fvdxdy. Then we replace the exact solution(q, u) by its approximation (qN , uN ) in the element space MN × VN, where MN := q ∈ L2(Ω ) : q|k ∈ S(K )d , ∀K ∈ Γ , VN := u ∈ L2(Ω ) : u|k ∈ S(K ), ∀K ∈ Γ , S(K ) : = polynomials of degree at most k in each variables on K . The method involves finding (qN , uN ) ∈ (MN × VN ) such that Ω Ω qN · rdxdy = uˆN r · nKds − uN ∇ · rdxdy, qN · ∇vdxdy = vqˆ N · nKds + fvdxdy. Ω Ω problems [13], complex/moving boundary problems [14], incompressible flows [15], and natural convection problems [16]. The above numerical calculation method adopts the finite difference method, the finite volume method or the finite element method. In this paper, a new IBM to solve the second-order partial equation applied to the local discontinuous Galerkin method (LDG) is presented and we analyzed the causes of error variation for the adaptive Cartesian grid. The LDG metho (...truncated)