Three‐dimensional immersed finite element methods for electric field simulation in composite materials

Abstract

SUMMARYThis paper presents two immersed finite element (IFE) methods for solving the elliptic interface problem arising from electric field simulation in composite materials. The meshes used in these IFE methods can be independent of the interface geometry and position; therefore, if desired, a structured mesh such as a Cartesian mesh can be used in an IFE method to simulate 3-D electric field in a domain with non-trivial interfaces separating different materials. Numerical examples are provided to demonstrate that the accuracies of these IFE methods are comparable to the standard linear finite element method with unstructured body-fit mesh.

A large jump asymptotic framework for solving elliptic and parabolic equations with interfaces and strong coefficient discontinuities

“…(11)- (13) are to be solved in succession, on region Ω − , then Ω + and then back to Ω − . Problem (12) uses Dirichlet data extracted from u − 0 on Γ and problem (13) uses Neumann data extracted from u + 0 on Γ .…”

“…This type of problem is relevant in many applications including bubbly flow [24], oil reservoir simulation [20], groundwater flow [3], electromagnetics [5], plasmas [13], biomaterials [23], etc., where coefficients may have large to very large discontinuities across an interface between regions with different material properties. In many of these problems the quantity β(x)∇u, the flux field of u, is physically important and in fact Eqs.…”

A Higher Order Immersed Discontinuous Galerkin Finite Element Method for the Acoustic Interface Problem

“…Piraux and Lombard [17,18] developed explicit interface method to solve acoustic wave propagation in nonhomogeneous media on non-fitted meshes. On the other hand higher order immersed finite element (IFE) methods have been developed for diffusive problems [19][20][21][22][23][24][25]. In this manuscript we present a higher order interface discontinuous Galerkin finite element (IDGFE) method for solving the acoustic problem in nonhomogeneous media on non-fitted meshes.…”

The convergence of the bilinear and linear immersed finite element solutions to interface problems

“…This feature becomes clearly advantageous when the interfaces are moving in a simulation or a physical model requires structured meshes for an interface problem [8][9][10][11]. Since 1998, several types of IFE spaces have been developed, such as 1D linear IFE [12], 2D linear IFE [1,4,13], 3D linear IFE [14], bilinear IFE [2], quadratic IFEs [15], and 1D IFEs of arbitrary order [16]. Error analysis in [3,4,16,17] show that these IFE spaces have the approximation capability similar to that of standard finite element spaces using polynomials of same degree.…”

“…Error analysis in [3,4,16,17] show that these IFE spaces have the approximation capability similar to that of standard finite element spaces using polynomials of same degree. Based on these results, the IFE spaces have been employed in Galerkin methods, finite volume methods, and discontinuous Galerkin methods [1,2,12,[14][15][16][18][19][20][21] to solve interface problems. The usage of IFE methods with structured meshes for solving some engineering problems can be seen in [8][9][10]22].…”