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An adaptive edge finite element method for the Maxwell's equations in metamaterials

  • Received: 01 February 2020 Revised: 01 April 2020
  • 78M10, 65N30

  • In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. A-posteriori error estimators based on the recovery type and residual type are proposed, respectively. Based on our a-posteriori error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed a-posteriori error estimations for the adaptive method.

    Citation: Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials[J]. Electronic Research Archive, 2020, 28(2): 961-976. doi: 10.3934/era.2020051

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  • In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. A-posteriori error estimators based on the recovery type and residual type are proposed, respectively. Based on our a-posteriori error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed a-posteriori error estimations for the adaptive method.



    Free-surface flow over a submerged obstacles is one of the well-known classical problem in fluid mechanics. Many researchers have investigated free surface flows over an obstacle for different bottoms topography, for examples, Forbes and Schwartz [13] used the boundary integral method to find fully non-linear solutions of subcritical and supercritical flows over a semi-circular obstacle. Supercritical and critical flows over a submerged triangular obstacle were investigated by Dias and Vanden-Broeck [11]. They employed a series truncation methods to calculate the solutions. Abd-el-Malek and Hanna [1] studied flow over a triangular obstacle by using the Hilbert method with gravity effect. When the fluid is subjected to the interaction of gravity and surface tension, in this case, the problem is generally difficult to solve. Forbes [12] was among the first to propose numerical solutions of non-linear flows over a semi-circular obstruction under the effect of gravity and surface tension. Later, many authors have studied this problem, for example, Grandison [14], Vanden-Broeck [24]. In the case of flows over two obstacles, Pratt [23] investigated this problem experimentally and theoretically using weakly non-linear analysis. Later, Belward [3] computed numerical solutions of a critical flow for which the hydraulic fall occurred at the leftmost obstacle with downstream supercritical flow. Recently, Binder, Vanden-Broeck and Dias [9] showed that there exist two types of solution in subcritical flow regime, and one type in supercritical flow regime. This paper is concerned with the numerical calculations of flow of finite depth over a successive obstacles. The purpose of this work is to examine further flows with many obstacles in order to classify the possible solutions. The effects of surface tension and gravity are included in the boundary conditions, the problem is solved numerically by using the boundary integral equation methods. These methods are based on a reformulation of the problem as a system of non-linear integro-differential equations for the unknown quantities on the free surface. These equations are then discretized and the resultant non-linear algebraic equations is solved by iteration. Such boundary integral equation methods have been used extensively by many researchers [3,4,6,7,15,17] and others. It is assumed that there is uniform flow far upstream where the flow approaches a uniform stream with constant velocity U and depth H (see Figure 1). The problem is characterized by the two parameters the Froude number Fr defined by

    Fr=UgH (1.1)

    and the inverse Weber number δ where

    δ=TρU2H (1.2)

    Here T is the surface tension, g is the gravity and ρ is the fluid density. When Fr<1, the flow is called subcritical and for Fr>1 the flow far upstream is called supercritical. In this work, we calculate waveless solutions for both supercritical and subcritical flows by introducing the effects of surface tension.

    Figure 1.  Sketch of the flows over a successive obstacles in the physical plane z=x+iy.

    Formulation of the problem and numerical procedure are given in section 2 and section 3 respectively. In section 4 we discuss the numerical results of free surface flows over a successive triangular obstacles with different angles γi, i=1,...,m2 and for various values of the two parameters Fr and δ. Solution diagrams for all flow regimes are presented.

    We consider steady two-dimensional potential free surface flows past a submerged obstacles at the bottom of a channel (see Figure 1). The flow is assumed to be inviscid and incompressible. Fluid domain is bounded below by a horizontal rigid wall A0Am and the successive obstructions forming the angles γi, i=1,...,m2 with the horizontal, where 0<|γi|<π2, and above by the free surface EF. Let us introduce Cartesian coordinates with the x axis along the bottom and the y axis directed vertically upwards, gravity g is acting in the negative ydirection. Let's introduce the velocity potential ϕ(x,y) and the stream function ψ(x,y) by defining the complex potential function f as

    f(x,y)=ϕ(x,y)+iψ(x,y) (2.1)

    The complex velocity w can be written as

    w=dfdz=uiv (2.2)

    Here u and v are velocity components in the x and y directions, and z=x+iy. For convenience, we define dimensionless variables by taking H as the reference length and U as the reference velocity. Without loss of generality, we choose ψ=0 on the free surface EF. By the choice of our dimensionless variables, we have ψ=1 on the bottom A0Am and ϕ=0 at the point Am2 (see Figure 2).

    Figure 2.  Sketch of the flow in the potential f-plane f=ϕ+iψ.

    The problem is formulated in terms of the velocity potential ϕ(x,y). This function satisfies Laplace's equation

    Δϕ=0    in the fluid domain

    The Bernoulli's equation on the free surface EF can be written

    12(u2+v2)+δK+1Fr2(y1)=12. (2.3)

    Here K is curvature of the free surface, Fr and δ are defined in (1.1) and (1.2) respectively.

    The kinematic boundary conditions in f-plane are given by

    {v=0 on ψ=1 and <ϕ<ϕA1 and ϕAm1<ϕ<+v=utan|γi| on ψ=1 and ϕAi<ϕ<ϕAi+1,i=1,...,m2   (2.4)

    Now we reformulate the problem as an integral equation. We define the function τiθ by

    w=uiv=eτiθ (2.5)

    and we map the flow domain onto the upper half of the ζplane by the transformation

    ζ=α+iβ=eπf=eπϕ(cosπψisinπψ). (2.6)

    The flow in the ζplane is shown in Figure 3. The curvature K of a streamline, in terms of θ, is given by

    K=eτ|θϕ|. (2.7)
    Figure 3.  The upper half ζplane ζ=α+iβ.

    Substituting (2.7) into (2.3), Bernoulli's equation becomes

    12e2τδeτ|θϕ|+1Fr2(y1)=12 on  EF. (2.8)

    We apply the Cauchy's integral formula to the function τiθ in the complex ζ-plane with a contour consisting of the αaxis and the semicircle of arbitrary large radius R in the upper half plane. After taking the real part and R+, we obtain

    τ(α0)=1π+θ(α)αα0dα. (2.9)

    Where τ(α0) and θ(α) denote the value of τ and θ on the free surface. The integral in (2.9) is a Cauchy principal value type.

    The kinematic boundary conditions (2.4) in ζ-plane become

    {θ=0for <α<αA1and αAm1<α<αAmθ=γifor αAi<α<αAi+1,i=1,...,m2θ= unknown  0<α<+   (2.10)

    By using (2.10), Eq (2.9) becomes :

    τ(α0)=1πi=1i=m2γilog|αAi+1α0αAiα0|1π0+θ(α)αα0dα. (2.11)

    Rewriting this equation in terms of ϕ by substituting α=eπϕ, α0=eπϕ0,

    this gives

    τ(ϕ0)=1πi=1i=m2γilog|eπϕAi+1+eπϕ0eπϕAi+eπϕ0|++θ(ϕ)eπϕeπϕeπϕ0dϕ. (2.12)

    Here τ(ϕ0)=τ(eπϕ0) and θ(ϕ)=θ(eπϕ). The Eq (2.8) is now rewritten in terms of τ and θ as

    12e2τ(ϕ0)δeτ(ϕ0)|θ(ϕ)ϕ|+1Fr2(y1)=12 on  EF. (2.13)

    Integrating the identity

    d(x+iy)df=w1. (2.14)

    We obtain the following parametric representation of the free surface EF

    x(ϕ)=ϕeτ(ϕ0)cosθ(ϕ0)dϕ0 for  <ϕ<+ (2.15)
    y(ϕ)=1+ϕeτ(ϕ0)sinθ(ϕ0)dϕ0 for <ϕ<+ (2.16)

    By substituting (2.16) into (2.13), an integro-differential equation is created and it is solved numerically.

    In this section, we describe numerical approach for the nonlinear problem derived in previous section. This numerical procedure has been successfully used by B. J. Binder [9], P. Guayjarernpanishk [15] and others for solving nonlinear integral equations. Firstly, the free surface must be truncated at ϕ1 and ϕN for the corresponding far upstream x, and far downstream x+, respectively. The truncated free surface is then discretized into N equally segments with

    ϕI=[(N1)2+(I1)]Δ,I=1,...,N,<ϕ<+ (3.1)

    and the unknown variables on the free surface are

    θI=θ(ϕI),I=1,...,N.

    Here Δ>0 is the mesh spacing. We evaluate the values τ (ϕ0) at the midpoints

    ϕM=ϕI+1+ϕI2,I=1,...,N1 (3.2)

    by applying the trapezoidal rule to the integral in (2.12) with summations over ϕI such that ϕ0 is the midpoints. We evaluate yI=y(ϕI) and xI=x(ϕI) by applying the Euler's method and by using (2.14). This yields

    {y1=1yI+1=yI+ΔeτMsinθM,I=1,...,N1

    and

    {x1=0xI+1=xI+ΔeτMcosθM,I=1,...,N1.

    Here θM=θI+1+θI2. We now satisfy (2.13) at the midpoints (3.2). This yields N non-linear algebraic equations for the N unknowns θI,I=1,...,N. The derivative, θϕ, at the mesh points (3.1), is approximated by a finite difference, whereby

    θϕ=θI+1θIΔ,I=1,...,N1.

    The system of N equations with N unknowns is solved by Newton's method.

    The numerical procedure of section 3 is used to compute solutions for free surface flows over a successive triangular obstacles. For simplicity, we assume that the triangles are isosceles forming the angles γi,i=1,...,m2 with the horizontal (see Figure 1). Also, we choose ϕAm2=0. Most of the calculations in this paper are obtained with N=401 and Δ=0.15. For a given values of ϕ at the points Ai,i=1,..,m1, we compute waveless solutions for various values of the angles γi, Froude number Fr and the inverse Weber number δ. We denote by L=|ϕAi+1ϕAi|,i=1,...,m2 which represents the length of the sides of the triangles. The problem is essentially characterized by four parameters; The Froude number Fr, the inverse of Weber δ, |γi| and L. In supercritical (Fr>1) or subcritical flow (Fr<1) and a fixed values of L=2.5 and |γi|=π4, the effect of surface tension on the shape of free surface, is shown in Figures 4 and 5. It should be noted that the free surface elevation increases when the inverse Weber number δ decreases. The Figures 6 and 7 show the effect of the Froude number Fr for fixed values of δ=0.5, |γi|=π6 and L=3. It can be seen that the elevation of the free surfaces increases as Froude number Fr increases. When the surface tension is neglected (δ=0) and Fr (without gravity) and for an arbitrary values of δ and γi; the problem has an exact solution that can be computed via the streamline method due to Kirchhoff [2], in this case, the elevation of free surfaces reaches its maximum. The effect of varying the length L, whilst γi, δ and Fr are fixed is shown in Figure 8. Figure 9 illustrates the effect of varying the angles γi where δ=0.7 and Fr=2 are fixed.

    Figure 4.  The shapes of free surface for Fr=,|γi|=π4,L=2.5 and various values of the inverse Weber number δ.
    Figure 5.  The shapes of free surface for Fr=0.8,L=2.5,|γi|=π4 and various values of the inverse Weber number δ.
    Figure 6.  The shapes of free surface for δ=0.5,L=3,|γi|=π6 and various values of the Froude number Fr.
    Figure 7.  The shapes of free surface for δ=0.5,L=3,|γi|=π6 and various values of the Froude number Fr.
    Figure 8.  The shapes of free surface for δ=1,Fr=1.5,ϕA6=0,|γi|=π4 and various values of the length L (L=2,L=2.5,L=3) (from top to bottom).
    Figure 9.  The shapes of free surface for δ=0.7,Fr=2,L=3 and various values of the angles γi where |γi|=π100,π18,6π45 (from top to bottom).

    In this paper, the problem of irrotational, two-dimensional free-surface flow over a successive obstacles has been presented. The fluid is assumed to be incompressible and inviscid. The fully non-linear problem is formulated by using a boundary integral equation technique. The numerical solutions are obtained, in the presence of surface tension and gravity. For supercritical flow (Fr>1), there is a three-parameters family of solutions depending on the height of obstacle, the inverse Weber number δ and the Froude number Fr which is similar to the case of subcritical flow (Fr<1). We have seen the effect of surface tension on free surface profiles for supercritical and subcritical flows. It noted that when the inverse Weber number decreases or the Froude number increases, the free surface elevation increases. The same observation is made when γi or L decreases the elevation of the free surface decreases and vice versa. The maximum free-surface elevation is obtained in the absence of the surface tension and gravity, in this case, the exact solution can be found via the hodograph transform due to Kirchhoff [2].

    The authors declare no conflicts of interest in this paper.



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