Information Teaching Model of Preschool Art Education in Colleges and Universities Based on Finite Element Higher-Order Fractional Differential Equation

Two-phase flow problems for immiscible and incompressible fluids. Let’s call it a phase function, which is a function of space and time. Let us usually take two kinds of constants, such as 1 and −1 to represent two different kinds of fluid, and the part between 1 and −1 represents the interface of two kinds of immiscible fluid. By observing the change of phase function, the phase-field model can indirectly reflect the change of the interface of two kinds of fluids so as to avoid directly tracking the location of the interface in the process of microstructure evolution [6]. Therefore, the phase-field model has a natural advantage in simulating the situation where the width of the interface is equivalent to the characteristic length of the system. Make F(Φ)=14η2(φ2−φ)2F(\Phi)=\frac{1}{4 \eta^{2}}\left(\varphi^{2}-\varphi\right)^{2} to Ginzburg-Landau double potential well and define the following mixing energy: (1) E(φ)=∫Ω(12|∇φ|2+F(Φ))dx E(\varphi)=\int_{\Omega}\left(\frac{1}{2}|\nabla \varphi|^{2}+F(\Phi)\right) d x

The potential function F (ϕ) has two minima, and they represent two pure phases. Let V represent the Hilbert space, whose inner product is defined as (·, ·)v. Then the phase function ϕ satisfies the following gradient flow: (2) φt=γ∇vE(φ) \varphi_{t}=\gamma \nabla v \mathrm{E}(\varphi)

where γ represents permeability parameter and ∇vE(ϕ) represents the gradient of energy functional E(ϕ). The gradient of E(ϕ) at point ϕ ∈ V is: (3) (∇vE(φ),ψ)v=ddsE(φ+sψ)|s=0 (\nabla v \mathrm{E}(\varphi), \psi) v=\left.\frac{d}{d s} \mathrm{E}(\varphi+s \psi)\right|_{s}=0

According to Formula (1), if V is selected as L² (Ω), then we can get: (4) ddsE(φ+sψ)|s=0=[(∇φ,∇ψ)L2+(f(φ),ψ)L2]=(−∇φ+f(φ),ψ)L2 \left.\frac{d}{d s} \mathrm{E}(\varphi+s \psi)\right|_{s=0}=\left[(\nabla \varphi, \nabla \psi)_{L^{2}}+(f(\varphi), \psi)_{L^{2}}\right]=(-\nabla \varphi+f(\varphi), \psi)_{L^{2}}

where f (ϕ) = F^′ (ϕ). Therefore, we can obtain the following classical Allen-Cahn equation of integer order (ignoring the boundary conditions): (5) φt+γ(−∇φ+f(φ))=0,(x,t)∈Ω×(0,T) \varphi_{t}+\gamma(-\nabla \varphi+f(\varphi))=0,(x, t) \in \Omega \times(0, T)

For the Allen-Cahn equation, we have: (6) ddt∫Ωφdx=∫Ωφtdx=−γ∫Ω(−∇φ+f(φ))dx \frac{d}{d t} \int_{\Omega} \varphi d x=\int_{\Omega} \varphi_{t} d x=-\gamma \int_{\Omega}(-\nabla \varphi+f(\varphi)) d x

In general, we cannot guarantee that the right end of Equation (6) is always 0, so the Allen-Cahn equation generally does not satisfy the conservation of mass.

The Cahn-Hilliard equation was originally used to describe the time evolution of a conserved field that is a continuous, spatially sufficiently differentiable function. If we pick V as H⁻¹, then the classical Cahn-Hilliard equation can be obtained as follows: (7) φt=γΔ(f(φ)−Δφ) \varphi_{t}=\gamma \Delta(f(\varphi)-\Delta \varphi)

In general, periodic boundary conditions or the following non-flux boundary conditions are used for closure (7): (8) n⋅∇φ=0.n⋅∇(f(φ)−Δφ)=0 n \cdot \nabla \varphi=0 . n \cdot \nabla(f(\varphi)-\Delta \varphi)=0

where n is the unit normal vector. For the Cahn-Hilliard equation, we have: (9) ddt∫Ωφdx=∫Ωφtdx=γ∫ΩΔ(f(φ)−Δφ)dx=0 \frac{d}{d t} \int_{\Omega} \varphi d x=\int_{\Omega} \varphi_{t} d x=\gamma \int_{\Omega} \Delta(f(\varphi)-\Delta \varphi) d x=0

This means that the Cahn-Hilliard equation satisfies the conservation of mass. For both Allen-Cahn and Cahn-Hilliard equations, we have: (10) dE(φ)dt=δE(φ)δφ∂φ∂t=−λ|∇vE(φ)|2≤0 \frac{d \mathrm{E}(\varphi)}{d t}=\frac{\delta \mathrm{E}(\varphi)}{\delta \varphi} \frac{\partial \varphi}{\partial t}=-\lambda|\nabla v \mathrm{E}(\varphi)|^{2} \leq 0

Therefore, both Allen-Cahn and Cahn-Hilliard equations satisfy the law of energy dissipation. However, if we consider the transport process with a large number of pores in the porous media, then the molecular mean free path is equivalent to the size of the characteristic pores, thus leading to a large Knudsen number. The resulting flow behaviour deviates from the traditional macroscopic flow behaviour and makes molecular diffusion play a dominant role in the transport process. The analysis of this phenomenon can be started from the assumption that Einstein and Pearson shared in the original derivation of Fick’s diffusion equation: (i) there is a mean free path and (ii) there is an average waiting time for particle motion. Based on these assumptions, it is rare for a particle to travel long distances in the same direction, and the motion of the particle can be represented by an independent jump that has no spatial correlation. In highly heterogeneous and space-limited porous media, particles may be absorbed by the porous media due to the heterogeneity of the porous media or the dominant interaction between fluid and rock, resulting in a sub-diffusion process which can be described by a fractional partial differential equation of time [7].

In this section, we study, for example, the finite element high-order fractional Allen-Cahn equation: (11) {0CDtαφ=γ(Δφ−f(φ)),(x,t)∈Ω×(0,T),φ(x,0)=φ0(x),x∈Ω,φ(x,t)=0,(x,t)∈∂Ω×[0,T] \left\{\begin{array}{l} { }_{0}^{C} D_{t}^{\alpha} \varphi=\gamma(\Delta \varphi-f(\varphi)),(x, t) \in \Omega \times(0, T), \\ \varphi(x, 0)=\varphi_{0}(x), x \in \Omega, \\ \varphi(x, t)=0,(x, t) \in \partial \Omega \times[0, T] \end{array}\right.

where Ω is a two-dimensional square region, ϕ₀ (x) is the initial condition. 0CDtα{ }_{0}^{C} D_{t}^{\alpha} represents the Caputo fractional derivative of order 0〈α ≤ 1. The Riemann-Liouville fractional derivative and Caputo fractional derivative are related as follows: (12) 0RDt1−α0CDtαφ(x,t)=0RDt1−α0It1−βφt(x,t)=φt(x,t) { }_{0}^{R} D_{t}^{1-\alpha}{ }_{0}^{C} D_{t}^{\alpha} \varphi(x, t)={ }_{0}^{R} D_{t}^{1-\alpha}{ }_{0} I_{t}^{1-\beta} \varphi_{t}(x, t)=\varphi_{t}(x, t)

Applying the fractional Riemann-Liouville derivative of order 1 − α to both sides of equation (11), we can get: (13) ddt∫Ωφdx=∫dφdtdx=γ0RLDt1−α∫(Δφ−f(φ))dx \frac{d}{d t} \int_{\Omega} \varphi d x=\int \frac{d \varphi}{d t} d x=\gamma_{0}^{R L} D_{t}^{1-\alpha} \int(\Delta \varphi-f(\varphi)) d x

It can be seen that, consistent with the Allen-Cahn equation of integer order, the above equation is not always 0 in general, which indicates that the time-fractional Allen-Cahn equation cannot maintain the conservation of mass [8].

Several numerical experiments are conducted to verify the accuracy of the fast finite difference (FFD) scheme for solving the time-fractional Allen-Cahn equation. Meanwhile, we will also test the energy dissipation problem of the time-fractional Allen-Cahn equation numerically.

Example 1. Let the reference solution of Equation (11) be ϕ (x,t) = t² sin (x), other parameters are Ω = [0π], T = 1, γ = 1.0, η = 1.0. In this case, the right-hand side of the equation can be given accordingly (14) f(x,t)=2sin⁡(x)t2−αΓ(3−α)+γsin⁡(x)t2+γη2φ(φ−1)(φ−12) f(x, t)=\frac{2 \sin (x)^{t^{2-\alpha}}}{\Gamma(3-\alpha)}+\gamma \sin (x) t^{2}+\frac{\gamma}{\eta^{2}} \varphi(\varphi-1)\left(\varphi-\frac{1}{2}\right)

Let’s say Δt = 2⁻¹⁶, h = 2⁻¹², to test the spatial and temporal convergence order of the numerical scheme.

In Tables 1 and 2, we give the cortical error of the numerical solution and the corresponding order of convergence (CoV). It can be seen that the finite-difference scheme has the first order precision in time and the second-order precision in space. We compare the performance of the FFD scheme and the time-catch-up method in solving the equation in Table 3.3, wherein FFD we use the fast method of finding the Caputo fractional derivative in Section 3.3.2. In Figure 3.1, fix M = 2" as well. = 0.7, we show the relationship between CPU time and time step for the FFD algorithm and direct finite difference (DFD) scheme to solve the equation. We can clearly see that the FFD algorithm consumes almost linear CPU time as time steps decrease, so it is much more efficient than the DFD method [9].

When M = 2⁶, α = 0.7 and the time and space step size is b, and the initial value of h = 2⁻⁹ is: (15) φ0(x)=12(1+tanh⁡(0.5−|x|2η)) \varphi_{0}(x)=\frac{1}{2}\left(1+\tanh \left(\frac{0.5-|x|}{\sqrt{2 \eta}}\right)\right)

We found that when the permeability parameters γ were 1.0, 1.2, 1.5 and 1.8, the greater the γ, the decay behaviour of the time-graded Allen-Calm equation appears earlier, which is consistent with the case of its integer order.

In addition, we also investigate the effects of interface parameter cutters and volume ratio of two different phases (VOF) on the attenuation behaviour of the time-fractional Allen-Calm equation. In Figure 2, fix γ = 5.0, we show how the energy changes over time for different VOFs (20%, 50% and 70%) and different knives, where. Take 0.5, 0.7, 0.9 and 1.0. In Figure 3.3(d), the Fourier spectrum method is used to verify the other three results obtained by the finite difference method. For different VOFs, it is observed that the time-fractional Allen-Calm equation still exhibits variable attenuation behaviour for the small ding, while for slightly larger η, this behaviour disappears. We have observed an interesting phenomenon that when the fractional order α becomes smaller, the energy dissipation of the time-fractional order Allen-Calm equation becomes faster. In Figure 2, for random initial values and fixed γ = 1.0, when η is 0.09, 0.08, 0.05 and 0.02, we give the time evolution process of energy and mass of one-dimensional time-fractional Allen-Cahn equation. From Figure 2, we find that the smaller the fractional order of α is the energy dissipation process corresponding to the time-fractional Allen-Cahn equation is accelerated, which seems physically unreasonable since the time-fractional derivative describes a sub-diffusion process. The specific reasons need further research [10].

Fig. 1

Fig. 2

To verify the conclusion in Example 1, we next consider the two-dimensional time-fractional Allen-Cahn equation. The data corresponding to the results in Figure 2 are as follows: (16) η=0.05γ=0.1Δt=1×10−4,Nx=Ny=256,Ω=[0,π]2 \eta=0.05 \gamma=0.1 \Delta t=1 \times 10^{-4}, N_{x}=N_{y}=256, \Omega=[0, \pi]^{2}

We take the initial value as an ellipse centred at (π2π2)\left(\frac{\pi}{2} \frac{\pi}{2}\right), and its half-length and half-short axes are α = 0.2π and b = 0.1π. At the same time, we rotate the ellipse so that its principal axis lies on the diagonal y = x. Therefore, this initial value can be expressed as: (17) φ0(x,y)=0.9tanh⁡(1−R(x,y)2η) \varphi_{0}(x, y)=0.9 \tanh \left(\frac{1-R(x, y)}{\sqrt{2 \eta}}\right)

In which (18) R(x,y)=((x−x0)cos⁡θ+(y−y0)sin⁡θ)2a2 R(x, y)=\frac{\left(\left(x-x_{0}\right) \cos \theta+\left(y-y_{0}\right) \sin \theta\right)^{2}}{a^{2}}

In Figure 3, with fixed γ = 0.1 and η = 0.05, we have drawn the outline of the phase function and the energy changes when α is 0.5, 0.7, 0.9 and 1.0. Meanwhile, let γ = 0.1 and η = 0.5. We can clearly see that the fractional order can affect the interface width of the equation and the speed of energy dissipation, and the smaller the order a is, the faster the energy decay is, which is consistent with the one-dimensional result.

Fig. 3

Here, 0CDtα{ }_{0}^{C} D_{t}^{\alpha} is the fractional derivative of Caputo of order 0〈α ≤ 1 defined in the equation. Unlike the Allen-Calm equation, the Cahn-Hilliard equation contains the fourth derivative term of space, and so it is relatively tough to solve. This indicates that the time-fractional Cahn-Hilliard equation also satisfies the conservation of mass, which is consistent with the result of its integer order form.