h | ‖y−yh‖ | ‖p−ph‖ | CPU time (s) |
14 | 0.1095 | 0.0856 | 0.7031 |
116 | 0.0079 | 0.0045 | 8.8702 |
164 | 0.0005 | 0.0002 | 2253.6396 |
Aiming at the problem that the extraction effect of abnormal vibration characteristics of current engineering electric drive construction machinery is poor, an automatic detection method of abnormal vibration of engineering electric drive construction machinery is proposed. Firstly, the abnormal data of mechanical abnormal vibration are collected and identified, and based on the identification results, the dynamic characteristic model of engineering electric drive construction machinery is constructed. The empirical mode decomposition and Hilbert spectrum are used to decompose the abnormal vibration of machinery, calculate the response amplitude and time lag value generated by the operation of the engineering electric drive construction machinery to simplify the diagnosis steps of the abnormal vibration of the engineering electric drive construction machinery and realize the positioning and detection of the transverse and torsional vibration characteristics. Finally, through experiments, it was confirmed that the automatic detection method of the abnormal vibration of the engineering electric drive construction machinery has high accuracy, which can better ensure the healthy operation of mechanical equipment. This endeavor aims to establish scientific methodologies and standards for fault detection techniques in construction machinery, ultimately forging a versatile solution better suited for detecting and resolving issues across various categories of construction equipment.
Citation: Jian Yuan, Hao Liu, Yang Zhang. Automatic detection method of abnormal vibration of engineering electric drive construction machinery[J]. Electronic Research Archive, 2023, 31(10): 6327-6346. doi: 10.3934/era.2023320
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Aiming at the problem that the extraction effect of abnormal vibration characteristics of current engineering electric drive construction machinery is poor, an automatic detection method of abnormal vibration of engineering electric drive construction machinery is proposed. Firstly, the abnormal data of mechanical abnormal vibration are collected and identified, and based on the identification results, the dynamic characteristic model of engineering electric drive construction machinery is constructed. The empirical mode decomposition and Hilbert spectrum are used to decompose the abnormal vibration of machinery, calculate the response amplitude and time lag value generated by the operation of the engineering electric drive construction machinery to simplify the diagnosis steps of the abnormal vibration of the engineering electric drive construction machinery and realize the positioning and detection of the transverse and torsional vibration characteristics. Finally, through experiments, it was confirmed that the automatic detection method of the abnormal vibration of the engineering electric drive construction machinery has high accuracy, which can better ensure the healthy operation of mechanical equipment. This endeavor aims to establish scientific methodologies and standards for fault detection techniques in construction machinery, ultimately forging a versatile solution better suited for detecting and resolving issues across various categories of construction equipment.
It is well known that optimal control problems play a very important role in the fields of science and engineering. In the operation of physical and economic processes, optimal control problems have a variety of applications. Therefore, highly effective numerical methods are key to the successful application of the optimal control problem in practice. The finite element method is an important method for solving optimal control problems and has been extensively studied in the literature. Many researchers have made various contributions on this topic. A systematic introduction to the finite element method for partial differential equations (PDEs) and optimal control problems can be found in [1,2]. For example, a priori error estimates of finite element approximation were established for the optimal control problems governed by linear elliptic and parabolic state equations, see [3,4]. Using adaptive finite element method to obtain posterior error estimation; see [5,6]. Furthermore, some superconvergence results have been established by applying recovery techniques, see [7,8].
The two-grid method based on two finite element spaces on one coarse and one fine grid was first proposed by Xu [9,10,11]. It is combined with other numerical methods to solve many partial differential equations, e.g., nonlinear elliptic problems [12], nonlinear parabolic equations [13], eigenvalue problems [14,15,16] and fractional differential equations [17].
Many real applications, such as heat conduction control of storage materials, population dynamics control and wave control problems governed by integro-differential equations, need to consider optimal control problems governed by elliptic integral equations and parabolic integro-differential equations. More and more experts and scholars began to pay attention to the numerical simulation of these optimal control problems. In [18], the authors analyzed the finite element method for optimal control problems governed by integral equations and integro-differential equations. In [19], the authors considered the error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro-differential equations. As far as we know, there is no research on a two-grid finite element method for parabolic integro-differential control problems in the existing literature.
In this paper, we design a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. It is shown that when the coarse and fine mesh sizes satisfy h=H2, the two-grid method achieves the same convergence property as the finite element method. We are interested in the following optimal control problems:
minu∈K⊂U{12∫T0‖y−yd‖2+‖u‖2dt}, | (1.1) |
yt−div(A∇y)+∫t0div(B(t,s)∇y(s))ds=f+u, ∀ x∈Ω, t∈J, | (1.2) |
y(x,t)=0, ∀ x∈∂Ω, t∈J, | (1.3) |
y(x,0)=y0(x), ∀ x∈Ω, | (1.4) |
where Ω is a bounded domain in R2 and J=(0,T]. Let K be a closed convex set in U=L2(J;L2(Ω)), f∈L2(J;L2(Ω)), yd∈H1(J;L2(Ω)) and y0∈H1(Ω). K is a set defined by
K={u∈U:∫Ωu(x,t)dx≥0}; | (1.5) |
A=A(x)=(aij(x)) is a symmetric matrix function with aij(x)∈W1,∞(Ω), which satisfies the ellipticity condition
a∗|ξ|2≤2∑i,j=1ai,j(x)ξiξj≤a∗|ξ|2, ∀(ξ,x)∈R2×ˉΩ, 0<a∗<a∗. |
Moreover, B(t,s)=B(x,t,s) is also a 2×2 matrix; assume that there exists a positive constant M such that
‖B(t,s)‖0,∞+‖Bt(t,s)‖0,∞≤M. |
In this paper, we adopt the standard notation Wm,p(Ω) for Sobolev spaces on Ω with a norm ‖⋅‖m,p given by ‖v‖pm,p=∑|α|≤m‖Dαv‖pLp(Ω), as well as a semi-norm |⋅|m,p given by |v|pm,p=∑|α|=m‖Dαv‖pLp(Ω). We set Wm,p0(Ω)={v∈Wm,p(Ω):v|∂Ω=0}. For p=2, we denote Hm(Ω)=Wm,2(Ω), Hm0(Ω)=Wm,20(Ω), and ‖⋅‖m=‖⋅‖m,2, ‖⋅‖=‖⋅‖0,2.
We denote by Ls(J;Wm,p(Ω)) the Banach space of all Ls integrable functions from J into Wm,p(Ω) with the norm ‖v‖Ls(J;Wm,p(Ω))=(∫T0||v||sWm,p(Ω)dt)1s for s∈[1,∞) and the standard modification for s=∞. For simplicity of presentation, we denote ‖v‖Ls(J;Wm,p(Ω)) by ‖v‖Ls(Wm,p). Similarly, one can define the spaces H1(J;Wm,p(Ω)) and Ck(J;Wm,p(Ω)). In addition C denotes a general positive constant independent of h and Δt, where h is the spatial mesh size and Δt is a time step.
The outline of this paper is as follows. In Section 2, we first construct a fully discrete finite element approximation scheme for the optimal control problems (1.1)–(1.4) and give its equivalent optimality conditions. In Section 3, we derive a priori error estimates for all variables, and then analyze the global superconvergence by using the recovery techniques. In Section 4, we present a two-grid scheme and discuss its convergence. In Section 5, we present a numerical example to verify the validity of the two-grid method.
In this section, we shall construct a fully discrete finite element approximation scheme for the control problems (1.1)–(1.4). For sake of simplicity, we take the state space Q=L2(J;V) and V=H10(Ω).
We recast (1.1)–(1.4) in the following weak form: find (y,u)∈Q×K such that
minu∈K⊂U{12∫T0‖y−yd‖2+‖u‖2dt}, | (2.1) |
(yt,v)+(A∇y,∇v)=∫t0(B(t,s)∇y(s),∇v)ds+(f+u,v), ∀ v∈V, t∈J, | (2.2) |
y(x,0)=y0(x), ∀ x∈Ω, | (2.3) |
where (⋅,⋅) is the inner product of L2(Ω).
Since the objective functional is convex, it follows from [2] that the optimal control problems (2.1)–(2.3) have a unique solution (y,u), and that (y,u) is the solution of (2.1)–(2.3) if and only if there is a co-state p∈Q such that (y,p,u) satisfies the following optimality conditions:
(yt,v)+(A∇y,∇v)=∫t0(B(t,s)∇y(s),∇v)ds+(f+u,v), ∀ v∈V, t∈J, | (2.4) |
y(x,0)=y0(x), ∀ x∈Ω, | (2.5) |
−(pt,q)+(A∇p,∇q)=∫Tt(B∗(s,t)∇p(s),∇q)ds+(y−yd,q), ∀ q∈V, t∈J, | (2.6) |
p(x,T)=0, ∀ x∈Ω, | (2.7) |
(u+p,˜u−u)≥0, ∀ ˜u∈K, t∈J. | (2.8) |
As in [20], the inequality (Eq 2.8) can be expressed as
u=max{0,ˉp}−p, | (2.9) |
where ˉp=∫Ωpdx∫Ωdx denotes the integral average on Ω of the function p.
Let Th denote a regular triangulation of the polygonal domain Ω, hτ denote the diameter of τ and h=maxτ∈Thhτ. Let Vh⊂V be defined by the following finite element space:
Vh={vh∈C0(ˉΩ)∩V,vh|τ∈P1(τ), ∀ τ∈Th}. | (2.10) |
And the approximated space of control is given by
Uh:={˜uh∈U:∀ τ∈Th, ˜uh|τ=constant}. | (2.11) |
Set Kh=Uh∩K.
Before the fully discrete finite element scheme is given, we introduce some projection operators. First, we define the Ritz-Volterra projection [21] Rh: V→Vh, which satisfies the following: for any y,p∈V
(A(∇(y−Rhy),∇vh)−∫t0(B(t,s)∇(y−Rhy),∇vh)ds=0, ∀ vh∈Vh, | (2.12) |
‖∂i(y−Rhy)∂ti‖+h‖∇∂i(y−Rhy)∂ti‖≤Ch2i∑m=0‖∂my∂tm‖2, i=0,1. | (2.13) |
(A(∇(p−Rhp),∇vh)−∫Tt(B∗(s,t)∇(p−Rhp),∇vh)ds=0, ∀ vh∈Vh, | (2.14) |
‖∂i(p−Rhp)∂ti‖+h‖∇∂i(p−Rhp)∂ti‖≤Ch2i∑m=0‖∂mp∂tm‖2, i=0,1. | (2.15) |
Next, we define the standard L2-orthogonal projection [22] Qh: L2(Ω)→Uh, which satisfies the following: for any ϕ∈L2(Ω)
(ϕ−Qhϕ,wh)=0, ∀ wh∈Uh, | (2.16) |
‖ϕ−Qhϕ‖−s,2≤Ch1+s‖u‖1, s=0,1, ∀ ϕ∈H1(Ω), | (2.17) |
At last, we define the element average operator [7] πh:L2(Ω)→Uh by
πhψ|τ=∫τψdx∫τdx, ∀ ψ∈L2(Ω), τ∈Th. | (2.18) |
We have the approximation property
‖ψ−πhψ‖−s,r≤Ch1+s‖ψ‖1,r, s=0,1, ∀ ψ∈W1,r(Ω). | (2.19) |
We now consider the fully discrete finite element approximation for the control problem. Let Δt>0, N=T/Δt∈Z and tn=nΔt, n∈Z. Also, let
ψn=ψn(x)=ψ(x,tn),dtψn=ψn−ψn−1Δt,δψn=ψn−ψn−1. |
Like in [23], we define for 1≤s≤∞ and s=∞, the discrete time dependent norms
|||ψ|||Ls(J;Wm,p(Ω)):=(N−l∑n=1−lΔt‖ψn‖sm,p)1s, |||ψ|||L∞(J;Wm,p(Ω)):=max1−l≤n≤N−l‖ψn‖m,p, |
where l=0 for the control variable u and the state variable y, and l=1 for the co-state variable p.
Then the fully discrete approximation scheme is to find (ynh,unh)∈Vh×Kh, n=1,2,⋯,N, such that
minunh∈Kh{12N∑n=1Δt(‖ynh−ynd‖2+‖unh‖2)}, | (2.20) |
(dtynh,vh)+(A∇ynh,∇vh)=(n∑i=1ΔtB(tn,ti−1)∇yih,∇vh)+(fn+unh,vh), ∀ vh∈Vh, | (2.21) |
y0h=Rhy0. | (2.22) |
Again, we can see that the above optimal control problem has a unique solution (ynh,unh), and that (ynh,unh)∈Vh×Kh is the solution of (2.20)–(2.22) if and only if there is a co-state pn−1h∈Vh such that (ynh,pn−1h,unh) satisfies the following optimality conditions:
(dtynh,vh)+(A∇ynh,∇vh)=(n∑i=1ΔtB(tn,ti−1)∇yih,∇vh)+(fn+unh,vh), ∀ vh∈Vh, | (2.23) |
y0h=Rhy0, | (2.24) |
−(dtpnh,qh)+(A∇pn−1h,∇qh)=(N∑i=nΔtB∗(ti,tn−1)∇pi−1h,∇qh)+(ynh−ynd,qh), ∀ qh∈Vh, | (2.25) |
pNh=0, | (2.26) |
(unh+pn−1h,˜uh−unh)≥0, ∀ ˜uh∈Kh. | (2.27) |
Similarly, employing the projection (2.9), the optimal condition (2.27) can be rewritten as follows:
unh=max{0,¯pn−1h}−πhpn−1h, | (2.28) |
where ¯pn−1h=∫Ωpn−1h∫Ω1.
In the rest of the paper, we shall use some intermediate variables. For any control function ˜u∈K satisfies the following:
(dtynh(˜u),vh)+(A∇ynh(˜u),∇vh)=(n∑i=1ΔtB(tn,ti−1)∇yih(˜u),∇vh)+(fn+˜un,vh), ∀ vh∈Vh, | (2.29) |
y0h(˜u)=Rhy0, | (2.30) |
−(dtpnh(˜u),qh)+(A∇pn−1h(˜u),∇qh)=(N∑i=nΔtB∗(ti,tn−1)∇pi−1h(˜u),∇qh)+(ynh(˜u)−ynd,qh), ∀ qh∈Vh, | (2.31) |
pNh(˜u)=0. | (2.32) |
In this section, we will discuss a priori error estimates and superconvergence of the fully discrete case for the state variable, the co-state variable and the control variable. In order to do it, we need the following lemmas.
Lemma 3.1. Let (ynh(u),pn−1h(u)) be the solution of (2.29)–(2.32) with ˜u=u and (y,p) be the solution of (2.4)–(2.8). Assume that the exact solution (y,p) has enough regularities for our purpose. Then, for Δt small enough and 1≤n≤N, we have
|||y−yh(u)|||L∞(L2)+|||p−ph(u)|||L∞(L2)≤C(Δt+h2), | (3.1) |
|||∇(y−yh(u))|||L∞(L2)+|||∇(p−ph(u))|||L∞(L2)≤C(Δt+h). | (3.2) |
Proof. For convenience, let
yn−ynh(u)=yn−Rhyn+Rhyn−ynh(u)=:ηny+ξny,pn−pnh(u)=pn−Rhpn+Rhpn−pnh(u)=:ηnp+ξnp. |
Taking t=tn in (2.4), subtracting (2.29) from (2.4) and then using (2.12), we have
(dtξny,vh)+(A∇ξny,∇vh)=(dtyn−ynt,vh)−(dtηny,vh)+[∫tn0(B(tn,s)∇Rhy(s),∇vh)ds−(n∑i=1ΔtB(tn,ti−1)∇yih(u),∇vh)]. | (3.3) |
Choosing vh=dtξny in (3.3), we get
(dtξny,dtξny)+(A∇ξny,dt∇ξny)=(dtyn−ynt,dtξny)−(dtηny,dtξny)+[∫tn0(B(tn,s)∇Rhy(s),dt∇ξny)ds−(n∑i=1ΔtB(tn,ti−1)∇yih(u),dt∇ξny)]. | (3.4) |
Notice that
(dt∇ξny,A∇ξny)≥12Δt(‖A12∇ξny‖2−‖A12∇ξn−1y‖2). | (3.5) |
Multiplying Δt and summing over n from 1 to l (1≤l≤N) on both sides of (3.4), and by using (3.5) and ξ0y=0, we find that
12‖A12∇ξly‖2+l∑n=1‖dtξny‖2Δt≤l∑n=1(dtyn−ynt,dtξny)Δt−l∑n=1(dtηny,dtξny)Δt+l∑n=1[∫tn0(B(tn,s)∇Rhy(s),dt∇ξny)ds−(n∑i=1ΔtB(tn,ti−1)∇yih(u),dt∇ξny)]Δt=:3∑i=1Ai. | (3.6) |
Now, we estimate the right-hand terms of (3.6). For A1, from the results given in [24], we have
A1≤Cl∑n=1(∫tntn−1‖ytt‖dt)2Δt+‖l∑n=1‖dtξny‖2Δt≤C(Δt)2∫tl0‖ytt‖2dt+14l∑n=1‖dtξny‖2Δt≤C(Δt)2‖ytt‖2L2(L2)+14l∑n=1‖dtξny‖2Δt. | (3.7) |
For A2, using (2.13), the Hölder inequality and the Cauchy inequality, we have
A2≤Cl∑n=1‖ηny−ηn−1yΔt‖2Δt+14l∑n=1‖dtξny‖2Δt≤Cl∑n=11Δt‖∫tntn−1(ηy)tdt‖2+14l∑n=1‖dtξny‖2Δt≤Cl∑n=11Δt((∫tntn−1‖(ηy)t‖2dt)12(∫tntn−112dt)12)2+14l∑n=1‖dtξny‖2Δt≤Ch4∫tl0‖yt‖22dt+14l∑n=1‖dtξny‖2Δt≤Ch4‖yt‖2L2(H2)+14l∑n=1‖dtξny‖2Δt. | (3.8) |
At last, for A3, it follows from the Cauchy inequality, Cauchy mean value theorem and assumptions on A and B that
A3=l∑n=1[∫tn0(B(tn,s)∇Rhy(s),dt∇ξny)ds−(n∑i=1ΔtB(tn,ti)∇yih(u),dt∇ξny)+(n∑i=1ΔtB(tn,ti)∇yih(u),dt∇ξny)−(n∑i=1ΔtB(tn,ti−1)∇yih(u),dt∇ξny)]Δt≤C(Δt)2(‖∇Rhyt‖2L2(L2)+‖∇Rhy‖2L2(L2))+Cl∑n=1‖∇ξny‖2Δt+Cl∑n=1Δtn∑i=1‖∇ξiy‖2Δt+a∗4‖∇ξly‖2, | (3.9) |
where
l∑n=1[∫tn0(B(tn,s)∇Rhy(s),dt∇ξny)ds−(n∑i=1ΔtB(tn,ti)∇yih(u),dt∇ξny)]Δt=(∫tl0B(tl,s)∇Rhy(s)ds−l∑i=1B(tl,ti)∇RhyiΔt,∇ξly)+l∑i=1(ΔtB(tl,ti)∇ξiy,∇ξly)+l−1∑n=1(∫tn0(B(tn,s)−B(tn+1,s))∇Rhyds,∇ξny)−l−1∑n=1(∫tn+1tnB(tn+1,s)(∇Rhy−∇Rhyn+1)ds,∇ξny)−l−1∑n=1(n∑i=1Δt(B(tn,ti)−B(tn+1,ti))∇Rhyi,∇ξny)−l−1∑n=1(ΔtB(tn+1,tn+1)∇ξn+1y,∇ξny) |
+l−1∑n=1(n∑i=1Δt(B(tn,ti)−B(tn+1,ti))∇ξiy,∇ξny)=(∫tl0B(tl,s)∇Rhy(s)ds−l∑i=1B(tl,ti)∇RhyiΔt,∇ξly)+(l∑i=1ΔtB(tl,ti)∇ξiy,∇ξly))+l−1∑n=1(∫tn0Bt(t∗n+1,s)Δt∇Rhyds,∇ξny)−l−1∑n=1(∫tn+1tnΔtB(tn+1,s)∇Rhyn+1tds,∇ξny)−l−1∑n=1(n∑i=1(Δt)2Bt(t∗n+1,ti)∇Rhyids,∇ξny)−l−1∑n=1(ΔtB(tn+1,tn+1)∇ξn+1y,∇ξny)+l−1∑n=1(n∑i=1(Δt)2Bt(t∗n+1,ti)∇ξiy,∇ξny)≤C(Δt)2(‖∇Rhyt‖2L2(L2)+‖∇Rhy‖2L2(L2))+Cl∑n=1‖∇ξny‖2Δt+Cl∑n=1Δtn∑i=1‖∇ξiy‖2Δt+a∗8‖∇ξly‖2 |
and
l∑n=1[(n∑i=1ΔtB(tn,ti)∇yih(u),dt∇ξny)−(n∑i=1ΔtB(tn,ti−1)∇yih(u),dt∇ξny)]Δt=(l∑i=1Δt(B(tl,ti)−B(tl,ti−1))∇Rhyi,∇ξly)−(l∑i=1Δt(B(tl,ti)−B(tl,ti−1))∇ξiy,∇ξly)+l−1∑n=1(n∑i=1Δt(B(tn,ti)−B(tn,ti−1))∇Rhyi,∇ξny)−l−1∑n=1(n∑i=1Δt(B(tn,ti)−B(tn,ti−1))∇ξiy,∇ξny)−l−1∑n=1(n+1∑i=1Δt(B(tn+1,ti)−B(tn+1,ti−1))∇Rhyi,∇ξny)+l−1∑n=1(n+1∑i=1Δt(B(tn+1,ti)−B(tn+1,ti−1))∇ξiy,∇ξny)=(l∑i=1(Δt)2Bt(tl,t∗i)∇Rhyi,∇ξly)−(l∑i=1(Δt)2Bt(tl,t∗i)∇ξiy,∇ξly) |
+l−1∑n=1(n∑i=1(Δt)2Bt(tn,t∗i)∇Rhyi,∇ξny)−l−1∑n=1(n∑i=1(Δt)2Bt(tn,t∗i)∇ξiy,∇ξny)−l−1∑n=1(n+1∑i=1(Δt)2Bt(tn+1,t∗i)∇Rhyi,∇ξny)+l−1∑n=1(n+1∑i=1(Δt)2Bt(tn+1,t∗i)∇ξiy,∇ξny)≤C(Δt)2‖∇Rhy‖2L2(L2)+Cl∑n=1‖∇ξny‖2Δt+Cl∑n=1Δtn∑i=1‖∇ξiy‖2Δt+a∗8‖∇ξly‖2, |
where t∗i is located between ti−1 and ti, and we also used
‖∫tn0B(tn,s)∇Rhy(s)ds−n∑i=1B(tn,ti)∇RhyiΔt‖≤CΔt(‖∇Rhyt‖L2(L2)+‖∇Rhy‖L2(L2)). |
From (3.7)–(3.9), we have
12‖A12∇ξly‖2+12l∑n=1‖dtξny‖2Δt≤Ch4‖yt‖2L2(H2)+C(Δt)2(‖ytt‖2L2(L2)+‖∇Rhyt‖2L2(L2)+‖∇Rhy‖2L2(L2))+Cl∑n=1‖∇ξny‖2Δt+Cl∑n=1Δtn∑i=1‖∇ξiy‖2Δt+a∗4‖∇ξly‖2. | (3.10) |
Adding ∑ln=1‖∇ξny‖2Δt to both sides of (3.10), by use of the assumption on A and discrete Gronwall's inequality, we have
|||∇(Rhy−yh(u))|||L∞(L2)≤C(Δt+h2). | (3.11) |
Using (2.13), the Poincare inequality and the triangle inequality, we get
|||y−yh(u)|||L∞(L2)≤C(Δt+h2), |||∇(y−yh(u))|||L∞(L2)≤C(Δt+h). | (3.12) |
Taking t=tn−1 in (2.6), subtracting (2.31) from (2.6) and then using (2.14), we have
−(dtξnp,qh)+(A∇ξn−1p,∇qh)=−(dtpn−pn−1t,qh)+(dtηnp,qh)+∫Ttn−1(B∗(s,tn−1)∇Rhp(s),∇qh)ds−(N∑i=nΔtB∗(ti,tn−1)∇pi−1h(u),∇qh)+(δynd−δyn+yn−ynh(u),qh). | (3.13) |
Choosing qh=−dtξnp in (3.13), multiplying by Δt and summing over n from l+1 to N (0≤l≤N−1) on both sides of (3.13), since ξNp=0, we find that
12‖A12∇ξlp‖2+N∑n=l+1‖dtξnp‖2Δt≤N∑n=l+1(dtpn−pn−1t,dtξnp)Δt−N∑n=l+1(dtηnp,dtξnp)Δt−N∑n=l+1[∫Ttn−1(B∗(s,tn−1)∇Rhp(s),dt∇ξnp)ds−(N∑i=nΔtB∗(ti,tn−1)∇pi−1h(u),dt∇ξnp)]Δt−N∑n=l+1(δynd−δyn+yn−ynh(u),dtξnp)Δt=:4∑i=1Bi. | (3.14) |
Notice that
−(A∇ξn−1p,dt∇ξnp)≥12Δt(‖A12∇ξn−1p‖2−‖A12∇ξnp‖2). | (3.15) |
Now, we estimate the right-hand terms of (3.14). Similar to (3.7), we have
B1≤C(Δt)2‖ptt‖2L2(L2)+14N∑n=l+1‖dtξnp‖2Δt. | (3.16) |
For B2, using (2.15) and the Cauchy inequality, we have
B2≤Ch4‖pt‖2L2(H2)+14N∑n=l+1‖dtξnp‖2Δt. | (3.17) |
For B3, applying the same estimates as A3, we conclude that
B3=−N∑n=l+1[∫Ttn−1(B∗(s,tn−1)∇Rhp(s),dt∇ξnp)ds−(N∑i=nΔtB∗(ti−1,tn−1)∇pi−1h(u),dt∇ξnp)+(N∑i=nΔtB∗(ti−1,tn−1)∇pi−1h(u),dt∇ξnp)−(N∑i=nΔtB∗(ti,tn−1)∇pi−1h(u),dt∇ξnp)]Δt≤C(Δt)2(‖∇Rhpt‖2L2(L2)+‖∇Rhp‖2L2(L2))+CN∑n=l+1‖∇ξnp‖2Δt+CN∑n=l+1ΔtN∑i=n‖∇ξip‖2Δt+a∗4‖∇ξlp‖2, | (3.18) |
where
‖∇Rhpt‖L2(L2)+‖∇Rhp‖L2(L2)≤‖∇(pt−Rhpt)‖L2(L2)+‖∇pt‖L2(L2)+‖∇(p−Rhp)‖L2(L2)+‖∇p‖L2(L2). |
For B4, using the Cauchy inequality and the smoothness of y and yd, we have
B4=−N∑n=l+1(δynd−δyn+yn−ynh(u),dtξnp)Δt≤C(Δt)2(‖yt‖2L2(L2)+‖(yd)t‖2L2(L2))+C‖yn−ynh(u)‖2L2(L2)+14N∑n=l+1‖dtξnp‖2Δt. | (3.19) |
Combining (3.16)–(3.19), we have
12‖A12∇ξlp‖2+14N∑n=l+1‖dtξnp‖2Δt≤C(Δt)2(‖ptt‖2L2(L2)+‖∇Rhpt‖2L2(L2)+‖∇Rhp‖2L2(L2)+‖yt‖2L2(L2)+‖(yd)t‖2L2(L2))+Ch4‖pt‖2L2(H2)+C‖yn−ynh(u)‖2L2(L2)+a∗4‖∇ξlp‖2+CN∑n=l+1‖∇ξnp‖2Δt+CN∑n=l+1ΔtN∑i=n‖∇ξip‖2Δt. | (3.20) |
By adding ∑Nn=l+1‖∇ξnp‖2Δt to both sides of (3.20) and applying the assumption on A, discrete Gronwall's inequality and (3.12), we conclude that
|||∇(Rhp−ph(u))|||L∞(L2)≤C(Δt+h2). | (3.21) |
Using (2.15) and the triangle inequality, we get
|||p−ph(u)|||L∞(L2)≤C(Δt+h2), |||∇(p−ph(u))|||L∞(L2)≤C(Δt+h); | (3.22) |
we have completed the proof of the Lemma 3.1.
Lemma 3.2. Choose ˜un=Qhun and ˜un=un in (2.29)–(2.32) respectively. Then, for Δt small enough and 1≤n≤N, we have
|||∇(yh(u)−yh(Qhu))|||L∞(L2)+|||∇(ph(u)−ph(Qhu))|||L∞(L2)≤Ch2. | (3.23) |
Proof. For convenience, let
λny=ynh(u)−ynh(Qhu), λnp=pnh(u)−pnh(Qhu). |
Taking ˜un=un and ˜un=Qhun in (2.29), we easily get
(dtλny,vh)+(A∇λny,∇vh)=n∑i=1Δt(B(tn,ti−1)∇λiy,∇vh)+(un−Qhun,vh). | (3.24) |
By choosing vh=dtλny in (3.24), multiplying by Δt and summing over n from 1 to l (1≤l≤N) on both sides of (3.24), we find that
12‖A12∇λly‖2+l∑n=1‖dtλny‖2Δt≤l∑n=1(n∑i=1Δt(B(tn,ti−1)∇λiy,dt∇λny)Δt+l∑n=1(un−Qhun,λny−λn−1y)=(l∑i=1ΔtB(tl,ti−1)∇λiy,∇λly)+l−1∑n=1(n∑i=1ΔtB(tn,ti−1)∇λiy−n+1∑i=1ΔtB(tn+1,ti−1)∇λiy,∇λny)+(ul−Qhul,λly)−l−1∑n=1(un+1−Qhun+1−(un−Qhun),λny)=(l∑i=1ΔtB(tl,ti−1)∇λiy,∇λly)+l−1∑n=1(n∑i=1(Δt)2Bt(t∗n+1,ti−1)∇λiy,∇λny)−l−1∑n=1(ΔtB(tn+1,tn)∇λn+1y,∇λny)+C‖ul−Qhul‖−1‖∇λly‖+l−1∑n=1‖(u−Qhu)t(θn)‖−1‖∇λny‖Δt≤Cl∑n=1‖∇λny‖2Δt+Cl∑n=1Δtn∑i=1‖∇λiy‖2Δt+a∗4‖∇λly‖2+Ch4(‖ul‖21+‖ut‖2L2(H1)), | (3.25) |
where we use (2.17) and the assumption on B; additionally, θn is located between tn and tn+1.
Add ∑ln=1‖∇λny‖2Δt to both sides of (3.25); then for sufficiently small Δt, combining (3.25) and the discrete Gronwall inequality, we have
‖|∇(yh(u)−yh(Qhu))‖|L∞(L2)≤Ch2. | (3.26) |
Similar to (3.24), we have
−(dtλnp,qh)+(A∇λn−1p,∇qh)=(N∑i=nΔtB∗(ti,tn−1)∇λi−1p,∇qh)+(λny,qh), ∀ qh∈Vh. | (3.27) |
By choosing qh=−dtλnp in (3.27), multiplying by Δt and summing over n from l+1 to N (0≤l≤N−1) on both sides of (3.27), combining (3.26) and Poincare inequality gives
12‖A12∇λlp‖2+l∑n=1‖dtλnp‖2Δt≤−N∑n=l+1(N∑i=nΔtB∗(ti,tn−1)∇λi−1p,dt∇λnp)Δt−N∑n=l+1(λny,dtλnp)Δt=(N∑i=l+1ΔtB∗(ti,tl)∇λi−1p,∇λlp)−N−1∑n=l+1(N∑i=nΔtB∗(ti,tn−1)∇λi−1p,∇λnp)+N−1∑n=l+1(N∑i=n+1ΔtB∗(ti,tn)∇λi−1p,∇λnp)−N∑n=l+1(λny,dtλnp)Δt=(N∑i=l+1ΔtB∗(ti,tl)∇λi−1p,∇λlp)−N−1∑n=l+1(N∑i=n(Δt)2B∗t(ti,t∗n)∇λi−1p,∇λnp)−N−1∑n=l+1(ΔtB∗(tn,tn)∇λn−1p,∇λnp)−N∑n=l+1(λny,dtλnp)Δt≤Ch4+a∗4‖∇λlp‖2+CN∑n=l+1‖∇λn−1p‖2Δt+CN−1∑n=l+1ΔtN∑i=n‖∇λip‖2Δt+12N∑n=l+1‖dtλnp‖2Δt. | (3.28) |
Add ∑Nn=l+1‖∇λn−1p‖2Δt to both sides of (3.28); then for sufficiently small Δt, applying the discrete Gronwall inequality and the assumptions on A and B, we have
|||∇(ph(u)−ph(Qhu))|||L∞(L2)≤Ch2. | (3.29) |
Using the stability analysis as in Lemma 3.2 yields Lemma 3.3.
Lemma 3.3. Let (ynh,pnh) and (ynh(Qhu),pnh(Qhu)) be the discrete solutions of (2.29)−(2.32) with ˜un=unh and ˜un=Qhun, respectively. Then, for Δt small enough and 1≤n≤N, we have
|||∇(yh(Qhu)−yh)|||L∞(L2)+|||∇(ph(Qhu)−ph)|||L∞(L2)≤C|||Qhu−uh|||L2(L2). | (3.30) |
Next, we derive the following inequality.
Lemma 3.4. Choose ˜un=Qhun and ˜un=unh in (2.29)−(2.32) respectively. Then, we have
N∑n=1(Qhun−unh,pn−1h(Qhu)−pn−1h)Δt≥0. | (3.31) |
Proof. For n=0,1,…,N, let
rnp=pnh(Qhu)−pnh, rny=ynh(Qhu)−ynh. |
From (2.29)–(2.32), we have
(dtrny,vh)+(A∇rny,∇vh)−n∑i=1Δt(B(tn,ti−1)∇riy,∇vh)=(Qhun−unh,vh), ∀ vh ∈Vh, | (3.32) |
−(dtrnp,qh)+(A∇rn−1p,∇qh)−N∑i=nΔt(B∗(ti,tn−1)∇ri−1p,∇qh)=(rny,qh), ∀ qh ∈Vh. | (3.33) |
Notice that
N∑n=1(Δtn∑i=1B(tn,ti−1)∇riy,∇rn−1p)=N∑n=1(ΔtN∑i=nB∗(ti,tn−1)∇ri−1p,∇rny) |
and
N∑n=1(dtrny,rn−1p)Δt+N∑n=1(dtrnp,rny)Δt=0. |
By choosing vh=−rn−1p in (3.32), qh=rny in (3.33), and then multiplying the two resulting equations by Δt and summing it over n from 1 to N, we have
N∑n=1(Qhun−unh,pn−1h(Qhu)−pn−1h)Δt=N∑n=1‖rny‖2Δt, | (3.34) |
which completes the proof of the lemma.
Lemma 3.5. Let u be the solution of (2.4)–(2.8) and unh be the solution of (2.23)–(2.27). Assume that all of the conditions in Lemmas 3.1–3.4 are valid. Then, for Δt small enough and 1≤n≤N, we have
|||Qhu−uh|||L2(L2)≤C(h2+Δt). | (3.35) |
Proof. Take ˜u=unh in (2.8) and ˜uh=Qhun in (2.27) to get the following two inequalities:
(un+pn,unh−un)≥0 | (3.36) |
and
(unh+pn−1h,Qhun−unh)≥0. | (3.37) |
Note that unh−un=unh−Qhun+Qhun−un. Adding the two inequalities (3.36) and (3.37), we have
(unh+pn−1h−un−pn,Qhun−unh)+(un+pn,Qhun−un)≥0. | (3.38) |
Thus, by (3.38), (2.16), (2.8) and Lemma 3.4, we find that
|||Qhu−uh|||2L2(L2)=N∑n=1(Qhun−unh,Qhun−unh)Δt≤N∑n=1(Qhun−un,Qhun−unh)Δt+N∑n=1(pn−1h−pn,Qhun−unh)Δt+N∑n=1(un+pn,Qhun−un)Δt=N∑n=1(pn−1h−pn−1h(Qhu),Qhun−unh)Δt+N∑n=1(pn−1−pn,Qhun−unh)Δt+N∑n=1(pn−1h(u)−pn−1,Qhun−unh)Δt+N∑n=1(un+pn,Qhun−un)Δt+N∑n=1(pn−1h(Qhu)−pn−1h(u),Qhun−unh)Δt≤N∑n=1(pn−1−pn,Qhun−unh)Δt+N∑n=1(pn−1h(u)−pn−1,Qhun−unh)Δt+N∑n=1(pn−1h(Qhu)−pn−1h(u),Qhun−unh)Δt=:3∑i=1Fi. | (3.39) |
It follows from the Cauchy inequality, Lemma 3.1, Lemma 3.2 and Poincare's inequality that
F1≤C(Δt)2‖pt‖2L2(L2)+14N∑n=1‖Qhun−unh‖2Δt, | (3.40) |
F2≤C(h4+(Δt)2)+14N∑n=1‖Qhun−unh‖2Δt, | (3.41) |
F3≤Ch4+14N∑n=1‖Qhun−unh‖2Δt. | (3.42) |
Substituting the estimates for F1–F3 into (3.39), we derive (3.35).
Using (3.11), (3.21), Lemmas 3.2–3.5 and the triangle inequality, we derive the following superconvergence for the state variable.
Lemma 3.6. Let u be the solution of (2.4)–(2.8) and unh be the solution of (2.23)–(2.27). Assume that all of the conditions in Lemmas 3.1–3.5 are valid. Then, for Δt small enough and 1≤n≤N, we have
|||∇(Rhy−yh)|||L∞(L2)+|||∇(Rhp−ph)|||L∞(L2)≤C(h2+Δt). | (3.43) |
Now, the main result of this section is given in the following theorem.
Theorem 3.1. Let (y,p,u) and (ynh,pn−1h,unh) be the solutions of (2.4)–(2.8) and (2.23)–(2.27), respectively. Assume that y, p and u have enough regularities for our purpose; then, for Δt small enough and 1≤n≤N, we have
|||y−yh|||L∞(L2)+|||p−ph|||L∞(L2)≤C(h2+Δt), | (3.44) |
|||∇(y−yh)|||L∞(L2)+|||∇(p−ph)|||L∞(L2)≤C(h+Δt), | (3.45) |
|||u−uh|||L2(L2)≤C(h+Δt). | (3.46) |
Proof. The proof of the theorem can be completed by using Lemmas 3.1–3.5, (2.17) and the triangle inequality.
To provide the global superconvergence for the control and state, we use the recovery techniques on uniform meshes. Let us construct the recovery operators Ph and Gh. Let Phv be a continuous piecewise linear function (without the zero boundary constraint). The value of Phv on the nodes are defined by a least squares argument on element patches surrounding the nodes; the details can be found in [25,26].
We construct the gradient recovery operator Ghv=(Phvx,Phvy) for the gradients of y and p. In the piecewise linear case, it is noted to be the same as the Z-Z gradient recovery (see [25,26]). We construct the discrete co-state with the admissible set
ˆunh=max{0,¯pn−1h}−pn−1h. | (3.47) |
Now, we can derive the global superconvergence result for the control variable and state variable.
Theorem 3.2. Let u and unh be the solutions of (2.4)–(2.8) and (2.29)–(2.32), respectively. Assume that all of the conditions in Lemmas 3.1–3.5 are valid. Then we have
|||u−ˆuh|||L2(L2)≤C(h2+Δt). | (3.48) |
Proof. Using (2.9), (3.47) and Theorem 3.1, we have
|||u−ˆuh|||2L2(L2)=N∑n=1‖un−ˆunh‖2Δt≤CN∑n=1‖max{0,¯pn}−max{0,¯pn−1h}‖2Δt+CN∑n=1‖pn−pn−1h‖2Δt≤CN∑n=1‖¯pn−¯pn−1h‖2Δt+CN∑n=1‖pn−pn−1h‖2Δt≤CN∑n=1‖pn−pn−1h‖2Δt≤CN∑n=1‖pn−pn−1‖2Δt+CN∑n=1‖pn−1−pn−1h‖2Δt≤C(h4+(Δt)2). | (3.49) |
Theorem 3.3. Let (y,p) and (ynh,pn−1h) be the solutions of (2.4)–(2.8) and (2.29)–(2.32), respectively. Assume that all of the conditions in Lemmas 3.1–3.5 are valid. Then we have
|||Ghyh−∇y|||L∞(L2)+|||Ghph−∇p|||L∞(L2)≤C(h2+Δt). | (3.50) |
Proof. Notice that
|||Ghyh−∇y|||L∞(L2)≤|||Ghyh−GhRhy|||L∞(L2)+|||GhRhy−∇y|||L∞(L2). | (3.51) |
It follows from Lemma 3.6 that
|||Ghyh−GhRhy|||L∞(L2)≤C|||∇(yh−Rhy)|||L∞(L2)≤C(h2+Δt). | (3.52) |
It can be proved by the standard interpolation error estimate technique (see [1]) that
|||GhRhy−∇y|||L∞(L2)≤Ch2. | (3.53) |
Therefore, it follows from (3.52) and (3.53) that
|||Ghyh−∇y|||L∞(L2)≤C(h2+Δt). | (3.54) |
Similarly, it can be proved that
|||Ghph−∇p|||L∞(L2)≤C(h2+Δt). | (3.55) |
Therefore, we complete the proof.
In this section, we will present a two-grid scheme and analyze a priori error estimates. Now, we present our two-grid algorithm which has the following two steps:
Step 1. On the coarse grid TH, find (ynH,pn−1H,unH)∈V2H×KH that satisfies the following optimality conditions:
(dtynH,vH)+(A∇ynH,∇vH)=(n∑i=1ΔtB(tn,ti−1)∇yiH,∇vH)+(fn+unH,vH), ∀ vH ∈VH, | (4.1) |
y0H=RHy0, | (4.2) |
−(dtpnH,qH)+(A∇pn−1H,∇qH)=(N∑i=nΔtB∗(ti,tn−1)∇pi−1H,∇qH)+(ynH−ynd,qH), ∀ qH ∈VH, | (4.3) |
pNH=0, | (4.4) |
(unH+pn−1H,u∗H−unH)≥0, ∀ u∗H∈KH. | (4.5) |
Step 2. On the fine grid Th, find (¯˜ynh,¯˜pn−1h,¯˜unh)∈V2h×Kh such that
(dt¯˜ynh,vh)+(A∇¯˜ynh,∇vh)=(n∑i=1ΔtB(tn,ti−1)∇¯˜yih,∇vh)+(fn+ˆunH,vh), ∀ vh ∈Vh, | (4.6) |
¯˜y0h=Rhy0, | (4.7) |
−(dt¯˜pnh,qh)+(A∇¯˜pn−1h,∇qh)=(N∑i=nΔtB∗(ti,tn−1)¯˜pi−1h,∇qh)+(¯˜ynh−ynd,qh), ∀ qh ∈Vh, | (4.8) |
¯˜pNh=0, | (4.9) |
(¯˜unh+¯˜pn−1h,u∗h−¯˜unh)≥0, ∀ u∗h∈Kh. | (4.10) |
Combining Theorem 3.1 and the stability estimates, we easily get the following results.
Theorem 4.1. Let (y,p,u) and (¯˜ynh,¯˜pnh,¯˜unh) be the solutions of (2.4)–(2.8) and (4.1)–(4.10), respectively. Assume that y, yd, p, pd and u have enough regularities for our purpose; then, for Δt small enough and 1≤n≤N, we have
|||∇(y−¯˜yh)|||L∞(L2)+|||∇(p−¯˜ph)|||L∞(L2)≤C(h+H2+Δt), | (4.11) |
|||u−¯˜uh|||L2(L2)≤C(h+H2+Δt). | (4.12) |
Proof. For convenience, let
yn−¯˜ynh=yn−Rhyn+Rhyn−¯˜ynh=:ηny+eny,pn−¯˜pnh=pn−Rhpn+Rhpn−¯˜pnh=:ηnp+enp. |
Taking t=tn in (2.4), subtracting (4.6) from (2.4) and then using (2.12), we have
(dteny,vh)+(A∇eny,∇vh)=(∫tn0B(tn,s)Rh∇y(s)ds−n∑i=1ΔtB(tn,ti−1)∇¯˜yih,∇vh)+(dtyn−ynt,vh)−(dtηny,vh)+(un−ˆunH,vh), ∀ vh∈Vh. | (4.13) |
Selecting vh=dteny in (4.13), multiplying by Δt and summing over n from 1 to l (1≤l≤N) on both sides of (4.13), we find that
12‖A12∇ely‖2+l∑n=1‖dteny‖2Δt≤−l∑n=1(dtηny,dteny)Δt+l∑n=1(dtyn−ynt,dteny)Δt+l∑n=1(∫tn0B(tn,s)Rh∇y(s)ds−n∑i=1ΔtB(tn,ti−1)∇¯˜yih,dt∇eny)Δt+l∑n=1(un−ˆunH,dteny)Δt=:4∑i=1Ii. | (4.14) |
Similar to Lemma 3.1, it is easy to show that
I1+I2≤Ch4‖yt‖2L2(H2)+C(Δt)2‖ytt‖2L2(L2)+12l∑n=1‖dteny‖2Δt. | (4.15) |
Similar to A3, we find that
I3≤C(Δt)2(‖∇Rhyt‖2L2(L2)+‖∇Rhy‖2L2(L2))+Cl∑n=1‖∇eny‖2Δt+Cl∑n=1Δtn∑i=1‖∇eiy‖2Δt+a∗4‖∇ely‖2. | (4.16) |
For I4, using Theorem 3.2, we have
I4≤C(H4+(Δt)2)+14l∑n=1‖dteny‖2Δt. | (4.17) |
Combining (4.15)–(4.17), the discrete Gronwall inequality, the triangle inequality and (2.13), we get
|||∇(y−¯˜yh)|||L∞(L2)≤C(h+H2+Δt). | (4.18) |
By taking t=tn−1 in (2.6), subtracting (4.8) from (2.6) and using (2.12), we have
−(dtenp,qh)+(A∇en−1p,∇qh)=(∫Ttn−1B∗(s,tn−1)∇Rhp(s)ds−N∑i=nB∗(ti,tn−1)¯˜pi−1hΔt,∇qh)−(dtpn−pn−1t,qh)+(dtηnp,qh)+(δynd−δyn,qh)+(yn−¯˜ynh,qh), ∀ qh∈Vh. | (4.19) |
By selecting qh=−dtenp in (4.19), multiplying by Δt and summing over n from l+1 to N (0≤l≤N−1) on both sides of (4.19), we find that using (2.15), (4.18) and the triangle inequality, similar to (3.14), gives
|||∇(p−¯˜ph)|||L∞(L2)≤C(h+H2+Δt). | (4.20) |
Note that
¯˜unh=max{0,¯¯˜pn−1h}−πh¯˜pn−1h,un=max{0,¯pn}−pn. |
Using (2.19), (4.20) and the mean value theorem, we have
|||u−¯˜uh|||2L2(L2)=N∑n=1‖un−¯˜unh‖2Δt≤CN∑n=1‖max{0,¯pn}−max{0,¯¯˜pn−1h}‖2Δt+CN∑n=1‖pn−πh¯˜pn−1h‖2Δt≤CN∑n=1‖¯pn−¯¯˜pn−1h‖2Δt+CN∑n=1‖pn−pn−1‖2Δt+CN∑n=1‖pn−1−πhpn−1‖2Δt+CN∑n=1‖πhpn−1−πh¯˜pn−1h‖2Δt≤CN∑n=1‖pn−¯˜pn−1h‖2Δt+CN∑n=1‖pn−pn−1‖2Δt+CN∑n=1‖pn−1−πhpn−1‖2Δt+CN∑n=1‖πhpn−1−πh¯˜pn−1h‖2Δt≤CN∑n=1‖pn−pn−1‖2Δt+CN∑n=1‖pn−1−πhpn−1‖2Δt+CN∑n=1‖pn−1−¯˜pn−1h‖2Δt≤C(h2+H4+(Δt)2), | (4.21) |
which completes the proof.
In this section, we present the following numerical experiment to verify the theoretical results. We consider the following two-dimensional parabolic integro-differential optimal control problems
minu∈K{12∫10(‖y−yd‖2+‖u‖2)dt} |
subject to
(yt,v)+(∇y,∇v)=∫t0(∇y(s),∇v)ds+(f+u,v), ∀ v∈V,y(x,0)=y0(x), ∀ x∈Ω, |
where Ω=(0,1)2.
We applied a piecewise linear finite element method for the state variable y and co-state variable p. The stopping criterion of the finite element method was chosen to be the abstract error of control variable u between two adjacent iterates less than a prescribed tolerance, i.e.,
‖ul+1h−ulh‖≤ϵ, |
where ϵ=10−5 was used in our numerical tests. For the linear system of equations, we used the algebraic multigrid method with tolerance 10−9.
The numerical experiments were conducted on a desktop computer with a 2.6 GHz 4-core Intel i7-6700HQ CPU and 8 GB 2133 MHz DDR4 memory. The MATLAB finite element package iFEM was used for the implementation [27].
Example: We chose the following source function f and the desired state yd as
f(x,t)=(2e2t+4π2e2t+4π2+sin(πt))sin(πx)sin(πy)−4π2sin(πt),yd(x,t)=(πcos(πt)−8π2sinπt+8π2(cos(πt)π)−cosπTπ+e2t)sin(πx)sin(πy) |
such that the exact solutions for y, p, u are respectively,
y=e2tsin(πx)sin(πy),p=sin(πt)sin(πx)sin(πy),u=sin(πt)(4π2−sin(πx)sin(πy)). |
In order to see the convergence order with respect to time step size △t and mesh size h, we choose △t=h or △t=h2 with h=14,116,164. To see the convergence order of the two-grid method, we choose the coarse and fine mesh size pairs (12,14),(14,116),(18,164). Let us use yh,ph and uh as two-grid solutions in the following tables. In Tables 1 and 2, we let △t=h2 and present the errors of the finite element method and two-grid method for y and p in the L2-norm. Next, in Tables 3 and 4, we set △t=h and show the errors of the two methods for y and p in the H1-norm and u in the L2-norm. We can see that the two-grid method maintains the same convergence order as the finite element method. Moreover, we also display the computing times of the finite element method and the two-grid method in these tables. By comparison, we find that the two-grid method is more effective for solving the optimal control problems (1.1)–(1.4).
h | ‖y−yh‖ | ‖p−ph‖ | CPU time (s) |
14 | 0.1095 | 0.0856 | 0.7031 |
116 | 0.0079 | 0.0045 | 8.8702 |
164 | 0.0005 | 0.0002 | 2253.6396 |
(h,H) | ‖y−yh‖ | ‖p−ph‖ | CPU time (s) |
(14,12) | 0.1059 | 0.0853 | 0.4335 |
(116,14) | 0.0056 | 0.0043 | 5.0842 |
(164,18) | 0.0006 | 0.0002 | 1027.9740 |
h | ‖y−yh‖1 | ‖p−ph‖1 | ‖u−uh‖ | CPU time (s) |
14 | 1.6604 | 1.1385 | 0.1358 | 0.4720 |
116 | 0.6187 | 0.2143 | 0.0367 | 0.6320 |
164 | 0.1687 | 0.0578 | 0.0090 | 24.0800 |
(h,H) | ‖y−yh‖1 | ‖p−ph‖1 | ‖u−uh‖ | CPU time (s) |
(14,12) | 1.6755 | 1.1375 | 0.0988 | 0.2880 |
(116,14) | 0.6288 | 0.2142 | 0.0346 | 0.3870 |
(164,18) | 0.1716 | 0.0579 | 0.0089 | 7.3120 |
In this paper, we presented a two-grid finite element scheme for linear parabolic integro-differential control problems (1.1)–(1.4). A priori error estimates for the two-grid method and finite element method have been derived. We have used recovery operators to prove the superconvergence results. These results seem to be new in the literature. In our future work, we will investigate a posteriori error estimates. Furthermore, we shall consider a priori error estimates and a posteriori error estimates for optimal control problems governed by hyperbolic integro-differential equations.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare there is no conflict of interest.
[1] | J. Lian, S. Fang, Y. Zhou, Model predictive control of the fuel cell cathode system based on state quantity estimation, Comput. Simul., 37 (2020), 119-122. |
[2] |
H. M. Numanoğlu, H. Ersoy, B. Akgöz, O. Civalek, A new eigenvalue problem solver for thermos-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method, Math. Methods Appl. Sci., 45 (2022), 2592-2614. https://doi.org/10.1002/mma.7942 doi: 10.1002/mma.7942
![]() |
[3] |
V. Chaturvedi, T. Talapaneni, Effect of mechanical vibration and grain refiner on microstructure and mechanical properties of AZ91Mg alloy during solidification, J. Mater. Eng. Perform., 30 (2021), 3187-3202. https://doi.org/10.1007/s11665-021-05471-3 doi: 10.1007/s11665-021-05471-3
![]() |
[4] |
W. Booyse, D. N. Wilke, S. Heyns, Deep digital twins for detection, diagnostics and prognostics, Mech. Syst. Signal Process., 140 (2020), 106612. https://doi.org/10.1016/j.ymssp.2019.106612 doi: 10.1016/j.ymssp.2019.106612
![]() |
[5] | F. Tao, X. Sun, J. Cheng, Y. Zhu, W. Liu, Y. Wang, et al., 2023, MakeTwin: a reference architecture for digital twin software platform, Chin. J. Aeronaut., in press, 2023. https://doi.org/10.1016/j.cja.2023.05.002 |
[6] |
Q. Qi, F. Tao, T. Hu, N. Anwer, A. Liu, Y. Wei, et al., Enabling technologies and tools for digital twin, J. Manuf. Syst., 58 (2021), 3-21. https://doi.org/10.1016/j.jmsy.2019.10.001 doi: 10.1016/j.jmsy.2019.10.001
![]() |
[7] |
S. Liu, Y. Lu, P. Zheng, H. Shen, J. Bao, Adaptive reconstruction of digital twins for machining systems: a transfer learning approach, Rob. Comput. Integr. Manuf., 78 (2022), 102390. https://doi.org/10.1016/j.rcim.2022.102390 doi: 10.1016/j.rcim.2022.102390
![]() |
[8] | C. Gao, H. Park, A. Easwaran, An anomaly detection framework for digital twin driven cyber-physical systems, in ICCPS '21: Proceedings of the ACM/IEEE 12th International Conference on Cyber-Physical Systems, (2021), 44-54. https://doi.org/10.1145/3450267.3450533 |
[9] |
X. Wang, Y. Wang, F. Tao, A. Liu, New paradigm of data-driven smart customisation through digital twin, J. Manuf. Syst., 58 (2021), 270-280. https://doi.org/10.1016/j.jmsy.2020.07.023 doi: 10.1016/j.jmsy.2020.07.023
![]() |
[10] |
V. S. Vishnu, K. G. Varghese, B. Gurumoorthy, A data-driven digital twin of CNC machining processes for predicting surface roughness, Procedia CIRP, 104 (2021), 1065-1070. https://doi.org/10.1016/j.procir.2021.11.179 doi: 10.1016/j.procir.2021.11.179
![]() |
[11] |
C. Zhang, G. Zhou, J. He, Z. Li, W. Cheng, A data- and knowledge-driven framework for digital twin manufacturing cell, Procedia CIRP, 83 (2019), 345-350. https://doi.org/10.1016/j.procir.2019.04.084 doi: 10.1016/j.procir.2019.04.084
![]() |
[12] |
Y. Sun, Y. Lu, J. Bao, F. Tao, Prognostics and health management via long short-term digital twins, J. Manuf. Syst., 68 (2023), 560-575. https://doi.org/10.1016/j.jmsy.2023.05.023 doi: 10.1016/j.jmsy.2023.05.023
![]() |
[13] |
K. Feng, J. C. Ji, Q. Ni, Y. Li, W. Mao, L. Liu, A novel vibration-based prognostic scheme for gear health management in surface wear progression of the intelligent manufacturing system, Wear, 522 (2023), 204697. https://doi.org/10.1016/j.wear.2023.204697 doi: 10.1016/j.wear.2023.204697
![]() |
[14] |
L. Ma, B. Jiang, L. Xiao, N. Lu, Digital twin-assisted enhanced meta-transfer learning for rolling bearing fault diagnosis, Mech. Syst. Signal Process., 200 (2023), 110490. https://doi.org/10.1016/j.ymssp.2023.110490 doi: 10.1016/j.ymssp.2023.110490
![]() |
[15] |
L. Li, Y. Ren, Q. Jin, Electro-mechanical vibration and stress field of piezoelectric nanobeam with symmetrical FGM core under the low-velocity impact, Eur. Phys. J. Plus, 137 (2022), 1-20. https://doi.org/10.1140/epjp/s13360-022-02934-x doi: 10.1140/epjp/s13360-022-02934-x
![]() |
[16] |
M. Rigacci, R. Sato, K. Shirase, Power consumption simulation of servo motors focusing on the influence of mechanical vibration on motor efficiency, Int. J. Autom. Technol., 16 (2022), 104-116. https://doi.org/10.20965/ijat.2022.p0104 doi: 10.20965/ijat.2022.p0104
![]() |
[17] |
P. Ewert, C. T. Kowalski, M. Jaworski, Comparison of the effectiveness of selected vibration signal analysis methods in the rotor unbalance detection of PMSM drive system, Electronics, 11 (2022), 1748. https://doi.org/10.3390/electronics11111748 doi: 10.3390/electronics11111748
![]() |
[18] | Y. W. Zhang, G. L. She, Wave propagation and vibration of FG pipes conveying hot fluid, Steel Compos. Struct., 42 (2022), 397-405. |
[19] |
Y. Kumar, A. Gupta, A. Tounsi, Size-dependent vibration response of porous graded nanostructure with FEM and nonlocal continuum model, Adv. Nano Res., 11 (2021), 1-17. https://doi.org/10.12989/anr.2021.11.1.001 doi: 10.12989/anr.2021.11.1.001
![]() |
[20] |
S. K. Barman, M. Mishra, D. K. Maiti, D. Maity, Vibration-based damage detection of structures employing Bayesian data fusion coupled with TLBO optimization algorithm, Struct. Multidiscip. Optim., 64 (2021), 2243-2266. https://doi.org/10.1007/s00158-021-02980-6 doi: 10.1007/s00158-021-02980-6
![]() |
[21] |
F. L. Zhang, C. W. Kim, Y. Goi, Efficient Bayesian FFT method for damage detection using ambient vibration data with consideration of uncertainty, Struct. Control Health Monit., 28 (2021), e2659. https://doi.org/10.1002/stc.2659 doi: 10.1002/stc.2659
![]() |
[22] |
A. Turnbull, J. Carroll, A. McDonald, Combining SCADA and vibration data into a single anomaly detection model to predict wind turbine component failure, Wind Energy, 24 (2021), 197-211. https://doi.org/10.1002/we.2567 doi: 10.1002/we.2567
![]() |
[23] |
S. K. Barman, D. K. Maiti, D. Maity, Vibration-based delamination detection in composite structures employing mixed unified particle swarm optimization, AIAA J., 59 (2021), 386-399. https://doi.org/10.2514/1.J059176 doi: 10.2514/1.J059176
![]() |
[24] |
C. Tarawneh, J. Montalvo, B. Wilson, Defect detection in freight railcar tapered-roller bearings using vibration techniques, Railway Eng. Sci., 29 (2021), 42-58. https://doi.org/10.1007/s40534-020-00230-x doi: 10.1007/s40534-020-00230-x
![]() |
[25] |
Z. Mousavi, S. Varahram, M. M. Ettefagh, H. M. Sadeghi, N. S. Razavi, Deep neural networks-based damage detection using vibration signals of finite element model and real intact state: An evaluation via a lab-scale offshore jacket structure, Struct. Health Monit., 20 (2021), 379-405. https://doi.org/10.1177/1475921720932614 doi: 10.1177/1475921720932614
![]() |
[26] |
N. Wu, S. Haruyama, The 20k samples-per-second real time detection of acoustic vibration based on displacement estimation of one-dimensional laser speckle images, Sensors, 21 (2021), 2938. https://doi.org/10.3390/s21092938 doi: 10.3390/s21092938
![]() |
[27] |
M. H. M. Ghazali, W. Rahiman, Vibration-based fault detection in drone using artificial intelligence, IEEE Sensors J., 22 (2022), 8439-8448. https://doi.org/10.1109/JSEN.2022.3163401 doi: 10.1109/JSEN.2022.3163401
![]() |
[28] |
B. R. F. Rende, A. A. Cavalini, I. F. Santos, Fault detection using vibration data-driven models—a simple and well-controlled experimental example, J. Braz. Soc. Mech. Sci. Eng., 44 (2022), 1-11. https://doi.org/10.1007/s40430-022-03462-6 doi: 10.1007/s40430-022-03462-6
![]() |
[29] |
X. Huang, Q. Huang, H. Cao, W. Yan, L. Cao, Q. Zhang, Optimal design for improving operation performance of electric construction machinery collaborative system: Method and application, Energy, 263 (2023), 125629. https://doi.org/10.1016/j.energy.2022.125629 doi: 10.1016/j.energy.2022.125629
![]() |
[30] | J. L. Conradi Hoffmann, L. P. Horstmann, M. Martínez Lucena, M. G. de Araujo, A. A. Fröhlich, H. M. Napoli Nishioka, Anomaly detection on wind turbines based on a deep learning analysis of vibration signals, Appl. Artif. Intell., 35 (2021), 893-913. https://doi.org/10.1080/08839514.2021.1966879 |
[31] |
Y. Zhu, F. Li, W. Bao, Fatigue crack detection under the vibration condition based on ultrasonic guided waves, Struct. Health Monit., 20 (2021), 931-941. https://doi.org/10.1177/1475921719860772 doi: 10.1177/1475921719860772
![]() |
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h | ‖y−yh‖ | ‖p−ph‖ | CPU time (s) |
14 | 0.1095 | 0.0856 | 0.7031 |
116 | 0.0079 | 0.0045 | 8.8702 |
164 | 0.0005 | 0.0002 | 2253.6396 |
(h,H) | ‖y−yh‖ | ‖p−ph‖ | CPU time (s) |
(14,12) | 0.1059 | 0.0853 | 0.4335 |
(116,14) | 0.0056 | 0.0043 | 5.0842 |
(164,18) | 0.0006 | 0.0002 | 1027.9740 |
h | ‖y−yh‖1 | ‖p−ph‖1 | ‖u−uh‖ | CPU time (s) |
14 | 1.6604 | 1.1385 | 0.1358 | 0.4720 |
116 | 0.6187 | 0.2143 | 0.0367 | 0.6320 |
164 | 0.1687 | 0.0578 | 0.0090 | 24.0800 |
(h,H) | ‖y−yh‖1 | ‖p−ph‖1 | ‖u−uh‖ | CPU time (s) |
(14,12) | 1.6755 | 1.1375 | 0.0988 | 0.2880 |
(116,14) | 0.6288 | 0.2142 | 0.0346 | 0.3870 |
(164,18) | 0.1716 | 0.0579 | 0.0089 | 7.3120 |