Research article

A nonparametric copula distribution framework for bivariate joint distribution analysis of flood characteristics for the Kelantan River basin in Malaysia

  • The joint distribution analysis of multidimensional flood characteristics i.e., flood peak flow, volume and duration, often facilitates a comprehensive understanding in the hydrologic risk assessments. Copula-based methodology are frequently incorporated via parametric approach to model dependence structure of parametric based univariate marginal distributions. But, if the targeted copulas and univariate marginal distributions belongs to some specific parametric families, it might be problematic, if the underlying assumption are violated. Also, no universal rules and literatures are imposed to model any hydrologic vectors and their joint dependence structure through any fixed or pre-defined distributions. In this literature, a nonparametric copula simulation are incorporated and applied as a case study for 50 years annual maximum flood samples of the Kelantan River basin at the Gulliemard bridge station in Malaysia. In this study, a combination of both parametric and nonparametric marginal distribution separately conjoined by a nonparametric copulas framework, which is based on the Beta kernel function. The Beta kernel copula function are incorporated to estimate bivariate copula density which further used to derived joint cumulative density of flood peak-volume, volume-duration and peak-duration pairs and their associated joint as well as conditional return periods.

    Citation: Shahid Latif, Firuza Mustafa. A nonparametric copula distribution framework for bivariate joint distribution analysis of flood characteristics for the Kelantan River basin in Malaysia[J]. AIMS Geosciences, 2020, 6(2): 171-198. doi: 10.3934/geosci.2020012

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  • The joint distribution analysis of multidimensional flood characteristics i.e., flood peak flow, volume and duration, often facilitates a comprehensive understanding in the hydrologic risk assessments. Copula-based methodology are frequently incorporated via parametric approach to model dependence structure of parametric based univariate marginal distributions. But, if the targeted copulas and univariate marginal distributions belongs to some specific parametric families, it might be problematic, if the underlying assumption are violated. Also, no universal rules and literatures are imposed to model any hydrologic vectors and their joint dependence structure through any fixed or pre-defined distributions. In this literature, a nonparametric copula simulation are incorporated and applied as a case study for 50 years annual maximum flood samples of the Kelantan River basin at the Gulliemard bridge station in Malaysia. In this study, a combination of both parametric and nonparametric marginal distribution separately conjoined by a nonparametric copulas framework, which is based on the Beta kernel function. The Beta kernel copula function are incorporated to estimate bivariate copula density which further used to derived joint cumulative density of flood peak-volume, volume-duration and peak-duration pairs and their associated joint as well as conditional return periods.


    As a basic structural unit, plates are widely used in many places, such as spacecrafts and aircrafts, ships, buildings, containers, etc. The vibration of plates caused by external forces can lead to serious damage to the entire structure of the machinery or building. One way to reduce the damage caused by vibration is by applying the viscous damping strategy. The vibration of damped plates is described by fourth-order differential equations, whose analytical solutions are often excessively difficult to obtain. Thus, the theoretical analysis and numerical calculation of the vibration of damped plates are of great research interest.

    So far, a great number of studies have been conducted on the vibration problems of damped plates. Leissa et al. studied the free vibration of rectangular plates [1] and the vibrations of cantilevered shallow cylindrical shells of rectangular platforms [2]. Nair et al. discussed the quasi-degeneracies in plate vibration problems [3]. Wang et al. studied the vibration problems of flexible circular plates with initial deflection[4]. Hui Li et al. studied the vibration of foam core[5,6], considered the nonlinear vibration analysis of fiber metal laminated plates with multiple viscoelastic layers[7] and considered the vibration damping of multifunctional grille composite sandwich plates[8,9].

    The numerical methods studied for the plate vibration problems include the integration method, finite difference method, finite element method, mixed finite element method, etc. For example, Rock et al. used the finite element method in the study of the free vibration and dynamic response of thick and thin plates [10]. Bezine proposed a mixed boundary integral as a finite element approach to plate vibration problems [11]. Qian et al. studied the vibration characteristics of cracked plates [12]. Xu et al. analyzed the vibration problems of thin plates using the integral equation method [13]. Duran et al. conducted the finite element analysis of the vibration problem of a plate coupled with a fluid [14]. Xiong et al. conducted an analysis of free vibration problems for a thin plate by the local Petrov-Galerkin method [15]. Dawe discussed a finite element approach to plate vibration problems [16]. Wu et al. utilized the mesh-free least-squares-based finite difference method for large-amplitude free vibration analysis of arbitrarily shaped thin plates [17]. Mora et al. analyzed the buckling and the vibration problems of thin plates using a piecewise linear finite element method [18]. Werfalli et al. analyzed the vibration of rectangular plates using Galerkin-based finite element method [19]. Yang et al. discussed a differential quadrature hierarchical finite element method and its application to thin plate free vibration [20]. The mixed finite element method is effective in solving differential equations. The general theory of this method was established by Brezzi and Babuska in 1970s to solve second order elliptic problems [21,22].

    Later, Brezzi et al. used the mixed finite element method to solve second order elliptic problems in three variables [23]. Diegel et al. discussed the stability and convergence of a second order mixed finite element method for the Cahn-Hilliard equation [24]. Singh et al. performed the compositional flow modeling using a multi-point flux mixed finite element method [25]. Burger et al. studied a mixed finite element method for nonlinear diffusion equations [26].

    The mixed finite element method is also effective in simulating fourth-order differential equations, including both biharmonic equations and vibration equations. For biharmonic equations, Monk et al. utilized a stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems [27]. Stein et al. proposed a mixed finite element method with piecewise linear elements for the biharmonic equation on surfaces [28]. Meng et al. studied the optimal order convergence for the lowest order mixed finite element method of the biharmonic eigenvalue problem [29]. For vibration equations, Meng et al. studied a mixed virtual element method for the vibration problem of clamped Kirchhoff plate [30].

    As far as we know, the current literature lacks studies that utilize the mixed finite element method to solve vibration equations for viscously damped plates. Therefore, this work seeks to establish the mixed finite element scheme for the initial boundary conditions of damped plate vibration problems and to verify the existence and uniqueness of the approximate solution for the semi-discrete and backward Euler fully discrete schemes. An error analysis is conducted, and numerical case studies are conducted to validate the effectiveness and precision of the mixed finite element scheme, as well as to quantify the influence of the damping coefficient on plate vibrations.

    According to the theory of elasticity, there is a vibration equation of thin plate in [31],

     D(4wx4+24wx2y2+4wy4)+m2wt2=f(x,y,t).

    In this article, we add the damped term and consider the damped plate vibration problem:

    {(a) D(4wx4+24wx2y2+4wy4)+m2wt2+λwt=f(x,y,t),(x,y,t)Ω×(0,T],(b) w(x,y,0)=Φ(x,y),wt(x,y,0)=Ψ(x,y),(x,y)Ω,(c) w|Ω=0,Δw|Ω=0,t(0,T]. (1.1)

    Where D is the flexural rigidity, m=ρh is the mass per unit area, ρ is the mass density of the plate, and h is the thickness of the plate. λ is the damping factor, f is the smooth function, w(x,y,t) is the flexible surface function, Ω is the piecewise smooth bounded polygon region, (0,T] is the time interval, Ψ(x,y),Φ(x,y) are known functions.

    In this paper, the damping plate vibration equation is analyzed by the mixed finite element method. The advantage of the mixed finite element method lies in its ability to reduce the order of the high order differential equations by introducing intermediate variables, which often have physical meaning by themselves. Consequently, it can reduce the requirement for smoothness of the finite element space and hence simplify the interpolation space of the finite elements. Moreover, by using the mixed finite element method, both the unknown variables and the intermediate variables with realistic meaning can be obtained, hence increasing the precision of the discrete solutions. Compared to other methods, the mixed finite element method is easier to apply and more likely to yield meaningful solutions.

    This article is divided into five sections. The first section introduces the research background of the plate vibration problems. The second section provides the variational formulation for the initial boundary conditions of damped plate vibration problems. The third and the fourth sections discuss the construction of the semi-discrete and fully discrete mixed finite element schemes for the initial boundary conditions of the damped plate vibration problems, respectively, followed by the verification of the existence and uniqueness of such schemes and the error analyses. Finally, the fifth section presents the numerical case studies aimed at validating the theoretical discussions in the previous sections.

    Introducing auxiliary variables Δw=u,v=wt, where Δ=x2+y2, we first rewrite Eq.(1.1) into the following coupled system:

    {(a) DΔu+mvt+λv=f(x,y,t),(x,y,t)Ω×(0,T],(b) ut+Δv=0,(x,y,t)Ω×(0,T],(c) u(x,y,0)=ΔΦ(x,y),v(x,y,0)=Ψ(x,y),(x,y)Ω,(d) u|Ω=0,v|Ω=0,t(0,T]. (2.1)

    Multiplying both sides of (2.1)(a) by φH10(Ω) and using Green's formula, we have

    D(u,φ)+m(vt,φ)+λ(v,φ)=(f,φ),φH10(Ω).

    Multiplying both sides of (2.1)(b) by ψH10(Ω) and using Green's formula, we obtain

    (ut,ψ)(v,ψ)=0,ψH10(Ω).

    Therefore, we have the following mixed weak formulation of (2.1) : find {u,v}:[0,T]H10(Ω)×H10(Ω), such that

    {(a) D(u,φ)+m(vt,φ)+λ(v,φ)=(f,φ),φH10(Ω),(b) (ut,ψ)(v,ψ)=0,ψH10(Ω),(c) u(x,y,0)=ΔΦ(x,y),v(x,y,0)=Ψ(x,y),(x,y)Ω,(d) u|Ω=0,v|Ω=0,t(0,T]. (2.2)

    First, we define the finite element space. Let Ω be a rectangular region whose boundaries are parallel to the two axes. The region Ω is divided into regular triangulation. ȷh is a triangulation family whose region satisfies the regular hypothesis, K represents the triangulation unit, and h is the maximum diameter of the subdivision unit. Ω=KϵȷhK, Sh={vhvhKPk(K),Kϵȷh}H1(Ω) is the finite element space composed of piecewise linear degree polynomials on ȷh.Then, the corresponding semi-discrete finite element scheme of (2.2) is to find {uh,vh}:[0,T]S0h×S0h, S0h=ShH10(Ω), such that

    {(a) D(uh,φh)+m(vht,φh)+λ(vh,φh)=(f,φh),φhS0h(Ω),(b) (uht,ψh)(vh,ψh)=0,ψhS0h(Ω),(c) uh(x,y,0)=Rhu(x,y,0),vh(x,y,0)=Rhv(x,y,0),(x,y)Ω,(d) uh|Ω=0,vh|Ω=0,t(0,T]. (3.1)

    Rh is an elliptic projection operator, which will be given below. The existence and uniqueness of semi-discrete finite element approximation scheme solutions and error analysis are given below.

    Theorem 3.1. Existence and uniqueness of the solution of the semi-discrete finite element approximation scheme (3.1).

    Proof. {ϕi}Mi=1 be a set of bases of S0h. Then uh = Mj=1ujϕj, vh = Mj=1vjϕj. According to (3.1)(a) and (3.1)(b), we have the following equalities

    DAU(t)+mBdV(t)dt+λBV(t)=F, (3.2)
    BdU(t)dtAV(t)=0. (3.3)

    Where U(t)=(u1(t),u2(t),,uN(t))T,

    V(t)=(v1(t),v2(t),,vN(t))T, A=(ϕj,ϕi),

    B=(ϕj,ϕi), F=(f,ϕi).

    According to (3.3), we deduce that

    V(t)=A1BdU(t)dt. (3.4)

    Substituting (3.4) into (3.2), we arrive at

    mBA1Bd2U(t)dt+λBA1BdU(t)dt+DAU(t)=F(t). (3.5)

    U(0) can be determined by uh(x,y,0), and (3.5) is an ordinary differential equation about vector U(t). A,BA1B are symmetric positive definite matrices. According to the theory of ordinary differential equations, it is easy to know that the solution of the semi-discrete finite element approximation scheme is existent and unique.

    In the following discussion, we will derive the proof of the error estimates for semi-discrete schemes. For carrying out an analysis, we need to introduce a useful lemma. First, to give the error analysis, for 0tT, we consider the elliptic projection operator Rh:H10S0h such that ((uRhu),vh)=0,vhS0h, which leads to the following estimate inequality.

    Lemma 3.1. [32] uHk+10, such that

    uRhu+huRhu1Chk+1uk+1. (3.6)

    Corollary 3.1. uHk+10, such that

    utRhut∥≤Chk+1utk+1. (3.7)

    Lemma 3.2. [33] The family Sh is based on a family of quasiuniform triangulations ȷh and Sh consists of piecewise polynomials of degree at most k1, and then one may show the inverse inequality:

    uh∥≤Ch1uh,uϵSh. (3.8)

    In the next analysis, we will discuss the proof of semi-discrete error estimates based on the elliptic projection in detail.

    Theorem 3.2. Let {u,v} and {uh,vh} be the solutions of (2.2)(a) and (2.2)(b) and (3.1)(a) and (3.1)(b), respectively, we have L2-mode and H1-mode error estimations of variable {u,v}:

    uuh2Ch2k+2(t0(vt2k+1+v2k+1+ut2k+1)ds+u2k+1), (3.9)
    vvh2Ch2k+2(t0(vt2k+1+v2k+1+ut2k+1)ds+v2k+1), (3.10)
    (uuh)∥≤Chk(uk+1+t0(vtk+1+vk+1+utk+1)ds), (3.11)
    (vvh)∥≤Chk(vk+1+t0(vtk+1+vk+1+utk+1)ds). (3.12)

    Proof. To simplify, we now rewrite the errors as uuh=(uRhu)+(Rhuuh)=ρ+θ,vvh=(vRhv)+(Rhvvh)=η+ξ.

    φh,ψhS0h, subtracting (2.2)(a) from (3.1)(a), subtracting (2.2)(b) from (3.1)(b), and applying the elliptic projection operator, we have the error equation:

    (θ,φh)+m(ηt,φh)+m(ξt,φh)+λ(η,φh)+λ(ξ,φh)=0, (3.13)
    (ρt,ψh)+(θt,ψh)(ξ,ψh)=0. (3.14)

    Choosing φh=ξ,ψh=θ, add (3.13) and D×(3.14), we have

    m2ddtξ2+D2ddtθ2+λξ2=(m(ηt,ξ)+λ(η,ξ)+D(ρt,θ)). (3.15)

    The Young inequality with ε and corollary 3.1 being applied to (3.15), we easily obtain

    m2ddtξ2+D2ddtθ2Ch2k+2(12λm2vt2k+1+λ2v2k+1+D2ut2k+1)+D2θ2. (3.16)

    Integrating from 0 to t on both sides of (3.16), because ξ(0)=θ(0)=0, we have

    mξ2+Dθ2Ch2k+2+Dt0θ2dst01λm2vt2k+1+λv2k+1+Dut2k+1ds.

    We use Gronwall inequality to get

    mξ2+Dθ2Ch2k+2t01λm2vt2k+1+λv2k+1+Dut2k+1ds. (3.17)

    Thus, we have L2-mode error estimation of variable {u,v}:

    Rhuuh2+Rhvvh2Ch2k+2t0vt2k+1+v2k+1+ut2k+1ds. (3.18)

    Using lemma 3.1 and triangle inequality, we finish the proof of (3.9)and(3.10).

    Theorem 3.3. Let {u,v} and {uh,vh} be the solutions of (2.1) and (2.2), respectively. When {u,v} is smooth enough, we have the error estimation of variable {ut,vt}:

    Rhutuht2+Rhvtvht2Ch2k+2t0vtt2k+1+utt2k+1+vt2k+1ds. (3.19)

    Similar to Theorem 3.2, we give a simple proof.

    Proof. First, taking the derivative of the variable t of the error equation (3.13)(3.14), we obtain

    D(θt,φh)+m(tηt,φh)+m(tξt,φh)+λ(ηt,φh)+λ(ξt,φh)=m(Rvt,φh), (3.20)
    (tρt,ψh)+(tθt,ψh)(ξt,ψh)=(Rut,ψh). (3.21)

    Choosing φh=ξt in (3.20), ψh=θt in (3.21), we have

    D(θt,ξt)+m(tηt,ξt)+m(tξt,ξt)+λ(ηt,ξt)+λ(ξt,ξt)=m(Rvt,ξt),(tρt,θt)+(tθt,θt)(ξt,θit)=(Rut,θt).

    With the same method as theorem 3.2, we easily obtain

    Rhutuht2+Rhvtvht2Ch2k+2t0vtt2k+1+utt2k+1+vt2k+1ds. (3.22)

    Using lemma 3.1 and inverse inequality, we have H1-mode error estimation of variable u:

    (uuh)∥≤∥ρ+θ∥≤Chkuk+1+Ch1θChkuk+1+Chk(t0vt2k+1+v2k+1+ut2k+1ds)Chk(uk+1+t0vtk+1+vk+1+utk+1ds).

    In the same way, we have H1-mode error estimation of variable v:

    (vvh)∥≤Chk(vk+1+t0vtk+1+vk+1+utk+1ds). (3.23)

    Hence, we finish the proof of theorem 3.2.

    Let 0=t0<t1<<tN=T be the subdivision of step τ=TN in time interval [0,T], tn=nτ,n=0,1,N, UnS0h stand for the approximation of u(tn), when t=tn=nτ. For any function ϕ on [0,T], define:

    ϕn=ϕ(tn),tϕn=(ϕnϕn1)/τ,

    Choosing t=tn, we have a format equivalent to (2.1):

    {(a) D(un,φ)+m(tvn,φ)+λ(vn,φ)=(fn,φ)+m(Rnv,φ),φH10(Ω),(b) (tun,ψ)(vn,ψ)=(Rnu,ψ),ψH10(Ω),(c) u(x,y,0)=ΔΦ(x,y),v(x,y,0)=Ψ(x,y),(x,y)Ω,(d) u|Ω=0,v|Ω=0,t(0,T]. (4.1)

    Where Rnu=tununt=1τtntn1(tn1s)utt(s)ds,Rnv=tvnvnt=1τtntn1(tn1s)vtt(s)ds.

    Then, the fully discrete finite element approximation scheme is described as: find {Un,Vn}: [0,T]S0h×S0h, S0h=ShH10(Ω), such that

    {(a) D(Un,φh)+m(tVn,φh)+λ(Vn,φh)=(fn,φh),φhS0h(Ω),(b) (tUn,ψh)(Vn,ψh)=0,ψhS0h(Ω),(c) U0(x,y)=Rhu(x,y,0),V0(x,y)=Rhv(x,y,0),(x,y)Ω,(d) U|Ω=0,V|Ω=0,t(0,T]. (4.2)

    Similarly, we give proof of the existence and uniqueness of the fully discrete finite element scheme solution and error analysis.

    Theorem 4.1. Existence and uniqueness of the solution of the fully discrete finite element approximation scheme (4.2).

    Proof. Let {ϕi}Mi=1 be a set of bases of S0h. We have Un=Mi=1uniϕi,Vn=Mi=1vniϕi. According to (4.2)(a) and (4.2)(b), we have

    τDAUn+(mB+τλB)VnmBVn1=τFn, (4.3)
    BUnτAVnBUn1=0, (4.4)

    where

    Un=(un1,un2,,unN)T,Vn=(vn1,vn2,,vnN)T,A=(ϕj,ϕi),B=(ϕj,ϕi),F=(fn,ϕi).

    According to (4.4), we easily arrive at

    Vn=1τA1B(UnUn1). (4.5)

    Substitute (4.5) into (4.3) to obtain

    (τDA+1τmBA1B+λBA1B)Un=τFn1τmBA1BUn2+(1τmBA1B+λBA1B)Un1. (4.6)

    U0 can be determined by Rhu(x,y,0). A,BA1B are symmetric positive definite matrices, so the solution of (4.6) is existent and unique, and the solution of (4.5) is existent and unique. The existence and uniqueness of the solution are equivalent to problem (4.2)(a) and (4.2)(b).

    Theorem 4.2. Let {un,vn} and {Un,Vn} be the solutions of (4.1) and (4.2), respectively, we have L2-mode error estimation of variable {un,vn}:

    unUn2+vnVn2Ch2k+2(t0vt2k+1+ut2k+1+v2k+1ds+u2k+1+v2k+1)+Cτ2t0vtt2+utt2ds, (4.7)

    Proof. To simplify, we now rewrite the errors as uiUi=(uiRhui)+(RhuiUi)=ρi+θi,viVi=(viRhvi)+(RhviVi)=ηi+ξi.

    φh,ψhS0h, subtracting (4.1)(a) from (4.2)(a), subtracting (4.1)(b) from (4.2)(b), and applying elliptic projection operator, we have the error equation:

    D(θi,φh)+m(tηi,φh)+m(tξi,φh)+λ(ηi,φh)+λ(ξi,φh)=m(Riv,φh), (4.8)
    (tρi,ψh)+(tθi,ψh)(ξi,ψh)=(Riu,ψh). (4.9)

    Let φh=ξi,ψh=θi. Adding (4.8) and D×(4.9), we have

    m(tξi,ξi)+λ(ξi,ξi)+D(tθi,θi)=(m(tηi,ξi)+λ(ηi,ξi)+D(tρi,θi))+m(Riv,ξi)+D(Riu,θi)=5i=1Mi. (4.10)

    Where

    m(tξi,ξi)=m2τ(ξi2ξi12+ξiξi12),D(tθi,θi)=D2τ(θi2θi12+θiθi12),λ(ξi,ξi)=∥ξi2.

    Let's estimate Mi in turn:

    Using the Young inequality with ε, lemma 3.1 and corollary 3.1, we obtain

    M13m24λτ2titi1ηtds2+λ3ξi23m24λτtiti1ηt2ds+λ3ξi2C3m24λτh2k+2titi1vt2k+1ds+λ3ξi2,M23λ4ηi2+λ3ξi2C3λ4h2k+2v2k+1+λ3ξi2,M3Dτ2titi1ρtds2+D4θi2CDτh2k+2titi1ut2k+1ds+D4θi2.

    Using Cauchy-Schwarz inequality and Young inequality with ε, we have

    M43m24λRiv2+λ3ξi23m24λτ1titi1(ti1s)vttds2+λ3ξi23m24λτ1[titi1(ti1s)2ds]12[titi1v2ttds]122+λ3ξi23m24λτtiti1vtt2ds+λ3ξi2,M5DRiu2+D4θi2Dτtiti1u2ttds+D4θi2.

    Substituting them into (4.10), we have

    m(ξi2ξi12+D(θi2θi12)Ch2k+2(titi1vt2k+1+ut2k+1ds)+Dτθi2+Cτ2(titi1vtt2+utt2ds)+Cτh2k+2v2k+1.

    Sum the above formula about i from 1 to n. Noticing that ξ(0)=θ(0)=0, we have

    mξn2+(DDτ)θn2Ch2k+2(t0vt2k+1+ut2k+1ds)+Dτni=1θi12+Ch2k+2v2k+1+Cτ2t0vtt2+utt2ds.

    Using Gronwall Lemma, we have τ sufficiently small

    mξn2+(DDτ)θn2Ch2k+2(t0vt2k+1+ut2k+1ds+v2k+1)+Cτ2(t0vtt2+utt2ds). (4.11)

    Thus, we have L2-mode error estimation of variable {un,vn}:

    ξn2+θn2Cτ2(t0vtt2+utt2ds)+Ch2k+2(t0vt2k+1+ut2k+1ds+v2k+1). (4.12)

    Using lemma 3.1 and the triangle inequality, we finish the proof of theorem 4.2.

    Next, we give the H1-mode error estimate of {un,vn}.

    Theorem 4.3. Letting {un,vn} and {Un,Vn} be the solutions of (4.1) and (4.2), respectively, we have H1-mode error estimation of variable {un,vn}:

    uiUi∥≤Chk+Chk+1+Cτ, (4.13)
    viVi∥≤Chk+Chk+1+Cτ. (4.14)

    Proof. Choosing φh=θi in (4.8), we have

    D(θi,θi)+m(tηi,θi)+m(tξi,θi)+λ(ηi,θi)+λ(ξi,θi)=m(Riv,θi).

    Which leads to

    Dθi2=m(tηi,θi)m(tξi,θi)λ(ηi,θi)λ(ξi,θi)+m(Riv,θi)=5j=1Mj. (4.15)

    The estimate of Mj is as follows:

    using Cauchy-Schwarz inequality, Young inequality with ε and corollary 3.1, we obtain

    M15m24τ2titi1ηtds2+15θi25m24τtiti1ηt2ds+15θi2C5m24τh2k+2titi1vt2k+1ds+15θi2C5m24h2k+2vt2k+1+15θi2. (4.16)

    Using Cauchy-Schwarz inequality, Young inequality with ε and Theorem 3.3, we get

    M25m24τ2titi1ξtds2+15θi25m24τtiti1ξt2ds+15θi215θi2+C5m24h2k+2t0vtt2k+1+utt2k+1+vt2k+1ds. (4.17)

    Using Young inequality with ε and Lemma 3.1, we have

    M35λ24ηi2+15θi2C5λ24h2k+2v2k+1+15θi2. (4.18)

    Using Young inequality with ε, we deduce that

    M45λ24ξi2+15θi2. (4.19)

    Using Cauchy-Schwarz inequality and Young inequality with ε, we get

    M55m24Riv2+15θi25m24τ1titi1(ti1s)vttds2+15θi25m24τ1[titi1(ti1s)2ds]12[titi1v2ttds]122+15θi25m24τtiti1vtt2ds+15θi25m24τ2vtt2+15θi2. (4.20)

    Combining (4.16)(4.20) and using Theorem 4.2, we have

    θi2Ch2k+2(vt2k+1+t0vtt2k+1+utt2k+1+vt2k+1dsv2k+1+t0vt2k+1+ut2k+1ds)+Cτ2vtt2 (4.21)

    Using lemma 3.1 and (4.21), we get

    uiUi∥≤∥ρ+θ∥≤Chk+Chk+1+Cτ. (4.22)

    Choosing ψh=ξi in (4.9), we obtain

    (tρi,ξi)+(tθi,ξi)(ξi,ξi)=(Riu,ξi).

    Which leads to

    ξi2=(tρi,ξi)+(tθi,ξi)(Riu,ξi). (4.23)

    We estimate the terms on the right-hand side of (4.23) one by one. Using Cauchy-Schwarz inequality, Young inequality with ε and corollary 3.1, we obtain

    (tρi,ξi)34τ2titi1ρtds2+13ξi234τtiti1ρt2ds+13ξi234τCh2k+2titi1ut2k+1ds+13ξi234Ch2k+2ut2k+1+13ξi2. (4.24)

    Using Cauchy-Schwarz inequality, Young inequality with ε and Theorem 3.3, we obtain

    (tθi,ξi)34τ2titi1θtds2+13ξi213ξi2+34τtiti1θt2ds13ξi2+34τCh2k+2titi1t0vtt2k+1+utt2k+1+vt2k+1dsdt34Ch2k+2t0vtt2k+1+utt2k+1+vt2k+1ds+13ξi2. (4.25)
    (Riu,ξi)34Riu2+13ξi234τ1titi1(ti1s)uttds2+13ξi234τ1[titi1(ti1s)2ds]12[titi1u2ttds]122+13ξi234τtiti1utt2ds+13ξi234τ2utt2+13ξi2. (4.26)

    Combining (4.24)(4.26) and using Theorem 4.2, it holds that

    ξi234Ch2k+2t0vtt2k+1+utt2k+1+vt2k+1ds+34Ch2k+2ut2k+1+34τ2utt2+Ch2k+2t0vt2k+1+ut2k+1+v2k+1dsCh2k+2+Cτ2. (4.27)

    Using corollary 3.2 and (4.27), we have

    viVi∥≤∥ηi+ξi∥≤Chk+Chk+1+Cτ. (4.28)

    In this section, we provide numerical examples to validate the backward Euler full discretization mixed finite element scheme (4.2) for the vibration problems of damped plates (2.1). We not only validate the convergence order of the error estimate, but also simulate the vibration of damped plates to quantify the influence of damping coefficient on the frequency and amplitude of vibration.

    Example 1

    For the numerical calculation, let the space domain be Ω=[0,4]×[0,4] and let the time domain be [0,T]=[0,1]. Let D=1,m=1,λ=1. The exact solution to the vibration problem of the damped plate (2.1) is w = costsin(π4x)sin(π4y). The source term f(x,y,t) can be obtained by inserting the given exact solution into the vibration equation (2.1). The mixed finite element space is a double linear first-order polynomial. Keep the time step size τ=1100000 constant while varying the space step size hx=hy=12,14,18,116. Tables 1 and 2 show the space errors and convergence orders, respectively, of the L2norm and H1norm of the solutions to the backward Euler full discretization mixed finite element scheme (4.2). Keep the space step size hx=hy=11024 constant while varying the time step size τ=14,18,116,132. Tables 3 and 4 show the time errors and convergence orders, respectively, of the L2norm and H1norm of the solutions to the backward Euler full discretization mixed finite element scheme (4.2). The second and third columns in Tables 1 and 2 show the space errors of the L2norm and H1norm for the solutions to the backward Euler full discretization mixed finite element scheme (4.2), respectively. The fourth and fifth columns show their corresponding space convergence orders. The second and third columns in Tables 3 and 4 show the time errors of the L2norm and H1norm for the solutions to the backward Euler full discretization mixed finite element scheme (4.2), respectively. The fourth and fifth columns show their corresponding time convergence orders.

    Table 1.  H1-mode and L2-mode errors of u.
    h1 L2norm H1norm convergence order of L2 convergence order of H1
    2 1.1563e-01 2.9644e-01
    4 2.9373e-02 1.4610e-01 1.9770 1.0208
    8 7.3786e-03 7.2778e-02 1.9931 1.0054
    16 1.8518e-03 3.6355e-02 1.9944 1.0013

     | Show Table
    DownLoad: CSV
    Table 2.  H1-mode and L2-mode errors of v.
    h1 L2norm H1norm convergence order of L2 convergence order of H1
    2 3.9781e-02 3.6893e-01
    4 1.0492e-02 1.8383e-01 1.9228 1.0050
    8 2.6594e-03 9.1799e-02 1.9801 1.0018
    16 6.6963e-04 4.5884e-02 1.9897 1.0005

     | Show Table
    DownLoad: CSV
    Table 3.  H1-mode and L2-mode errors of u.
    τ1 L2norm H1norm convergence order of L2 convergence order of H1
    4 1.3495e-01 1.4990e-01
    8 7.5280e-02 8.3643e-02 0.84209 0.84168
    16 3.9851e-02 4.4319e-02 0.91765 0.91632
    32 2.0514e-02 2.2895e-02 0.95801 0.95289

     | Show Table
    DownLoad: CSV
    Table 4.  H1-mode and L2-mode errors of v.
    τ1 L2norm H1norm convergence order of L2 convergence order of H1
    4 1.6961e-01 1.8840e-01
    8 8.9051e-02 9.8948e-02 0.92918 0.92906
    16 4.5534e-02 5.0652e-02 0.96769 0.96605
    32 2.3006e-02 2.5710e-02 0.98493 0.97829

     | Show Table
    DownLoad: CSV

    The tables illustrate that the space convergence orders are 2 or 1, while the time convergence orders are uniformly 1, for the L2norm and H1norm of the solutions to the backward Euler full discretization mixed finite element scheme (4.2) for the vibration problems of damped plates (2.1). This is consistent with the theoretical results, and hence the conclusions of the theorem are validated.

    When spatial step h = 11024, w = costsin(π4x)sin(π4y), we have

    Example 2

    In this numerical example, we not only simulate the vibration of damped plates, but also validate the influence of damping coefficient on the frequency and amplitude of the vibration.

    First, let D=100,m=5,λ=40, and the external force f=0. Let the non-zero initial displacement of plate vibration be w = sin(π4x)sin(π4y). Vibrations at different moments are simulated. The vibration patterns at t=0.05,t=0.2,t=0.3,t=1,t=3andt=5 are shown in Figure 1, respectively.

    Figure 1.  Simulation at different time under free vibration.

    By comparing Figure 1, it is noticed that the amplitude of vibration decreases over time. From t=3, the amplitude changes increasingly slowly until it stabilizes at a fixed value.

    Then, let D=10,m=20,λ=40, the initial vibration displacement w=0, and the duration of external force = 0.1, i.e.

    {f=100t0.1,f=00.1<t5. (5.1)

    The change in vibration amplitude over time is studied. The vibration patterns at t=0.05,t=0.1,t=0.2,t=0.5,t=2andt=2.5 are shown in Figure 2, respectively.

    Figure 2.  Vibration simulation at different time when external force is applied.

    By comparing Figure 2, it is observed that the amplitude increases from t=0.05 to t=0.5, and then starts to decrease and eventually stabilizes.

    Finally, let D=10,m=20, the initial vibration displacement w=0, and the duration of external force = 0.1, i.e.

    {f=100t0.1,f=00.1<t5. (5.2)

    The influence on vibration amplitude by changing the damping coefficient is studied. The damping coefficient is set at 10, 20,160 and 640. When t=0.15, the vibration patterns when the damping coefficient is 10, 20,160 and 640 are shown in Figure 3, respectively. The influence of changing the damping coefficient on the vibration frequency is also studied. When the damping coefficient is 10, 20, 40 and 80, the changes of a certain point on the plate as a function of time are shown in Figure 4, respectively.

    Figure 3.  Simulation of plate vibration under different damping coefficients.
    Figure 4.  Simulation of plate center vibration under different damping coefficients.

    By comparing Figure 3, it is observed that when the external force is constant, a greater damping coefficient leads to a smaller vibration amplitude. The comparison between Figure 4 suggests that when the external force is constant, a greater damping coefficient leads to a lower frequency.

    In this article, we propose the semi-discrete and fully discrete finite element approximation schemes for the vibration equations of damped plates. The existence and the uniqueness of the solution are verified, and the order of convergence of errors is deduced. Moreover, the theoretical analysis is validated by numerical case studies, the pattern of plate vibration is simulated, and the influence of the damping coefficient on the frequency and amplitude of the plate vibration is elucidated. In the future, we attempt to discretize the time using the C-N scheme and approximate the space using elements of higher orders to obtain numerical solutions of higher precision while reducing the calculation load, in order to further improve the simulation of vibration problems of damped plates.

    The research was supported by the NSFC of China (No. 12171287) and the NSFC of Shandong Province (No. ZR2021MA063).

    The authors declare that they have no conflicts of interest to this work.



    [1] Drainage and Irrigation Department Malaysia (2004) Annual flood report of DID for Peninsular Malaysia. DID: Kuala Lumpur. Available from: http://www.statistics.gov.my/eng/images/stories/files/journalDOSM/V104ArticleJamaliah.pdf.
    [2] Malaysian Meteorological Department (2007) Report on Heavy Rainfall that Caused Floods in Kelantan and Terengganu. MMD: Kuala Lumpur. Available from: https://reliefweb.int/sites/reliefweb.int/files/resources/EE19DAFDE99078B649257266001FED46-Full_Report.pdf.
    [3] Adnan NA, Atkinson PM (2011) Exploring the impact of climate and land use changes on streamflow trends in a monsoon catchment. Int J Clim 31: 815-831. doi: 10.1002/joc.2112
    [4] Chan NW (1997) Institutional arrangement of flood hazard management in Malaysia: an evaluation using criteria approach. Disasters 21: 206-222. doi: 10.1111/1467-7717.00057
    [5] Hussain STPR, Ismail H (2013) Flood frequency analysis of Kelantan River Basin, Malaysia. World Appl Sci J 28: 1989-1995.
    [6] Nashwan MS, Ismail T, Ahmed K (2018) Flood susceptibility assessment in Kelantan river basin using copula. Int J Eng Technol 7: 584-590. doi: 10.14419/ijet.v7i2.10447
    [7] Zhang L, Singh VP (2006) Bivariate flood frequency analysis using copula method. J Hydrol Eng 11: 150-164. doi: 10.1061/(ASCE)1084-0699(2006)11:2(150)
    [8] Zhang L (2005) Multivariate hydrological frequency analysis and risk mapping. Doctoral dissertation, Beijing Normal University.
    [9] Reddy MJ, Ganguli P (2012) Bivariate Flood Frequency Analysis of Upper Godavari River Flows Using Archimedean Copulas. Water Resour Manage 26: 3995-4018. doi: 10.1007/s11269-012-0124-z
    [10] Bobee B, Rasmussen PF (1994) Statistical analysis of annual flood series, In: Menon J (Ed.). Trend in Hydrology, 1. Council of Scientific Research Integration, India, 117-135.
    [11] Krstanovic PF, Singh VP (1987) A multivariate stochastic flood analysis using entropy. In: Singh VP (Ed.). Hydrologic Frequency Modelling, Reidel, Dordrecht, 515-539. doi: 10.1007/978-94-009-3953-0_37
    [12] Yue S (2000) The bivariate lognormal distribution to model a multivariate flood episode. Hydrol Process 14: 2575-2588. doi: 10.1002/1099-1085(20001015)14:14<2575::AID-HYP115>3.0.CO;2-L
    [13] Sandoval CE, Raynal-Villasenor J (2008) Trivariate generalized extreme value distribution in flood frequency analysis. Hydrol Sci J 53: 550-567. doi: 10.1623/hysj.53.3.550
    [14] Song S, Singh VP (2010) Metaelliptical copulas for drought frequency analysis of periodic hydrologic data. Stoch Environ Res Risk Assess 24: 425-444. doi: 10.1007/s00477-009-0331-1
    [15] De Michele C, Salvadori G (2003) A generalized Pareto intensity-duration model of storm rainfall exploiting 2-copulas. J Geophys Res 108: 4067. doi: 10.1029/2002JD002534
    [16] Saklar A (1959) Functions de repartition n dimensions et leurs marges. ublications de l'Institut Statistique de l'Université de Paris, 8: 229-231.
    [17] Nelsen RB (2006) An introduction to copulas. Springer, New York.
    [18] Salvadori G (2004) Bivariate return periods via-2 copulas. Stat Methodol 1:129-144. doi: 10.1016/j.stamet.2004.07.002
    [19] Salvadori G, De Michele C (2004) Frequency analysis via copulas: theoretical aspects and applications to hydrological events. Water Resour Res 40: W12511. doi: 10.1029/2004WR003133
    [20] Salvadori G, De Michele C (2006) Statistical characterization of temporal structure of storms. Adv Water Resour 29: 827-842. doi: 10.1016/j.advwatres.2005.07.013
    [21] Cong RG, Brady M (2011) The interdependence between Rainfall and Temperature: copula Analyses. Sci World J 2012: 405675.
    [22] Karmakar S, Simonovic SP (2008) Bivariate flood frequency analysis. Part 1: Determination of marginal by parametric and non-parametric techniques. J Flood Risk Manag 1: 190-200.
    [23] Adamowski K (1989) A monte Carlo comparison of parametric and nonparametric estimations of flood frequencies. J Hydrol 108: 295-308. doi: 10.1016/0022-1694(89)90290-4
    [24] Silverman BW (1986) Density Estimation for Statistics and Data Analysis, 1st edition. Chapman and Hall, London.
    [25] Kim KD, Heo JH (2002) Comparative study of flood quantiles estimation by nonparametric models. J Hydrol 260: 176-193. doi: 10.1016/S0022-1694(01)00613-8
    [26] Botev ZI, Grotowski JF, Kroese DP (2010) Kernel Density Estimation via Diffusion. Ann Stat 38: 2916-2957. doi: 10.1214/10-AOS799
    [27] Dooge JCE (1986) Looking for hydrologic laws. Water Resour Res 22: 46-58. doi: 10.1029/WR022i09Sp0046S
    [28] Bardsley WE (1988) Toward a General Procedure for Analysis of Extreme Random Events in the Earth Sciences. Math Geol 20: 513-528. doi: 10.1007/BF00890334
    [29] Lall U, Moon YI, Bosworth K (1993) kernel flood frequency estimators: Bandwidth selection and kernel choice. Water Resour Res 29: 1003-1015. doi: 10.1029/92WR02466
    [30] Santhosh D, Srinivas V (2013) Bivariate frequency analysis of flood using a diffusion kernel density estimators. Water Resour Res 49: 8328-8343. doi: 10.1002/2011WR010777
    [31] Moon YI, Lall U (1994) Kernel function estimator for flood frequency analysis. Water Resour Res 30: 3095-3103. doi: 10.1029/94WR01217
    [32] Lall U (1995) Nonparametric function estimation: recent hydrologic contributions, U.S. National Republic. International Union of Geodesy and Geophysics, 1991-1994. Rev Geophys 33: 1093-1099.
    [33] Karmakar S, Simonovic SP (2009) Bivariate flood frequency analysis. Part 2: A copula-based approach with mixed marginal distributions. J Flood Risk Manag 2: 32-44.
    [34] Chen SX, Huang TM (2007) Nonparametric estimation of copula functions for dependence modelling. Can J Stat 35: 265-282. doi: 10.1002/cjs.5550350205
    [35] Latif S, Mustafa F (2020) Trivariate distribution modelling of flood characteristics using copula function-A case study for Kelantan River basin in Malaysia. AIMS Geosci 6: 92-130. doi: 10.3934/geosci.2020007
    [36] Hosking JRM, Walis JR (1987) Parameter and quantile estimations for the generalized Pareto distributions. Technometrics 29: 339-349. doi: 10.1080/00401706.1987.10488243
    [37] Yue S, Rasmussen P (2002) Bivariate frequency analysis: discussion of some useful concepts in hydrological applications. Hydrol Process 16: 2881-2898. doi: 10.1002/hyp.1185
    [38] Rao AR, Hamed KH (2000) Flood frequency analysis. CRC Press, Boca Raton, Fla.
    [39] Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27: 832-837. doi: 10.1214/aoms/1177728190
    [40] Scott DW (1992) Multivariate Density estimation: Theory, Practice and Visualization. Wiley, New York.
    [41] Härdle W (1991) Smoothing Technique with Implementation in S. Springer, New York.
    [42] Kim KD, Heo JH (2002) Comparative study of flood quantiles estimation by nonparametric models. J Hydrol 260: 176-193. doi: 10.1016/S0022-1694(01)00613-8
    [43] Shabri A (2002) Nonparametric Kernel Estimation of Annual Maximum Stream Flow Quantiles, Matematika, 18: 99-107.
    [44] Miladinovic B (2008) Kernel density estimation of reliability with applications to extreme value distribution. Graduate Theses and Dissertations. Available from: https://scholarcommons.usf.edu/etd/408.
    [45] Azzalini A (1981) A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68: 326-328. doi: 10.1093/biomet/68.1.326
    [46] Shiau JT (2006) Fitting drought duration and severity with two dimensional copulas. Water Resour Manag 20: 795-815. doi: 10.1007/s11269-005-9008-9
    [47] Harrell FE, Davis CE (1982) A new distribution-free quantile estimator. Biometrika 69: 635-640. doi: 10.1093/biomet/69.3.635
    [48] Brown BM, Chen SX (1999) Beta-bernstein smoothing for regression curves with compact support. Scand J Stat 26: 47-59. doi: 10.1111/1467-9469.00136
    [49] Chen SX (2000) Beta kernel estimators for density functions. Comput Stat Data Anal 31: 131-145. doi: 10.1016/S0167-9473(99)00010-9
    [50] Bounezmarni T, Rombouts JVK (2009) Nonparametric density estimation for positive time series. Comput Stat Data Anal 54: 245-261. doi: 10.1016/j.csda.2009.08.016
    [51] Charpentier A, Fermanian JD, Scaillet O (2006) The estimation of copulas: Theory and practice. In Rank J, editor. Copulas: From theory to application in finance. London: Risk Books, 35-64.
    [52] Kim TW, Valdés JB, Yoo C (2006) Nonparametric approach for bivariate drought characterisation using Palmer drought index. J Hydrol Eng 11: 134-143. doi: 10.1061/(ASCE)1084-0699(2006)11:2(134)
    [53] Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22: 79-86. doi: 10.1214/aoms/1177729694
    [54] Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19: 716-723. doi: 10.1109/TAC.1974.1100705
    [55] Schwarz GE (1978) Estimating the dimension of a model. Ann Stat 6: 461-464. doi: 10.1214/aos/1176344136
    [56] Hannan EJ, Quinn BG (1979) The Determination of the Order of an Autoregression. J R Stat Soc Ser B 41: 190-195.
    [57] Shiau JT (2003) Return period of bivariate distributed extreme hydrological events. Stoch Environ Res Risk Assess 17: 42-57. doi: 10.1007/s00477-003-0125-9
    [58] Brunner MI, Seibert J, Favre AC (2016) Bivariate return periods and their importance for flood peak and volume estimations. WIREs Water 3: 819-833. doi: 10.1002/wat2.1173
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