Loading [MathJax]/jax/output/SVG/jax.js
Research article Special Issues

Higher-order convergence analysis for interior and boundary layers in a semi-linear reaction-diffusion system networked by a k-star graph with non-smooth source terms

  • Received: 15 July 2024 Revised: 13 September 2024 Accepted: 30 September 2024 Published: 09 October 2024
  • We investigated a nonlinear singularly perturbed elliptic reaction-diffusion coupled system having non-smooth data networked by a k-star graph. We considered all possible boundary conditions at the free boundary located at the tail of the edge and imposed the continuity condition with Kirchhoff's junction law at the junction point of the k-star graph to obtain a continuous solution for this coupled system. First, we showed the existence and uniqueness of the solution using the variational formulation approach. Then, we reformulated it into a minimization problem over a function space to conclude the uniqueness of the solution. For the approximation of the continuous problem, note that the upwind scheme for the flux condition at the free boundary leads to a parameter uniform first-order approximation. To obtain a higher-order uniform accuracy, we utilized a three-point scheme for first-order derivatives and a five-point approximation at the point of discontinuity. These approximations typically did not yield an M-matrix or strict diagonally dominant structure of the stiffness matrix. Hence, we provided a suitable transformation that could lead to a sufficient condition for preserving the strict diagonally dominant structure of the stiffness matrix. We performed a comprehensive convergence analysis to demonstrate the almost second-order uniform accuracy on each edge of the k-star graph. Numerical experiments highly validate the theory on the k-star graph.

    Citation: Dilip Sarkar, Shridhar Kumar, Pratibhamoy Das, Higinio Ramos. Higher-order convergence analysis for interior and boundary layers in a semi-linear reaction-diffusion system networked by a k-star graph with non-smooth source terms[J]. Networks and Heterogeneous Media, 2024, 19(3): 1085-1115. doi: 10.3934/nhm.2024048

    Related Papers:

    [1] Marcelo Menezes Morato, Vladimir Stojanovic . A robust identification method for stochastic nonlinear parameter varying systems. Mathematical Modelling and Control, 2021, 1(1): 35-51. doi: 10.3934/mmc.2021004
    [2] Anil Chavada, Nimisha Pathak, Sagar R. Khirsariya . A fractional mathematical model for assessing cancer risk due to smoking habits. Mathematical Modelling and Control, 2024, 4(3): 246-259. doi: 10.3934/mmc.2024020
    [3] Wen Zhang, Jinjun Fan, Yuanyuan Peng . On the discontinuous dynamics of a class of 2-DOF frictional vibration systems with asymmetric elastic constraints. Mathematical Modelling and Control, 2023, 3(4): 278-305. doi: 10.3934/mmc.2023024
    [4] Ihtisham Ul Haq, Nigar Ali, Hijaz Ahmad . Analysis of a chaotic system using fractal-fractional derivatives with exponential decay type kernels. Mathematical Modelling and Control, 2022, 2(4): 185-199. doi: 10.3934/mmc.2022019
    [5] Yulin Guan, Xue Zhang . Dynamics of a coupled epileptic network with time delay. Mathematical Modelling and Control, 2022, 2(1): 13-23. doi: 10.3934/mmc.2022003
    [6] Xue Zhang, Bo Sang, Bingxue Li, Jie Liu, Lihua Fan, Ning Wang . Hidden chaotic mechanisms for a family of chameleon systems. Mathematical Modelling and Control, 2023, 3(4): 400-415. doi: 10.3934/mmc.2023032
    [7] Hassan Alsuhabi . The new Topp-Leone exponentied exponential model for modeling financial data. Mathematical Modelling and Control, 2024, 4(1): 44-63. doi: 10.3934/mmc.2024005
    [8] Anusmita Das, Kaushik Dehingia, Nabajit Ray, Hemanta Kumar Sarmah . Stability analysis of a targeted chemotherapy-cancer model. Mathematical Modelling and Control, 2023, 3(2): 116-126. doi: 10.3934/mmc.2023011
    [9] M. Sathish Kumar, M. Deepa, J Kavitha, V. Sadhasivam . Existence theory of fractional order three-dimensional differential system at resonance. Mathematical Modelling and Control, 2023, 3(2): 127-138. doi: 10.3934/mmc.2023012
    [10] Abdul-Fatawu O. Ayembillah, Baba Seidu, C. S. Bornaa . Mathematical modeling of the dynamics of maize streak virus disease (MSVD). Mathematical Modelling and Control, 2022, 2(4): 153-164. doi: 10.3934/mmc.2022016
  • We investigated a nonlinear singularly perturbed elliptic reaction-diffusion coupled system having non-smooth data networked by a k-star graph. We considered all possible boundary conditions at the free boundary located at the tail of the edge and imposed the continuity condition with Kirchhoff's junction law at the junction point of the k-star graph to obtain a continuous solution for this coupled system. First, we showed the existence and uniqueness of the solution using the variational formulation approach. Then, we reformulated it into a minimization problem over a function space to conclude the uniqueness of the solution. For the approximation of the continuous problem, note that the upwind scheme for the flux condition at the free boundary leads to a parameter uniform first-order approximation. To obtain a higher-order uniform accuracy, we utilized a three-point scheme for first-order derivatives and a five-point approximation at the point of discontinuity. These approximations typically did not yield an M-matrix or strict diagonally dominant structure of the stiffness matrix. Hence, we provided a suitable transformation that could lead to a sufficient condition for preserving the strict diagonally dominant structure of the stiffness matrix. We performed a comprehensive convergence analysis to demonstrate the almost second-order uniform accuracy on each edge of the k-star graph. Numerical experiments highly validate the theory on the k-star graph.



    During the vehicle movement, the performance of the vehicle is affected by various vehicle structures and functions, such as power steering system, suspension system, braking system, etc. Moreover, in complex and high-speed environment, the vehicle's vertical, roll and pitch displacements contain a strong coupling relationship. Therefore, considering the motion and coupling characteristics of vehicle structures is meaningful to investigate vehicle dynamics. On the basis of geometric structure parameters of vehicle system and the nonlinear characteristics of shock absorber and leaf spring, the authors in [1] establish a nonlinear dynamic model for heavy vehicle. The correctness of the dynamic model is verified by testing the vertical acceleration data of the driver's seat, front wheel, middle wheel and rear wheel. To investigate the longitudinal driving behaviors of vehicle dynamics in the platoons, by taking the acceleration capability of heavy-duty vehicle into account, numerous heavy-duty vehicle platoon models are proposed [2,3,4,5,6,7,8,9,10]. Furthermore, by considering the lateral and longitudinal displacement characteristics of the vehicle, [11] represents a 2 DOF model of the vehicle and two diver cab models with time delays. In order to further describe the vehicle dynamics characteristics, in accordance with the two driver cab models in [11]. [15] and [16] further investigate the nonlinear lateral dynamics of a 2 DOF vehicle model. Based on longitudinal vehicle dynamics and by analyzing the dynamic of engine, torque converter, tire and capacitor pack, the authors of [17] present a dynamic model for a heavy-duty vehicle.

    On the other side of research, the above mentioned vehicle models are mostly used to evaluate vehicle lateral and longitudinal dynamics characteristics, the influence of vehicle lateral and yaw dynamics characteristics are not considered enough. In practice, the tires not only provide horizontal and vertical forces to the vehicle, but also give vertical forces to the suspension system, especially in complex driving situations such as lane changes, cornering, or obstacle avoidance. In these cases, the vehicle's vertical, roll, and pitch dynamics are clearly coupled with lateral and yaw motion. Due to large inertia, high center of gravity and high roll center, heavy vehicles have poor stability when entering a turn or lane change, and the three-way coupling effect is large. Therefore, it is necessary to establish a three dimensional coupled vehicle model and study the influence of steering process on vehicle dynamics. More recently, more and more works focus on the coupling property of the vehicle. To reflect the steering influence on the overall response of the vehicle, [18] designs a novel 4 DOF hydraulic power steering (HPS) system. Simultaneously, [18] develops a 24 DOF model by taking the HPS system, the steering hand wheel angle, rack displacement, and hand wheel angles into account. According to the nonlinear characteristics of suspension damping and tire stiffness, [19] establishes a nonlinear three-way coupled lumped parameter model, and an improved nonlinear delay preview driver model was proposed based on [11], which was connected with the TCLP model to form a driver-vehicle closed-loop system. [20] establishes a complete vehicle model of a heavy truck, which not only investigates the nonlinear characteristics of suspension damping and tire stiffness, but also contains a modified preview driver model with nonlinear time delays to calculate the right front wheel steering angle for driving the vehicle along the desired route. In this paper, the kinematics and dynamics equations of cab and body are established by analyzing the three-way coupling effect of cab and body, as well as the dynamic characteristics of tire and suspension. Firstly, the dynamic relation of the tyre with deflection angle is introduced. Secondly, the coupling dynamics equation of cab was established by analyzing the three-way coupling effect of cab. Then, considering the dynamic characteristics of the vehicle suspension, the three-way coupling dynamic equation of the vehicle body is established. Finally, the kinematic and dynamic equations of cab and body are established based on the dynamic characteristics of tire and suspension and the Euler rotation theorem.

    Table 1.  The symbols of the heavy-duty vehicle.
    Definition Symbol
    forward traction (lateral traction) of the ith tire Fxi(i=1,,6)(Fyi)
    steering angle of the ith wheel δi
    steering angular speed of the front axle tires ωt
    transverse (longitudinal) component of ith tire along the coordinate system {B} FXi(FYi)
    suspension force, damping coefficient and spring constant of the jth spring between cab and body Fcj,Hcj and Kcj (j=1,2,3,4)
    vertical displacement of the cab (body) zc(zb)
    pitching angle of the cab (body) φc(φb)
    roll displacements of the cab (body) ϕc(ϕb)
    longitudinal distance between origin of coordinates {C} and cab rear (front) spring l5(l6)
    the distance between the origin of coordinates {C} and {B} l4
    the angle between the origin of coordinate {B} and the sprung mass bar center of suspension φ0
    transverse distance betweencab front spring and rear spring bc
    resultant force of the cab inthe direction of axes XC, YC and ZC Fxc,Fyc and Fzc
    resultant moment of the cab in the direction of axes XC, YC and ZC Nxc,Nyc and Nzc
    total mass of the vehicle, cab and body ms,mt and mb
    velocity vectors of the cab in the coordinate system {C} and {B} uc,vc and wc, ub,vb and wb
    roll angle rate, pitch angle rate and yaw angle rate of the coordinate system {C} and {B} pc,qc and rc, pb,qb and rb
    vertical and transverse distance from the origin of {C} and {B} to the center of gravity of cab hos and eos, hob and eob
    moment of inertia of a vehicle about axle XC,YC, ZC,XB,YB and ZB Ixxc,Iyyc and Izzc, Ixxb,Iyyb and Izzb
    moment of inertia of the cab about the axis XC and YC (XB and YB) Ixxsc and Iyysc (Ixxsb and Iyysb)
    integral of the product of the XC and YC(XB and YB) deviation of an area element in a vehicle Ixzc(Ixzb)
    compression displacement of the jth spring between cab and body xcj
    cab center of gravity (body center of gravity) to cab front and rear spring transverse distance l7 and l8 (l15 and l16)
    transverse (longitudinal) distance of cab center of gravity to set 1, set 2 and set 3 tires l9,l10 and l11 (l12,l13 and l14)
    distance between the front axle and the rear axle in a suspension system l3
    resultant force of the cab in the direction of axes XB,YB and ZB Fxb,Fyb and Fzb
    suspension force, damping coefficient and springconstant of the jth spring between the front axle or rear axle and the body of the suspension system Fsj,Hsj and Ksj
    longitudinal transverse distance from the center of gravity of suspension system to the front axis of the suspension system l1
    pitching angle of the left and right balance bars of the suspension system φp1 and φp2
    vertical displacement of the jth axle zuj
    angle of inclination of the jth wheel shaft ϕuj
    lateral distance between the left and right springs of the suspension system bs1,bs2 and bs3
    damping forces of front suspension left and right springs Fd1 and Fd2
    resultant moment of the cab in the direction of axes XB,YB and ZB Nxb,Nyb and Nzb
    transverse (longitudinal) distance of cab center of gravity to set 1, set 2 and set 3 bearing spring l17,l18 and l19 (l22,l23 and l24)
    transverse (longitudinal) distance of body center of gravity to set 1, set 2 and set 3 tires l25,l26 and l27 (l28,l29 and l30)
    transverse distance (longitudinal distance) between the body center of gravity and the cab front (rear) spring l20(l21)
    longitudinal distance from the center of gravity of the suspension system to the center of gravity of the rear axle of the suspension l2

     | Show Table
    DownLoad: CSV

    In this paper, the kinematic characteristics of a heavy-duty vehicle are considered to construct a 26 DOF vehicle body and a cab model. As shown in Figs. 1-4, the considered heavy vehicle has one front axle and two rear axles, which is called a three-axial vehicle. The degrees of freedom are vertical, roll and pitch displacements of the diver cab, vehicle body, the vertical and roll motion of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire. To further study the coupling property with each part, the vertical, roll and pitch motion of cab and body is modeled independently. Before introducing the related coordinate systems, the Euler's laws of motion is firstly given.

    Figure 1.  The established reference frame.
    Figure 2.  The top view of the three-axle heavy-duty vehicle.
    Figure 3.  The lateral view of the three-axle heavy-duty vehicle.
    Figure 4.  The top view of the three-axle heavy-duty vehicle.

    Lemma 3.1. Observed from an inertial reference frame, the force applied to a rigid body is equal to the product of the mass of the rigid body and the acceleration of the center of mass, i.e.

    Fe=mac

    where Fe is the resultant external force of the rigid body, m is the rigid body mass, and ac is the acceleration of center of mass.

    Lemma 3.2. . The fixed point O (for example, the origin) of an inertial reference frame is set as the reference point. The net external moment applied to the rigid body is equal to the time rate of change of the angular momentum, i.e.

    MO=dLOdt

    where MO is the is the external torque at point O, LOis the angular momentum at point O.

    To analyze the motion of heavy-duty vehicle, the corresponding coordinate frames are elaborated to describe the movement of the vehicle and indicated in Fig. 1. The moving coordinate frame {B} is fixed to the vehicle's body and is called the body-fixed reference frame. The second coordinate frame {C} is fixed to the cab and is called the cab-fixed reference frame. The third coordinate frame {E} is fixed to the earth and is called the earth-fixed reference frame. The last coordinate frame {Ti} is fixed to the ith, (i=1,2,3,4,5,6) tire and is called the tire-fixed reference frame. In this paper, we assume that the body axes XB,YB,ZB, the tire axes XTi,YTi, and the cab axes XC,YC,ZC, of heavy vehicle coincide with the principal axes of inertia, which are usually defined as:

    XB/XTi/XC -longitudinal axis (directed from aft of the body/tire/cab to front).

    YB/YTi/YC -transverse axis (directed to right side of body/tire/cab).

    ZB/ZC -normal axis (directed from top to bottom).

    To extract the kinetic model for the considered three-axial heavy-duty vehicle, the coordinate frame Ti is designed for each tire, the corresponding schematic diagram is shown in Fig. 2. By taking the yaw angle into account, the forces produced by the engine are transformed into the forward traction and longitudinal traction on the suspension of heavy-duty vehicle. Based on the coordinate frame and Fig. 2, the forward and lateral traction of each tire can be expressed as

    FXi=FxicosδiFyisinδi,FYi=FyisinδiFyicosδi,˙δ1=˙δ2=ωt,δ3,4,5,6=0,i=1,,6. (3.1)

    In this subsection, the vertical, roll and pitch motion of the cab are considered to further accurately reflect the performance of spring suspension force between the cab and the body in the actual scenario. In accordance with the coordinate frame {C} and Figs. 3-4, the spring force between the cab and body can be given as

    Fc1=Kc1(zcφcl5zb+(φbφ0)(l4+l5)(ϕbϕc)bc2)+Hc1(˙zc˙φcl5˙zb+˙φb(l4+l5)(˙ϕb˙ϕc)bc2) (3.2)
    Fc2=Kc2(zcφcl5zb+(φbφ0)(l4+l5)+(ϕbϕc)bc2)+Hc2(˙zc˙φcl5˙zb+˙φb(l4+l5)+(˙ϕb˙ϕc)bc2) (3.3)
    Fc3=Kc3(zcφcl5zb+(φbφ0)(l4+l5)(ϕbϕc)bc2)+Hc3(˙zc˙φcl5˙zb+˙φb(l4+l5)(˙ϕb˙ϕc)bc2) (3.4)
    Fc4=Kc4(zcφcl5zb+(φbφ0)(l4+l5)+(ϕbϕc)bc2)+Hc4(˙zc˙φcl5˙zb+˙φb(l4+l5)+(˙ϕb˙ϕc)bc2) (3.5)

    Furthermore, by viewing the diver cab as a rigid body and employing the Lemmas 3.1 and 3.2, the kinematical equation of diver cab model is

    Fxc=mt(˙ucvcrc)+ms(wcqc˙qchospcrchoseosq2c), (3.6)
    Fyc=mt(˙vcucrc)+ms(˙pchoswcpc+qc(pceosrchos)) (3.7)
    Fzc=ms(˙wc+vcpcucqc+hos(q2c+p2c)+eos(pcrc˙qc)), (3.8)
    Nxc=Ixxc˙pcIxzc(˙rc+pcqc)+(IxxscIyyscmsh2os)qcrc+mshos(˙vc+ucrcwcpc), (3.9)
    Nyc=Iyyc˙qcIxzc(p2cr2c)+(IyyscIxxsc)pcrcmshos(˙ucvcrc+wcqc)mseos(˙wc+vcpcucqc), (3.10)
    Nzc=Izzc˙rc+Ixzc(qcrc˙pc)+(IxxscIyysc+mse2os)pcqcmseoswcpc, (3.11)

    When the vehicle is moving, the spring will produce spring force whether it is in a state of compression or tension. However, the direction of the force is opposite, so this paper considers the sign function and the direction of the spring displacement to determine the direction of the spring force. In order to obtain the dynamic force equation of the vehicle cab and body, we assume that the cab and body mass are evenly distributed, that is, the transverse distance between the cab and body center of gravity from the left and right tires is the same. By considering the definition of resultant force and resultant moment, the kinetic formula of longitudinal, transverse and vertical forces acting on the cab, as well as the yaw, pitch and roll moments is described as

    Fxc=c(δ1)c(φc)6i=1FXi+(s(ϕc)s(φc)c(δ1)s(δ1)c(ϕc))6i=1FYi+(s(δ1)s(ϕc)+c(ϕc)s(φc)c(δ1))(Fc1+sign(xc2)Fc2+sign(xc3)Fc3+sign(xc4)Fc4) (3.12)
    Fyc=s(δ1)c(φc)6i=1FXi+(s(ϕc)s(φc)s(δ1)+c(δ1)c(ϕc))6i=1FYi+(c(ϕc)s(ϕc)s(δ1)c(δ1)s(ϕc))(Fc1+sign(xc2)Fc2+sign(xc3)Fc3+sign(xc4)Fc4) (3.13)
    Fzc=s(φc)6i=1FXi+(s(ϕc)c(φc))6i=1FYi+(c(ϕc)c(φc)(Fc1+sign(xc2)Fc2) (3.14)
    Nxc=(Fc1+Fc2)l7(Fc3+Fc4)l8, (3.15)
    Nyc=(Fc1Fc2)l5+(Fc3Fc4)l6, (3.16)
    Nzc=(FX2FX1)l9+(FX4FX3)l10(FX5FX6)l11+(FY1+FY2)l12(FY3+FY4)l13(FY5+FY6)l14, (3.17)

    where sign() denotes the symbolic function, c()=cos() and s()=sin().

    For the considered heavy duty vehicle, two hydraulic dampers are fixed to the left and right front suspensions, and the balanced suspension does not have any shock absorbers. Thus, to represent the force situation of leaf spring in suspension system, the damping force of the two hydraulic dampers is considered for the front axle. The kinetic equation is given by

    Fs1=Ks1(zb(φbφ0)l1zu1+(ϕbϕu1)bs12)+Fd1, (3.18)
    Fs2=Ks2(zb(φbφ0)l1zu1(ϕbϕu1)bs12)+Fd2, (3.19)
    Fs3=Ks3(zb+(φbφ0)l2φp1l32zu2+(ϕbϕu2)bs22)+Hs3(˙zb+˙φbl2˙φp1l32˙zu2+(˙ϕb˙ϕu2)bs22), (3.20)
    Fs4=Ks4(zb+(φbφ0)l2φp2l32zu2+(ϕbϕu2)bs22)+Hs4(˙zb+˙φbl2˙φp2l32˙zu2(˙ϕb˙ϕu2)bs22), (3.21)
    Fs5=Ks5(zb+(φbφ0)l2φp1l32zu3+(ϕbϕu3)bs32)+Hs5(˙zb+˙φbl2+˙φp1l32˙zu3+(˙ϕb˙ϕu3)bs32), (3.22)
    Fs6=Ks6(zb+(φbφ0)l2φp2l32zu3+(ϕbϕu3)bs32)+Hs6(˙zb+˙φbl2+˙φp2l32˙zu3(˙ϕb˙ϕu3)bs32), (3.23)

    This is analogous to the diver cab part, taking the body as a rigid body and according to the Lemmas 3.1 and 3.2, the force equation of body model can be expressed as

    Fxb=mt(˙ubvbrb)+mb(wbqb˙qbhobpbrbhobeobq2b), (3.24)
    Fyb=mt(˙vbubrb)+mb(˙pbhobwbpb+qb(pbeobrbhob)), (3.25)
    Fzb=mb(˙wb+vbpbubqb)+hob(q2b+p2b)+eob(pbrb˙qb)), (3.26)
    Nxb=Ixxb˙pbIxzb(˙rb+pbqb)+(IxxsbIyysbmbh2ob)qbrb+mbhob(˙vb+ubrbwbpb), (3.27)
    Nyb=Iyyb˙qbIxzb(p2br2b)+(IyysbIxxsb)pbrbmbhob(˙ubvbrb+wbqb)mbeob(˙wb+vbpbubqb), (3.28)
    Nzb=Izzb˙rb+Ixzb(qbrb˙pb)+(IyysbIxxsb+mbe2ob)pbqbmbeobwbpb, (3.29)

    Recalling the problem of spring force direction and the assumption of uniform distribution of body mass, the longitudinal, transverse and vertical forces acting on the body are:

    Fxb=c(δ1)c(φb)6i=1FXi+(s(ϕb)s(φb)c(δ1)s(δ1)c(ϕb))6i=1FYi+(s(δ1)s(ϕb)+c(ϕb)s(φb)c(δ1))(Fc1sign(xc2)Fc2sign(xc3)Fc3sign(xc4)Fc4+Fs1+sign(zu2)Fs2+sign(zu3)Fs3+sign(zu4)Fs4+sign(zu5)Fs5+sign(zu6)Fs6), (3.30)
    Fyb=s(δ1)c(φb)6i=1FXi+(s(ϕb)s(φb)s(δ1)+c(δ1)c(ϕb))6i=1FYi+(c(ϕb)s(φb)s(δ1)c(δ1)s(ϕb))(Fc1sign(xc2)Fc2sign(xc3)Fc3sign(xc4)Fc4+Fs1+sign(zu2)Fs2+sign(zu3)Fs3+sign(zu4)Fs4+sign(zu5)Fs5+sign(zu6)Fs6), (3.31)
    Fzb=s(φb)6i=1FXi+(s(ϕb)c(φb)6i=1FYi+(c(ϕb)c(φb)(Fc1sign(xc2)Fc2sign(xc3)Fc3sign(xc4)Fc4+Fs1+sign(zu2)Fs2+sign(zu3)Fs3+sign(zu4)Fs4+sign(zu5)Fs5+sign(zu6)Fs6), (3.32)
    Nxb=(Fc1+Fc2)l15+(Fc3+Fc4)l16+(Fs1+Fs2)l17+(Fs3+Fs4)l18(Fs5+Fs6)l19, (3.33)
    Nyb=(Fc2Fc1)l20+(Fc4Fc3)l21+(Fs1Fs2)l22+(Fs3Fs4)l23+(Fs5Fs6)l24, (3.34)
    Nzb=(FX2FX1)l25+(FX4FX3)l26+(FX5FX6)l27+(FY1+FY2)l28(FY3+FY4)l29(FY5+FY6)l30, (3.35)

    In this final subsection, a common type of diver cab and body model for heavy-duty vehicle is proposed.

    In accordance with 3.1-3.17 and invoking the Euler rotation theorem, the dynamic and kinetic equation of driver cab are designed as

    ˙ηc=Jc(ηc)υc,˙υc=Gc(ηc,υc)[FXi,FYi]T+Fc(υ)+uc, (3.36)

    where ηc=[xc,yc,zc,ϕc,φc,δ1]T,

    Jc(ηc)=[J1(ηc)03×303×3J2(ηc)],

    J2(ηc)=[1s(ϕc)t(φc)c(ϕc)t(φc)0s(ϕc)s(ϕc)0s(ϕc)/c(φc)c(ϕc)/c(φc)],

    Gc(ηc,υc)=[J3(ηc)03×2],

    J3(ηc)=[c(δ1)c(φc)mts(δ1)c(ϕc)+s(ϕc)s(φc)c(δ1)mts(δ1)c(φc)mt+c(δ1)c(ϕc)+s(ϕc)s(φc)s(δ1)mts(φc)mss(ϕc)c(φc)ms],

    Fc(υc)=[Fc1(υc),Fc2(υc),Fc3(υc),Fc4(υc),Fc5(υc),Fc6(υc)]T,

    J1(ηc)=[c(δ1)c(φc)s(δ1)c(ϕc)+s(ϕc)s(φc)c(δ1)s(δ1)c(φc)c(δ1)c(ϕc)+s(ϕc)s(φc)s(δ1)s(φc)s(ϕc)c(φc)

    s(δ1)s(ϕc)+c(ϕc)s(φc)c(δ1)c(δ1)s(ϕc)+c(ϕc)s(φc)s(δ1)c(ϕc)c(φc)],

    υc=[uc,vc,wc,pc,qc,rc]T, FXi=6i=1FXi, FYi=6i=1FYi,

    uc=[05×1((FX2FX1)l9+(FX4FX3)l10+(FX5FX6)l11Izzc+(FY1+FY2)l12(FY3+FY4)l13(FY5+FY6)l14))], t() represents the tangent function. The expansion equation of the matrix Fc(υc) is

    Fc1(υc)=1mt[(s(δ1)s(ϕc)+c(ϕc)s(φc)c(δ1))(Fc1+sign(xc2)Fc2+sign(xc3)Fc3+sign(xc4)Fc4)ms(wcqc˙qchospcrchoseosq2c)+mtvcrc], (3.37)
    Fc2(υc)=1mt[(c(ϕc)s(φc)s(δ)c(δ)s(ϕc))(Fc1+sign(xc2)Fc2+sign(xc3)Fc3+sign(xc4)Fc4)ms(˙pchoswcpc+qc(pceosrchos))+mtucrc], (3.38)
    Fc3(υc)=1ms[c(ϕc)c(φc)(Fc1+sign(xc2)Fc2+sign(xc3)Fc3+sign(xc4)Fc4)](vcpcucqc+hos(q2c+p2c)+eos(pcrc˙qc)), (3.39)
    Fc4(υc)=1Ixxc(Fc1+Fc2)l7(Fc3+Fc4)l8mshos(˙vc+ucrcwcpc)+Ixzc(˙rc+pcqc)(IxxscIyyscmsh2os)qcrc, (3.40)
    Fc5(υc)=1Iyyc(Ixzc(p2cr2c)(IyyscIxxsc)pcrc+mshos(˙ucvcrc+wcqc)+mseos(˙wc+vcpcucqc)+(Fc1Fc2)l5+(Fc3Fc4)l6), (3.41)
    Fc6(υc)=1Izzc[mseoswcpcIxzc(qcrc˙pc)(IyyscIxxsc+mse2os)pcqc], (3.42)

    According to kinetic equations (18)-(42) and employing the Euler rotation theorem, the dynamic and kinetic equation of body are designed as

    ˙ηb=Jb(ηb)υb,˙υb=Gb(ηb,υb)[FXi,FYi]T+Fb(υ)+ub, (3.43)

    where ηb=[xb,yb,z,bϕb,φb,δ1]T, Jb(ηb)=[J1(ηb)03×303×3J2(ηb)], Gb(ηb,υb)=[J3(ηb)03×2], υb=[ub,vb,wb,pb,qb,rb]T, FYi=6i=1FXi, Fb(υb)=[Fb1(υb),Fb2(υb),Fb3(υb),Fb4(υb),Fb5(υb),Fb6(υb)]T, J2(ηb)=[1s(ϕb)t(φb)c(ϕb)t(φb)0s(ϕb)s(ϕb)0s(ϕb)/c(φb)c(ϕb)/c(φb)], FXi=6i=1FXi, J3(ηb)=[c(δ1)c(φb)mts(δ1)c(ϕb)+s(ϕb)s(φb)c(δ1)mts(δ1)c(φb)mt+c(δ1)c(ϕb)+s(ϕb)s(φb)s(δ1)mts(φb)mbs(ϕb)c(φb)mb], ub=[05×1((FX2FX1)l25+(FX4FX3)l26+(FX5FX6)l27Izzb+(FY1+FY2)l28(FY3+FY4)l29(FY5+FY6)l30)], J1(ηb)=[c(δ1)c(φb)s(δ1)c(ϕcb)+s(ϕb)s(φb)c(δ1)s(δ1)c(φb)c(δ1)c(ϕb)+s(ϕb)s(φb)s(δ1)s(φb)s(ϕb)c(φb)

    s(δ1)s(ϕb)+c(ϕb)s(φb)c(δ1)c(δ1)s(ϕb)+c(ϕc)s(φb)s(δ1)c(ϕb)c(φb)],

    The expansion equation of the matrix Fb(υb) is

    Fb1(υb)=1mt[(s(δ1)s(ϕb)+c(ϕb)s(φb)c(δ1))(Fc1sign(xc2)Fc2sign(xc3)Fc3sign(xc4)Fc4)+Fs1+sign(zu2)Fs2+sign(zu3)Fs3+sign(zu4)Fs4+sign(zu5)Fs5+sign(zu6)Fs6)mb(wbqb˙qbhobpbrbhobeobq2b)+mtvbrb], (3.44)
    Fb2(υb)=1mt[(c(ϕb)s(φb)s(δ)c(δ1)s(ϕb))(Fc1sign(xc2)Fc2sign(xc3)Fc3sign(xc4)Fc4)+Fs1+sign(zu2)Fs2+sign(zu3)Fs3+sign(zu4)Fs4+sign(zu5)Fs5+sign(zu6)Fs6)mb(˙pbhobwbpb+qb(pbeobrbhob))+mtubrb], (3.45)
    Fb3(υb)=1mb[c(ϕb)c(φb)(Fc1sign(xc2)Fc2sign(xc3)Fc3sign(xc4)Fc4)+Fs1+sign(zu2)Fs2+sign(zu3)Fs3+sign(zu4)Fs4+sign(zu5)Fs5+sign(zu6)Fs6)+mb(ubqbvbpbhob(q2b+p2b)eob(pbrb˙qb)), (3.46)
    Fb4(υb)=1Ixxb[(Fc3+Fc4)l16(Fc1+Fc2)l15+(Fs1+Fs2)l17+(Fs3+Fs4)l18(Fs5+Fs6)l19+Ixzb(˙rb+pbqb)(IxxsbIyysbmbh2ob)qbrbmbhob(˙vb+ubrbwbpb)], (3.47)
    Fb5(υb)=1Iyyb[(Fc2Fc1)l20+(Fc4Fc3)l21+(Fs1Fs2)l22+(Fs3Fs4)l23(Fs5Fs6)l24+Ixzb(p2br2b)(IyysbIxxsb)pbrb+mbhob(˙ubvbrb+wbqb)+mbeob(˙wb+vbpbubqb)], (3.48)
    Fb6(υb)=1Izzb[mbeobwbpbIxzb(qbrb˙pb)(IyysbIxxsb+mbe2ob)pbqb], (3.49)

    In complex working conditions, there is a coupling relationship of the vertical, lateral and longitudinal dynamics of vehicles. By considering the kinetic character of the vertical, roll and pitch motion of the diver cab, vehicle body, the vertical and roll behavior of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire. In this paper, a common model of three-axles heavy-duty vehicle with 26 DOF have been proposed to extrude the kinetic characterization diver cab and vehicle body.

    This work was supported by the National Natural Science Foundation of China under Grants U22A2043 and 62173172.

    The author declares that there is no conflicts of interest in this paper.



    [1] Y. V. Pokornyi, A. V. Borovskikh, Differential equations on networks (geometric graphs), J. Math. Sci., 119 (2004), 691–718. https://doi.org/10.1023/B:JOTH.0000012752.77290.fa doi: 10.1023/B:JOTH.0000012752.77290.fa
    [2] B. S. Pavlov, M. D. Faddeev, Model of free electrons and the scattering problem, Theor. Math. Phys., 55 (1983), 485–492. https://doi.org/10.1007/BF01015809 doi: 10.1007/BF01015809
    [3] T. Nagatani, Traffic flow on star graph: Nonlinear diffusion, Physica A, 561 (2021), 125251. https://doi.org/10.1016/j.physa.2020.125251 doi: 10.1016/j.physa.2020.125251
    [4] D. B. West, Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.
    [5] W. C. Connor, J. Wengong, R. Luke, F. J. Timothy, S. J. Tommi, H. G. William, et al., A graph-convolutional neural network model for the prediction of chemical reactivity, Chem. Sci., 10 (2019), 370–377. https://doi.org/10.1039/C8SC04228D doi: 10.1039/C8SC04228D
    [6] J. D. Murray, Mathematical biology: Ⅱ: Spatial models and biomedical applications, Interdiscip. Appl. Math., 2003.
    [7] L. O. Müller, G. Leugering, P. J. Blanco, Consistent treatment of viscoelastic effects at junctions in one-dimensional blood flow models, J. Comput. Phys., 314 (2016), 167–193. https://doi.org/10.1016/j.jcp.2016.03.012 doi: 10.1016/j.jcp.2016.03.012
    [8] I. Rodriguez-Iturbe, R. Muneepeerakul, E. Bertuzzo, S. A. Levin, A. Rinaldo, River networks as ecological corridors: A complex systems perspective for integrating hydrologic, geomorphologic, and ecologic dynamics, Water Resour. Res., 45 (2009), 1–22. https://doi.org/10.1029/2008WR007124 doi: 10.1029/2008WR007124
    [9] J. V. Below, A. J. Lubary, Instability of stationary solutions of reaction-diffusion equations on graphs, Results Math., 68 (2015), 171–201. https://doi.org/10.1007/s00025-014-0429-8 doi: 10.1007/s00025-014-0429-8
    [10] S. Iwasaki, S. Jimbo, Y. Morita, Standing waves of reaction-diffusion equations on an unbounded graph with two vertices, SIAM J. Appl. Math., 82 (2022), 1733–1763. https://doi.org/10.1137/21M1454572 doi: 10.1137/21M1454572
    [11] H. M. Srivastava, A. K. Nain, R. K. Vats, P. Das, A theoretical study of the fractional-order p-Laplacian nonlinear Hadamard type turbulent flow models having the Ulam–Hyers stability, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117 (2023), 1–19. https://doi.org/10.1007/s13398-023-01488-6 doi: 10.1007/s13398-023-01488-6
    [12] V. Mehandiratta, M. Mehra, G. Leugering, Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph, J. Math. Anal. Appl., 477 (2019), 1243–1264. https://doi.org/10.1016/j.jmaa.2019.05.011 doi: 10.1016/j.jmaa.2019.05.011
    [13] D. G. Gordeziani, M. Kupreishvili, H. V. Meladze, T. D. Davitashvili, On the solution of boundary value problem for differential equations given in graphs, Appl. Math. Inform. Mech., 13 (2008), 80–91.
    [14] G. M. Gie, M. Hamouda, C. Y. Jung, R. M. Temam, Singular Perturbations and Boundary Layers, Springer International Publishing, 2018. https://doi.org/10.1007/978-3-030-00638-9 doi: 10.1007/978-3-030-00638-9
    [15] P. Das, S. Rana, J. Vigo-Aguiar, Higher-order accurate approximations on equidistributed meshes for boundary layer originated mixed type reaction-diffusion systems with multiple scale nature, Appl. Numer. Math., 148 (2020), 79–97. https://doi.org/10.1016/j.apnum.2019.08.028 doi: 10.1016/j.apnum.2019.08.028
    [16] P. Das, Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems, J. Comput. Appl. Math., 290 (2015), 16–25. https://doi.org/10.1016/j.cam.2015.04.034 doi: 10.1016/j.cam.2015.04.034
    [17] P. Das, An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh, Numerical Algorithms, 81 (2019), 465–487. https://doi.org/10.1007/s11075-018-0557-4 doi: 10.1007/s11075-018-0557-4
    [18] P. Das, A higher order difference method for singularly perturbed parabolic partial differential equations, J. Differ. Equations Appl., 24 (2018), 452–477. https://www.tandfonline.com/doi/full/10.1080/10236198.2017.1420792 doi: 10.1080/10236198.2017.1420792
    [19] D. Shakti, J. Mohapatra, P. Das, J. Vigo-Aguiar, A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction-diffusion problems with arbitrary small diffusion terms, J. Comput. Appl. Math., 404 (2022), 113167. https://doi.org/10.1016/j.cam.2020.113167 doi: 10.1016/j.cam.2020.113167
    [20] S. Kumar, P. Das, K. Kumar, Adaptive mesh-based efficient approximations for Darcy scale precipitation–dissolution models in porous media, Int. J. Numer. Methods Fluids, 96 (2024), 1415–1444. https://doi.org/10.1002/fld.5294 doi: 10.1002/fld.5294
    [21] S. Saini, P. Das, S. Kumar, Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction-diffusion multiple scale problems with large delay in time, Appl. Numer. Math., 196 (2024), 1–21. https://doi.org/10.1016/j.apnum.2023.10.003 doi: 10.1016/j.apnum.2023.10.003
    [22] S. Sain, P. Das, S. Kumar, Computational cost reduction for coupled system of multiple scale reaction-diffusion problems with mixed type boundary conditions having boundary layers, Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 117 (2023), 1–27. https://doi.org/10.1007/s13398-023-01397-8 doi: 10.1007/s13398-023-01397-8
    [23] S. Kumar, P. Das, A uniformly convergent analysis for multiple scale parabolic singularly perturbed convection-diffusion coupled systems: Optimal accuracy with less computational time, Appl. Numer. Math., 207 (2025), 534–557. https://doi.org/10.1016/j.apnum.2024.09.020 doi: 10.1016/j.apnum.2024.09.020
    [24] B. P. Andreianov, G. M. Coclite, C. Donadello, Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network, Discrete Contin. Dyn. Syst., 37 (2017), 5913–5942. http://dx.doi.org/10.3934/dcds.2017257 doi: 10.3934/dcds.2017257
    [25] M. Musch, U. S. Fjordholm, N. H. Risebro, Well-posedness theory for nonlinear scalar conservation laws on networks, Networks Heterogen. Media, 17 (2022), 101–128. https://doi.org/10.3934/nhm.2021025 doi: 10.3934/nhm.2021025
    [26] G. M. Coclite, C. Donadello, Vanishing viscosity on a star-shaped graph under general transmission conditions at the node, Networks Heterogen. Media, 15 (2020), 197–213. https://doi.org/10.3934/nhm.2020009 doi: 10.3934/nhm.2020009
    [27] J. D. Towers, An explicit finite volume algorithm for vanishing viscosity solutions on a network, Networks Heterogen. Media, 17 (2022), 1–13. https://doi.org/10.3934/nhm.2021021 doi: 10.3934/nhm.2021021
    [28] S. F. Pellegrino, On the implementation of a finite volumes scheme with monotone transmission conditions for scalar conservation laws on a star-shaped network, Appl. Numer. Math., 155 (2020), 181–191. https://doi.org/10.1016/j.apnum.2019.09.011 doi: 10.1016/j.apnum.2019.09.011
    [29] F. R. Guarguaglini, R. Natalini, Vanishing viscosity approximation for linear transport equations on finite star-shaped networks, J. Evol. Equations, 21 (2021), 2413–2447. https://doi.org/10.1007/s00028-021-00688-0 doi: 10.1007/s00028-021-00688-0
    [30] H. Egger, N. Philippi, On the transport limit of singularly perturbed convection-diffusion problems on networks, Math. Methods Appl. Sci., 44 (2021), 5005–5020. https://doi.org/10.1002/mma.7084 doi: 10.1002/mma.7084
    [31] H. Egger, N. Philippi, A hybrid discontinuous Galerkin method for transport equations on networks, in Finite volumes for complex applications IX, Bergen, Norway, Springer, 323 (2020), 487–495. https://doi.org/10.1007/978-3-030-43651-3_45
    [32] H. Egger, N. Philippi, A hybrid-DG method for singularly perturbed convection-diffusion equations on pipe networks, ESAIM Math. Model. Numer. Anal., 57 (2023), 2077–2095. https://doi.org/10.1051/m2an/2023044 doi: 10.1051/m2an/2023044
    [33] V. Kumar, G. Leugering, Singularly perturbed reaction-diffusion problems on a k-star graph, Math. Methods Appl. Sci., 44 (2021), 14874–14891. https://doi.org/10.1002/mma.7749 doi: 10.1002/mma.7749
    [34] P. A. Farrell, J. J. H. Miller, E. O'Riordan, G. I. Shishkin, Singularly perturbed differential equations with discontinuous source terms, in Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems (eds. J.J.H. Miller, G.I. Shishkin and L. Vulkov), Nova Science Publishers, New York, (2000), 23–32.
    [35] Z. Cen, A hybrid difference scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient, Appl. Math. Comput., 169 (2005), 689–699. https://doi.org/10.1016/j.amc.2004.08.051 doi: 10.1016/j.amc.2004.08.051
    [36] S. Kumar, S. Kumar, P. Das, Second-order a priori and a posteriori error estimations for integral boundary value problems of nonlinear singularly perturbed parameterized form, Numerical Algorithms, 2024. https://doi.org/10.1007/s11075-024-01918-5
    [37] S. Santra, J. Mohapatra, P. Das, D. Choudhari, Higher-order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis, Comput. Math. Appl., 150 (2023), 87–101. https://doi.org/10.1016/j.camwa.2023.09.008 doi: 10.1016/j.camwa.2023.09.008
    [38] V. Kumar, G. Leugering, Convection dominated singularly perturbed problems on a metric graph, J. Comput. Appl. Math. 425 (2023), 115062. https://doi.org/10.1016/j.cam.2023.115062 doi: 10.1016/j.cam.2023.115062
    [39] H. Zhu, Z. Li, Z. Yang, Analysis and computation for a class of semilinear elliptic boundary value problems, Comput. Math. Appl., 64 (2012), 2735–2743. https://doi.org/10.1016/j.camwa.2012.08.004 doi: 10.1016/j.camwa.2012.08.004
    [40] J. J. Nieto, J. M. Uzal, Nonlinear second-order impulsive differential problems with dependence on the derivative via variational structure, J. Fixed Point Theory Appl., 22 (2020), 1–19. https://doi.org/10.1007/s11784-019-0754-3 doi: 10.1007/s11784-019-0754-3
    [41] K. Atkinson, W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer, New York, 2009.
    [42] G. V. Alekseev, R. V. Brizitskii, Z. Y. Saritskaya, Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation, J. Appl. Ind. Math., 10 (2016), 155–167. https://doi.org/10.17377/sibjim.2016.19.201 doi: 10.17377/sibjim.2016.19.201
    [43] M. Manikandan, R. Ishwariya, Robust computational technique for a class of singularly perturbed nonlinear differential equations with Robin boundary conditions, J. Math. Model., 11 (2023), 411–423. https://doi.org/10.22124/jmm.2023.23515.2100 doi: 10.22124/jmm.2023.23515.2100
    [44] R. Shiromani, V. Shanthi, P. Das, A higher order hybrid-numerical approximation for a class of singularly perturbed two-dimensional convection-diffusion elliptic problem with non-smooth convection and source terms, Comput. Math. Appl., 142 (2023), 9–30. https://doi.org/10.1016/j.camwa.2023.04.004 doi: 10.1016/j.camwa.2023.04.004
    [45] M. Chandru, T. Prabha, V. Shanthi, A hybrid difference scheme for a second-order singularly perturbed reaction-diffusion problem with non-smooth data, Int. J. Appl. Comput. Math., 1 (2015), 87–100. https://doi.org/10.1007/s40819-014-0004-8 doi: 10.1007/s40819-014-0004-8
    [46] J. J. H. Miller, E. O'Riordan, G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, 1996.
    [47] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan, G. I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman and Hall, CRC Press, Boca Raton, USA, 2000.
    [48] N. Kopteva, M. Stynes, Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions, Appl. Numer. Math., 51 (2004), 273–288. https://doi.org/10.1016/j.apnum.2004.07.001 doi: 10.1016/j.apnum.2004.07.001
    [49] S. Kumar, R. Ishwariya, P. Das, Impact of mixed boundary conditions and non-smooth data on layer originated non-premixed combustion problems: Higher order convergence analysis, Stud. Appl. Math., (2024), e12673. https://doi.org/10.1111/sapm.12763 doi: 10.1111/sapm.12763
  • This article has been cited by:

    1. Shiwei Xu, Junqiu Li, Xiaopeng Zhang, Daikun Zhu, Research on Optimal Driving Torque Control Strategy for Multi-Axle Distributed Electric Drive Heavy-Duty Vehicles, 2024, 16, 2071-1050, 7231, 10.3390/su16167231
    2. Li Li, Hsin Guan, Chunguang Duan, Le Jiang, Jun Zhan, A driving feel oriented dynamic model for commercial vehicles, 2025, 15, 2045-2322, 10.1038/s41598-025-95775-4
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1574) PDF downloads(79) Cited by(31)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog