Central Finite Difference Method Research Articles

Nonlinear Dynamic Properties of Rigid–Elastic–Liquid Coupled Ball Bearings Considering External Load Excitation

In this study, a novel dynamic model of a ball bearing based on rigid–elastic–liquid (REL) coupling structure considering external load excitation is established, and the dynamic response of its outer ring under external load excitation is examined. First, the differential equation of the journal motion (i.e. the outer ring) is derived based on Newton’s law, and the oil film force induced by the REL coupling structure is determined by using the central finite difference method, planar bending theory of thin ring, and Simpson’s rule. Further, the fourth and fifth-order variable step length Runge–Kutta method is used to solve the kinematic differential equation. Second, the dynamic characteristics of the outer ring of the bearing under different speeds and elastic support structural parameters are examined. Finally, the feasibility of the theoretical model is verified through experiments. The results show that compared with the traditional squeeze film damper (SFD) system, the ball bearing with REL coupling structure is more stable, and the vibration damping effect is better under higher bearing speed, lower squirrel cage stiffness, and smaller elastic ring flexibility coefficient. The above findings provide useful insights into the dynamic characteristics of REL coupled ball bearings under unbalanced excitation, which can serve as a reference for the optimization design of such ball bearings.

A second-order fitted scheme combined with time two-grid technique for two-dimensional nonlinear time fractional telegraph equations involving initial singularity

In this paper, we derive the improved regularity for two-dimensional nonlinear time fractional telegraph equations by virtue of the technic of decomposition at first. Then, the famous L2-1σ formula is adopted to approximate the Caputo derivative and the central finite difference method is used for spatial discretization. The convergence accuracy of the proposed method is second order in both temporal and spatial direction. Meanwhile, for the sake of reducing the time of solving the high dimensional nonlinear problems, an efficient time two-grid algorithm is proposed. Furthermore, stability analysis of the proposed scheme is studied by the energy method. At last, numerical experiments are presented to verify the validity of the theoretical statements.

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

NUMERICAL SOLUTION OF HYPERBOLIC GOURSAT PARTIAL DIFFERENTIAL EQUATIONS WITH HYBRID CENTRAL DIFFERENCE - TAYLOR SERIES EXPANSIONS METHOD

This paper investigates a new method for solving the Goursat partial differential equation (PDE) using a combination of the central finite difference method (FDM) and Taylor series expansion. The study evaluates the effectiveness and accuracy of this new approach, analyzing linear Goursat problems and conducting multiple numerical experiments. The simulation study demonstrates that the suggested approach surpasses the existing method in terms of performance and accuracy. Applying this proposed scheme will minimize the cost, especially for engineers that might apply this model in solving their real-life problems.

CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t=0, we use the non-uniform meshes to capture the rapid change of the solution at t=0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L2-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O(τ2+hx2+hy2+hz2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method.

Efficient hybridized numerical scheme for singularly perturbed parabolic reaction–diffusion equations with Robin boundary conditions

An efficient numerical technique for a singularly perturbed parabolic reaction–diffusion problem with Robin type boundary conditions is presented in this work. The governing problem is discretized using the implicit Euler technique in time direction and a hybrid numerical technique that comprises a central finite difference method in the outer region and a cubic spline in compression method in the boundary layer regions in space direction. We use Shishkin-type meshes in the space domain to resolve the layers. The Robin type boundary conditions are handled using the second-order method. The totally discretized problem is well examined for stability and convergence. The use Bakhvalov-Shishkin and Vulanović-Shishkin meshes yields a more efficient and second-order convergence whereas Shishkin mesh produces almost second-order convergent solutions. Two test examples are computed. The current method has been compared to other methods in the scientific community.

An efficient numerical technique for investigating the generalized Rosenau–KDV–RLW equation by using the Fourier spectral method

<abstract><p>In this article, the generalized Rosenau-Korteweg-de Vries-regularized long wave (GR–KDV–RLW) equation was numerically studied by employing the Fourier spectral collection method to discretize the space variable, while the central finite difference method was utilized for the time dependency. The efficiency, accuracy, and simplicity of the employed methodology were tested by solving eight different cases involving four examples of the single solitary wave with different parameter values, interactions between two solitary waves, interactions between three solitary waves, and evolution of solitons with Gaussian and undular bore initial conditions. The error norms were evaluated for the motion of the single solitary wave. The conservation properties of the GR-Rosenau–KDV–RLW equation were studied by computing the momentum and energy. Additionally, the numerical results were compared with the previous studies in the literature.</p></abstract>

Upwind reproducing kernel collocation method for convection-dominated problems

Because of the best approximation property, traditional Bubnov–Galerkin numerical methods have proven immensely successful in modeling self-adjoint problems, such as heat conduction, elasticity, and so on. However, a numerical instability arises in these (and central finite difference) methods for problems with strong convection. In this class of problems, the convective transport term can lead to large spurious oscillations but can be handled by the class of Petrov–Galerkin methods. In particular, the upwind-type schemes and their variational and subgrid descendants have been substantially developed over the years for an effective weak-form Galerkin solution that precludes these instabilities. Nevertheless, the scale of development of upwind methods for strong-form collocation is substantially smaller, where numerical oscillations are also observed when they are straightforwardly applied to convection-dominated problems without special treatment. To this end, this paper presents a new upwind collocation method. First, the connection between the upwind finite difference scheme and the gradient smoothing technique in meshfree methods is established. It is then shown that selecting the collocation points as meshfree nodal points is not optimal; selecting the collocation points according to the flow direction and Péclet number is then studied. The upwind effect is achieved without introducing artificial parameters and is trivial to generalize for multi-dimensional cases. Cross-wind diffusion is also not observed in the solution. An error analysis is presented, and the effectiveness of the proposed methodology is well demonstrated by the steady and unsteady numerical examples.

Pricing a resettable convertible bond based on decomposition method and PDE models

In this paper, a partial differential equation approach based on the underlying stock price path decomposition is developed to price an American-style resettable convertible bond. The American-style resettable convertible bond is viewed as a mixture of three simple securities, which can be used to replicate the feature of payoffs of the resettable convertible bond completely. The partial differential equations under the Black–Scholes framework are established to price these simple securities. An implicit Euler method is used to discretize the first-order time derivative while a central finite difference method on a piecewise uniform mesh is used to discretize the spatial derivatives. The error estimates are developed by using the maximum principle in two mesh sets both for the time semi-discretization scheme and the spatial discretization scheme, respectively. It is proved that the scheme is first-order convergent for the time variable and second-order convergent for the spatial variable. Numerical experiments support these theoretical results.

Evaluation of Neural Network-based Derivatives for Topology Optimization

Abstract Neural networks have gained popularity for modeling complex non-linear relationships. Their computational efficiency has led to their growing adoption in optimization methods, including topology optimization. Recently, there have been several contributions towards improving derivatives of neural network outputs, which can improve their use in gradient-based optimization. However, a comparative study has yet to be conducted on the different derivative methods for the sensitivity of the input features on the neural network outputs. This paper aims to evaluate four derivative methods: analytical neural network's Jacobian, central finite difference method, complex step method, and automatic differentiation. These methods are implemented into density-based and homogenization-based topology optimization using multilayer perceptrons (MLPs). For density-based topology optimization, the MLP approximates Young's modulus for the solid-isotropic-material-with-penalization (SIMP) model. For homogenization-based topology optimization, the MLP approximates the homogenized stiffness tensor of a representative volume element, e.g., square cell microstructure with a rectangular hole. The comparative study is performed by solving two-dimensional topology optimization problems using the sensitivity coefficients from each derivative method. Evaluation includes initial sensitivity coefficients, convergence plots, and the final topologies, compliance, and design variables. The findings demonstrate that neural network-based sensitivity coefficients are sufficient for density-based and homogenization-based topology optimization. The neural network's Jacobian, complex step method, and automatic differentiation produced identical sensitivity coefficients to working precision. The study's open-source code is provided through an included Python repository.

Dynamic Characteristics Study of Elastic Ring Squeeze Film Damper with Rigid–Elastic–Oil Coupled Model

Due to the coupling of the damper journal with the elastic ring and oil film, the elastic ring squeeze film damper (ERSFD) shows better dynamic performance in comparison with the traditional squeeze film damper (SFD). Therefore, a novel rigid–elastic–oil coupled mathematical model was established. The elastic ring deformation, as the key point, is solved according to the planar bending theory. Then, based on the pressure governing equation of the oil film, using the central finite difference method, the oil film pressure was addressed. Meanwhile, the Simpson method was implemented to calculate the dynamic characteristic coefficients (equivalent stiffness and damping Ce) of ERSFD (DCCEs). Also, we analyzed the influence of journal eccentricity, oil film radius clearance, flexibility coefficient and damping hole diameter on the DCCEs, and the results were compared and verified with the existing literature. The sensitivity of each parameter to the DCCEs was analyzed by using the linear regression method. According to the results, the flexibility coefficient has the greatest effect on the DCCEs, followed by the oil film radius clearance. The eccentricity of the journal and damping hole diameter have the least impact. This work will provide a theoretical basis for reflecting on the bearing dynamic characteristics more truly and accurately.

Lubrication analysis of hydrodynamic-static hybrid bearing with deep-shallow chambers considering thermal conduction

The oil film separates the shaft-bearing system to reduce the frictional force of contact surfaces with large temperature variations owing to thermal conduction. Multiple impact factors seriously affected the lubrication behavior of the bearing during actual working process. An applicable model regarding the lubrication analysis of a hydrodynamic-static hybrid bearing with deep-shallow chambers is proposed and validated with published literature. The lubrication behavior includes film pressure, load capacity, bearing stiffness and flow rate is numerically calculated with centered finite difference method by solving the Reynolds equation. The influence of film thickness and working temperature are considered, over a range of eccentricity ratio. Moreover, the temperature variation caused by thermal conduction is analyzed in details. To investigate the influence of thermal conduction on the lubrication behavior, the results show that the load capacity, bearing stiffness, friction power, and flow power significantly decline with the consideration of thermal conduction. This study proposes an accurate model that is helpful for the design of hydrodynamic-static hybrid bearings with deep-shallow chambers under low rotational speed, heavy load, and high temperature conditions.

Nonlinear transient responses of lattice core sandwich plates under a sudden heat flux

In this study, we investigated the nonlinear thermally induced transient response of lattice core sandwich plates under the sudden application of heat flux for the first time. First, the equivalent thermal conductivity of the lattice core sandwich plates is obtained by considering the thermal conduction within the trusses and heat radiation in the cells. The one-dimensional transient heat transfer equation was implemented and solved using the central finite difference method to calculate the transient temperature field along the direction of the thickness of the plate. The displacement fields of the facesheets and lattice core layer were modeled using the Kirchhoff plate theory and the first-order shear deformation theory, respectively. Based on the Lagrange method and the assumed modal method, the governing equations, with mechanical-thermal coupling, for the simply supported lattice core sandwich plates under sudden heat flux are derived. Through a series of examples, the influences of the different lattice core cell types, thermal boundary conditions, truss radii, and cavity heat radiation on the nonlinear thermally induced transient responses of the sandwich plates are explored.

A modified and efficient phase field model for the biological transport network

This paper aims to establish the biological transport network based on the phase field model. In order to ensure that the topological shape is formed under the guidance of electrical conductivity, we generate the biological networks with sufficient information based on the distributions of the venation of the leaf represented by the reaction-diffusion model. We modify the original energy of the network generating model by considering the auxin gradient property. By applying the gradient flow method to minimize the modified energy, we derive the Poisson type equation for pressure, the reaction-diffusion type equation for the network conductance, and the Allen-Cahn type equation for the phase field. The proposed model is significant on the investigation of phase transitions by considering the gradient properties on the boundaries. We have innovatively added conductivity and phase-field coupling terms that inhibit perpendicular transport of nutrients, making it easy to generate thin branches from the trunk through this model. In order to obtain the second-order temporal accuracy, we take the Crank–Nicolson method for the governing system. To obtain the second-order spatial accuracy, we discretize the coupling system with the central finite difference method and linearize the nonlinear terms semi-explicitly to form a linear system at each time step. The discrete energy dissipation is provably preserved and we can use a larger time step. We apply the preconditioned conjugate gradient method with the multigrid method as a preconditioner to implement a practical algorithm with only linear algebraic complexity. The proposed algorithm is easy to implement and achieves a fast convergence. Various numerical tests are demonstrated to verify the efficiency, stability, and robustness of the proposed method.

An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions

The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.

Persistent currents and electronic properties of Mandelbrot quantum rings

In this study, we investigate the persistent current, and electronic energy levels of Mandelbrot quantum rings. For this purpose, three types of Mandelbrot quantum rings are proposed. Furthermore, Mandelbrot equation is generalized by introducing parameter m, which makes Mandelbrot’s shape more symmetric by adding new branches to it, on the other hand, the iteration parameter M, controls geometrical deficiencies. We explain the procedure needed to form these structures, including a padding scheme, then we solve the resulting two-dimensional Schrodinger equation using the central finite difference method with uniform distribution of the mesh points. Thereafter, we obtain the persistent current in different situations including different Mandelbrot orders and quantum ring shapes. We show that the persistent current can have different shapes and intensities by changing the described geometrical parameters of Mandelbrot quantum rings. We explain this phenomenon by considering symmetries in the potential, and consequently the wavefunction.

Nonlinear aeroelastic stability analysis of a folding wing by using geometrically exact fully intrinsic beam equations

In this paper, the aeroelastic instability of folding wings by using the geometrically exact fully intrinsic beam equations is investigated. The important advantages of these equations in comparison with other structural beam equations are complete modeling without simplifying assumptions in large deformations, low-order nonlinearities, and thus less complexity. For the first time, folding angles have been implemented in the geometrically exact fully intrinsic beam equations and hence this is the main novelty of this study. The applied aerodynamic loads in an incompressible flow regime are determined using Peter’s unsteady aerodynamic model. In order to check the stability of the system, first the resulting non-linear partial differential equations are discretized by employing the central finite difference method, and then linearized about the nonlinear steady-state condition. By obtaining the eigenvalues of the linearized system, the stability of the wing is evaluated. Furthermore, investigation of the effects of some important parameters such as stiffness ratio and length ratio on the flutter speed of the folding wing for various folding angles, is another achievement of this study. It is observed that the geometrically exact fully intrinsic beam equations can model the folding angles for the aeroelastic analysis more accurately and the capabilities of these equations became more specific.

Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

AbstractThe Allen‐Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time‐discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen‐Cahn equation obtained based on two first‐order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order.

An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator

In this study, we investigate a high-order accurate conservative finite difference scheme by utilizing a fourth-order fractional central finite difference method for the two-dimensional Riesz space-fractional nonlinear Schrödinger wave equation. The conservation laws of the discrete difference scheme are shown. Meanwhile, the exactness, uniqueness, and prior estimate of the numerical solution are rigorously established. Then, it is proved that the proposed scheme is unconditionally convergent in the discrete L2 and Hγ/2 norm, where γ is a fractional order. Furthermore, we demonstrate that when the fractional order γ and the spatial grid number J increase, the block-Toeplitz coefficient matrix generated by the spatial discretization becomes ill-conditioned. As a result, we adopt an effective linearized iteration method for the nonlinear system, allowing it to be solved efficiently by the Krylov subspace solver with an appropriate circulant preconditioner, in which the fast Fourier transform is applied to speed up the computational cost at each iterative step. Finally, numerical experiments are presented to validate the theoretical findings and the efficiency of the fast algorithm.

Error Estimation of the Relaxation Finite Difference Scheme for the Nonlinear Schrödinger Equation

We consider an initial- and boundary-value problem for the nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in the one space dimension case. We discretize the problem in space by a central finite difference method and in time by the Relaxation Scheme proposed by C. Besse [C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), pp. 1427–1432]. We provide optimal order error estimates, in the discrete -norm, for the approximation error at the time nodes and at the intermediate time nodes. In the context of the nonlinear Schrödinger equation, this is the first time that the derivation of an error estimate, for a fully discrete method based on the Relaxation Scheme, is completely addressed.