Finite Difference Formulation Research Articles

Efficient finite difference formulation of a geometrically nonlinear beam element

AbstractThe article is focused on a two‐dimensional geometrically nonlinear formulation of a Bernoulli beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which are combined with the kinematic equations and generalized material equations, leading to a set of three first‐order differential equations. These equations are then discretized by finite differences and the boundary value problem is converted into an initial value problem using a technique inspired by the shooting method. Accuracy of the numerical approximation is conveniently increased by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant, which leads to high computational efficiency. The element has been implemented into an open‐source finite element code. Numerical examples show a favorable comparison with standard beam elements formulated in the finite‐strain framework and with analytical solutions.

On the discrete version of the Kerr geometry

In this paper, a Kerr-type solution in the Regge calculus is considered. It is assumed that the discrete general relativity, the Regge calculus, is quantized within the path integral approach. The only consequence of this approach used here is the existence of a length scale at which edge lengths are loosely fixed, as considered in our earlier paper. In addition, we previously considered the Regge action on a simplicial manifold on which the vertices are coordinatized and the corresponding piecewise constant metric is introduced, and found that for the simplest periodic simplicial structure and in the leading order over metric variations between four-simplices, this reduces to a finite-difference form of the Hilbert–Einstein action. The problem of solving the corresponding discrete Einstein equations (classical) with a length scale (having a quantum nature) arises as the problem of determining the optimal background metric for the perturbative expansion generated by the functional integral. Using a one-complex-function ansatz for the metric, which reduces to the Kerr–Schild metric in the continuum, we find a discrete metric that approximates the continuum one at large distances and is nonsingular on the (earlier) singularity ring. The effective curvature [Formula: see text], including where [Formula: see text] (gravity sources), is analyzed with a focus on the vicinity of the singularity ring.

Characterization of bifurcations in the Zeeman catastrophe machine using singular value decomposition

Stable limit cycles of the dynamic Zeeman Catastrophe Machine are calculated and classified. As the geometry of the device is changed, they evolve into different types at discrete bifurcation points. The process is normally presented as a bifurcation diagram using a stroboscopic map, but this does not provide enough information to uniquely identify the bifurcation event. Three types of bifurcation are common: period-doubling, symmetry-breaking, and turning points. A method is developed to distinguish between these types by calculating the response to a change in device geometry. Assuming that the limit cycle has already been calculated, the response to a change in geometry is defined as the solution of a linear second-order differential equation. The equation is solved in finite-difference form by defining a matrix which contains the second-order response to a change in the limit cycle, as well as an augmented matrix which contains the response to a change in device geometry. At a bifurcation, the solution can be analyzed using singular value decomposition of these two matrices. The resulting singular values contain enough information to uniquely identify, and characterize, each type of event. The corresponding singular vectors can also be used to uniquely characterize the symmetry changes that occur in the response to change at each event.

Development of a mathematical model of vibratory non-lift movement of light seeds taking into account the aerodynamic forces and moments

The modern practice of using vibratory machines when working with fine-size light-weight seeds is faced with such an undesirable phenomenon as the impact of aerodynamic forces and moments on the kinematics of vibrational movement of particles of the seed mixture fractions. According to the results of scientific studies devoted to the solution of this problem, only mathematical models of vibrational movement are used, where the aerodynamic factor is taken into account as taking the seeds by airflow. This is typical only for cleaning modes with the rebound of seeds from the vibrating surface. Aerodynamic forces and moments are present in them only as a force of aerodynamic resistance. The action of lateral aerodynamic forces and their moments are not taken into account. Their consideration allows to extend the range of action of the aerodynamic factor on the modes of vibration cleaning (vibroseparation) without rebound (but with sliding and rolling) which are of greater interest in terms of improving the efficiency of processing fine-size seeds. A mathematical model of seed vibration movement taking into account the action of a complete set of aerodynamic forces (dynamic resistance forces and lateral aerodynamic forces) and moments was proposed. This makes it possible to simulate non-lifting modes of vibrational movement of seeds. A system of algebraic equations that are linear with respect to the kinematic parameters of seed movement which was obtained by translating differential equations of movement into a finite-difference form was presented. The possibility of numerical solution of equations of movement by the Euler method was shown. The results of the evaluation of the model adequacy for the processes of vibration separation of tobacco seeds and false flax were presented. As shown by the results of calculations and experiments, the developed model provides an increase in the adequacy of the simulation results by 30 % in comparison with the model where the aerodynamic factor is not taken into account

Hybrid mimetic finite-difference and virtual element formulation for coupled poromechanics

We present a hybrid mimetic finite-difference and virtual element formulation for coupled single-phase poromechanics on unstructured meshes. The key advantage of the scheme is that it is convergent on complex meshes containing highly distorted cells with arbitrary shapes. We use a local pressure-jump stabilization method based on unstructured macro-elements to prevent the development of spurious pressure modes in incompressible problems approaching undrained conditions. A scalable linear solution strategy is obtained using a block-triangular preconditioner designed specifically for the saddle-point systems arising from the proposed discretization. The accuracy and efficiency of our approach are demonstrated numerically on two-dimensional benchmark problems.

High performance 3D pin-by-pin neutron diffusion calculation based on 2D/1D decoupling method for accurate pin power estimation

The methods and performance of a 3D pin-by-pin neutronics code based on the 2D/1D decoupling method are presented. The code was newly developed as an effort to achieve enhanced accuracy and high calculation performance that are sufficient for the use in practical nuclear design analyses. From the 3D diffusion-based finite difference method (FDM) formulation, decoupled planar formulations are established by treating pre-determined axial leakage as a source term. The decoupled axial problems are formulated with the radial leakage source term. To accelerate the pin-by-pin calculation, the two-level coarse mesh finite difference (CMFD) formulation, which consists of the multigroup node-wise CMFD and the two-group assembly-wise CMFD is implemented. To enhance the accuracy, both the discontinuity factor method and the super-homogenization (SPH) factor method are examined for pin-wise cross-section homogenization. The parallelization is achieved with the OpenMP package. The accuracy and performance of the pin-by-pin calculations are assessed with the VERA and APR1400 benchmark problems. It is demonstrated that pin-by-pin 2D/1D alternating calculations within the two-level 3D CMFD framework yield accurate solutions in about 30 s for the typical commercial core problems, on a parallel platform employing 32 threads.

Time-Domain Studies of General Dispersive Anisotropic Media by the Complex-Conjugate Pole–Residue Pairs Model

A novel finite-difference time-domain formulation for the modeling of general anisotropic dispersive media is introduced in this work. The method accounts for fully anisotropic electric or magnetic materials with all elements of the permittivity and permeability tensors being non-zero. In addition, each element shows an arbitrary frequency dispersion described by the complex-conjugate pole–residue pairs model. The efficiency of the technique is demonstrated in benchmark numerical examples involving electromagnetic wave propagation through magnetized plasma, nematic liquid crystals and ferrites.

An immersed boundary approach for high order weighted essentially non-oscillatory schemes

A new immersed boundary approach for high order Weighted Essentially non-Oscillatory (WENO) schemes is proposed. The schemes is based on the main ideas from both the general immersed boundary algorithms and the level-set approach and can be easily applied to both finite difference and finite volume formulation. Although formally only second order accurate, numerical tests prove that the use of higher order approximation for the Eulerian fluxes can be very convenient to capture flow details and to obtain low uncertainty also with very coarse grids.

Numerical solution of coupled radiative and conductive heat transfer’s equations in PolyMethylMethAcrylate by the finite difference method

ABSTRACT This study has been conducted to illustrate combined heat transfer in a PolyMethylMethAcrylate (PMMA) sample. PMMA is a promising polymer utilized to optical, pneumatic actuation, sensor, analytical separation, and conductive devices. Combined radiation and conduction is considered, corresponding to the problem of one-dimensional transient heat transfer. Numerical results have been obtained by computer implementation, considering the effect of radiation, based on implicit finite difference formulation. Data have been provided from a nonlinear heat conduction equation using Kirchhoff transformation, centered the finite difference scheme by considering temperature conditions applied to the boundaries and monochromatic radiation intensity. In the present study, the temperature distribution and combined heat transfer equation are derived for PMMA. The temperature distribution is verified by the data of other studies that were done before. A comparison of the results showed a very good agreement with others. The results showed a great temperature gradient in the primary positions which cause a large amount of Conductive heat flux. The conductive heat flux was similar to radiative heat flux in terms of behavior but approximately twice in amount. On the other hand, total heat flux was constant at a specific time when it was an approximated steady-state condition.

Efficient computation of 1D and 2D nonlinear Viscous Burgers’ equation

Model order reduction (MOR) has emerged as a de-facto solution for simulation, control, and optimization of large-scale and complex dynamical systems from the past decades. This paper demonstrates two comprehensive MOR approaches to overcome the overall computational cost required to simulate the nonlinear, viscous Burger’s equation. The former scheme is the approximate proper orthogonal decomposition (APOD), which involves capturing the approximate snapshot ensemble of the states by performing multiple linearizations along the state-trajectory. The latter technique is based on the notion of nonlinear moment matching (NLMM), in which the reduced projection basis is generated by numerically solving the simplified, nonlinear, Sylvester equation. Both the reduction schemes are tested on the high-fidelity finite difference (FD) formulation of 1D and the 2D Burgers’ equation. The results show substantial computational savings in extracting the orthonormal projection basis required to generate the reduced manifold. Furthermore, a comparative study of both methods is presented with proper orthogonal decomposition (POD) for various test signals.

Energy-consistent finite difference schemes for compressible hydrodynamics and magnetohydrodynamics using nonlinear filtering

In this paper, an energy-consistent finite difference formulation for the compressible hydrodynamic and magnetohydrodynamic (MHD) equations is introduced. For the compressible magnetohydrodynamics, an energy-consistent finite difference formulation is derived using the product rule for the spatial difference. The conservation properties of the internal, kinetic, and magnetic energy equations can be satisfied in the discrete level without explicitly solving the total energy equation. The shock waves and discontinuities in the numerical solution are stabilized by nonlinear filtering schemes. An energy-consistent discretization of the filtering schemes is also derived by introducing the viscous and resistive heating rates. The resulting energy-consistent formulation can be implemented with the various kinds of central difference, nonlinear filtering, and time integration schemes. The second- and fifth-order schemes are implemented based on the proposed formulation. The conservation properties and the robustness of the present schemes are demonstrated via one- and two-dimensional numerical tests. The proposed schemes successfully handle the most stringent problems in extremely high Mach number and low beta conditions.

СХОДИМОСТЬ ЛОКАЛЬНО АВТОМОДЕЛЬНЫХ РЕШЕНИЙ К ТОЧНЫМ ЧИСЛЕННЫМ РЕШЕНИЯМ УРАВНЕНИЙ ПОГРАНИЧНОГО СЛОЯ НА ПЛАСТИНЕ

This paper considers a possibility of using locally self-similar solutions for a stationary boundary layer in linear stability problems. The solutions, obtained at various boundary conditions for a vibrationally excited gas, are compared with finite-difference calculations of the corresponding flows. An initial system of equations for a plane boundary layer of the vibrationally excited gas is derived from complete equations of two-temperature relaxation aerodynamics. Relaxation of vibrational modes of gas molecules is described in the framework of the Landau – Teller equation. Transfer coefficients depend on the static flow temperature. In a complete problem statement, the flows are calculated using the Crank – Nicolson finite-difference scheme. In all the considered cases, it is shown that the locally self-similar velocity and temperature profiles converge to the corresponding profiles for a fully developed boundary-layer flow calculated in a finite-difference formulation. The obtained results justify the use of locally self-similar solutions in problems of the linear stability theory for boundary-layer flows of a vibrationally excited gas.

Redefined Extended Cubic B-Spline Functions for Numerical Solution of Time-Fractional Telegraph Equation

This work is concerned with the application of a redefined set of extended uniform cubic B-spline (RECBS) functions for the numerical treatment of time-fractional Telegraph equation. The presented technique engages finite difference formulation for discretizing the Caputo time-fractional derivatives and RECBS functions to interpolate the solution curve along the spatial grid. Stability analysis of the scheme is provided to ensure that the errors do not amplify during the execution of the numerical procedure. The derivation of uniform convergence has also been presented. Some computational experiments are executed to verify the theoretical considerations. Numerical results are compared with the existing schemes and it is concluded that the present scheme returns superior outcomes on the topic.

Numerical analyses of stress and deformation in sandwich-structured composites

Sandwich-structured composites are extensively used in aerospace, transportation, marine, and automotive applications where mechanical performance and weight-saving features are extremely critical. Consequently, analyzing the mechanical properties including the elastic behavior of the sandwich structure is incumbent for the precise application of such composites. This paper deals with the stress analysis of a two-dimensional (2-D) rectangular sandwich-structured composite with the help of a robust numerical model. A finite difference (FD) scheme based on the displacement potential function is developed to investigate the stress-strain behavior of the sandwich structure under different loading conditions. The 2-D mixed-boundary-value elastic problem is solved by implementing special FD formulations inside the domain and at the boundaries. The mathematical intricacy of the interfaces of the sandwich structure is handled by employing modified FD formula structures. Results obtained from the FD model show that the two interfacial regions constitute the critical stress zones, and the material having a higher modulus of elasticity experiences higher stress. The proposed FD model is validated by comparing the FD results for stress distribution and displacement fields with the results obtained from a commercial finite element (FE) package. The comparison between the FE and FD results shows a good agreement which constructively establishes the proposed FD scheme as an accurate and reliable technique for conducting the structural analysis.

Investigation of the thermal performance of a solar absorber and nocturnal radiator (SAANR) hybrid panel for summer and winter seasons

Transient analysis and performance prediction of a solar absorber and nocturnal radiator (SAANR) hybrid panel for water heating and cooling in the summer and winter seasons of Gaborone, Botswana is presented. Transient heat transfer models were developed for the panel and the models were then transformed into explicit finite difference forms for numerical analysis using MATLAB. The SAANR panel added up to 7 kWh diurnal heating energy in winter and removed up to 0.5 kWh at night through radiation cooling in summer. The resulting numerical solutions of the model were validated by comparison with experimental results and they showed excellent strength of association with a correlation coefficient of 0.832 and fairly acceptable goodness of fit at an R-squared value of 69%. Sensitivity analysis carried out showed the model to react linearly to panel area changes. From the results, it is concluded that the hybrid panel is robust enough to perform in both the summer and winter seasons of Gaborone, and areas with similar climatic conditions. This implies that the system can be used simultaneously for space cooling and domestic water heating in both summer and winter. This technology can therefore be used instead of electric water heaters and air-conditioners and thus save energy.

An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model

This paper adopts an efficient meshless approach for approximating the nonlinear fractional fourth-order diffusion model described in the Riemann–Liouville sense. A second-order difference technique is applied to discretize temporal derivatives, while the radial basis function meshless generated the finite difference scheme approximates the spatial derivatives. One key advantage of the local collocation method is the approximation of the derivatives via the finite difference formulation, for each local-support domain, by deriving the basis functions expansion. Another advantage of this method is that it can be applied in problems with non-regular geometrical domains. For the proposed time discretization, the unconditional stability is examined and an error bound is obtained. Numerical results illustrate the applicability and validity of the scheme and confirm the theoretical formulation.

Analysis of C5G7-TD benchmark with a multi-group pin homogenized SP3 code SPHINCS

The transient capability of a SP3 based pin-wise core analysis code SPHINCS is developed and verified through the analyses of the C5G7-TD benchmark. Spatial discretization is done by the fine mesh finite difference method (FDM) within the framework of the coarse mesh finite difference (CMFD) formulation. Pin size fine meshes are used in the radial fine mesh kernels. The time derivatives of the odd moments in the time-dependent SP3 equations are neglected. The pin homogenized group constants and Super Homogenization (SPH) factors generated from the 2D single assembly calculations at the unrodded and rodded conditions are used in the transient calculations via proper interpolation involving the approximate flux weighting method for the cases that involve control rod movement. The simplifications and approximations introduced in SPHINCS are assessed and verified by solving all the problems of C5G7-TD and then by comparing with the results of the direct whole core calculation code nTRACER. It is demonstrated that SPHINCS yields accurate solutions in the transient behaviors of core power and reactivity.

Tensorial elastodynamics for isotropic media

Numerical solutions of 3D isotropic elastodynamics form the key computational kernel for many isotropic elastic reverse time migration and full-waveform inversion applications. However, real-life scenarios often require computing solutions for computational domains characterized by non-Cartesian geometry (e.g., free-surface topography). One solution strategy is to compute the elastodynamic response on vertically deformed meshes designed to incorporate irregular topology. Using a tensorial formulation, we have developed and validated a novel system of semianalytic equations governing 3D elastodynamics in a stress-velocity formulation for a family of vertically deformed meshes defined by Bézier interpolation functions between two (or more) nonintersecting surfaces. The analytic coordinate definition also leads to a corresponding analytic free-surface boundary condition (FSBC) as well as expressions for wavefield injection and extraction. Theoretical examples illustrate the utility of the tensorial approach in generating analytic equations of 3D elastodynamics and the corresponding FSBCs for scenarios involving free-surface topography. Numerical examples developed using a fully staggered grid with a mimetic finite-difference formulation demonstrate the ability to model the expected full-wavefield behavior, including complex free-surface interactions.

Characterization of Water Dynamics and Modelling of an Open Channel Irrigation System

Management of open channel canals is very essential to avoid overflows and overcome water demand. Automatic control of delivery canals has been gaining significance in recent years with the purpose of improving the efficiency of water management. This paper presents the development of single reach irrigation canal with the objective of controlling the water level. Upstream gate opening of an irrigation canal reach is operated based upon a constant-level reservoir height. The canal reach between two gates is divided into N nodes, and the finite-difference forms of the continuity and the momentum equations (saint venant equations) are written for each node. The Taylor series expansion is applied to linearize the equations around the equilibrium conditions. The state space model of the canal system is developed and the same will further be used to design an efficient controller for maintaining the level of water in the canal.

Implicit meshless method to solve 2D fractional stochastic Tricomi‐type equation defined on irregular domain occurring in fractal transonic flow

AbstractIn the current paper, we develop a new method based on the finite difference formulation and meshless method to solve 2D time fractional stochastic Tricomi‐type equation with Caputo derivative in temporal direction and Neumann boundary condition defined on irregular domain, numerically. In this method, first the fractional order derivative is discretized by a finite difference scheme and then meshless method based on radial basis functions is employed to approximate the functions in the spatial directions. So, the solution of considered problem is transformed to the solution of a linear system of algebraic equations which will be solved at each time step. The most advantage of this method is that it has no triangular, quadrangular, or other type of meshes and therefore is very applicable to solve various high dimensional problems. Finally, some test problems with different domains have been solved via present method to verify the accuracy and efficiency of the newly developed meshless formulation. Numerical results confirm the present technique is very effective to approximate the solution of 2D fractional stochastic Tricomi‐type equation with Neumann boundary condition.