Central Difference Method Research Articles

Gravity and magnetic fast inversion method with cross-gradient based on function fitting

SUMMARY The underground density and magnetic susceptibility structure obtained by cross-gradient inversion of gravity and magnetic data can provide an important basis for the evaluation of mineral resource potential. The inversion is realized by discretizing the subsurface into a series of cells and calculating the physical properties of each cell. This restricts the application in the inversion of large-area data because of the large memory usage and low computational efficiency owing to the large solution matrix. We proposed a high-efficiency cross-gradient inversion method of gravity and magnetic data based on function fitting, which uses a parametrized function to fit the physical properties of the central points of a number of cells in certain space based on the assumption that the physical property in each cell is uniform in the inversion. Therefore, the calculation of physical properties is replaced by the calculation of function coefficients. The number of cells that can be fitted by a function is more than the number of function coefficients. Thus, the new method requires less memory and can improve computational efficiency. In addition, with the function fitting method, the gradient terms can be directly obtained by the function coefficients instead of using the traditional central difference or multipoint fitting method, which could further improve the computational efficiency. The effectiveness and high computational efficiency of the proposed method were verified by model tests, and the accuracy of the inversion results was not lower than that of the traditional inversion method. We also proved that the proposed method is suitable for uniform hexahedral and unstructured tetrahedral cells. The real application area is located in eastern China and contains skarn-type magnetite deposits, which are characterized by high density and high magnetic susceptibility. The proposed method was used to obtain subsurface high-resolution density and magnetic susceptibility structures. We obtained the distribution range of mineral resources based on the ratio of density to magnetic susceptibility, which provides an important basis for further exploration. The model tests and real data applications show that the proposed method is more suitable for large-scale precise inversion and has better practicability.

Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with Hookean and neo-Hookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation , hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. The SDME bases are accurate and provide better conditioned equations for iterative solutions in general, leading to substantial reduction in processing time over an order of magnitude compared to other modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard modal basis. Results for a two-dimensional impact problem approximated with the SDME bases showed that the implicit Newmark method performed much better in terms of processing time and contact stress error when compared to the standard explicit central difference method with lumped mass matrix . • Application of simultaneous diagonalization procedure and minimum energy highorder bases to non-linear structural and impact transient problems. • Simple modifications of high-order bases improve conditioning of system of equations obtained from the high-order approximation of the considered problems, with remarkable gains in performance of iterative solvers. • Light variation on the number of pre-conditioned gradient iterations in terms of the polynomial order. • Speed-ups over 40 when compared to the standard modal Jacobi high-order bases used in high-order interpolation.

A fully explicit staggered algorithm for near‐field wave propagation of fluid‐saturated porous media based on u‐p dynamic formulation

AbstractNear‐field wave propagation of the fluid‐saturated porous media (FSPM) is an important issue in soil dynamics and geotechnical earthquake engineering. In the current study, a fully explicit staggered algorithm for the near‐field wave propagation analysis of the FSPM in the time domain is developed based on the u‐p dynamic formulation. The decoupling of dynamic formulation is implemented by the diagonalization of the mass matrix and pore fluid compressibility matrix. The central difference method and Newmark constant average acceleration method are used for the solution of the displacement and velocity of solid skeleton, and the backward difference method is used for the solution of pore fluid pressure in the time domain. The effectiveness and feasibility of the matrix diagonalization are verified by the comparison with the classical numerical algorithm. The proposed algorithm is also validated by the comparison of the numerical results with the corresponding analytical solution results. Combing with the transmitting artificial boundary condition, the proposed algorithm is then applied to investigate the seismic response of a FSPM free field, and its applicability to the typical near‐field wave propagation problems of the FSPM is indicated. The stability analysis of the proposed algorithm is finally conducted, and the corresponding stability criterion is presented. For the proposed algorithm, all the dynamic variables of the FSPM are solved in an iterative way and the coupling dynamic equations need not to be solved at each time step. The computational effort can be reduced considerably, and the computational efficiency can be improved remarkably.

Dynamic non-local damage analysis using an octree pattern-based massively parallel explicit solver

This paper presents the development of a massively parallel explicit solver based on the central difference method (CDM) for the dynamic analysis of damage processes. In this context, the material degradation is incorporated using an integral-type non-local isotropic damage model. A fully automatic preprocessing is enabled by following the octree-based mesh generation paradigm. To handle neighbouring elements with different sizes and to avoid hanging nodes in the generated mesh, polyhedral elements are implemented within the framework of the scaled boundary finite element method (SBFEM). A pre-computation approach is advocated to develop a highly efficient solver, where the limited number of master cells in a balanced octree grid are exploited. The parallelisation is carried out using a mesh-partitioning technique and message-passing-interface (MPI) directives, which is highly complementary to the use in high-performance computing (HPC) facilities. In order to showcase the performance of the proposed solver, a problem of practical interest is selected, featuring the analysis of an XCT image-based model of concrete containing about 1 billion voxels. The simulated results have been obtained within minutes by exploiting the computational power of approximately 12k cores.

Design of reverse flow ambient air heated vaporizer under different ambient temperatures and optimization of the fin arrangement

• Reverse flow system is preferable for the ambient temperature above 281 K. • Heat transfer is augmented by providing lower number of fins in the frosted region. • Higher number of fins to be used to augment heat transfer in no-frost segment. • Optimized number of fins and fin height vary depending on the location of the tube. • Optimized multi-fin system can augment the heat transfer throughout the vaporizer. Ambient air heated cryogenic vaporizers utilize aerothermal energy from the environment and re-gasify stored liquefied industrial gases. In a conventional system, there are two vaporizer blocks: one remains active, while the other gets defrosted. There is often an idle time in the defrosting process that the reverse flow vaporizer proposes to eliminate. As the thickness of frost over the surface of finned tubes varies from inlet to outlet points, optimized number of fins and their height also vary throughout the vaporizer block. In the literature, optimization process keeps the fin specification identical for all the tubes. However, a varying frost thickness along the vaporizer length provides an opportunity to augment the heat transfer by optimizing each finned tube based on the frost deposition over it. One dimensional heat transfer model of defrosting is developed and solved implicitly using the central difference method. Genetic Algorithm optimizes each tube of the vaporizer based on its location inside the block and minimizes its weight which is 16% lower than that of the single-fin reverse flow system and 35% lower than that of the single-fin conventional ambient air heated vaporizer. This would lead to large savings in the capital cost of the vaporizers.

Weak damped wave equations defined by a class of self-similar measures with overlaps

We study weak damped wave equations defined by fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. First, we prove the weak well-posedness of these wave equations and some regularity results. Second, by using some important information about the structure of the three measures above, we set up a framework for one-dimensional measures to discretize the wave equations. Then we use the finite element and central difference methods to obtain numerical approximations of the weak solutions for such equations. Finally, we show that the numerical solutions converge to the actual weak solution and obtain the rate of convergence.

Positivity-preserving and unconditionally energy stable numerical schemes for MEMS model

In this paper, we propose and analyze the positivity-preserving, unconditionally energy stable, and linear second order fully discrete schemes for Micro-Electromechanical system (MEMS). More precisely, we use the first order backward difference formulation (BDF1) and second order Crank-Nicolson (CN) formulation for the temporal discretization, and the central finite difference method for spatial discretization. A variant of the exponential scalar auxiliary variable (ESAV) approach is involved in our numerical schemes to deal with the singular nonlinear term. The unconditional energy stability of the numerical schemes is rigorously proved, without any restriction for the time step sizes. Furthermore, we derive that the numerical solutions always preserve the positivity property of the MEMS model, that is the distance variable is always between 0 and the steady solution, at a point-wise level. A series of numerical simulations are presented to demonstrate the positivity and energy stability of our numerical schemes.

Numerical prediction of vortex-induced vibrations of a long flexible riser with an axially varying tension based on a wake oscillator model

A numerical study based on the wake oscillator model has been conducted to determine the vortex-induced vibration (VIV) responses of a flexible riser with an axial time-varying tension. Three different types of flows, viz., linear shear, exponential shear, and real stepped flows, have been considered. The coupling equations of a structural oscillator and a wake oscillator have been solved using a standard central finite difference method of the second order. The VIV response characteristics including the structural displacement, structural frequency, displacement envelope, and displacement evolution for three different flow profiles have been systematically compared. Subsequently, for each flow profile, the effect of the different tension models on the VIV characteristics has been investigated. The numerical results indicated that, both the VIV displacement and VIV frequency in real stepped flow had diverse characteristics from those in linear shear flow. Compared with the corresponding VIV characteristics in real stepped flow, the VIV displacement in exponential shear flow was similar, however, the VIV frequency in exponential shear flow was disparate. The VIV displacement with the constant tension model had larger values than that with the real tension model.

Micro-Wear Simulation of Braking Interface Based on Particle Discrete Element Modeling

For material fracture and severe wear in braking conditions, the discrete element method (DEM) is used to simulate the wear process of the braking interface explicitly. Based on the central difference method, particle motion equations are established considering the influence of elemental damping on particle contact. Combined with the Particle Flow Code (PFC) software, a DEM wear model of the braking interface is established using the parallel bond modeling method. The braking wear process is simulated, and the material damage process is investigated. The simulation results demonstrate that with the increase of the initial braking load and the initial braking speed, the wear depth increased by 24.75% and 16.22%, respectively. The increase in the number of detached particles leads to an increasing trend of fracture force chains, which increases the thickness of the flowing particle layer, revealing the micro-wear mechanism of the braking interface.

Meshfree Collocation Framework for Multi-Phase Coupling Nonlinear Dynamic System in a Porous Enclosure

This study is aimed at establishing an effective incremental-iterative framework, for the first time, using reproducing kernel collocation method for solving multi-phase dynamic coupling problems in a porous enclosure. The central difference method is utilized in the temporal discretization while direct collocation with reproducing kernel approximation is adopted in the spatial discretization, in particular, with Newton–Raphson method for nonlinear iteration. Under the proposed numerical framework, a determined and well-conditioned nonlinear system is ensured during nonlinear iteration. In addition to the traditional mesh-based weak-form methods, it is shown that the present study provides an alternative effective approach to solve unsteady-state double-diffusive convection problems.

Magneto-Optical Imaging of Arbitrarily Distributed Defects in Welds under Combined Magnetic Field

Using the traditional magneto-optical detection methods, micro-weld defects parallel with the magnetic field direction may be overlooked. In order to overcome this, a non-destructive testing method based on magneto-optical imaging under a vertical combined magnetic field (VCMF) is proposed. To demonstrate this, the experimental results of the magneto-optical imaging of weld defects excited by a vertical combined magnetic field (VCMF) or parallel combined magnetic field (PCMF) are compared with those of traditional magnetic fields (constant magnetic field (CMF), alternating magnetic field (AMF), and rotating magnetic field (RMF)). It is found that the magneto-optical imaging under a VCMF can accurately detect weld defects of any shape and distribution. In addition, the center difference method is used to eliminate the influence of noise on the defect contour extraction of magneto-optical images, and the active contour of weld defects in the magneto-optical images is extracted. The results show that many noises can be identifiedby the robustness of the level set method, operating in low-pass filtering, so that much information that is usually lost can be retained.

Analysis of the pantograph’s mass distribution affecting the contact quality in high-speed railway

ABSTRACT The pantograph–catenary system is an essential device, transmitting electric energy to high-speed trains. The sliding contact quality between the collector strip and the contact wire must be strictly required, keeping the good current collection quality. In this work, the displacements of two independent tension wires are described by the Fourier sine-series expansions separately. The Lagrange equation of the second kind is used to establish the dynamic equation of the catenary, which is solved by the central difference method. A multi-rigid body dynamics theory based on the relative coordinates is adopted, describing the dynamic behaviour of the pantograph. Three types of mass distribution strategies applied to the main structure of the pantograph are discussed. The mass distribution of the main structure has a large effect on the pantograph–catenary interaction, and the maximum reduction ratio of the standard deviation (STD) of the dynamic contact force reaches −35.62%, which is significant for the pantograph’s structural optimization in high-speed railways.

Development and assessment of the interface lattice Boltzmann flux solvers for multiphase flows

Unlike the lattice Boltzmann method (LBM), the lattice Boltzmann flux solver (LBFS) is free from the limitation of uniform meshes, coupled time step and mesh spacing, and high memory cost. Specifically, the existing interface LBFS (LBFS-CH0) is reconstructed by locally applying the Cahn–Hilliard-based (CH-based) LBM with additional artificial terms. In the framework of LBM, the influence of the additional artificial terms has been assessed. Besides, the comparative study of the CH-based and the Allen–Cahn-based (AC-based) LBM shows that the latter has higher accuracy and stability. Though the LBFS is originated from the LBM, their performances may not be exactly the same. In this study, the AC-based and CH-based LBFS (LBFS-AC and LBFS-CH1) models which can accurately recover the phase field equations are developed from the interface LBM models without any additional terms. The gradient term in the LBFS-AC is discretized by the second-order central difference method. Compared with the LBFS-CH0, the LBFS-CH1 can eliminate the additional terms and recover the true CH equation. Then these interface LBFS models are compared to evaluate their performance and the influence of the additional terms. Different from the LBM, it is found that the CH-based LBFS is superior to the AC-based one in terms of the numerical accuracy and stability. In addition, numerical results show that the additional terms have minor effect on the ability to model the phase interface. This study can provide guidance to ensure the accuracy and stability of interface modeling. Taking into account the accuracy, stability and simplicity, we can conclude that the LBFS-CH0 model could be the first choice to simulate the phase interface in the framework of LBFS.

Dynamic snap-through of functionally graded shallow arches under rapid surface heating

In this paper, dynamic buckling response of a shallow arch constructed from functionally graded materials under thermal shock is investigated. It is assumed that one surface of the beam is subjected to sudden temperature elevation while the other surface is kept at reference temperature. To solve the one-dimensional Fourier heat transfer equation, Crank–Nicolson numerical method along with the central finite difference method are employed. In order to obtain the strain–deflection relations, Donnell theory of shallow arches with von Kármán simplifications are used. The governing equations are acquired based on well-known Hamilton’s principle that are converted to a set of non-linear algebraic equations using the Ritz method. Since the governing equations are non-linear equations, the Newton–Raphson method is applied and after that temporal evolution of displacements are obtained employing the Newmark time marching scheme. In the final stage, a theory to estimate the buckling load is required, which is the Bodiansky–Roth criterion in this research. To validate the formulation and solution method of this project, this work is compared with previous studies and good agreement is obtained. In addition, impacts of power law index, arch thickness, geometrical parameters, and temperature dependency on the dynamic buckling temperature are reported in this study.

Production Simulation of Oil Reservoirs with Complex Fracture Network Using Numerical Simulation

This paper established a numerical simulation model to analyze the pressure transient and rate transient behaviors in reservoir with complex fracture network. Firstly, the fractures are introduced into the coordinate system through the position, angle, and length. Secondly, a mathematical model is established by using unstable seepage model. Thirdly, the central difference method was used to solve the model and local grid refinement method is introduced to describe the network fractures. Finally, we compared the results obtained from this paper’s model with the production data. The results show acceptable and reasonable matches for typical well. Meanwhile, the sensitivity of two properties is discussed. The model solution is verified with an analytical method thoroughly. The novelty of this paper is to introduce each fracture in fracture network into the coordinate system. Then, the grid refinement is achieved according to the fracture information. The presented new model simplifies the analysis of the pressure transient and rate transient of the reservoir with complex fracture network, and it is more efficient than the conventional numerical method. Compared with the analytical methods, the new model describes the fractures system in more detail. However, the new model treats fractures as reservoirs with higher permeability in the central difference method, which is simpler and rougher than traditional numerical methods.

Study on the Stabilities of RTH Tests Using the CDM Method considering Two Kinds of Systematic Errors

Real-time hybrid (RTH) test is a promising test method to investigate seismic responses of structures and is essentially a time history analysis. Therefore, an in-depth understanding of the control algorithm, such as a central difference method (CDM), is important for this method. In this paper, stabilities of the CDM considering two kinds of errors are investigated theoretically and experimentally. It turns out that the convergence criterion of the CDM cannot be changed by constant errors but can be changed by proportional errors. The lowest structural natural period to keep the results stable increases with the increase of the ratio of the displacement error to the interface displacement and the integration time step. At last, the theoretical result of the stability of the CDM considering two kinds of systematic errors are confirmed by the proposed test system with a natural rubber bearing.

Simulation analysis and experimental study on impact fall test of forklift overhead guard

Abstract This paper introduces the safety performance test standards and requirements of forklift overhead guard. According to the requirements of the impact fall test in the standard, an explicit dynamic calculation model of the impact of overhead guard under load was established based on the LS-DYNA display central difference method, and the field impact test of the overhead guard designed in this paper was carried out. The results show that the hourglass energy of the impact falling simulation test is less than 5% of the total energy, which conforms to the law of energy conservation, so the simulation model can be considered correct. At the same time, the simulation analysis results are consistent with the field impact falling test results. The connection position of riser and position of base bolting appear failure fracture, and the structural strengthening suggestions are given.

Nonlinear aeroelastic stability analysis of a two-stage axially moving telescopic wing by using fully intrinsic equations

During the process of span extension for an aircraft wing equipped with a telescopic morphing mechanism, the wing aspect ratio increases, and hence, the geometrical nonlinearities might become more significant. In this regard, this paper aims to investigate the effect of structural nonlinearity on the aeroelasticity of span morphing wings using the exact fully intrinsic equations for the first time. Furthermore, the effects of various parameters such as thrust force, engine location, chord size, flight altitude, initial angle of attack, and overlapping mass on the aeroelasticity of the wing are studied. The applied aerodynamic loads in an incompressible flow regime are determined using Peters’ unsteady aerodynamic model. In order to check the stability of the system, first the resulting nonlinear partial differential equations are discretized by using the central finite difference method and then linearized about the static equilibrium. Finally, by obtaining the eigenvalues of the linearized system, the stability of the wing is evaluated. It is observed that by using the fully intrinsic equations, the instability of the axially moving telescopic wing can be determined more accurately. Moreover, the results show that the morphing length and overlapping mass have significant effects on the aeroelastic stability of the telescopic wing.

Wetting boundary schemes in modified phase-field lattice Boltzmann method for binary fluids with large density ratios

In this work, wetting boundary schemes in a modified phase-field lattice Boltzmann (LB) method are developed to simulate the contact-line motion and contact angle hysteresis of binary fluids with large density ratios. In the modified phase-field LB method, two multiple-relaxation-time (MRT) LB equations are utilized, one with a nondiagonal relaxation matrix used to solve the conserved Allen-Cahn (A-C) equation and the other with a modified discrete source term used to solve the incompressible Navier-Stokes (N-S) equations. Based on the idea of geometric formulation, two wetting boundary schemes corresponding to the modified phase-field LB model are proposed. In the first scheme, a layer of virtual nodes is utilized to realize the geometric formulation. The values of the order parameter at the virtual nodes are determined by the central difference method. In the second scheme, the geometric formulation is realized through a local scheme for the gradient terms of the order parameter where the nonequilibrium parts of the distribution function are utilized. Compared with the first wetting boundary scheme and some previous models, the second wetting boundary scheme does not require the virtual nodes and simplifies the numerical implementation. The developed two schemes are validated through a series of benchmark simulations, including droplet spreading on idea substrate, capillary intrusion and droplet shearing on nonideal substrate. It is found that satisfactory accuracy can be achieved by both two schemes, and the second scheme can obtain better stability while the first scheme shows better performance in simulating contact angle hysteresis.

Deployment modeling for soft cable networks from slack to tension

Cable networks are widely used in space deployable devices and other fields for their large spans, lightweight, and small storage volumes. Among them, the deployable cable network usually turns from a stowed state to a fully deployed configuration and finally forms the working shape. The accurate and effective deployment analysis from slack to tension concerns the stable and efficient service of these cable networks. However, there are still no effective ways to analyze the deployment of cable networks with varying topologies and deployments modes. This paper proposed a general dynamic method for the deployment analysis of cable networks. The cable network is divided into a set of particles, and the deployment process is regarded as a series of time steps. Each node's equations about the force and displacement are formulated and assembled by applying a topological matrix to form the dynamic system model. The cable node's coordinates in the stowed state are calculated and further used to solve the coordinates and tension in the following time step by central difference method. Repeat this process until obtaining all the nodes’ positions and cable forces during the deployment. The cables’ slack/tension states are evaluated real-timely, and cable tension and node coordinates are updated accordingly, which makes the dynamic coupling with supporting mechanisms possible. The presented method applies to the deployment dynamic modeling and analysis of cable networks with various topologies, deployment modes, and geometric dimensions, with effective, easy operation, good convergence, and high-efficiency features.