International Journal of Computational Methods

Pressure piping is the most productive way for large-volume compressed natural gas (CNG) transportation. In pipeline constructions, the thickness at the point where two pipes join together is often not consistent due to the mismatch in dimensions, and thus stress concentrations can often occur at the pipe joints, causing safety concerns. Therefore, it is important to accurately analyze the key influencing factors of dimensional mismatch defects, providing a theoretical basis for the preliminary design and post-repair of pipelines. This work uses the smoothed finite element method (S-FEM) that has been proven accurate in stress analysis compared with the traditional FEM. Since geometry and load of the pressure piping are axisymmetric, a novel axisymmetric S-FEM element is first developed, coded and integrated in ABAQUS using the User-defined Element Library (UEL). Intensive studies are then carried out to examine the effects of different levels of mismatch in the thicknesses of two joined pipes and the effects of the radius of the transitional fillet used to bridge the mismatches. It is found that the maximum hoop stress reduces as the radius of the transitional fillet increases. For the thinner section of the pipe, the maximum hoop stress is only affected by the thickness mismatch.

The causes of fatal and serious injuries in rollover accidents have yet to be fully identified. This study aimed to reconstruct a complex accident, based on both video analysis and vehicle coupling simulation. Besides, the major fatal injury causation and curtain airbag’s effectiveness were analyzed. The coupling model, including finite element tire and multi-body vehicle body, was developed to estimate the vehicle trajectory. A validated curtain airbag (CAB) module, was introduced for occupant protection effectiveness evaluation. The vehicle trajectory from the simulation correlated well with that from video recordings and simulation result indicated that passenger’s head contact with road (maximum contact force 9834.9N) was assumed to be the main reason of the victim death. After introducing the CAB, the maximum head acceleration was reduced from 129.1g to 24.9g, and the neck Fz was reduced by 80.0% as well. The coupling simulation method turned out to be efficient to reconstruct certain complex accident case. Such study is beneficial for further similar accident reconstruction and restraint system evaluation.

In this paper, the meshless smoothed particle hydrodynamic (SPH) method is applied for solving the Black–Scholes model for European and American options, which are governed by a generalized Black–Scholes partial differential equation. We use the $𝜃$-method and SPH for discretizing the governing equation in time variable and option pricing, respectively. To validate our SPH method, we compare it with the analytical solution and also the finite difference method. The numerical tests demonstrate the accuracy and robustness of our method.

Thermal actuators use thermal expansion and contraction of an elastic body to produce motion at its output port. In the present study, a thermal actuator comprises an elastic body and heating/cooling devices. Such devices have a thin-layer shape and are installed on the surface of the elastic body. The design optimization of thermal actuator is a multiphysics problem, including both heat conduction and elastic deformation. The design variables include multiple types of boundaries, i.e., the temperature boundary (high temperature and low temperature) and the free boundary. In order to solve such a multiphysics optimization problem involving multiple types of boundaries, the level set-based multiple-type boundary method is employed. In the analysis for the shape derivative of the temperature boundary, the constrained variational principle is employed to explicitly include the temperature boundary condition into the weak form of heat conduction equation. Numerical examples in two dimensions are investigated.

On rockfall slopes, trees can play a natural and efficient protective role in preventing the occurrence of rockfall disasters. This paper presents a study on the tree barrier effect on rockfall based on a three-dimensional discontinuous deformation analysis (3D DDA) method. The tree barrier effect on rockfall is studied considering the 3D spatial effect of rockfall movement and the 3D spatial distribution of trees. The basic movement forms and collision modes of rockfall related to the tree barrier are classified, and the impact factors of tree arrangements on rockfall are proposed. The contact mechanism employed in the 3D DDA method is illustrated, and the accuracy of the 3D DDA rockfall simulation is verified by comparing with the results obtained from laboratory experiments. Numerical examples are conducted to study the effect of the tree barrier on rockfall, and general laws are derived by analyzing the variation of kinetic energies and motion trajectories of moving blocks. The results show that the 3D DDA method can effectively predict the 3D motion behaviors of rockfall and assess the tree barrier effect. Furthermore, a reasonable arrangement of trees or a combination of protective facilities and trees can prevent or mitigate the rockfall hazards.

In this work, a modified Newton–Özban composition of convergence order six for solving nonlinear systems is presented. The first two steps of proposed scheme are based on third-order method given by Özban [Özban, A. Y. [2004] “Some new variants of Newton’s method,” Appl. Math. Lett. 17, 677–682.] for solving scalar equations. Computational efficiency of the presented method is discussed and compared with well-known existing methods. Numerical examples are studied to demonstrate the accuracy of the proposed method. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

Two families of higher-order accurate time integration algorithms are numerically tested by using various nonlinear problems of structural dynamics, and the numerical results obtained from them are compared. To be specific, the higher-order algorithms of Kim and Reddy and the higher-order algorithms of Fung are used for this study. In linear analyses, these two different families of higher-order algorithms do not present noticeable differences. However, performances of these algorithms are quite different when they are applied to various nonlinear dynamic problems. For the numerical tests, well-known nonlinear problems are selected from the past studies. For the completeness, the two families of algorithms are briefly reviewed, and their advantageous computational structures are also explained.

This work concerns mainly the finite element (FE) implementation of polyconvex incompressible hyperelastic models. A user material subroutine (UMAT) has been developed and can be utilized to define the aforementioned material behaviors in the FE system ABAQUS. The subroutine is written using a novel strategy in order to maximally simplify the relations for the analytical material Jacobian (MJ). The UMAT code is attached in the appendix. The developed subroutine allows to significantly decrease the time of computations and to avoid possible convergence difficulties. The structure of the code enables modifications which may lead to a rheological, damage or growth models, for instance.

Based on two-grid discretizations, local and parallel finite element algorithms are studied for the Stokes equations with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Thereafter, the variational inequality can be transform into the variational identity as a multiplier in a convex set. The main idea of our algorithms is to approximate the low frequencies of the finite element solution using a coarse grid and use a fine grid to correct the resulted residual (that includes mostly high frequencies of the solution) by some local and parallel procedures. Error bounds for the approximate solutions are estimated. Numerical results are also given to demonstrate the effectiveness of the algorithms.

The stilling basin has been one of the most powerful hydraulic structures for the dissipation of the flow energy. Meshfree and particle methods have special advantages in modeling incompressible flows with free surfaces. In this paper, an integrated smoothed particle hydrodynamics (SPH) method is developed to model energy dissipation process of stilling basins. The integrated SPH includes the kernel gradient correction (KGC) technique, the dynamic solid boundary treatment, $\delta$-SPH model and density reinitialization. We first conducted the simulations of dam-breaking and hydraulic jump to validate the accuracy of the present method. The present simulation results agree well with the experimental observations and numerical results from other sources. Then the discharge process of stilling basin with baffle-blocks is simulated with the integrated SPH. It is demonstrated that the detailed discharge process can be well captured by this method. The energy dissipation effect of stilling basin could be significantly improved by the baffle-blocks. The structure and position of the baffle-block directly affect the energy dissipation effect, while the height of the baffle-block has big influence on the drainage capacity.

The physical response of acoustic metamaterials may change due to the variation of the material properties in the manufacture process. Thus, an interval perturbation finite element method is formulated to study the mechanical response of acoustic metamaterials with the interval parameters, which includes the uncertainty effects on the band structure, resonance mode and frequency response of acoustic metamaterials. By virtue of the first-order Taylor series expansion and sensitivity analysis of dynamic properties of acoustic metamaterials with respect to the interval parameters, the interval perturbation finite element method established in this work can predict the upper and lower bounds of the dynamic properties of the acoustic metamaterials. Three numerical examples are studied to validate the effectiveness of the interval perturbation finite element method to analyze the physical response of acoustic metamaterials with the interval parameters, and the results calculated by Monte Carlo method are regarded as the reference results to validate the interval perturbation finite element method. The uncertainty model constructed by interval perturbation finite element method provides a great help in the design of acoustic metamaterials.

We present a stabilized finite volume element method for the coupled Stokes–Darcy problem with the lowest order $P_1-P_0$ element for the Stokes region and $P_1$ element for the Darcy region. Based on adding a jump term of discrete pressure to the approximation equation, a discrete inf-sup condition is established for the proposed method. The optimal error estimates in the $H^1$-norm for the velocity and piezometric head and in the $L^2$-norm for the pressure are proved. And they are also verified through some numerical experiments. Two figures are given to show the full comparison for the local mass conservation between the proposed method and the stabilized finite element method. And this method can also be computed directly in the irregular domain according to the last experiment.

In this paper, the cell-based smoothed finite element method is extended to solve stochastic partial differential equations with uncertain input parameters. The spatial field of Young’s Modulus and the corresponding stochastic results are represented by Karhunen-Loéve expansion and polynomial chaos expansion, respectively. Young’s Modulus of structure is considered to be random for stochastic static as well as free vibration problems. Mathematical expressions and the solution procedure are articulated in detail to evaluate the statistical characteristics of responses in terms of the static displacements and the free vibration frequencies. The feasibility and the effectiveness of the proposed SGCS–FEM method in terms of accuracy and lower demand on the mesh size in the solution domain over that of conventional FEM for stochastic problems are demonstrated by carefully chosen numerical examples. From the numerical study, it is inferred that the proposed framework yields accurate results.

In this work, an indoor soil-bin is designed to investigate the tire–terrain interaction mechanisms for the off-road tires rolling on the gravel terrain. The soil-bin test is carried out by the indoor soil-bin experimental device and the three-dimensional (3D) finite element (FE) and discrete element (DE) coupling method under the same particles conditions, respectively. First, with the indoor soil-bin measurement system, the repeatability of the soil-bin experiments is employed to validate the experimental device and the numerical models. Moreover, the tractive performance experiments of the off-road tires with two tread patterns, smooth and grooved interacting with gravel terrain, are performed at the slip of 10%, 20% and 30%, respectively, to obtain the tractive force and the rim sinkage. Second, the corresponding numerical models are also established, and simulated by the FE–DE coupling method, where the FEM and the DEM are used to describe the off-road tires and the gravel terrain, respectively. The tractive mechanisms of the off-road tires in interaction with the gravel terrain such as the tractive force and the rim sinkage are investigated numerically. Meanwhile, The dynamics and discontinuity of the gravel assembly are described by the presented approach. Besides, both the results of the simulations and experiments indicate that tread patterns and slip conditions have great influence on the tire tractive performance. Finally, the numerical simulations and the experimental results qualitatively show good agreements, which certifies the effectiveness of the FE–DE coupling method in the tractive performance analysis of tire–gravel terrain interactions.

The reanalysis method to obtain the second derivatives of static displacement is innovatively proposed in this paper. This method is based on the combined approximations method. The reanalysis formulations of the second derivative of static displacement are derived to provide a programmatic procedure of formulations construction. Besides, the normalized error and the number of algebraic operations are considered to evaluate the accuracy and efficiency, respectively. Finally, three typical numerical results verify the accuracy and robustness of the proposed method.

So far, the carbon nanotube (CNT) distributions have been still limited in the linear forms, which may lead to the limitations in maximizing the strength and potential of carbon nanotube reinforced composite (CNTRC) structures. This study, hence proposes a type of novel CNTs distribution for improving the stiffness of CNTRC beams. The distributions are in the nonlinear forms and ensure the same total CNT volume fraction along the thickness of structures. For demonstrating, the effectiveness of the proposed CNT distributions, the static, free vibration and buckling analyses of functionally graded carbon nanotube (FG-CNT) reinforced composite beams using the new CNT distributions are conducted and one-dimensional NURSB basis functions based on the third-order shear deformation theory (TSDT) are utilized to describe the exact geometry and to approximate the unknown solution in finite element model of the beam. The numerical investigations of the geometric and material parameters reveal that the new nonlinear CNT distributions can help increase the normalized frequency, buckling load and the nondimensional central deflection to the maxima of 8%, 16% and 16% respectively, in some conditions of geometric parameters.

A novel method of order reduction is proposed to the high-dimensional convection-diffusion-reaction equation with Robin boundary condition based on the multiquadric radial basis function-generated finite difference method (MQ RBF-FD). The main motivation is to get not only a second-order accurate solution but also a second-order accurate gradient. Key to the proposed method is introducing the intermediate variables representing the first-order derivatives to reduce the original second-order problem into an equivalent system of first-order partial differential equations. Then a discrete scheme for the latter is constructed, in which MQ RBF-FD method is applied to approximate the first-order derivatives of the original variable at the center point with decoupled method. Moreover, we can obtain an equivalent discrete scheme about the original variable and intermediate variables which can be proven all second-order convergent, that is, the convergence rate of the gradient of solution is also second-order. Finally numerical examples are presented to show the efficiency and accuracy of the proposed method.

The aeroelastic coupling dynamic equation of helicopter rotor is essentially a set of nonlinear and inhomogeneous equations with large rigidity, in which the inhomogeneous term is a function of blade motion and aerodynamic load. In this paper, the precise time integration method and its derived formats are introduced to solve the rotor blade dynamic equation, and the Duhamel integral item can be calculated by various numerical methods. In terms of computational accuracy and numerical stability, the precise Kutta method and high precision direct integration method (HPD method) are carefully selected to compare with classical Runge–Kutta method numerically. HPD method is used to solve the rotor blade dynamic equation, and the transient response of the rotor blade is examined by Newmark and implicit trapezoidal methods. Results indicate that HPD method dominates the classical Runge–Kutta method in step size independence, and gets close to implicit methods in numerical stability and accuracy for dynamic equation of helicopter rotor blade.