A stable implicit nodal integration-based particle finite element method(N-PFEM)for modelling saturated soil dynamics

In this study,we present a novel nodal integration-based particle finite element method(N-PFEM)designed for the dynamic analysis of saturated soils.Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner(HR)variational principle,creating an implicit PFEM formulation.To mitigate the volumetric locking issue in low-order elements,we employ a node-based strain smoothing technique.By discretising field variables at the centre of smoothing cells,we achieve nodal integration over cells,eliminating the need for sophisticated mapping operations after re-meshing in the PFEM.We express the discretised governing equations as a min-max optimisation problem,which is further reformulated as a standard second-order cone programming(SOCP)problem.Stresses,pore water pressure,and displacements are simultaneously determined using the advanced primal-dual interior point method.Consequently,our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation.Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy,obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches.This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

THE SUPERCLOSENESS OF THE FINITE ELEMENT METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM ON A BAKHVALOV-TYPE MESH IN 2D

For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of the mesh in the layer adjacent to the transition point,resulting in a suboptimal estimate for convergence.Existing analysis techniques cannot handle these difficulties well.To fill this gap,here a novel interpolation is designed delicately for the smooth part of the solution,bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method.Our theoretical result is uniform in the singular perturbation parameterεand is supported by the numerical experiments.

A BICUBIC B-SPLINE FINITE ELEMENT METHOD FOR FOURTH-ORDER SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations.Its key feature is the selection of bicubic B-splines as trial functions to approximate the state and costate variables in two space dimensions.A Crank-Nicolson difference scheme is constructed for time discretization.The resulting numerical solutions belong to C2in space,and the order of the coefficient matrix is low.Moreover,the Bogner-Fox-Schmit element is considered for comparison.Two numerical experiments demonstrate the feasibility and effectiveness of the proposed method.

A Deep Learning Approach to Shape Optimization Problems for Flexoelectric Materials Using the Isogeometric Finite Element Method

A new approach for flexoelectricmaterial shape optimization is proposed in this study.In this work,a proxymodel based on artificial neural network(ANN)is used to solve the parameter optimization and shape optimization problems.To improve the fitting ability of the neural network,we use the idea of pre-training to determine the structure of the neural network and combine different optimizers for training.The isogeometric analysis-finite element method(IGA-FEM)is used to discretize the flexural theoretical formulas and obtain samples,which helps ANN to build a proxy model from the model shape to the target value.The effectiveness of the proposed method is verified through two numerical examples of parameter optimization and one numerical example of shape optimization.

A New Isogeometric Finite Element Method for Analyzing Structures

High-performance finite element research has always been a major focus of finite element method studies.This article introduces isogeometric analysis into the finite element method and proposes a new isogeometric finite element method.Firstly,the physical field is approximated by uniform B-spline interpolation,while geometry is represented by non-uniform rational B-spline interpolation.By introducing a transformation matrix,elements of types C^(0)and C^(1)are constructed in the isogeometric finite element method.Subsequently,the corresponding calculation formats for one-dimensional bars,beams,and two-dimensional linear elasticity in the isogeometric finite element method are derived through variational principles and parameter mapping.The proposed method combines element construction techniques of the finite element method with geometric construction techniques of isogeometric analysis,eliminating the need for mesh generation and maintaining flexibility in element construc-tion.Two elements with interpolation characteristics are constructed in the method so that boundary conditions and connections between elements can be processed like the finite element method.Finally,the test results of several examples show that:(1)Under the same degree and element node numbers,the constructed elements are almost consistent with the results obtained by traditional finite element method;(2)For bar problems with large local field variations and beam problems with variable cross-sections,high-degree and multi-nodes elements constructed can achieve high computational accuracy with fewer degrees of freedom than finite element method;(3)The computational efficiency of isogeometric finite element method is higher than finite element method under similar degrees of freedom,while as degrees of freedom increase,the computational efficiency between the two is similar.

Extended finite element-based cohesive zone method for modeling simultaneous hydraulic fracture height growth in layered reservoirs

In this study,a fully coupled hydromechanical model within the extended finite element method(XFEM)-based cohesive zone method(CZM)is employed to investigate the simultaneous height growth behavior of multi-cluster hydraulic fractures in layered porous reservoirs with modulus contrast.The coupled hydromechanical model is first verified against an analytical solution and a laboratory experiment.Then,the fracture geometry(e.g.height,aperture,and area)and fluid pressure evolutions of multiple hydraulic fractures placed in a porous reservoir interbedded with alternating stiff and soft layers are investigated using the model.The stress and pore pressure distributions within the layered reservoir during fluid injection are also presented.The simulation results reveal that stress umbrellas are easily to form among multiple hydraulic fractures’tips when propagating in soft layers,which impedes the simultaneous height growth.It is also observed that the impediment effect of soft layer is much more significant in the fractures suppressed by the preferential growth of adjoining fractures.After that,the combined effect of in situ stress ratio and fracturing spacing on the multi-fracture height growth is presented,and the results elucidate the influence of in situ stress ratio on the height growth behavior depending on the fracture spacing.Finally,it is found that the inclusion of soft layers changes the aperture distribution of outmost and interior hydraulic fractures.The results obtained from this study may provide some insights on the understanding of hydraulic fracture height containment observed in filed.

A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities

The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.

Multiscale Finite Element Method for Coupling Analysis of Heterogeneous Magneto-Electro-Elastic Structures in Thermal Environment

Magneto-electro-elastic (MEE) materials, a new type of composite intelligent materials, exhibit excellent multifield coupling effects. Due to the heterogeneity of the materials, it is challenging to use the traditional finite element method (FEM) for mechanical analysis. Additionally, the MEE materials are often in a complex service environment, especially under the influence of the thermal field with thermoelectric and thermomagnetic effects, which affect its mechanical properties. Therefore, this paper proposes the efficient multiscale computational method for the multifield coupling problem of heterogeneous MEE structures under the thermal environment. The method constructs a multi-physics field with numerical base functions (the displacement, electric potential, and magnetic potential multiscale base functions). It equates a single cell of heterogeneous MEE materials to a macroscopic unit and supplements the macroscopic model with a microscopic model. This allows the problem to be solved directly on a macroscopic scale. Finally, the numerical simulation results demonstrate that compared with the traditional FEM, the multiscale finite element method (MsFEM) can achieve the purpose of ensuring accuracy and reducing the degree of freedom, and significantly improving the calculation efficiency.

Hermite Finite Element Method for Vibration Problem of Euler-Bernoulli Beam on Viscoelastic Pasternak Foundation

Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Euler-Bernoulli beam on viscoelastic Pasternak foundation can be used to analyze the deformation and response of buildings under complex geological conditions. In this paper, we use Hermite finite element method to get the numerical approximation scheme for the vibration equation of viscoelastic Pasternak foundation beam. Convergence and error estimation are rigourously established. We prove that the fully discrete scheme has convergence order O(τ2+h4), where τis time step size and his space step size. Finally, we give four numerical examples to verify the validity of theoretical analysis.

Gradient Recovery Based Two-Grid Finite Element Method for Parabolic Integro-Differential Optimal Control Problems

In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.

OpenIFEM:A High Performance Modular Open-Source Software of the Immersed Finite Element Method for Fluid-Structure Interactions

We present a high performance modularly-built open-source software-OpenIFEM.OpenIFEM is a C++implementation of the modified immersed finite element method(mIFEM)to solve fluid-structure interaction(FSI)problems.This software is modularly built to perform multiple tasks including fluid dynamics(incompressible and slightly compressible fluid models),linear and nonlinear solid mechanics,and fully coupled fluid-structure interactions.Most of open-source software packages are restricted to certain discretization methods;some are under-tested,under-documented,and lack modularity as well as extensibility.OpenIFEM is designed and built to include a set of generic classes for users to adapt so that any fluid and solid solvers can be coupled through the FSI algorithm.In addition,the package utilizes well-developed and tested libraries.It also comes with standard test cases that serve as software and algorithm validation.The software can be built on cross-platform,i.e.,Linux,Windows,and Mac OS,using CMake.Efficient parallelization is also implemented for high-performance computing for large-sized problems.OpenIFEM is documented using Doxygen and publicly available to download on GitHub.It is expected to benefit the future development of FSI algorithms and be applied to a variety of FSI applications.

Cracking analysis of fracture mechanics by the finite element method of lines(FEMOL)

The Finite Element Method of Lines （FEMOL） is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to a set of examples such as edge-crack plate, the central-crack plate, the plate with cracks emanating from a hole under tensile or under combination loads of tensile and bending. Their dimensionless stress distribution, the stress intensify factor （SIF） and crack opening displacement （COD） are obtained, and comparison with known solutions by other methods are reported. It is found that a good accuracy is achieved by FEMOL. The method is successfully modified to remarkably increase the accuracy and reduce convergence difficulties. So it is a very useful and new tool in studying fracture mechanics problems.

3D finite element numerical simulation of advanced detection in roadway for DC focus method

Within the roadway advanced detection methods, DC resistivity method has an extensive application because of its simple principle and operation. Numerical simulation of the effect of focusing current on advanced detection was carried out using a three-dimensional finite element method （FEM）, meanwhile the electric-field distribution of the point source and nine-point power source were calculated and analyzed with the same electric charges. The results show that the nine-point power source array has a very good ability to focus, and the DC focus method can be used to predict the aquifer abnormality body precisely. By comparing the FEM modelling results with physical simulation results from soil sink, it is shown that the accuracy of forward simulation meets the requirement and the artificial disturbance from roadway has no impact on the DC focus method.

NONLINEAR BUCKLING ANALYSIS OF TUBING IN DEVIATED WELLS BY FINITE ELEMENT METHOD

The equilibrium equations and the functional for tubing buckling in arbitrary straight wells are derived. The entire buckling process of tubing in deviated wells is analyzed for the first time by utilizing the finite element method. The effects of gravity and torques on the buckling are included in the analyses and the calculated results are well compared with existing solutions. It is shown that the buckling only occurs at the lower portion of the tubing where the axial load is the largest, and the contact force of the well, the bending moment of the tubing and the buckling displacement of this portion vary periodically. The buckling spreads upwards from the bit with the increase of axial load. There is no buckling at the upper portion of the tubing where the bending moment is zero. And the contact force of this section increases only slightly with the increase of the axial load. With the increase of the deviation angle, the length of buckling portion and buckling displacement amplitude decrease, the contact force increases with the increase of load at the upper portion and its amplitude decreases at the lower buckling section, the bending moment remains zero at the upper portion and its amplitude decreases at the lower buckling portion. The buckling displacement increases with the increase of the torque, but the increment is very small.

NUMERICAL SIMULATION OF UNSTEADY-STATE UNDEREXPANDED JET USING DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

A discontinuous Galerkin finite element method （DG-FEM） is developed for solving the axisymmetric Euler equations based on two-dimensional conservation laws. The method is used to simulate the unsteady-state underexpanded axisymmetric jet. Several flow property distributions along the jet axis, including density, pres- sure and Mach number are obtained and the qualitative flowfield structures of interest are well captured using the proposed method, including shock waves, slipstreams, traveling vortex ring and multiple Mach disks. Two Mach disk locations agree well with computational and experimental measurement results. It indicates that the method is robust and efficient for solving the unsteady-state underexpanded axisymmetric jet.

Phase-field modeling of dendritic growth under forced flow based on adaptive finite element method

A mathematical model combined projection algorithm with phase-field method was applied. The adaptive finite element method was adopted to solve the model based on the non-uniform grid, and the behavior of dendritic growth was simulated from undercooled nickel melt under the forced flow. The simulation results show that the asymmetry behavior of the dendritic growth is caused by the forced flow. When the flow velocity is less than the critical value, the asymmetry of dendrite is little influenced by the forced flow. Once the flow velocity reaches or exceeds the critical value, the controlling factor of dendrite growth gradually changes from thermal diffusion to convection. With the increase of the flow velocity, the deflection angle towards upstream direction of the primary dendrite stem becomes larger. The effect of the dendrite growth on the flow field of the melt is apparent. With the increase of the dendrite size, the vortex is present in the downstream regions, and the vortex region is gradually enlarged. Dendrite tips appear to remelt. In addition, the adaptive finite element method can reduce CPU running time by one order of magnitude compared with uniform grid method, and the speed-up ratio is proportional to the size of computational domain.

NUMERICAL INVESTIGATION OF TOROIDAL SHOCK WAVES FOCUSING USING DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

A numerical simulation of the toroidal shock wave focusing in a co-axial cylindrical shock tube is inves- tigated by using discontinuous Galerkin （DG） finite element method to solve the axisymmetric Euler equations. For validating the numerical method, the shock-tube problem with exact solution is computed, and the computed results agree well with the exact cases. Then, several cases with higher incident Mach numbers varying from 2.0 to 5.0 are simulated. Simulation results show that complicated flow-field structures of toroidal shock wave diffraction, reflection, and focusing in a co-axial cylindrical shock tube can be obtained at different incident Mach numbers and the numerical solutions appear steep gradients near the focusing point, which illustrates the DG method has higher accuracy and better resolution near the discontinuous point. Moreover, the focusing peak pres- sure with different grid scales is compared.

Combined method for fast 3-D finite element modeling of nondestructive testing signal

A combined method for the fast 3-D finite element modeling of defect responses in nondestructive testing of electromagnetics is presented. The method consists of three numerical techniques: zoom-in technique, difference field technique and iterative solution technique. Utilizing the zoom-in technique, the computational zone focuses on a relatively small domain around the defect. Employing the difference field technique, the axisymmetrical field solution corresponding to the case with no defect can be used to simplify the mesh generation and obtain the modeling results quickly. Using the iterative solution technique, the matrix equation system in the 3-D finite element modeling of nondestructive probe signals can easily be solved. The sample calculation shows that the presented method is highly effective and can consequently save significant computer resources.

PARTITION OF UNITY FINITE ELEMENT METHOD FOR SHORT WAVE PROPAGATION IN SOLIDS

A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.

Finite Element Analysis for Magneto-Convection Heat Transfer Performance in Vertical Wavy Surface Enclosure:Fin Size Impact

The goal of this paper is to represent a numerical study of magnetohydrodynamic mixed convection heat transfer in a lid-driven vertical wavy enclosure with a fin attached to the bottomwall.We use a finite elementmethod based on Galerkin weighted residual(GWR)techniques to set up the appropriate governing equations for the present flow model.We have conducted a parametric investigation to examine the impact of Hartmann and Richardson numbers on the flow pattern and heat transmission features inside a wavy cavity.We graphically represent the numerical results,such as isotherms,streamlines,velocity profiles,local and mean Nusselt numbers,and average surface temperature.Comparisons between the results of this work and previously published work in a literature review have been produced to examine the reliability and consistency of the data.The different sizes of the fin surface significantly impact flow creation and temperature fields.Additionally,the long fin size is necessary to enhance the heat transfer rate on the right surface at large Richardson numbers and low Hartmann numbers.Fin surfaces can significantly increase the mixing of fluid inside the enclosure,which can mean reductions in reaction times and operating costs,along with increases in heat transfer and efficiency.