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 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 construction.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.

A Computational Framework for Parachute Inflation Based on Immersed Boundary/Finite Element Approach

A computational framework for parachute inflation is developed based on the immersed boundary/finite element approach within the open-source IBAMR library.The fluid motion is solved by Peskin's diffuse-interface immersed boundary(IB)method,which is attractive for simulating moving-boundary flows with large deformations.The adaptive mesh refinement technique is employed to reduce the computational cost while retain the desired resolution.The dynamic response of the parachute is solved with the finite element approach.The canopy and cables of the parachute system are modeled with the hyperelastic material.A tether force is introduced to impose rigidity constraints for the parachute system.The accuracy and reliability of the present framework is validated by simulating inflation of a constrained square plate.Application of the present framework on several canonical cases further demonstrates its versatility for simulation of parachute inflation.

Modularized and Parametric Modeling Technology for Finite Element Simulations of Underground Engineering under Complicated Geological Conditions

The surrounding geological conditions and supporting structures of underground engineering are often updated during construction,and these updates require repeated numerical modeling.To improve the numerical modeling efficiency of underground engineering,a modularized and parametric modeling cloud server is developed by using Python codes.The basic framework of the cloud server is as follows:input the modeling parameters into the web platform,implement Rhino software and FLAC3D software to model and run simulations in the cloud server,and return the simulation results to the web platform.The modeling program can automatically generate instructions that can run the modeling process in Rhino based on the input modeling parameters.The main modules of the modeling program include modeling the 3D geological structures,the underground engineering structures,and the supporting structures as well as meshing the geometric models.In particular,various cross-sections of underground caverns are crafted as parametricmodules in themodeling program.Themodularized and parametric modeling program is used for a finite element simulation of the underground powerhouse of the Shuangjiangkou Hydropower Station.This complicatedmodel is rapidly generated for the simulation,and the simulation results are reasonable.Thus,this modularized and parametric modeling program is applicable for three-dimensional finite element simulations and analyses.

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.

Probabilistic Analysis of Slope Using Finite Element Approach and Limit Equilibrium Approach around Amalpata Landslide of West Central, Nepal

The stability study of the ongoing and recurring Amalpata landslide in Baglung in Nepal’s Gandaki Province is presented in this research. The impacted slope is around 200 meters high, with two terraces that have different slope inclinations. The lower bench, located above the basement, consistently fails and sets others up for failure. The fluctuating water level of the slope, which travels down the slope masses, exacerbates the slide problem. The majority of these rocks are Amalpata landslide area experiences several structural disruptions. The area’s stability must be evaluated in order to prevent and control more harm from occurring to the nearby agricultural land and people living along the slope. The slopes’ failures increase the damages of house existing in nearby area and the erosion of the slope. Two modeling techniques the finite element approach and the limit equilibrium method were used to simulate the slope. The findings show that, in every case, the terrace above the basement is where the majority of the stress is concentrated, with a safety factor of near unity. Using probabilistic slope stability analysis, the failure probability was predicted to be between 98.90% and 100%.

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.

ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS

This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments.First,for the analytical solutions of the equations,we derive their expressions and asymptotical stability criteria.Second,for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations,we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable.Finally,with a typical example with numerical experiments,we illustrate the applicability of the obtained theoretical results.

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

基于SBFEM的剪切型裂纹动态开裂模拟

剪切型断裂是岩土工程中常见的破坏模式,了解剪切破坏机理并准确预测剪切型裂纹的萌生、扩展过程对保障工程结构的安全性与稳定性具有重要意义.文章建立了基于比例边界有限元法(scaled boundary finite element methods,SBFEM)和非局部宏-微观损伤模型的剪切型裂纹动态开裂模拟方法,定义了基于偏应变概念的物质点对的正伸长量,可作为预测剪切型裂纹扩展行为的动态开裂准则,一点的损伤定义为该点影响域范围内连接的物质键损伤的加权平均值,而物质键的损伤则与基于偏应变概念的物质点对的正伸长量相关联,并引入能量退化函数建立结构域几何拓扑损伤与能量损失之间的关系,将拓扑损伤与应力应变联系起来,通过能量退化函数修正了SBFEM的刚度系数矩阵,得到了子域在损伤状态下的刚度矩阵,推导了考虑结构损伤的SBFEM动力控制方程,采用Newmark隐式算法对控制方程进行时间离散.最后,通过3个典型算例验证了建议的模型可较好地模拟剪切型断裂问题,能够很好地捕捉剪切型裂纹的扩展路径,并得到较为准确的载荷-位移曲线.

Finite Element Simulation of Radial Tire Building and Shaping Processes Using an Elasto-Viscoplastic Model

The comprehensive tire building and shaping processes are investigated through the finite element method(FEM)in this article.The mechanical properties of the uncured rubber from different tire components are investigated through cyclic loading-unloading experiments under different strain rates.Based on the experiments,an elastoviscoplastic constitutive model is adopted to describe themechanical behaviors of the uncured rubber.The distinct mechanical properties,including the stress level,hysteresis and residual strain,of the uncured rubber can all be well characterized.The whole tire building process(including component winding,rubber bladder inflation,component stitching and carcass band folding-back)and the shaping process are simulated using this constitutive model.The simulated green tire profile is in good agreement with the actual profile obtained through 3D scanning.The deformation and stress of the rubber components and the cord reinforcements during production can be obtained fromthe FE simulation,which is helpful for judging the rationality of the tire construction design.Finally,the influence of the parameter“drum width”is investigated,and the simulated result is found to be consistent with the experimental observations,which verifies the effectiveness of the simulation.The established simulation strategy provides some guiding significance for the improvement of tire design parameters and the elimination of tire production defects.

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

Investigation on mechanism of magnetization reversal for nanocrystalline Pr-Fe-B permanent magnets by micromagnetic finite element methods

Magnetization configurations were calculated under various magnetic fields for nanocrystalline Pr-Fe-B permanent magnets by micromagnetic finite element method.According to the configurations during demagnetization process, the mechanism of magnetization reversal was analyzed.For the Pr2Fe14B with 10 nm grains or its composite with 10vol.% α-Fe, the coercivity was determined by nucleation of reversed domain that took place at grain boundaries.However, for Pr2Fe14B with 30 nm grains, coercivity was controlled by pinning of the nucle-ated domain.For Pr2Fe14B/α-Fe with 30vol.% α-Fe, the demagnetization behavior was characterized by continuous reversal of α-Fe moment.

Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.

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.