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.

A dynamic large-deformation particle finite element method for geotechnical applications based on Abaqus

In this paper,the application of Abaqus-based particle finite element method(PFEM)is extended from static to dynamic large deformation.The PFEM is based on periodic mesh regeneration with Delaunay triangulation to avoid mesh distortion.Additional mesh smoothing and boundary node smoothing techniques are incorporated to improve the mesh quality and solution accuracy.The field variables are mapped from the old to the new mesh using the closest point projection method to minimize the mapping error.The procedures of the proposed Abaqus-based dynamic PFEM(Abaqus-DPFEM)analysis and its implementation in Abaqus are detailed.The accuracy and robustness of the proposed approach are examined via four illustrative numerical examples.The numerical results show a satisfactory agreement with published results and further confirm the applicability of the Abaqus-DPFEM to solving dynamic large-deformation problems in geotechnical engineering.

Implementation of a particle-in-cell method for the energy solver in 3D spherical geodynamic modeling

The thermal evolution of the Earth’s interior and its dynamic effects are the focus of Earth sciences.However,the commonly adopted grid-based temperature solver is usually prone to numerical oscillations,especially in the presence of sharp thermal gradients,such as when modeling subducting slabs and rising plumes.This phenomenon prohibits the correct representation of thermal evolution and may cause incorrect implications of geodynamic processes.After examining several approaches for removing these numerical oscillations,we show that the Lagrangian method provides an ideal way to solve this problem.In this study,we propose a particle-in-cell method as a strategy for improving the solution to the energy equation and demonstrate its effectiveness in both one-dimensional and three-dimensional thermal problems,as well as in a global spherical simulation with data assimilation.We have implemented this method in the open-source finite-element code CitcomS,which features a spherical coordinate system,distributed memory parallel computing,and data assimilation algorithms.

OPTIMIZATION OF AUTOBODY PANEL STAMPING PROCESS BASED ON DYNAMIC EXPLICIT FINITE ELEMENT METHOD

Taking CPU time cost and analysis accuracy into account, dynamic explicit finite ele- ment method is adopted to optimize the forming process of autobody panels that often have large sizes and complex geometry. In this paper, for the sake of illustrating in detail how dynamic explicit finite element method is applied to the numerical simulation of the autobody panel forming process,an example of optimization of stamping process pain meters of an inner door panel is presented. Using dynamic explicit finite element code Ls-DYNA3D, the inner door panel has been optimized by adapting pa- rameters such as the initial blank geometry and position, blank-holder forces and the location of drawbeads, and satisfied results are obtained.

Transient analysis of 1D inhomogeneous media by dynamic inhomogeneous finite element method

The dynamic inhomogeneous finite element method is studied for use in the transient analysis of one dimensional inhomogeneous media. The general formula of the inhomogeneous consistent mass matrix is established based on the shape function. In order to research the advantages of this method, it is compared with the general finite element method. A linear bar element is chosen for the discretization tests of material parameters with two fictitious distributions. And, a numerical example is solved to observe the differences in the results between these two methods. Some characteristics of the dynamic inhomogeneous finite element method that demonstrate its advantages are obtained through comparison with the general finite element method. It is found that the method can be used to solve elastic wave motion problems with a large element scale and a large number of iteration steps.

Predicting the Dynamic Behavior of Asphalt Concrete Using Three-dimensional Discrete Element Method

A user-defined three-dimensional （3D） discrete element model was presented to predict the dynamic modulus and phase angle of asphalt concrete （AC）. The 3D discrete element method （DEM） model of AC was constructed employing a user-defined computer program developed using the ＂Fish＂ language in PFC3D. Important microstructural features of AC were modeled, including aggregate gradation, air voids and mastic. The irregular shape of aggregate particle was modeled using a clump of spheres. The developed model was validated through comparing with experimental measurements and then used to simulate the cyclic uniaxial compression test, based on which the dynamic modulus and phase angle were calculated from the output stress- strain relationship. The effects of air void content, aggregate stiffness and volumetric fraction on AC modulus were further investigated. The experimental results show that the 3D DEM model is able to accurately predict both dynamic modulus and phase angle of AC across a range of temperature and loading frequencies. The user- defined 3D model also demonstrated significant improvement over the general existing two-dimensional models.

Three-dimensional forward modeling for magnetotelluric sounding by finite element method

A finite element algorithm combined with divergence condition was presented for computing three-dimensional(3D) magnetotelluric forward modeling. The finite element equation of three-dimensional magnetotelluric forward modeling was derived from Maxwell's equations using general variation principle. The divergence condition was added forcedly to the electric field boundary value problem, which made the solution correct. The system of equation of the finite element algorithm was a large sparse, banded, symmetric, ill-conditioned, non-Hermitian complex matrix equation, which can be solved using the Bi-CGSTAB method. In order to prove correctness of the three-dimensional magnetotelluric forward algorithm, the computed results and analytic results of one-dimensional geo-electrical model were compared. In addition, the three-dimensional magnetotelluric forward algorithm is given a further evaluation by computing COMMEMI model. The forward modeling results show that the algorithm is very efficient, and it has a lot of advantages, such as the high precision, the canonical process of solving problem, meeting the internal boundary condition automatically and adapting to all kinds of distribution of multi-substances.

An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

In this paper,Runge-Kutta Discontinuous Galerkin（RKDG） finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ？ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ？ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.

Application of scaled boundary finite element method in static and dynamic fracture problems

The scaled boundary finite element method （SBFEM） is a recently developed numerical method combining advantages of both finite element methods （FEM） and boundary element methods （BEM） and with its own special features as well. One of the most prominent advantages is its capability of calculating stress intensity factors （SIFs） directly from the stress solutions whose singularities at crack tips are analytically represented. This advantage is taken in this study to model static and dynamic fracture problems. For static problems, a remeshing algorithm as simple as used in the BEM is developed while retaining the generality and flexibility of the FEM. Fully-automatic modelling of the mixed-mode crack propagation is then realised by combining the remeshing algorithm with a propagation criterion. For dynamic fracture problems, a newly developed series-increasing solution to the SBFEM governing equations in the frequency domain is applied to calculate dynamic SIFs. Three plane problems are modelled. The numerical results show that the SBFEM can accurately predict static and dynamic SIFs, cracking paths and load-displacement curves, using only a fraction of degrees of freedom generally needed by the traditional finite element methods.

Simulation of three-dimensional tension-induced cracks based on cracking potential function-incorporated extended finite element method

In the finite element method,the numerical simulation of three-dimensional crack propagation is relatively rare,and it is often realized by commercial programs.In addition to the geometric complexity,the determination of the cracking direction constitutes a great challenge.In most cases,the local stress state provides the fundamental criterion to judge the presence of cracks and the direction of crack propagation.However,in the case of three-dimensional analysis,the coordination relationship between grid elements due to occurrence of cracks becomes a difficult problem for this method.In this paper,based on the extended finite element method,the stress-related function field is introduced into the calculation domain,and then the boundary value problem of the function is solved.Subsequently,the envelope surface of all propagation directions can be obtained at one time.At last,the possible surface can be selected as the direction of crack development.Based on the aforementioned procedure,such method greatly reduces the programming complexity of tracking the crack propagation.As a suitable method for simulating tension-induced failure,it can simulate multiple cracks simultaneously.

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.

Three-Dimensional Thermo-Elastic-Plastic Finite Element Method Modeling for Predicting Weld-Induced Residual Stresses and Distortions in Steel Stiffened-Plate Structures

The objective of the present paper is to develop nonlinear finite element method models for predicting the weld-induced initial deflection and residual stress of plating in steel stiffened-plate structures. For this purpose, three-dimensional thermo-elastic-plastic finite element method computations are performed with varying plate thickness and weld bead length (leg length) in welded plate panels, the latter being associated with weld heat input. The finite element models are verified by a comparison with experimental database which was obtained by the authors in separate studies with full scale measurements. It is concluded that the nonlinear finite element method models developed in the present paper are very accurate in terms of predicting the weld-induced initial imperfections of steel stiffened plate structures. Details of the numerical computations together with test database are documented.

Dynamic Crack Propagation Analysis Using Scaled Boundary Finite Element Method

The prediction of dynamic crack propagation in brittle materials is still an important issue in many engineering fields. The remeshing technique based on scaled boundary finite element method(SBFEM) is extended to predict the dynamic crack propagation in brittle materials. The structure is firstly divided into a number of superelements, only the boundaries of which need to be discretized with line elements. In the SBFEM formulation, the stiffness and mass matrices of the super-elements can be coupled seamlessly with standard finite elements, thus the advantages of versatility and flexibility of the FEM are well maintained. The transient response of the structure can be calculated directly in the time domain using a standard time-integration scheme. Then the dynamic stress intensity factor(DSIF) during crack propagation can be solved analytically due to the semi-analytical nature of SBFEM. Only the fine mesh discretization for the crack-tip super-element is needed to ensure the required accuracy for the determination of stress intensity factor(SIF). According to the predicted crack-tip position, a simple remeshing algorithm with the minimum mesh changes is suggested to simulate the dynamic crack propagation. Numerical examples indicate that the proposed method can be effectively used to deal with the dynamic crack propagation in a finite sized rectangular plate including a central crack. Comparison is made with the results available in the literature, which shows good agreement between each other.

VECTOR BOND GRAPH REPRESENTATION OF FINITE ELEMENT METHOD IN STRUCTURAL DYNAMICS

In this paper we have shown that the invariance of energy(kinetic energy,potential energy)and virtual work is the common feature of vector bond graph and finite element method in struc-tural dynamics.Then we have discussed the vector bond graph representation of finite elementmethod in detail,there are:(1)the transformation of reference systems,(2)the transformation ofinertia matrices,stiffness matrices and vectors of joint force,(3)verctor bond graph representationof Lagrangian dynamic equation of structure.

PENALTY FINITE ELEMENT METHOD FOR NONLINEAR DYNAMIC RESPONSE OF VISCOUS FLUID-SATURATED BIPHASE POROUS MEDIA

The governing equations as well as boundary land initial conditions for nonlinear dynamic response problems of viscous fluid-saturated biphase porous medium model, based on mixture theory, are presented. With Galerkin weighted residual method the corresponding nonlinear dynamic penalty finite element equation, in which the dependencies of volume fraction and permeation coefficients an deformation are included, is obtained. The iteration solution method of the nonlinear system equation is also discussed. As a numerical example, the dynamic response of a porous medium column under impulsive loading action is analyzed with the developed finite element program. The numerical results demonstrate the efficiency and correctness of the method.

An explicit finite element method for dynamic analysis in three-medium coupling system and its application

In this paper, an explicit finite element method to analyze the dynamic responses of three-medium coupled systems with any terrain is developed on the basis of the numerical simulation of the continuous conditions on the bounda-ries among fluid saturated porous medium, elastic single-phase medium and ideal fluid medium. This method is a very effective one with the characteristic of high calculating speed and small memory needed because the formulae for this explicit finite element method have the characteristic of decoupling, and which does not need to solve sys-tem of linear equations. The method is applied to analyze the dynamic response of a reservoir with considering the dynamic interactions among water, dam, sediment and basement rock. The vertical displacement at the top point of the dam is calculated and some conclusions are given.

FINITE ELEMENT METHOD COMBINED WITH DYNAMIC PHOTOELASTIC ANALYSIS TO DETERMINE DYNAMIC STRESS-INTENSITY FACTORS

The present paper is addressed to the finite element method combined with dynamic photoelastic analysis of propagating cracks, that is, on the basis of [1] by Chien Wei-zang, finite elements which incorporate the propagating crack-tip singularity intrinsic to two-dimensional elasticity are employed. THe relation between crack opening length and time step obtained from dynamic photoelaslie analysis is used as a definite condition for solving the dynamic equations and simulating the crack propagations as well As an example, the impact response of dynamie-bending-test specimen is investigated and the dynamic stress-intensity factor obtained from the mentioned finite element analysis and dynamic photoelasticity is in reasonable agreement with each other.

Three-dimensional anisotropic induced polarization modeling using finite element method

With the development of geophysical exploration technology,the anisotropy of underground media has got more and more attention.At present,there are few studies on the anisotropy of the induced polarization method.This article explores the effect of anisotropy on the underground media of the induced polarization method under three-dimensional complex terrain.The research work transforms the underground electric field control equation into a variational problem,and use the unstructured finite element method to construct a large linear equation system for solving electric potentials.By the sparse matrix compression technique and symmetric successive over-relaxation preconditioned conjugate gradient algorithm(SSOR-PCG)to solve the equation system.Finally,the article uses the classic central gradient array method to obtain the forward apparent polarizability value.The calculation results of the model find that different anisotropic conditions will significantly affect the forward results which show a strong directional correlation,revealing the importance of considering anisotropy in practical work.

THE APPLICATION OF COMPATIBLE STRESS ITERATIVE METHOD IN DYNAMIC FINITE ELEMENT ANALYSIS OF HIGH VELOCITY IMPACT

There is a common difficulty in elastic-plastic impact codes such as EPIC[2,3] NONSAP[4], etc.. Most of these codes use the simple linear functions usually taken from static problem to represent the displacement components. In such finite element formulation, the stress components are constant in each element and they are discontinuous in any two neighboring elements. Therefore, the bases of using the virtual work principle in such elements are unreliable. In this paper, we introduce a new method, namely, the compatible stress iterative method, to eliminate the above-said difficulty. The calculated examples show that the calculation using the new method in dynamic finite element analysis of high velocity impact is valid and stable, and the element stiffness can be somewhat reduced.

A symplectic finite element method based on Galerkin discretization for solving linear systems

We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems.For the dynamic responses of continuous medium structures,the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation.Thus,a symplectic finite element method with energy conservation is constructed in this paper.A linear elastic system can be discretized into multiple elements,and a Hamiltonian system of each element can be constructed.The single element is discretized by the Galerkin method,and then the Hamiltonian system is constructed into the Birkhoffian system.Finally,all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme.Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate,it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm.The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic systems.