Finite Element Mesh Refinement Research Articles

Finite element mesh refinement for in-plane and out-of-plane vibration of variable geometrical Timoshenko beams based on superconvergent vibration modes

PurposeIn this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.Design/methodology/approachAn adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.FindingsThe results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.Originality/valueThe proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

The surface effects on solid and hollow nanowires under diametral loading

The size effects of energetic surfaces on the elastic behavior of solid and hollow nanowires are investigated under diametral loading for the first time. In order to model the diametral compression test on nanowires, the structural version of the locally-exact theory is developed for solid and hollow nanowires (LETNW) by considering energetic surfaces attached at the boundaries of bulk materials and modeled using the Gurtin-Murdoch model. The developed analytical solutions employ Fourier series representations of displacement fields in the cylindrical coordinate system that exactly satisfy the governing differential equations, facilitating rapid convergence of full-field stress distributions. The solution's accuracy is validated against the in-house finite-element (FE) program which is currently the computational standard, where excellent agreement is obtained. The surface effects on the local stresses are then illustrated by varying the surface parameters, loading angle, dimension of nanowires, as well as residual surface stress. The efficiency of the LETNW is attributed to its analytical nature and thus consumes much less execution time relative to the FE implementation. What's more, instabilities associated with the solution of stress fields are observed with surface properties in certain crystallographic orientations for both LETNW and FE approaches. Contrary to the LETNW technique, the bifurcation phenomenon is shown to depend on the finite-element mesh refinement. The locally-exact theory provides an alternative and independent way of identifying surface bifurcations in solid and hollow nanowires and hence is a good tool in assessing the finite-element method as well as other approaches’ predictive capability for this class of materials.

Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross–Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of a novel adaptive finite element mesh refinement technique, which does not rely on any a posteriori error estimates, and a recently proposed new gradient flow. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of degrees of freedom.

Assessing the applicability of a smeared crack approach for simulating the behaviour of concrete beams flexurally reinforced with GFRP bars and failing in shear

Numerical simulation of beams failing in shear is still a challenge. With the scope of verifying the applicability of smeared crack approaches to simulate the behavior of reinforced concrete (RC) beams failing in shear, a set of concrete beams reinforced with longitudinal glass fiber reinforced polymer (GFRP) bars, experimentally tested up to their failure, and comprehensibly monitored, are numerically simulated. The simulations are carried out with a multi-directional fixed smeared crack model available in the FEMIX computer program that has several options for modeling the crack shear stress transfer, which is a critical aspect when simulating RC elements failing in shear. The predictive performance of the numerical simulations is assessed in term of load vs deflection, crack pattern at failure, concrete strains in critical shear regions, and moment–curvature relationship. The influence on the predictive performance of the following modeling aspects is also investigated: finite element mesh refinement; simulation of the crack shear stress transfer by using the classical shear retention factor and a crack shear-softening diagram; bond conditions between flexural reinforcement and surrounding concrete. The simulations carried out demonstrate that small dependence of the results on the finite element mesh refinement and adequate crack patterns can be obtained with refinement levels suitable for design purposes and taking into account the actual computer performances, as long as a crack shear-softening diagram is used. However, the predictive performance of the simulations depends significantly on the values adopted for the parameters that define this diagram, as demonstrated by the performed parametric studies.

Computational simulation of damage accumulation processes in cracked bodies by the UMAT procedure of SIMULIA Abaqus

The paper presents the experience of using the user subroutine UMAT for finite element package SIMULIA Abaqus/CAE for damage accumulation processes in the vicinity of the crack. A continuum damage mechanics model based on the constitutive relations of linear elastic isotropic materials with the incorporated damage tensor components is used to describe the material behavior. The material nonlinearity arising from the deformation process is modeled by introducing an anisotropic damage tensor of the second rank into the constitutive equation. The material model is described by means of user procedure UMAT of SIMULIA Abaqus. The finite element (FE) mechanical constitutive model is implemented in Abaqus/Standard via a UMAT routine. Numerical experiments for a large series of cracked specimens have been performed. Computed stress and damage tensor components were found. It is shown that they are not dependent on the FE mesh refinement. Distributions of the damage tensor components in the vicinity of the crack tip in cracked specimens of different configurations under mixed mode loading in a wide range of mixed mode loadings are found. The configurations of active damage accumulation process zone in the cracked specimens are obtained. It is shown that the damage accumulation process has substantial influence on the stress-strain state in the vicinity of the crack tip and leads to decrease of the stress concentration in cracked specimens.

КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ ПРОЦЕССОВ НАКОПЛЕНИЯ ПОВРЕЖДЕНИЙ В ТВЕРДЫХ ТЕЛАХ С ТРЕЩИНАМИ С ПОМОЩЬЮ ПОЛЬЗОВАТЕЛЬСКОЙ ПРОЦЕДУРЫ UMAT ВЫЧИСЛИТЕЛЬНОГО КОМПЛЕКСА SIMULIA ABAQUS

The paper presents the experience of using the user subroutine UMAT for FEM package SIMULIA Abaqus/CAE for damage accumulation processes in the vicinity of the crack. A continuum damage mechanics model based on the constitutive relations of linear elastic isotropic materials with the incorporated damage tensor components is used to describe the material behavior. The material nonlinearity arising from the deformation process is modelled by introducing an anisotropic damage tensor of the second rank into the constitutive equation. The material model is described by means of user procedure UMAT of SIMULIA Abaqus. The finite element (FE) mechanical constitutive model is implemented in Abaqus/Standard via a UMAT routine. Numerical experiments for a large series of cracked specimens have been performed. Computed stress and damage tensor components were found. It is shown that they are not dependent on the FE mesh refinement. Distributions of the damage tensor components in the vicinity of the crack tip in cracked specimens of different configurations under mixed mode loading in a wide range of mixed mode loadings are found. The configurations of active damage accumulation process zone in the cracked specimens are obtained. It is shown that the damage accumulation process has a substantial influence on the stress-strain state in the vicinity of the crack tip and leads to a decrease of the stress concentration in cracked specimens.

A comparison of the primal and semi-dual variational formats of gradient-extended crystal inelasticity

In this paper we discuss issues related to the theoretical as well as the computational format of gradient-extended crystal viscoplasticity. The so-called primal format uses the displacements, the slip of each slip system and the dissipative stresses as the primary unknown fields. An alternative format is coined the semi-dual format, which in addition includes energetic microstresses among the primary unknown fields. We compare the primal and semi-dual variational formats in terms of advantages and disadvantages from modeling as well as numerical viewpoints. Finally, we perform a series of representative numerical tests to investigate the rate of convergence with finite element mesh refinement. In particular, it is shown that the commonly adopted microhard boundary condition poses a challenge in the special case that the slip direction is parallel to a grain boundary.

A numerical study on the symmetrization of tangent stiffness matrix in non-linear analysis using corotational approach

In the corotational kinematics (EIRC), the movement is decomposed in deformational and rigid body components using projection operators. Large rigid body translations and rotations, and infinitesimal deformation tensors are adopted. In this way, a non-symmetric stiffness matrix is obtained for the finite element. Literature points that this matrix may be symmetrized in the usual way and the original non-symmetric matrix tends to be in a symmetric form as equilibrium is approached. This paper performed symmetrization studies using Frobenius norm and Absolute Maximum Coefficient of the anti-symmetric part of stiffness matrix in several numerical analyses with high non-linearity solved with an incremental-iterative scheme. An element independent corotational (EICR) kinematics is used to analyze non-linear spatial frames with 3D Euler–Bernoulli beam elements. The results show that this symmetrization not always occurs and depends significantly on the magnitude of displacements and finite-element mesh refinement. The authors demonstrated that, in some cases, the symmetrization would only take place with very refined meshes when the discretized structure approaches the continuum. In this way, the current literature and general mathematical proof of that symmetrization have to be restated.

The Henry-Saltwater Intrusion Benchmark – Alternatives in Multiphysics Formulations and Solution Strategies

In a classical paper Henry set up a conceptual model for simulating saltwater intrusion into coastal aquifers. Up to now the problem has been taken up by software developers and modellers as a benchmark for codes simulating coupled flow and transport in porous media. The Henry test case has been treated using different numerical methods based on various formulations of differential equations. We compare several of these approaches using multiphysics software. We model the problem using Finite Elements, utilizing the primitive variables and the streamfunction approach, both with and without using the Oberbeck-Boussinesq assumption. We compare directly coupled solvers with segregated solver strategies. Changing finite element orders and mesh refinement, we find that models based on the streamfunction converge 2-4 times faster than runs based on primitive variables. Concerning the solution strategy, we find an advantage of Picard iterations compared to monolithic Newton iterations.

J0330203 ポーラスアルミニウムのボクセル有限要素弾塑性大変形圧縮解析

Compression behavior of a porous aluminum made by the friction stir processing was investigated through the finite element analysis. The finite element model was automatically constructed by the image-based modeling using a micro-focus X-ray CT. The voxel modeling was employed to robustly and quickly generate the finite element mesh. To examine the validity of finite element simulation, the nominal stress and nominal strain relationship was compared between the finite element analysis and the experiment. The simulation result overestimated the total stiffness and nominal stress due to mainly the inadequate approximation of contact boundary condition with the compressing plate. However, the error in plateau stress was about thirty percent, and it may be reduced by the rigorous contact analysis and the refinement of finite element mesh.

On numerical solution of elastic–plastic problems by using configurational force driven adaptive methods

In this paper, the concept of configurational forces is introduced in the context of finite element mesh refinement for elastic–ideally plastic problems. This paper also includes the numerical computation of configurational forces in the elastic and plastic domains. Methods are demonstrated on three plane problems where the analytical solution is available. The first example is a thick-walled tube loaded by internal pressure. This simple, one dimensional problem allows computation of configurational volume forces analytically to validate the finite element (FE) results. The second example is Galin׳s problem that involves an infinite plate with a circular hole loaded by biaxial tension at the infinity. This is a two dimensional problem for which the analytical solution is known with some restrictions for elastic–ideally plastic case when Tresca yield criterion is considered. The last example introduces another plane problem that follows Naghdi׳s solution on infinite wedges. For this, a new analytical solution is presented for plane stress state using von Mises yield criterion with a uniform shear loading along the boundary. R-, h-, and combined adaptive procedures are demonstrated on the above examples. Since exact stress fields are known, the norm of the difference between numerical and analytical solutions is used as the measure of error.

Overview and modeling of mechanical and thermomechanical impact of underground coal gasification exploitation

From an economic point of view Underground Coal Gasification (UCG) is a promising technology that can be used to reach coal resources that are difficult or expensive to by conventional mining methods. Furthermore, the process addresses safety concerns, by avoiding the presence of workers underground. An optimal UCG process requires the integration of various scientific fields (chemistry, geochemistry, geomechanics) and the demonstration of limited of environmental impacts. This paper focuses on the mechanical component of the UCG operation and its impact on the surrounding environment in terms of stability and land subsidence. The mechanical components are also considered. Underground mining by coal combustion UCG challenges include the mechanical behavior of the site and of stability of the overburden rock layers. By studying the underground reactor, its inlet and outlet, we confirm the key role played by mechanical damage and thermo-mechanical phenomena are identified. Deformation or collapse above the cavity may cause a collapse in the overlying layers or subsidence at the surface level. These phenomena are highly dependent on the thermoporomechanical behavior of the rock surrounding the cavity (the host rocks). Unlike conventional methods, the UCG technology introduces an additional variable into the physical problem: the high temperatures, which evolve with time and space. In this framework, we performed numerical analyses of the coal site that could be exploited using this method. The numerical results presented in this paper are derived from models based on different assumptions describing a raw geological background. Several 3D (3 dimensional) and 2D (2 dimensional, plane) nonlinear finite element modelings are performed based on two methods. The first assumes a rock medium as a perfect thermo-elastoplastic continuum. In the second, in order to simulate large space scale crack propagation explicitly, we develop a method based upon finite element deactivation. This method is built on a finite element mesh refinement and uses Mohr-Coulomb failure criterion. Based on the analysis of the numerical results, we can highlight two main factors influencing the behavior and the mechanical stability of the overburden, and consequently the UCG process evolution. The first is the size of the cavity. This geometrical parameter, which is common to all types of coal exploitation, is best controlled using the classic exploitation method. We show that in the case of UCG, the shape of the cavity and its evolution over time can be modified considerably by the thermomechanical behavior of the host rocks. The second is the presence of a heat source whose location and intensity evolve over time. Even if thermal diffusivity of the rock is low and only a small distance from the coal reactor is thermally affected, we show that the induced mechanical changes extend significantly in the overburden, and that subsidence can therefore be estimated at the surface. We conclude the integration of the mechanical analysis into a risk analysis process mechanical analysis can be integrated in a thorough risk analysis.

Interlaminar shear strength of marine composite laminates: Tests and numerical simulations

This paper deals with an experimental/numerical comparison of the interlaminar shear stress in typical marine composite laminates manufactured according to two different fabrication processes, namely vacuum infusion and pre-impregnated lay-up. Experimental tests were carried out at the Marine Structures Testing Lab of the University of Genova. Actually, among widely applied empirical formulations for the characterization of composites suggested by Classification Societies rules, the ones for the interlaminar shear strength appear the most scattered because of the complexity in assessing failure phenomena. Composite manufacturers generally carry out rather extensive tests for the complete characterization of the laminates.Due to non-linearities involved in the captioned study, the support of nonlinear numerical models was required. The aim of the comprehensive use of numerical approaches is the reduction of time-consuming and expensive experimental campaigns.Three different FE modeling strategies are proposed in order to verify the reliability of the numerical results in comparison with experimental targets. The merits and shortcomings of each are discussed, also comparing different types of finite element formulations and mesh refinement sensitivity.

Modelling Crack Growth within Graphite Bricks due to Irradiation and Radiolytic Oxidation

During operation, nuclear graphite structures are exposed to gradients of neutron radiation and temperature. They are also prone to radiolytic oxidation, subsequent porosity changes and weight loss. These processes cause significant changes in the material properties of graphite and create complicated stress fields within graphite bricks. Significant efforts have been made to analyse these stresses over recent years. In this paper, the influence of stresses within a typical graphite brick, during operation, on crack growth is studied. A finite element (FE) mechanical constitutive model is implemented in ABAQUS as a user-subroutine (UMAT) in conjunction with a damage model to simulate crack growth within the graphite brick. The damage model is a linear traction separation model and it is applied with XFEM enrichment to model crack initiation and growth. The results indicate that the cracks initiate in the vicinity of the keyway roots where the maximum principal stress is at its peak. Complete failure was not observed from the simulation of the graphite brick under 30 years of irradiation conditions with a maximum brick temperature of 500°C. Computed stresses and crack lengths were found to be largely dependent on the FE mesh refinement and the results of mesh sensitivity studies are also presented in this paper.

Adaptive stochastic Galerkin FEM

A framework for residual-based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic PDEs with random coefficients is presented. A parametric, deterministic elliptic boundary value problem on an infinite-dimensional parameter space is discretized by means of a Galerkin projection onto finite generalized polynomial chaos (gpc) expansions, and by discretizing each gpc coefficient by a FEM in the physical domain.An anisotropic residual-based a posteriori error estimator is developed. It contains bounds for both contributions to the overall error: the error due to gpc discretization and the error due to Finite Element discretization of the gpc coefficients in the expansion. The reliability of the residual estimator is established.Based on the explicit form of the residual estimator, an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic Galerkin discretization, and, embedded in the gpc adaptation loop, also the Finite Element mesh refinement of the gpc coefficients in the physical domain. Asynchronous mesh adaptation for different gpc coefficients is permitted, subject to a minimal compatibility requirement on meshes used for different gpc coefficients.Details on the implementation with the open-source software framework ALEA are presented; it is generic, and is based on available stiffness and mass matrices of a FEM for the deterministic, nonparametric nominal problem evaluated in the FEniCS environment.Preconditioning of the resulting matrix equation and iterative solution are discussed. Numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented. They indicate substantial savings in total computational complexity due to FE mesh coarsening in high gpc coefficients.

Finite element simulation of welding residual stresses in martensitic steel pipes

Finite element (FE) simulations of the welding of two high grade steel pipes are described. The first is a P91 steel pipe welded with a similar P91 weld consumable, and the second is a P92 steel pipe welded with dissimilar nickel–chromium based weld consumables. Both welds are multipass circumferential butt welds, having 73 weld beads in the P91 pipe and 36 beads in the P92 pipe. Since the pipes and welds are symmetric around their axes, the FE simulations are axisymmetric, allowing high FE mesh refinement and residual stress prediction accuracy. The FE simulations of the welding of the P91 and P92 pipes comprise thermal and sequentially coupled structural analyses. The thermal analyses model the heat evolution produced by the welding arc, determining the temperature history throughout the FE models. Structural analyses use the computed temperature history as input data to predict the residual stress fields throughout the models. Post-weld heat treatment (PWHT) of both pipes has also been numerically simulated by assuming that the FE models obey the Norton creep law during the hold time period at 760°C. The residual stresses presented here have all been validated by corresponding experimental measurements. Before PWHT, it has been found that, at certain locations in the weld region and heat affected zone (HAZ) in the pipes, tensile hoop and axial residual stresses approach the tensile strength of the material, presenting a high risk of failure. It has also been found that PWHT substantially reduces the magnitude of residual stresses by varying degrees depending on the material.

Numerical Simulation of a Pyramid Steel Sheet Formed by Single Point Incremental Forming Using Solid-Shell Finite Elements

Single Point Incremental Forming (SPIF) is an interesting manufacturing process due to its dieless nature and its increased formability compared to conventional forming processes. Nevertheless, the process suffers from large geometric deviations when compared to the original CAD profile. One particular example arises when analyzing a truncated two-slope pyramid [. In this paper, a finite element simulation of this geometry is carried out using a newly implemented solid-shell element [, which is based on the Enhanced Assumed Strain (EAS) and the Assumed Natural Strain (ANS) techniques. The model predicts the shape of the pyramid very well, correctly representing the springback and the through thickness shear (TTS). Besides, the effects of the finite element mesh refinement, the EAS and ANS techniques on the numerical prediction are presented. It is shown that the EAS modes included in the model have a significant influence on the accuracy of the results.

Determination of local mesh parameters in finite element problems

Two basic cases of convergence of finite element solution in the static problems of the theory of elasticity are considered. Based on Saint-Venant’s principle, the sufficiency of finite element mesh refinement for local regions with stress concentration is determined. An algorithm is presented for evaluation of point-by-point convergence of solution for local mesh refinement. To illustrate the method, the results of computation of a cylindrical three-layered composite shell under local loads are given. For stresses reaching the maximum value in the loaded area, the mesh density parameters yielding an initial approximation are found.

Coarse-grained atomistic simulations of dislocations in Al, Ni and Cu crystals

This paper presents the application of a recently developed concurrent atomistic continuum (CAC) methodology in coarse-grained (CG) atomistic simulations of dislocation nucleation and migration in face-centered cubic (fcc) Al, Ni and Cu crystals, using an EAM force field and an adaptive mesh refinement strategy. The CAC method is based on recently developed atomistic field theory (AFT) that frames the problem in terms of a continuum field representation of lattice and sublattice atomic arrangements. Four sets of CG models with different finite element mesh refinement are constructed to test the accuracy and efficiency of the CG method relative to full molecular dynamics (MD). Simulation results show that the CG method is able to reproduce key phenomena of dislocation dynamics in initially defect free fcc crystals, including strain bursts caused by dislocation nucleation and migration, the structure of leading and trailing partial dislocations separated by equilibrium stacking faults in Al, formation of intrinsic stacking fault ribbons in Ni and Cu, and 3D migration of curved dislocations, all comparable with results of MD simulations. CG simulations have also revealed that the yield strength of Al depends on the thickness of the specimens as well as the operative deformation mechanisms. The effects of mesh size and adaptive mesh refinement on CG simulation results are analyzed and discussed.

Power Aware Parallel 3-D Finite Element Mesh Refinement Performance Modeling and Analysis With CUDA/MPI on GPU and Multi-Core Architecture

Software power performance tuning handles the critical design constraints of software running on hardware platforms composed of large numbers of power-hungry components. The power dissipation of a Single Program/Instruction Multiple Data (SPMD/SIMD) computation such as finite element method (FEM) mesh refinement is highly dependent on the underlying algorithm and the power-consuming features of hardware Processing Elements (PE). This contribution presents a practical methodology for modeling and analyzing the power performance of parallel 3-D FEM mesh refinement on CUDA/MPI architecture based on detailed software prototypes and power parameters in order to predict the power functionality and runtime behavior of the algorithm, optimize the program design and thus achieve the best power efficiency. In detail, we have proposed approaches for GPU parallelization, dynamic CPU frequency scaling and dynamic load scheduling among PEs. The performance improvement of our designs has been demonstrated and the results have been validated on a real multi-core and GPU cluster.