Element Mesh Refinement Research Articles

Numerical formulation for advanced analysis of semi-rigid steel-concrete composite frames

The present work aims at the implementation and validation of a displacement-based two-dimensional numerical formulation including several sources of non-linearities in steel-concrete composite frames, such as second-order effects, plasticity and beam-to-column semi-rigid connections. The co-rotational-based approach is used to describe the finite element formulation, allowing large displacements and rotations in the numerical model. Two rotational pseudo-springs in series are positioned in the finite elements ends. One of them are used to include the gradual loss of stiffness determined by the cross-sectional plastification. The limiting of the uncracked, elastic and plastic regimes is defined in the Normal Force-Bending Moment diagram. In the cross-sectional analysis, the Strain Compatibility Method (SCM) is used to capture the axial strains in the section components. In this way, the constitutive models of the materials are described by continuous functions. The cracked effect is considered by the effective moment of inertia of the concrete cross-section. The other spring include the effects of the semi-rigid beam-to-column connections through the moment-rotation relationship. A multi-linear model for beam-to-column connections is used. To validate the proposed numerical formulation, the results obtained are compared with numerical and experimental data available in the literature. Since the model proposed here starts with the concentrated simulation of nonlinear effects, an examination of the finite element mesh refinement is also carried out. These comparisons indicated for the validation of the numerical procedure proposed and implemented here, highlighting the precision of the formulation in both the pre- and post-critical structures behavior.

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.

A comparative assessment of different adaptive spatial refinement strategies in phase-field fracture models for brittle fracture

For the smeared approximation of a discrete crack, phase-field fracture simulations of brittle materials require suitable finite element meshes in regions where crack propagation is expected to get an accurate resolution of the phase-field function. The straightforward option is to pre-refine the mesh in regions of the expected crack paths. However, this could lead to very computationally intensive simulations due to the high number of elements. Alternatively, adaptive spatial refinement of the finite element mesh is utilized based on appropriate error indicators to obtain the required accuracy in the areas of crack propagation. Different error indicators can be used: the most common one for phase-field fracture simulations is the threshold-based approach, in which elements are refined depending on the value of the phase-field function. Alternatively, the Kelly error indicator can be used as a criterion for spatial adaptivity. It considers the jumps in the gradients of the phase-field function between the elements. We additionally introduce here an error indicator based on configurational forces, that depend on the Eshelby stress tensor. For mode I loading in linear elastic fracture mechanics, the configurational forces have a close connection to the J-Integral and the critical fracture energy Gc, respectively. Therefore, a suitable norm of the configurational forces is introduced as an error indicator here. These three error indicators are introduced and compared to each other in terms of accuracy and efficiency by means of numerical examples for crack growth in the single edge notched shear test.

Influences on unstable hydraulic fractures: propagation of adjacent multiple perforations and bedded interfaces in multilayered reservoir via FE-DE model

PurposeSimultaneous hydrofracturing of multiple perforation clusters in vertical wells has been applied in the stimulation of hydrocarbon resources reservoirs. This technology is significantly impeded due to the challenges in its application to the multilayered reservoirs that comprise multiple interlayers. One of the challenges is the accurate understanding and characterization of propagation and deflection of the multiple hydraulic fractures between reservoirs and embedded interlayers.Design/methodology/approachNumerical models of the tight multilayered reservoirs containing multiple interlayers were established to study hydrofracturing of multiple perforation clusters and its influencing factors on unstable propagation and deflection of hydraulic fractures. Brittle and plastic multilayered reservoirs fully considering the influences of different in situ stress ratio and physical attributes for reservoir and interlayer strata on propagations of hydraulic fractures were investigated. The combined finite element–discrete element method and mesh refinement strategy were adopted to guarantee the accuracy of stress solutions and reliability of fracture path in computation.FindingsResults show that the shear stress fields between adjacent multiple hydraulic fractures are superposed to cause fractures deflection. Stress shadows induce the shielding effects of hydraulic fractures and inhibit fractures growth to emerge unstable propagation behaviors, and a main single fracture and several minor fractures develop. As the in situ stress ratio increases, hydraulic fractures more easily deflect toward the direction of maximum in situ stress, and stress shadow and mutual interaction effects between them are intensified. Compared to brittle reservoir, plastic-enhanced reservoir may limit fracture growth and cannot form long fracture length; nevertheless, plastic properties of reservoir are prone to induce more microseismic events with larger magnitude.Originality/valueThe obtained fracturing behaviors and mechanisms based on engineering-scale multilayered reservoir may provide effective schemes for controlling and estimating the unstable propagation of multiple hydraulic fractures.

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.

Optimization of Convergence of Mixed Finite Element Approximations Analysis

The numerical approximation of partial differential equations (PDEs) plays a vital role in many scientific and engineering applications. Mixed finite element methods have emerged as powerful techniques for solving a wide range of PDEs due to their ability to handle problems with mixed variables, such as fluid flow and elasticity. However, ensuring the convergence of mixed finite element approximations remains a challenging task. It presents a comprehensive analysis of the optimization strategies employed to enhance the convergence of mixed finite element approximations. We investigate the key factors that impact the convergence behaviour of these methods and propose various techniques to mitigate convergence issues. we discuss the importance of appropriate discretization strategies for mixed finite element approximations. We analyze the impact of element types, mesh refinement, and stabilization techniques on the convergence rates. By examining the properties of the underlying mixed variational formulations, we identify the optimal discretization choices that lead to improved convergence behaviour. We delve into the analysis of numerical stability and consistency in the context of mixed finite element methods. We explore the role of stabilization techniques, such as bubble functions and penalty terms, in mitigating instabilities and achieving optimal convergence rates. We investigate the effect of different stabilization parameters and establish guidelines for their selection to ensure both stability and convergence. We address the issue of error estimation and adaptivity in mixed finite element approximations. We review error indicators and adaptive mesh refinement strategies that enable the refinement of regions with high solution gradients, thus enhancing the convergence rates. We discuss the interplay between error estimation and adaptive refinement and present numerical examples illustrating their effectiveness.
 We highlight recent advancements in optimization algorithms specifically tailored for enhancing convergence in mixed finite element approximations. We explore strategies like multigrid methods, preconditioning techniques, and domain decomposition methods, which accelerate the convergence rates and enable the solution of large-scale problems. This paper provides a comprehensive analysis of the optimization of convergence for mixed finite element approximations. It serves as a valuable resource for researchers and practitioners seeking to improve the efficiency and accuracy of numerical solutions obtained through mixed finite element methods.

Conforming to interface structured adaptive mesh refinement: 3D algorithm and implementation

A new non-iterative mesh generation algorithm named conforming to interface structured adaptive mesh refinement (CISAMR) is introduced for creating 3D finite element models of problems with complex geometries. CISAMR transforms a structured mesh composed of tetrahedral elements into a conforming mesh with low element aspect ratios. The construction of the mesh begins with the structured adaptive mesh refinement of elements in the vicinity of material interfaces. An r-adaptivity algorithm is then employed to relocate selected nodes of nonconforming elements, followed by face-swapping a small fraction of them to eliminate tetrahedrons with high aspect ratios. The final conforming mesh is constructed by sub-tetrahedralizing remaining nonconforming elements, as well as tetrahedrons with hanging nodes. In addition to studying the convergence and analyzing element-wise errors in meshes generated using CISAMR, several example problems are presented to show the ability of this method for modeling 3D problems with intricate morphologies.

Numerical effects on notch fatigue strength assessment of non-welded and welded components

Local fatigue strength assessment based on threshold values obtained by linear-elastic notch stress calculation is commonly utilized due to its applicability to complex geometries with an acceptable effort. As the result of a finite element computation is affected by the element type and mesh refinement, this paper investigates the numerical influence on the notch stress based fatigue strength assessment of non-welded and welded components. Based on extensive finite element studies employing the software package Abaqus it is concluded that quadratic shape functions with a number of sixteen for non-welded parts and twelve elements over a semicircle for welded joints should be at least applied to minimize the numerical impact.

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.

Improved displacement based alternative to force based finite element for nonlinear analysis of framed structures

This paper presents an improved displacement based element (iDBE) for nonlinear finite element analysis of framed structures, which, like the force based elements (FBE), needs neither high order shape functions nor the refinement of the finite element mesh in straight members to attain an accurate solution. To achieve this objective, to the strain fields based on the displacements shape functions of the conventional displacement based element (cDBE) corrective fields are added, which are determined at the element level by the principle of virtual forces. Simple examples involving tapered elements, nonlinear axial-bending interaction and elasto-plastic behavior, are included to illustrate the application of iDBE and compare its performance to cDBE and FBE.

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.

CAE applications in a thermoforming mould design

Preparation of honeycomb layer is a critical step for successful fabrications of thermoformed based sandwiched structures. This paper deals with an initial investigation on the rapid manufacturing process of corrugated sheet with 120° dihedral angles. Time history of local displacements and thickness, assuming viscous dominated material model for a 1mm thick thermoformable material, was computed by using ANSYS® Polyflow solver. The quality of formed surfaces was evaluated for selection of mould geometry and assessment of two common variants of thermoforming process. Inadequate mesh refinement of a membrane elements produces satisfactorily detailing and incomplete forming. A perfectly uniform material distribution was predicted using drape forming process. However, the geometrical properties of vacuum formed part are poorly distributed and difficult to control with increasing inflation volumes. Details of the discrepancies and the contributions of the CAE tool to complement traditional trial and error methodology in the process and design development are discussed.

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.

A multiply parallel implementation of finite element-based discrete dislocation dynamics for arbitrary geometries

Discrete dislocation dynamics (DD) approaches have proven useful in modeling the dynamics of large ensembles of dislocations. Continuing interest in finite body effects via image stresses has extended DD numerical approaches to improve the handling of surfaces. However, a physically accurate, yet computationally scalable, implementation has been elusive. This paper presents a new framework and implementation of a finite element-based discrete DD code that (1) treats arbitrarily shaped non-convex surfaces through image tractions, (2) allows for systematic refinement of the finite element mesh both in the bulk and on the surface and (3) provides a platform to scale to relatively larger and lengthier simulations. The approach is based on the capabilities of the Parallel Dislocation Simulator coupled through a distributed shared memory implementation for the calculation of large numbers of dislocation segments interacting with an independently large number of surface finite elements. Surface tracking approaches enable topological features at surfaces to be modeled. We verify the computed results via comparisons with analytical solutions for an infinite screw dislocation and prismatic loop near a surface and examine surface effects on a Frank–Read source. Convergence of the image force error with h- and p-refinement is shown to indicate the computational robustness. Additionally, through larger numerical experiments, we demonstrate the new capabilities in a three-dimensional elastic body of finite extent.

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.

Computation of flows with shocks using the Spectral Difference method with artificial viscosity, II: Modified formulation with local mesh refinement

The current work focuses on applying an artificial viscosity (AV) approach to the Spectral Difference (SD) method to enable high-order computation of compressible fluid flows with discontinuities. The study improves on the AV approach proposed in our prior work [1] to obtain a formulation that performs better on curvilinear and non-cartesian unstructured grids. A dilatation sensor for the AV coefficient, combined with a dilatation-based switch and filter for smoothing, is found to reduce the oscillatory behavior and enable sharp capturing of discontinuities. The work also incorporates the capability for local mesh refinement, which is used in conjunction with AV to obtain improved shock profiles. A mortar element method is used to handle the non-conforming interfaces generated from mesh-refinement of quadrilateral elements. Promising results are demonstrated for 2D test problems.

H — P Finite element approximation for full-potential electronic structure calculations

The (continuous) finite element approximations of different orders for the computation of the solution to electronic structures was proposed in some papers and the performance of these approaches is becoming appreciable and is now well understood. In this publication, the author proposes to extend this discretization for full-potential electronic structure calculations by combining the refinement of the finite element mesh, where the solution is most singular with the increase of the degree of the polynomial approximations in the regions where the solution is mostly regular. This combination of increase of approximation properties, done in an a priori or a posteriori manner, is well-known to generally produce an optimal exponential type convergence rate with respect to the number of degrees of freedom even when the solution is singular. The analysis performed here sustains this property in the case of Hartree-Fock and Kohn-Sham problems.

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.