Mesh Dependency Research Articles

Element‐based coupling modeling of peridynamics and classical continuum mechanics for dynamic brittle fracture

AbstractIn this work, the hybrid peridynamic and classical continuum mechanical model based on the element‐type numerical discretization is used to study the dynamic fracture problem of brittle materials. The weak form of the motion equation for the hybrid model is derived by applying Hamilton's principle. An element‐based spatial discretization scheme is set up using the peridynamics‐based finite element method (PeriFEM) (Han and Li). The ‐convergence and m‐convergence of the hybrid model are analyzed to verify the numerical convergence and the mesh dependency. The influence of the area of the peridynamic model subdomain and the hybrid model subdomain on the calculation efficiency and accuracy is discussed. And the relationship between the crack propagation speed and the crack branching phenomenon is also investigated. Good agreements are obtained by comparing the crack path and the crack speed profile predicted by the hybrid model with those in the literature. Numerical results verify the ability and efficiency of the hybrid model for dynamic fracture problem.

Numerical implementation of tension–compression asymmetry in a gradient‐extended two‐surface damage–plasticity model

AbstractBased on the spectral decomposition of the strain tensor, the tension‐compression asymmetry (TCA) in a gradient‐extended two‐surface damage‐plasticity model is considered in this study. More specifically, the gradient extension is applied in the current model as a suitable damage regularization technique to prevent pathological mesh dependence issues. Furthermore, plasticity and damage are treated as distinct but strongly coupled dissipative mechanisms by using two different types of damage and yield criteria as well as the related loading/unloading conditions, respectively. Finally, a specific material parameter is adopted to control the degree of TCA which results in a unique damage evolution in compression.

Finite element modeling of high-speed orthogonal cutting: A contribution to understanding the influence of mesh parameters on saw-toothed chip formation

Modeling of metal cutting using finite element (FE) method has been investigated in many studies. They make it possible to better understand the chip formation. However, the mesh dependence of the numerical solution is still unsolved. The purpose of this paper is the modeling and simulation of saw-toothed chip formation during high-speed orthogonal cutting. Further, the influences of mesh parameters on saw-toothed chip formation are investigated as an innovative highlight of this study. The investigation of high-speed cutting (HSC) is based on FE analysis, where the Johnson-Cook (JC) constitutive model is combined with an energy-based fracture criterion. The mesh parameters considered include element dimension, element orientation, and hourglass treatment. The results show that the peak and valley values of saw-toothed chip obtained through the model mesh10 × 10 are close to the measurements, but the pitch value is predicted with a larger error than those of peak and valley values. Square elements of dimension 10 μm can be also used to avoid the influence of element orientation on cutting forces with an error about 10%. When using square elements, the hourglass treatment causes slightly differences in chip morphology and cutting force. On the other hand, the enhanced is the more efficient method to prevent the hourglass effect for rectangular elements. The model mesh8 × 10 gives an error less than 15% for cutting force. Moreover, the effect of element dimension on cutting force can be reduced by applying the fracture energy criterion controlled through a characteristic length to different mesh dimensions.

A continuum damage model for creasing and folding of paperboard

AbstractA continuum damage framework is proposed for modelling the delamination process occurring in paperboard during mechanical loading. The main application of interest is line creasing and subsequent line folding used in package forming. To adequately capture creasing and folding, a continuum damage framework that uses a single isotropic damage variable that evolves with the plastic strains associated with out‐of‐plane shearing is used. The damage evolution is calibrated against folding experiments for a specific reference mesh, and a simple scaling strategy is proposed to reduce the inherent mesh dependency. To highlight the potential of the proposed model, an illustrative 3D example is considered where a paperboard sheet is creased by two plates and folded to resemble the corner of a package.

Meso to macro connections to capture fatigue damage in cemented materials

Fatigue-induced damage is a common issue in cemented materials, involving the progressive formation and complicated propagation of fatigue cracks. These cracks typically localise on weak or fracture planes, leading to inhomogeneous deformation within the material. Thus, accurately predicting the fatigue phenomenon becomes challenging due to the material’s inhomogeneity and the complex evolution of cracks, from initiation to propagation and eventual failure. To address this challenge, this paper presents a constitutive model that accounts for strain discontinuity across fracture planes by employing kinematic enrichment. This enhancement facilitates interaction between the material responses of cracks and the outer bulk, thereby contributing to the overall macro behaviour of the materials. Moreover, the proposed model incorporates a new cohesive-frictional fatigue model that couples damage mechanics and bounding surface plasticity to describe the fatigue behaviour of fracture planes/cracks. Since the proposed model features a characteristic length scale, it exhibits size-dependent behaviour and helps overcome the issue of mesh dependence. The model's validity is demonstrated through its ability to capture nonlinear fatigue damage under constant/variable cyclic loading and to simulate the propagation of fatigue fracture process zones. Furthermore, the model effectively captures the significant influence of stress amplitudes on the fatigue lives of materials, making it an essential tool for predicting and mitigating fatigue-induced damage in cemented materials.

A regularized higher-order beam elements for damage analysis of reinforced concrete beams

This work presents a numerical method for damage analysis of reinforced concrete beams using the higher-order beam theory based on Carrera unified formulation. The component-wise approach is employed to model the concrete and steel reinforcing bars as two independent one-dimensional finite elements. A modified Mazars damage model with tensile and compressive damage propagation laws is utilized for concrete, and an elastic-perfectly plastic law is used for steel rebars. To address the instability and mesh dependence caused by the strain-softening behavior of concrete, a fracture energy regularization technique based on the crack band model is developed, especially for the higher-order beam theory. The proposed method is validated by comparing its numerical results with three experimental benchmark results. The comparison indicates that the method accurately predicts the damage distribution of concrete and the flexural behavior of RC beams under quasi-static loading conditions while remaining computationally efficient.

A novel peridynamic solution for modelling saturated soil-pore fluid interaction in liquefaction analysis

Saturated soil-pore fluid interaction under earthquake shaking can cause extensive damages to critical infrastructure systems. As such, accurate modeling of saturated soil-pore fluid interaction and its consequences (e.g., seismically-induced large deformation) are of particular importance for liquefaction analysis. Peridynamics (PD) has emerged as an effective method for solving large deformation problems by overcoming the disadvantage of finite element method caused by mesh dependency. However, the PD has not been extended to model soil-pore fluid interaction, limiting its application in liquefaction analysis. To bridge this gap, a novel solution strategy is presented for modeling soil-pore fluid interaction (i.e., the u-p equations herein) in liquefaction analysis of saturated granular soils, based on a hybrid PD (HPD) method recently proposed by the authors and co-workers. An explicit-explicit method is used to solve the coupled u-p equations. Liquefiable soil on level and sloped ground are analyzed and compared with experimental data and finite element simulations to validate availability and efficiency of the method, demonstrating the method to be effective in solving the soil-pore fluid equations and applicable to a broad range of geotechnical earthquake engineering problems associated with liquefaction.

Nonlocal implicit gradient enhancements for strain localization informed by controllability criteria for plastic solids

Nonlocal implicit gradient enhancements are widely used to suppress mesh dependency in simulations involving strain localization. For example, nonlocality is often introduced through internal variables that account for possible material softening. This work, however, shows that this approach may become ineffective if the solid displays plastic non-normality (i.e., non-associated plastic flow). For this purpose, we consider an over-nonlocal formulation and mathematically inspect the conditions at which regularization is lost in the presence of plastic non-normality. Specifically, such loss of regularization is linked to the loss of uniqueness and/or existence of the incremental plastic response that is kinematically compatible with the development of a deformation band. By doing so, we find a lower limit for the admissibility of the parameters controlling the effectiveness of nonlocal implicit gradient regularization. Furthermore, we show that such a lower limit is regulated by a plastic modulus reflecting the loss of controllability of the constitutive response, and, hence, depends on the degree of plastic non-normality. We also derive a closed-form expression relating the thickness of the deformation band to both the controllability modulus and gradient regularization constants, which suggests that the thickness of the process zone may change in response to the prevailing plastic flow characteristics and evolve during active plastic deformation. The proposed nonlocal enhancement is applied to a non-associated elasto-plastic model for porous sedimentary rocks, which is capable of displaying both shear-dominated and compaction-dominated bands. Numerical simulations reveal that effective regularization can be enforced only when the over-nonlocal weighting coefficient is larger than the above-mentioned lower limit.

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

Comparison of local and nonlocal regularization approaches in Eulerian-based finite element analyses

Finite element (FE) results might suffer from significant mesh dependency, especially when modeling shear bands in strain-softening soils. On the other hand, the shear bands in the field are generally thin. The computational costs might dramatically increase when modeling such thin shear bands, especially for large-scale problems, such as progressive landslides. To overcome these issues, FE analyses are generally performed with larger shear band thicknesses by adopting ‘softening scaling’ rules based on local and nonlocal (averaged) strains at the shear bands. The present study compares the performances of local and nonlocal approaches, with a specific focus on softening scaling and large deformation. Analyses are performed using a Eulerian-based FE program, which can model large strains in the shear band without numerical issues related to mesh distortion. Implementing strain-softening behavior in local and nonlocal methods, two idealized cases are simulated: (i) biaxial compression test, (ii) slope failure due to upslope surface loading. FE simulations show that the macroscopic response (i.e., load–displacement behavior) can be modeled using both local and nonlocal regularization techniques. Shear band thickness increases with the progress of shearing. In local analysis, mesh orientation has a considerable effect on shear band thickness. The existence of neighboring shear bands could affect nonlocal strain calculation and the simulation results.

A non-local approach for Reissner–Mindlin shell elements in dynamic simulations: Application with a Gurson model

Numerical prediction of ductile tearing during a car crash is necessary to ensure the structural integrity of the vehicle. Recently, encouraging results were obtained by Davaze et al., (2021) with a non-local damage approach, compatible with dynamic explicit simulations and preserving parallel computation: an implicit gradient model enabled to reproduce experimental results with DP450 steel sheets with 3D solid elements. However, in the automotive industry, thin sheets of metals are mostly modeled with shell elements for computation time efficiency. Shell elements have a more complex formulation: their kinematic description is enriched accounting for displacements but also rotations, leading to two different strain contributions: membrane strain and bending strain. The transverse strain is computed independently in a post-processing step from local internal variables with a zero normal stress assumption. This formulation is known to lead to strain localization after the onset of necking, while this is not the case with 3D solid elements. The problem is even more challenging when considering shells in association with a softening material accounting for damage as this also leads to strong mesh dependency, as with 3D solid elements. This complex problem is solved using non-local formulations for the material behavior. The proposed solution accounts for both in-plane and in-thickness damage gradients. It consists of a two-step regularization procedure introducing nested wire elements in the shell thickness. Using non-local variables to evaluate the transverse strain enables to obtain a continuous spatial distribution for strains, in agreement with the 3D solid case. An updated Lagrangian formulation is used to account for large strains. The performance of the strategy is demonstrated in three test cases, with a comparison with an experimental result for one of them. The results obtained following the proposed strategy lead are equivalent to the ones obtained with the non-local method proposed by Davaze et al., (2021) and are thus consistent with the experimental data.

A gradient-based non-local GTN model: Explicit finite element simulation of ductile damage and fracture

Non-local models have over the years been established as an effective approach to solve the pathological mesh dependency problem observed in finite element simulation of strain-softening materials. This paper presents the formulation, implementation, and application of a gradient-based non-local Gurson–Tvergaard–Needleman (GTN) model for explicit finite element analysis. The porosity is taken as the non-local variable where the increment in porosity is averaged over the volume using an implicit gradient model. The gradient model is implemented in Abaqus/Explicit by utilising the coupled thermal–mechanical solver, which proves to be both a simple and computationally efficient approach. Due to the use of an explicit integration scheme, a transient term is introduced to the partial differential equation of the gradient formulation. The non-local GTN model is compared to the local counterpart for increasing mesh refinements using a plane strain shear band specimen, a plane strain tension specimen, and a plane strain compact tension specimen. The proposed approach can remedy the pathological mesh dependency problem. For the plane strain tension specimen, it is shown that the non-local GTN model will preserve the fracture mode (slant versus cup–cup fracture mode) when the mesh is refined. The non-local GTN model is also able to predict the same fracture mode as observed in ductile tearing experiments. However, non-local averaging can over-smooth the fields and exclude the slant fracture mode from occurring if the material length Lc is too large.

Large deformation and brittle failure calculated using the dual-domain material point method

The dual domain material point (DDMP) method is explored as a candidate to be implemented in a general purpose code to perform simulations of materials with complex geometry that undergo large history-dependent deformation and failure. To test its candidacy, we study its mesh convergence, its sensitivity to mesh orientation, and its ability to handle softening and failure of a material. Simulations of large deformation and simulations of mechanical failure are performed using both DDMP and the material point method (MPM). When cell-crossing of material points is not an issue and when there are a sufficient number of material points in each computation cell, the numerical error decreases with the square of the cell size as expected for both MPM and DDMP. DDMP has reduced error compared with MPM when there are many instances of material points crossing cell boundaries due to the continuous nature of the modified gradient of the shape functions. Simulations of a specimen under tension are also performed where the background mesh is aligned and misaligned with the tension direction. MPM displays a significant mesh-dependent stress field, DDMP shows negligible mesh dependency. Despite a mesh orientation-dependent stress field from MPM, the critical tension and failure mode from both MPM and DDMP calculations have negligible mesh dependency when using a non-local failure model. If only the failure mode is important (i.e., local stresses are unimportant), MPM with a non-local failure model is a suitable method for modeling failure with small deformations. However, if local stresses are also important or if there are large deformations with many cell-crossings before failure, DDMP should be the method that is used. A needed improvement for DDMP is identified from our numerical simulations.

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, our experiments showed that the relaxation by rank-one convexification delivering an approximation to the quasiconvex envelope prevents mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

Singularity and mesh divergence of inviscid adjoint solutions at solid walls

• Comprehensive review of the mesh dependence anomaly observed in two and three dimensional numerical adjoint solutions for subsonic and transonic inviscid flows past wings, airfoils and blunt bodies. • Review of the analytic adjoint solution obtained for the 2D incompressible Euler equations. • Identification of the origin of the mesh dependence anomaly as a singularity of the analytic solution. • Cross-comparison of analytic and numerical adjoint solutions for lift and drag.

Non-local and local criteria based on the extended finite element method (XFEM) for fracture simulation of anisotropic 3D-printed polymeric components

PurposeThe purpose of this study is to develop an efficient numerical procedure for simulating the effect of printing orientation, as one of the primary sources of anisotropy in 3D-printed components, on their fracture properties.Design/methodology/approachThe extended finite element method and the cohesive zone model (XFEM-CZM) are used to develop subroutines for fracture simulation. The ability of two prevalent models, i.e. the continuous-varying fracture properties (CVF) model and the weak plane model (WPM), and a combination of both models (WPM-CVF) are evaluated to capture fracture behavior of the additively manufactured samples. These models are based on the non-local and local forms of the anisotropic maximum tangential stress criterion. The numerical models are assessed by comparing their results with experimental outcomes of 16 different configurations of polycarbonate samples printed using the material extrusion technique.FindingsThe results demonstrate that the CVF exaggerates the level of anisotropy, and the WPM cannot detect the mild anisotropy of 3D-printed parts, while the WPM-CVF produces the best results. Additionally, the non-local scheme outperforms the local approach in terms of finite element analysis performance, such as mesh dependency, robustness, etc.Originality/valueThis paper provides a method for modeling anisotropic fracture in 3D-printed objects. A new damage model based on a combination of two prevalent models is offered. Moreover, the developed subroutines for implementing the non-local anisotropic fracture criterion enable a reliable crack propagation simulation in media with varying degrees of complication, such as anisotropy.

Exploring the numerical performance of node-based smoothed finite elements in coupled hydro-mechanical problems

Node-based smoothed finite element (NS-FEM) formulations for the coupled hydro-mechanical problem are now becoming popular in soil mechanics, but there has been limited comparative study of their performance. In this work, a node-based formulation is developed for the coupled hydro-mechanical problem (u−pw formulation) using low-order, equal-order interpolants and an implicit time marching scheme for the global problem. The numerical properties of the resulting formulation are then compared to different implementations of classical finite elements. The use of nodal smoothing does not render by itself the solution stable at the undrained limit, on the contrary, the range of undrained instability is increased. Stabilizing terms previously employed in FEM are equally successful in stabilizing NS-FEM solutions, but the computational cost of the stabilized solution is much higher for NS-FEM. It is also shown that NS-FEM does not present by itself any particular advantage in terms of mesh dependency over classic FEM: when strain localization is present the solution obtained is strongly dependent on mesh size.

Leveraging Full-Field Deformation Measurements in Computational Modeling of Damage

Abstract Advances in sensing and nondestructive evaluation methods have increased the interest in developing data-driven modeling and associated computational workflows for model-updating, in relation also to a variety of emerging digital twin applications. In this context, of particular interest in this investigation are transient effects that lead to or are caused by deformation instabilities, typically found in the cases of complex material behavior or interactions between material and geometry. In both cases, deformation localizations are observed which are typically also related to damage effects. This paper describes a novel framework to incorporate deformation data into a finite element model (FEM) that has been formulated using non-local mechanics and is capable of receiving such data and using it to describe the development of localizations. Specifically, experimentally measured full-field displacement data is used as an input in FEM as an ad-hoc boundary condition at any or every node in the body. To achieve this goal, a plasticity model which includes a spatially averaged non-local hardening parameter in the yield criterion is used to account for associated numerical instabilities and mesh dependence. Furthermore, the introduction of a length scale parameter into the constitutive law allows the connection between material behavior, geometry, and localizations. Additional steps remove the experimental data and evolve the computational predictions forward in time. Both one and three-dimensional boundary value problems are used to present results obtained by the proposed framework, while comments are made in terms of its potential uses.

A thermodynamically consistent anisotropic damage model for metallic glasses

AbstractThe recent success in manufacturing large‐size, also called bulk metallic glasses (BMGs) using 3D‐printing based on laser powder bed fusion (LPBF) opens an avenue for the broad application of this material class. To explore the great potential of both as‐cast and 3D‐printed BMGs, a comprehensive understanding and an accurate prediction of the plastic deformation and damage behaviour of this material class are indispensable. In this study, we develop a thermodynamically consistent anisotropic damage model incorporating tension‐compression asymmetry (TCA) to describe the unique inelastic deformation and damage behaviours of metallic glasses. The widely observed normal stress sensitivity and plastic dilatancy in metallic glasses are considered by using an extended Mohr‐Coulomb criterion in the constitutive description. Furthermore, a second‐order damage tensor is adopted for describing the anisotropic damage behaviour. By augmenting the Helmholtz free energy function to be dependent on the gradient of the free volume concentration and the gradient of the nonlocal damage parameter, the governing equations for the corresponding internal variables are derived within the framework of finite deformation. The damage localisation and mesh dependency are correspondingly alleviated. The simulation result shows that the shear band patterns are in good agreement with the experimental observations from literature.

Cross-scale optimization of advanced materials for micro and nano structures based on strain gradient theory

Recently developed cross-scale optimization methods are mainly the macro equivalent calculation of performances based on the traditional homogenization method. However, the traditional homogenization is limited to the classic continuum of Cauchy–Boltzmann. Therefore, it is inadequate to interpret the size dependence of the optimal result. Hence, a new cross-scale optimization is proposed based on Wei–Hutchinson strain gradient theory by employing the non-local homogenization model, which could describe and explain the size dependence during optimization process when considering micro structures. The topological optimization procedure simultaneously has the ability of coupled computing by using subdomain parameterized coarse meshes. The numerical computations involved in the entire model can be solved in one iteration, which helps to eliminate mesh dependencies and greatly reduce the computation time. It is shown that the final stiffness of the optimized periodic structure can be significantly increased by considering the strain gradient theory compared with the classic homogenization scheme in the process of cross-scale optimization.