Nodal Enrichment Research Articles

A cracked zone clustering method for discrete fracture with minimal enhanced degrees of freedom

Many state-of-the-art approaches for predicting discrete fracture within the finite element framework were designed by enhancing the continuous displacement field with a discontinuous part. Broadly categorised into elemental and nodal enrichment approaches, they provided unprecedented versatility in discretising domains with prescribed and evolving boundaries. However, there can be an additional computational burden when global tracking schemes are used to enforce the continuity of tractions/jumps along crack paths. Many current approaches also entail involved integration procedures over cut subdomains, which pose challenges of efficiency and accuracy. In addition, although cracks do not need to conform to the geometry of the mesh, the discretisation of cracks is still linked to that of the underlying elements, which can lead to the inaccurate representation of traction profiles along stiff interfaces. Aiming at overcoming these issues, the present work introduces a new formulation with minimal enhanced degrees of freedom. This is achieved by decoupling the discretisation of bulk and cracks, a feature that is enabled by a novel clustered rigid body definition of enhancement field in the set of the variational equations. A new method, termed Cracked Zone Clustering Method (CZCM), is then proposed for discrete fracture propagation by deploying only one rotational degree of freedom per set of cracked elements within the same zone. The stability of the method is first studied using element examples. The method is also shown to resolve the issue of traction oscillations over stiff interfaces. The accuracy and efficiency of the independent discretisation strategy are studied using several known crack propagation benchmarks. Results are compared with the Extended/Generalised Finite Element Method (XFEM/GFEM) and the Discrete Strong Discontinuity Approach (DSDA). In summary, it is shown that the cracked zones can encompass a multitude of elements, thus significantly decreasing the required number of enhancement degrees of freedom by orders of magnitude and without compromising accuracy.

A consistent finite element approach for dynamic crack propagation with explicit time integration

The concept of the partition of unity (PU) enabled the development of nodal enrichment strategies, such as the Extended Finite Element Method (XFEM) and Generalised Finite Element Method (GFEM), for realistic simulations of structural behaviour with applications to both static and dynamic problems. Nonetheless, the majority of existing methodologies still inherit instability issues for arbitrary discontinuity geometries when using an explicit time integration scheme. To address this, the discrete strong discontinuity approach is herein developed for the simulation of dynamic crack propagation. The formulation is fully variationally consistent and a new mapping approach is introduced to embed the rigid body movements associated with discontinuities while keeping the critical time step bounded. Multiple discontinuities within a single element are also considered for the accurate modelling of crack branching. The stability of the new technique is first verified in one- and two-dimensional elements. Next, the accuracy and efficiency are validated with structural examples including tensile and mixed-mode loadings. A concrete L-specimen, where different loading rates produce significant variations in failure patterns and strength, is also considered. Results show the good overall agreement with experimental data and other numerical studies available in the literature. The new formulation, however, is able to capture complex crack propagation phenomena, such as crack branching, without any specific additional criterion (e.g. based on crack tip velocity). The formulation presented in this paper is, to the best of the authors’ knowledge, the first PU-based consistent finite element with intrinsically bounded critical time step for explicit time integration.

Cracking elements method with 6-node triangular element

The cracking elements method (CEM) is a novel Galerkin-based numerical approach for simulating cracking and fracturing processes. It is a crack-opening approach that avoids precise descriptions of the mechanical states of crack tips and captures the initiations and propagations of multiple cracks without nodal enrichment or crack tracking. The CEM requires element types with nonlinear interpolation of the displacement field to avoid stress-locking. In the 2D condition, the 6-node triangular element (T6) and 8-node quadrilateral element (Q8) are potential candidates. However, despite the success of the formerly proposed CEM with Q8, the CEM with T6 showed considerable mesh dependencies. In this work, to solve this problem, the CEM with T6 is further investigated. The mesh dependencies are shown to be eliminated with simple modification to the real characteristic length of the T6 element in the CEM framework. Several numerical examples with regular and irregular Q8 and T6 mixed meshes are provided, indicating the effectiveness and robustness of this approach.

Protein Subdomain Enrichment of NUP155 Variants Identify a Novel Predicted Pathogenic Hotspot.

Functional variants in nuclear envelope genes are implicated as underlying causes of cardiopathology. To examine the potential association of single nucleotide variants of nucleoporin genes with cardiac disease, we employed a prognostic scoring approach to investigate variants of NUP155, a nucleoporin gene clinically linked with atrial fibrillation. Here we implemented bioinformatic profiling and predictive scoring, based on the gnomAD, National Heart Lung and Blood Institute-Exome Sequencing Project (NHLBI-ESP) Exome Variant Server, and dbNSFP databases to identify rare single nucleotide variants (SNVs) of NUP155 potentially associated with cardiopathology. This predictive scoring revealed 24 SNVs of NUP155 as potentially cardiopathogenic variants located primarily in the N-terminal crescent-shaped domain of NUP155. In addition, a predicted NUP155 R672G variant prioritized in our study was mapped to a region within the alpha helical stack of the crescent domain of NUP155. Bioinformatic analysis of inferred protein-protein interactions of NUP155 revealed over representation of top functions related to molecular transport, RNA trafficking, and RNA post-transcriptional modification. Topology analysis revealed prioritized hubs critical for maintaining network integrity and informational flow that included FN1, SIRT7, and CUL7 with nodal enrichment of RNA helicases in the topmost enriched subnetwork. Furthermore, integration of the top 5 subnetworks to capture network topology of an expanded framework revealed that FN1 maintained its hub status, with elevation of EED, CUL3, and EFTUD2. This is the first study to report novel discovery of a NUP155 subdomain hotspot that enriches for allelic variants of NUP155 predicted to be clinically damaging, and supports a role for RNA metabolism in cardiac disease and development.

A new approach for physically nonlinear analysis of continuum damage mechanics problems using the generalized/extended finite element method with global-local enrichment

The Generalized/Extended Finite Element Method (G/XFEM) is a numerical technique suitable to solve a wide range of continuum mechanics problems. It applies hierarchical nodal enrichment strategies within the Finite Element Method framework, allowing more flexibility in the numerical solution of boundary value problems (BVP) and being a powerful mesh-reduced method that takes advantage of the state-of-art of the standard FEM. One of the enrichment mechanisms is the so-called global-local enrichment (G/XFEMGL), in which the solution of a refined local BVP generates enrichment functions for the global domain. In this context, this work presents a new two-scale nonlinear strategy, associated with the G/XFEMGL, able to solve the material nonlinear analysis of continuum damage mechanics problems, in which quasi-brittle media degrade and soften, using a versatile mesh refinement strategy. A comprehensive description of the proposed strategy is registered and the aforementioned computational technique is validated throughout a set of plane stress numerical experiments that employ smeared crack constitutive model formulations, with well-known stress-strain laws. The results indicate that the implemented methodology could be used to solve a reasonable range of problems with considerable flexibility.

On the crack opening and energy dissipation in a continuum based disconnected crack model

All crack models developed to date can be classified into discrete- and continuum-based approaches. While discrete models are advantageously capable of capturing the kinetics of fractures, continuum-based approaches still pique considerable interest due to their straightforward implementation within the finite element method (FEM) framework. The cracking element method (CEM), a recently developed numerical approach for simulating quasi-brittle fracturing, is based on the FEM and does not need remeshing, a nodal cover algorithm, nodal enrichment or a crack-tracking strategy. The CEM takes self-propagating disconnected cracking segments to represent crack paths and naturally captures crack initiation and propagation processes. However, as with other types of continuum-based approaches that employ discontinuous crack paths, one critical question remains: Can the crack openings can be reliably and accurately obtained? To answer this question, a detailed study on the released energy, kinetic model, and displacement between the upper and lower facets of a crack is required. Multiple tests are conducted in this paper, and the results of the CEM are compared with those of the interface element method (IEM), which explicitly describes the crack openings. For reference and comparison purposes, an a priori crack path obtained by using an equivalent crack path (cracked elements) previously obtained from the CEM is implemented for the IEM. The crack openings obtained by the CEM and IEM are subsequently compared, and the results indicate that the crack openings and dissipated energy obtained by the CEM generally agree well with those obtained by the IEM. These findings highlight the effectiveness of utilizing disconnected cracking segments and further demonstrate the robustness and reliability of the CEM.

Cracking Elements Method for Simulating Complex Crack Growth

The cracking elements method (CEM) is a novel numerical approach for simulating fracture of quasi-brittle materials. This method is built in the framework of conventional finite element method (FEM) based on standard Galerkin approximation, which models the cracks with disconnected cracking segments. The orientation of propagating cracks is determined by local criteria and no explicit or implicit representations of the cracks' topology are needed. CEM does not need remeshing technique, cover algorithm, nodal enrichment or specific crack tracking strategies. The crack opening is condensed in local element, greatly reducing the coding efforts and simplifying the numerical procedure. This paper presents numerical simulations with CEM regarding several benchmark tests, the results of which further indicate the capability of CEM in capturing complex crack growths referring propagations of existed cracks as well as initiations of new cracks.

Cracking elements method for dynamic brittle fracture

The cracking elements (CE) method is a recently presented self-propagating strong discontinuity embedded approach with the statically optimal symmetric (SDA-SOS) formulation for simulating the fracture of quasi-brittle materials. CE uses disconnected cracking segments to represent cracks, and the crack openings are condensed locally; hence, this method does not require remeshing, a cover algorithm or nodal enrichment. Furthermore, local criteria for determining the crack orientations are presented, thus making crack tracking unnecessary. In this paper, we propose a CE numerical procedure for dynamic fractures. Several benchmark tests regarding irregular discretizations are performed, and the results indicate that the CE method is capable of naturally capturing complicated crack patterns in dynamic fractures, including crack branchings.

An improved crack tracking algorithm with self‐correction ability of the crack path and its application in a continuum damage model

SummaryIn this paper, a new procedure for predicting the crack propagation and fracture behavior in quasi‐brittle materials is presented based on continuum damage mechanics. Several crack tracking algorithms are widely used for failure analysis, among which the local tracking algorithm is very simple and easy to implement with low computational cost. However, in the prediction of crack growth with the traditional local tracking algorithm, it can be often seen that the sharp changes in the crack path such as U‐turns can occur. An improved local tracking algorithm with self‐correction ability of the crack path is proposed to overcome these difficulties of the traditional crack tracking algorithms. A continuum damage model, in addition to the present crack tracking algorithm, can greatly enhance the performance of failure prediction in quasi‐brittle materials. The present approach has the advantages that it can well predict the crack propagation path and can avoid mesh size and mesh bias dependence without the embedment of discontinuities or nodal enrichment. The present model is incorporated into ANSYS by the ANSYS Parameter Design Language to simulate the initiation and propagation of the discrete crack in engineering applications. The numerical results by this model are compared with the ones by the extended finite element method and experiments, and good agreements are achieved.

An Improved XFEM for the Poisson Equation with Discontinuous Coefficients

Abstract Discontinuous coefficients in the Poisson equation lead to the weak discontinuity in the solution, e.g. the gradient in the field quantity exhibits a rapid change across an interface. In the real world, discontinuities are frequently found (cracks, material interfaces, voids, phase-change phenomena) and their mathematical model can be represented by Poisson type equation. In this study, the extended finite element method (XFEM) is used to solve the formulated discontinuous problem. The XFEM solution introduce the discontinuity through nodal enrichment function, and controls it by additional degrees of freedom. This allows one to make the finite element mesh independent of discontinuity location. The quality of the solution depends mainly on the assumed enrichment basis functions. In the paper, a new set of enrichments are proposed in the solution of the Poisson equation with discontinuous coefficients. The global and local error estimates are used in order to assess the quality of the solution. The stability of the solution is investigated using the condition number of the stiffness matrix. The solutions obtained with standard and new enrichment functions are compared and discussed.

Improving accuracy and efficiency of stress analysis using scaled boundary finite elements

The scaled boundary finite element method (SBFEM) is a fundamental-solution-less boundary element method, which leads to semi-analytical solutions for stress fields. As only the boundary is discretized, the spatial dimension is reduced by one. In this paper, the SBFEM based polygon elements are utilized to improve the accuracy and efficiency of stress analysis. It retains the attractive feature of the SBFEM in solving problems with unbounded media and singularities. In addition, polygon elements are more flexible in meshing and mesh transition. Various measures which help improving accuracy or efficiency of the stress analysis, i.e. refining polygon mesh, nodal enrichment, appropriate placing of the scaling center, merging polygon elements and NURBS enhanced curved boundaries are discussed and compared. As a result, further insight into the refinement and improvement strategies for stress analysis is provided.

Recent efforts toward modeling interactions of matrix cracks and delaminations: an integrated XFEM-CE approach

This paper presents a novel integrated approach using the extended finite element method (XFEM) and cohesive elements (CEs) for modeling three-dimensional (3D) delaminations, matrix cracks, and their interactions in progressive failure of composite laminates. In the proposed XFEM-CE approach, the matrix cracks are explicitly modeled by the XFEM through nodal enrichment and element partition, and the inter-ply delaminations are modeled by cohesive elements through traction–separation law. An XFEM-based cohesive element enrichment scheme is developed in order to properly model the interactions between the delaminations and matrix cracks. It is critical to enrich the cohesive elements at the local juncture where cracks meet, in order to obtain the correct crack path interactions. Examples are presented for failure analysis of double-cantilever-beam and end-notch-flexure laminates with different layups. The results show strong explicit matrix crack–delamination interactions in these laminates. This work presents another significant development of a computational platform for realistic modeling of progressive damage in composites.

Simulation of Primary Fracture Propagation around Compressive Cavity with the Extended Finite Element Method

The fracture distribution around circular cavities has been widely observed in experiments and well documented in literature from 1980. Most of the works are based on experiments’ results, which cost considerable time and money. With varied numerical methods developed more and more researcher employ numerical experiments on computers instead of physical experiments. Firstly, the nodal enrichment functions for Extended Finite Element Method in conjunction with additional degrees are presented. Moreover, we describe the cohesive segments method, which is followed by the damage initiation and evolution laws. In the last a borehole numerical model is built up and the simulation results of the primary fracture propagation are presented.

A COMPARATIVE STUDY ON 2D CRACK MODELLING USING EXTENDED FINITE ELEMENT METHOD

In this study the Extended Finite Element Method (XFEM) was used for modelling 2D cracks in plane and plate problems. The aim was to investigate the significance of the crack Tip functions in the nodal enrichment procedure. A new set of Tip functions were extracted from analytical solutions of Kirchhoff plates and various nodal enrichment schemes were examined for plane and plate elements. The stress intensity factors (SIF) for typical benchmark problems were calculated using the J-Integral and the Interaction Integral methods. The results indicated that elimination of the Tip functions had no noticeable effect on the accuracy of the computed stress intensity factors for the plane problems. However, the precision of the computed values for the plate problems were quite affected by this procedure. Moreover, the Interaction Integral method was found to be more efficient for the computation of crack parameters in such problems. DOI: http://dx.doi.org/10.5755/j01.mech.19.4.5043

A novel continuum-decohesive finite element for modeling in-plane fracture in fiber reinforced composites

A new finite element to seamlessly model the transition from a continuum to a non-continuum (through fracture) is introduced in this paper, motivated by the variational multi-scale cohesive (VMCM) method. In-plane fiber–matrix fracture (also referred to as splitting) is frequently observed in tensile failure of fiber reinforced polymer matrix composites (FRPCs). By considering a single lamina, this mechanism is modeled through the development of a continuum-decohesive finite element (CDFE). In the CDFE method, the transition from a continuum to a non-continuum is modeled directly (physically) without resorting to enrichment of the shape functions of the element, as is done in other methods, such as the VMCM or through nodal enrichment, as is done with the extended finite element method (XFEM). The CDFE is a natural merger between cohesive elements and continuum elements. The predictions of the CDFE method were found to be in very good agreement with corresponding experimental data.

Dynamic crack propagation simulation with scaled boundary polygon elements and automatic remeshing technique

An efficient methodology for automatic dynamic crack propagation simulations using polygon elements is developed in this study. The polygon mesh is automatically generated from a Delaunay triangulated mesh. The formulation of an arbitrary n-sided polygon element is based on the scaled boundary finite element method (SBFEM). All kind of singular stress fields can be described by the matrix power function solution of a cracked polygon. Generalised dynamic stress intensity factors are evaluated using standard finite element stress recovery procedures. This technique does not require local mesh refinement around the crack tip, special purpose elements or nodal enrichment functions. An automatic local remeshing algorithm that can be applied to any polygon mesh is developed in this study to accommodate crack propagation. Each remeshing operation involves only a small patch of polygons around the crack tip, resulting in only minimal change to the global mesh structure. The increase of the number of degrees-of-freedom caused by crack propagation is moderate. The method is validated using four dynamic crack propagation benchmarks. The predicted dynamic fracture parameters show good agreement with experiment observations and numerical simulations reported in the literature.

Analysis of the Crack Propagation Based on Extended Finite Method

For solve efficiently the problem of crack propagation, the extended finite method(XFEM) was applied to analyse the problem. The extended finite method introduced nodal enrichment functions based on usual nodal shape functions, and traced crack propagation with the level set method. The extended finite method can model arbitrary crack growth without remeshing. This paper simulated the problem of the compact tension specimen with circular hole by extended finite method. The calculation results show that the extended finite method can solve efficiently the problem of crack growth , and the circular hole of compact tension specimen effect the trajectory of mode I crack propagation within certain distance.

Polygon scaled boundary finite elements for crack propagation modelling

SUMMARYAn automatic crack propagation modelling technique using polygon elements is presented. A simple algorithm to generate a polygon mesh from a Delaunay triangulated mesh is implemented. The polygon element formulation is constructed from the scaled boundary finite element method (SBFEM), treating each polygon as a SBFEM subdomain and is very efficient in modelling singular stress fields in the vicinity of cracks. Stress intensity factors are computed directly from their definitions without any nodal enrichment functions. An automatic remeshing algorithm capable of handling any n‐sided polygon is developed to accommodate crack propagation. The algorithm is simple yet flexible because remeshing involves minimal changes to the global mesh and is limited to only polygons on the crack paths. The efficiency of the polygon SBFEM in computing accurate stress intensity factors is first demonstrated for a problem with a stationary crack. Four crack propagation benchmarks are then modelled to validate the developed technique and demonstrate its salient features. The predicted crack paths show good agreement with experimental observations and numerical simulations reported in the literature. Copyright © 2012 John Wiley & Sons, Ltd.

Enhanced solid element for modelling of reinforced concrete structures with bond-slip

Since its invention in the <TEX>$19^{th}$</TEX> century, Reinforced Concrete (RC) has been widely used in the construction of a lot of different structures, as buildings, bridges, nuclear central plants, or even ships. The details of the mechanical response for this kind of structures depends directly upon the material behavior of each component: concrete and steel, as well as their interaction through the bond-slip, which makes a rigorous engineering analysis of RC structures quite complicated. Consequently, the practical calculation of RC structures is done by adopting a lot of simplifications and hypotheses validated in the elastic range. Nevertheless, as soon as any RC structural element is working in the inelastic range, it is possible to obtain the numerical prediction of its realistic behavior only through the use of non linear analysis. The aim of this work is to develop a new kind of Finite Element: the "Enhanced Solid Element (ESE)" which takes into account the complex composition of reinforced concrete, being able to handle each dissipative material behavior and their different deformations, and on the other hand, conserving a simplified shape for engineering applications. Based on the recent XFEM developments, we introduce the concept of nodal enrichment to represent kinematics of steel rebars as well as bonding. This enrichment allows to reproduce the strain incompatibility between concrete and steel that occurs because of the bond degradation and slip. This formulation was tested with a couple of simple examples and compared to the results obtained from other standard formulations.

Meshless method for shallow water equations with free surface flow

Solving problems with free surface often encounters numerical difficulties related to excessive mesh distortion as is the case of dambreak or breaking waves. In this paper the Natural element method (NEM) is used to simulate a 2D shallow water flows in the presence of theses strong gradients. This particle-based method used a fully Lagrangian formulation based on the notion of natural neighbors. In the present study we consider the full non-linear set of Shallow Water Equations, with a transient flow under the Coriolis effect. For the numerical treatment of the nonlinear terms we used a Lagrangian technique based on the method of characteristics. This will allow avoiding divergence of Newton–Raphson scheme, when dealing with the convective terms. We also define a thin area close to the boundaries and a computational domain dedicated for nodal enrichment at each time step. Two numerical test cases were performed to verify the well-founded hopes for the future of this method in real applications.