Showing papers on "Extended finite element method published in 2023"

The peridynamics-based finite element method (PeriFEM) with adaptive continuous/discrete element implementation for fracture simulation

Abstract: The classical finite element method has been successfully applied to many engineering problems but not for the cases with space discontinuity. In our previous work, a peridynamics-based finite element method was presented according to the principle of minimum potential energy, which enables discontinuity. As a continuation of the previous work, on the one hand, we derive the peridynamics-based finite element formulation from a new perspective, i.e., the principle of virtual work. On the other hand, we propose an adaptive continuous/discrete element conversion technique, thus the cracks could be described explicitly without increasing the computational cost significantly. Finally, numerical results are executed to verify the proposed method, including the computational cost, the influence of discretization strategy and critical stretch on crack paths, and the comparison of predicted crack paths with experimental results. Numerical results show the efficiency and accuracy of the proposed method.

Nonconforming finite element Stokes complexes in three dimensions

Abstract: Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming P1-P0 element for the Stokes equation in three dimensions are studied. Commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order H(gradcurl)-nonconforming finite element only has 14 degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The H(gradcurl)-nonconforming elements are applied to solve the quad-curl problem, and the optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming P1-P0 element method for the Stokes equation, based on which a fast solver is discussed. Numerical results are provided to verify the theoretical convergence rates.

A hydro‐mechanically‐coupled XFEM model for the injection‐induced evolution of multiple fractures

Abstract: In this paper, a two‐dimensional eXtended Finite Element Method (XFEM) solution is presented for the hydro‐mechanically coupled hydro‐frac‐induced propagation of multiple fractures in rocks. Fractures are considered one‐dimensional objects, and the rock matrix is regarded as a two‐dimensional medium. Rules for improving the accuracy of the solution are included to deal with the transfer of variables between fractures and matrix in the coupling process. The main variables include fluid pressure and fracture aperture. The following assumptions are made: (1) the fluid pressure inside a fracture is applied as a net pressure; (2) fluid leak‐off is ignored; (3) the fluid front is regarded as a fracture front provided that the area of fluid lag has a small effect on fracture propagation compared to the area at which the fluid pressure is applied. The fluid flow is governed by the lubrication equation, while the XFEM is used to describe the behaviour of the elastic medium. The simulation results are compared with analytical solutions for a single hydraulic fracture model to verify and validate the proposed algorithm. A simulation with multiple hydraulic fractures is also performed to investigate the interference effect of multiple fractures and fracture propagation. The shadow effect caused by multiple fractures is analysed to manifest the stress variation along the propagation path.

Mesh refinement of finite element method for free vibration analysis of variable geometrical rotating cylindrical shells

Abstract: PurposeThis study aims to provide a reliable and effective algorithm that is suitable for addressing the problems of continuous orders of frequencies and modes under different boundary conditions, circumferential wave numbers and thickness-to-length ratios of moderately thick circular cylindrical shells. The theory of free vibration of rotating cylindrical shells is of utmost importance in fields such as structural engineering, rock engineering and aerospace engineering. The finite element method is commonly used to study the theory of free vibration of rotating cylindrical shells. The proposed adaptive finite element method can achieve a considerably more reliable high-precision solution than the conventional finite element method.Design/methodology/approachOn a given finite element mesh, the solutions of the frequency mode of the moderately thick circular cylindrical shell were obtained using the conventional finite element method. Subsequently, the superconvergent patch recovery displacement method and high-order shape function interpolation techniques were introduced to obtain the superconvergent solution of the mode (displacement), while the superconvergent solution of the frequency was obtained using the Rayleigh quotient computation. Finally, the superconvergent solution of the mode was used to estimate the errors of the finite element solutions in the energy norm, and the mesh was subdivided to generate a new mesh in accordance with the errors.FindingsIn this study, a high-precision and reliable superconvergent patch recovery solution for the vibration modes of variable geometrical rotating cylindrical shells was developed. Compared with conventional finite element method, under the challenging varying geometrical circumferential wave numbers, and thickness–length ratios, the optimised finite element meshes and high-precision solutions satisfying the preset error limits were obtained successfully to solve the frequency and mode of continuous orders of rotating cylindrical shells with multiple boundary conditions such as simple and fixed supports, demonstrating good solution efficiency. The existing problem on the difficulty of adapting a set of meshes to the changes in vibration modes of different orders is finally overcome by applying the adaptive optimisation.Originality/valueThe approach developed in this study can accurately obtain the superconvergent patch recovery solution of the vibration mode of rotating cylindrical shells. It can potentially be extended to fine numerical models and high-precision computations of vibration modes (displacement field) and solid stress (displacement derivative field) for general structural special value problems, which can be extensively applied in the field of engineering computations in the future. Furthermore, the proposed method has the potential for adaptive analyses of shell structures and three-dimensional structures with crack damage. Compared with conventional finite element methods, significant advantages can be achieved by solving the eigenvalues of structures with high precision and stability.

A study on XFEM simulation of tensile, fracture toughness, and fatigue crack growth behavior of Al 2024 alloy through fatigue crack growth rate models using genetic algorithm

Abstract: This study aims to numerically validate and simulate the fatigue crack growth rate (FCGR), tensile, and fracture toughness properties of T3 and T351 tempered aerospace grade Al 2024 alloy. The extended finite element method (XFEM) coupled with a genetic algorithm (GA) was used to predict FCGR behavior in the framework of an in-house developed MATLAB code. The numerical simulation was performed using three different FCGR models, i.e., Paris, Forman, and Nasgro. The FCGR prediction ability of simulation models was improved and compared further using GA. The Nasgro model with genetically optimized parameters shows the highest FCGR prediction accuracy compared to other FCGR models. The critical stress intensity factor and von Mises stress distribution under static loading were also predicted for C(T) specimen. A similar XFEM approach was used for analyzing the tensile behavior using a center cracked specimen at the gage section of a dog bone-shaped specimen.

Recovery strategies, a posteriori error estimation, and local error indication for second‐order G/XFEM and FEM

Abstract: This article presents a computationally efficient and straightforward to implement a posteriori error estimator for second‐order G/XFEM and FEM approximations. The formulation is based on the recently proposed block‐diagonal Zienkiewicz–Zhu (ZZ‐BD) a posteriori error estimator. The focus is on linear elastic fracture mechanics (LEFM) problems but the proposed error estimator formulation is general and can be adapted to other types of problems such as those involving material interfaces. The proposed ZZ‐BD error estimator is based on a new strategy to recover stress fields from second‐order G/XFEM and FEM approximations. This recovery procedure involves locally weighted L2$$ {L}^2 $$ projections of raw stresses on an approximation space for discontinuous and singular stress fields. The basis functions for these stress approximations are defined using a low‐order partition of unity together with polynomial, discontinuous, and singular recovery enrichment functions. These singular enrichments are provided by the gradient of G/XFEM enrichments for LEFM problems and are, thus, available in most G/XFEM implementations. They also lead to a more computationally efficient stress recovery procedure than the one in the original ZZ‐BD error estimator. The adopted G/XFEM are optimally convergent with the error in the energy norm of 𝒪(h2) . Numerical experiments show that the error of the recovered stress field converges at the same rate as the discretization error. They also show that the effectivity index of the error estimator is close to the unity and that it provides good local error indicators for mesh adaptivity algorithms. Second‐order FEM for elasticity problems is a special case of the G/XFEM adopted here. This is exploited to define a block‐diagonal ZZ error estimator for the FEM as a special case of the proposed estimator for the G/XFEM. This brings the computational efficiency, simplicity, and accuracy of the ZZ‐BD error estimator, originally developed for the G/XFEM, to second‐order standard FEM.

Fatigue Life Prediction of Notched Details Using SWT Model and LEFM-Based Approach

Abstract: The fatigue crack initiation life of unwelded steel components accounts for the majority of the total fatigue life, and the accurate prediction of it is of vital importance. In this study, a numerical model utilizing the extended finite element method (XFEM) and Smith–Watson–Topper (SWT) model is established to predict the fatigue crack initiation life of notched details extensively used in orthotropic steel deck bridges. Using the user subroutine UDMGINI in Abaqus, a new algorithm was proposed to calculate the damage parameter of SWT under high-cycle fatigue loads. The virtual crack-closure technique (VCCT) was introduced to monitor crack propagation. Nineteen tests were performed, and the results were used to validate the proposed algorithm and XFEM model. The simulation results show that the proposed XFEM model with UDMGINI and VCCT can reasonably predict the fatigue lives of the notched specimens within the regime of high-cycle fatigue with a load ratio of 0.1. The error for the prediction of fatigue initiation life ranges from −27.5% to 41.1%, and the prediction of total fatigue life has a good agreement with the experimental results with a scatter factor of around 2.

Implementation into OpenSees of XFEM for Analysis of Crack Propagation in Brittle Materials

Abstract: Fracture propagation simulations by means of the traditional Finite Element Method require progressive remeshing to match the geometry of the discontinuity, which heavily increases the computational effort. To overcome this limitation, methods like the eXtended Finite Element Method (XFEM), in which element nodes are enriched through the medium of Heaviside step function multiplied by nodal shape functions, may be used. The addition of a discontinuous field allows the full crack geometry to be modelled independently of the mesh, eliminating the need to remesh altogether. In this paper OpenSees framework has been used to evaluate crack propagation in brittle materials by means of the XFEM method. Two shell-type XFEM elements have been implemented into OpenSees: a three-node triangular element and a four-node quadrangular element. These elements are an improvement of the elements with drilling degrees of freedom lately suggested by the Authors [6]. The implementation of XFEM elements implied some major modifications directly into OpenSees code to take into account the rise of number of degrees of freedom in the enriched element nodes during the analysis. The developed XFEM elements have been used to evaluate crack propagation into a plane shell subject to monotonically increasing loads. Moreover, with due tuning, the modified XFEM OpenSees code can be used to study also other problems such as material discontinuities, complex geometries and contact problems.

An enriched phase-field method for the efficient simulation of fracture processes

Abstract: Abstract The efficient simulation of complex fracture processes is still a challenging task. In this contribution, an enriched phase-field method for the simulation of 2D fracture processes is presented. It has the potential to drastically reduce computational cost compared to the classical phase-field method (PFM). The method is based on the combination of a phase-field approach with an ansatz transformation for the simulation of fracture processes and an enrichment technique for the displacement field as it is used in the extended finite element method (XFEM) or generalised finite element method (GFEM). This combination allows for the application of significantly coarser meshes than it is possible in PFM while still obtaining accurate solutions. In contrast to classical XFEM / GFEM, the presented method does not require level set techniques or explicit representations of crack geometries, considerably simplifying the simulation of crack initiation, propagation, and coalescence. The efficiency and accuracy of this new method is shown in 2D simulations.

Variable node higher-order XFEM for fracture modeling in orthotropic material

Abstract: A novel computational approach presented in this work to improve the accuracy and efficiency of fracture modeling in an orthotropic material medium. Extended finite element method (XFEM) with higher-order enrichment functions was employed at the different scale mesh topology. The approach combined variable node element concepts for different scale mesh connections and higher-order XFEM for accuracy and completeness of discontinuity domain. The proposed computational methodology was employed with in-house developed MATLAB code. Further, stochastic fracture studies were discussed for reliability of the cracked structures. Few numerical examples with multiple geometrical discontinuities were simulated to check the computational efficiency and accuracy.

Compatible finite element methods for geophysical fluid dynamics

Abstract: This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

Abstract: Wormhole propagation plays a very important role in the product enhancement of oil and gas reservoir. A new combined hybrid mixed finite element method is proposed to solve incompressible wormhole propagation problem with discontinuous Galerkin finite element procedure, in which, the new hybrid mixed finite element algorithm is established for pressure equation, while the discontinuous Galerkin finite element method is considered for concentration equation, and then the porosity function is computed straightly by the approximate value of the concentration. This new combined method can keep local mass balance, meantime it also keeps the boundedness of the porosity. The convergence of the proposed method is analyzed and the optimal error estimate in L2-norm is derived. Finally, by the first-order backward Euler scheme in time, numerical examples are presented to verify the validity of the algorithm and the correctness of the theoretical results.

Fracture toughness and fatigue crack growth behaviour of laser powder bed fusion (LPBF) built Ti-6Al-4V alloy through XFEM

Abstract: Additively manufactured (AMed) components are rapidly gaining popularity over conventional-subtractive manufacturing techniques in aerospace, automobile, biomedical, sports, and electronics industries because of design flexibility, reduced industrial waste, economical and joint free components. Despite appreciable progress in AM technologies, as-built parts pose issues such as residual stresses and micro/macro defects, which affect their mechanical properties. Numerical methods are beneficial to understand materials (Ti-6Al-4V alloy) deformation behaviour under static and cyclic load to avoid trial and error experimentation. In the present work, the fracture and fatigue behaviour of LPBF fabricated Ti6Al4V were analysed using XFEM and validated it with experimental fatigue data reported in the literature. The extended finite element method (XFEM) was efficiently utilized to approximate the crack initiation and propagation in LPBF built Ti-6Al-4V alloy for estimating its fatigue life. The level set method was used to discretise the discontinuity (crack) implicitly in the displacement field. The nodes at crack front are approximated through the crack tip enrichment function and incorporated in the overall shape function along with Heaviside enrichment to capture the crack path. The fracture morphology and fatigue life were modelled through Abaqus and FE-safe commercial packages, respectively. The model was used to predict the cyclic life of LPBF fabricated Ti alloy specimens governed by Basquin's Law. The influence of microstructural characteristics on fracture and fatigue properties of Ti alloy are discussed.

Fatigue Analysis of an Inclined Crack Propagation Problem by the X-FEM Method

Abstract: The extended finite element method (X-FEM) has been used to solve fracture mechanics, problems in materials with various behavior laws (for example, isotropic, orthotropic or piezoelectric materials... For each type of material, it is necessary to obtain “enrichment functions” which model the behavior of the fields of displacement and stresses in the vicinity of the front of crack. In this paper, fatigue crack propagation analysis was modeled, by the extended finite element method X-FEM to evaluate the total energy and strain energy at the angled crack length, and to have the development of the increment time concerning the different values of α which is equal to 15°, 30° and 45, this development has been studied numerically by solving the problem of finite elements by the computer code ABAQUS. Quadratic 4-node elements (CPS4R) were used.

General Finite Element Methods with Special Focus on XFEM

Abstract: This chapter summarizes the principles related to a wide spectrum of mesh-based methods that belong to the category of “general finite element methods (GFEMs)”. They are the most promising advanced CMs in explicitly handling local failure phenomena and meso-scale components (e.g. deformation localization bands, concrete cracks, structural planes, etc.) to give their detailed description using simple “finite element (FE)” mesh. Such a FE mesh may be generated beforehand to discretize the engineering structure concerned, where the deployment and size of FEs are dominated by the structure configuration and the gradient of field functions (e.g. strains and flow rates). The existence and evolution of local failure phenomena and meso-scale components are allocated within enriched elements. The basic variables within enriched element may be interpolated from the correspondent “enriched” nodal variables only (CEM), or from “enriched” nodal variables and basis functions together (XFEM). According to the virtual work or variational principle, the governing equations are established to solve these nodal variables. In this manner, less restraint is imposed on the mesh generation with considerable amount of heterogeneous components, which allows for a great simplification in the pre-process work towards the analysis for complex engineering structures. By the validation example of the 1-D discontinuous bar containing joint, it is clear that the FEM with joint element, the XFEM and the CEM with different enrichments will give identical outcomes. By the engineering example, it is shown that the GFEMs specified in this chapter already possess high ability to handle complex engineering issues, but the enriched sub-models for crack-tips are additionally demanded, which is cumbersome because of the repeated data mapping between the global mesh of arch dam and the local mesh containing crack-tip.

Finite element method

Abstract: The last numerical method we will study is the so-called finite element method. Compared to the finite difference and finite volume method, the finite element method is the most stable numerical scheme which is why it is the most widely used method in computational fluid dynamics. For this method, the computational domain is first split up into small elements. For each element a so-called localized support function is constructed which is a function that is only defined within the respective element. The overall solution sought is obtained by “patching” together the individual solutions. We will illustrate this method using the velocity profile of the Hagen-Poiseuille flow.