Singular Finite Element Research Articles

A singular finite element of multi-material crack problems under elastic antiplane shear deformation

Multi-material antiplane crack problem is a typical mode III crack problem, of which the symplectic eigen-solutions under polar coordinate system and the stress singularity order at the crack tip have been reported in existing literatures. Based on the existing works, the special solution corresponding to the non-homogeneous boundary condition on crack surfaces is derived in this study, and then a symplectic analytical singular element (SASE) is constructed using the symplectic eigen-solutions and the special solution to deal with the related problems. Combining the proposed SASE with conventional finite elements, the multi-material mode III crack problems can be solved and the numerical results of stress intensity factors (SIF) can be calculated directly. Numerical examples show that the proposed method possesses high solving accuracy as well as high efficiency for the analysis of the discussed problem.

Crack-Tip Singularity Growth Modeling without Remeshing

On the basis of fracture mechanics, a numerical study has been considered in order to model the crack growth without re-meshing. Firstly, we have carried out a numerical comparative investigation on the finite elements for capturing crack-tip singularity. The aim is to numerically assess the values of the stress intensity factors by using different crack-tip finite elements (QPE, DQPE and IQPE). Secondly, we display a new numerical scheme to model the crack propagation without re-meshing. The fundamental idea is to couple the level set method (LSM) with several kinds of singular finite elements to solve the cracks problem: the special elements are employed to evaluate the crack-tip displacement fields, whereas the level sets are utilized to represent the crack location. By the end, a physical problem of crack growth simulations is supplied to show the robustness and versatility of the crack-tip finite element modelling.

Estimation of the local convective heat transfer coefficient in pipe flow using a 2D thermal Quadrupole model and Truncated Singular Value Decomposition

The techniques for solving the Inverse Heat Conduction Problem represent useful tools for designing heat transfer apparatuses. One of their most challenging applications derives from the necessity of catching what happens inside a heat transfer apparatus by monitoring the temperature distribution on the external wall of the device, possibly by means of contactless experimental methodologies. The research presented here deals with the application of a solution strategy of the Inverse Heat Conduction Problem (IHCP) aimed at estimating the local heat transfer coefficient on the internal wall surface of a pipe, under a forced convection problem. The solution strategy, formulated for a 2D model, is based on the Quadrupole Method (QM) coupled to the Truncated Singular Value Decomposition approach, used to cope with the ill-conditioning of the problem. QM presents some advantages over the more classical domain or boundary discretization methods as for instance the fact that, being meshless, brings to a reduction of the computational cost. The analytical model, built under the QM, is validated by means of numerical simulations and the numerical outputs are then used as synthetic data inputs to solve the IHCP. The estimation methodology is also applied to experimental data regarding a forced convection problem in coiled pipes. Moreover, the adopted solution technique is compared to other two well-known and consolidated approaches: Finite Element Method coupled to the Tikhonov Regularization Method and Gaussian Filtering Technique. The comparison highlights that, for the problem here investigated, the Quadrupole Method coupled to the Truncated Singular Value Decomposition and Finite Element Method coupled to the Tikhonov Regularization Method perform better than the Gaussian Filtering Technique when the noise level is low, while, for higher noise level values, their efficiency is almost comparable, as it happens in the considered experimental study case.

A singular finite element technique for calculating continuum damping of Alfvén eigenmodes

Damping due to continuum resonances can be calculated using dissipation-less ideal magnetohydrodynamics provided that the poles due to these resonances are properly treated. We describe a singular finite element technique for calculating the continuum damping of Alfvén waves. A Frobenius expansion is used to determine appropriate finite element basis functions on an inner region surrounding a pole due to the continuum resonance. The location of the pole due to the continuum resonance and mode frequency is calculated iteratively using a Galerkin method. This method is used to find the complex frequency and mode structure of a toroidicity-induced Alfvén eigenmode in a large aspect ratio circular tokamak and is shown to agree closely with a complex contour technique.

A singular element for reissner plate bending problem with V-shaped notches

A novel singular finite element is presented to study Reissner plate bending problem with V-shaped notches. Firstly, the expressions of the first four order asymptotic displacement field around the tip of V-shaped notch are derived systematically using William’s eigenfunction series expansion method. Subsequently, the expressions are used to construct a novel displacement mode singular finite element, which can well depict the characteristic of singular stress field around the tip of V-shaped notch having arbitrary opening angle. Combining with conventional finite elements, the novel element can be applied to solve Reissner plate bending problem with V-shaped notches, and mode I, mode II and mode III stress intensity factors can all be determined directly by the coefficients of the asymptotic expansion terms. Finally, four numerical examples are performed to demonstrate the performance of the present method, and the influences of the size and number of export nodes of the singular element on the calculation of bending stress intensity factors are also discussed. Numerical results show that the singular element method is a practical and effective numerical technique for obtaining directly and synchronously mode I, mode II and mode III stress intensity factors without other numerical techniques, such as extrapolation.

Determination of Stress Intensity Factors in Low Pressure Turbine Rotor Discs

An attention in this paper is focused on the stress analysis and the determination of fracture mechanics parameters in low pressure (LP) turbine rotor discs and on developing analytic expressions for stress intensity factors at the critical location of LP steam turbine disc. Critical locations such as keyway and dovetail area experienced stress concentration leading to crack initiation. Major concerns for the power industry are determining the critical locations with one side and fracture mechanics parameters with the other side. For determination of the critical locations in LP turbine rotor disc conventional finite elements are used here. For this initial crack length and during crack growth it is necessary to determine SIFs. In fatigue crack growth process it is necessary to have analytic formulas for the stress intensity factor. To determine analytic formula for stress intensity factor (SIF) of cracked turbine rotor disc special singular finite elements are used. Using discrete values of SIFs which correspond to various crack lengths analytic formula of SIF in polynomial forms is derived here. For determination of SIF in this paper, combinedJ-integral approach and singular finite elements are used. The interaction of mechanical and thermal effects was correlated in terms of the fracture toughness.

Mixed mode crack growth simulation with/without overloads

In the present paper, mathematical models/procedures are developed in order to investigate the mixed mode crack growth process. The proposed procedures examine the fatigue life estimation and the crack path simulation under either constant amplitude loading or single overload. The fatigue life up to failure is calculated by employing two-parameter driving force model together with the equivalent stress intensity factor. The overload effect is theoretically analyzed by introducing the Wheeler’s retardation parameter. Further, the crack extension under Mode I/II conditions is experimentally investigated for two cracks emanating from two holes in the plate. Then, for the same mixed mode situation with two cracks, the crack growth paths are modeled by using the maximum principal stress criterion and the finite element method. The authors employ analytical and numerical approaches for the stress intensity factor calculations. The numerical investigation is performed by applying quarter-point (Q-P) singular finite elements. The efficiency of proposed procedures is verified through available crack growth data. The different mixed mode calculations are in a good correlation with crack growth observations.

Application of the LU Recombination Method to the FETI-DP Method for Solving Low-Frequency Multiscale Electromagnetic Problems

The LU recombination method is applied within a dual-primal finite element tearing and interconnecting (FETI-DP) framework to solve low-frequency multiscale electromagnetic problems. The FETI-DP method, a robust domain decomposition method, is adopted to deal with the large-scale linear system resulting from finite element modeling of multiscale problems. When frequency decreases or a highly nonuniform mesh is used, the FETI-DP method encounters a low-frequency breakdown problem. The LU recombination method is adopted to identify and eliminate dependent degrees of freedom (DOFs), so that nearly singular finite element matrices are regularized. Since it deals with matrices directly, the LU recombination method is easily combined with different types and orders of basis functions. A nested scheme is adopted to apply LU recombination within the FETI-DP framework. The proposed algorithm is applied to large-scale eddy-current, switched reluctance motor, and logging-while-drilling tool problems. With the LU recombination method, the FETI-DP method is extended down to DC, making it possible to deal with a variety of problems of practical interest.

Fatigue crack growth modeling of attachment lugs

In the present paper, computational procedures for the strength analysis of two cracked lug configurations are proposed. The authors consider the lug with single through-the-thickness crack as well as with single quarter-elliptical corner crack. The stress analysis and the fatigue life evaluations are tackled for both configurations. Numerical approaches based on quarter-point (Q-P) singular finite elements are developed for stress intensity factor calculations. Furthermore, the finite element analysis is applied in order to evaluate the polynomial expression for the stress intensity factor calculation of the lug with one through-the-thickness crack. The residual life estimation of cracked lugs is analyzed by employing the crack growth law based on a two-parameter driving force model. Moreover, the crack growth path for the lug with quarter-elliptical crack is simulated. The reliability of the proposed procedures is verified using experimental fatigue crack growth data. Fatigue life calculations and crack path simulations show a very good agreement with experimental observations.

A Singular Finite Element on the Mixed-Mode Bimaterial Interfacial Cracks

A singular finite element is presented to study the mixed-mode Dugdale-model-based bimaterial interfacial cracks. Firstly, the bimaterial interfacial crack problem is led into the symplectic space, and the symplectic dual equation is obtained and solved analytically. The cohesive stresses of the Dugdale model are treated as special solutions. Subsequently, the analytical solution is employed to develop a novel singular finite element, which depicts accurately the characteristic of displacements and singular stress fields near the crack tip. Finally, combining the singular finite element and conventional finite element method, the length of plastic zone, crack tip opening, and/or sliding displacement can be solved by iteration. Numerical examples are given to illustrate the validity of the present method.

Effect of the thickness on the mixed mode crack front fields

Results pertaining to 3D investigations on the effect of the thickness on the stress fields at the crack front are presented. A 3D finite element analysis is performed using a modified single edge-notched tension specimen configuration, with an inclined crack to include mixed mode I-II. A technique to mesh the crack front (3D) with singular finite elements in ANSYS without using third party software is introduced and used in this study. The effect of the specimen thickness is explicitly investigated for six thicknesses ranging from 1 to 32 mm. In addition, three crack inclination angles, including pure Mode-I, are used to study the effect of mixed-mode I-II fracture. An attempt is made to correlate the extent of a particular stress state along the crack front to thickness. In addition, <TEX>${\sigma}_{zz}/{\nu}({\sigma}_{xx}+{\sigma}_{yy})$</TEX> contours at the crack front are presented as a useful means to analyze the stress state.

A Novel Singular Finite Element on Mixed-Mode Dugdale Model Based Crack

The symplectic dual approach is employed to obtain the analytical solutions of displacements and stresses around the mixed-mode Dugdale crack tip. Based on the analytical solutions, a novel singular finite element is developed to study the problem. The singular finite element can be applied to determine the sizes of crack tip opening/sliding displacement of a mixed-mode Dugdale model. Numerical results obtained by the present method show excellent agreement with the existing analytical solutions.

A novel singular finite element on mixed-mode bimaterial interfacial cracks with arbitrary crack surface tractions

This paper investigates interfacial cracks with arbitrary crack surface tractions. A novel singular finite element which is constructed with the analytical solution around interfacial cracks is presented. Interfacial crack problems can be analyzed numerically using the singular finite element, and Mode I and/or Mode II stress intensity factors can be obtained directly. Unlike other enriched elements for cracks, neither extra unknowns nor transition elements are required. Numerical examples are given to illustrate the validity of present method.

Computation of Mixed Mode Stress Intensity Factor for Parallel Edge Cracks

This paper presents the extensions of newly developed finite element (FE) formulation to evaluate fracture behavior of parallel edge cracks problems. The numerical formulation used Barsoum singular finite elements to compute fracture parameters in two dimensional finite element models subjected to different crack-width ratio and cracks interval ratio. Mixed mode stress intensity factors (SIFs) of parallel edge cracks are computed in extending of FE formulation for pure Mode I formulation proposed by authors. In 2D linear elastic problem under mixed mode condition, the variation of SIF value near crack tips are discussed comprehensively. The newly finite element formulations are resulted with remarkable agreement with energy release rate based method compared to analytical solution available in the literatures.

A novel singular finite element of mixed-mode crack problems with arbitrary crack tractions

A novel singular finite element is presented to study cracked plates with arbitrary traction acting on crack surfaces. Firstly, the analytical solution around crack tips is determined using the symplectic dual approach. Subsequently, the solution is used to develop a novel singular finite element, which depicts accurately the characteristic of singular stresses field near crack tips. And the novel element can be applied to solve cracked plates, and both Mode I and Mode II stress intensity factors can be determined directly and accurately. Lastly, two numerical examples are given to illustrate the present method.

Analysis of the crack growth propagation process under mixed-mode loading

In the present paper, a computational model for crack growth analysis under Mode I/II conditions is formulated. The focus is on two issues – crack path simulation and fatigue life estimation. The finite element method is used together with the maximum principal stress criterion and the crack growth rate equation based on the equivalent stress intensity factor. To determine the mixed-mode stress intensity factors, quarter-point (Q-P) singular finite elements are employed. For verification purposes, a plate with crack emanating from the edge of a hole is examined. The crack path of the plate made of 2024 T3 Al Alloy is investigated experimentally and simulated by using the finite element method with the maximum tangential stress criterion. Then, the validation of the procedure is illustrated by applying the numerical evaluation of the curvilinear crack propagation in the polymethyl methacrylate (PMMA) beam and the Arcan specimen made of Al Alloy for which experimental results are available in the literature. In order to estimate fatigue life up to failure of the plate with crack emanating from the edge of a hole, the polynomial expression is evaluated for the equivalent stress intensity factor using values of stress intensity factors obtained from the finite element analysis. Additionally, the fatigue life up to failure of the Arcan specimen is analyzed for different loading angles and compared with experimental data. Excellent correlations between the computed and experimental results are obtained.

Computation of Stress Intensity Factor for Multiple Cracks Using Singular Finite Element

The simplification of two dimensional approaches in singular finite elements has promoted the method to be used in the formulation of stress intensity factor (SIF) of multiple cracks in finite body. The effect of shielding and amplification are considered in defining the SIF. As been observed, the current available analytical approximations are more restricted to several assumptions. The more accurate and less restricted method has motivated this study. This paper presents the investigation of singular finite elements applied in two dimensional finite element models subjected to different crack-width ratio and cracks interval ratio. The newly finite element formulations are resulted with good agreement with theoretical statement compared to analytical solution. The weak points of presented analytical solution are discussed regards to the influence of crack width ratio and cracks interval ratio.

J-Integral Evaluation in Two Dimensional Interacting Cracks

This paper investigates crack interaction of multiple edge cracks in elastic solid finite body under pure Mode I loading. The cracks are located in parallel to another in 2D plate model. The stress intensity factors (SIFs) are determined based on strain energy release rate. The J-integral path independent is employed to study the interaction between cracks in regards to the effect of crack shielding and amplification of various cracks arrangement. The aim of present work is to test several numerical techniques reported in literature. J-integral approach are applied in 2D ANSYS finite element models subjected to different crack-width ratio (a/w) and cracks interval ratio (a/b). For validation, the results are compared to singular finite element approach and related analytical formulation. The results obtained by these methods are found in good agreement with singular finite element. Some discrepancies between analytical solutions are discussed. Nevertheless, since strain energy release is concern to characterize the near crack tip field, the J-integral method seems to be more applicable and accurate for interacting cracks analysis.

Fatigue Life Estimation of Notched Structural Components

This work considers the analytical/numerical methods and procedures for obtaining the stress intensity factors and for predicting the fatigue crack growth life of notched structural components. Many efforts have been made during the past two decades to evaluate the stress intensity factor for corner cracks and for through cracks emanating from fastener holes. A variety of methods have been used to estimate the stress intensity factor (SIF), values, such as approximate analytical methods, finite element method (FEM), finite element alternating, weight function, photo elasticity and fatigue tests. In this paper the analytic/numerical methods and procedures were used to obtain SIF, and predict the fatigue crack growth life for cracks at attachment lugs. Single through crack in the attachment lug analysis is considered. For this purpose analytic expressions are evaluated for SIF of cracked lug structures. For validation of the analytic stress intensity factors of cracked lugs, FEM with singular finite elements is used. Good agreement between computation and experimental results for fatigue life of aircraft cracked lugs was obtained. To determine crack trajectory of cracked structural components under mixed modes,, conventional singular finite elements and X-FEM are used.

Determination of Second-Order term Coefficients for the Inclined Crack in Orthotropic Plate using Singular Finite Elements

The finite element method with quarter-point crack-tip elements is used and a simple formula for obtaining the coefficients of the second-order terms in the series expansion for near crack tip stresses in orthotropic materials under biaxial loading is presented. This formula is obtained by comparing the variation of the displacements along the crack tip element with the elastic field solution for the crack tip. Numerical examples are given for the validity of the present formulation. The results obtained are compared with the theoretical ones and a good agreement between the two solutions is obtained.