Global Local Finite Element Method Research Articles

Proposal of analysis method of global-local finite element method for finite strain elastic-plastic fracture mechanics analysis

The purpose of this study is to make it possible to perform a large number of paratric analyzes on the position and shape of cracks under finite elasto-plastic deformation. In this report, we propose a one-way coupled analysis method using displacement interpolation between a global finite element method analysis model created without assuming cracks and a local finite element method analysis model assuming cracks. The proposed method is based on the superconvergent patch recovery, and uses the least-squares method to apply the nodal displacements of the global finite element model to the nodes of the local finite element model as displacement boundary conditions. In addition, we will create an merged finite element method analysis model assuming a crack directly in the global finite element method analysis model and identify the application conditions that can be applied to the problem without loss of accuracy through relative comparison by J-integral calculation at the crack tip which is nonlinear fracture mechanics parameter.

Coupled FEM–DBEM approach on multiple crack growth in cryogenic magnet system of nuclear fusion experiment 'Wendelstein 7-X'

At the Max Planck Institute for plasma physics in Greifswald, Germany, the world's largest nuclear fusion experiment of modular stellarator type Wendelstein 7-X has started plasma operation. The hot hydrogen plasma is confined in a plasma vessel by an electromagnetic field generated by 50 non-planar and 20 planar superconducting coils. The superconducting coils are encased in cast stainless steel housings. The coils are bolted onto a central support ring and welded together by so called lateral support elements (LSEs). In this paper, a procedure, based on a global–local finite element method (FEM)–dual boundary element method (DBEM) approach, is developed to simulate the propagation of multiple cracks detected in LSEs and undergoing a fatigue load spectrum. The global stress analysis on the superconducting coils is performed by FEM whereas the sub-modelling approach is adopted to solve the crack propagation in the DBEM environment. The boundary conditions applied on the DBEM submodel are the displacements calculated by the FEM global analysis, in correspondence of the cut surfaces (there are no body forces nor external loads applied on the submodel volume). Two cracks are simultaneously introduced, and a linear elastic fracture mechanics analysis is performed. Results in terms of cracks growth rates and evolving crack shapes are provided, and the residual life of the component is forecast.

Coupled FEM–DBEM approach on multiple crack growth in cryogenic magnet system of nuclear fusion experiment ‘Wendelstein 7‐X’

AbstractAt the Max Planck Institute for plasma physics in Greifswald, Germany, the world's largest nuclear fusion experiment of modular stellarator type Wendelstein 7‐X has started plasma operation. The hot hydrogen plasma is confined in a plasma vessel by an electromagnetic field generated by 50 non‐planar and 20 planar superconducting coils. The superconducting coils are encased in cast stainless steel housings. The coils are bolted onto a central support ring and welded together by so called lateral support elements (LSEs). In this paper, a procedure, based on a global–local finite element method (FEM)–dual boundary element method (DBEM) approach, is developed to simulate the propagation of multiple cracks detected in LSEs and undergoing a fatigue load spectrum. The global stress analysis on the superconducting coils is performed by FEM whereas the sub‐modelling approach is adopted to solve the crack propagation in the DBEM environment. The boundary conditions applied on the DBEM submodel are the displacements calculated by the FEM global analysis, in correspondence of the cut surfaces (there are no body forces nor external loads applied on the submodel volume). Two cracks are simultaneously introduced, and a linear elastic fracture mechanics analysis is performed. Results in terms of cracks growth rates and evolving crack shapes are provided, and the residual life of the component is forecast.

Global–Local Finite Element Analysis of Adhesive Joints and Crack Propagation

Accurately capturing the stress distribution in an adhesive joint of aircraft structures requires discretizing the adhesive layer with a very fine finite element mesh. Because such a simulation requires high computational processing unit time, researchers are looking for alternative methods to simulate adhesive joints of aircraft structures for saving the computational processing unit time. Another high computational processing unit requiring the study of aircraft structures is the evaluation of delamination growth in adhesive joints and crack propagation in brittle materials using cohesive zone modeling along with the very fine finite element mesh for the bulk material. To reduce these computational times, a possible alternative is to use a global–local finite element method. Therefore, both the crack propagation and the characteristics of adhesive joints were studied using a global–local finite element method. Three cases were studied using the proposed global–local finite element method, including 1) an adhesively bonded double cantilever beam, 2) an adhesive lap joint, and 3) a three-point bending test specimen. Using global–local methods, in a crack propagation problem of an adhesively bonded double cantilever beam, more than 80% data storage space and more than 65% computational processing unit time requirement could be saved. In the adhesive lap joints, around 70% data storage space and 70% computational processing unit time requirement could be saved using the global–local method. For the three-point bending test specimen case, more than 90% for both data storage space and computational processing unit time requirement could be saved using the global–local method.

Fracture Analysis of a Curvilinearly Stiffener Panel Subjected to Multiple Combined Loadings

A global–local finite element analysis was performed to study the damage tolerance of curvilinearly stiffened panels, fabricated using the modern additive-manufacturing process that can create so-called unitized structures. In the first step, a buckling analysis of panels was performed to determine whether the panels satisfied the buckling constraint in an undamaged state. In the second step, stress distributions in the panel were analyzed to determine the location of the critical stress under combined shear and compression loadings. Then, a fracture analysis of the curvilinearly stiffened panel with a crack designed at the earlier-obtained location of the critical stress, which was the common location with the maximum magnitude of the principal stresses and von Mises stress, was performed under combined shear and tensile loadings. A mesh-sensitivity analysis was performed to validate the choice of the mesh density near the crack tip. All analyses were performed using the global–local finite element method using MSC.Marc, and the global finite element methods using MSC.Marc and ABAQUS. A negligible difference in results and 94% savings in the CPU time were achieved using the global–local finite element method over the global finite element method by using a mesh density of 8.4 element/mm ahead of the crack tip. To study the influence of different loads on basic modes of fracture, the shear and normal (tensile) loads were varied differently. It was observed that the case with the fixed shear load but variable normal loads, and the case with the fixed normal load but variable shear loads were mode I. Under the maximum combined-loading condition, the largest effective stress-intensity factor was much smaller than the critical-stress-intensity factor.

Analysis of through-the-thickness stress distribution in thick laminate multi-bolt joints using global-local method

The stress distribution surrounding the fastener hole in thick laminate mechanical joints is complex. It is time-consuming to analyze the distribution using finite element method. To accurately and efficiently obtain the stress state around the fastener hole in multi-bolt thick laminate joints, a global-local approach is introduced. In the method, the most seriously damaged zone is 3D modeled by taking the displacement field got from the 2D global model as boundary conditions. Through comparison and analysis there are the following findings: the global-local finite element method is a reliable and efficient way to solve the stress distribution problem; the stress distribution around the fastener hole is quite uneven in through-the-thickness direction, and the stresses of the elements close to the shearing plane are much higher than the stresses of the elements far away from the shearing plane; the out-of-plane stresses introduced by the single-lap joint cannot be ignored due to the delamination failure; the stress state is a useful criterion for further more complex studies involving failure analysis.

Analysis and improvements of global–local enrichments for the Generalized Finite Element Method

Partition of unity methods like the GFEM, rely on closed-form analytical enrichment functions. However, only limited a priori knowledge about the solution is available for many problems of engineering relevance. Control of discretization errors in this class of problems requires mesh refinement which offsets some of the advantages associated with these methods. This has led to the development of the GFEM with global–local enrichments (GFEMgl), which uses ideas of the classical global–local finite element method to numerically build enrichment functions for the GFEM. It involves the solution of local boundary value problems using boundary conditions from the solution of the global problem discretized with a coarse mesh. These local solutions are in turn used to enrich the solution space of the global problem with the help of the partition of unity framework.This paper presents an a priori error estimate accounting for the effect of inexact boundary conditions applied to local problems. Two strategies to control the error of the GFEMgl solution due to these boundary conditions are also presented. The first one is based on the use of a buffer or over-sampling zone in the local problems. The second strategy investigated is based on multiple global–local iterations. Several numerical experiments performed on three-dimensional fracture mechanics problems illustrate the effectiveness of these strategies in controlling the error of the GFEMgl solution. The problems are selected such that the boundary conditions on the local problems range from smooth to singular functions.

Multiscale analysis of stiffness and fracture of nanoparticle-reinforced composites using micromechanics and global–local finite element models

A combined micromechanics analysis and global–local finite element method is proposed to study the interaction of particles and matrix at the nano-scale near a crack tip. An analytical model is used to obtain the effective elastic modulus of nanoparticle-reinforced composites, then a global–local multi-scale finite element model with effective homogeneous material properties is used to study the fracture of a compact tension sample. For SiO 2 particle-reinforced epoxy composites with various volume fractions, the simulation results for effective elastic modulus, fracture toughness, and critical strain energy release rate show good agreement with previously published experimental data. It is demonstrated that the proposed parametric multi-scale model can be used to efficiently study the toughness mechanisms at both the macro and nano-scale.

Fatigue life prediction for radial truck tires using a global-local finite element method

A global-local finite element modeling technique is employed in this paper to predict the fatigue life of radial truck tires. This paper assumes that a flaw exists inside the tire, in the local model. The local model uses an FEM fracture analysis in conjunction with a global-local technique in ABAQUS. A 3D finite element local model calculates the energy release rate at the belt edge. Using the analysis of the local model, a study of the energy release rate is performed in the crack region and used to determine the crack growth rate analysis. The result considers how different driving conditions contribute to the detrimental effects of belt separation in truck tire failure. The calculation of the total mileage on four sizes of radial truck tires has performed on the belt edge separation. The effect of the change of belt width design on the fatigue lifetime of tire belt separation is discussed.

Investigation of the delamination mechanism of the thin film dielectric structure in flip chip packages

As the electronics industry continues its efforts in miniaturizing the integrated circuit (IC), an IC chip with copper/low-k stacked Back End of Line (BEoL) structures has been developed for reducing R– C delay in order to obtain high-speed signal communication. However, its reliability might become a concern owing to the considerably lower adhesive strength, as well as the greater coefficient of thermal expansion (CTE) of the low-k materials. In this paper, the global–local finite element method, specified boundary condition (SBC) method, is employed as a bridge to estimate the impact from package level to the deep submicron BEoL structure of the flip chip package. The results show that the defect in the stacking structure at the center of the silicon has a lower tendency to crack than that at the corner region. In addition, the higher underfill CTE shows the disadvantage of the defect.

Analysis of Cu/Low- k structure under back end of line process

As the progress of the semiconductor process develops to achieve miniaturization and attain better performances for the electronic device, next-generation IC chips with deep sub-micron Cu/low-k stacked structures adopting the fabrication of (dual) damascene are developed to meet the urgent requirements of reducing high RC delay; the purpose of this is to obtain high-speed signal communication. However, due to poor adhesion and intrinsically lower fracture toughness of low-k materials as well as process loading that introduces flaws and delaminations, the phenomenon of crack growth is observed. To investigate the large scale difference problem, such as the back end of line (BEoL) structure to the silicon chip, a special multi-scale finite element simulation technology, global–local finite element method, is used to deal with this issue. The interfacial crack in the BEoL structure is modeled using the global–local technique. The chemical vapor deposition (CVD) process that induced loading to a micro crack in the interface between etch stop layer and metal track layer (ESL/Mx interface) will also be discussed through a statistical factorial design method in order to understand the crack growth phenomena that might occur during the BEoL process.

Analysis and applications of a generalized finite element method with global–local enrichment functions

This paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method. The procedure combines classical global–local finite element method concepts with the partition of unity approach. It involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization. The local solutions are in turn used to enrich the global space using the partition of unity framework. The computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently.The effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper. We also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution.Key aspects of the computational implementation, in particular, the numerical integration of generalized FEM approximations built with global–local enrichment functions, are presented.The method is applied to fracture mechanics problems with multiple cracks in the domain. Our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems. Our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global–local enrichment functions.

Reliability of interfacial adhesion in a multi-level copper/low- k interconnect structure

As the electronics industry continues its efforts in miniaturizing the integrated circuit (IC), an IC chip with copper/low- k stacked back end of line (BEoL) structures has been developed for reducing RC delay in order to obtain high-speed signal communication. However, its reliability might become a concern due to the considerably lower adhesive strength as well as greater coefficient of thermal expansion (CTE) of the low- k materials. In this paper, the four-point bending technique is used to quantify the adhesion energy ( G C). The global local finite element method, including the specified boundary condition (SBC) method and the multi-point constrain (MPC) method, is validated by the four-point bending experiment and is employed as a bridge to estimate the impact from package level to the deep sub-micron BEoL structure of the plastic ball grid array (PBGA) package. A crack driving force of 1.8 J/m 2 acting on the BEoL structure is obtained when the curing process of the moulding compound is performed.

Stress analysis of ply drop-off in composite structures

A combination of analytical and numerical method is employed to solve the ply drop-off problem. The singular stress fields in the drop-off region are characterized by using the eigenfunction expansion method. A global element capable of capturing the singular behavior is developed and incorporated into a general finite element analysis to provide an overall accurate solution of the stress fields in the ply drop-off region. This facilitates the use of a coarse mesh at the ply drop-off location even there are singular fields. The validation of the global–local finite element method has been carried out with the help of numerical examples.

104 Global Local 有限要素法による歯車の応力解析 : 歯面膜要素の適用

When the finite element method is applied to the stress analysis of meshing tooth pair, a large difference between dimensions of tooth flank modification and finite elements should be one of the main issues. In the present study, as a solution to the issue, a new element for representing boundary condition at contact area, a tooth flank film element, is proposed. This new element can include particular information on a tooth flank; e. g., tooth modification, geometrical overlap, and radius of curvature, with general information included in a finite element; e. g., coordinate, and displacement. The tooth flank film element enables the boundary condition at contact area to be given as distributed load. Meanwhile, Global Local Finite Element Method, in which a classical analytical solution is combined with finite element solutions on the basis of energy principle, is also applied to the stress analysis of meshing tooth pair. Using the proposed analytical method, the contact problem of semi-infinite body and sphere is analyzed. As a result, validity of the method can be confirmed. Furthermore, the application of the proposed method can also induce reasonable results in the stress analysis of meshing spur gear pair.

Fracture analysis of interfacial crack by global-local finite element

An eigenfunction expansion is used to formulate the global element on the crack tip. The global-local finite element method employs both conventional finite element and classical Rayleigh-Ritz kinematic approach. The hybrid Ritz method not only preserves the finite element modelling capability but adds the advantage of using prior information regarding the anticipate behaviour of the particular problem. Thus, it is able to achieve better accuracy with fewer elements in comparison with conventional finite element. Several examples relative to crack problems are presented to demonstrate the global-local finite element method.

Postbuckling of Delaminated Composites Under Compressive Loads Using Global-Local Approach

A global-local finite element method is developed to analyze postbuckling behaviors of composite laminates with multiple delaminations. In the delaminated zone, layerwise element are used whereas in the undelaminated zone first order beam bending elements are used for the computational efficiency. A transition element are used between global and local zone interface. A postprocess method is utilized to convert the layerwise variables to those of first order shear model. By introducing this novel matching idea, the laminated composites with multiple delaminations can be analyzed without significant increase of memory size. The present global-local approach and layerwise models are used to solve multiple delamination problems. The postbuckling buckling behaviors of the present global-local method agree well with other known results.

Scattering of Lamb waves from a rivet hole with edge cracks

Lamb waves propagating in an infinite plate containing a circular hole, with or without edge cracks, are investigated both theoretically and experimentally in this paper. The theoretical analysis is accomplished by means of a hybrid method called the global local finite element method, in which a bounded region enclosing the hole and the cracks is discretized into finite elements, while the field external to this region is represented analytically in terms of Lamb wave modes. The scattered field is calculated through the application of the boundary conditions at the interface between the discretized region and the unbounded exterior. In the experiments the incident Lamb wave of a specific mode is generated by means of a wedge transducer and the waves scattered by the hole is recorded in conventional contact type transducers located on the plate surface. The measured time histories and amplitude spectra of the transmitted and reflected waves are compared with those calculated from the hybrid model. The agreement between the theoretical and experimental results is found to be excellent in the cases considered. Application of the technique to non-destructive evaluation (NDE) of corrosion and fatigue induced defects in aging structural components is discussed.

Gear Stress Analysis by the Hybrid Finite Element Method. Deviation Technique for the Global Local Finite Element Method.

A new numerical technique using the global local finite element method (GLFEM) has been proposed for analyzing a gear subjected to a single concentrated load. The GLFEM is a kind of deviational technique which combines the conventional finite element solutions and the analytical solution like Kelvin's one. One of the most interesting features of this method is considerable reduction in the data required in the modeling process in comparison to that based only on conventional finite elements. Clearly, the nature of the solution and the degree of approximation depend on the characteristics of the element equations. In this study, although there are many ways of formulation, the deviation technique and hybrid method combining the domain type and boundary type solutions are especially used for formulating the properties of elements. Then the GLFEM formulated in this study is applied to the gear tooth stress analysis to show the validity of this technique.

Bifurcation Buckling Analysis of Delaminated Composites Using Global-Local Approach

A global-local finite element method to analyze multiple delamination in composite laminates has been developed. To enhance computational efficiency, layerwise elements are used in the delaminated zone and first order beam bending elements are used in the undelaminated zone. A transition element is used between global and local zones. A postprocess method is utilized to convert the layerwise variables to those of first order shear model. By introducing this novel matching scheme, the laminated composites with multiple delaminations can be analyzed without a significant increase in memory capacity. When applied to multiple delamination problems, the buckling predictions of the the global-local method agree well with other known results.