Automatic generation of stiffness matrices for finite element analysis

Procedures for the automatic generation of symbolic stiffness matrices are presented that greatly reduce the amount of work involved in the construction of finite element stiffness matrices. The related system of programs is presented which completely automates the matrix generation and an example is given.     

The specified boundary stiffness/force SBSF method for finite element subregion analysis

It is difficult to analyse a large, complex structure in sufficient detail to obtain accurate results everywhere. One approach is simply to refine the whole structure model in the regions of interest. Another approach is to identify a subregion of the structure and develop a separate refined model of the subregion. It is difficult to assure accuracy in this subregion model because of uncertainties in specifying boundary conditions and loading from the whole structure model solution. In the current literature, three methods other than whole model refinement have been described to deal with this problem. These are called the specified boundary displacement method, the linear constraint method and the zooming method. This paper describes a new approach to this problem of modelling subregions. This approach uses the stiffnesses and forces from the whole model solution at the nodes on the boundary of the sub-region model. Accurate displacement and stress solutions are obtained with this method as it takes into account the interaction between the new stiffness of the subregion and the rest of the structure. This approach is similar to substructuring; however, the equations outside the subregion are discarded rather than condensed out, which results in much less computation effort. Examples of the application of this method to the problem of a plate with a centre hole in tensile loading are presented. The results compared favourably with the results of the same problem solved using other methods, with significant improvement in accuracy over the specified boundary displacement method. Also presented are some results from design modification of the subregion which illustrate the potential of this method for redesign application.     

A dual homogenization and finite element approach for material characterization of textile composites

A novel procedure for predicting the effective nonlinear elastic moduli of textile composites through a combined approach of the homogenization method and the finite element formulation is presented. The homogenization method is first applied to investigate the meso-microscopic material behavior of a single fiber yarn based on the properties of the constituent phases. The obtained results are compared to existing analytical and experimental results to validate the homogenization method. Very good agreements have been obtained. A unit cell is then built to enclose the characteristic periodic pattern in the textile composites. Various numerical tests such as uni-axial and bi-axial extension and trellising tests are performed by 3D finite element analysis on the unit cell. Characteristic behaviors of force versus displacement are obtained. Meanwhile, trial mechanical elastic constants are imposed on a four-node shell element with the same outer size as the unit cell to match the force–displacement curves. The effective nonlinear mechanical stiffness tensor is thus obtained numerically as functions of elemental strains. The procedure is exemplified on a plain weave glass composite and is validated by comparing to experimental data. Using the proposed approach, the nonlinear behavior of textile composites can be anticipated accurately and efficiently.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

A fast stiffness formulation for finite element analysis of two-dimensional solids

This paper describes and efficient method for computing two-dimensional element stiffness matrices. It is a generalization of the algorithm given by Taylor. ¹Theory is developed for elements which are loaded under plane stress, plane strain or axisymmetric conditions, and two FORTRAN IV subroutines are included. These subroutines, which are presented in a modular form, are applicable to any type of serendipity, Lagrangian or triangular element, and may be incorporated into a user's library. Alternatively, if desired, the relevant code may be embedded directly into existing stiffness generation routines.     

Mechanical analysis of 3-D braided composites by the finite multiphase element method

A finite multiphase element method (FMEM), in which the element comprises more than one kind of material, has been proposed to predict the effective elastic properties of 3-D braided composites. This method is based on the variational principle and our previous geometric model that assumes the existence of different types of unit cells in the three regions in a 3-D braided composite, i.e. the interior, surface and corner. The numerical procedure involved two steps. First, a fine local mesh at the unit cell level is used to analyze the stress/strain of each unit cell. Then, a relatively coarse global mesh is used to obtain the overall responses of the composite at macroscopic level. By using the stress volume averaging method, the effective elastic properties of the composite can be calculated under the prescribed uniform strain boundary conditions. Finally, the predicted stress/strain curves are compared with experimental results, demonstrating the applicability of the FME method.     

Dynamic analysis of fixed cracks in composites by the extended finite element method

This paper is dedicated to simulation of dynamic analysis of fixed cracks in orthotropic media using an extended finite element method. This work is in fact an extension to dynamic problems of the recently developed orthotropic extended finite element method for fracture analysis of composites. In this method, the Heaviside and near-tip enrichment functions are used in the framework of the partition of unity for modeling crack discontinuity and crack-tip singularities within the classical finite element method. In this procedure, elements that include a crack are not required to conform to crack edges. Therefore, mesh generation can be performed without any need to comply to crack edges and the method is capable of modeling the crack propagation without any remeshing. To determine the fracture properties, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated by means of domain separation integral (J^′-integral) method. Results of the proposed method are compared with other available analytical and computational results.     

Anisotropic creep modeling of coated textile membrane using finite element analysis

Coated textile membranes (CTMs) form a class of flexible textile composites undergoing viscoelastic deformation because they consist of a polymeric reinforcement and matrix and are tensioned in service. In most CTMs, woven fabrics are frequently used as a reinforcement structure, causing anisotropic mechanical behavior including time dependent viscoelastic deformation. To describe such anisotropic and nonlinear time dependent deformation, the creep potential with three orthotropic parameters was introduced and incorporated into finite element software through a user material subroutine. The three parameters included in the creep potential were determined by carrying out off-axis coupon creep tests and using various mathematical formulae for the effective creep compliance. To validate the current creep modeling and its implementation in finite element software, off-axis coupon creep tests were re-simulated and compared with the experiments, showing that the present modeling can describe the anisotropic and nonlinear creep deformation of CTM with acceptable accuracy.     

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 SiO₂ 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.     

An h-p- multigrid method for finite element analysis

The finite element method generates solutions to partial differential equations by minimizing a strain energy based functional. Strain energy based techniques for adaptive mesh refinements are not always effective, however. The adaptive refinement technique proposed in this paper uses strain energy but also incorporates advantages from the h- and p- finite element methods, the multigrid method and a Delaunay based mesh generation method. The refinement technique converged rapidly and was numerically efficient when applied to determining stress concentrations around the circular hole of a thick plate under tension.     

An efficient zooming method for finite element analysis

In this paper, an efficient, exact zooming technique is developed which employs static condensation and exact structural reanalysis methods. For a multiple level of zooming such that the successive level of zooming is contained within the prior levels of zoom, repeated application of static condensation will reduce the system to one that is associated only with the degrees-of-freedom (dof) of the original model. Then, application of an exact static reanalysis technique permits the displacements at the dof of the original model that are contained in the final zoomed portion of the structure to be obtained first. Next, the response external to the zoom, as well as the response of additional dof within various levels of zooming, can be computed. With the triangular factor of the stiffness matrix of the original system available, this approach involves only the solution of a system of equations of small order. The proposed method is demonstrated by a numerical example.     

Performance study of the domain decomposition method with direct equation solver for parallel finite element analysis

A parallel performance of the domain decomposition method with directLDU algorithm of condensation and solution is studied. Typical subdomains arising after division of a square domain are considered, and operation count equations for all steps of the numerical procedure are derived. The parallel efficiency model is developed using operation count equations and message passing estimates. It is shown how to achieve interprocessor load balancing by partitioning a domain into unequal subdomains. The evaluation of the parallel efficiency model and performance studies for a square finite element domain are performed on the IBM SP2 computer with 4, 6 and 8 processor nodes. It is found that proper load balancing of the domain decomposition algorithm with direct solution of equation systems provides acceptable parallel efficiency for multiprocessor computers: 95% for the 6-processor configuration and 85% for the 8-processor configuration.     

Stress transfer analysis of unidirectional composites with randomly distributed fibers using finite element method

A three-dimensional representative volume element (RVE) of unidirectional composites with both randomly distributed fibers and periodic geometry was generated using DIGIMAT-FE software. Finite element analysis of the stress transfer mechanisms around a fiber break in the RVE was performed via ABAQUS/Standard. The influences of distance to the broken fiber, fiber/matrix stiffness ratio and fiber volume fraction on the stress transfer process of intact fibers were discussed for the case of perfect fiber/matrix adhesion. The study shows that the nearest fibers and the second nearest fibers share the stress released from the broken fiber. The stress transfer coefficient and the ineffective stress transfer length of the nearest fibers was found to increase with the increasing distance to the broken fiber and the stiffness ratio, while decrease with the increasing fiber volume fraction. However, the trends in the two stress transfer parameters of the second nearest fibers are slightly different from those of the nearest fibers due to the random distribution of other intact fibers.     

The p-version of finite element method for shell analysis

A new quadrature scheme and a family of hierarchical assumed strain elements have been developed to enhance the performance of the displacement-based hierarchical shell elements. Various linear iterative procedures have been examined for their suitability to solve system of equations resulting from hierarchic shell formulations.     

采用等效刚度有限元模型的复合材料机翼颤振分析

采用等效刚度方法,研究了一种适用于机翼初步设计阶段的动力学和颤振分析的结构有限元模型。该方法首先计算不同布局形式的加筋壁板的刚度矩阵,然后将其赋予与加筋壁板平面形状相同的光板（等效板）上,使加筋壁板和等效板具相同的力学性能。该方法的优点是避免了加强筋的有限元建模,从而使有限元模型的复杂程度大大降低,但同时等效刚度结构有限元模型仍能反映机翼加筋壁板的结构特性。以某客机概念方案的机翼为例,建立了反映实际结构详细有限元及其等效刚度有限元模型。计算结果和对比分析表明,两种模型的固有频率、振动模态和颤振分析结果吻合得很好,从而验证了等效刚度方法在机翼结构动力学和颤振分析方面的准确性。由于该方法具有简单快速和准确的优点,可用于机翼初步设计阶段对颤振特性的评估。     

Strength prediction of multi-layer plain weave textile composites using the direct micromechanics method

Previously developed micromechanical methods for stiffness and strength prediction are adapted for analysis of multi-layer plain weave textile composites. Utilizing the direct micromechanics method (DMM) via finite element modeling, three methods are presented: (a) direct simulation of a multi-layer plain weave textile composite; (b) micromechanical analysis of a single layer of interest from the force and moment resultants acting on that layer; and (c) application of the previously developed quadratic stress-gradient failure theory to the layer of interest. In comparison to direct modeling, the other two techniques show only 5% difference over a number of random test cases. Several practical design examples of strength prediction are included to illustrate the importance and accuracy of method implementation.     

A material model for the finite element analysis of metal matrix composites

A finite element based procedure is presented which accounts for micromechanical nonlinear behavior of the matrix material in continuous fiber reinforced composites. The micromechanical model is a periodic hexagonal array of elastic fibers embedded in an elastic-plastic matrix material. This model is used to calculate the overall instantaneous material matrix at material points of a macromechanical finite element model of the structure being analyzed. The procedure is applied to a number of metal matrix composite systems subjected to thermomechanical loads.     

Bounds for element size in a variable stiffness cohesive finite element model

The cohesive finite element method (CFEM) allows explicit modelling of fracture processes. One form of CFEM models integrates cohesive surfaces along all finite element boundaries, facilitating the explicit resolution of arbitrary fracture paths and fracture patterns. This framework also permits explicit account of arbitrary microstructures with multiple length scales, allowing the effects of material heterogeneity, phase morphology, phase size and phase distribution to be quantified. However, use of this form of CFEM with cohesive traction–separation laws with finite initial stiffness imposes two competing requirements on the finite element size. On one hand, an upper bound is needed to ensure that fields within crack‐tip cohesive zones are accurately described. On the other hand, a lower bound is also required to ensure that the discrete model closely approximates the physical problem at hand. Both issues are analysed in this paper within the context of fracture in multi‐phase composite microstructures and a variable stiffness bilinear cohesive model. The resulting criterion for solution convergence is given for meshes with uniform, cross‐triangle elements. A series of calculations is carried out to illustrate the issues discussed and to verify the criterion given. These simulations concern dynamic crack growth in an Al₂O₃ ceramic and in an Al₂O₃/TiB₂ ceramic composite whose phases are modelled as being hyperelastic in constitutive behaviour. Copyright © 2004 John Wiley & Sons, Ltd.