Tensor objects in finite element programming

This paper describes a novel programming tool, nDarray, which is designed using an Object Oriented Paradigm (OOP) and implemented in the C++ programming language. Finite element equations, represented in terms of multidimensional tensors are easily manipulated and programmed. The usual matrix form of the finite element equations are traditionally coded in FORTRAN, which makes it difficult to build and maintain complex program systems. Multidimensional data systems and their implementation details are seldom transparent and thus not easily dealt with and usually avoided. On the other hand, OOP together with efficient programming in C++ allows building new concrete data types, namely tensors of any order, thus hiding the lower level implementation details. These concrete data types prove to be quite useful in implementing complicated tensorial formulae associated with the numerical solution of various elastic and elastoplastic problems in solid mechanics. They permit implementing complex nonlinear continuum mechanics theories in an orderly manner. Ease of use and the immediacy of the nDarray programming tool in constitutive driver programming and in building finite element classes will be shown. © 1998 John Wiley & Sons, Ltd.     

A finite element method for stability problems in finite elasticity

In this paper a finite element method is developed to treat stability problems in finite elasticity. For this purpose the constitutive equations are formulated in principal stretches which allows a general representation of the derivatives of the strain energy function with respect to the principal stretches. These results can then be used to derive an efficient numerical scheme for the computation of singular points.     

Stress‐based finite element analysis of plane plasticity problems

A stress‐based model of the finite element method is evolved for two‐dimensional quasi‐static plasticity problems. The self‐equilibrating fields of stresses are constructed by means of the Airy stress function, which is approximated by three types of elements: the Bogner–Fox–Schmit rectangle, the Hsieh–Clough–Tocher triangle and its reduced variant. Traction boundary conditions are imposed by the use of the Lagrange multiplier method which gives the possibility of calculation of displacements for boundary points. The concept of multi‐point‐constraints elements is applied in order to facilitate the application of this technique. The iterative algorithm, analogous to the closest‐point‐projection method commonly used in the displacement‐based finite element model, is proposed for solving non‐linear equations for each load increment. Two numerical examples with stress‐ and displacement‐controlled load are considered. The results are compared with those obtained by the displacement model of FEM. Bounds for limit loads are obtained. Copyright © 1999 John Wiley & Sons, Ltd.     

A novel three‐dimensional contact finite element based on smooth pressure interpolations

This article proposes a new three‐dimensional contact finite element which employs continuous and weakly coupled pressure interpolations on each of the interacting boundaries. The resulting formulation circumvents the geometric bias of one‐pass methods, as well as the surface locking of traditional two‐pass node‐on‐surface methods. A Lagrange multiplier implementation of the proposed element is validated for frictionless quasi‐static contact by a series of numerical simulations. Published in 2001 by John Wiley & Sons, Ltd.     

A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

This paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two‐dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g., at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. Copyright © 2014 John Wiley & Sons, Ltd.     

CLP(ℛ) as a general finite element model definition language

There are today as many finite element input languages as there are finite element programs. These input languages were developed to serve one purpose: to provide the analyst with the means to convey the necessary analysis and design criteria to specific finite element programs. They were not developed as general finite element model-definition languages, and as a result their syntax and semantics are often unnatural and contrived. In this work we do not invent a new input language. Instead, we propose and evaluate the Contraint Logic Programming language CLP(?) as a general finite element model-definition and input language.     

Uncertainty analysis of elastostatic problems incorporating a new hybrid stochastic-spectral finite element method

This article aims to present a combination of stochastic finite element and spectral finite element methods as a new numerical tool for uncertainty quantification. One of the well-established numerical methods for reliability analysis of engineering systems is the stochastic finite element method. In this article, a commonly used version of the stochastic finite element method is combined with the spectral finite element method. Furthermore, the spectral finite element method is a numerical method employing special orthogonal polynomials (e.g., Lobatto) and quadrature schemes (e.g., Gauss-Lobatto-Legendre), leading to suitable accuracy, and much less domain discretization with excellent convergence as well. The proposed method of this article is a hybrid method utilizing efficiencies of both methods for analysis of stochastically linear elastostatic problems. Moreover, a spectral finite element method is proposed for numerical solution of a Fredholm integral equation followed by the present method, to provide further efficiencies to accelerate stochastic computations. Numerical examples indicate the efficiency and accuracy of the proposed method.     

Variational approach to the analysis of steady‐state thermo‐elastic wear regimes

A transient wear process on frictional interface of two thermo‐elastic bodies in a relative steady sliding motion induces shape evolution of contact interface and tends to a steady state for which the wear process occurs at fixed contact stress and strain distribution. The temperature field generated by frictional and wear dissipation on the contact surface is assumed to reach a steady state. This state is assumed to correspond to minimum of the wear dissipation power and the temperature field corresponds to maximum of the heat entropy production. The stationarity conditions of the response functionals provide the contact pressure distribution and the corresponding temperature field. The present approach extends the authors previous analyses of optimal or steady‐state contact shapes by accounting for coupled wear and thermal distortion effects. The wear rule is assumed as a non‐linear relation of wear rate to shear stress and relative sliding velocity. The analysis of disk and drum brakes is presented with account for thermal distortion effect. It is shown that the contact shape in a steady thermo‐elastic state essentially differs from that specified for mechanical loading with neglect of thermal effects. The thermal instability regimes are not considered in the paper. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient indirect boundary element technique for multi‐frequency acoustic analysis

An efficient indirect boundary element solution procedure for the analysis of multi‐frequency acoustic problems is developed by incorporating techniques that improve the efficiency of the integration and matrix solution phases of the computing process. The integration phase is made efficient by computing the system matrices at few predetermined key frequencies only and then evaluating the matrices at other intermediate frequencies by quadratic interpolation. The matrix solution process is made efficient by iterating the solutions using the factored form of the key frequency matrices. The effectiveness of the present development is confirmed by solving a number of example problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Error bounds for the reliability index in finite element reliability analysis

This work presents an extension of the goal‐oriented error estimation techniques to the reliability analysis of a linear elastic structure. We use a first‐order reliability method in conjunction with a finite element analysis (FEA) to compute the failure probability of the structure. In such a situation the output of interest that is computed from the FEA is the reliability index β. The accuracy of this output, and thus of the reliability analysis, depends, in particular, on the accuracy of the FEA. In this paper, upper and lower bounds of the reliability index are proposed, as well as simple bounds of the failure probability. An application to linear fracture mechanics is presented. Copyright © 2011 John Wiley & Sons, Ltd.     

An over‐deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis

An over‐deterministic method has been employed for calculating the stress intensity factors (SIFs) as well as the coefficients of the higher‐order terms in the Williams series expansions in cracked bodies, using the conventional finite element analysis. For a large number of nodes around the crack tip, an over‐determined set of simultaneous linear equations is obtained, and using the fundamental concepts of the least‐squares method, the coefficients of the Williams expansion can be calculated for pure mode I, pure mode II and mixed mode I/II conditions. A convergence study has been conducted to examine the effects of the number of nodes used, the number of terms in Williams expansion and the distance of the selected nodes from the crack tip, on the accuracy of the results. It is shown that the simple method presented in this paper, yields accurate results even for coarse finite element meshes or in the absence of singular elements. The accuracy of SIFs and the coefficients of higher‐order terms are validated by using the available results in the literature.     

3D‐FEM formulations of limit analysis methods for porous pressure‐sensitive materials

The first purpose of this paper is the numerical formulation of the three general limit analysis methods for problems involving pressure‐sensitive materials, that is, the static, classic, and mixed kinematic methods applied to problems with Drucker–Prager, Mises–Schleicher, or Green materials. In each case, quadratic or rotated quadratic cone programming is considered to solve the final optimization problems, leading to original and efficient numerical formulations. As a second purpose, the resulting codes are applied to non‐classic 3D problems, that is, the Gurson‐like hollow sphere problem with these materials as matrices. To this end are first presented the 3D finite element implementations of the static and kinematic classic methods of limit analysis together with a mixed method formulated to give also a purely kinematic result. Discontinuous stress and velocity fields are included in the analysis. The static and the two kinematic approaches are compared afterwards in the hydrostatic loading case whose exact solution is known for the three cases of matrix. Then, the static and the mixed approaches are used to assess the available approximate criteria for porous Drucker–Prager, Mises–Schleicher, and Green materials. Copyright © 2013 John Wiley & Sons, Ltd.     

Weighted integral method in multi-dimensional stochastic finite element analysis

This paper, using the weighted integral method, proposes a new stochastic finite element method for estimating the response variability of multi-dimensional stochastic systems. Young's modules is considered to have spatial variation and is idealized as a multi-dimensional stochastic field. An essential feature of the proposed method is that the continuous stochastic field is rigorously taken care of by means of weighted integrations to construct element stiffness matrices, as the results, the issue involving the stochastic field is transformed into a problem involving only a few random variables. This may lead to substantial improvement in computational efficiency. Numerical examples show that the proposed SFEM is concluded as an efficient and accurate method.     

Elasto-plastic finite element analysis of a crack in an infinite plate

The elastic support method was recently developed to simulate the effects of unbounded solids in the finite element analysis of stresses and displacements. The method eliminates all the computational disadvantages encountered in the use of `infinite' elements or coupled finite element boundary element methods while retaining all the computational advantages of the finite element method. In this paper, the method is extended to the elasto-plastic analysis of fracture in infinite solids by using the load increment approach and including the effects of strain hardening. Numerical tests and parametric study are conducted by analysing a straight crack in an infinite plate. Present results for J integrals and plastified zones are compared, respectively, with analytical solutions and available results obtained by using the body force method. The agreement between the results is found to be very good even if the truncation boundary of the finite element model is located very close to the crack tip or the plastified zone.     

Nonlinear finite element analysis of masonry wall strengthened with CFRP strips

In many experimental studies, it has been proved that unreinforced masonry (URM) brick walls have high strength against lateral forces acting in plane. However, out-of-plane strength of URM brick walls against lateral forces has found to be quite low. According to the experiences that were obtained from the major earthquakes, the low out-of-plane performance of URM brick walls resulted in excessive loss of human lives during an earthquake, hence the strengthening of URM brick walls with CFRP strips has been appeared to be a very important subject. However, very limited literature has been found. Especially, the data obtained from experimental studies must be increased for the true understanding of the behavior of strengthened brick walls under out-of-plane lateral forces. However, in most cases, this procedure required large number of expensive experiments. At this stage, numerical analysis can be an appropriate choice, thus in this paper a finite element model is presented for modeling URM brick walls that are strengthened with CFRP strips. The numerical results are compared with the experimental ones and consistent results are obtained from the finite element model. General purpose finite element analysis software ANSYS is used throughout this study. Contact elements are used along the masonry wall–CFRP strip interfaces for the investigation of the stress distribution and load – strain behavior.     

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.     

Application of the unsymmetric Lanczos method to radionuclide decay‐chain transport in porous media

Using either a finite element or finite‐difference method to solve radionuclide waste transport problems involving advection, dispersion, decay and transformation simultaneously is numerically complicated. Most applications have used only one‐dimensional transport and extending this approach to higher‐dimensional problems is a formidable task. This paper uses the ULR method to reduce the size of the semi‐discretized linear system and enables this extension. The development of this application includes a new method for choosing a common starting vector for all species in order to ensure the ULR method is convergent. The results of numerical simulations of application of this method to practical field contaminant transport problems show the efficiency of the method. Copyright © 1999 John Wiley & Sons, Ltd.     

A methodology for adaptive finite element analysis: Towards an integrated computational environment

This work introduces a methodology for self-adaptive numerical procedures, which relies on the various components of an integrated, object-oriented, computational environment involving pre-, analysis, and post-processing modules. A basic platform for numerical experiments and further development is provided, which allows implementation of new elements/error estimators and sensitivity analysis. A general implementation of the Superconvergent Patch Recovery (SPR) and the recently proposed Recovery by Equilibrium in Patches (REP) is presented. Both SPR and REP are compared and used for error estimation and for guiding the adaptive remeshing process. Moreover, the SPR is extended for calculating sensitivity quantities of first and higher orders. The mesh (re-)generation process is accomplished by means of modern methods combining quadtree and Delaunay triangulation techniques. Surface mesh generation in arbitrary domains is performed automatically (i.e. with no user intervention) during the self-adaptive analysis using either quadrilateral or triangular elements. These ideas are implemented in the Finite Element System Technology in Adaptivity (FESTA) software. The effectiveness and versatility of FESTA are demonstrated by representative numerical examples illustrating the interconnections among finite element analysis, recovery procedures, error estimation/adaptivity and automatic mesh generation.     

On stability and reflection‐transmission analysis of the bipenalty method in contact‐impact problems: A one‐dimensional,homogeneous case study

The stability and reflection‐transmission properties of the bipenalty method are studied in application to explicit finite element analysis of one‐dimensional contact‐impact problems. It is known that the standard penalty method, where an additional stiffness term corresponding to contact boundary conditions is applied, attacks the stability limit of finite element model. Generally, the critical time step size rapidly decreases with increasing penalty stiffness. Recent comprehensive studies have shown that the so‐called bipenalty technique, using mass penalty together with standard stiffness penalty, preserves the critical time step size associated to contact‐free bodies. In this paper, the influence of the penalty ratio (ratio of stiffness and mass penalty parameters) on stability and reflection‐transmission properties in one‐dimensional contact‐impact problems using the same material and mesh size for both domains is studied. The paper closes with numerical examples, which demonstrate the stability and reflection‐transmission behavior of the bipenalty method in one‐dimensional contact‐impact and wave propagation problems of homogeneous materials.