Adaptive Kronrod–Patterson integration of non‐linear finite element matrices

Efficient simulation of unsaturated moisture flow in porous media is of great importance in many engineering fields. The highly non‐linear character of unsaturated flow typically gives sharp moving moisture fronts during wetting and drying of materials with strong local moisture permeability and capacity variations as result. It is shown that these strong variations conflict with the common preference for low‐order numerical integration in finite element simulations of unsaturated moisture flow: inaccurate numerical integration leads to errors that are often far more important than errors from inappropriate discretization. In response, this article develops adaptive integration, based on nested Kronrod–Patterson–Gauss integration schemes: basically, the integration order is adapted to the locally observed grade of non‐linearity. Adaptive integration is developed based on a standard infiltration problem, and it is demonstrated that serious reductions in the numbers of required integration points and discretization nodes can be obtained, thus significantly increasing computational efficiency. The multi‐dimensional applicability is exemplified with two‐dimensional wetting and drying applications. While developed for finite element unsaturated moisture transfer simulation, adaptive integration is similarly applicable for other non‐linear problems and other discretization methods, and whereas perhaps outperformed by mesh‐adaptive techniques, adaptive integration requires much less implementation and computation. Both techniques can moreover be easily combined. Copyright © 2009 John Wiley & Sons, Ltd.     

Reduced modified quadratures for quadratic membrane finite elements

Reduced integration is frequently used in evaluating the element stiffness matrix of quadratically interpolated finite elements. Typical examples are the serendipity (Q8) and Lagrangian (Q9) membrane finite elements, for which a reduced 2 × 2 Gauss–Legendre integration rule is frequently used, as opposed to full 3 × 3 Gauss–Legendre integration. This ‘softens’ these element, thereby increasing accuracy, albeit at the introduction of spurious zero energy modes on the element level. This is in general not considered problematic for the ‘hourglass’ mode common to Q8 and Q9 elements, since this spurious mode is non‐communicable. The remaining two zero energy modes occurring in the Q9 element are indeed communicable. However, in topology optimization for instance, conditions may arise where the non‐communicable spurious mode associated with the elements becomes activated. To effectively suppress these modes altogether in elements employing quadratic interpolation fields, two modified quadratures are employed herein. For the Q8 and Q9 membrane elements, the respective rules are a five and an eight point rule. As compared to fully integrated elements, the new rules enhance element accuracy due to the introduction of soft, higher‐order deformation modes. A number of standard test problems reveal that element accuracy remains comparable to that of the under‐integrated counterparts. Copyright © 2004 John Wiley & Sons, Ltd.     

Alternative time marching process for BEM and FEM viscoelastic analysis

Based on the weighted residual technique, both Finite Element and Boundary Element alternative procedures for viscoelastic analysis are proposed. After imposing the space approximations, applying the kinematical relations for material and strain velocities at the approximation level, the time integration is carried out using appropriate operators. The Kelvin‐Voigt viscoelastic model is implemented in order to validate the idea. The Newmark β time integral scheme is applied to the Finite Element procedure while the Houbolt scheme is applied to the Boundary Elements, allowing the consideration of dynamic analysis in future works. Copyright © 2001 John Wiley & Sons, Ltd.     

A TORSIONAL SPRING-LIKE BEAM ELEMENT FOR THE DYNAMIC ANALYSIS OF FLEXIBLE MULTIBODY SYSTEMS

A new finite element beam formulation for modelling flexible multibody systems undergoing large rigid-body motion and large deflections is developed. In this formulation, the motion of the ‘nodes’ is referred to a global inertial reference frame. Only Cartesian position co-ordinates are used as degrees of freedom. The beam element is divided into two subelements. The first element is a truss element which gives the axial response. The second element is a torsional spring-like bending element which gives the transverse bending response. D'Alembert principle is directly used to derive the system's equations of motion by invoking the equilibrium, at the nodes, of inertia forces, structural (internal) forces and externally applied forces. Structural forces on a node are calculated from the state of deformation of the elements surrounding that node. Each element has a convected frame which translates and rotates with it. This frame is used to determine the flexible deformations of the element and to extract those deformations from the total element motion. The equations of motion are solved along with constraint equations using a direct iterative integration scheme. Two numerical examples which were presented in earlier literature are solved to demonstrate the features and accuracy of the new method.     

An XFEM method for modeling geometrically elaborate crack propagation in brittle materials

We present a method for simulating quasistatic crack propagation in 2‐D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple yet general and flexible quadrature rule based on the same geometric algorithm. The combination of these methods gives several advantages. First, the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, our integration scheme is straightforward to implement and accurate, without requiring a triangulation that incorporates the new crack edges or the addition of new degrees of freedom to the system. The use of this cutting algorithm and integration rule allows for geometrically complicated domains and complex crack patterns. Copyright © 2011 John Wiley & Sons, Ltd.     

3D dynamics of discrete element systems comprising irregular discrete elements—integration solution for finite rotations in 3D

An algorithm for transient dynamics of discrete element systems comprising a large number of irregular discrete elements in 3D is presented. The algorithm is a natural extension of contact detection, contact interaction and transient dynamics algorithms developed in recent years in the context of discrete element methods and also the combined finite‐discrete element method. It complements the existing algorithmic procedures enabling transient motion including finite rotations of irregular discrete elements in 3D space to be accurately integrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Degenerated shell element with composite implicit time integration scheme for geometric nonlinear analysis

A degenerated shell element with composite implicit time integration scheme is developed in the present paper to solve the geometric nonlinear large deformation and dynamics problems of shell structures. The degenerated shell element is established based on the eight‐node solid element, where the nodal forces, mass matrices, and stiffness matrices are firstly obtained upon virtual velocity principle and then translated to the shell element. The strain field is modified based on the mixed interpolation of tensorial components method to eliminate the shear locking, and the constitutive relation is modified to satisfy the shell assumptions. A simple and practical computational method for nonlinear dynamic response is developed by embedding the composite implicit time integration scheme into the degenerated shell element, where the composite scheme combines the trapezoidal rule with the three‐point backward Euler method. The developed approach can not only keep the momentum and energy conservation and decay the high frequency modes but also lead to a symmetrical stiffness matrix. Numerical results show that the developed degenerated shell element with the composite implicit time integration scheme is capable of solving the geometric nonlinear large deformation and dynamics problems of the shell structures with momentum and energy conservation and/or decay. Copyright © 2015 John Wiley & Sons, Ltd.     

Stabilized finite element methods with reduced integration techniques for miscible displacements in porous media

The objective of this work is to study some techniques to increase computational performance of stabilized finite element simulations of miscible displacements. We propose the use of a reduced integration technique for bilinear quadrilateral elements in the determination of the pressure and concentration fields. We also study the evaluation of pressure gradient (Darcy's velocity) by differentiation at super‐convergent points. Numerical examples are shown to validate our approach, accessing its efficiency and accuracy. Copyright © 2003 John Wiley & Sons, Ltd.     

Quadrature rules for triangular and tetrahedral elements with generalized functions

Quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains. These quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries. Specifically, the integrands considered here consist of a quadratic polynomial multiplied by a Heaviside or Dirac delta function operating on a linear polynomial. This form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function. Exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values. Compared with methods involving explicit integration over subdomains, the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical Jacobian information describing the dependence of the integrals on the nodal level set values. The accuracy of the method is demonstrated for a simple conduction problem with the Neumann and Robin‐type boundary conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

A 3D mortar method for solid mechanics

A version of the mortar method is developed for tying arbitrary dissimilar 3D meshes with a focus on issues related to large deformation solid mechanics. Issues regarding momentum conservation, large deformations, computational efficiency and bending are considered. In particular, a mortar method formulation that is invariant to rigid body rotations is introduced. A scheme is presented for the numerical integration of the mortar surface projection integrals applicable to arbitrary 3D curved dissimilar interfaces. Here, integration need only be performed at problem initialization such that coefficients can be stored and used throughout a quasi‐static time stepping process even for large deformation problems. A degree of freedom reduction scheme exploiting the dual space interpolation method such that direct linear solution techniques can be applied without Lagrange multipliers is proposed. This provided a significant reduction in factorization times. Example problems which touch on the aforementioned solid mechanics related issues are presented. Published in 2003 by John Wiley & Sons, Ltd.     

Analysis of the integration of cohesive elements in regard to utilization of coarse mesh in laminated composite materials

An analysis of the solver convergence difficulties and erroneous results when large cohesive elements are utilized in delamination propagation simulations in laminated composites is presented. Special focus is given to the numerical integration of the cohesive element force vector and stiffness matrix. The magnitude and variation of the integration error are analyzed, and the results show that contrary to statements found elsewhere in the literature, the 2 × 2 point Newton‐Cotes quadrature, commonly used in commercial software, results in large errors and is one of the limiting factors for using larger elements. By reducing the integration error in the damage process zone of the interface, more accurate results can be obtained, and larger elements can be utilized with less iterations, thereby decreasing the computational cost. Copyright © 2014 John Wiley & Sons, Ltd.     

Explicit finite element perfectly matched layer for transient three‐dimensional elastic waves

The use of a perfectly matched layer (PML) model is an efficient approach toward the bounded‐domain modelling of wave propagation on unbounded domains. This paper formulates a three‐dimensional PML for elastic waves by building upon previous work by the author and implements it in a displacement‐based finite element setting. The novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration, thus making it practicable for use in large‐scale three‐dimensional dynamic analyses. An efficient method of calculating the strain terms in the PML is developed in order to take advantage of the lack of the overhead of solving equations at each time step. The PML formulation is studied and validated first for a semi‐infinite bar and then for the classical soil–structure interaction problems of a square flexible footing on a (i) half‐space, (ii) layer on a half‐space and (iii) layer on a rigid base. Numerical results for these problems demonstrate that the PML models produce highly accurate results with small bounded domains and at low computational cost and that these models are long‐time stable, with critical time step sizes similar to those of corresponding fully elastic models. Copyright © 2008 John Wiley & Sons, Ltd.     

Fast integration and weight function blending in the extended finite element method

Two issues in the extended finite element method (XFEM) are addressed: efficient numerical integration of the weak form when the enrichment function is self‐equilibrating and blending of the enrichment. The integration is based on transforming the domain integrals in the weak form into equivalent contour integrals. It is shown that the contour form is computationally more efficient than the domain form, especially when the enrichment function is singular and/or discontinuous. A method for alleviating the errors in the blending elements is also studied. In this method, the enrichment function is pre‐multiplied by a smooth weight function with compact support to allow for a completely smooth transition between enriched and unenriched subdomains. A method for blending step function enrichment with singular enrichments is described. It is also shown that if the enrichment is not shifted properly, the weighted enrichment is equivalent to the standard enrichment. An edge dislocation and a crack problem are used to benchmark the technique; the influence of the variables that parameterize the weight function is analyzed. The resulting method shows both improved accuracy and optimum convergence rates and is easily implemented into existing XFEM codes. Copyright © 2008 John Wiley & Sons, Ltd.     

A stabilized nodally integrated tetrahedral

A stabilized, nodally integrated linear tetrahedral is formulated and analysed. It is well known that linear tetrahedral elements perform poorly in problems with plasticity, nearly incompressible materials, and acute bending. For a variety of reasons, low‐order tetrahedral elements are preferable to quadratic tetrahedral elements; particularly for nonlinear problems. But the severe locking problems of tetrahedrals have forced analysts to employ hexahedral formulations for most nonlinear problems. On the other hand, automatic mesh generation is often not feasible for building many 3D hexahedral meshes. A stabilized, nodally integrated linear tetrahedral is developed and shown to perform very well in problems with plasticity, nearly incompressible materials and acute bending. The formulation is analytically and numerically shown to be stable and optimally convergent for the compressible case provided sufficient smoothness of the exact solution u ∈ C² ∩ (H¹)³. Future work may extend the formulation to the incompressible regime and relax the regularity requirements; nonetheless, the results demonstrate that the method is not susceptible to locking and performs quite well in several standard linear and nonlinear benchmarks. Published in 2006 by John Wiley & Sons, Ltd.     

Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues

The finite element analysis of delamination in laminated composites is addressed using interface elements and an interface damage law. The principles of linear elastic fracture mechanics are indirectly used by equating, in the case of single‐mode delamination, the area underneath the traction/relative displacement curve to the critical energy release rate of the mode under examination. For mixed‐mode delamination an interaction model is used which can fulfil various fracture criteria proposed in the literature. It is then shown that the model can be recast in the framework of a more general damage mechanics theory. Numerical results are presented for the analyses of a double cantilever beam specimen and for a problem involving multiple delamination for which comparisons are made with experimental results. Issues related with the numerical solution of the non‐linear problem of the delamination are discussed, such as the influence of the interface strength on the convergence properties and the final results, the optimal choice of the iterative matrix in the predictor and the number of integration points in the interface elements. Copyright © 2001 John Wiley & Sons, Ltd.     

CONSISTENT LINEARIZATION FOR THE EXACT STRESS UPDATE OF PRANDTL–REUSS NON-HARDENING ELASTOPLASTIC MODELS

This paper presents a consistent algorithm, which combines the advantages of the exact time integration of Prandtl–Reuss elastoplastic models and the quadratic asymptotic convergence of Newton–Raphson iteration strategies. The consistent modulus is evaluated by a full linearization of the exact stress update procedure. Numerical tests for a thin wall tube subjected to combined loads of tension and torsion are performed to illustrate the accuracy and efficiency of the consistently linearized exact stress update algorithm described in the paper. For comparison purpose numerical results of the radial return method are also given.     

A parametric study on the fractal finite element method for two‐dimensional crack problems

The Fractal Finite Element Method for calculating 2D stress intensity factors is modified by making the similarity ratio in the construction of the fractal mesh a variable. A parametric study is then carried out to examine the effects of the similarity ratio, the number of transformation terms, reduced integration and the initial crack opening angle on the quality of the numerical solutions. It is concluded that a large similarity ratio should be used to create the fractal mesh, and that reduced integration and a small initial crack opening angle may be used without producing significant errors in the solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element implementation of a macromolecular viscoplastic polymer model

This paper presents a time‐integration method for a viscoplastic physics‐based polymer model at finite strains. The macromolecular character of the model resides in (i) the viscoplastic law based on a double‐kink molecular mechanism, and (ii) a full chain network model inspired by rubber elasticity to describe the large‐strain orientation hardening. A back stress enters the constitutive model formulation. Essential aspects of a three‐dimensional finite‐element implementation are outlined, the main novelty being in the back stress formulation. The computational efficiency and accuracy of the algorithm are examined in a series of parameter studies. In addition, because a co‐rotational formulation of the constitutive equations is employed using the Jaumann rate in the hypoelastic equation and the back stress evolution equation a detailed analysis of stress oscillations is carried out up to very large strains in simple shear. Subsequently, three‐dimensional FE analyses of compression with friction and instability propagation in tension are used as a means to demonstrate the robustness of the implementation and the potential occurrence of stress oscillations and shear bands in large‐strain analyses. Copyright © 2013 John Wiley & Sons, Ltd.     

The 3-D thermoelastic boundary element method: semi-analytical integration for subparametric triangular elements

In this paper a semi-analytical integration scheme is described which is designed to reduce the errors that can result with the numerical evaluation of integrals with singular integrands. The semi-analytical scheme can be applied to quadratic subparametric triangular elements for use in thermoelastic problems. Established analytical solutions for linear isoparametric triangular elements are combined with standard quadrature techniques to provide an accurate integration scheme for quadratic subparametric triangular elements. The use of subparametric elements provides an efficient means for coupling thermal and elastostatic analyses. It is possible for the same mesh to be employed, with linear isoparametric elements used for thermal analysis and quadratic subparametric elements used for deformation analysis. Numerical tests are performed on simple test problems to demonstrate the advantages of the semi-analytical approach which is shown to be orders of magnitude more accurate than standard quadrature techniques. Moreover, the expected increased accuracy with subparametric elements is also demonstrated. © 1998 John Wiley & Sons, Ltd.     

Numerical comparison of some explicit time integration schemes used in DEM,FEM/DEM and molecular dynamics

Discontinua simulations are becoming an important part of Computational Mechanics to the extent that Computational Mechanics of Discontinua is becoming a separate subdiscipline of Computational Mechanics. Among the most widely used methods of Computational Mechanics of Discontinua are Discrete Element Methods, Molecular Dynamics Methods, Combined Finite‐Discrete Element Methods, DDA, Manifold Methods, etc. The common feature of all these methods is time discretization of the governing equations and the resulting mostly explicit time integration schemes. A wide range of time integration schemes is available in the literature. In this paper a comparative study of some of the most commonly used explicit time integration schemes is made in terms of accuracy, stability and CPU efficiency. The study has been performed using numerical experiments based on a one degree of freedom mass‐spring system. The results are presented as charts that can be used when deciding which scheme to use for a particular discontinua problem. Copyright © 2004 John Wiley & Sons, Ltd.