Time‐step constraints in transient coupled finite element analysis

In transient finite element analysis, reducing the time‐step size improves the accuracy of the solution. However, a lower bound to the time‐step size exists, below which the solution may exhibit spatial oscillations at the initial stages of the analysis. This numerical ‘shock’ problem may lead to accumulated errors in coupled analyses. To satisfy the non‐oscillatory criterion, a novel analytical approach is presented in this paper to obtain the time‐step constraints using the θ‐method for the transient coupled analysis, including both heat conduction–convection and coupled consolidation analyses. The expressions of the minimum time‐step size for heat conduction–convection problems with both linear and quadratic elements reduce to those applicable to heat conduction problems if the effect of heat convection is not taken into account. For coupled consolidation analysis, time‐step constraints are obtained for three different types of elements, and the one for composite elements matches that in the literature. Finally, recommendations on how to handle the numerical ‘shock’ issues are suggested. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and accuracy of finite element methods for flow acoustics. II: Two‐dimensional effects

This is the second of two articles that focus on the dispersion properties of finite element models for acoustic propagation on mean flows. We consider finite element methods based on linear potential theory in which the acoustic disturbance is modelled by the convected Helmholtz equation, and also those based on a mixed Galbrun formulation in which acoustic pressure and Lagrangian displacement are used as discrete variables. The current paper focuses on the effects of numerical anisotropy which are associated with the orientation of the propagating wave to the mean flow and to the grid axes. Conditions which produce aliasing error in the Helmholtz formulation are of particular interest. The 9‐noded Lagrangian element is shown to be superior to the more commonly used 8‐noded serendipity element. In the case of the Galbrun elements, the current analysis indicates that isotropic meshes generally reduce numerical error of triangular elements and that higher order mixed quadrilaterals are generally less effective than an equivalent mesh of lower order triangles. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Many finite elements exhibit the so‐called ‘volumetric locking’ in the analysis of incompressible or quasi‐incompressible problems.In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective–diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi‐implicit and explicit 2D and 3D non‐linear transient dynamic analysis of an impact problem and a bulk forming process are presented. Copyright © 2004 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.     

Linear finite elements with orthogonal discontinuous basis functions for explicit earthquake ground motion modeling

The dynamic explicit finite element method is commonly used in earthquake ground motion modeling. In this method, the element mass matrix is approximately lumped, which may lead to numerical dispersion. On the other hand, the orthogonal finite element method, based on orthogonal polynomial basis functions, naturally derives a lumped diagonal mass matrix and can be applied to dynamic explicit finite element analysis. In this paper, we propose finite elements based on orthogonal discontinuous basis functions, the element mass matrices of which are lumped without approximation. Orthogonal discontinuous basis functions are used to improve the accuracy and reduce the numerical dispersion in earthquake ground motion modeling. We present a detailed formulation of the 4‐node tetrahedral and 8‐node hexahedral elements. The relationship between the proposed finite elements and conventional finite elements is investigated, and the solutions obtained from the conventional explicit finite element method are compared with analytical solutions to verify the numerical dispersion caused by the lumping approximation. Comparison of solutions obtained with the proposed finite elements to analytical solutions demonstrates the usefulness of the technique. Examples are also presented to illustrate the effectiveness of the proposed method in earthquake ground motion modeling in the actual three‐dimensional crust structure. Copyright © 2010 John Wiley & Sons, Ltd.     

Goal‐oriented adaptive refinement for phase field modeling with finite elements

Phase field modeling is very often performed with the finite‐difference method for equally spaced grids. Typically its solutions are highly non‐homogenous; and, therefore, non‐equally spaced grids with dense meshes at interfaces between different phases and coarse meshes in homogenous regions would be more advantageous with respect to both, efficiency and reliability of the numerical solutions. To this end, in the present work, an adaptive strategy with finite elements for phase field modeling is adopted, where the time step and the grid size are selected on the basis of goal‐oriented error estimation. In order to account for nonlinearity of the variational equations, we introduce a secant form for the dual problem, which for practical purposes is approximated by a tangent form. In a numerical example, we investigate transformation and retransformation for a two‐phase system in a square region subjected to thermal loading. Copyright © 2013 John Wiley & Sons, Ltd.     

The partition of unity finite element method for short wave acoustic propagation on non‐uniform potential flows

A novel numerical method is proposed for modelling time‐harmonic acoustic propagation of short wavelength disturbances on non‐uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non‐uniform. Many wavelengths can be included within a single element leading to ultra‐sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid‐based schemes. Numerical results for lined uniform two‐dimensional ducts and for non‐uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the ‘pollution’ effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated. Copyright © 2005 John Wiley & Sons, Ltd.     

On completeness of shape functions for finite element analysis

Some elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n-dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three-dimensional spaces (e.g. space–time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third-degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher-order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.     

Time continuity in cohesive finite element modeling

We introduce the notion of time continuity for the analysis of cohesive zone interface finite element models. We focus on ‘initially rigid’ models in which an interface is inactive until the traction across it reaches a critical level. We argue that methods in this class are time discontinuous, unless special provision is made for the opposite. Time discontinuity leads to pitfalls in numerical implementations: oscillatory behavior, non‐convergence in time and dependence on nonphysical regularization parameters. These problems arise at least partly from the attempt to extend uniaxial traction–displacement relationships to multiaxial loading. We also argue that any formulation of a time‐continuous functional traction–displacement cohesive model entails encoding the value of the traction components at incipient softening into the model. We exhibit an example of such a model. Most of our numerical experiments concern explicit dynamics. Copyright © 2003 John Wiley & Sons, Ltd.     

Least square-finite element for elasto-static problems. Use of ‘reduced’ integration

A least square based finite element algorithm is developed for some elasto-static problems. In the formulation both stresses and displacements appear as simultaneous variables. In two dimensional (plane) analysis, parabolie isoparametric elements are used. Considerable improvement of performance is obtained with a numerical integration based on 2 × 2 Gauss point distribution over more accurate integration schemes. Reasons for this are presented. The formulation is extended in the section ‘General least square formulation’ to beams and plates with a similar success of ‘reduced’ integration.     

Assumed strain finite element methods for conserving temporal integrations in non‐linear solid dynamics

This paper presents a new class of assumed strain finite elements to use in combination with general energy‐momentum‐conserving time‐stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time. The case of interest corresponds to nearly incompressible material responses, in the fully non‐linear finite strain elastic and elastoplastic ranges. The new elements consider the classical scaling of the deformation gradient with an assumed Jacobian (its determinant) defined locally through a weighted averaging procedure at the element level. The key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or B‐bar operator. The developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta, and the conservation/dissipation law of energy for elastic and inelastic problems, respectively. Care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws, and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration. With these developments at hand, a new general B‐bar operator is introduced that satisfies these conditions. The new operator not only accounts for the spatial interpolations (e.g. bilinear displacements with piece‐wise constant volume) but also depends on the discrete structure of the equations in time. The aforementioned conservation/dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes, unconditionally of the time step size and the material model (elastic or elastoplastic). Different finite elements are considered in this framework, including quadrilateral and triangular elements for plane problems and brick elements for three‐dimensional problems. Several representative numerical simulations are presented involving, in particular, the use of energy‐dissipating momentum‐conserving time‐stepping schemes recently developed by the author and co‐workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements, including these conservation/dissipation properties in time and their locking‐free response in the quasi‐incompressible case. Copyright © 2007 John Wiley & Sons, Ltd.     

Conservation properties of a time FE method—part II: Time‐stepping schemes for non‐linear elastodynamics

In the present paper one‐step implicit integration algorithms for non‐linear elastodynamics are developed. The discretization process rests on Galerkin methods in space and time. In particular, the continuous Galerkin method applied to the Hamiltonian formulation of semidiscrete non‐linear elastodynamics lies at the heart of the time‐stepping schemes. Algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals. We newly introduce the ‘assumed strain method in time’ which enables the design of energy–momentum conserving schemes and which can be interpreted as temporal counterpart of the well‐established assumed strain method for finite elements in space. The numerical examples deal with quasi‐rigid motion as well as large‐strain motion. Copyright © 2001 John Wiley & Sons, Ltd.     

Gate elements at injection locations in numerical simulations of flow through porous media: applications to mold filling

Mold filling in polymer and composite processing is usually modelled as a special case of Darcy flow in porous media. The flow pattern and the time necessary to fill the mold depend on the ‘gate’ locations where resin is injected into the closed mold. In composite manufacturing, these are commonly outlets of small tubes transporting resin from a reservoir and their diameters are several orders of magnitude smaller than the mold dimensions. Similar size issue is also encountered in other applications of flow through porous media, such as oil and water pumping and drilling. Traditionally, these inlets are modelled by pressure or flow rate boundary condition as applied at a node of the finite element mesh that represents the injection gate. The omission of the inlet radius in the model results in a mathematical singularity as the mesh gets refined. The computed pressure or flow field depends on the mesh size and does not converge to the accurate solution, as the finite element mesh is refined. It is possible to deal with this phenomenon by modelling the inlet geometry more accurately but this approach is inefficient, as it requires additional degrees of freedom and, above all, significantly complicates the modelling process if the inlet location is not fixed a priori. This paper presents a more efficient alternate solution. It uses special ‘gate’ elements embedded in the mesh around the injection locations. Instead of adjusting the geometrical modelling of the injection location, the adjacent elements use modified shape functions to accurately model pressure field in the neighbourhood of small radial inlet. The proper pressure field shape‐functions for ‘gate’ elements based on linear finite elements are derived. The implementation in an existing mold filling simulation and how the ‘gate elements’ are automatically selected is described. An example to demonstrate the use of ‘gate’ elements and convergence towards the accurate solution with mesh refinement is presented. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Equivalent localization element for crack band approach to mesh‐sensitivity in microplane model

The paper presents in detail a novel method for finite element analysis of materials undergoing strain‐softening damage based on the crack band concept. The method allows applying complex material models, such as the microplane model for concrete or rock, in finite element calculations with variable finite element sizes not smaller than the localized crack band width (corresponding to the material characteristic length). The method uses special localization elements in which a zone of characteristic size, undergoing strain softening, is coupled with layers (called ‘springs’) which undergo elastic unloading and are normal to the principal stress directions. Because of the coupling of strain‐softening zone with elastic layers, the computations of the microplane model need to be iterated in each finite element in each load step, which increases the computer time. Insensitivity of the proposed method to mesh size is demonstrated by numerical examples. Simulation of various experimental results is shown to give good agreement. Copyright © 2004 John Wiley & Sons, Ltd.     

Dispersion error reduction for acoustic problems using the edge‐based smoothed finite element method (ES‐FEM)

The paper reports a detailed analysis on the numerical dispersion error in solving 2D acoustic problems governed by the Helmholtz equation using the edge‐based smoothed finite element method (ES‐FEM), in comparison with the standard FEM. It is found that the dispersion error of the standard FEM for solving acoustic problems is essentially caused by the ‘overly stiff’ feature of the discrete model. In such an ‘overly stiff’ FEM model, the wave propagates with an artificially higher ‘numerical’ speed, and hence the numerical wave‐number becomes significantly smaller than the actual exact one. Owing to the proper softening effects provided naturally by the edge‐based gradient smoothing operations, the ES‐FEM model, however, behaves much softer than the standard FEM model, leading to the so‐called very ‘close‐to‐exact’ stiffness. Therefore the ES‐FEM can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems. Results of both theoretical and numerical studies will support these important findings. It is shown clearly that the ES‐FEM suits ideally well for solving acoustic problems governed by the Helmholtz equations, because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model. Copyright © 2010 John Wiley & Sons, Ltd.     

A fractal patch test

Element consistency is generally checked using the patch test on an element patch of finite size. This condition may in certain cases be too restrictive, and disqualifies elements that appear to be convergent. A method termed ‘fractal patch test’ is presented, in which the patch size is maintained constant while the distorted mesh is refined. Examples are given for four-node quadrilateral elements used in plane stress and strain analysis, and for plate bending elements.     

MLS‐based variable‐node elements compatible with quadratic interpolation. Part II: application for finite crack element

Two‐dimensional finite ‘crack’ elements for simulation of propagating cracks are developed using the moving least‐square (MLS) approximation. The mapping from the parental domain to the physical element domain is implicitly obtained from MLS approximation, with the shape functions and their derivatives calculated and saved only at the numerical integration points. The MLS‐based variable‐node elements are extended to construct the crack elements, which allow the discontinuity of crack faces and the crack‐tip singularity. The accuracy of the crack elements is checked by calculating the stress intensity factor under mode I loading. The crack elements turn out to be very efficient and accurate for simulating crack propagations, only with the minimal amount of element adjustment and node addition as the crack tip moves. Numerical results and comparison to the results from other works demonstrate the effectiveness and accuracy of the present scheme for the crack elements. Copyright © 2005 John Wiley & Sons, Ltd.     

Isogeometric finite element data structures based on Bézier extraction of T‐splines

We develop finite element data structures for T‐splines based on Bézier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bézier elements are defined in terms of a fixed set of polynomial basis functions, the so‐called Bernstein basis. The Bézier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T‐junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T‐splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.