Showing papers in "International Journal for Numerical Methods in Engineering in 2000"

Analysis of functionally graded plates

Abstract: Theoretical formulation, Navier's solutions of rectangular plates, and finite element models based on the third-order shear deformation plate theory are presented for the analysis of through-thickness functionally graded plates. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The formulation accounts for the thermomechanical coupling, time dependency, and the von Karman-type geometric non-linearity. Numerical results of the linear third-order theory and non-linear first-order theory are presented to show the effect of the material distribution on the deflections and stresses. Copyright © 2000 John Wiley & Sons, Ltd.

Extended finite element method for three-dimensional crack modelling

Abstract: An extended finite element method (X-FEM) for three-dimensional crack modelling is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three-dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.

Arbitrary branched and intersecting cracks with the eXtended Finite Element Method

Abstract: SUMMARY Extensions of a new technique for the nite element modelling of cracks with multiple branches, multiple holes and cracks emanating from holes are presented. This extended nite element method (X-FEM) allows the representation of crack discontinuities and voids independently of the mesh. A standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method. A methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed. Computation of the stress intensity factors (SIF) in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique. Copyright ? 2000 John Wiley & Sons, Ltd.

Subdivision surfaces: a new paradigm for thin-shell finite-element analysis

Abstract: We develop a new paradigm for thin-shell finite-element analysis based on the use of subdivision surfaces for (i) describing the geometry of the shell in its undeformed configuration, and (ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the Kirchhoff–Love theory of thin shells. The particular subdivision strategy adopted here is Loop's scheme, with extensions such as required to account for creases and displacement boundary conditions. The displacement fields obtained by subdivision are H2 and, consequently, have a finite Kirchhoff–Love energy. The resulting finite elements contain three nodes and element integrals are computed by a one-point quadrature. The displacement field of the shell is interpolated from nodal displacements only. In particular, no nodal rotations are used in the interpolation. The interpolation scheme induced by subdivision is non-local, i.e. the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements. However, the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface. Numerical tests, including the Belytschko et al. [10] obstacle course of benchmark problems, demonstrate the high accuracy and optimal convergence of the method.

Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations

Abstract: Smooth particle hydrodynamics (SPH) is a robust and conceptually simple method which suffers from unsatisfactory performance due to lack of consistency. The kernel function can be corrected to enforce the consistency conditions and improve the accuracy. For simplicity in this paper the SPH method with the corrected kernel is referred to as corrected smooth particle hydrodynamics (CSPH). The numerical solutions of CSPH can be further improved by introducing an integration correction which also enables the method to pass patch tests. It is also shown that the nodal integration of this corrected SPH method suffers from spurious singular modes. This spatial instability results from under integration of the weak form, and it is treated by a least-squares stabilization procedure which is discussed in detail in Section 4. The effects of the stabilization and improvement in the accuracy are illustrated via examples. Further, the application of CSPH method to metal-forming simulations is discussed by formulating the governing equation associated with the process. Finally, the numerical examples showing the effectiveness of the method in simulating metal-forming problems are presented. Copyright © 2000 John Wiley & Sons, Ltd.

Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems

Abstract: The purpose of this work is twofold. First, we demonstrate analytically that the classical Newmark family as well as related integration algorithms are variational in the sense of the Veselov formulation of discrete mechanics. Such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior. This analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well. Second, we develop algorithms for mechanical systems with forcing, and in particular, for dissipative systems. In this case, we develop integrators that are based on a discretization of the Lagrange d'Alembert principle as well as on a variational formulation of dissipation. It is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run.

A unified stability analysis of meshless particle methods

Abstract: A uni ed stability analysis of meshless methods with Eulerian and Lagrangian kernels is presented. Three types of instabilities were identi ed in one dimension: an instability due to rank de ciency, a tensile instability and a material instability which is also found in continua. The stability of particle methods with Eulerian and Lagrangian kernels is markedly di erent: Lagrangian kernels do not exhibit the tensile instability. In both kernels, the instability due to rank de ciency can be suppressed by stress points. In two dimensions the stabilizing e ect of stress points is dependent on their locations. It was found that the best approach to stable particle discretizations is to use Lagrangian kernels with stress points. The stability of the least-squares stabilization was also studied. Copyright ? 2000 John Wiley & Sons, Ltd.

The generalized finite element method: an example of its implementation and illustration of its performance

Abstract: The generalized finite element method (GFEM) was introduced in Reference 1 as a combination of the standard FEM and the partition of unity method The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions) The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation The essential boundary conditions can be imposed exactly as in the standard FEM Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM Copyright © 2000 John Wiley & Sons, Ltd

Numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms

Abstract: Previous studies by the authors and their co-workers show that the structure of equations representing shape Memory Alloy (SMA) constitutive behaviour can be very similar to those of rate-independent plasticity models. For example, the Boyd–Lagoudas polynomial hardening model has a stress-elastic strain constitutive relation that includes the transformation strain as an internal state variable, a transformation function determining the onset of phase transformation, and an evolution equation for the transformation strain. Such a structure allows techniques used in rate-independent elastoplastic behaviour to be directly applicable to SMAs. In this paper, a comprehensive study on the numerical implementation of SMA thermomechanical constitutive response using return mapping (elastic predictor-transformation corrector) algorithms is presented. The closest point projection return mapping algorithm which is an implicit scheme is given special attention together with the convex cutting plane return mapping algorithm, an explicit scheme already presented in an earlier work. The closest point algorithm involves relatively large number of tensorial operations than the cutting plane algorithm besides the evaluation of the gradient of the transformation tensor in the flow rule and the inversion of the algorithmic tangent tensor. A unified thermomechanical constitutive model, which does not take into account reorientation of martensitic variants but unifies several of the existing SMA constitutive models, is used for implementation. Remarks on numerical accuracy of both algorithms are given, and it is concluded that both algorithms are applicable for this class of SMA constitutive models and preference can only be given based on the computational cost. Copyright © 2000 John Wiley & Sons, Ltd.

A survey of recent shell finite elements

Abstract: Since the mid-1960s when the forms of curved shell finite elements were originated, including those pioneered by Professor Gallagher, the published literature on the subject has grown extensively. The first two present authors and Liaw presented a survey of such literature in 1990 in this journal. Professor Gallagher maintained an active interest in this subject during his entire academic career, publishing milestone research works and providing periodic reviews of the literature. In this paper, we endeavor to summarize the important literature on shell finite elements over the past 15 years. It is hoped that this will be a befitting tribute to the pioneering achievements and sustained legacy of our beloved Professor Gallagher in the area of shell finite elements. This survey includes: the degenerated shell approach; stress-resultant-based formulations and Cosserat surface approach; reduced integration with stabilization; incompatible modes approach; enhanced strain formulations; 3-D elasticity elements; drilling d.o.f. elements; co-rotational approach; and higher-order theories for composites. Copyright © 2000 John Wiley & Sons, Ltd.

Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II—A framework for volume mesh optimization and the condition number of the Jacobian matrix

Abstract: Three-dimensional unstructured tetrahedral and hexahedral finite element mesh optimization is studied from a theoretical perspective and by computer experiments to determine what objective functions are most effective in attaining valid, high quality meshes. The approach uses matrices and matrix norms to extend the work in Part I to build suitable 3D objective functions. Because certain matrix norm identities which hold for 2 x 2 matrices do not hold for 3 x 3 matrices. significant differences arise between surface and volume mesh optimization objective functions. It is shown, for example, that the equivalence in two-dimensions of the Smoothness and Condition Number of the Jacobian matrix objective functions does not extend to three dimensions and further. that the equivalence of the Oddy and Condition Number of the Metric Tensor objective functions in two-dimensions also fails to extend to three-dimensions. Matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions. The concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements. Non-dimensional objective functions having barriers are emphasized as the most logical choice for mesh optimization. The performance of a number of objective functions in improving mesh quality was assessed on a suite of realistic test problems, focusing particularly on all-hexahedral ''whisker-weaved'' meshes. Performance is investigated on both structured and unstructured meshes and on both hexahedral and tetrahedral meshes. Although several objective functions are competitive, the condition number objective function is particularly attractive. The objective functions are closely related to mesh quality measures. To illustrate, it is shown that the condition number metric can be viewed as a new tetrahedral element quality measure.

Infinite elements for wave problems: a review of current formulations and an assessment of accuracy

Abstract: Infinite element schemes for unbounded wave problems are reviewed and a procedure is presented forassessing their performance. A general computational scheme is implemented in which orthogonal functions are used for the transverse interpolation within the infinite element region. This is used as a basis for numerical studies of the effectiveness of various combinations of the radial test and trial functions which give rise to different conjugated and unconjugated formulations. Results are presented for the test case of a spherical radiator to which infinite elements are directly attached. Accuracy of the various schemes is assessed for pure multipole solutions of arbitrary order. Previous studies which have indicated that the conjugated and unconjugated schemes are more effective in the far and near fields, respectively, are confirmed by the current results. All of the schemes tested converge to the exact solution as radial order increases. All are however susceptible to ill conditioning. This places practical restrictions on their effectiveness at high radial orders. A close relationship is demonstrated between the discrete equations which arise from first-order infiniteelement schemes and those derived from the application of more traditional, local non-reflecting boundary conditions. Copyright © 2000 John Wiley & Sons, Ltd.

New concepts in meshless methods

Abstract: Meshless methods have been extensively popularized in literature in recent years, due to their flexibility in solving boundary value problems. Two kinds of truly meshless methods, the meshless local boundary integral equation (MLBIE) method and the meshless local Petrov–Galerkin (MLPG) approach, are presented and discussed. Both methods use the moving least-squares approximation to interpolate the solution variables, while the MLBIE method uses a local boundary integral equation formulation, and the MLPG employs a local symmetric weak form. The two methods are truly meshless ones as both of them do not need a ‘finite element or boundary element mesh’, either for purposes of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. Numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable. In essence, the present meshless method based on the LSWF is found to be a simple, efficient and attractive method with a great potential in engineering applications. Copyright © 2000 John Wiley & Sons, Ltd.

Normalized SPH with stress points

Abstract: Smoothed particle hydrodynamics is extended to a normalized, staggered particle formulation with boundary conditions. A companion set of interpolation points is introduced that carry the stress, velocity gradient, and other derived field variables. The method is stable, linearly consistent, and has an explicit treatment of boundary conditions. Also, a new method for finding neighbours is introduced which selects a minimal and robust set and is insensitive to anisotropy in the particle arrangement. Test problems show that these improvements lead to increased accuracy and stability. Published in 2000 by John Wiley & Sons, Ltd.

A point collocation method based on reproducing kernel approximations

Abstract: A reproducing kernel particle method with built-in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations. The design and implementation of a Galerkin-based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions. In this paper we present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher-order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The point collocation method is shown to be accurate for several one and two-dimensional problems and the convergence rate of the point collocation method is addressed. Copyright © 2000 John Wiley & Sons, Ltd.

Application of the finite volume method and unstructured meshes to linear elasticity

Abstract: SUMMARY A recent emergence of the nite volume method (FVM) in structural analysis promises a viable alternative to the well-established nite element solvers. In this paper, the linear stress analysis problem is discretized using the practices usually associated with the FVM in uid ows. These include the second-order accurate discretization on control volumes of arbitrary polyhedral shape; segregated solution procedure, in which the displacement components are solved consecutively and iterative solvers for the systems of linear algebraic equations. Special attention is given to the optimization of the discretization practice in order to provide rapid convergence for the segregated solution procedure. The solver is set-up to work eciently on parallel distributed memory computer architectures, allowing a fast turn-around for the mesh sizes expected in an industrial environment. The methodology is validated on two test cases: stress concentration around a circular hole and transient wave propagation in a bar. Finally, the steady and transient stress analysis of a Diesel injector valve seat in 3-D is presented, together with the set of parallel speed-up results. Copyright ? 2000 John Wiley & Sons, Ltd.

Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes

Abstract: A family of uniform strain elements is presented for three-node triangular and four-node tetrahedral meshes. The elements use the linear interpolation functions of the original mesh, but each element is associated with a single node. As a result, a favorable constraint ratio for the volumetric response is obtained for problems in solid mechanics. The uniform strain elements do not require the introduction of additional degrees of freedom and their performance is shown to be significantly better than that of three-node triangular or four-node tetrahedral elements. In addition, nodes inside the boundary of the mesh are observed to exhibit superconvergent behavior for a set of example problems.

Enrichment and coupling of the finite element and meshless methods

Abstract: A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown. Copyright © 2000 John Wiley & Sons, Ltd.

Penalty function method for combined finite–discrete element systems comprising large number of separate bodies

Abstract: Large-scale discrete element simulations, the combined finite–discrete element method, DDA as well as a whole range of related methods, involve contact of a large number of separate bodies. In the context of the combined finite–discrete element method, each of these bodies is represented by a single discrete element which is then discretized into finite elements. The combined finite–discrete element method thus also involves algorithms dealing with fracture and fragmentation of individual discrete elements which result in ever changing topology and size of the problem. All these require complex algorithmic procedures and significant computational resources, especially in terms of CPU time. In this context, it is also necessary to have an efficient and robust algorithm for handling mechanical contact. In this work, a contact algorithm based on the penalty function method and incorporating contact kinematics preserving energy balance, is proposed and implemented into the combined finite element code. Copyright © 2000 John Wiley & Sons, Ltd.

Local optimization-based simplicial mesh untangling and improvement.

Abstract: We present an optimization-based approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh. These functions are linear with respect to the free vertex position; thus the problem can be formulated as a linear program that is solved by using the computationally inexpensive simplex method. We prove that the function level sets are convex regardless of the position of the free vertex, and hence the local subproblem is guaranteed to converge. Maximizing the minimum area or volume of mesh elements, although well suited for mesh untangling, is not ideal for mesh improvement, and its use often results in poor quality meshes. We therefore combine the mesh untangling technique with optimization-based mesh improvement techniques and expand previous results to show that a commonly used two-dimensional mesh quality criterion can be guaranteed to converge when starting with a valid mesh. Typical results showing the effectiveness of the combined untangling and smoothing techniques are given for both two- and three- dimensional simplicial meshes.

Automated 3‐D crack growth simulation

Abstract: SUMMARY Automated simulation of arbitrary, non-planar, 3D crack growth in real-life engineered structures requires two key components: crack representation and crack growth mechanics. A model environment for representing the evolving 3D crack geometry and for testing various crack growth mechanics is presented. Reference is made to a specific implementation of the model, called FRANC3D. Computational geometry and topology are used to represent the evolution of crack growth in a structure. Current 3D crack growth mechanics are insufficient; however, the model allows for the implementation of new mechanics. A specific numerical analysis program is not an intrinsic part of the model; i.e., finite and boundary elements are both supported. For demonstration purposes, a 3D hypersingular boundary element code is used for two example simulations. The simulations support the conclusion that automatic propagation of a 3D crack in a real-life structure is feasible. Automated simulation lessens the tedious and time-consuming operations that are usually associated with crack growth analyses. Specifically, modifications to the geometry of the structure due to crack growth, re-meshing of the modified portion of the structure after crack growth, and re-application of boundary conditions proceeds without user intervention.

Qualitative measures for initial meshes

Abstract: This paper reviews geometric measures used to assess the shape of finite elements in two- and three-dimensional meshes Measures have been normalized and made scale invariant whenever possible This paper also introduces a Universal Similarity Region that enhances comparisons of triangles and their measures As a byproduct, the USR provides a dynamic way to compare improved triangular meshes Copyright © 2000 John Wiley & Sons, Ltd

Finite element simulation of metal casting

Abstract: Finite element algorithms are presented for the entire casting process from the mould filling stage to the prediction of the final distorted shape. The various algorithms available in the literature for solidification modelling are discussed in detail. Special emphasis is given to the coupling of the solidification analysis based on the heat conduction equation with fluid flow or thermal stress analysis. Finally, some results are presented to demonstrate the capabilities of the numerical models. Copyright © 2000 John Wiley & Sons, Ltd.

Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I—a framework for surface mesh optimization

Abstract: Structured mesh quality optimization methods are extended to optimization of unstructured triangular, quadrilateral, and mixed finite element meshes. N"ew interpretations of well-known nodally-bssed objective functions are made possible using matrices and matrix norms. The matrix perspective also suggests several new objective functions. Particularly significant is the interpretation of the Oddy metric and the Smoothness objective functions in terms of the condition number of the metric tensor and Jacobian matrix, respectively. Objective functions are grouped according to dimensionality to form weighted combinations. A simple unconstrained local optimum is computed using a modiiied N-ewton iteration. The optimization approach was implemented in the CUBIT mesh generation code and tested on several problems. Results were compared against several standard element-based quaIity measures to demonstrate that good mesh quality can be achieved with nodally-based objective functions.

Achievements and some unsolved problems of the finite element method

Abstract: After outlining the early history of the finite element method the paper concentrates on to (1) some important achievements of the last decade and (2) presents an outline of some problems still requiring treatement. In the first part, the developments of the patch test for assessment of elements, new developments of adaptive refinement and the principles of the general CBS algorithm for fluid dynamics are presented. Copyright © 2000 John Wiley & Sons, Ltd.

Rotation-free triangular plate and shell elements

Abstract: SUMMARY The paper describes how the nite element method and the nite volume method can be successfully combined to derive two new families of thin plate and shell triangles with translational degrees of freedom as the only nodal variables. The simplest elements of the two families based on combining a linear interpolation of displacements with cell centred and cell vertex nite volume schemes are presented in detail. Examples of the good performance of the new rotation-free plate and shell triangles are given. Copyright ? 2000 John Wiley & Sons, Ltd.

Three‐dimensional finite‐element simulation of the dynamic Brazilian tests on concrete cylinders

Abstract: We investigate the feasibility of using cohesive theories of fracture, in conjunction with the direct simulation of fracture and fragmentation, in order to describe processes of tensile damage and compressive crushing in concrete specimens subjected to dynamic loading. We account explicitly for microcracking, the development of macroscopic cracks and inertia, and the effective dynamic behaviour of the material is predicted as an outcome of the calculations. The cohesive properties of the material are assumed to be rate-independent and are therefore determined by static properties such as the static tensile strength. The ability of model to predict the dynamic behaviour of concrete may be traced to the fact that cohesive theories endow the material with an intrinsic time scale. The particular configuration contemplated in this study is the Brazilian cylinder test performed in a Hopkinson bar. Our simulations capture closely the experimentally observed rate sensitivity of the dynamic strength of concrete in the form of a nearly linear increase in dynamic strength with strain rate. More generally, our simulations give accurate transmitted loads over a range of strain rates, which attests to the fidelity of the model where rate effects are concerned. The model also predicts key features of the fracture pattern such as the primary lens-shaped cracks parallel to the load plane, as well as the secondary profuse cracking near the supports. The primary cracks are predicted to be nucleated at the centre of the circular bases of the cylinder and to subsequently propagate towards the interior, in accordance with experimental observations. The primary and secondary cracks are responsible for two peaks in the load history, also in keeping with experiment. The results of the simulations also exhibit a size effect. These results validate the theory as it bears on mixed-mode fracture and fragmentation processes in concrete over a range of strain rates.

A variational principle for the formulation of partitioned structural systems

Abstract: A continuum-based variational principle is presented for the formulation of the discrete governing equations of partitioned structural systems. This application includes coupled substructures as well as subdomains obtained by mesh decomposition. The present variational principle is derived by a series of modifications of a hybrid functional originally proposed by Atluri for finite element development. The interface is treated by a displacement frame and a localized version of the method of Lagrange multipliers. Interior displacements are decomposed into rigid-body and deformational components to handle floating subdomains. Both static and dynamic versions are considered. An important application of the present principle is the treatment of nonmatching meshes that arise from various sources such as separate discretization of substructures, independent mesh refinement, and global–local analysis. The present principle is compared with that of a globalized version of the multiplier method. Copyright © 2000 John Wiley & Sons, Ltd.