Showing papers in "International Journal for Numerical Methods in Engineering in 1994"

Element‐free Galerkin methods

Abstract: An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.

Efficient three‐dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints

Abstract: A method is described which constructs three-dimensional unstructured tetrahedral meshes using the Delaunay triangulation criterion. Several automatic point creation techniques will be highlighted and an algorithm will be presented which can ensure that, given an initial surface triangulation which bounds a domain, a valid boundary conforming assembly of tetrahedra will be produced. Statistics of measures of grid quality are presented for several grids. The efficiency of the proposed procedure reduces the computer time for the generation of realistic unstructured tetrahedral grids to the order of minutes on workstations of modest computational capabilities.

Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept

Abstract: Conventional shell formulations, such as 3- or 5-parameter theories or even 6-parameter theories including the thickness change as extra parameter, require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness. This means that the normal stress in thickness direction has to either vanish or be constant. In general, these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions. The main objective of the present study is to introduce directly a complete 3-D constitutive law without modification. Therefore, a 7-parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity

Abstract: Tangent operators and design sensitivities are derived for transient non-linear coupled problems. The solution process and the formation of tangent operators are presented in a systematic manner and sensitivities for a generalized response functional are formulated via both the direct differentiation and adjoint methods. The derived formulations are suitable for finite element implementations. Analyses of systems, with materials that exhibit history dependent response, may be obtained directly by applying the analyses of transient non-linear coupled systems. Rate-independent elastoplasticity is investigated as a case study and a problem with an analytical solution is analysed for demonstration purposes.

Aspects of the formulation and finite element implementation of large strain isotropic elasticity

Abstract: The paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation. On the theoretical side, an Eulerian setting of isotropic elasticity is discussed exclusively in terms of the Finger tensor as a strain measure. Noval aspects are a direct representation of the Eulerian elastic moduli in terms of the Finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the Finger tensor into spherical and unimodular parts. The isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity. On the computational side, the implementation of the constitutive model in three possible finite element formulations is discussed. After pointing out algorithmic techniques for the treatment of incompressible elasticity, several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity.

Asymptotic-numerical methods and pade approximants for non-linear elastic structures

Abstract: In this paper, we apply asymptotic–numerical methods for computing non-linear equilibrium paths of elastic beam, plate and shell structures. The non-linear branches are sought in the form of asymptotic expansions, and they are determined by solving numerically (FEM) several linear problems with a single stiffness matrix. A large number of terms of the series can be easily computed by using recurrence formulas. In comparison with a more classical step-by-step procedure, the method is rapid and automatic. We show, with some examples, that the choice of the expansion's parameter and the use of Pade approximants play an important role in the determination of the size of the domain of convergence.

On stability and convergence of finite element approximations of biot's consolidation problem

Abstract: Stability and convergence analysis of finite element approximations of Biot's equations governing quasistatic consolidation of saturated porous media are, discussed. A family of decay functions, parametrized by the number of time steps, is derived for the fully discrete backward Euler–Galerkin formulation, showing that the pore-pressure oscillations, arising from an unstable approximation of the incompressibility constraint on the initial condition, decay in time. Error estimates holding over the unbounded time domain for both semidiscrete and fully discrete formulations are presented, and a post-processing technique is employed to improve the pore-pressure accuracy.

Wavelet–Galerkin solutions for one‐dimensional partial differential equations

Abstract: In this paper we describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multi-grid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies1) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. In order to demonstrate the wavelet technique, we consider the one-dimensional counterpart of Helmholtz's equation. By comparison with a simple finite difference solution to this problem with periodic boundary conditions, we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund2 and others. The convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method, at least for problems with simple geometries.

A new energy and momentum conserving algorithm for the non‐linear dynamics of shells

Abstract: A numerical time-integration scheme for the dynamics of non-linear elastic shells is presented that simultaneously and independent of the time-step size inherits exactly the conservation laws of total linear, total angular momentum as well as total energy The proposed technique generalizes to non-linear shells recent work of the authors on non-linear elastodynamics and is ideally suited for long-term/large-scale simulations The algorithm is second-order accurate and can be immediately extended with no modification to a fourth-order accurate scheme The property of exact energy conservation induces a strong notion of non-linear numerical stability which manifests itself in actual simulations

Thin-layer method: Formulation in the time domain

Abstract: The thin-layer method is a semi-discrete numerical technique that may be used for the dynamic analysis of laminated solids or fluids. In its classical implementation, the method is normally formulated in the frequency domain and requires the solution of a complex-valued quadratic eigenvalue problem; in this paper we present an alternative time-domain formulation which can offer advantages in some cases, such as avoiding the use of complex algebra. The proposed method entails expressing the governing equations in the frequency-wavenumber domain, solving a linear real-valued eigenvalue problem in the frequency variable, carrying out an analytical integration over frequencies, and performing a numerical transform over wavenumbers. This strategy allows obtaining the Green's functions for impulsive sources directly in the time domain, even when the system has little or no damping. We first develop the algorithm in its most general form, allowing fully anisotropic materials and arbitrary expansion orders; then we consider a restricted class of anisotropic materials for which the required linear eigenvalue problem involves only real, narrowly banded symmetric matrices and finally, we demonstrate the method by means of a simple problem involving a homogeneous stratum subjected to an antiplane impulsive source.

Introduction to wavelets in engineering

Abstract: The aim of this paper is to provide an introduction to the subject of wavelet analysis for engineering applications. The paper selects from the recent mathematical literature on wavelets the results necessary to develop wavelet-based numerical algorithms. In particular, we provide extensive details of the derivation of Mallat's transform and Daubechies' wavelet coefficients, since these are fundamental to gaining an insight into the properties of wavelets. The potential benefits of using wavelets are highlighted by presenting results of our research in one-and two-dimensional data analysis and in wavelet solutions of partial differential equations.

Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements

Abstract: The superconvergent patch derivative recovery method of Zienkiewicz and Zhu is enhanced by adding the squares of the residuals of the equilibrium equation and natural boundary conditions. In addition, a new conjoint polynomial for interpolating the local patch stresses over the element which significantly improves the local projection scheme is presented. Results show that in the 4-node quadrilateral, the equilibrium and boundary condition residuals usually improve accuracy but not the rate of convergence, whereas in the 9-node quadrilateral, results are mixed. The conjoint polynomial always improves the accuracy of the derivative field within the element as compared to the standard nodal interpolation, particularly in 4-node quadrilaterals.

Application of generalized differential quadrature method to structural problems

Abstract: SUMMARY This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis. Firstly, drawbacks existing in the method of differential quadrature (DQ) are evaluated and discussed. Then, an improved and simpler generalized differential quadrature method (GDQ) is introduced to overcome the existing drawback and to simplify the procedure for determining the weighting coefficients. Subsequently, the generalized differential quadrature is systematically employed to solve problems in structural analysis. Numerical examples have shown the superb accuracy, efficiency, convenience and the great potential of this method. Numerical approximation methods for solving partial differential equations have been widely used in various engineering fields. Classical techniques such as finite element and finite difference methods are well developed and well known. These methods can provide very accurate results by using a large number of grid points. However, in a large number of cases, reasonably approximate solutions are desired at only a few specific points in the physical domain. In order to get results even at or around a point of interest with acceptable accuracy, conventional finite element and finite difference methods still require the use of a large number of grid points. Consequently, the requirement for computer capacity is often unnecessarily large in such cases. In seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy, the method of differential quadrature (DQ) was introduced by Bellman et a!.'. The method of DQ is a global approximate method. This method is based on the ideas that the derivative of a function with respect to a co-ordinate direction can be expressed as a weighted linear sum of all the function values at all mesh points along that direction and that a continuous function can be approximated by a higher-order polynomial in the overall domain. The DQ method differs from the finite element method (FEM) in two aspects. Firstly, the FEM uses lower-order polynomials to approximate a function on a local element level, while the DQ method approximates a function on the global area using higher-order polynomials. Secondly, the DQ method directly approximates the derivatives of a function at a point, while the FEM approximates a function over a local element and the derivatives can then be derived from the approximate function. In this aspect, the DQ method is more similar to the finite difference method (FDM). However, the FDM is also a local approximation method based on lower-order polynomial approximation. In fact, it can be shown that the FDM is just a special case of the DQ method where it is applied locally on the range [.xi- xi + l]. Owing to the higher-order

Enhanced Superconvergent Patch Recovery incorporating equilibrium and boundary conditions

Abstract: Patch recovery based on superconvergent derivatives and equilibrium (SPRE), an enhancement of the Superconvergent Patch Recovery (SPR), is studied for linear elasticity problems. The paper also presents a further improvement for recovery of derivatives near boundaries, SPREB, where either tractions or displacements are prescribed. This is made by inclusion of weighted residual errors at boundary points in the patch recovery. A pronounced improvement in the post processed gradients of the finite element solution is observed by this method.

A new family of explicit time integration methods for linear and non‐linear structural dynamics

Abstract: A new family of explicit single-step time integration methods with controllable high-frequency dissipation is presented for linear and non-linear structural dynamic analyses. The proposed methods are second-order accurate and completely explicit with a diagonal mass matrix, even when the damping matrix is not diagonal in the linear structural dynamics or the internal force vector is a function of velocities in the non-linear structural dynamics. Stability and accuracy of the new explicit methods are analysed for the linear undamped/damped cases. Furthermore, the new methods are compared with other explicit methods.

On error estimates and adaptivity in elastoplastic solids: Applications to the numerical simulation of strain localization in classical and Cosserat continua

Abstract: The a posteriori error estimates based on the post-processing approach are introduced for elastoplastic solids. The standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy. This is known to represent a natural metric for the class of elastoplastic problems of evolution. In addition, the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates. A posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem. As a model problem, the constitutive equations describing a generalization of standard J2-elastoplasticity within the Cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post-instability region. The proposed a posteriori error estimates are appropriately modified to account for the Cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems. Superior behaviour of the Cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material, resulting in convergent solutions with an h-refinement and almost uniform error distribution in all considered error norms.

The advancing-front mesh generation method revisited

Abstract: This paper presents a mesh generation method of the advancing-front type which is designed in such a way that the well-known difficulties of the classical advancing-front method are not present. The retained solution consists of using the first steps of a Voronoi–Delaunay method to construct a background mesh which is then used to govern the algorithm. The two-dimensional case is considered in detail and possible extensions to adaption problems and three dimensions are indicated.

Finite volumes and finite elements: Two ‘good friends’

Abstract: In this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic, convective–diffusion and fluid flow problems. The paper shows that both procedures share a number of features, like mesh discretization and approximation. Moreover, it is shown that in many cases both techniques are completely equivalent.

Topology design with optimized, self‐adaptive materials

Abstract: Significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems. We study the optimal shape design of a two-dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material. The macroscopic version of this problem is not well-posed if no restrictions are placed on the structure topology; relaxation of the optimization problem via quasiconvexification or homogenization methods is required. The effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material

A finite volume format for structural mechanics

Abstract: SUMMARY A general Finite Volume Method (FVM) for the analysis of structural problems is presented. It is shown that the FVM can be considered to be a particular case of finite elements with a non-Galerkin weighting. For structural analysis this can readily be interpreted as equivalent to the unit displacement method which involves mainly surface integrals. Both displacement and mixed FV formulations are presented for static and dynamic problems. The Finite Volume Method (FVM) evolved in the early seventies via finite difference approxima- tions on non-orthogonal grids. The popularity of the FVM has been extensive in the field of Computational Fluid Dynamics (CFD) and heat transfer.'-' On the contrary, in the field of Computational Solid Mechanics (CSM) the use of the FVM has never achieved such acceptance. An early attempt to use FV concepts in CSM is due to Wilkins6 as an alternative approximation to derivatives in a cell. In this he defines the average gradient of a function u in a volume l2 as using the well-known divergence theorem. Such a definition of gradients can be written entirely in terms of function values at the boundary of a volume and has been used in the early 'hydrocodes' of the Lawrence Livermore Laboratory. The reasons for the unpopularity of the FVM in structural mechanics is understandable, as finite volumes are well known to be less accurate than Galerkin-based finite elements for self-adjoint (elliptic) problems. A comparison between FVM and FEM has been recently pre- sented by Zienkiewicz and Oiiate.' Here the authors show that FVM and FEM share concepts such as mesh discretization and interpolation, giving precisely the same discretized systems of equations for some particular cases. This coincidence also shows clearly for 2-D and 3-D structural problems as detailed in this work. Here, however, surface integrals are mostly involved and the number of computations can be shown to be proportional to the number of 'sides' in the mesh. This leads to an overall solution cost very similar to that of FE computations (note that for a refined mesh of three-node triangles the number of sides is 1.5 times that of elements). This fact suggests that computational speed is not 'a prior? one of the keys to the possible success of FVM in structural problems. However, the possibility of obtaining the element matrices and vectors in terms of computations along the element sides opens new possibilities for the solution of some structural problems which, in the authors' opinion, may be worth exploring in detail. Indeed, it

A transient FETI methodology for large‐scale parallel implicit computations in structural mechanics

Abstract: Explicit codes are often used to simulate the nonlinear dynamics of large-scale structural systems, even for low frequency response, because the storage and CPU requirements entailed by the repeated factorizations traditionally found in implicit codes rapidly overwhelm the available computing resources. With the advent of parallel processing, this trend is accelerating because explicit schemes are also easier to parallelize than implicit ones. However, the time step restriction imposed by the Courant stability condition on all explicit schemes cannot yet -- and perhaps will never -- be offset by the speed of parallel hardware. Therefore, it is essential to develop efficient and robust alternatives to direct methods that are also amenable to massively parallel processing because implicit codes using unconditionally stable time-integration algorithms are computationally more efficient when simulating low-frequency dynamics. Here we present a domain decomposition method for implicit schemes that requires significantly less storage than factorization algorithms, that is several times faster than other popular direct and iterative methods, that can be easily implemented on both shared and local memory parallel processors, and that is both computationally and communication-wise efficient. The proposed transient domain decomposition method is an extension of the method of Finite Element Tearing and Interconnecting (FETI) developed by Farhat and Roux for the solution of static problems. Serial and parallel performance results on the CRAY Y-MP/8 and the iPSC-860/128 systems are reported and analyzed for realistic structural dynamics problems. These results establish the superiority of the FETI method over both the serial/parallel conjugate gradient algorithm with diagonal scaling and the serial/parallel direct method, and contrast the computational power of the iPSC-860/128 parallel processor with that of the CRAY Y-MP/8 system.

A co‐rotational element/time‐integration strategy for non‐linear dynamics

Abstract: New procedures are proposed for implicit dynamic analysis using the finite element method. The main aim is to give stable solutions with significant rigid-body motions, in particular rotations. In contrast to most conventional approaches, the time-integration strategy is closely linked to the ‘element technologies’ with the latter involving a form of co-rotational procedure. For the undamped situation, one of the solution procedures leads to an algorithm that exactly conserves energy when constant external forces are applied (i.e. with gravity loading).

Numerical models of steady rolling for non‐linear viscoelastic structures in finite deformations

Abstract: A Lagrangian formulation of constitutive laws for a viscoelastic material based on irreversible thermodynamics is first presented. These laws are expressed by a non-linear differential equation governing the evolution of an internal variable. Then equations describing the steady rolling of an axisymmetric viscoelastic structure are obtained from the conservation laws of continuum mechanics. A finite element approximation and a solution technique of the algebraic system is proposed. The eiimination of the internal variable allows one to keep an elastic-like algorithm with an independent solution of the viscoelastic equation. Numerical tests show the feasibility and the efficiency of the proposed methods in large three-dimensional situations.

Isoparametric Hermite elements

Abstract: Isoparametric Hermite elements are created using Bogner–Fox–Schmit rectangles on a reference domain and mapping these numerically onto the computational domain. The difficulties involved in devising explicit C1 shape functions for isoparametric elements are thus avoided, and the resulting elements have all the benefits of full C1 continuity, the simplicity of the Bogner–Fox–Schmit element and the geometrical flexibility expected from higher-order isoparametric elements. The numerical mapping consists in the finite element solution of a linear boundary value problem, which is inexpensive and is carried out as a preprocessing operation—the required derivatives of the mapping then being supplied to the main analysis as data. Some care is required in defining the differential boundary conditions, and guidance on this is provided. Examples are given showing the success of the mapping procedure, and the use of the resulting elements in the solution of some boundary value problems. The numerical results confirm a convergence analysis provided for the new isoparametric Hermite element.

Boundary integral equations for thin bodies

Abstract: SUMMARY A boundary integral equation formulation for thin bodies which uses CBIE (conventional BIE) only is well known to be degenerate. A mixed formulation for a thin rigid scatterer which combines CBIE and HBIE (hypersingular BIE) is motivated by examining the discretized form of the integral equations, and this formulation is shown to be nondegenerate for thin non-rigid inclusion problems. A near-singular integration procedure, useful for singular integrals as well, is presented. Finally, numerical examples for acoustic wave scattering from rigid and soft scatterers are presented.

The position code algorithm for contact searching

Abstract: A new global search method for general contact systems is developed and implemented in the DYNA3D program, along with a recent contact interface algorithm. The concept of «position codes» for efficient global contact searching is presented. With the position code algorithm, the problem of sorting and searching in three dimensions is transformed to a process of sorting and searching within a one-dimensional array. The cost of contact searching is of the order of N log 2 N, where N is the number of nodes in the system. The proposed algorithms are uncomplicated and the implementation into any finite element code is straightforward

A multiharmonic method for non‐linear vibration analysis

Abstract: A multiharmonic method for analysis of non-linear dynamic systems submitted to periodic loading conditions is presented. The approach is based on a systematic use of the fast Fourier transform. The exact linearization of the resulting equations in the frequency domain allows to obtain full quadratic convergence rate

A finite element model for the simulation of pile driving

Abstract: The feasibility of conducting a detailed analysis of pile driving using a finite element technique is examined in this paper, taking into account the non-linear behaviour of undrained clayey soil and tracing the penetration of the pile into the soil. A three-dimensional model is used for this purpose, which is handled by two-dimensional analysis due to the axisymmetric nature of the problem. A non-linear time-domain dynamic analysis is performed in which the hammer blows on the pile are represented by a periodic forcing function, and the pile penetration is treated using a frictional contact slideline algorithm. The model is applied 10 the driving of a concrete pile in a clayey soil.

Charge iteration: A procedure for the finite element computation of unbounded electrical fields

Abstract: An iterative procedure is presented for the finite element computation of unbounded electrical fields created by voltaged conductors. The procedure is based on successive evaluations of the potential on a fictitious boundary enclosing all the conductors, according to the charge lying on their surface. The convergence of the procedure to the solution of the unbounded field problem is demonstrated. Indications for the optimal placement of the fictitious boundary are provided, also taking into account the accuracy of the solution. The way in which computational efficiency can be reached is also discussed. The main advantage of this procedure lies in its simplicity of implementation in the context of a standard FE code for bounded problems, because a very limited amount of additional software is required; moreover, 2-D and axisymmetric versions can be implemented with minor changes from a suitable 3-D one. Examples of application are given in order to illustrate the practical use of the procedure and to validate it by comparisons with available solutions.