Comparison of boundary element and finite element methods for dynamic analysis of elastoplastic plates

Boundary and finite element methodologies for the determination of the response of inelastic plates are compared and critically discussed. Flexural dynamic plate bending problems are considered and a hardening elastoplastic constitutive model is used to describe material behaviour. The domain/boundary element methodology using linear boundary and quadratic interior elements and the finite element method with quadratic Mindlin plate elements are used in this work. The discretized equations of motion in both methodologies are solved by an efficient step-by-step time integration algorithm. Numerical results obtained are presented and compared in order to access the accuracy and computational efficiency of the two methods. In order to make the comparison as meaningful as possible, boundary and finite element computer codes developed by the author are used in this paper. In general, boundary elements appear to be a better choice than finite elements with respect to computational efficiency for the same level of accuracy.     

Beam and plate stability by boundary elements

The direct boundary element method is used for the linear elastic stability analysis of Bernoulli-Euler beams and Kirchhoff thin plates. The formulation is based on the reciprocal work theorem of Betti and utilizes either fundamental solutions which incorporate the effect of axial and in-plane forces on bending, or fundamental solutions which correspond to pure flexure. In the former case. only a boundary discretization of the structure is required, while in the latter case discretization of the boundary as well as of the interior is necessary. However, the fundamental solutions in the latter case are less complicated than the ones in the former case. Numerical examples are subsequently presented to illustrate the methodology. The basic conclusion is that the simpler fundamental solutions are adequate and, by virtue of being more general, greatly expand the versatility of the boundary element method.     

Field/boundary element approach to the large deflection of thin flat plates

The problem of large deflections of thin flat plates is rederived here using a novel integral equation approach. These plate deformations are governed by the von Karman plate theory. The numerical solution that is implemented combines both boundary and interior elements in the discretization of the continuum. The formulation also illustrates the adaptability of the boundary element technique to nonlinear problems. Included in the examples here are static, dynamic and buckling applications.     

A simple method to impose rotations and concentrated moments on ANC beams

Recently introduced ANC beam elements furnish a simple formulation that allows to solve nonlinear problems of beams, including those with large displacements and strains, as well as complex nonlinear (inelastic) materials. The success and simplicity of these finite elements is mainly due to the fact that the only nodal degrees of freedom that they employ are displacements, and rotations are thus completely avoided. This in turn makes it very difficult to apply concentrated moments or to impose rotations at specific nodes of a finite element mesh. In this article, we present a simple enhancement to this beam formulation that allows to apply these two types of boundary conditions in a simple manner, making ANC beam elements more versatile for both multibody and structural applications.     

Identification of nonlinear elastic solids by a finite element method

The problem of utilizing experimental data to characterize the stress constitutive function for a nonlinear elastic solid is formulated as an inverse boundary value problem. The use of finite element discretization is extended by introducing a technique of material parameterization that utilizes finite elements defined over the domain of the stress constitutive function. The discretized identification problem is then reduced to a system of nonlinear algebraic equations that couples the data set and the discretized boundary value problem. The effect of errors in measured data is minimized by employing a weighted least squares error norm to generate the equations from which the unknown material parameters are obtained. An illustrative numerical experimental is included.     

Semi-Unstructured Grids

We present a novel automatic grid generator for the finite element discretization of partial differential equations in 3D. The grids constructed by this grid generator are composed of a pure tensor product grid in the interior of the domain and an unstructured grid which is only contained in boundary cells. The unstructured component consists of tetrahedra, each of which satisfies a maximal interior angle condition. By suitable constructing the boundary cells, the number of types of boundary subcells is reduced to 12 types. Since this grid generator constructs large structured grids in the interior and small unstructured grids near the boundary, the resulting semi-unstructured grids have similar properties as structured tensor product grids. Some appealing properties of this method are computational efficiency and natural construction of coarse grids for multilevel algorithms. Numerical results and an analysis of the discretization error are presented. Received July 17, 2000; revised October 27, 2000     

Post-processing and visualization techniques in 2D boundary element analysis

Most post-processors for boundary element (BE) analysis use an auxiliary domain mesh to display domain results, working against the profitable modelling process of a pure boundary discretization. This paper introduces a novel visualization technique which preserves the basic properties of the boundary element methods. The proposed algorithm does not require any domain discretization and is based on the direct and automatic identification of isolines. Another critical aspect of the visualization of domain results in BE analysis is the effort required to evaluate results in interior points. In order to tackle this issue, the present article also provides a comparison between the performance of two different BE formulations (conventional and hybrid). In addition, this paper presents an overview of the most common post-processing and visualization techniques in BE analysis, such as the classical algorithms of scan line and the interpolation over a domain discretization. The results presented herein show that the proposed algorithm offers a very high performance compared with other visualization procedures.     

Dynamic elastoplastic analysis of 3-D structures by the domain/boundary element method

The main objective of this paper is to present a general three-dimensional boundary element methodology for solving transient dynamic elastoplastic problems. The elastostatic fundamental solution is used in writing the integral representation and this creates in addition to the surface integrals, volume integrals due to inertia and inelasticity. Thus, an interior discretization in addition to the usual surface discretization is necessary. Isoparametric linear quadrilateral elements are used for the surface discretization and isoparametric linear hexahedra for the interior discretization. Advanced numerical integration techniques for singular and nearly singular integrals are employed. Houbolt's step-by-step numerical time integration algorithm is used to provide the dynamic response. Numerical examples are presented to illustrate the method and demonstrate its accuracy.     

Volumetric mesh generation from T-spline surface representations

A new approach to obtain a volumetric discretization from a T-spline surface representation is presented. A T-spline boundary zone is created beneath the surface, while the core of the model is discretized with Lagrangian elements. T-spline enriched elements are used as an interface between isogeometric and Lagrangian finite elements. The thickness of the T-spline zone and thereby the isogeometric volume fraction can be chosen arbitrarily large such that pure Lagrangian and pure isogeometric discretizations are included. The presented approach combines the advantages of isogeometric elements (accuracy and smoothness) and classical finite elements (simplicity and efficiency).Different heat transfer problems are solved with the finite element method using the presented discretization approach with different isogeometric volume fractions. For suitable applications, the approach leads to a substantial accuracy gain.     

A comprehensive generalized mesh system for CFD applications

The numerical approaches used for the solution of governing equations of fluid flow are dictated highly by the topology of the domain discretization. Two of the most commonly used discretization approaches are the structured and unstructured topologies. This paper describes the discretization of the domain using generalized elements with an arbitrary number of nodes to combine the advantages of both the structured and unstructured methodologies. Numerical algorithms for the solution of the governing equations for generalized mesh, an approach for handling mesh movement applicable to rotating machineries, and the application of this framework for overset meshes to handle moving body problems are discussed. A library-based approach has been adopted for the implementation of overset capability for the framework. The results from the application of this framework for various applications are presented.     

A Domain Decomposition Preconditioner for p-FEM Discretizations of Two-dimensional Elliptic Problems

In this paper, a uniformly elliptic second order boundary value problem in 2-D discretized by the p-version of the finite element method is considered. An inexact Dirichlet-Dirichlet domain decomposition pre-conditioner for the system of linear algebraic equations is investigated. Two solvers for the problem in the sub-domains, a pre-conditioner for the Schur-complement and an extension operator operating from the edges of the elements into the interior are proposed as ingredients for the inexact DD-pre-conditioner. In the main part of the paper, several numerical experiments on a parallel computer are given.     

A coupled formulation of finite and boundary element methods in two-dimensional elasticity

A symmetric stiffness formulation based on a boundary element method is studied for the structural analysis of a shear wall, with or without cutouts. To satisfy compatibility requirements with finite beam elements and to avoid problems due to the eventual discontinuities of the traction vector, different interpolation schemes are adopted to approximate the boundary displacements and tractions. A set of boundary integral equations is obtained with the collocation points on the boundary, which are selected by the error minimization technique proposed in this paper, and the stiffness matrix is formulated with those equations and symmetric coupling techniques of finite and boundary element methods. The newly developed plane stress element can have the openings in its interior domain and can be easily linked with the finite beam/column elements.     

Effective approach for the identification of boundary conditions in distributed-parameter systems

This paper is concerned with the identification of boundary conditions in parabolic-type distributed systems with boundaries of irregular shape. In the present approach, finite element discretization in the spatial domain and orthogonal functions expansion in the time domain are adopted to reduce the partial differential equation to a set of algebraic equations. The boundary conditions are then estimated by the method of least squares using state observations taken at a few interior points. The present approach is very straightforward and, at least as shown in illustrative examples, the results are in excellent agreement with exact results.     

Local quality of smoothening based a-posteriori error estimators for laminated plates under transverse loading

The local and global quality of various smoothening based a-posteriori error estimators is tested in this paper, for symmetric laminated composite plates subjected to transverse loads. Smoothening based on strain recovery and displacement-field recovery is studied here. Effect of ply orientation, laminate thickness, boundary conditions, mesh topology, and plate model is studied for a rectangular plate. It is observed that for interior patches of elements, both the estimators based on strain or displacement smoothening are reliable. For element patches at the boundary of the domain, all estimators tend to be unreliable (especially for angle-ply laminates). However, the strain recovery based estimator is clearly more robust for element patches at the boundary, as compared to displacement-recovery based error estimators. Globally, all the estimators tested here were found to be very robust.     

Stable Boundary Treatment for the Wave Equation on Second-Order Form

A stable and accurate boundary treatment is derived for the second-order wave equation. The domain is discretized using narrow-diagonal summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a stable and efficient scheme. The analysis is verified by numerical simulations in one-dimension using high-order finite difference discretizations, and in three-dimensions using an unstructured finite volume discretization.     

Study of indentation by using domain decomposition and boundary elements methods on a microcomputer

In this paper, we study the problem of indentation of an elastic support by a sphere, using a domain decomposition method coupled with the boundary elements method. The depcomosition method has the advantage of solving the problem on each solid separately. The contact is governed only by boundary conditions. Therefore, the Hennizel method, a non-overlapping technique is particularly well adapted. Indeed only information on interface is transmitted from one sub-domain to another. This leads us naturally to associate it with the boundary elements method, since it needs only the discretization of the boundaries of solids. We have implemented three programes on a microcomputer (boundary elements, domain decomposition and solution of contact). The latter gives results in conformity with the Hertz theory and the analytical solution of Spence.     

Numerical solution of time-fractional fourth-order partial differential equations

A quintic B-spline collocation technique is employed for the numerical solution of time-fractional fourth-order partial differential equations. These equations occur in many applications in real-life problems such as modelling of plates and thin beams, strain gradient elasticity and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical and aerospace engineering. The time-fractional derivative is described in the Caputo sense. Backward Euler scheme is used for time discretization and the quintic B-spline-based numerical method is used for space discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. The given problem is solved with three different boundary conditions, including clamped-type condition, simply supported-type condition, and a transversely supported-type condition. Numerical results are considered to investigate the accuracy and efficiency of the proposed method.     

A finite element formulation for large amplitude flexural vibrations of thin rectangular plates

Large amplitude flexural vibrations of rectangular plates are studied in this paper using a direct finite element formulation. The formulation is based on an appropriate linearisation of strain displacement relations and uses an iterative method of solution. Results are presented for rectangular plates with various boundary conditions using a conforming rectangular element. Whenever possible the present solutions are compared with those of earlier work. This comparison brings out the superiority of the proposed formulation over the earlier finite element formulation.     

A posteriori error estimates for fem–bem couplings of three-dimensional electromagnetic problems

We present residual based and p-hierarchical a posteriori error estimators for a Galerkin method coupling finite elements and boundary elements for time–harmonic interface problems in electromagnetics; special emphasis is taken for the eddy current problem. The Galerkin discretization uses lowest order Nédélec elements in the interior domain and vectorial surface rotations of continuous, piecewise linear functions on the interface boundary. Singular, weakly singular and hypersingular boundary integral operators appearing in the variational formulation show up in the terms of the error estimators as well. The estimators are derived from the defect equation using Helmholtz and Hodge decompositions. Numerical tests underline reliability and efficiency of the given error estimators yielding reasonable mesh refinements.     

A Least-Squares Spectral Element Formulation for the Stokes Problem

Least-squares spectral element methods seem very promising since they combine the generality of finite element methods with the accuracy of the spectral methods and also the theoretical and computational advantages in the algorithmic design and implementation of the least-squares methods. The new element in this work is the choice of spectral elements for the discretization of the least-squares formulation for its superior accuracy due to the high-order basis-functions. The main issue of this paper is the derivation of a least-squares spectral element formulation for the Stokes equations and the role of the boundary conditions on the coercivity relations. The numerical simulations confirm the usual exponential rate of convergence when p-refinement is applied which is typical for spectral element discretization.