Three-dimensional solid-to-beam transition elements for structural dynamics analysis

Complex structural components such as those encountered in many industrial applications may generally be considered as being composed of shell- or beam-like portions linked to three-dimensional solid continua. When discretized into finite elements, these structures present geometrical and mathematical difficulties at the connections between the different element types since the nodal degrees of freedom allocated to the solid, shell and beam elements are incompatible with each other. The development of specific and reliable transition finite elements is, thus, of outstanding practical importance. This paper presents efficient C⁰ compatible transition elements with a variable number of nodes for modelling solid to beam junctions. Based upon the standard isoparametric solid and beam formulations, the current approach includes the properties of both solids and beams, verifies the basic continuity, smoothness and completeness criteria inherent in the finite element convergence requirements, and avoids the shear locking phenomenon typical of C⁰ elements by using a strain-projection method. Several numerical examples which compare this formulation to analytical and experimental solutions are provided in order to show the applicability and efficiency of this approach.     

Implementation of a lamina-based constitutive relationship for analysing thick composites

The finite element implementation of a lamina-based elastic constitutive relation appropriate for describing the three-dimensional (3-D) behaviour of fibre composite laminated media is presented. Unconstrained by the two-dimensional restrictions accompanying plate and shell theories, this approach resolves the macro 3-D deformation fields found in 'thick' laminated or filament wound composites without requiring each lamina to be individually discretized to assign material properties. In rectangular isoparametric solid elements where laminae locally parallel an element 'face', the implementation represents 'exactly' the elastic laminate behaviour by replacing the discrete lamina stiffnesses with continuous polynomial moduli functions. Attributable directly to the lamina constitutive relationship used, appropriate for fibre-dominated material systems, the implementation automatically preserves interlaminar continuity of tractions and displacements within each element for laminates assembled from a single lamina type. The order of the effective moduli functions is determined by the element kinematics and by requiring identical net mid-surface elemental forces and moments. In general, this representation substantially reduces the number of stored variables and through-thickness integrations required, and allows 'exact' integration by higher-order Gaussian quadrature. Additionally a single element may span any portion or number of laminae, thus allowing nearly arbitrary meshing and solution resolution. Several numerical examples using 3-D 8-node isoparametric solid elements demonstrate the approach's convergence and overall behaviour.     

Improving the modified XFEM for optimal high-order approximation

This paper investigates the accuracy of high-order extended finite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modified XFEM, a specific form of the stable generalised finite element method, is found to offer advantages in cost and complexity over other approaches, but suffers from suboptimal rates of convergence due to spurious higher-order contributions to the approximation space. An improved modified XFEM is presented, with basis functions “corrected” by projecting out higher-order contributions that cannot be represented by the standard finite element basis. The resulting corrections are independent of the equations being solved and need be pre-computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order p ≤ 4. These are the first truly optimal convergence results achieved using the XFEM for a curved weak discontinuity and are also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order p>1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order p ≤ 4.     

The fourth-dimention concept in the finite element analysis of transient heat transfer problems

This study presents the finite element analysis for the transient heat transfer problems by introducing the Fourth-dimension concept. Time is treated as an additional dimension in the solution domain thereby Increasing the number of dimensions by one. For instance, a three-dimensional transient problem can be considered as a four-dimensional problem in the x, y, z, t domain and a two-dimensional transient problem can be considered as a three-dimensional steady-state problem in the x, y, t domain, respectively. The variational principle of the finite element method and the techniques existing for steady-state problems can be directly utilized. Numerical calculations were performed for heat conduction problems, laminar-turbulent convective heat transfer problems, and radiative heat transfer problems.     

Use of the over-relaxation technique in the simulation of large groundwater basins by the finite element method

The over-relaxation technique has been used in conjunction with the finite element method in a regional time-dependent simulation of the subsidence of Venice. One year of computer experiments have shown that in basins with a ratio h/R ? 10^?2 between the vertical and the horizontal dimension, the over-relaxation technique can lead to unsatisfactory results. In single precision and for relatively large time steps, the solution of the final linear system can be inaccurate if the optimum over-relaxation factor ω is not correctly assessed. In transient analyses, instability can also occur. The latter may be avoided by properly reducing the time step or by switching to double precision. Steady state simulations can also require double precision to provide accurate results even when the best ω is used. Instability and inaccuracy disappear in basins for which h/R ? 10^?1. In addition the processor time diminishes significantly as in this case the number of iterations necessary to obtain a good solution is considerably smaller than the order of the matrix.     

A well‐conditioned and optimally convergent XFEM for 3D linear elastic fracture

A variation of the extended finite element method for three‐dimensional fracture mechanics is proposed. It utilizes a novel form of enrichment and point‐wise and integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy, optimal convergence rates, and improved conditioning for two‐dimensional and three‐dimensional crack problems. A bespoke benchmark problem is introduced to determine the method's accuracy in the general three‐dimensional case where it is demonstrated that the proposed approach improves the accuracy and reduces the number of iterations required for the iterative solution of the resulting system of equations by 40% for moderately refined meshes and topological enrichment. Moreover, when a fixed enrichment volume is used, the number of iterations required grows at a rate which is reduced by a factor of 2 compared with standard extended finite element method, diminishing the number of iterations by almost one order of magnitude. Copyright © 2015 John Wiley & Sons, Ltd.     

Using the sequential linear integer programming method as a post‐processor for stress‐constrained topology optimization problems

This paper deals with topology optimization of load‐carrying structures defined on discretized continuum design domains. In particular, the minimum compliance problem with stress constraints is considered. The finite element method is used to discretize the design domain into n finite elements and the design of a certain structure is represented by an n‐dimensional binary design variable vector. In order to solve the problems, the binary constraints on the design variables are initially relaxed and the problems are solved with both the method of moving asymptotes and the sparse non‐linear optimizer solvers for continuous optimization in order to compare the two solvers. By solving a sequence of problems with a sequentially lower limit on the amount of grey allowed, designs that are close to ‘black‐and‐white’ are obtained. In order to get locally optimal solutions that are purely {0, 1}ⁿ, a sequential linear integer programming method is applied as a post‐processor. Numerical results are presented for some different test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Convergence studies of eigenvalue solutions using two finite plate bending elements

The convergence rates of eigenvalue solutions using two finite plate bending elements are studied. The elements considered are the well-known 12 degree of freedom, non-conforming rectangular element and the 16 degree of freedom, conforming rectangular element. Three problems are analysed, a square plate simply supported on two opposite sides with the other two sides clamped, simply supported, or free. Closed form, finite element solutions for these problems are obtained by using shifting E-operators. With few exceptions, eigenvalue solutions found with the non-conforming element converge from below the exact answers at an asymptotic rate of n^?2, where n is the number of elements on a side. However, since the array size needed for such convergence is very large, little can be said about the convergence rates for practical arrays. The conforming element solutions converge from above at an asymptotic rate of n^?4. A comparison of the errors involved in using these two elements shows that the conforming element is far superior to the non-conforming element.     

Performance of a Petrov–Galerkin algebraic multilevel preconditioner for finite element modeling of the semiconductor device drift‐diffusion equations

This study compares the performance of a relatively new Petrov–Galerkin smoothed aggregation (PGSA) multilevel preconditioner with a nonsmoothed aggregation (NSA) multilevel preconditioner to accelerate the convergence of Krylov solvers on systems arising from a drift‐diffusion model for semiconductor devices. PGSA is designed for nonsymmetric linear systems, Ax=b, and has two main differences with smoothed aggregation. Damping parameters for smoothing interpolation basis functions are now calculated locally and restriction is no longer the transpose of interpolation but instead corresponds to applying the interpolation algorithm to A^T and then transposing the result. The drift‐diffusion system consists of a Poisson equation for the electrostatic potential and two convection–diffusion‐reaction‐type equations for the electron and hole concentration. This system is discretized in space with a stabilized finite element method and the discrete solution is obtained by using a fully coupled preconditioned Newton–Krylov solver. The results demonstrate that the PGSA preconditioner scales significantly better than the NSA preconditioner, and can reduce the solution time by more than a factor of two for a problem with 110 million unknowns on 4000 processors. The solution of a 1B unknown problem on 24 000 processor cores of a Cray XT3/4 machine was obtained using the PGSA preconditioner. Copyright © 2010 John Wiley & Sons, Ltd.     

Polytope finite elements

In this paper, finite elements based on arbitrary convex and non‐convex polytopes are introduced. Polytopes in combination with natural element coordinates (NECs) permit a uniform element formulation of interpolation functions that are independent of the dimension of space, localization and the number of vertices. NECs based on the natural neighbor interpolation are restricted to the polytope and can be understood as an extension of the barycentric coordinates on simplexes. The differentiation and integration of these interpolation functions on the basis of NECs is essential for finite element approximations. The accuracy of the finite element interpolation or approximation can be controlled by either applying the h‐version or by utilizing the p‐version of the finite element method (FEM). Advantages in the handling of hanging nodes are discussed. Furthermore, we present construction methods for Lagrangian as well as for hierarchical interpolation functions based on NECs. Numerical experiments on different convex and non‐convex decompositions will show the usability, accuracy and convergence of the developed polytope FEM. Copyright © 2007 John Wiley & Sons, Ltd.     

H(div) finite elements based on nonaffine meshes for 3D mixed formulations of flow problems with arbitrary high order accuracy of the divergence of the flux

Effects of nonaffine elements on the accuracy of 3D H(div)-conforming finite elements are studied. Instead of convergence order k+1 for the flux and the divergence of the flux obtained with Raviart-Thomas or Nédélec spaces with normal traces of degree k, based on affine hexahedra or triangular prisms, reduced orders k for the flux and k−1 for the divergence of the flux may occur for distorted elements. To improve this scenario, a hierarchy of enriched flux approximations is considered, by adding internal shape functions up to a higher degree k+n, n>0, while keeping the original normal traces of degree k. The resulting enriched approximations, using multilinear transformations, keep the original flux accuracy (of order k+1 with affine elements or reduced order k otherwise), but enhanced divergence (of order k+n+1, in the affine case, or k+n−1 otherwise) can be reached. The reduced flux accuracy due to quadrilateral face distortions cannot be corrected by including higher order internal functions. The enriched spaces are applied to the mixed finite element formulation of Darcy's model. The computational cost of matrix assembly increases with n, but the condensed system to be solved has the same dimension and structure as the original scheme.     

Three-dimensional finite element contact algorithm for metal forming

This work presents a three-dimensional rigid plastic finite element formulation. The workpiece is discretized with eight node hexahedral isoparametric elements. Friction is included in the formulation by means of a shear stress depending on the relative velocity between the workpiece and the tool. Special attention is given to the contact problems, and a three-dimensional contact algorithm based on a discretization of the tool surface with triangular elements is presented. Finally, some selected examples are solved, in order to show the capabilities of the formulation.     

Postbuckling analysis of layered composites using p- version finite strips

This paper presents the results of a study on the buckling and postbuckling analyses of layered shear-deformable composite plates using p-version finite strips. The plates are considered to be sufficiently long so that the effect of boundary conditions along the shorter edges is of little significance; this makes it possible to use the ‘exact’ trigonometric functions in the longitudinal direction. The displacement field in the transverse direction is discretized by p-version finite strips and convergence is studied by p-extension, i.e. keeping the number of strips relatively small whilst increasing the polynomial order till convergence of the buckling load or the postbuckling stiffness is achieved. Detailed convergence studies are presented and comparison with results of earlier investigations are shown wherever appropriate. The results demonstrate the superior convergence characteristics of the p-version approach.     

Analysis of two‐grid methods for reaction‐diffusion equations by expanded mixed finite element methods

We present two efficient methods of two‐grid scheme for the approximation of two‐dimensional semi‐linear reaction‐diffusion equations using an expanded mixed finite element method. To linearize the discretized equations, we use two Newton iterations on the fine grid in our methods. Firstly, we solve an original non‐linear problem on the coarse grid. Then we use twice Newton iterations on the fine grid in our first method, and while in second method we make a correction on the coarse grid between two Newton iterations on the fine grid. These two‐grid ideas are from Xu's work (SIAM J. Sci. Comput. 1994; 15 :231–237; SIAM J. Numer. Anal. 1996; 33 :1759–1777) on standard finite element method. We extend the ideas to the mixed finite element method. Moreover, we obtain the error estimates for two algorithms of two‐grid method. It is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy H =??(h^¼) in the first algorithm and H =??(h^?) in second algorithm. Copyright © 2006 John Wiley & Sons, Ltd.     

A 3-D turbulent flow analysis using finite elements withk-ε model

This paper describes the finite element turbulent flow analysis, which is suitable for three-dimensional large scale problems. Thek- turbulence model as well as the conservation equations of mass and momentum are discretized in space using rather low order elements. Resulting coefficient matrices are evaluated by one-point quadrature in order to reduce the computational storage and the CPU cost. The time integration scheme based on the velocity correction method is employed to obtain steady state solutions. For the verification of this FEM program, two-dimensional plenum flow is simulated and compared with experiment. As the application to three-dimensional practical problems, the turbulent flows in the upper plenum of the fast breeder reactor are calculated for various boundary conditions.     

Optimal discretizations in adaptive finite element electromagnetics

One of the primary objectives of adaptive finite element analysis research is to determine how to effectively discretize a problem in order to obtain a sufficiently accurate solution efficiently. Therefore, the characterization of optimal finite element solution properties could have significant implications on the development of improved adaptive solver technologies. Ultimately, the analysis of optimally discretized systems, in order to learn about ideal solution characteristics, can lead to the design of better feedback refinement criteria for guiding practical adaptive solvers towards optimal solutions efficiently and reliably. A theoretical framework for the qualitative and numerical study of optimal finite element solutions to differential equations of macroscopic electromagnetics is presented in this study for one-, two- and three-dimensional systems. The formulation is based on variational aspects of optimal discretizations for Helmholtz systems that are closely related to the underlying stationarity principle used in computing finite element solutions to continuum problems. In addition, the theory is adequately general and appropriate for the study of a range of electromagnetics problems including static and time-harmonic phenomena. Moreover, finite element discretizations with arbitrary distributions of element sizes and degrees of approximating functions are assumed, so that the implications of the theory for practical h-, p-, hp- and r-type finite element adaption in multidimensional analyses may be examined. Copyright © 2001 John Wiley & Sons, Ltd.     

二维泊松方程问题Lagrange型有限元p型超收敛算法

该文针对二维泊松方程问题的Lagrange型有限元法提出了一种p型超收敛算法。该法受有限元线法对二维问题降维思想的启发,基于网格结点位移的天然超收敛性,通过从网格中取出一行对边相邻的单元作一子域,将子域内各单元另一对边解答取为原有限元解答,在子域上建立真解近似满足的局部偏微分方程边值问题,对该局部边值问题,沿对边方向单向提高单元阶次进行有限元求解获得单元对边上的超收敛解。单元另一对边上的超收敛解可通过另一方向的单元行类似获得。在单元边超收敛解的基础上,依次取出各个单元,以单元边位移超收敛解为Dirichlet边界条件,双向提高单元阶次对原泊松方程问题进行有限元求解即可获得全域超收敛解。数值算例表明,通过简单的后处理计算本法可显著提高解答的精度和收敛阶。     

An <Emphasis Type="Italic">rp</Emphasis>-adaptive finite element method for the deformation theory of plasticity

In this paper, we present an rp-discretization strategy for physically non-linear problems based on a high order finite element formulation. In order to achieve convergence, the p-version leaves the mesh unchanged and increases the polynomial degree of the shape functions locally or globally, whereas the r-method moves nodes and edges of an existing FE-mesh. The basic idea of our rp-version approach is to adjust the finite element mesh to the shape of the elastic–plastic interface in order to take into account the loss of regularity which arises along the curve of the plastic front. Numerical examples will demonstrate that this approach leads to an exponential rate of convergence and highly accurate results.     

Nodal DG-FEM solution of high-order Boussinesq-type equations

A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.