Standard FEM Research Articles

Exponential convergence in a Galerkin least squares hp-FEM for Stokes flow

A stabilized hp‐finite element method (FEM) of Galerkin least squares (GLS) type is analysed for the Stokes equations in polygonal domains. Contrary to the standard Galerkin FEM, the GLSFEM admits the implementationally attractive equal‐order interpolation in the velocity and the pressure. In conjunction with geometrically refined meshes and linearly increasing approximation orders it is shown that the hp‐GLSFEM leads to exponential rates of convergence for solutions exhibiting singularities near corners. To obtain this result a novel hp‐interpolation result is proved that allows the approximation of pressure functions in certain weighted Sobolev spaces in a conforming way and at exponential rates of convergence on geometric meshes.

Modeling of Dislocations in an Epitaxial Island Structure

ABSTRACTWe present calculations of dislocations in CoSi2 islands grown by reactive epitaxy on a Si(111) substrate. The stress fields due to the lattice mismatch are calculated with standard FEM techniques, and are converted into a structured, multi-level and multi-grid stress table that is imported into the PARANOID code to study the dislocation dynamics. Single and multiple dislocations in the island have been simulated, and the predicted patterns are strikingly similar to those observed experimentally. By looking at the growth behavior of very small loops we also find that dislocation-loop nucleation becomes easier as the islands become larger, and that thick islands are dislocated at smaller sizes than thin ones. These results are also in good agreement with experimental observations. We conclude that current modeling techniques are sufficient to treat this type of problem at a useful level of accuracy.

The first locking-free plane-elastic finite element: historia mathematica

We consider a two-parameter family of conforming bilinear quadrilateral finite element formulations for plane-elastic problems with homogeneous isotropic material. The approach, taken earlier by Belytschko et al. is based on numerical strain reductions and stabilization and may be viewed as an extension & reformulation of the classical Turner rectangle of 1956. We present the full theory of this approach in the energy norm framework. The analysis is based on splitting the error in two orthogonal components, the approximation error and the equilibrium or consistency error. The theory covers the full parametric dependence of the plane-elastic problem, emphasizing in particular the case of beam-like bodies with arbitrarily small aspect ratios. The theory shows that, depending on the mesh, the stabilized reduced-strain formulation is capable of avoiding some, or even all, the parametric locking effects characteristic to 2D elastic problems. We analyze first these locking effects in the context of standard bilinear FEM.

Hybrid coupled finite–boundary element methods for elliptic systems of second order

In this hybrid method, we consider, in addition to traditional finite elements, the Trefftz elements for which the governing equations of equilibrium are required to be satisfied a priori within the subdomain elements. If the Trefftz elements are modelled with boundary potentials supported by the individual element boundaries, this defines the so-called macro-elements. These allow one to handle in particular situations involving singular features such as cracks, inclusions, corners and notches providing a locally high resolution of the desired stress fields, in combination with a traditional global variational FEM analysis. The global stiffness matrix is here sparse as the one in conventional FEM. In addition, with slight modifications, the macro-elements can be incorporated into standard commercial FEM codes. The coupling between the elements is modelled by using a generalized compatibility condition in a weak sense with additional elements on the skeleton. The latter allows us to relax the continuity requirements for the global solution field. In particular, the mesh points of the macro-elements can be chosen independently of the nodes of the FEM structure. This approach permits the combination of independent meshes and also the exploitation of modern parallel computing facilities. We present here the formulation of the method and its functional analytic setting as well as corresponding discretizations and asymptotic error estimates. For illustration, we include some computational results in two- and three-dimensional elasticity.

Dynamical Influence Lines At Drop Impact OfTransport Masses

The safety enhancement measures of the Paks NPP (Hungary) included the investigation of the drop impact check of reinforced concrete structures strength. A FE model of the structures was generated and a standard FEM computer program was used to calculate the displacements and internal forces. However, the problem is that the place of force vector and initial velocity vector on the slab are unknown. If we want to obtain the maximum of displacements and different internal forces, we have to drop the body on different points of the slab. To decide the maximum of an internal force we have to use dynamic influence lines of the different internal force. On this basis the strength of the existing structures were assessed. The results showed that above the reactor hermetic containment in the case of a minimal drop height the strength capacity is adequate, but in the case of higher drop heights special measures should be taken.

Beam analogy for optimal control of linear dynamic systems

Optimal control problems for linear dynamic systems with quadratic performance index are solved using the beam analogy. The governing equations for the optimal maneuver are derived in the form of coupled fourth order differential equations in the time domain. These equations are uncoupled using modal variables. Next, each independent equation is made analogous to the corresponding problem of a beam on an elastic foundation. The beam problem in the spatial domain is solved using standard FEM software. Finally the FEM results are transferred back to the time domain where they represent the optimal trajectories and controls for the dynamic system.

Numerical experiments with preconditioning by gram matrix approximation for non-linear elliptic equations

The paper analyzes the numerical performances of the Gram matrix approximations preconditioning for solving non-linear elliptic equations. The two test problems are non-linear 2-D elliptic equations which describe: (1) the Plateau problem and (2) the general pseudohomogeneous model of the catalytic chemical reactor. The standard FEM with piecewise linear test and trial functions is used for discretization. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the preconditioned generalized conjugate gradient methods (CGS and GMRES) as inner iteration. The Gram matrix approximations consist in replacing the exact solution of the equation with the preconditioner by few iterations of an appropriate iterative scheme. Two iterative algorithms are tested: incomplete Cholesky and multigrid. Numerical experiments indicate that preconditioners improve the convergence properties of the algorithms for both test problems. At the second test problem the numerical performances deteriorate at relatively high values of the Pe numbers.

A Reflection Free Boundary Condition for Propagation in Uniform Flow Using Mapped Infinite Wave Envelope Elements

Variable order mapped infinite wave envelope elements are developed for finite element modeling of acoustic radiation in a uniformly moving medium. These elements are used as a nonreflecting boundary condition for computations on an infinite domain in which a radiating body is immersed in a moving medium which is essentially undisturbed outside of the near field. The mapped elements provide a boundary condition equivalent to element stiffness, mass, and damping matrices appended to an inner standard FEM mesh. A demonstration of the performance of mapped elements as influenced by element order is given in the context of acoustic radiation from a turbofan inlet and exhaust.

A REFLECTION FREE BOUNDARY CONDITION FOR PROPAGATION IN UNIFORM FLOW USING MAPPED INFINITE WAVE ENVELOPE ELEMENTS

Variable order mapped infinite wave envelope elements are developed for finite element modeling of acoustic radiation in a uniformly moving medium. These elements are used as a nonreflecting boundary condition for computations on an infinite domain in which a radiating body is immersed in a moving medium which is essentially undisturbed outside of the near field. The mapped elements provide a boundary condition equivalent to element stiffness, mass, and damping matrices appended to an inner standard FEM mesh. A demonstration of the performance of mapped elements as influenced by element order is given in the context of acoustic radiation from a turbofan inlet and exhaust.

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

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.

Standard and nonstandard Galerkin FEM for numerical acoustics

Over the last decade, several modern nonstandard modifications of the standard Galerkin finite-element method have been developed for, or applied to, the numerical solution of the Helmholtz equation. Most of these concepts, like Galerkin least-squares or bubble methods, have been first developed and demonstrated in one dimension, where also a rigorous analysis of the so-called preasymptotic error behavior of the standard Galerkin FEM is available. In this talk, a short overview of these topics will be given and the extension of the methods/results to higher dimensions will be discussed, where practical problems are formulated.

A reflection free boundary condition for propagation in uniform flow using mapped infinite wave envelope elements

A new reflection free boundary condition based on mapped infinite wave envelope elements is developed for finite-element modeling of acoustic radiation in a uniformly moving medium. These elements are useful for obtaining closure for computations on an infinite domain in which a radiating body is immersed in a moving medium which is essentially undisturbed outside of the near field. The application described is for harmonic excitation which leads to a model independent of time. An additional result of this study shows that the mapped wave envelope elements provide a boundary condition equivalent to stiffness, mass, and damping matrices appended to the inner mesh. By choosing the transition between the standard FEM mesh and the mapped WE as a surface of constant phase, the mass matrix is caused to vanish identically. This has implications for transient FEM modeling of acoustic radiation. A demonstration of the characteristics of the mapped WE elements is given in the context of acoustic radiation from a turbofan inlet for which benchmark results are known. It is found that for similar computational cost mapped infinite wave envelope elements are superior to conventional wave envelope elements for time-independent formulations.

The application of a laser scanning vibrometer for dynamic characterization of large impellers

The application of a laser scanning vibrometer to detect resonance frequencies and mode shapes of large impeller rotors is illustrated. The dynamic analysis of the structure has been previously performed by means of a standard FEM code. The key to success was the ability of the laser vibrometer to detect very low vibration amplitudes (of the order of nanometers) with a high spatial resolution, inside an industrial environment, in very low testing time; all fundamentals for this application. The experimental and numerical activity presented in this work has been carried out at Nuovo Pignone together with the University of Ancona with the final purpose of establishing a new, fast measurement technique to identify, in a production industrial environment, the dynamic behaviour of large dimension impellers for centrifugal compressors.

SHAPE OPTIMIZATION OF A 2D BODY SUBJECTED TO SEVERAL LOADING CONDITIONS

An algorithm for shape optimization based on simultaneous solution of the equations and inequalities arising from Kuhn-Tucker necessary conditions is presented. Regular triangular FE assembly is proposed. Element vertices are associated with design variables directly or through spline parameters defining the boundary of the optimized body. This way, during the iteration procedure, FE assembly is automatically remeshed together with the motion of the optimized boundary. Multiple loading conditions are represented in the problem as equality conditions in the form of a set of equilibrium equations for each loading condition separately. From the necessary condition equations an additional, important relation between cost function, Lagrange multipliers associated with inequality constraints and their limit values is derived. The algorithm combines standard professional FEM programs with an optimizer proposed in the paper which is illustrated with shape optimization of several 2D bodies. The proposed approach is theoretically rigorous and relatively simple for practical applications, and allows considerations sensitivities, adjoint systems and constraints linearization to be avoided.

Analysis of intergranular creep crack growth by means of grain-elements

A new computational technique is discussed to analyze intergranular creep crack growth in a discrete manner. A standard continuum FEM description is matched to a crack tip region in which grains are represented individually by so-called grain-elements and grain boundaries by special interface-elements. These grain boundary elements incorporate grain boundary sliding and the nucleation and creep-diffusive growth of cavities until coalescence and microcracking. Some preliminary results are presented for C * controlled, Mode I crack growth under small-scale damage conditions.

Finite element analysis of unbounded non-linear transient magnetic fields

This paper presents an iterative FEM procedure for the solution of unbounded non-linear transient magnetic field problems. The boundlessness of the domain is dealt with by introducing a fictitious boundary defining a bounded region. At each discrete time, by assuming a first guess for the Dirichlet condition on such a boundary a standard FEM solution for the bounded non-linear transient problem is carried out; the results of the FEM analysis are used to update the Dirichlet condition on the fictitious boundary; the above steps are then iterated until a solution to the unbounded problem is reached. The procedure is rapidly convergent, accurate and easy to implement.

RECOVERY BY EQUILIBRIUM IN PATCHES (REP)

A new recovery technique is developed in this paper. It is shown that, for many elements, the behaviour of the method is very similar to that of SPR. Because it does not need to identify super-convergent points, it is applicable for any form of element in which these points are not defined. The formulation is very simple and is based on equilibrating the recovered stresses, in the patch, in the same way that the standard FEM does. This procedure leads to a weak form of equilibrium equations of new stresses on the patch and consequently to answers satisfying the discrete equilibrium conditions. The formulation is consistent with non-linear formulations which iteratively equilibrate the problem. Therefore, this method can be used to project the Gauss points values to nodal points, with minimum disturbance of the global equilibrium. © 1997 by John Wiley & Sons, Ltd.

Elimination of ill-posedness associated with shell problems

After identifying the reason for the numerical difficulty associated with the treatment of shell type potential problems by standard FEM, the paper introduces the gap-jump-float (GJF) method, an extension of gap element method (GEM), which eliminates the ill-posedness for arbitrary /spl sigma//sub i/. The applied float and jump functions are linear combinations of conventional shape functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Electronic band structure for two‐dimensional periodic lattice quantum configurations by the finite element method

AbstractA finite element method formulation is given for solving Schrodinger's wave equation for a single electron in a crystal lattice cell subject to a known periodic potential. The formulation has been implemented for a two‐dimensional lattice, with an arbitrary potential profile, modelled by quadratic isoparametric elements, The FEM solver returns a specified number of electronic energy states, En, and nodal values of the complex wavefunctionψn Input data is generated by a standard FEM mesh generator. The postprocessing, given n, for reproducing a full 2‐D E‐k Brillouin diagram and given k, the electronic distribution, has been implemented. Tests on a 2‐D generalized Kronig‐Penney energy band model showed excellent agreement; between FEM results and analysis. The solver was further satisfactorily checked against published augmented plane wave calculations for a circular potential well within a square lattice. Specimen results are presented for the same circular well but with graded potential distributions and for a rectangular potential barrier set askew in a square lattice. Two‐dimensional energy band solvers have application to superlattice nanostruc‐ tures, whilst a general, full 3‐D FEM quantum solver seems feasible.

Combined structures-controls optimization of lattice trusses

The purpose of this paper is to demonstrate concretely the role that distributed parameter models can play in CSI, in particular in combined structures-controls optimization problems of importance in preliminary design. Closed form solutions can be obtained for performance criteria such as rms attitude error, making possible analytical solutions of the optimization problem. This in contrast to the need for numerical computer solution involving the inversion of large matrices in traditional finite element model use. Another advantage of the analytic solution is that it can provide much needed insight into phenomena that can otherwise be obscured or difficult to discern from numerical computer results. As a compromise in level of complexity between a toy laboratory model and a real space structure we have chosen the lattice truss used in the earth pointing satellite (EPS). The optimization problem chosen is a generic one: of minimizing the structure mass subject to a specified stability margin and to a specified upper bound on the rms attitude error (tip response), using a co-located controller and sensors. Standard FEM, treating each bar as a truss element is used, while the continuum model is an anisotropic Timoshenko beam model. Performance criteria are derived for either model, except that for the distributed parameter model we obtain explicit closed form solutions. Numerical results obtained by the two models show complete agreement. Based on the continuum model we obtain a solution to the problem of optimal placement of actuators to minimize mean square attitude error. A canonical optimization problem is examined and shown to be trivial, and even capable of analytical solution, using the continuum model performance criteria formulas in contrast to the complex computer solutions based on FEM or truncated modal models currently in vogue.