C0 finite element discretization of Kirchhoff's equations of thin plate bending

An alternative formulation of Kirchhoff's equations is given which is amenable to a standard C⁰ finite element discretization. In this formulation, the potential energy of the plate is formulated entirely in terms of rotations, whereas the deflections are the outcome of a subsidiary problem. The nature of the resulting equations is such that C⁰ interpolation can be used on both rotations and deflections. In particular, general classes of triangular and quadrilateral isoparametric elements can be used in conjunction with the method. Unlike other finite element methods which are based on three-dimensional or Mindlin formulations, the present approach deals directly with Kirchhoff's equations of thin plate bending. Excellent accuracy is observed in standard numerical tests using both distorted and undistorted mesh patterns.     

Hybrid stress finite element model for non-linear shell problems

The present paper describes a hybrid stress finite element formulation for geometrically non-linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger-Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi-field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non-linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton-Raphson iteratioin. Numerical examples presented in the paper include complete snap-through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non-linear shell problems of high complexity.     

Three-dimensional finite element for analysing thin plate/shell structures

A three-dimensional (3-D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange-Germain-Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher-order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2-D elements, or to higher-order 3-D continuum elements. The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.     

Numerical manifold space of Hermitian form and application to Kirchhoff's thin plate problems

For second‐order problems, where the behavior is described by second‐order partial differential equations, the numerical manifold method (NMM) has gained great success. Because of difficulties in the construction of the H ²‐regular Lagrangian partition of unity subordinate to the finite element cover; however, few applications of the NMM have been found to fourth‐order problems such as Kirchhoff's thin plate problems. Parallel to the finite element methods, this study constructs the numerical manifold space of the Hermitian form to solve fourth‐order problems. From the minimum potential principle, meanwhile, the mixed primal formulation and the penalized formulation fitted to the NMM for Kirchhoff's thin plate problems are derived. The typical examples indicate that by the proposed procedures, even those earliest developed elements in the finite element history, such as Zienkiewicz's plate element, regain their vigor. Copyright © 2013 John Wiley & Sons, Ltd.     

The application of the least squates finite element method to Abel's integral equation

Abel's integral equation is the governing equation for certain problems in physics and engineering, such as radiation from distributed sources. The finite element method for the soultion of this non-linear equation is presented for problems with cylindrical symmetry and the extension to more general integral equations is indicated. The technique was applied to an axisymmetric glow discharge problem and the results show excellent agreement with previously obtained solutions.     

Modelling of muscle behaviour by the finite element method using Hill's three-element model

We present a numerical algorithm for the determination of muscle response by the finite element method. Hill's three-element model is used as a basis for our analysis. The model consists of one linear elastic element, coupled in parallel with one non-linear elastic element, and one non-linear contractile element connected in series. An activation function is defined for the model in order to describe a time-dependent character of the contractile element with respect to stimulation. Complex mechanical response of muscle, accounting for non-linear force–displacement relation and change of geometrical shape, is possible by the finite element method. In an incremental-iterative scheme of calculation of equilibrium configurations of a muscle, the key step is determination of stresses corresponding to a strain increment. We present here the stress calculation for Hill's model which is reduced to the solution of one non-linear equation with respect to the stretch increment of the serial elastic element. The muscle fibers can be arbitrarily oriented in space and we give a corresponding computational procedure of calculation of nodal forces and stiffness of finite elements. The proposed computational scheme is built in our FE package PAK, so that real muscles of complex three-dimensional shapes can be modelled. In numerical examples we illustrate the main characteristic of the developed numerical model and the possibilities of solution of real problems in muscle functioning. © 1998 John Wiley & Sons, Ltd.     

p-version least-squares finite element formulation of Burgers' equation

A p-version least-squares finite element formulation for non-linear problems is presented and applied to the steady-state, one-dimensional Burgers' equation. The second-order equation is recast as a set of first-order equations which permit the use of C⁰ elements. The primary and auxiliary variables are approximated using equal-order p-version hierarchical approximation functions. The system of non-linear simultaneous algebraic equations resulting from the least-squares process is solved using Newton's method with a line search. The use of ‘exact’ and ‘reduced’ quadrature rules is investigated and the results are compared. The formulation is found to produce excellent results when the ‘exact’ integration rule is used. The combination of least-squares finite element formulation and p-version works extremely well for Burgers' equation and appears to have great potential in fluid dynamics problems.     

Locking of thin plate/shell elements

Often, finite element solutions of thin plate/shell elements become very stiff and the displacement field solutions diverge from those predicted by Kirchhoff's theory. This phenomenon is known as the locking phenomenon. A theoretical fomulation demonstrating its existence is developed, and results of finite element analysis of a single element and mesh are discussed. This leads to a sufficient and necessary criterion which must be satisfied to avoid the locking phenomenon.     

Boundary element solution of heat conduction problems in multizone bodies of non-linear material

A novel algorithm for handling material non-linearities in bodies consisting of subregions having different, temperature dependent heat conductivities is developed. The technique is based on Kirchhoff's transformation. The material non-linearity is reduced to a solution dependent function of unified form added to unknown nodal Kirchhoff's transforms. Assembling of element contributions brings the non-linearity to the right hand sides of the global set of equations. The first step of the solution of this set is the Gaussian pre-elimination (condensation) of linear degrees of freedom. At this stage efficient block solvers can be used. Then, a set of non-linear equations is extracted from the condensed one and solved employing the Newton-Raphson technique. The iteratively solved set consists of the least possible number of equations and its Jacobian matrix is calculated efficiently by taking advantage of the specific form of the equations.     

Fluid-structure interaction analysis by the finite element method–a variational approach

We have developed a finite element method for analysing non-linear and linear fluid-structure interaction problems by working directly from a variational indicator based on Hamilton's principle. We restrict our analyses to inviscid, irrotational and isentropic fluid flows. The variational indicator includes the fluid potential energy due to gravity, which is often ignored. This and the fact that we consider our domain to be variable provide us with the capability to model free surfaces. We demonstrate the effectiveness of both linear and non-linear finite element formulations in analysing a variety of fluid-structure interaction problems.     

3D modeling of electromagnets fed by alternating voltage sources

A method allowing 3D simulation of AC voltage driven electromagnets is described. Since the harmonic current in the exciting coil depends on the air gap width, the electromotive force in the windings has to be taken into account. A computation method based on the finite element method was developed in order to obtain the static attractive force characteristics of AC electromagnets used for circuit breakers. This method is based on nodal finite elements and loop equations of the electric circuit. Both magnetic field and electric circuit equations are solved simultaneously in the frequency domain. Computation results are compared with measurements on a 32-A contractor/circuit breaker     

Computational aspects of vector-like parametrization of three-dimensional finite rotations

Theoretical and computational aspects of vector-like parametrization of three-dimensional finite rotations, which uses only three rotation parameters, are examined in detail in this work. The relationship of the proposed parametrization with the intrinsic representation of finite rotations (via an orthogonal matrix) is clearly identified. Careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of Reissner's non-linear beam theory. Pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2-node three-dimensional beam element. A number of numerical simulations in three-dimensional finite rotation analysis are presented in order to illustrate the proposed approach.     

Hygrothermomechanical evaluation of porous media under finite deformation. Part I - finite element formulations

The coupled thermomechanical responses of fluid-saturated porous continua subjected to finite deformation are investigated. Field equations governing the transient response of the media are derived from a continuum thermodynamics mixture theory based on mass balance, momentum balance and energy balance laws as well as the Clausius-Duhem inequality. Finite element procedures for the two-dimensional response, employing updated Lagrangian formulations for the solid skeleton deformation and the weak formulations for fluid and thermal transport equations, are implemented in a fully implicit form. Temperature-dependent mechanical properties for the non-linear solid matrix, characterized by Perzyna's viscoplastic model, are assumed. An iterative scheme based on the full Newton-Raphson method is presented for simultaneously solving the coupled non-linear equations.     

A finite element formulation based on Nadai's deformation theory for elasto-plastic analysis

A numerical method of the Newton-Raphson type is presented for elasto-plastic analysis using the finite element method. The method is developed from Nadai's deformation theory and Hooke's law. Numerical examples are used to show that the method provides very rapid solution convergence.     

Superposition method of finite element and analytical solutions for transient creep analysis

Based on the Lagrange multiplier's concept, a superposition method of analytical and finite element solutions has been developed to solve efficiently various non-linear and/or time-dependent problems in structural mechanics. According to the theory, the transient creep behaviour of a cantilever beam is analysed as an expository example.     

A normalization scheme for comparing eddy current differential probe signals using a finite element formulation

A first-order finite element formulation is used to model an eddy current differential bobbin coil probe scanning a tube with axisymmetric flaws. A multifrequency signal normalization scheme is developed to allow direct comparison between experimental measurements of the differential bobbin coil probe signal and finite element calculations of the probe coil impedance. Results demonstrate that both magnitude and phase of the differential bobbin coil impedance are useful in characterizing flaws in tubing for multifrequency scans.     

Thin shells with finite rotations formulated in biot stresses: Theory and finite element formulation

A bending theory for thin shells undergoing finite rotations is presented, and its associated finite element model is described. The kinematic assumption is based on a shear elastic Reissner-Mindlin theory. The starting point for the derivation of the strain measures are the resultant equilibrium equations and the associated principle of virtual work. Within this formulation the polar decomposition of the shell material deformation gradient leads to symmetric strain measures. The associated work-conjugate stress resultants and stress couples are integrals of the Biot stress tensor. This tensor is invariant with respect to rigid body motions and, therefore, appropriate for the formulation of constitutive equations. Finite rotations are introduced via Eulerian angles. The finite element discretization of arbitrary shells is based on the isoparametric concept formulated with respect to a plane reference configuration. The numerical model is applied to different non-linear plate and shell problems and compared with existing formulations. Due to a consistent linearization, the step size of a load increment is only limited by the local convergence behaviour of Newton's method.     

A fully non-linear axisymmetrical quasi-kirchhoff-type shell element for rubber-like materials

An axisymmetrical shell element for large deformations is developed by using Ogden's non-linear elastic material law. This constitutive equation, however, demands the neglect of transverse shear deformations in order to yield a consistent theory. Therefore, the theory can be applied to thin shells only. Eventually a ‘quasi-Kirchhoff-type theory’ emerges. Within this approach the computation of the deformed director vector d is a main assumption which is essential to describe the fully non-linear bending behaviour. Furthermore, special attention is paid to the linearization procedure in order to obtain quadratic convergence behaviour within Newton's method. Finally, the finite element formulation for a conical two-node element is given. Several examples show the applicability and performance of the proposed formulation.     

Least-squares finite elements for first-order hyperbolic systems

A class of least-squares finite element methods has been developed for first-order systems and here we study this approach for hyperbolic problems. The formulation of the least-squares method is developed in detail and compared with the Petrov-Galerkin and Taylor-Galerkin procedures. A stability analysis is carried out and the extension to the non-linear problem described. Numerical comparison studies demonstrate the performance of the method and suggest that it is a promising alternative to existing schemes. Applications considered include the convection equation, inviscid Burger's equation and shallow-water equations.     

Magnetic resonance imaging gradient coil design by combiningoptimization techniques with the finite element method

In this paper, the optimization techniques of complex method, steepest descent, and conjugate gradient are investigated in terms of their convergence behaviors. The conjugate gradient method is then combined with finite element analysis techniques to develop a magnetic resonance imaging (MRI) G_z gradient coil design strategy which maximizes the field linearity within a specified region of interest. It is found that conjugate gradient optimization in conjunction with the finite element method is a powerful and flexible coil design approach with the potential to incorporate complex coil geometries, inhomogeneous media, and transient current excitation