Finite Element Assembly Research Articles

Simulation of two‐ and three‐dimensional internal subsonic flows using a finite element method

AbstractIn this paper a finite element method is presented to predict internal subsonic flows. Using a low‐Mach‐number approximation, the pressure is decomposed into a mean thermodynamic contribution and a dynamic fluctuation to deal with the complex role of the pressure in internal aerodynamics. A semi‐implicit time integration and a finite element method with a moving mesh are described to take into account complex geometries and moving boundaries. An Uzawa algorithm accelerated by a preconditioned residual method is introduced to solve the coupled non‐symmetric linear system for the velocity components and the pressure. An efficient conjugate gradient method combined with an incomplete LU preconditioning is used to solve the non‐symmetric linear systems arising from the discretization. The implementation of the numerical scheme on parallel supercomputers is also discussed. Efficient algorithms for the finite element assembly phase and for the solution of linear systems are described which take advantage of the parallel architecture of the new generation of supercomputers. With this technique a global speed‐up of 10 is achieved on a supercomputer with eight processors. To illustrate the capabilities of the numerical method, 2D and 3D simulations of flows in the combustion chamber of a reciprocating engine and around the combustor dome of a gas turbine engine are presented.

Heating and melting deformable models

AbstractWe develop physically‐based graphics models of non‐rigid objects capable of heat conduction, thermoelasticity, melting and fluid‐like behaviour in the molten state. These deformable models feature non‐rigid dynamics governed by Lagrangian equations of motion and conductive heat transfer governed by the heat equation for non‐homogeneous, non‐isotropic media. In its solid state, the discretized model is an assembly of hexahedral finite elements in which thermoelastic units interconnect particles situated in a lattice. The stiffness of a thermoelastic unit decreases as its temperature increases, and the unit fuses when its temperature exceeds the melting point. The molten state of the model involves a molecular dynamics simulation in which ‘fluid’ particles that have broken free from the lattice interact through long‐range attraction forces and short‐range repulsion forces. We present a physically‐based animation of a thermoelastic model in a simulated physical world populated by hot constraint surfaces.

Moment coefficients of incompressible-flow seals with conically whirling rotors

A computational model is developed to investigate the dynamic behavior of a rotor that is in contact with a fluid, to infinitesimally small disturbance modes. The rotor here is that of an annular seal, commonly used in turbomachinery applications, with an incompressible flow in the annular clearance gap. Under consideration is an angular excitation of the rotor axis, coupled with a whirling motion around the housing centerline at a finite whirl frequency. The fluid response in this case is quantified in the form of direct and cross-coupled moment coefficients, which constitute a measure of the stiffness, damping and inertia effects of the rotor/fluid interaction. Uniqueness of the computational model stems from the manner in which the rotor eccentricity is physically perceived and subsequently incorporated. It is first established that the fluid reaction components are the result of infinitesimally small deformations of varied magnitudes that are experienced by an assembly of finite elements in the rotor-to-housing gap as the gap becomes distorted due to the rotor virtual eccentricity. The idea is then cast into a perturbation model in which the perturbation equations emerge from the flow-governing equations in their discrete finite-element form, as opposed to the differential form, which is traditionally the case. The computational model is potentially applicable to a wide range of rotor-to-housing pasage configurations where ellipticity of the passage flow field is too significant to ignore. The numerical results are compared to those obtained through an existing bulk-flow model which is particularly suited for the leakage passage in annular seals and other similarly simple passage configurations.

Use of incompatible displacement modes for the calculation of element stiffnesses or stresses

The addition of incompatible displacement modes to lower-order displacement-based elements is re-evaluated. Recent research has indicated that a simple numerical correction can be applied to the shape functions in order that the constant-strain patch test is passed. In this paper a new method of stress recovery is presented in which incompatible modes are introduced. A least-square approximation is used to calculate element stresses which are in microscopic equilibrium and tend to be in global equilibrium with the applied nodal loads on the finite element assemblage. Also, a consistent and robust method for the evaluation of thermal stresses is presented. The numerical methods presented are general and can be applied to all displacement-based finite elements. The basic formulation and examples which are presented in this paper are in three-dimensional elasticity using the eight-node isoparametric elements. Accuracy of the element is illustrated and it is demonstrated that both displacements and stresses are almost identical to those produced by Pian's hybrid stress elements.

Variational formulation of a mixed method for second order frame analysis

A variational finite element formulation is developed for a mixed method of analysis of plane frames. The governing equilibrium and compatibility equations for the complete structure are formed through a finite element assembly procedure using matrices and vectors derived at the member level. In particular, modifications required to include the second order effects of stiffness reduction due to compressive axial forces and P-delta effects are shown. The procedures required to deal with plastic hinge formation are also discussed.

Two-dimensional semiconductor device analysis based on new finite-element discretization employing the S-G scheme

To make the application of the finite-element method practical in semiconductor device simulation, the authors have applied the Scharfetter-Gummel (S-G) scheme in conjunction with an accurate seven-point Gaussian quadrature rule to the assembly of the finite-element stiffness matrices and the right-hand-side vector of the semiconductor equations. The key of this method lies in accurate interpolation rules, which are derived on the basis of simple device physics considerations. The inherent simplicity and flexibility in the finite element formulation make the new method applicable to multidimensional problems. The simplicity of embedding the S-G scheme in the quadrature of finite-element assembly lends itself to all kinds of finite-element methods using various elements, shape functions, and weightings. The resultant exponential functional fitting avoids high discretization errors usually incurred by the classical finite-element discretization method. Solutions with high accuracy, even on coarse mesh, and a significant speed-up of convergence rate are obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Finite element modeling of large capacity stiffened steel tee-hanger connections

This paper presents the development of a mathematical model describing the behavior of a stiffened steel tee-hanger connection. The particular tee-hanger connection has two rows of pretensioned high-strength bolts on either side of the tee-stem (eight bolts total). This geometric configuration results in a highly indeterminate problem as the bolt forces cannot be determined directly. Thus, an analytical study with modeling of the connection as an assemblage of finite elements was conducted. Only one-quarter of a symmetric section of the tee-hanger was analysed with eight-noded isoparametric brick elements used for the tee-flange plate elements. Bi-linear stress-strain behavior is considered for both the bolt and plate material. To consider the inelastic steel behavior in each cycle, the elastic moduli of the yielded elements are reset to their secant values. Analytical results are compared with results from tee-hanger connection tests. Based on comparison of experimental and analytical results, it is concluded that the model developed adequately explains the connection behavior. The study is restricted to A36 steel and A325 bolts with maximum diameter of 1 1 2 inches.

Degenerate Elements for Combined Flexural and Torsional Analysis of Thin‐Walled Structures

Finite element idealization of thin‐walled structural members under combined flexural and torsional deformation, using the degeneration concept, is presented. The warping displacement field is discretized by an assembly of finite elements which together forms a specific thin‐walled cross section. Based on the assumption that the distortion in the plane of the thin‐walled cross section is negligible, only three “sectional” degrees of freedom are employed in common for all elements of the same cross section. This greatly reduced the overall size of the degrees of freedom. Several thin‐walled structural problems with torsional restraints are solved to illustrate the versatility and the efficiency of the proposed elements.

A Multiple Degree of Freedom Cavity Element for Acoustic Transmission Lines

Mathematical modelling of acoustic transmission lines containing elements which can be represented by their acoustic modes is considered. Of particular interest are tapered elements and acoustic cavities which operate at frequencies above which a simple volume expansion representation or a plane wave representation is adequate. An element matrix formulation is given which is suitable for inclusion in the Acoustic Wave Finite Element assembly procedure, or with suitable modifications, to the more traditional transfer matrix approach. Numerical examples are provided demonstrating the accuracy of the procedure.

On the application of an element-by-element lanczos solver to large offshore structural engineering problems

Iterative methods for solving large sets of linear equations have been used as an alternative to direct methods of solution since the early beginning of numerical analysis. The conjugate gradient method (CCM), one of the most widely used, seeks a solution that minimizes the potential energy of the finite element assemblage. Recently, the use of Lanczos algorithm for the solution of large sets of linear equations has been re-examined. Lanczos biorthogonalization procedure is an oblique projection method that provides a solution approximation whose residual is orthogonal to a Krylov subspace. It has been shown that Lanczos and CGM share several properties but the former has the advantage of not being necessary to compute the approximated solution at each iteration. Jacobi preconditioning can also be employed in order to accelerate convergence. The Lanczos procedure was implemented using an element-by-element (EBE) scheme. The applications spread over typical offshore engineering problems encompassing regular and irregular meshes. These problems are normally ill-conditioned when compared with continuum problems. For all the analyses addressed the element-by-element Lanczos procedures presented outstanding computational efficiency.

Finite element analysis of mixed-mode thermoelastic fracture problems

This paper presents a rigorous hybrid finite element procedure for the analysis of the thermoelastic problems with mixed-mode cracks. The singular character of the temperature gradient which significantly affects the distribution of thermal stresses near the crack-tip is precisely described. The calculations of temperature and thermal stress fields are carried out by the finite element assemblage in which hybrid singular elements are used around the crack-tip and high-order isoparametric regular elements are taken elsewhere. To determine the mixed-mode stress intensity factors, both the direct extrapolation method based on the finite element solutions and the indirect method using the modified J k integrals accounting for thermal effects have been proposed. To avoid the underestimation of stress intensity factors obtained by quarter-point singular elements, the important role of the hybrid singular elements developed is also demonstrated. For verification purposes, several pure mode and mixed-mode problems are solved. Excellent correlations between the computer results and available referenced solutions are drawn.

An assembled surface normal algorithm for interior node removal in three-dimensional finite element meshes

An algorithm is developed for identifying interior nodes and edges in finite element meshes of three-dimensional solids. The algorithm is extremely simple and quite fast. The interior nodes are identified by assembling the normals to all surfaces of the model according to the usual finite element assembly procedure. Comparison with a sort-search algorithm shows a tenfold reduction in running time; running time is a linear function of the number of nodes or elements.

Seismic analysis of three-dimensional soil-structure interaction system on a rectangular base

An analytical procedure is presented for determining the three-dimensional interaction effects of a soil-structure system under seismic excitation. The structure is modeled as an assembly of finite elements representing the three-dimensional superstructure that is attached to a rectangular base on a half-space. Using triple Fourier transform techniques, dynamic displacement equations of the three-dimensional elastic half-space are solved. Boundary conditions on the surface of the half-space are expanded into a Fourier series to generate Dirac delta functions that simplify greatly the inversion of the Fourier transforms. Foundation impedances are then obtained from the dynamic displacement functions of the rectangular bases on the halfspace with water table and elastic half-space subjected to harmonic loading. Solutions are presented in closed form. Based on specific values of some of the basic parameters of the problem, numerical results are obtained and presented in the form of a series of curves associated with the dynamic response of a rectangular base. Solutions are also obtained for the soilstructure interaction analysis of three-dimensional structures subjected to earthquake excitation.

A mixed-mode crack analysis of rotating disk using finite element method

This paper presents a rigorous elastodynamic hybrid-displacement finite element procedure for a safety analysis of fast rotating disks with mixed-mode cracks. Based on a modified Hamilton's principle, the finite element model is derived such that the proper crack-tip singularities are taken into consideration and the interelement displacement compatibility conditions are still satisfied. Thus, the specimen can be represented by a finite element assemblage in which “singular” elements are used around the crack-tip and high-order isoparametric “regular” elements are taken elsewhere. To determine the mixed-mode stress intensity factors, the modified J ̃ k integrals for rotating cracked disks have been established taking into account the effect of centrifugal force. Using the “strain-energy-density factor” concept, the direction of crack growth of a rotating disk with an arbitrary internal crack is predicted. To provide a method of non-destructive testing in evaluating the integrity of structures, natural vibrations of cracked disk are then studied. Lastly, the influence of inertia effects due to rotating speed changes in determining the dynamic stress intensity factors is examined. For verification purposes, the simple case of a rotating disk with radial cracks is first solved. Excellent correlations between the computed results and available referenced solutions are drawn. New solutions for the circular disk with circumferential or arbitrarily-oriented cracks are also presented.

Nonlinear dynamic analysis of complex structures

AbstractA general step‐by‐step solution technique is presented for the evaluation of the dynamic response of structural systems with physical and geometrical nonlinearities. The algorithm is stable for all time increments and in the analysis of linear systems introduces a predictable amount of error for a specified time step. Guidelines are given for the selection of the time step size for different types of dynamic loadings. The method can be applied to the static and dynamic analysis of both discrete structural systems and continuous solids idealized as an assemblage of finite elements. Results of several nonlinear analyses are presented and compared with results obtained by other methods and from experiments.

Theoretical Formulation of Finite-Element Methods in Linear-Elastic Analysis of General Shells∗

Abstract A systematic classification of the variational functional whose stationarity conditions (Euler equations) can be used alternately to solve for the various unknowns in a boundary-value problem in linear-shell theory is made. The application of these alternate variational principles to a finite-element assembly of a shell and thus, the development of the properties of an individual discrete element are studied in detail. A classification of the finite-element methods, formulated from the variational principles by systematically relaxing the continuity requirements at the interelement boundaries of adjoining discrete elements is made.

ASKA — Automatic System for Kinematic Analysis: A universal system for structural analysis based on the matrix displacement (finite element) method

The system ASKA (Automatic System of Kinematic Analysis) implements the matrix displacement method, using a discretisation technique. Thereby the complete system is idealised into an assembly of finite elements to which a simplified strain pattern is ascribed. The kinematic compatibility conditions are in this procedure satisfied everywhere, but the static compatibility conditions are as a rule only satisfied at the so-called nodal points where the grid-lines of adjoining elements meet. Finally, a system of equations is set up in the degrees of freedom at the nodal points. This method effectively solves a physically idealised system in an as exact a manner as the finite bits operations on the digital computer allow. Through the years ASKA has been kept up to date by implementing the latest state of the art, be it structural or numerical innovations. For example, the element library, containing about 30 families of elements, certainly represents one of the superior systems today.