Transition finite elements with temperature and temperature gradients as primary variables for axisymmetric heat conduction

This paper presents finite element formulation for a special class of elements referred to as “transition finite elements” for axisymmetric heat conduction. The transition elements are necessary in applications requiring the use of both axisymmetric solid elements and axisymmetric shell elements. The elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodal temperatures as well as nodal temperature gradients are retained as primary variables. The weak formulation of the Fourier heat conduction equation is constructed in the cylindrical co-ordinate system (r, z). The element geometry is defined in terms of the co-ordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The element temperature field is approximated in terms of element approximation functions, nodal temperatures and the nodal temperature gradients. The properties of the transition elements are then derived using the weak formulation and the element temperature approximation. The formulation presented here permits linear temperature distribution through the element thickness. Convective boundaries as well as distributed heat flux is permitted on all four faces of the element. Furthermore, the element formulation also permits distributed heat flux and orthotropic material behaviour. Numerical examples are presented, first to illustrate the accuracy of the formulation and second to demonstrate its usefulness in practical applications. Numerical results are also compared with the theoretical solutions.     

Shear flexible element based on coupled displacement field for large deflection analysis of laminated plates

A shear deformable theory accounting for the transverse-shear (in the sense of Reissner-Mindlin’s thick plate theory) and large deflections (in the sense of von Karman theory) is employed in the construction of variational statement. A four-node, lock-free, shear-flexible rectangular plate element based on the coupled displacement field is developed in this paper to carry out the large deflection analysis. The displacement field of the element is derived by making use of the linearized equations of static equilibrium. A bi-cubic polynomial distribution is assumed for the transverse displacement ‘w’. The field distribution for the in-plane displacements (u,v) and plate normal rotations (θ_x, θ_y) and twist (θ_xy) is derived using equilibrium of composite strips parallel to the plate edges. The displacement fields so derived are coupled through material couplings. The transverse shear strain fields of the proposed element do not contain inconsistent terms, so that the element predicts even shear-rigid bending accurately.The element is validated for a series of numerical problems and results for deflections and stresses are presented for rectangular composite plates with various boundary conditions, loading and lay-ups. The influence of the sign of the loading on the deflection of unsymmetrically laminated plates, in the large deflection regime is also investigated.     

Geometrically non-linear formulation for two dimensional curved beam elements

This paper presents a geometrically non-linear formulation using a total lagrangian approach for the two dimensional curved beam elements. The beam element is derived using linear, paralinear and cubic-linear plane stress elements. The basic element geometry is constructed using the coordinates of the nodes on the element center line (η = 0) and the nodal point normals. The element displacement field is described using two translations of the node on the center line and a rotation about the axes normal to the plane containing the center line of the element. The existing beam element formulations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such a restriction. This is accomplished by retaining non-linear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting appropriate non-linear functions representing the effects of nodal rotations. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the behavior and the accuracy of the two dimensional beam elements for geometrically non-linear applications. In all cases comparisons made with theory and/or other published data show that the beam elements product accurate results and permit large load increments with good convergence characteristics.     

Three dimensional curved shell finite elements for heat conduction

This paper presents a finite element formulation for three dimensional curved shell heat conduction where nodal temperatures and nodal temperature gradients through the shell thickness are retained as primary variables. The three dimensional curved shell geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the nodal point normals. The element temperature field is defined in terms of the element approximation functions, nodal temperatures and nodal temperature gradients. The weak formulation of the three dimensional Fourier heat conduction equation is constructed in the Cartesian coordinate system. The properties of the curved shell elements are then derived using the weak formulation and the element temperature approximation. The element formulation permits linear temperature distribution through the element thickness.Distributed heat flux as well as convective boundaries are permitted on all six faces of the element. The element also has internal heat generation as well as orthotropic material capability. The superiority of the formulation in terms of applications, efficiency and accuracy is demonstrated. Numerical examples are presented and comparisons are made with theoretical solutions.     

Three dimensional solid-shell transition finite elements for heat conduction

This paper presents a finite element formulation for a special class of finite elements referred to as ‘Solid-Shell Transition Finite Elements’ for three dimensional heat conduction. The solid-shell transition elements are necessary in applications requiring the use of both three dimensional solid elements and the curved shell elements. These elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodel temperatures as well as nodal temperature gradients are retained as primary variables. The element geometry is defined in terms of coordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The temperature field with the element is approximated in terms of element approximation functions, nodal temperatures and nodal temperature gradients. The properties of the transition element are then derived using the weak formulation (or the quadratic functional) of the Fourier heat conduction equation in the Cartesian coordinate system and the element temperature approximation. The formulation presented here permits linear temperature distribution in the element thickness direction.

Convective boundaries as well as distributed heat flux is permitted on all six faces of the elements. Furthermore, the element formulation also permits internal heat generation and orthotropic material behavior. Numerical examples are presented firstly to illustrate the accuracy of the formulation and secondly to demonstrate its usefulness in practical application. Numerical results are also compared with the theoretical solutions.     

Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained using two different nonlinear finite element formulations is examined. Both formulations can correctly describe arbitrary rigid body displacements and can be used in the large deformation analysis. The first formulation is based on the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation. The second formulation, the absolute nodal coordinate formulation (ANCF), relaxes this assumption and introduces modes that couple the deformation of the cross section and the axial and bending deformations. In the absolute nodal coordinate formulation, four different models are developed; a beam model based on a general continuum mechanics approach, a beam model based on an elastic line approach, a beam model based on an elastic line approach combined with the Hellinger–Reissner principle, and a plate model based on a general continuum mechanics approach. The use of the general continuum mechanics approach leads to a model that includes the ANCF coupled deformation modes. Because of these modes, the continuum mechanics model differs from the models based on the elastic line approach. In both the geometrically exact beam and the absolute nodal coordinate formulations, the centrifugal forces are formulated in terms of the element nodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotating beam is examined, and the results obtained using the two formulations are compared for different values of the beam angular velocity. The numerical comparative study presented in this investigation shows that when the effect of some ANCF coupled deformation modes is neglected, the eigenvalue solutions obtained using the geometrically exact beam and the absolute nodal coordinate formulations are in a good agreement. The results also show that as the effect of the centrifugal forces, which tend to increase the beam stiffness, increases, the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes less significant. It is shown in this paper that when the effect of the Poisson ration is neglected, the eigenvalue solution obtained using the absolute nodal coordinate formulation based on a general continuum mechanics approach is in a good agreement with the solution obtained using the geometrically exact beam model.     

A quadratic mindlin element using shear constraints

“Shear constraints” are used to derive a displacement-based bending element for the analysis of thin and moderately thick plates of general plan form. As a starting point, the eight serendipity modes are adopted for the normal rotations and the nine Lagrangian modes for the transverse displacement, w. Subsequently, the shear constraints are used to eliminate the mid-side and central w variables so that the final element has three degrees-of-freedom at the corners and two at each mid-side. The bending energy is integrated using the standard formulation for the serendipity Mindlin element (with two-point Gaussian integration) so that the only modifications to that element involve the shear strain-displacement matrix. The constraints, which are used to implement these modifications, involve explicit algebraic expressions rather than numerical integration or matrix manipulation. A Fortran subroutine is provided for implementing these changes in a general quadrilateral. Using hierarchical displacement functions, the mid-side displacement variables Δw, that are missing from the standard serendipity element, may be simply constrained to zero as “boundary conditions”. Numerical experiments are presented which show that the element does not “lock” and that it gives excellent results for both thin and moderately thick plates. It also passes the patch test for a general quadrilateral.     

Accurate force evaluation with a simple,bi-linear,plate bending element

A great deal of attention has been given to the development of simple C ° continuous plate and shell elements based on the shear flexible theories for application to thick plates, sandwich or cellular plates and transversely isotropic or laminated plates. After considerable experimentation using unconventional approaches such as reduced integration, selective integration, mixed methods using discontinuous force fields, etc., it has been possible to develop simple displacement-type elements which can be reliably used. The stress recovery at nodes from such elements is often unreliable as the nodes are usually the points where strains or stresses are least accurate in the element domain. Further, nodal values can reflect severe oscillations at some difficult corner or edge conditions. In this paper, we focus attention on the optimal stress recovery from such an element. This is done after an interpretation of the displacement method as a procedure that obtains strains over the finite-element domain in a least-squares accurate fashion. If a shear flexible element is field-consistent, there are optimal locations at which bending moments and shear forces are accurate in a least-squares sense. These points are identified for the present element and used to study stresses in typical plate problems. Another difficulty faced is the rapid variation of twisting moments at free edges and corners of shear flexible plates and its influence on the shear forces at that edge. A related source of difficulty is the distinction made in Kirchhoff theory between shear forces and the effective shear reactions of that theory. The present study is seen to give accurate enough shear force and twisting moment predictions to allow one to draw the severe conclusion that the use of the Kirchhoff shear reaction at edges in classical plate theory is an ambiguous and unnecessary one and can be avoided. The findings confirm a recent suggestion that it may be more appropriate to have three (as introduced originally by Poisson) instead of the two boundary conditions (as modified by Kirchhoff) usually applied on the edge of a thin plate, especially if that edge is unsupported.     

The approximate Dirichlet Domain Decomposition method. Part I: An algebraic approach

We present a new approach to the construction of Domain Decomposition (DD) preconditioners for the conjugate gradient method applied to the solution of symmetric and positive definite finite element equations. The DD technique is based on a non-overlapping decomposition of the domain Ω intop subdomains connected later with thep processors of a MIMD computer. The DD preconditioner derived contains three block matrices which must be specified for the specific problem considered. One of the matrices is used for the transformation of the nodal finite element basis into the approximate discrete harmonic basis. The other two matrices are block preconditioners for the Dirichlet problems arising on the subdomains and for a modified Schur complement defined over all nodes on the coupling boundaries between the subdomains. The relative spectral condition number is estimated. Relations to the additive Schwarz method are discussed. In the second part of this paper, we will apply the results of this paper to two-dimensional, symmetric, second-order, elliptic boundary value problems and present numerical results performed on a transputer-network.     

An analysis of the petrov—galerkin finite element method

Petrov-Galerkin finite element methods, using different test and trial functions, are applied to the solution of an unsymmetric two-point boundary value problem intended to simulate certain aspects of convection-diffusion problems. For a specified space of trial functions we utilise an energy error bound to optimize this class of methods over a family of test spaces. The optimized method performs well provided the asymmetry in the differential operator does not lead to boundary layers in the solution. Following an analysis of the boundary layer behaviour of the continuous problem, L-splines are introduced, and, by studying their behaviour for coarse meshes, we are able to modify the original schemes to produce so-called “disconnected” finite element methods. Even for coarse meshes, when no nodes occur in the boundary layer, the accuracy at all nodal points is good. This would make them good candidates for application in more general situations.     

Accurate evaluation of support reactions in finite element plate-bending analysis

Two simple approaches are presented which allow the distribution of support reactions to be predicted with as high degree of accuracy as the displacements. In the first approach the plate element assembly is completed with special one-dimensional elastic support elements. If their Winkler coefficient is suitably tuned, an accurate prediction of reactions is obtained as a part of the finite element analysis without unduly affecting the displacements and moments of the plate. In the second approach, a standard finite element calculation (without elastic support elements) is performed first and the distribution of reactions is then evaluated based on the known nodal forces at boundary nodes of the plate.

The two approaches are indiscriminately applicable with Kirchhoff and Reissner-Mindlin plate bending elements. Their practical efficiency is illustrated by numerical examples.     

Isoparametric axisymmetric shell elements with temperature gradients for heat conduction

This paper presents a finite element formulation for axisymmetric shell heat conduction where temperature gradients through the shell thickness are retained as primary nodal variables. The element geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the middle surface nodal point normals. The element temperature field is approximated in terms of element approximation functions, the nodal temperature, and the nodal temperature gradients. The weak formulation of the two-dimensional Fourier heat conduction equation in cylindrical coordinate system is constructed. The finite element properties of the shell element are then derived using the weak formulation and the element temperature field approximation. The formulation permits linear temperature gradients through the shell thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the element. Furthermore, the element can also have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented and a comparison is made with the theoretical results.     

Penalty resolution of the babuska circle paradox

If the solution for a simply-supported circular plate is approximated by a sequence of finite element solutions on successively refined polygonal domains the finite element approximations converge to the solution of a different problem. In the present study we present a finite element formulation with boundary penalty that produces valid approximations to the simply-supported plate. The penalty term is shown to require the use of reduced integration. The dependence of the penalty parameter on mesh size h is also examined. Numerical experiments confirm the validity of the method and determine rates of convergence. A second approach involving a modified corner condition is also considered and error estimates determined. This scheme is implemented also using a discrete penalty technique. The results and rates are compared with the boundary penalty method and their relative merits discussed.     

Convergence and accuracy of the path integral approach for elastostatics

This paper addresses convergence rate and accuracy of a numerical technique for linear elastostatics based on a path integral formulation [Int. J. Numer. Math. Eng. 47 (2000) 1463]. The computational implementation combines a simple polynomial approximation of the displacement field with an approximate statement of the exact evolution equations, which is designated as functional integral method. A convergence analysis is performed for some simple nodal arrays. This is followed by two different estimations of the optimum parameter ζ: one is based on statistical arguments and the other on inspection of third order residuals. When the eight closest neighbors to a node are used for polynomial approximation the optimum parameter is found to depend on Poisson's ratio and lie in the range 0.5<ζ<1.5. Two well established numerical methods are then recovered as specific instances of the FIM. The strong formulation––point collocation––corresponds to the limit ζ=0 while bilinear finite elements corresponds exactly to the choice ζ=0.5. The use of the optimum parameter provides better precision than the other two methods with similar computational cost. Other nodal arrays are also studied both in two and three dimensions and the performance of the FIM compared with the corresponding finite element and collocation schemes. Finally, the implementation of FIM on unstructured meshes is discussed, and a numerical example solving Laplace equation is analyzed. It is shown that FIM compares favorably with FEM and offers a number of advantages.     

p-Version axisymmetric shell element for geometrically nonlinear analysis

This paper presents a p-version geometrically nonlinear (GNL) formulation based on total Lagrangian approach for a three-node axisymmetric curved shell element. The approximation functions and the nodal variables for the element are derived directly from the Lagrange family of interpolation functions of order p_ξ and p_η. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three- and one-node equivalent configurations that correspond to p_ξ + 1 and p_η+ 1 equally spaced nodes in the ξ and η directions and then taking their products. The resulting element approximation functions and the nodal variables are hierarchical and the element approximation ensures C⁰ continuity. The element geometry is described by the coordinates of the nodes located on the middle surface of the element and the nodal vectors describing top and bottom surfaces of the element.

The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete axisymmetric state of stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells. The formulation presented here removes virtually all of the drawbacks present in the existing GNL axisymmetric shell finite element formulations and has many additional benefits. First, the currently available GNL axisymmetric shell finite element formulations are based on fixed interpolation order and thus are not hierarchical and have no mechanism for p-level change. Secondly, the element displacement approximations in the existing formulations are either based on linearized (with respect to nodal rotation) displacement fields in which case a true Lagrangian formulation is not possible and the load step size is severely limited or are based on nonlinear nodal rotation functions approach in which case though the kinematics of deformation is exact but additional complications arise due to the noncummutative nature of nonlinear nodal rotation functions. Such limitations and difficulties do not exist in the present formulation. The hierarchical displacement approximation used here does not involve traditional nodal rotations that have been used in the existing shell element formulations, thus the difficulties associated with their use are not present in this formulation.

Incremental equations of equilibrium are derived and solved using the standard Newton method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the preset formulation. The results obtained from the present formulation are compared with those available in the literature.     

An unstructured finite elements method for solving the compressible RANS equations and the Spalart-Allmaras turbulence model

This paper deals with a stabilized finite element method for solving the compressible Navier-Stokes equations combined with the Spalart-Allmaras turbulence model. The aim is to assess the ability of this formulation to solve high Reynolds number turbulent flows over anisotropic meshes. This formulation lies in the framework of the Stream-Line-Upwind-Petrove-galerkin method. We propose a simple and efficient formulation for the stabilization τ matrix and develop a shock-capturing operator. Numerical tests on the 3D boundary layer over a flat plate and on the ONERA-M6 wing show the stability and robustness of the proposed method.     

Rigid body concept for geometric nonlinear analysis of 3D frames, plates and shells based on the updated Lagrangian formulation

In the nonlinear analysis of elastic structures, the displacement increments generated at each incremental step can be decomposed into two components as the rigid displacements and natural deformations. Based on the updated Lagrangian (UL) formulation, the geometric stiffness matrix [k_g] is derived for a 3D rigid beam element from the virtual work equation using a rigid displacement field. Further, by treating the three-node triangular plate element (TPE) as the composition of three rigid beams lying along the three sides, the [k_g] matrix for the TPE can be assembled from those of the rigid beams. The idea for the UL-type incremental-iterative nonlinear analysis is that if the rigid rotation effects are fully taken into account at each stage of analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. The present approach is featured by the fact that the formulation is simple, the expressions are explicit, and all kinds of actions are considered in the stiffness matrices. The robustness of the procedure is demonstrated in the solution of several benchmark problems involving the postbuckling response.     

Influence surfaces by boundary element/least square methods coupling

This work presents a new application for calculating the influence surfaces of transverse displacements, directional derivatives and bending moments for generic bridge decks. A plate bending boundary element method formulation is coupled with the application of a continuous field surface derived by the least square procedure. This original BE formulation permits calculating influence surfaces of plates with polygonal, curved or circular geometry, and several transverse load conditions. The proposal allows future analysis of building floors and single and continuous bridge trusses connected to longitudinal and transversal girders. Numerical examples are presented to demonstrate the potential of the present formulation and the results are compared with analytical values and with usual vehicular loads.     

A boundary element analysis of symmetric laminated composite shallow shells

This paper presents a boundary element formulation for the analysis of symmetric laminated composite shallow shells where only the boundary is discretized. Classical plate bending and plane elasticity formulations are coupled and effects of curvature are treated as body forces. Fundamental solutions for elastostatic formulations are used and body forces are written as a sum of approximation functions multiplied by unknown coefficients. Two approximation functions are used. Domain integrals which arise in the formulation are transformed into boundary integrals by the radial integration method. Results for the approximation functions are compared and the accuracy of the proposed formulation is assessed by results from literature. It was shown that results obtained with the approximation function called augmented thin plate spline present very good agreement with literature even for shells that are not so shallow.