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.     

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.     

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.     

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.     

Parameter identifiability for heat conduction with a boundary input

The identifiability (i.e. the unique identification) of conductivity in a heat conduction process is considered in the class of piecewise constant conductivities. The 1-D process may have nonzero boundary inputs as well as distributed inputs. Its measurements are collected at finitely many observation points. They are processed to obtain the first eigenvalue and a constant multiple of the first eigenfunction at the observation points. It is shown that the identification by the Marching Algorithm is continuous with respect to the mean convergence in the admissible set. The result is based on the continuous dependence of eigenvalues, eigenfunctions, and the solutions on the conductivities. Numerical experiments confirm the perfect identification for noiseless data. A numerical algorithm for the identification in the presence of noise is proposed and implemented.     

Difference schemes for time-dependent heat conduction models with delay

Different non-Fourier models of heat conduction, that incorporate time lags in the heat flux and/or the temperature gradient, have been increasingly considered in the last years to model microscale heat transfer problems in engineering. Numerical schemes to obtain approximate solutions of constant coefficients lagging models of heat conduction have already been proposed. In this work, an explicit finite difference scheme for a model with coefficients variable in time is developed, and their properties of convergence and stability are studied. Numerical computations showing examples of applications of the scheme are presented.     

Isoparametric line and transition finite elements with temperature and temperature gradients as primary variables for two dimensional heat conduction

This paper presents isoparametric line and transition finite element formulation for two dimensional heat conduction. The element properties are derived using weak formulation of the Fourier heat conduction equation and the element approximation where nodal temperatures and the nodal temperature gradients are retained as primary variables. The formulation permits linear temperature distribution through the element thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the elements. Furthermore, the elements can 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 to illustrate their applications, and a comparison is made with theoretical results.     

Noise reconstruction for the inverse heat conduction problem

A new automatic procedure to numerically recover the sample root mean square norm of the data error for the linear inverse heat conduction problem (IHCP)—when this information is not readily available—is presented. Numerical results are described which illustrate the accuracy of the algorithm.     

The best of the enhanced displacement non-conforming plate/ shell elements

Summary A state-of-the-art in the development of a series of new finite elements by the addition of non-conforming displacement modes is reviewed in this paper. These new plate finite elements are achieved by the combined use of the addition of non-conforming modes, the application of reduced (or selective) integration scheme, and the construction of substitute shear strain fields. The improvement achieved may be attributable to the fact that the merits of these techniques are merged into the new element in a complementary manner. It is shown that the results obtained by the new elements give reliable solutions without any defects for several benchmark problems.     

Curved finite elements for the analysis of shallow and deep arches

The paper considers the analysis of shallow and deep arches by curved beam finite elements, the interest in this problem being that it forms a limiting case of the shell problem. The individual elements are assumed to be shallow with respect to a local base line and different types of strain-displacement equations are utilised. Particular attention is focussed on the choice of the displacement patterns for the elements and models based both on independently-interpolated displacement components, of up to quintic order, and on assumed strain distributions are considered. Numerical applications using the various models show that of those tested only quintic-quintic models and assumed-strain models are generally satisfactory.     

MITC elements for a classical shell model

The formulation of nine-node mixed-interpolated shell elements based on a classical mathematical shell theory is presented, taking into account some fundamental considerations for the finite element analysis of shells. The elements are based on the mixed interpolation of tensorial components approach (MITC), but the assumed covariant strain fields are applied only for the membrane and shear components. Two different types of elements are considered, depending on whether or not geometric approximations are included in the formulation. The performance of the proposed elements is illustrated with a well-established test problem––the Scordelis-Lo roof.     

Comparison of shell theories in the analysis of cylindrical shells with discontinuity in the thickness

Cylindrical shells with discontinuity in the thickness and that are subjected to axisymmetric loading have been analysed. Two types of finite elements are used: the first is based on thin shell theory and the second on thick shell theory. The loadings considered are a uniform internal pressure and a circular ring load at the mid-section. The effect of these loads for various end conditions and various step-ratios in the thickness have been analysed. Numerical results are presented and compared for both the theories. It has been shown that the transverse normal stress acting along the thickness direction is not negligible compared to other stresses at places of discontinuity either in the thickness or in the loading. The weight of the shell is kept constant for various step-ratios.     

A multilayer composite triangular element for steady-state conduction/convection/radiation heat transfer in complex shells

Our latest study presents the theoretical formulation and computer implementation of a three-node six degrees of freedom multilayer flat triangular element intended for the study of the temperature fields in complex multilayer composite shells. Inherent in the formulation, in this first introductory and self-consistent systematic study, are the three modes of heat transfer, namely conduction, convection and radiation, the latter introducing in our theoretical model strong nonlinear effects. In the present discourse, all nonlinear terms are strictly due to radiation; the material properties are assumed independent of temperature but this in no way restricts the generality of the basic theory. The formulation is based on a first-order thermal lamination theory which assumes a linear through-the-thickness temperature variation. The following features are uniquely implemented in the computer model:

1. (1) Exact integration of all matrices including the highly nonlinear radiation matrix

2. (2) Exact integration of all derivative (Jacobian) matrices for efficient nonlinear analysis

3. (3) Geometrical generality achieved by an arbitrarily oriented inexpensive flat shell element

4. (4) Compatibility with structural elements

5. (5) Computational efficiency and simplicity

A predictor-corrector scheme in the form of the Newton-Raphson method is adopted for the solution of the stead-state nonlinear problem. Numerical examples, ranging from simple panels to complex anisotropic shells substantiate the theoretical formulation and show the potential of the present laminated triangular element in the computer simulation of temperature effects in complex geometries.     

Thermoelectric spherical shell with fractional order heat transfer

Microsystem Technologies - The model of fractional magneto-thermoelasticity is applied to one-dimensional problems of a thermoelectric spherical shell subjected to an arbitrary thermal loading of...     

Fast and robust techniques for the euclidean p-median problem with uniform weights

We discuss new solution techniques for the p-median problem, with the goal being to improve the solution time and quality of current techniques. In particular, we hybridize the discrete Lloyd algorithm and the vertex substitution heuristic. We also compare three starting point techniques and present a new solution method that provides consistently good results when appropriately initialized.     

Finite element of the heat conduction equation with temperature dependent coefficients

A finite element formulation is derived for the equation of heat conduction with temperature dependent conductivity and heat capacity. The derivation of the finite element model is based on a variational formulation of the heat conduction equation which, together with generalized coordinates, yields an equation of similar form to Lagrange's equation in mechanics. The obtained equation is of special interest for applying the finite element method to solve problems with temperature-dependent properties.     

On the use of deep,shallow or flat shell finite elements for the analysis of thin shell structures

This paper is confined to the study of thin shells. The aim is to summarize the different theories used and to examine the assumptions upon which each of them is based. The intention is to show when it is more suitable to use a particular approximation and to indicate the errors it introduces. Beginning with the general deep shell theory, some simplifications are introduced to obtain the shallow shell theories. The special implications of this theory for the finite element method are also examined. Finally the particular case of flat elements is discussed.     

Concurrent lattice infill with feature evolution optimization for additive manufactured heat conduction design

Additive manufacturing (AM) eliminates many of the geometric restrictions in conventional manufacturing, and hence complex geometry, such as lattice structures, can be produced with little additional cost. AM designs based on lattice structuring have become increasingly popular as it possesses tunable properties and can be designed to be self-supporting easily. For these reasons, lattice infill recently has been actively studied and a variety of lattice structure topology optimization methods have been developed. On the other hand, lattice infill cannot span the design domain when there are functional features in the mechanical design (e.g. assembly holes and cooling channels). Also, the geometric form of these functional features need to be maintained and cannot be replaced by the lattice structure. Thus far, lattice structure topology optimization considers these features fixed in space without design freedom and obviously, this treatment lacks overall optimality. To fill this critical gap, this work combines the feature evolution into the variable-density lattice structure topology optimization framework, which leads to a concurrent lattice density and feature layout optimization method. Parametric level set functions are employed for the feature representation and R-functions are adopted to combine the density and level set fields. Sensitivity information is calculated on both the lattice densities and feature parameters, in order to solve the problem through a unified gradient-based approach. Several 3D numerical examples are provided to demonstrate the efficiency and robustness of the proposed method.     

A modified simple heuristic for the p-median problem, with facilities design applications

Facilities design is closely related to efficient use of available resources. This paper presents a heuristic approach to solve two core problems of a good facilities design: facility location and facility layout. The latter group of problems will be solved for warehouse and production systems in particular. All these problems can be formulated as p-median clustering problems. Teitz and Bart (Oper. Res. 16 (1968) 955–961) developed the vertex substitution method to solve those problems. This paper introduces effective improvements on this heuristic. The approach is tested on a large number of randomly generated cases and on problems taken from the literature. The results demonstrate the effectiveness and superiority of our method.