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.     

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.     

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.     

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.     

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.     

Curved shell elements for heat conduction with p-approximation in the shell thickness direction

A finite element formulation is presented for the curved shell elements for heat conduction where the element temperature approximation in the shell thickness direction can be of an arbitrary polynomial order p. This is accomplished by introducing additional nodal variables in the element approximation corresponding to the complete Lagrange interpolating polynomials in the shell thickness direction. This family of elements has the important hierarchical property, i.e. the element properties corresponding to an approximation order p are a subset of the element properties corresponding to an approximation order p + 1. The formulation also enforces continuity or smoothness of temperature across the inter-element boundaries, i.e. C⁰ continuity is guaranteed.

The curved shell geometry is constructed using the co-ordinates of the nodes lying on the middle surface of the shell and the nodal point normals to the middle surface. The element temperature field is defined in terms of hierarchical element approximation functions, nodal temperatures and the derivatives of the nodal temperatures in the element thickness direction corresponding to the complete Lagrange interpolating polynomials. The weak formulation (or the quadratic functional) of the three-dimensional Fourier heat conduction equation is constructed in the Cartesian co-ordinate space. The element properties of the curved shell elements are then derived using the weak formulation (or the quadratic functional) and the hierarchical element approximation. The element matrices and the equivalent heat vectors (resulting from distributed heat flux, convective boundaries and internal heat generation) are all of hierarchical nature. The element formulation permits any desired order of temperature distribution through the shell thickness.

A number of numerical examples are presented to demonstrate the superiority, efficiency and accuracy of the present formulation and the results are also compared with the analytical solutions. For the first three examples, the h-approximation results are also presented for comparison purposes.     

A comparison of the finite element and finite difference methods for the analysis of steady two dimensional heat conduction problems

A comparison between the finite difference method and the finite element method for solving the linear Two-dimensional heat conduction equation is presented. In all areas except computer core storage, the finite element method is demonstrated to be superior to the finite difference method for this type of problem.     

Uncertainty in Thermal Basin Modeling: An Interval Finite Element Approach

Uncertainty assessment in basin modeling and reservoir characterization is traditionally treated by geostatistical methods which are normally based on stochastic probabilistic approaches. In this paper, we present an alternative approach which is based on interval arithmetic. Here, we discuss a fnite element formulation which uses interval numbers rather than real numbers to solve the transient heat conduction in sedimentary basins. For this purpose, a novel formulation was developed to deal with both the special interval arithmetic properties and the transient term in the differential Equation governing heat transfer. In this formulation, the “stiffness” matrix resulting from the discretization of the heat conduction equation is assembled with an element-by-element technique in which the elements are globally independent and the continuity is enforced by Lagrange multipliers. This formulation is an alternative to traditional Monte Carlo method, where it is necessary to run a simulation several times to estimate the uncertainty in the results.We have applied the newly developed techniques to a one-dimensional thermal basin simulation to assess their potential and limitations.We also compared the quality of our formulation with other solution methods for interval linear systems of equations.     

Analysis of a thermoviscoelastic model in large strain

This paper deals with a thermoviscoelastic model and its analysis. The mechanical formulation is based on the generalization to large strain of the Poynting–Thomson rheological model. The heat transfers are governed by the classical heat equation and the Fourier law. We briefly expose the finite element formulation, which takes into account the quasi-incompressibility constraint for the mechanical approach. The influence of several parameters is examined.     

Computational coupled non-associative thermo-plasticity

Computationally oriented formulation of the isothermal, rate-independent theory of non-associative elasto-plasticity is extended in this paper to describe coupled thermo-elastic-plastic and thermo-elastic-visco-plastic behaviour of materials. This is done by additionally considering thermal strains, assumming all material properties to be temperature dependent and accounting for the mechanical coupling terms in the non-stationary heat conduction equation. The finite deformation effects are included in the analysis. The theory is employed for the analysis of thermo-mechanical response of ductile metals with damage effects modelled by a generalization of the so-called Gurson approach. This constitutive model is known to generate equations typical of non-associative plasticity and hence it can be consistently incorporated into the present more general considerations. The finite element assessment of combined thermal and damage effects on the axisymmetric necking process illustrates the paper. Numerical aspects such as a ‘tangent’ stiffness for rate-dependent thermo-plasticity and the algorithmic (or consistent) tangent stiffness matrix for non-associative plasticity are discussed as well.     

2D and 3D transient heat conduction analysis by BEM via particular integrals

Particular integral formulations are presented for 2D and 3D transient potential flow (heat conduction) analysis. The results of the analysis are compared with an alternative formulation developed using the volume integral conversion approach. Although the mathematical foundation of the two methods are different both formulations are shown to produce almost identical results.For the particular integral formulation, the steady-state heat conduction equation is used as the complementary solution and two global shape functions (GSFs) are considered to approximate the transient term of the heat conduction equation.The numerical results for three example problems are given and compared with their analytical solutions.     

Thermal stresses of a wind turbine blade made of orthotropic material

This study is to investigate the thermal stress of a wind turbine blade made of wood composite material. First, the governing partial differential equation on heat conduction is stated, then, a finite element procedure using a variational approach is employed for the solution of the governing equation. Thus, the temperature field throughout the blade is determined. Next, based on the temperature field, a finite element procedure using potential energy approach is applied to determine the thermal stress field. A set of results is obtained through the use of a computer, which is considered to be satisfactory.     

Numerical simulation of a polysilicon thermal flexure actuator

 An electrothermal equation of a polysilicon thermal flexure actuator is presented, which takes heat conduction, air convection and radiation into account. A numerical method is developed to solve the equation. The deflection model based on the matrix displacement method, i.e. finite element method (FEM) in structural mechanics, is given. It transforms deflection equations into a matrix and is easy to calculate numerically. Simulation results for the actuator with typical dimensions are presented. Discussions are finally given. Received: 28 April 2000/Accepted: 9 January 2001     

Orthogonal collocation on finite elements for elliptic equations

The method of orthogonal collocation on finite elements (OCFE) combines the features of orthogonal collocation with those of the finite element method. The method is illustrated for a Poisson equation (heat conduction with source term) in a rectangular domain. Two different basis functions are employed: either Hermite or Lagrange polynomials (with first derivative continuity imposed to ensure equivalence to the Hermite basis). Cubic or higher degree polynomials are used. The equations are solved using an LU-decomposition for the Hermite basis and an alternating direction implicit (ADI) method for the Lagrange basis.     

A symmetric finite element formulation for the inverse problem of aquifer transmissivity

This paper presents a symmetric isoparametric finite element formulation for the inverse problem of aquifer transmissivity calculation with known piezometric head. An important aspect of the present formulation is that the groundwater flow equation describing the aquifer behavior is transformed into a second-order differential equation by introducing an artificial variable φ. The two-dimensional, line and transition elements derived based on the weak formulation of this transformed equation possess symmetric matrices. In the formulation of the line elements φ and its derivative in η direction are retained as primary variables. This permits modelling of sudden changes in aquifer width. The transition elements provide a natural connecting link between the two-dimensional elements and the line elements. The line elements provide an efficient means of modelling aquifers with unidirectional flow. Numerical examples are given. A comparison of the results obtained here with the Galerkin finite element solution (nonsymmetric formulation) clearly demonstrates the superiority of the formulation presented here.     

二维热传导方程有限容积法的MATLAB实现

通过偏微分方程描述了二维无扩散热传导现象。基于有限容积法推导了该方程的离散代数方程组,针对恒定热流强度、恒定温度、对流换热和绝热这四种不同边界条件,分别讨论了热传导代数方程的离散系数和源项。通过MATLAB编程,分析了一维具有均匀厚度无限大板和二维矩形区域的瞬态传热现象。采用图形显示方式使得偏微分方程求解更为直观和容易理解,计算结果证明了有限容积求解方法是可行、稳定的。     

Topology optimization involving thermo-elastic stress loads

Structural topology optimization of thermo-elastic problems is investigated in this paper. The key issues about the penalty models of the element stiffness and thermal stress load of the finite element model are highlighted. The penalization of thermal stress coefficient (TSC) measured by the product between thermal expansion coefficient and Young’s modulus is proposed for the first time to characterize the dependence of the thermal stress load upon the design variables defined by element pseudo-densities. In such a way, the element stiffness and the thermal stress load can be penalized independently in terms of element pseudo-density. This formulation demonstrates especially its capability of solving problems with multiphase materials. Besides, the comparison study shows that the interpolation model RAMP is more advantageous than the SIMP in our case. Furthermore, sensitivity analysis of the structural mean compliance is developed in the case of steady-state heat conduction. Numerical examples of two-phase and three-phase materials are presented.     

A finite element formulation for elastoplasticity based on a three-field variational equation

A three-field variational equation, which expresses the momentum balance equation, the plastic consistency condition, and the dilatational constitutive equation in a weak form, is proposed as a basis for finite element computations in hardening elastoplasticity. The finite element formulation includes algorithms for the integration of the elastoplastic rate constitutive equations which are similar to members of the “return mapping” family of algorithms employed in displacement formulations, except that the proposed algorithms are not required to explicitly satisfy the plastic consistency condition at the end of each time step. This condition is imposed globally by the inclusion of a variational equation that suitably constrains the solution. The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton's method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples are provided.     

Computation of subcritical compressible flows

A finite element analysis and iterative solution of steady plane inviscid compressible flows is developed. Specific attention is directed to subcritical flows in which the nonlinear governing equation is elliptic, and to slightly supercritical mixed flows. The underlying variational theory for this nonlinear flow problem and a corresponding finite element formulation are presented. A Newton-Raphson iteration is introduced within this approximate analysis to provide efficient solution of subcritical flows and also slightly supercritical flows in which the nonlinear flow equation is of mixed type. The performance of the algorithm is studied with particular reference to the effect of a two-parameter scheme in which incident Mach number $\text{M}{}_{\mathrm{x}}$ and specific heat ratio γ may be varied. By thus regularizing the operator during the early iteration history, more efficient computations can be realized in the mildly-transonic flow regime of higher incident Mach numbers.     

A transient finite element solution method for the Navier-Stokes equations

This paper is concerned with the discrete finite element formulation and numerical solution of transient incompressible viscous flow in terms of the primitive variables. A restricted variational principle is introduced as equivalent to the momentum equations and the Poisson equation for pressure. The latter is introduced to replace the continuity equation, and thus the incompressibility condition is realized only asymptotically; i.e. through the iterative process. An incomplete cubic interpolation function is used for both the velocities and pressure within a triangular finite element. The discrete equations are integrated in time with backward finite differences. We illustrate the similarity between the (ψ,ζ) finite difference method and the ($\text{u}\text{,p}$) finite element method by calculations on the driven square cavity problem.