Nodal Variable Operators Research Articles

A p-version curved shell element based on piecewise hierarchical displacement approximation for laminated composite plates and shells

This paper presents a piecewise hierarchical p-version three-dimensional nine-node curved shell element formulation with interlamina continuity of displacements for laminated composite plates and shells. The displacement field approximation for each lamina can be of arbitrary polynomial order in the plane of the lamina as well as in the transverse direction, and is based on p-version. The p-version hierarchical approximation functions and the corresponding nodal variable operators for each lamina are derived directly from the Lagrange family of interpolation functions by first constructing the one-dimensional p-version hierarchical approximation functions and the corresponding nodal variable operators for three- and one-node equivalent configurations in the ξ, η and ζ directions and then taking their products. The lamina stiffness matrix and the equivalent load vectors are derived using the principle of virtual work and the p-version lamina displacement approximation. The interlamina continuity conditions of displacements are imposed at the lamina interfaces. The interlamina continuity conditions are conveniently arranged in the form of transformation matrices, which allow the transformation of the lamina degrees of freedom to the laminate degrees of freedom. These transformation matrices are used to transform the lamina stiffness matrices and equivalent load vectors to the laminate stiffness matrix and equivalent load vector. The transformed lamina stiffness matrices and load vectors are summed to obtain the laminate stiffness matrix and equivalent load vectors. The interlamina continuity conditions of displacements permit condensation of [3( p ξ + 1)( p) η + 1)] degrees of freedom (where ζ is the direction of lamina lay up) for all laminae except the first. Numerical examples are presented to demonstrate the accuracy, efficiency and convergence characteristics of the present formulation.

A least squares finite element formulation for plane elasto-statics based on p-version with applications to laminated composites

This paper presents a weighted p-version least squares finite element formulation (LSFEF) for two-dimensional plane elasto-statics and its application for simulating linear static behavior of laminated composites. The second order partial differential equations of equilibrium in displacements are recast into a series of first order coupled partial differential equations using stresses as auxiliary variables. The primary (displacements: u and v) and the secondary (stresses: σ xx , τ xy and σ yy ) variables are interpolated using equal order, C 0 continuity, p-version hierarchical approximation functions. The p-version hierarchical approximation functions and the corresponding nodal variable operators are derived directly from the Lagrange family of interpolation functions by first constructing the one dimensional p-version hierarchical approximation functions and the corresponding nodal variable operators for three node equivalent configurations in the ξ and η directions and then taking their products. The least squares procedure yields a system of linear, symmetric simultaneous algebraic equations. The formulation provides inter-element continuity of displacements as well as stresses. Even though the element approximation is of type C 0, in the limit when the error functional I converges to zero (i.e., when I → 0) the formulation behaves more like a C 1 type. In addition, for an inadequate mesh with an inadequate order of approximation for the elements, the element error functional values provide a mechanism for h and/or p-adaptive refinements. This is a unique feature of the LSFEF as compared to variational based formulations. Many other unique features of the formulation are discussed and illustrated in the paper. A numerical example is presented to illustrate the effectiveness of the formulation in simulating accurate inter-lamina stress behavior as well as accurate stress values and stress gradients at and near singularities. The results obtrained from the present formulation are compared with the results obtained from the p- version variational formulation and with those reported in the literature.

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 C 0 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.

Piecewise hierarchical p-version curved shell finite element for heat conduction in laminated composites

This paper presents a nine-node three-dimensional curved shell finite element formulation for linear steady-state heat conduction in laminated composites where the temperature approximation for the laminate is piecewise hierarchical and is derived based on the P-version. The temperature approximation for the element is developed by first establishing a hierarchical temperature approximation for each lamina of the laminate and then by imposing inter-lamina continuity conditions of temperature at the interface between the laminas. The approximation functions and the nodal variables for a lamina are derived directly from the Lagrange family of interpolation functions of order pξ, pη and k Pζ. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ, η and k ζ directions for the equivalent (3, 3 and 1 node) configurations that correspond to pξ + 1, pη + 1 and k pζ + 1 equally spaced nodes in the ξ, η, and k ζ directions and then taking their products. The nodal variables for the laminated shell element are derived from the nodal variables of the laminas and the inter-lamina continuity conditions of temperature. The element formulation ensures C 0 continuity of temperature across inter-element as well as the inter-lamina boundaries.

Formula omitted] piecewise hierarchical axisymmetric shell element for heat conduction in laminated composites

This paper presents a three-node axisymmetric shell finite element formulation for axisymmetric steady-state heat conduction in laminated composites where the temperature approximation for the laminate is piecewise hierarchical and is derived based on p- version . The temperature approximation for the element is developed by first establishing a hierarchical temperature approximation for each lamina of the laminate and then by imposing inter-lamina continuity conditions of temperature at the interface between the laminas. The approximation functions and the nodal variables for each lamina are derived directly from the Lagrange family of interpolation functions of order p ξ and kp η . This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and kη directions for the three- and one-node equivalent configurations that correspond to p ξ + 1 and kp η + 1 equally spaced nodes in the ξ and kη directions and then taking their products. The nodal variables for the entire laminate are derived from the nodal variables of the laminas and the inter-lamina continuity conditions of temperature. The element formulation ensures C 0 continuity or smoothness of temperature across inter-element as well as the inter-lamina boundaries. Weak formulation of the Fourier heat conduction equation in the cylindrical coordinate system is constructed and the individual lamina matrices and equivalent nodal heat vectors are derived using this weak formulation and the hierarchical temperature approximation for the laminas. Inter-lamina continuity conditions are used to construct the transformation matrices for the laminas. These matrices permit transformation of the lamina degrees of freedom to the laminate degrees of freedom. Using these transformation matrices individual lamina matrices and the equivalent heat vectors are transformed and then summed to obtain the laminate element matrix and equivalent heat vectors. There is no restriction on the number of laminas and their lay-up pattern. Each layer can be generally orthotropic. The material directions and the layer thickness may vary from point to point within each lamina. The geometry of the axisymmetric shell element is defined by the nodes located at the middle surface of the element and the lamina thicknesses. Numerical examples are presented to demonstrate the accuracy, efficiency, convergence characteristics of the overall superiority of the present formulation. The results are compared with analytical solutions, h- models and those obtained from the formulations utilizing a single hierarchical temperature approximation for the entire laminate.

Piecewise hierarchical p-version axisymmetric shell element for non-linear heat conduction in laminated composites

This paper presents a three-node axisymmetric shell finite element formulation for non-linear axisymmetric steady-state heat conduction in laminated composites where the temperature approximation for the laminate is piecewise hierarchical and is derived based on the P-version. The temperature approximation for the element is developed first by establishing a hierarchical temperature approximation for each lamina of the laminate and then by imposing inter-lamina continuity conditions of temperature at the interface between the laminas. The approximation functions and the nodal variables for each lamina are derived directly from the Lagrange family of interpolation functions of order P ξ and kP η . This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and kη directions for the three- and one-node equivalent configurations that correspond to p ξ + 1 and kp η + 1 equally spaced nodes in the ξ and kη directions and then taking their products. The nodal variables for the entire laminate are derived from the nodal variables of the laminas and the inter-lamina continuity conditions of temperature. The element formulation ensures C° continuity or smoothness of temperature across inter-element as well as the inter-lamina boundaries. Weak formulation of the Fourier heat conduction equation in the cylindrical coordinate system with temperature-dependent thermal conductivities, internal heat generation, film coefficients and radiation effects is constructed and the individual lamina matrices and equivalent nodal heat vectors are derived using this weak formulation and the hierarchical temperature approximation for the laminas. Inter-lamina continuity conditions are used to construct the transformation matrices for the laminas. These matrices permit transformation of the lamina degrees of freedom to the laminate degrees of freedom. Using these transformation matrices individual lamina matrices and the equivalent heat vectors are transformed and then summed to obtain the laminate element matrix and equivalent heat vectors. There is no restriction on the number of laminas and their lay-up pattern. Each layer can be generally orthotropic. The material directions and the layer thickness may vary from point to point within each lamina. The geometry of the axisymmetric shell element is defined by the nodes located at the middle surface of the element and the lamina thicknesses. Due to the temperature dependence of thermal conductivities, internal heat generation, film coefficients and radiation parameters; the resulting discretized equations of equilibrium are non-linear. These equations are solved using the modified Newton-Raphson method. The tangent matrix is derived using the non-linear equations of equilibrium. The modified tangent matrix is symmetric and provides excellent convergence during the equilibrium iteration process. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation and the extremely good and fast convergence of the iteration process. Comparisons are made with the available results in the literature.

P-Version two-dimensional curved beam element using piecewise hierarchical approximation for laminated composites

A three node two-dimensional laminated composite curved beam finite element formulation for linear static analysis is presented where the displacement approximation for the laminate is piecewise hierarchical and is derived based on p-version. The displacement approximation for the element is developed by establishing a hierarchical displacement approximation for each lamina of the laminate and then by imposing interlamina continuity conditions of displacement at the interfaces between the laminas. The approximation functions and the nodal variables for each lamina are derived directly from the Lagrange family of interpolation functions of order pξand pη. 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 pξ+1 and pη+1 equally spaced nodes in the ξ and η directions and then taking their products. The nodal variables for the entire laminate are derived from the nodal variables of the laminas and the interlamina continuity conditions of displacements. The element formulation ensures C0 continuity of displacement across the interelement as well as interlamina boundaries.

Completely hierarchical p-version axisymmetric shell element for nonlinear heat conduction in laminated composites

This paper presents a completely hierarchical p-version finite element formulation for an axisymmetric shell element for nonlinear conduction in laminated composites where the element temperature field can be of arbitrary polynomial orders p ξ and P η in the longitudinal (ξ) and the transverse (η) directions of the shell element. The approximation functions and the corresponding nodal variables for the axisymmetric shell element are derived by first, constructing the one-dimensional hierarchical approximation functions of orders p ξ and p η and the corresponding hierarchical nodal variable operators for each of the two directions ξ and η and then taking their product. This procedure gives the approximation functions and the nodal variables for the axisymmetric shell element that correspond to the polynomial orders p ξ and p η . The element approximation functions as well as the nodal variables are hierarchical and therefore the element matrices and the vectors corresponding to orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). The formulation ensures C 0 continuity across the interelement boundaries. A weak formulation of the Fourier heat conduction equation with temperature dependent thermal conductivities, internal heat generation, film coefficients and radiation effects for globally orthotropic material is constructed. The element properties are derived using the weak formulation and the hierarchical element approximation. This formulation is extended for generally orthotropic material behavior where the material directions are not necessarily parallel to the global axes. Further extension of the formulation for laminated composites is accomplished by incorporating the material properties of each layer through numerical integration of the element matrix for each layer. The element matrices and the equivalent heat vectors (due to convection, distributed heat flux, radiation and internal heat generation) are all hierarchical. The formulation permits any desired order temperature distribution in the shell thickness direction without remodelling. There is no restriction on the number of layers and the lay up pattern of the layers. Each layer can be generally orthotropic. The material directions and the layer thicknesses may vary from point to point within each layer. Because of the temperature dependence of thermal conductivities, internal heat generation, film coefficients and radiation boundaries; the resulting discretized equations of equilibrium are nonlinear. These equations are solved using modified Newton-Raphson method. The tangent matrix is derived using the nonlinear equations of equilibrium. The modified tangent matrix is symmetric and provides excellent convergence during the equilibrium iteration process. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation and the extremely good and fast convergence of the iteration process. Comparisons are made with available results in the literature.

P‐version least‐squares finite element formulation for convection‐diffusion problems

AbstractThis paper presents a p‐version least‐squares finite element formulation of the convection‐diffusion equation. The second‐order differential equation describing convection‐diffusion is reduced to a series of equivalent first‐order differential equations for which the least‐squares formulation is constructed using the same order of approximation for each of the dependent variables. The hierarchical approximation functions and the nodal variable operators are established by first constructing the one‐dimensional hierarchical approximation functions of orders pξ and pη and the corresponding nodal variable operators in ξ and η‐direction and then taking their products. Numerical results are presented and compared with analytical and numerical solutions for a two‐dimensional test problem to demonstrate the accuracy and the convergence characteristics of the present formulation.The Gaussian quadrature rule used to calculate the numerical values of the element matrices, vectors as well as the error functional I(E), is established based on the highest degree of the polynomial in the integrands. It is demonstrated that this quadrature rule with the present p‐version formulation produces excellent results for very low as well as extremely high Peclet numbers (10‐106) and, furthermore, the error functional I (sum of the integrals of E2) is a monotonically decreasing function of the number of degrees of freedom as the p‐levels are increased for a fixed mesh. It is shown that exact integration with the h‐version (linear and parabolic elements) produces inaccurate solutions at high Peclet numbers. Results are also presented using reduced integration. It is shown that the reduced integration with p‐version produces accurate values of the primary variable even for relatively low p‐levels but the error functional I (when calculated using the proper integration rule) has a much higher value (due to errors in the derivatives of the primary variable) and is a non‐monotonic function of the degrees of freedom as p‐levels are increased for a fixed mesh. Similar behaviour of the error functional I is also observed for the h‐models using linear elements when reduced integration is used. Although the h‐models using parabolic elements produce monotonic error functional behaviour as the number of degrees of freedom are increased, the error values are inferior to the p‐version results using exact integration.

Three-dimensional curved shell element based on completely hierarchical p-approximation for heat conduction in laminated composites

This paper presents a finite element formulation for the laminated three-dimensional curved shell element for heat conduction where the temperature approximation for the element can be of arbitrary polynomial order p ξ , p n , and p ξ in the ξ, n, and ξ directions. This is accomplished by first constructing element approximation functions and the corresponding nodal variable operators for each of the three directions ξ, n, and ξ using Lagrange interpolating polynomials and then taking their products (sometimes also referred to as tensor products). This procedure yields the desired approximation functions as well as the nodal variables that correspond to the polynomial orders p ξ , p n, and p ξ . The element approximation functions as well as the nodal variables are hierarchical and therefore the element matrices and the equivalent nodal heat vectors due to distributed heat flux, convection and internal heat generation are hierarchical also, i.e., the element properties corresponding to the polynomial orders p ξ , p n and p ξ are a subset of those corresponding to the polynomial orders ( p ξ + 1), ( p h +1), and ( p ζ + 1). The element formulation retains continuity or smoothness of temperature across the inter-element boundaries, i.e., C 0 continuity is ensured. Since true derivatives of the temperature are retained as nodal variables in the thickness direction of the shell, the element formulation permits step change in the thickness at a mating boundary resulting in continuous temperature distribution along such a mating face. The curved shell geometry is constructed in the usual manner by the coordinates of the element nodes lying on the middle surface of the element (ζ = 0) and the lamina thicknesses at the nodes. The temperature approximation for the element is described by the hierarchical approximation functions and the nodal variables both obtained through the use of tensor product. The element properties, i.e., element matrices and the equivalent nodal heat vectors are derived using weak formulation (or the quadratic functional) of three-dimensional Fourier heat conduction equation and the hierarchical element temperature approximation. The properties of the composite laminate are incorporated in the element formulation through numerically integrating the element matrices for each lamina. The formulation has no restriction on the number of laminas or the layup pattern of the laminas. Each lamina can be generally orthotropic and the material conductivities and the lamina thicknesses may vary from point to point within each lamina. The element formulation is equally effective for very thin as well as very thick laminated curved shells. In fact, in most of the three-dimensional applications the element can be used to replace the hierarchical three-dimensional solid element without any loss of accuracy but significant gain in modeling convenience. Numerical examples are presented to establish the accuracy of the formulation and to demonstrate its efficiency, modeling convenience, and fast rate of convergence. The numerical results obtained from the present formulation are compared with analytical solutions whenever possible. The h-approximation results obtained using three dimensional solids and thick shells are also presented for comparison purposes.

P‐version hierarchical three dimensional curved shell element for elastostatics

AbstractThis paper presents a hierarchical three dimensional curved shell finite element formulation based on the p‐approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (ξ, η) and the transverse direction (ξ). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders pξ, pη and pξ and the corresponding nodal variable operators for each of the three directions ξ, η and ξ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders pξ, pη and pξ. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders pξ, pη and pξ are a subset of those corresponding to the orders (pξ + 1), (pη +1) and (pξ +1). The formulation guarantees C° continuity or smoothness of the displacement field across the interelement boundaries.The geometry of the element is described by the co‐ordinates of the nodes on its middle surface (ξ = 0) and the nodal vectors describing its bottom (ξ = −1) and top (ξ = +1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.

Three-dimensional curved beam element based on p-Version for dynamics

This paper presents a three-node curved three-dimensional beam element for linear dynamic analysis, where the element displacement approximation in the axial (ξ) and transverse directions (η and ζ) can be of arbitrary polynomial orders, p ξ , p η , and p ζ . This is accomplished by first constructing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in ξ, η and ζ directions using Lagrange interpolating polynomials, and then taking the products of these one-dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the corresponding nodal variables for the three-dimensional beam element are hierarchical. The formulation guarantees C 0 continuity. The element properties are established using the principle of virtual work. In formulating the properties of the element, all six components of the stress and strain tensors are retained. The geometry of the beam element is defined by the coordinates of the nodes located at the axis of the beam and node point vectors, representing the nodal cross-sections. The results obtained from the present formulation are compared with available analytical solutions and the h-models using isoparametric three-dimensional solid elements. The formulation is equally effective for very slender as well as deep beams, since no assumptions are made regarding such conditions during the formulation. To demonstrate the effectiveness of this hierarchical formulation in dynamics, numerical results are presented for eigenvalue analysis. The frequencies and the mode shapes are presented for beams with various boundary conditions to demonstrate the accuracy, efficiency, modelling convenience and overall superiority of the present formulation. Numerical results obtained from the present formulation are also compared with those given by the Timoshenko beam theory and other available analytical solutions.

P-Version axisymmetric solid element for heat conduction

A nine-node two-dimensional axisymmetric solid element is presented for linear steady state axisymmetric heat conduction where the temperature field in the ξ and η directions of the element can be of arbitrary polynomial orders p ξ and p η . This is accomplished by first constructing the one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions using Lagrange interpolating polynomials and then taking the product (sometimes also called tensor product) of these hierarchical one-dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the nodal variables are hierarchical, i.e. the element approximation functions and the nodal variables corresponding to the polynomial orders p ξ and p η are a subset of those corresponding to the polynomial orders p ξ + 1 and p η + 1 and as a consequence the element matrices and the equivalent nodal heat vectors are hierarchical also. The formulation ensures C 0 continuity, i.e. the continuity of the temperature across the interelement boundaries is guaranteed. The weak formulation of the Fourier heat conduction equation in the cylindrical coordinate system r, z is constructed and the element properties are derived using the weak formulation (or the associated quadratic functional) and the hierarchical temperature approximation for the element. The element formulation permits isotropic as well as generally orthotropic material properties. The formulation allows distributed heat flux, convective boundaries and internal heat generation. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, faster convergence rate and overall superiority of the present formulation over existing isoparametric axisymmetric solid elements. For each example the results obtained from the present formulation are compared with available analytical solutions and h-models employing isoparametric elements. Convergence graphs showing the rates at which the errors are decreasing are presented for both h and p-models. The results obtained from the present formulation exhibit excellent accuracy, faster convergence rate and require extremely simple models.

Hierarchical three dimensional curved beam element based on p-version

This paper presents a three node curved three dimensional beam element for linear static analysis where the element displacement approximation in the axial (ξ) and transverse directions (η and ζ) can be of arbitrary polynomial orders pξ, pη and pζ. This is accomplished by, first constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators in ξ, η and ζ directions using Lagrange interpolating polynomials and then taking the products (also called tensor product) of these hierarchical one dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the corresponding nodal variables for the three dimensional beam element were hierarchical. The formulation guarantees C0 continuity. The element properties are established using the principle of virtual work. In formulating the properties of the element all six components of the stress and strain tensor are ratained. The geometry of the beam element is defined by the coordinates of the nodes located at the axis of the beam and node point vectors representing the nodal cross-sections. The results obtained from the present formulation are compared with analytical solutions (when available) and the h-models using isoparametric three dimensional solid elements. The formulation is equally effective for very slender as well as deep beams since no assumptions are made regarding such conditions during the formulation.

A twenty-seven-node three-dimensional solid element for heat conduction based on p-version

This paper presents a finite element formulation for a three-dimensional 27-node hierarchical solid element for heat conduction where the element temperature approximation can be of arbitrary polynomial orders p ξ , p η , and p ζ in the ξ, η and ζ directions. The element approximation functions and the corresponding nodal variables are derived directly from the one-dimensional Lagrange interpolation functions of order p ξ , p η and p ζ . This is accomplished by first, establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ, η and ζ directions for the equivalent three-node configurations that correspond to p ξ +1, p η − 1 and p ζ + 1 equally spaced nodes in ξ, η and ζ directions between −1 and +1 and then taking their products. The resulting approximation functions and nodal variables are hierarchical and the element temperature approximation satisfies C 0 condition. The element properties, i.e., the element matrices and the equivalent heat vectors, are derived using the weak formulation (or the quadratic functional) of the three-dimensional Fourier heat conduction equation and the hierarchical temperature approximation. Since the element temperature approximation is hierarchical, the element matrices and the equivalent heat vectors are hierarchical also, i.e., the element properties corresponding to the polynomial orders p ξ , p η and p ζ are a subset of those corresponding to the polynomial orders ( p ξ + 1), ( p η + 1) and ( p ζ + 1). A novel feature of the present formulation is that the element approximation functions and the nodal variables are derived directly from the traditional one-dimensional Lagrange interpolation functions and thus they automatically satisfy completeness requirement. Numerical examples are presented to demonstrate the accuracy, efficiency, modelling convenience, faster rate of convergence and overall superiority of the present formulation. The h-models were also constructed using twenty-node isoparametric solid elements and the results obtained from them are presented for comparison purposes.

P-Version hierarchical three dimensional curved shell element for heat conduction

This paper presents a finite element formulation for a three dimensional nine node p-version hierarchical curved shell element for heat conduction where the element temperature approximation can be of arbitrary order pξ, pη, and pζ in the ξ, η and ζ directions. This is accomplished by first, constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators for each of the three directions ξ, η and ζ using Lagrange interpolating polynomials and then taking their products (sometimes also called tensor products). The element approximation functions as well as the nodal variables are hierarchical and therefore the element matrices and the equivalent heat vectors are hierarchical also i.e. the element properties corresponding to polynomial orders pξ, pη, and pζ are a subset of those corresponding to (pξ+1), (pη+1), and (pζ+1). The element formulation ensures C0 continuity. The curved shell geometry is constructed in the usual way by taking the coordinates of the nodes lying on the middle surface of the element (ζ=0) and the nodal thickness vectors. The element properties i.e. element matrices and the equivalent heat vectors are derived using weak formulation (or quadratic functional) of the three dimensional F ourier heat conduction equation and the hierarchical element temperature approximation. The element formulation is equally effective for very thin as well as extremely thick shells. Numerical examples are presented to demonstrate the accuracy, efficiency, modeling convenience, faster rate of convergence and over all superiority of the present formulation. The h-approximation results are presented for comparison purposes.

Completely hierarchical axisymmetric shell element based on p-version for heat conduction in laminated composites

A completely hierarchical p-version finite element formulation is presented for an axisymmetric shell element for heat conduction in laminated composites, where the element temperature field can be of arbitrary polynomial orders p ξ and p η in the longitudinal (ξ) and the transverse (η) directions of the shell element. The approximation functions and the corresponding nodal variables for the axisymmetric shell element are derived by first constructing the one-dimensional hierarchical approximation functions of orders p ξ and p η and the corresponding hierarchical nodal variable operators for each of the two directions ξ and η and then taking their product (sometimes called the tensor product). This procedure gives the approximation functions and the nodal variables for the axisymmetric shell element that correspond to the polynomial orders p ξ and p η . The element approximation functions and the nodal variables are both hierarchical and therefore the element matrices and the vectors corresponding to orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). The formulation ensures C 0 continuity across the interelement boundaries. The geometry of the element is described in the usual way by the Cartesian coordinates of the nodes located at the middle surface ( η = 0) of the element and the nodal vectors representing nodal values of the lamina thicknesses. Weak formulation of the Fourier heat conduction equation for globally orthotropic material is constructed in the cylindrical coordinate system rz. The element properties are derived using the weak formulation (or the associated quadratic functional) and the hierarchical element approximation. This formulation is extended for generally orthotropic material behavior where the material directions are not necessarily parallel to the global axes. Further extension of the formulation for laminated composites is accomplished by incorporating the material properties of each layer through numerical integration of the element matrix for each layer. The element matrices and the equivalent heat vectors (due to convection, distributed heat flux and internal heat generation) are all hierarchical. The formulation permits any desired order temperature distribution in the ξ and η directions without remodeling. There is no restriction on the number of layers and the lay up pattern of the layers. Each layer can be generally orthotropic. The material directions and the layer thicknesses may vary from point to point within each layer. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. For all examples, h-model results using isoparametric elements are also presented. Comparisons are made with analytical solutions wherever possible.

P-Version hierarchical two dimensional curved beam element for elastostatics

This paper presents a completely hierarchical two dimensional curved beam element formulation where the element displacement field can be of arbitrary polynomial orders p ξ and p η in the axial and the transverse directions of the element. The approximation functions and the corresponding nodal variables for the beam element are derived by first constructing the hierarchical one dimensional approximation functions of orders p ξ and p η , and the corresponding hierarchical nodal variable operators for each of the two directions ξ and η and then taking their product. This procedure yields approximation functions and nodal variables for the curved beam element that correspond to polynomial orders p ξ and p η in ξ and η directions. The element approximation is hierarchical, i.e. the approximation functions and the nodal variables are both hierarchical. Thus, the element matrix and the load vectors corresponding to the polynomial orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). The element formulation ensures C 0 continuity. The element properties are derived using the principle of virtual work and the hierarchical element displacement approximation. The element geometry is constructed using the coordinates of the nodes located on the elastic axis of the element and the node point vectors indicating nodal depths and the element width at the nodes. The orders of approximation along the length of the element as well as in the transverse direction can be chosen independently to obtain optimum (maximum) rate of convergence. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, effectiveness, faster rate of convergence and overall superiority of the present formulation. Results obtained from the present formulation are also compared with h-approximation models and available analytical solutions.