Karan S. Surana

Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements

Abstract This paper presents a geometrically non-linear formulation using total lagrangian approach for the solid-shell transition finite elements. Such transition finite elements are necessary in geometrically non-linear analysis of structures modelled with three dimensional solid elements and the curved shell elements. These elements are an essential connecting link between the solid elements and the shell elements. The element formulation presented here is derived using the properties of the three dimensional solid elements and the curved shell elements. No restrictions are imposed on the magnitude of the nodal rotations. Thus the element formulation is capable of handling large rotations between two successive load increments. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate their behavior, accuracy and applications in three dimensional stress analysis. It is shown that the selection of different stress and strain components at the integration points do not effect the overall linear response of the element. However, in geometrically non-linear applications it may be necessary to select appropriate stress and the strain components at the integration points for stable and converging element behavior. Numerical examples illustrate various characteristics of the element.

Two-dimensional curved beam element with higher-order hierarchical transverse approximation for laminated composites

Abstract A new two-dimensional curved beam element formulation with higher-order transverse shear deformation for linear static analysis of laminated composites is presented. The displacement approximation in the transverse direction can be of arbitrary desired polynomial order p, thereby permitting strains of at least order (p − 1). This is accomplished by introducing additional nodal degrees of freedom in the element displacement approximation that correspond to Lagrange interpolating polynomials in the transverse direction. The element has an important hierarchical property; the element stiffness matrix, generalized nodal displacement vector and the equivalent nodal load vector corresponding to a polynomial order p are a subset of those corresponding to the approximation order p + 1. The element formulation provides displacement continuity across the interelement boundaries, i.e. C0 continuity is ensured automatically at the mating element boundaries. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is presented for generally orthotropic material behavior, where the material directions may not be parallel to the global axes, as well as for the laminated composites. The laminated composite material behavior is incorporated in the element formulation through numerical integration for each lamina. For each lamina, the material behavior may be generally orthotropic, and the material direction and lamina thicknesses may vary from point to point within the lamina. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, effectiveness, faster rate of convergence and overall superiority of the present formulation for laminated composite material behavior. Results obtained from the present formulation are also compared with the available analytical solutions.

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

In this paper a new mathematical and computational framework for boundary value problems described by self-adjoint differential operators is presented. In this framework, numerically computed solutions, when converged, possess the same degree of global smoothness in terms of differentiability up to any desired order as the theoretical solutions. This is accomplished using spaces Ĥk,p that contain basis functions of degree p and order k - 1 (or the order of the space k). It is shown that the order of space k is an intrinsically important independent parameter in all finite element computational processes in addition to the discretization characteristic length h and the degree of basis functions p when the theoretical solutions are analytic. Thus, in all finite element computations, all quantities of interest (e.g., quadratic functional, error or residual functional, norms and seminorms, error norms, etc.) are dependent on h, p as well as k. Therefore, for fixed h and p, convergence of the finite element process can also be investigated by changing k, hence k-convergence and thus the k-version of finite element method. With h, p, and k as three independent parameters influencing all finite element processes, we now have k, hk, pk, and hpk versions of finite element methods. The issue of minimally conforming finite element spaces is reexamined and it is demonstrated that the definition of currently believed minimally conforming space which permit weak convergence of the highest-order derivatives of the dependent variables appearing in the bilinear form is not justifiable mathematically or from physics view point. A new criterion is proposed for establishing the minimally conforming spaces which is more in agreement with the physics and mathematics of the BVP. Significant features and merits of the proposed mathematical and computational framework are presented, discussed, illustrated, and substantiated mathematically as well as numerically with the Galerkin and least-squares finite element formulations for self-adjoint boundary-value problems.

Three dimensional curved shell finite elements for heat conduction

Abstract 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.

A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

This paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier–Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier–Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C0 element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the space–time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space–time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {δ} which makes gradient of error functional with respect to {δ} a null vector. This is accomplished by using Newtons method with a line search. A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space–time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space–time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space–time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.

THE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVP

In the companion papers [1,2], authors introduced the concepts of k-version of finite element method and k, hk, pk, hkp-processes of the finite element method for boundary value problems described by self-adjoint and non-self adjoint operators using Ĥk,p(Ω) spaces with specific details including numerical studies for weak forms and least square processes. It was demonstrated that a variationally consistent (VC) weak form is possible when the differential operator is self-adjoint, however, in case of non-self-adjoint operators the weak forms are variationally inconsistent (VIC) which lead to degenerate computational processes that can produce spurious oscillations in the computed solutions. In this paper we demonstrate that when the boundary value problems are described by non-linear differential operators, Galerkin processes and weak forms can never be variationally consistent and hence result in degenerate computational processes and suffer from same problems as in the case of non-self-adjoint operators ...

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

Abstract 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. C 0 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.

The k-Version of Finite Element Method for Initial Value Problems: Mathematical and Computational Framework

This paper presents a mathematical and computational framework for initial value problems (IVP) in which the numerical approximations can be of higher order global differentiability in space and time and the resulting computational processes are unconditionally stable. This is accomplished using Hk, p scalar product spaces containing basis functions of degree p = (p1,p2), p1 and p2 being the degrees of local approximation in space and time and order k = (k1, k2), k1 and k2 being orders of the scalar product space in space and time and ensuring that the integral forms are space-time integral forms that are space-time variationally consistent (STVC). It is shown that order of the scalar product space k (in space and time) is an intrinsically important independent parameter in all finite element computations for IVP in addition to the discretization length h and the degree p of the local approximations, thus in all finite element computations all quantities are dependent on h, p and k. Hence, we have k-version of finite element method and associated k, hk, pk and hpk processes in addition to h, p and hp processes for IVP. Space-time meshes as well as space-time, time marching approaches are discussed and it is demonstrated that space-time, time marching processes are superior in all aspects compared to space-time meshes. Significant features of the proposed mathematical and computational framework are presented and discussed mathematically in context with Galerkin method, Petrov Galerkin method, Weighted Residual method, Galerkin method with weak form and Least Squares Processes.

ON p-VERSION HIERARCHICAL INTERPOLATION FUNCTIONS FOR HIGHER-ORDER CONTINUITY FINITE ELEMENT MODELS

This paper presents a development of the p-version hierarchical interpolation functions in one, two three and higher dimensions. The interpolation functions may be used, for example, in obtaining strong solutions of differential equations. The concepts are presented first through the derivation of the Ci-functions in one dimension. Then the one-dimensional functions are used to derive the interpolation functions in two, three and higher dimensions. It is demonstrated that for rectangular finite elements the concepts presented herein allow the construction of any order hierarchical interpolation functions in one, two, three and higher dimensions. The necessary and sufficient conditions that must be satisfied by higher order continuity interpolation functions are presented. Theorems that insure that the proposed procedure of generating the higher order continuity interpolation functions are also presented. With this approach of constructing interpolations it is indeed possible to generate finite element interpolation functions with inter-element continuity in agreement with the strong solutions, which eliminate the need for the weak forms of differential equations.

NONWEAK/STRONG SOLUTIONS IN GAS DYNAMICS: A C11 p-VERSION STLSFEF IN LAGRANGIAN FRAME OF REFERENCE USING ρ, U, T PRIMITIVE VARIABLES

In this paper we present computations of the nonweak solutions of class C11 of the strong form of transient Navier–Stokes equations for compressible flow in Lagrangian frame of reference using space–time least squares finite element formulation (STLSFEF) with primitive variables ρ, u, T. For high speed compressible flows the solutions reported here possess the same orders of continuity as the governing differential equations (GDEs). It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations without any assumptions or approximations are possible. In the approach presented here SUPG, SUPG/DC, or SUPG/DC/LS operators are neither used nor needed. The role of diffusion, i.e., viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Merits of using ρ, u, T as primitive variables over ρ, u, p are discussed. Compression of air in a rigid cylinder by a rigid, massless and frictionless piston is used as a model problem. True time evolutions of class C11 are reported beginning with the first time step until steady shock conditions are achieved. Comparisons with analytical solutions are presented when possible.