Isoparametric finite element analysis for solid rocket motors

Several large scale finite element computerprograms have been written for the stress analysis of solid propellant rocket motors during the past several years. The displacement formulation is employed, and continuum and shell elements are coupled together along the idealized interface of the grain or insulation and the motor case. However, since the continuum elements exhibit a linear variation in displacements along each edge, and the shell elements have a cubic variation in displacements along their length, gaps can develop at the common interface of a coupled continuum and shell element. This incompatibility may lead to errors in stress predictions at a very important area in the solid rocket motor, the case/liner or case/ grain bondline.

In this present paper, the problem of grain/case finite element incompatibility is addressed within the framework of the conventional displacement formulation. Both the shell elements and the continuum elements are derived from the same quadratic isoparametric shape function, yielding finite element idealizations which are totally compatible along the interface of a continuum and shell element. A numerical example is presented which shows the differences between the present compatible formulation and the conventional incompatible formulations.     

A contact friction algorithm including nonlinear viscoelasticity and a singular yield surface provision

A finite element solution method for two-dimensional boundary value problems involving nonlinear viscoelasticity and contact friction is presented. The simulation of ceramic composites at elevated temperatures is the motivation of coupling viscoelasticity and contact friction. Three major topics are discussed; (i) the time-integration scheme developed for coupling the interface contact friction with nonlinear viscoelasticity in the surrounding continuum, (ii) the spatial discretization of a generalized two-dimensional deformation field using finite elements, and (iii) two methodologies for treating the nonlinearities introduced by the contact friction. The implementation of the contact friction utilizes a penalty method coupled with an incremental plasticity formulation. This formulation results in a highly nonlinear problem, and many solution techniques have difficulty with convergence due to a directional sensitivity arising from the unknown slip direction in the case of three-dimensional contact friction (or the unknown slip direction for hardening or dilatant two-dimensional friction problems). This directional sensitivity is illustrated along with an algorithm which alleviates this difficulty. Also, an algorithm based upon proportional stressing is developed to eliminate problems created by a singular yield surface for idealized Coulomb friction.     

A generalized geometrically nonlinear formulation with large rotations for finite elements with rotational degrees of freedoms

A generalized geometrically nonlinear formulation using total Lagrangian approach is presented for the finite elements with translational as well as rotational degrees of freedoms. An important aspect of the formulation presented here is that the restriction on the magnitude of the nodal rotations is eliminated by retaining true nonlinear nodal rotation terms in the definition of the element displacement field and the consistent derivation of the element properties based on this displacement field. The general derivation and the formulation steps are applicable to any element with translational and rotational nodal degrees of freedoms. The specific forms of the formulation for axisymmetric shells, two-dimensional isoparametric beams, curved shells, two-dimensional transition elements and solid-shell transition elements can be easily derived by considering the explicit forms of the nonlinear nodal rotations for the element at hand. The specific forms of this formulation have already been well tested and applied to various two- and three-dimensional elements, the results for some of which are presented here. Currently it is being applied to the three-dimensional isoparametric beam elements.     

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.     

A scaled thickness conditioning for solid- and solid-shell discretizations of thin-walled structures

The numerical discretization of thin shell structures yields ill-conditioned stiffness matrices due to an inherent large eigenvalue spectrum. Finite element parametrization that depends on shell thickness, like relative displacement shells, solid shells and other solid finite elements even add to the ill-conditioning by introducing high eigenmodes.To overcome this numerical issue we present a scaled thickness conditioning (STC) approach, a mechanically motivated preconditioner for thin-walled structures discretized with continuum based element formulations. The proposed approach is motivated by the scaled director conditioning (SDC) method for relative displacement shell elements. In contrast to SDC, the novel STC approach yields a preconditioner for the effective linear system. It is applicable independently of element technology employed, coupling to other physical fields, boundary conditions applied and additional algebraic constraints and can be easily extended to multilayer shell formulations.The effect of the proposed preconditioner on the conditioning of the effective stiffness matrix and its eigenvalue spectrum is studied. It is shown that the condition number of the modified system becomes almost independent from the aspect ratio of the employed elements. The improved conditioning has a positive influence on the convergence behavior of iterative linear solvers. In particular, in combination with algebraic multigrid preconditioners the number of iterations could be decreased by more than 85% for some examples and the computation time could be reduced by about 60%.     

Insight into finite element shell discretizations by use of the “basic shell mathematical model”

The objective of this paper is to gain insight into finite element discretizations of shells using the basic shell mathematical model and, in particular, regarding the sources of “locking”. We briefly review the “basic shell mathematical model” and present a formulation of shell finite elements based on this model. These shell finite elements are equivalent to the widely-used continuum mechanics based shell finite elements. We consider a free hyperboloid shell problem, which is known to be difficult to solve accurately. Using a fine mesh of MITC9 elements based on the basic shell mathematical model, a detailed analysis is performed giving the distributions of all strain terms. A similar analysis using the MITC6 shell element shows why this element locks when the shell thickness is very small.     

Three-dimensional fluid-structure coupling in transient analysis

The numerical simulation of nonlinear, transient fluid-structure interactions (FSI) is a current area of concern by researchers in various fields, including the field of nuclear reactor safety. This paper primarily discusses the formulation used in an algorithm that couples three-dimensional hydrodynamic and structural domains. Here, both the fluid and structure are discretized using finite elements. The semi-discretized equations of motion are solved using an explicit temporal integrator.Coupling is accomplished by satisfying interface mechanics. The structure imposes kinematic constraints to the moving fluid boundary, and the fluid in turn provides an external loading on the structure. At each interface node, normals are computed from the nodal basis functions of only the hydrodynamic nodes. By defining the interface normal in this manner, it becomes independent of the type of structural boundary (i.e. shell, plate, continuum, etc.) and thus makes this aspect of the coupling independent of the structure type. A penalty type gap-impact element is developed to model the impact region between the fluid and structure.Results for several problems are presented and these include a comparison between analytical results for a FSI problem and numerical predictions.     

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.     

A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis

In this paper, the development of a general three-dimensional L-section beam finite element for elastoplastic large deformation analysis is presented. We propose the generalized interpolation scheme for the isoparametric formulation of three-dimensional beam finite elements and the numerical procedure is developed for elastoplastic large deformation analysis. The formulation is general and effective for other thin-walled section beam finite elements. To show the validity of the formulation proposed, a 2-node three-dimensional L-section beam finite element is implemented in an analysis code. As numerical examples, we first perform elastic small and large deformation analyses of a cantilever beam structure subjected to various tip loadings, and elastoplastic large deformation analysis of the same structure under reversed cyclic tip loading. We then analyze the failures of simply supported beam structures of different lengths and slenderness ratios under elastoplastic large deformation. The same problems are solved using refined shell finite element models of the structures. The numerical results of the L-section beam finite element developed here are compared with the solutions obtained using shell finite element analyses. We also discuss the numerical solutions in detail.     

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

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

Resultant-stress degenerated-shell element

In this paper, a resultant-stress degenerated-shell element is described and a variety of numerical examples, including the post-buckling analysis of an axially loaded perfect cylinder, are presented. The general degenerated nonlinear shell theory of Hughes and Liu is employed in deriving this resultant-stress degenerated-shell element.Contrary to the traditional integration through the thickness approach, which assumes no coupling between the in-plane and transverse material and structural response matrices, the present approach can permit use of arbitrary, three-dimensional (3-D) nonlinear constitutive equations. Furthermore, explicit expressions of the element matrices for a 4-node shell element are developed. This rank-sufficient 4-node shell element, termed the resultant-stress degenerated-shell (RSDS) element, avoids the need for the costly numerical quadrature function evaluations of the element matrices and force vectors. And thus there are large increases in computational efficiency with this method. The comparisons of this RSDS element with six other shell elements are also given in this paper.     

A one field full discontinuous Galerkin method for Kirchhoff–Love shells applied to fracture mechanics

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the finite element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive elements. It is well known that, on one the hand, if these elements are present before fracture there is a modification of the structure stiffness, and that, on the other hand, their insertion during the simulation requires very complex implementation, especially with parallel codes. These drawbacks can be avoided by combining the cohesive method with the use of a discontinuous Galerkin formulation. In such a formulation, all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements. The recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture.The purpose of this paper is to develop this formulation for Kirchhoff–Love plates and shells. It is achieved by the establishment of a full DG formulation of shell combined with a cohesive model, which is adapted to the special thickness discretization of the shell formulation. In fact, this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations. Finally, numerical examples demonstrate the efficiency of the method.     

Viscoplasticity based on additive decomposition of logarithmic strain and unified constitutive equations: Theoretical and computational considerations with reference to shell applications

In contrast to multiplicative models of finite strain plasticity and viscoplasticity, a framework of additive nature is developed in this paper. The theory is based on the additive decomposition of the logarithmic strain tensor. The stress conjugate to the logarithmic strain then plays the role of the thermodynamically driving force. The approach in this paper is motivated by the search for numerically accessible structures which can be extended to incorporate anisotropy as well. Specifically in this region, multiplicative formulations become extremely tedious. The evolution equations are of the unified type due to Bodner and Partom, and are modified so as to fit into the theoretical framework adopted. The numerical treatment of the problem is fully developed. Specifically, the algorithmic aspects of the approach are discussed and various applications to shell problems are considered. A shell theory with seven degrees of freedom, together with a four-node enhanced strain finite element formulation, is used. A central feature of the shell formulation is its eligibility to the application of a three-dimensional constitutive law.     

Hybrid-interface element for thick laminated composite plates

A novel 27-node three-dimensional hexahedral hybrid-interface finite element (FE) model has been presented to analyze laminated composite plates and sandwich plates using the minimum potential energy principle. Fundamental elasticity relationship between components of stress, strain and displacement fields are maintained throughout the elastic continuum as the transverse stress components have been invoked as nodal degrees of freedom. Continuity of the transverse stresses at lamina interface has been maintained. Each lamina is modeled by using hybrid-interface elements at the top and the bottom interfaces and conventional displacement based elements sandwiched between these interfaces. Results obtained from the present formulation have found to be in excellent agreement with the elasticity solutions for thin and thick composite cross-ply, angle-ply laminates, as well as sandwich plates. Additional results have also been presented on the variation of the transverse strains to highlight magnitude of discontinuity in these quantities due to difference in properties of face and core materials of sandwich plates. Present formulation can be used effectively to interface hybrid formulation that uses transverse stresses and displacements as degrees of freedom with conventional purely displacement based formulation for realistic estimates of the transverse stresses.     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.     

Applications of isoparametric three-dimensional hybrid-stress finite elements with least-order stress fields

This paper investigates the factors relating to element stability, coordinate invariance and optimality in 8- and 20-noded three-dimensional brick elements in the context of hybrid-stress formulations with compatible boundary displacements and both a priori and a posteriori equilibrated assumed stresses. Symmetry group theory is used to guarantee the essential non-orthogonality of the stress and strain fields, resulting in a set of least-order selections of stable invariant stress polynomials. The performance of these elements is examined in numerical examples providing a broad range of analytical stress distributions, and the results are favourably compared to those of the displacement formulation and a stress-function based complete stress approach with regard to displacements, stresses, sampling, convergence and distortion sensitivity.     

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.     

Modeling of piezoelectric sensors adhesively bonded on trusses using a mathematical programming approach

In this study, piezoelectric sensors design adhesively bonded on truss elements is treated in the framework of mathematical programming. A numerical formulation based on the strength capacity of set structure, adhesive and piezoelectric sensor is proposed. Inside the formulations maximum strength capacity of the adhesive is considered as a limit value in the design. Two formulations are established to obtain the maximum strength of the set; the first one is built on the basis of finite differences and the other one on a formulation of finite elements both based on an admissible static field. The lower bound method applied to limit analysis is extended in this research to analyze trusses with sensors including the adhesive interface. Four examples are designed to assess the numerical methodologies in which the results are compared with other known data. The main contribution of this work is focused on finding the maximum coupling load that a piezoelectric sensor can read before being debonded based on the minimum size constraint of the sensor.     

Comparison of different formulations of 2D beam elements based on Bond Graph technique

Bond Graphs are well suited for modelling multibody systems. In this paper modelling of planar flexible beams undergoing large overall motions are studied based on finite element (FE) technique. Two well-known approaches are used – the co-rotational (CR) and absolute nodal coordinate (ANC) formulation. Two ANC formulations are analyzed – one in which elastic forces is described using classical beam theory in a local coordinate frame, and another based on a global continuum mechanics approach. Starting from these classical formulations velocity formulations are developed and used to develop Bond Graph FE components. The effect of gravity has been considered as well. These components can be put in libraries and used for systematic Bond Graph flexible body model development. It is shown that Bond Graph technique is capable of dealing with different flexible body formulations and can be used as a general approach in parallel to other modelling approaches. Models are developed and simulations are performed using the object oriented environment of BondSim. Owing to the object oriented approach, transformation from one to the other model is relatively simply. The results are illustrated by suitable examples and they confirm accuracy of the developed models. It was shown that the CR approach offers much better performance than the both ANC formulations.