An analysis of some mixed-enhanced finite element for plane linear elasticity

The paper investigates Mixed-Enhanced Strain finite elements developed within the context of the u/p formulation for nearly incompressible linear elasticity problems. A rigorous convergence and stability analysis is detailed, providing also L²-error estimates for the displacement field. Extensive numerical tests are developed, showing in particular the accordance of the computational results with the theoretical predictions.     

Vibration of thin shells of revolution based on a mixed finite element formulation

A modified Hellinger-Reissner functional for thin shells of revolution is presented. A mixed finite element formulation is developed from this functional which is free from line integrals and relaxed continuity terms. This formulation is applied to the problem of free vibration of spherical and conical shells. Bilinear trial functions are used for all field variables. The quadrilateral curved elements here presented satisfy the C⁰ continuity requirement of the functional. In all the results obtained the accuracy is quite good even for a reasonable     

A new mixed finite element based on different approximations of the minors of deformation tensors

Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element.     

Mixed finite elements of least-squares type for elasticity

In terms of stress and displacement, the linear elasticity problem is discretized by a least-squares finite element method. In the case of a convex polygonal domain, the stress is approximated by the lowest-order Raviart–Thomas–Nédélec flux element, and the displacement by the linear C⁰ element. We obtain coerciveness and optimal H¹, L² and H(div)-error bounds, uniform in Lamé constant λ, for displacement and stress, respectively. Our method also allows the use of any other combination of conforming elements for stress and displacement, e.g., C⁰ elements for all variables.     

Topology optimization of structures with gradient elastic material

Topology optimization of structures and mechanisms with microstructural length-scale effect is investigated based on gradient elasticity theory. To meet the higher-order continuity requirement in gradient elasticity theory, Hermite finite elements are used in the finite element implementation. As an alternative to the gradient elasticity, the staggered gradient elasticity that requires C ⁰-continuity, is also presented. The solid isotropic material with penalization (SIMP) like material interpolation schemes are adopted to connect the element density with the constitutive parameters of the gradient elastic solid. The effectiveness of the proposed formulations is demonstrated via numerical examples, where remarkable length-scale effects can be found in the optimized topologies of gradient elastic solids as compared with linear elastic solids.     

A least-squares variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells

A variational principle is formulated for the inverse problem of full-field reconstruction of three-dimensional plate/shell deformations from experimentally measured surface strains. The formulation is based upon the minimization of a least-squares functional that uses the complete set of strain measures consistent with linear, first-order shear-deformation theory. The formulation, which accommodates for transverse shear-deformation, is applicable for the analysis of thin and moderately thick plate and shell structures. The main benefit of the variational principle is that it is well-suited for C⁰-continuous displacement finite element discretizations, thus enabling the development of robust algorithms for application to complex civil and aeronautical structures. The methodology is especially aimed at the next generation of aerospace vehicles for use in real-time structural health monitoring systems.     

A class of C finite elements for the static and dynamic analysis of laminated plates

A general higher-order deformation theory is developed to analyse the behaviour of an arbitrary laminated fibre-reinforced composite plate. Three-dimensional effects such as the warping of sections and the presence of interlaminar stress field components are taken into account assuming a power series expansion of displacements along the thickness. A class of C⁰ finite element models based on this theory is then developed for mono- and bi-dimensional elements. Applications of the models to bending and vibration of laminated plates are then discussed. The present solutions are compared with those obtained using the three-dimensional elasticity theory, classical laminate theory and other higher-order theories.     

Implementation of a C1 triangular element based on the P-version of the finite element method

The implementation of a computer code CONE (for C¹ continuity) based on the p-version of the finite element method is described. A hierarchic family of triangular finite elements of degree p ≥ 5 is used. This family enforces C¹-continuity across inter-element boundaries, and the code is applicable to fourth order partial differential equations in two independent variables, in particular to the biharmonic equation. Applications to several benchmark problems in plate bending are presented. Sample results are examined and compared both with theoretical predictions and with the computations of other programs. Significant improvements are shown for the results obtained using CONE.     

A finite element model for the analysis of viscoelastic sandwich structures

In this work a finite element model is developed for vibration analysis of active–passive damped multilayer sandwich plates, with a viscoelastic core sandwiched between elastic layers, including piezoelectric layers. The elastic layers are modelled using the classic plate theory and the core is modelled using the Reissener–Mindlin theory. The finite element is obtained by assembly of N “elements” through the thickness, using specific assumptions on the displacement continuity at the interfaces between layers. The lack of finite element plate-shell models to analyse structures with passive and active damping, is the principal motivation for the present development, where the solution of some illustrative examples and the results are presented and discussed.     

A nonlinear visco-elastic constitutive model for large rotation finite element formulations

Although all known materials have internal damping that leads to energy dissipation, most existing large deformation visco-elastic finite element formulations are based on linear constitutive models or on nonlinear constitutive models that can be used in the framework of an incremental co-rotational finite element solution procedure. In this investigation, a new nonlinear objective visco-elastic constitutive model that can be implemented in non-incremental large rotation and large deformation finite element formulations is developed. This new model is based on developing a simple linear relationship between the damping forces and the rates of deformation vector gradients. The deformation vector gradients can be defined using the decomposition of the matrix of position vector gradients. In this paper, the decomposition associated with the use of the tangent frame that is equivalent to the QR decomposition is employed to define the matrix of deformation gradients that enter into the formulation of the viso-elastic constitutive model developed in this investigation. Using the relationship between the deformation gradients and the components of the Green–Lagrange strain tensor, it is shown that the damping forces depend nonlinearly on the strains and linearly on the classical strain rates. The relationship between the damping forces and strains and their rates is used to develop a new visco-elastic model that satisfies the objectivity requirements and leads to zero strain rates under an arbitrary rigid body displacement. The linear visco-elastic Kelvin–Voigt model frequently used in the literature can be obtained as a special case of the proposed nonlinear model when only two visco-elastic coefficients are used. As demonstrated in this paper, the use of two visco-elastic coefficients only leads to viscous coupling between the deformation gradients. The model developed in this investigation can be used in the framework of large deformation and large rotation non-incremental solution procedure without the need for using existing co-rotational finite element formulations. The finite element absolute nodal coordinate formulation (ANCF) that allows for straightforward implementation of general constitutive material models is used in the validation of the proposed visco-elastic model. A comparison with the linear visco-elastic model is also made in this study. The results obtained in this investigation show that there is a good agreement between the solutions obtained using the proposed nonlinear model and the linear model in the case of small deformations.     

Triangular element for analysis of perforated plates under inplane and transverse loads

A C⁰-type triangular element formulation in orthogonal curvilinear co-ordinates has been developed, based on assumptions of transverse inextensibility and constant shear angle through thickness for analysis of perforated plates subjected to inplane and transverse loads. The assumed quadratic displacement potential energy approach is utilized in obtaining an element stiffness matrix and consistent load vector, which are numerically integrated. Numerical results have been obtained using a straight-sided triangular version, which behaves like a subparametric element, for stretching and bending analyses of perforated plates.     

Application of a three-dimensional mixed finite element model to the flexure of sandwich plate

The bending analysis of sandwich plates consisting of very stiff face sheets and a comparatively flexible core material offers challenge due to large variation in the magnitude of stress and strain components in the face and in the core regions of the plate. Similarly, the displacement fields do vary in zigzag manner at the layer interface of stiff face sheet and the soft core, thereby making the transverse strains highly discontinuous at such layer interfaces. All these behavioural aspects indicate that only an individual layerwise model can appropriately analyze sandwich plates. A layerwise (three-dimensional), mixed, 18-node finite element (FE) model developed by Ramtekkar et al. [Mech. Adv. Mater. Struct. 9 (2002) 133] has been employed for the accurate evaluation of transverse stresses in sandwich laminates. The FE model consists of six degrees-of-freedom (three displacement components and three transverse stress components τ_xz, τ_yz, σ_z, where z is the thickness direction) per node which ensures the through thickness continuity of transverse stress and displacement fields. Results obtained by using the FE model have shown excellent agreement with the available elasticity solutions for sandwich plates. Additional results on the variation of transverse strains have also been presented to highlight the magnitude of discontinuity in these quantities due to difference in properties of the face and the core materials of sandwich plates.     

Complete stress field in sandwich panels with a new triangular finite element of single-layer theory

An alternative to conventional three-dimensional solid elements or elements based on the layerwise (zig-zag) theory is an element based on a single-layer plate theory in which the weighted-average field variables capture the panel displacement and stress fields. This study presents a new triangular finite element for modeling thick sandwich panels based on a {3, 2}-order single-layer plate theory. It utilizes seven weighted-average field variables arising from the cubic and quadratic representations of the in-plane and transverse displacement fields, respectively. In order to satisfy the C¹ interelement continuity requirement, this triangular sandwich element is developed by utilizing the hybrid energy functional.     

An analytical C3 continuous tool path corner smoothing algorithm for 6R robot manipulator

Tool path smoothness is important to guarantee good dynamic and tracking performance of robot manipulators. An analytical C³ continuous tool path corner smoothing algorithm is proposed for robot manipulators with 6 rotational (6R) joints. The tool tip position is smoothed directly in the workpiece coordinate system (WCS). The tool orientation is smoothed after transferring the tool orientation matrix as three rotary angles. Micro-splines of the tool tip position and tool orientation are constructed under the constraints of the maximum deviation error tolerances in the WCS. Then the tool orientation and tool tip position are synchronized to the tool tip displacement with C³ continuity by replacing the remaining linear segments using specially constructed B-splines. Control points of the locally inserted micro-splines are all evaluated analytically without any iterative calculations. Simulation and experimental results show that the proposed algorithm satisfies constraints of the preset tool tip position and the tool orientation tolerances. The proposed corner smoothing algorithm achieves smoother and lower jerks than C² continuous corner smoothing algorithm. Experimental results show that the tracking errors associated to the execution of the C³ continuous tool path are up to 10% smaller than C² continuous path errors.     

The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches

In this paper we present an isogeometric formulation for rotation-free thin shell analysis of structures comprised of multiple patches. The structural patches are C¹- or higher-order continuous in the interior, and are joined with C⁰-continuity. The Kirchhoff–Love shell theory that relies on higher-order continuity of the basis functions is employed in the patch interior as presented in Kiendl et al. [36]. For the treatment of patch boundaries, a method is developed in which strips of fictitious material with unidirectional bending stiffness and zero membrane stiffness are added at patch interfaces. The direction of bending stiffness is chosen to be transverse to the patch interface. This choice leads to an approximate satisfaction of the appropriate kinematic constraints at patch interfaces without introducing additional stiffness to the shell structure. The attractive features of the method include simplicity of implementation and direct applicability to complex, multi-patch shell structures. The good performance of the bending strip method is demonstrated on a set of benchmark examples. Application to a wind turbine rotor subjected to realistic wind loads is also shown. Extension of the bending strip approach to the coupling of solids and shells is proposed and demonstrated numerically.     

Compatibility Condition in Theory of Solid Mechanics (Elasticity,Structures, and Design Optimization)

The strain formulation in elasticity and the compatibility condition in structural mechanics have neither been understood nor have they been utilized. This shortcoming prevented the formulation of a direct method to calculate stress and strain, which are currently obtained indirectly by differentiating the displacement. We have researched and understood the compatibility condition for linear problems in elasticity and in finite element structural analysis. This has lead to the completion of the “method of force” with stress (or stress resultant) as the primary unknown. The method in elasticity is referred to as the completed Beltrami-Michell formulation (CBMF), and it is the integrated force method (IFM) in the finite element analysis. The dual integrated force method (IFMD) with displacement as the primary unknown had been formulated. Both the IFM and IFMD produce identical responses. The IFMD can utilize the equation solver of the traditional stiffness method. The variational derivation of the CBMF produced the existing sets of elasticity equations along with the new boundary compatibility conditions, which were missed since the time of Saint-Venant, who formulated the field equations about 1860. The CBMF, which can be used to solve stress, displacement, and mixed boundary value problems, has eliminated the restriction of the classical method that was applicable only to stress boundary value problem. The IFM in structures produced high-fidelity response even with a modest finite element model. Because structural design is stress driven, the IFM has influenced it considerably. A fully utilized design method for strength and stiffness limitation was developed via the IFM analysis tool. The method has identified the singularity condition in structural optimization and furnished a strategy that alleviated the limitation and reduced substantially the computation time to reach the optimum solution. The CBMF and IFM tensorial approaches are robust formulations because both methods simultaneously emphasize the equilibrium equation and the compatibility condition. The vectorial displacement method emphasized the equilibrium, while the compatibility condition became the basis of the scalar stress-function approach. The tensorial approach can be transformed to obtain the vector and the scalar methods, but the reverse course cannot be followed. The tensorial approach outperformed other methods as expected. This paper introduces the new concepts in elasticity, in finite element analysis, and in design optimization with numerical illustrations.     

Isogeometric analysis for strain field measurements

In this paper, the potential of isogeometric analysis for strain field measurement by digital image correlation is investigated. Digital image correlation (DIC) is a full field kinematics measurement technique based on gray level conservation principle and the formulation we adopt allows for using arbitrary displacement bases. The high continuity properties of Non-Uniform Rational B-Spline (NURBS) functions are exploited herein as an additional regularization of the initial ill-posed problem. k-Refinement is analyzed on an artificial test case where the proposed methodology is shown to outperform the usual finite element-based DIC. Finally a fatigue tensile test on a thin aluminum sheet is analyzed. Strain localization occurs after a certain number of cycles and combination of NURBS into a DIC algorithm clearly shows a great potential to improve the robustness of non-linear constitutive law identification.     

Influence of the order of polynomial on the convergence in ritz finite element formulation to nonlinear vibrations of beams

The influence of the order of inplane polynomial on the convergence of solution, when a Ritz finite element formulation is used to study nonlinear vibrations of beams, is investigated here. Three types of polynomial distributions for the inplane displacement “u” are considered while the polynomial distribution for transverse displacement “w” is retained as cubic always. A hinged-hinged beam on immovable ends with different discretization is chosen as an example for the convergence study on the nonlinear hardening parameter. From the results obtained, it has been concluded that for a chosen cubic polynomial distribution for transverse displacement, a cubic polynomial distribution for the inplane displacement will be a compatible mode shape satisfying the physical aspects of the convergence and nature of bound for the nonlinear hardening parameter.     

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 unified analysis for stress/strain hybrid methods of high performance

Several methods have been developed in the literatures of computational mechanics to improve the performance of the conventional lower-order displacement finite elements which yield poor results for problems with bending and for nearly incompressible medium. This paper is devoted to a unified analysis of convergence for Pian–Sumihara’s, Chen–Cheung’s and Piltner–Taylor’s enhanced stress/strain schemes. By virtue of the energy compatibility and the rank condition, error estimates for these typical finite elements of high performance are obtained in a unified framework, and especially, weakly locking-free error estimates with respect to the Poisson’s ratio ν in energy norms are obtained uniformly for ν⩽(1−Ch)/2 as h→0, where C is a constant independent of ν and the mesh size h. Very much the same about the three methods is pointed out.