Geometrically nonlinear analysis of shallow shells using higher order finite elements

The postbuckling behaviour of thin shells has fascinated researchers because the theoretical prediction and their experimental verification are often different. In reality, shell panels possess small imperfections and these can cause large reduction in static buckling strength. This is more relevant in thin laminated composite shells. To study the postbuckling behaviour of thin, imperfect laminated composite shells using finite elements, explicit incremental or secant matrices have been presented in this paper. These incremental matrices which are derived using Marguerre's shallow shell theory can be used in combination with any thin plate/shell finite element (Classical Laminated Plate Theory - CLPT) and can be easily extended to the First Order Shear deformation Theory (FOST). The advantage of the present formulation is that it involves no numerical approximation in forming total potential energy of the shell during large deformations as opposed to earlier approximate formulations published in the literature. The initial imperfection in shells could be modeled by simply adjusting the ordinate of the shell forms. The present formulation is very easy to implement in any existing finite element codes. The secant matrices presented in this paper are shown to be very accurate in tracing the postbuckling behaviour of thin isotropic and laminated composite shells with general initial imperfections.

Free Vibrations of Composite Shallow Circular Cylindrical Shell Panels Stiffened by A Bonded Non-Central Shell Strip

This paper presents recent work on developing methods for analyzing radiation heat transfer between diffuse-gray surfaces using p-version finite elements. The work was motivated by a thermal analysis of a High Speed Civil Transport (HSCT) wing structure which showed the importance of radiation heat transfer throughout the structure. The analysis also showed that refining the finite element mesh to accurately capture the temperature distribution on the internal structure led to very large meshes with unacceptably long execution times. Traditional methods for calculating surface-to-surface radiation are based on assumptions that are not appropriate for p-version finite elements. Two methods for determining internal radiation heat transfer are developed for one and two-dimensional p-version finite elements. In the first method, higher-order elements are divided into a number of sub-elements. Traditional methods are used to determine radiation heat flux along each sub-element and then mapped back to the parent element. In the second method, the radiation heat transfer equations are numerically integrated over the higher-order element. Comparisons with analytical solutions show that the integration scheme is generally more accurate than the sub-element method. Comparison to results from traditional finite elements shows that significant reduction in the number of elements in the mesh is possible using higher-order (p-version) finite elements.

Boundary Element Analysis of Cracks in Shear Deformable Plates and Shells. Topics in Engineering, Vol 43

State-of-the-art nonlinear computational analysis of shells, nonlinearities due to large deformations and nonlinear material behavior, alternative shell element formulations, algorithms and implementational aspects, composite and sandwich shells, local and global instabilities, optimization of shell structures and concepts of shape finding methods of free from shells. Furthermore, algorithms for the treatment of the nonlinear stability behavior of shell structures (including bifurcation and snap-through buckling) are presented in the book.

Spurious modes in finite-element discretizations of the wave equation may not be all that bad

Finite- and boundary-element linear and nonlinear analyses of shells and shell-like structures: A bibliography (1999–2001)

Vibrations of Moderately Thick Shallow Spherical Shells at Large Amplitudes

Nonlinear axisymmetric buckling analysis of the FGM sandwich shallow spherical shells under thermomechanical loads

This study deals with the geometrically non-linear vibration analysis of concrete shallow funicular shells of rectangular plan with four clamped edges under impulse loads. The shape of a concrete funicular shell is such that the shell is subjected to pure compression under its dead weight. Following the existing method presented for the linear vibration analysis, the geometrically non-linear vibration analysis is considered through the use of non-linear shallow shells theory. Each displacement component is expanded in a double Fourier series and the kinetic energy, the elastic strain energy and the virtual work done by external forces are calculated in terms of the displacement components. Then, the equations of motion are obtained using the Lagrangian approach and are solved with the Runge–Kutta fourth-order method. The solution is verified against the results obtained with the finite-element method. The difference between the results of the linear and non-linear vibration analyses has been considered. Furthermore, it is indicated that under the considered dynamic loads, internal moments and consequentially tensile stresses are formed in the funicular shell and the shell does not behave purely as a funicular element. Finally, the plan aspect ratio effect on the time response of funicular shells has been shown.

Procedures are investigated by which nonlinear finite element shell analysis algorithms can be simplified to provide more cost effective approximate analyses of orthogonally-reinforced concrete flat plate structures. Two alternative effective stiffness formulations, and an unbalanced force formulation, are described. These are then implemented into a nonlinear shell analysis algorithm. Nonlinear geometry, three-dimensional layered stress analyses, and other general formulations are bypassed to reduce the computational burden. In application to standard patch test problems, these simplified approximate analysis procedures are shown to provide reasonable accuracy while significantly reducing the computational effort. Corroboration studies using various simple and complex test specimens provide an indication of the relative accuracy of the constitutive models utilized. The studies also point to the limitations of the approximate formulations, and identify situations where one should revert back to full nonlinear shell analyses.

As demands for high-fidelity physics-based animations increase, the need for accurate methods for simulating deformable solids grows. While higherorder finite elements are commonplace in engineering due to their superior approximation properties for many problems, they have gained little traction in the computer graphics community. This may partially be explained by the need for finite element meshes to approximate the highly complex geometry of models used in graphics applications. Due to the additional perelement computational expense of higher-order elements, larger elements are needed, and the error incurred due to the geometry mismatch eradicates the benefits of higher-order discretizations. One solution to this problem is the embedding of the geometry into a coarser finite element mesh. However, to date there is no adequate, practical computational framework that permits the accurate embedding into higher-order elements. We develop a novel, robust quadrature generation method that generates theoretically guaranteed high-quality sub-cell integration rules of arbitrary polynomial accuracy. The number of quadrature points generated is bounded only by the desired degree of the polynomial, independent of the embedded geometry. Additionally, we build on recent work in the Finite Cell Method (FCM) community so as to tackle the severe ill-conditioning caused by partially filled elements by adapting an Additive-Schwarz-based preconditioner so that it is suitable for use with state-of-the-art non-linear material models from the graphics literature. Together these two contributions constitute a general-purpose framework for embedded simulation with higher-order finite elements. We finally demonstrate the benefits of our framework in several scenarios, in which second-order hexahedra and tetrahedra clearly outperform their first-order counterparts.

A refined non-conforming triangular plate/shell element for linear and geometrically nonlinear analysis of plates and shells is developed in this paper based on the refined non-conforming element method (RNEM). A conforming triangle membrane element with drilling degrees of freedom in Cartesian coordinates and the refined non-conforming triangular plate-bending element RT9, in which Kirchhoff kinematic assumption was adopted, are used to construct the present element. The displacement continuity condition along the interelement boundary is satisfied in an average sense for plate analysis, and the coupled displacement continuity requirement at the interelement is satisfied in an average sense, thereby improving the performance of the element for shell analysis. Selectively reduced integration with stabilization scheme is employed in this paper to avoid membrane locking. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plate bending problems and plane problems or for the geometrically nonlinear analysis of thin plates and shells with large displacement, moderate rotation but small strain.

The paper is concerned with eigen buckling analysis of curvilinear shells with and without cutouts by an effective meshfree method. In particular, shallow shell, cylinder and perforated cylinder buckling problems are considered. A Galerkin meshfree reproducing kernel (RK) approach is then developed. The present meshfree curvilinear shell model is based on Reissner-Mindlin plate formulation, which allows the transverse shear deformation of the curved shells. There are five degrees of freedom per node (i.e., three displacements and two rotations). In this setting, the meshfree interpolation functions are derived from the RK. A singular kernel is introduced to impose the essential boundary conditions because of the RK shape functions, which do not automatically possess the Kronecker delta property. The stiffness matrix is derived using the stabilized conforming nodal integration technique. A convected coordinate system is introduced into the formulation to deal with the curvilinear surface. More importantly, the RKs taken here are used not only for the interpolation of the curved geometry, but also for the approximation of field variables. Several numerical examples with shallow shells and full cylinder models are considered, and the critical buckling loads and their buckling mode shapes are calculated by the meshfree eigenvalue analysis and examined. To show the accuracy and performance of the developed meshfree method, the computed critical buckling loads and mode shapes are compared with reference solutions based on boundary domain element, finite element and analytical methods.

This paper provides information on hyperboloids of revolution surfaces and their classification, using methods of differential geometry and computer design. All surfaces are plotted by means of computer graphics. The geometric nonlinear numerical analysis of thin shells in the form of hyperboloid of revolution surfaces on buckling resistance and stress-strain state is given in this paper, using the finite elements method in a computer software. Comparison is done with the result of the finite elements linear analysis of their buckling resistance.