Nonlinear Dynamics Of Shells Research Articles

Nonlinear dynamics of shells conveying pulsatile flow with pulse-wave propagation. Theory and numerical results for a single harmonic pulsation

In deformable shells conveying pulsatile flow, oscillatory pressure changes cause local movements of the fluid and deformation of the shell wall, which propagate downstream in the form of a wave. In biomechanics, it is the propagation of the pulse that determines the pressure gradient during the flow at every location of the arterial tree. In this study, a woven Dacron aortic prosthesis is modelled as an orthotropic circular cylindrical shell described by means of the Novozhilov nonlinear shell theory. Flexible boundary conditions are considered to simulate connection with the remaining tissue. Nonlinear vibrations of the shell conveying pulsatile flow and subjected to pulsatile pressure are investigated taking into account the effects of the pulse-wave propagation. For the first time in literature, coupled fluid-structure Lagrange equations of motion for a non-material volume with wave propagation in case of pulsatile flow are developed. The fluid is modeled as a Newtonian inviscid pulsatile flow and it is formulated using a hybrid model based on the linear potential flow theory and considering the unsteady viscous effects obtained from the unsteady time-averaged Navier-Stokes equations. Contributions of pressure and velocity propagation are also considered in the pressure drop along the shell and in the pulsatile frictional traction on the internal wall in the axial direction. A numerical bifurcation analysis employs a refined reduced order model to investigate the dynamic behavior of a pressurized Dacron aortic graft conveying blood flow. A pulsatile time-dependent blood flow model is considered by applying the first harmonic of the physiological waveforms of velocity and pressure during the heart beating period. Geometrically nonlinear vibration response to pulsatile flow and transmural pulsatile pressure, considering the propagation of pressure and velocity changes inside the shell, is here presented via frequency-response curves, time histories, bifurcation diagrams and Poincaré maps. It is shown that traveling waves of pressure and velocity cause a delay in the radial displacement of the shell at different values of the axial coordinate. The effect of different pulse wave velocities is also studied. Comparisons with the corresponding ideal case without wave propagation (i.e. with the same pulsatile velocity and pressure at any point of the shell) are here discussed. Bifurcation diagrams of Poincaré maps obtained from direct time integration have been used to study the system in the spectral neighborhood of the fundamental natural frequency. By increasing the forcing frequency, the response undergoes very complex nonlinear dynamics (chaos, amplitude modulation and period-doubling bifurcation), here deeply investigated.

On fluid–shell coupling using an arbitrary Lagrangian–Eulerian fluid solver coupled to a positional Lagrangian shell solver

In this work, we develop a partitioned algorithm for three-dimensional geometric non-linear fluid–structure interaction analysis using the finite element method. The fluid solver is explicit and its time integration based on characteristics, which automatically introduces stabilising terms on stream direction. The Navier–Stokes equations are written using the arbitrary Lagrangian–Eulerian (ALE) description, in order to accept moving boundaries and coupling with Lagrangian shell elements. The structure is modelled using a novel finite element method formulation for geometric non-linear shell dynamics. Such shell formulation, so-called positional formulation, is based on the minimum potential energy theorem, written regarding nodal positions and generalised unconstrained vectors, not displacements and rotations. These characteristics avoid the use of large rotation approximations. The coupling between the two different meshes is done by mapping the fluid boundary nodes local positions over the shell elements and vice versa, avoiding the need for matching fluid and shell nodes. The fluid mesh is adapted using a simple approach based on shell positions and velocities. The efficiency and robustness of the proposed approach is demonstrated by examples.

Large amplitude free vibration of orthotropic shallow shells of complex shapes with variable thickness

The present formulation of the analysed problem is based on Donell’s nonlinear shallow shell theory, which adopts Kirchhoff’s hypothesis. Transverse shear deformations and rotary inertia of a shell are neglected. According to this theory, the non-linear strain-displacement relations at the shell midsurface has been proposed. The validity and reliability of the proposed approach has been illustrated and discussed, and then a few examples of either linear or non-linear dynamics of shells with variable thickness and complex shapes have been presented and

An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 2: shells

Following the approach developed for rods in Part 1 of this paper (Pimenta et al. in Comput. Mech. 42:715---732, 2008), this work presents a fully conserving algorithm for the integration of the equations of motion in nonlinear shell dynamics. We begin with a re-parameterization of the rotation field in terms of the so-called Rodrigues rotation vector, allowing for an extremely simple update of the rotational variables within the scheme. The weak form is constructed via non-orthogonal projection, the time-collocation of which ensures exact conservation of momentum and total energy in the absence of external forces. Appealing is the fact that general hyperelastic materials (and not only materials with quadratic potentials) are permitted in a totally consistent way. Spatial discretization is performed using the finite element method and the robust performance of the scheme is demonstrated by means of numerical examples.

Unconstrained Finite Element for Geometrical Nonlinear Dynamics of Shells

This paper presents a positional FEM formulation to deal with geometrical nonlinear dynamics of shells. The main objective is to develop a new FEM methodology based on the minimum potential energy theorem written regarding nodal positions and generalized unconstrained vectors not displacements and rotations. These characteristics are the novelty of the present work and avoid the use of large rotation approximations. A nondimensional auxiliary coordinate system is created, and the change of configuration function is written following two independent mappings from which the strain energy function is derived. This methodology is called positional and, as far as the authors′ knowledge goes, is a new procedure to approximated geometrical nonlinear structures. In this paper a proof for the linear and angular momentum conservation property of the Newmark β algorithm is provided for total Lagrangian description. The proposed shell element is locking free for elastic stress‐strain relations due to the presence of linear strain variation along the shell thickness. The curved, high‐order element together with an implicit procedure to solve nonlinear equations guarantees precision in calculations. The momentum conserving, the locking free behavior, and the frame invariance of the adopted mapping are numerically confirmed by examples.

Book Review: Nonclassical Thermoelastic Problems in nonlinear Dynamics of Shells

Book Review: Nonclassical Thermoelastic Problems in nonlinear Dynamics of Shells

Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. Applications of the Bubnov-Galerkin and Finite Difference Numerical Methods

Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. Applications of the Bubnov-Galerkin and Finite Difference Numerical Methods

Nonlinear dynamic analysis of shells with the triangular element TRIC

TRIC is a facet triangular shell element, which is based on the natural mode method. It has been shown that the TRIC shell element satisfies the individual element test and in the framework of the nonconsistent formulation the convergence requirements are fulfilled, while it has been proved to be very efficient in linear and nonlinear static problems. Moreover, another major advantage in the formulation of this element is the incorporation of the transverse shear deformations in a way that defies the shear-locking phenomenon. In this work the derivation of the consistent and lumped mass matrices of the TRIC element is presented so that it can be used in linear and nonlinear dynamic problems. Both translational and rotational inertia are included in the consistent mass matrix, which is conceived, using kinematical and geometrical arguments consistent with the assumed natural rigid body and straining modes of the element. All the kinematical and geometrical arguments that are invoked for the derivation of the consistent mass matrix are briefly presented. Moreover, two formulations of the lumped mass matrix of TRIC are derived. The first formulation is based entirely on geometrical considerations whereas the second is based on lumping the consistent mass matrix of TRIC. Finally, the element’s robustness and accuracy will be shown by applying it to properly selected benchmark examples of nonlinear shell dynamics, while its computational efficiency will be demonstrated by comparing the CPU performance of the element with the other available shell elements.

An energy decaying scheme for nonlinear dynamics of shells

A novel integration scheme for nonlinear dynamics of geometrically exact shells is developed based on the inextensible director assumption. The new algorithm is designed so as to imply the strict decay of the system total mechanical energy at each time step, and consequently unconditional stability is achieved in the nonlinear regime. Furthermore, the scheme features tunable high frequency numerical damping and it is therefore stiffly accurate. The method is tested for a finite element spatial formulation of shells based on mixed interpolations of strain tensorial components and on a two-parameter representation of director rotations. The robustness of the scheme is illustrated with the help of numerical examples.

An energy–momentum integration scheme and enhanced strain finite elements for the non-linear dynamics of shells

The paper is concerned with a dynamic formulation of shells and the development of a corresponding robust energy–momentum integration scheme within the framework of enhanced finite elements. Energy–momentum schemes preserve, by design, specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the system and the applied forces allow to. In a previous work, an energy–momentum scheme was developed by the authors which enjoys the feature of being applicable to any shell theory whatever the non-linearity in the strain–displacement relations may be. The method goes beyond a formulation by Simo and Tarnow which applies only when the non-linearity is of quadratic nature. In this paper, we build up on previous work and extend the formulation to encompass enhanced strain finite elements frequently used in structural analysis. The shell formulation used is characterized by seven degrees of freedom and the non-linearity in the strain–displacement relations is of cubic nature. Various examples of non-linear shell dynamics including free large overall motion and non-linear vibrations in conjunction with cases of dynamic stability are considered.

Time integration in the context of energy control and locking free finite elements

In the present paper two main research areas of computational mechanics, namely the finite element development and the design of time integration algorithms are reviewed and discussed with a special emphasis on their combination. The finite element techniques are designed to prevent locking and the time integration schemes to guarantee numerical stability in non-linear elastodynamics. If classical finite element techniques are used, their combination with time integration schemes allow to avoid any modifications on the element or algorithmic level. It is pointed out, that on the other hand Assumed Stress and Enhanced Assumed Strain elements have to be modified if they are combined with energy conserving or decaying time integration schemes, especially the Energy-Momentum Method in its original and generalized form. The paper focusses on the necessary algorithmic formulation of Enhanced Assumed Strain elements which will be developed by the reformulation of the Generalized Energy-Momentum Method based on a classical one-field functional, the extension to a modifiedHu-Washizu three-field functional including enhanced strains and a suitable time discretization of the additional strain terms. The proposed method is applied to non-linear shell dynamics using a shell element which allows for shear deformation and thickness change, and in which the Enhanced Assumed Strain Concept is introduced to avoid artificial thickness locking. Selected examples illustrate the locking free and numerically stable analysis.

Drape Simulation of Woven Fabrics by Using Explicit Dynamic Analysis

For the three-dimensional drape simulation of large samples of woven fabric, an explicit-dynamic analysis code is developed to predict the shape of woven fabric as it deforms into a natural shape under gravity. A contact algorithm is incorporated into the code, which is based on explicit algorithms for the non-linear dynamics of shells with a simple damping effect. For validation purposes, various simulations are performed, and the results are compared with those obtained in experiments.

Generalized Energy–Momentum Method for non-linear adaptive shell dynamics

In the present study, a generalization of the Energy–Momentum Method, denoted by Generalized Energy–Momentum Method, applied to the non-linear dynamics of shells will be developed within the framework of the Generalized- α Method. This algorithmic environment contains the unconditionally stable Energy–Momentum Method and its numerically damped version as well as the classical Newmark and α-methods as special cases. In order to control the size of the time steps of the integration scheme with respect to accuracy and efficiency, an adaptive time stepping procedure based on local a posteriori error estimation will be improved for non-linear dynamical systems and applied to the proposed class of algorithms. The spatial discretization is realized by an eight noded finite shell element of Reissner/Mindlin type including an extensible shell director field permitting the application of three-dimensional material laws. The original formulation of this finite element will be developed for non-linear dynamic analysis and adapted for the employment within the introduced energy conserving/decaying time integration scheme.

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.

Nonlinear Dynamics of Shells: Theory, Finite Element Formulation, and Integration Schemes

The paper is concerned with a dynamical formulation of a recently established shell theory capable to catch finite deformations and falls within the class of geometrically exact shell theories. A basic aspect is the design of time integration schemes which preserve specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the applied forces allow to. The integration method differs from the one recently proposed by Simo and Tarnow in being applicable without modifications to shell formulations with linear as well as nonlinear configuration spaces and in being independent of the nonlinearities involved in the strain-displacement relations. A finite element formulation is presented and various examples of nonlinear shell dynamics including large overall and chaotic motions are considered.

Constraint Energy Momentum Algorithm and its application to non-linear dynamics of shells

A modified time-integration method is developed for non-linear structural dynamics combining controllable algorithmic dissipation of higher modes and, at the same time, conservation of energy as well as linear and angular momentum. Although these features seem to be actually inconsistent, numerical stability with large time steps for dynamic buckling and snap-through problems as well as for dynamical systems with smooth solutions is guaranteed. Arbitrary implicit, one-step time-integration schemes with Newmark approximations are suitable for use as basic algorithms for the proposed method. The desired conservation attributes are obtained if the basic algorithm is augmented by energy and momentum constraints. The potential of the suggested algorithm is demonstrated by selected applications to non-linear dynamics of shell structures.

Fission-fusion adaptivity in finite elements for nonlinear dynamics of shells

An adaptive finite element procedure is developed for the transient analysis of nonlinear shells. The scheme is an h-method which employs fission and fusion of elements to adaptively refine and coarsen the mesh. Incremental work and deviation of the bilinear finite element approximation to the shell from a Kirchhoff-Love surface are used as error criteria for adaptivity. The example problems show that the adaptive schemes are capable of achieving substantial improvements in accuracy for a given computational effort. They include both material and geometric nonlinearities and local and global buckling.

Non-linear shell dynamics based on a mixed variational formulation

A simple, but efficient procedure for the investigation of the geometrically non-linear dynamic behaviour of thin elastic shells which can be described sufficiently accurate by the Donnell-Marguerre-Mushtari-Vlasov theory is proposed and tested. It is based on a mixed variational formulation generalizing D'Alembert's principle. The discretization with respect to the space is performed by Euler's method and that with respect to the time by means of the central difference quotient. Due to the regular and simple structure of the governing algebraic equations wave propagation phenomena can be studied adequately. The introduction of an (artificial) damping allows the approximation of pure static problems and includes as well the branching of equilibrium. Some illustrative examples are presented.

Explicit algorithms for the nonlinear dynamics of shells

A finite element formulation and algorithm for the nonlinear analysis of the large deflection, materially nonlinear response of impulsively loaded shells is presented. A unique feature of this algorithm is the use of a bilinear four-node quadrilateral element with single-point quadrature and a simple hourglass control which is orthogonal to rigid body modes on an element level and does not compromise the consistency of the equations. The geometric nonlinearities are treated by using a corotational description wherein a coordinate system that rotates with the material is embedded at the integration point; thus the algorithm is directly applicable to anisotropic materials without any corrections for frame invariance of material property tensors. This algorithm can treat about 200 element-time-steps per CPU second on a CYBER 170/730 computer in the explicit time integration mode. Numerous results are presented for both elastic and elastic-plastic problems with large strains that show that the method in most cases is comparable in accuracy with an earlier version of this algorithm employing a cubic triangular plate-shell element, but substantially faster.

Nonlinear Shell Dynamics—Intrinsic and Semi-Intrinsic Approaches

The intrinsic approach to the nonlinear dynamics of shells, which was introduced in [6], is reviewed and extended by the addition of appropriate initial and boundary conditions of the dynamic and kinematic types to the field equations. The alternative semi-intrinsic velocity approaches (where the velocity components supply the connection between the equations of motion and the time rates of the metric and curvature) are also presented. Both linear and rotational velocity forms are included. The relative merits of these approaches to shell dynamics are discussed and compared with extrinsic approaches.