Koiter Model Research Articles

On the numerical corroboration of an obstacle problem for linearly elastic flexural shells.

In this article, we study the numerical corroboration of a variational model governed by a fourth-order elliptic operator that describes the deformation of a linearly elastic flexural shell subjected not to cross a prescribed flat obstacle. The problem under consideration is modelled by means of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Sobolev space and is known to admit a unique solution. Qualitative and quantitative numerical experiments corroborating the validity of the model and its asymptotic similarity with Koiter's model are also presented.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Simulating Capillary Folding of Thin Elastic Sheets with Pinned Contact Lines

We consider the capillary folding of thin elastic sheets with pinned contact lines in three dimensions. The folding occurs due to the interaction between the elastic sheet and a droplet deposited on top of it. Firstly, we derive the equilibrium equations by minimizing the total energy of the system. This energy comprises the interfacial energies and the elastic energy, which is described by the nonlinear Koiter's model. Then we develop a time-splitting numerical scheme to solve the gradient flow dynamics. Following this dynamics, the system evolves towards the equilibrium state. At each time step, we first determine the droplet surface with a constant mean curvature by minimizing a squared area functional, subject to the boundary condition at the pinned contact line. The elastic sheet, discretized by the C1-conforming subdivision element method, is then evolved by taking into account the fluid pressure and the capillary force. We propose a novel remeshing strategy to avoid the need for interpolating the capillary force on the sheet boundary and to prevent simulations from breakdown due to distorted meshes. We perform numerical tests to demonstrate the accuracy of the numerical method, as well as its ability to predict folded structures for various types of sheets.

Derivation of an elastic shell model of order as tends to infinity

Starting from three-dimensional linear elasticity and performing the dimensional reduction by integration over the thickness, we derive a general form of the areal strain energy density for elastic shells. To obtain the new constitutive model, we do not approximate the deformation fields as polynomials in the thickness coordinate, but rather we keep all terms in the thickness-wise series expansions. As a result, we deduce the explicit form of the shell strain energy density in which the constitutive coefficients are expressed as integrals depending on the thickness h and on the initial curvature. Then, to obtain the shell model of order , we expand the integral coefficients in the strain energy function as power series of h and truncate them to the power . In the case , we recover the classical strain energy density for combined bending and stretching of linear shells, which leads to the Koiter model. Finally, we prove that the proposed shell strain energy function is coercive for any , as well as for the general case .

On the justification of the frictionless time-dependent Koiter's model for thermoelastic shells

Our first objective is to identify two-dimensional equations that model the displacement of a linearly elastic flexural shell subjected to the action of an external heat source. To this end, we embed the shell into a family of linearly elastic flexural shells, all sharing the same middle surface θ ( ω ‾ ) , where ω is a domain in R 2 and θ : ω ‾ → E 3 is a smooth enough immersion and whose thickness 2 ε > 0 is considered as a “small” parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as ε approaches zero, the corresponding “limit” two-dimensional variational problem. Our second objective is to identify and justify a set of two-dimensional equations that are meant to approximate the original three-dimensional model in the case where the shell under consideration is either an elliptic membrane shell or a flexural shell.

A C^1-continuous Trace-Finite-Cell-Method for linear thin shell analysis on implicitly defined surfaces

A Trace-Finite-Cell-Method for the numerical analysis of thin shells is presented combining concepts of the TraceFEM and the Finite-Cell-Method. As an underlying shell model we use the Koiter model, which we re-derive in strong form based on first principles of continuum mechanics by recasting well-known relations formulated in local coordinates to a formulation independent of a parametrization. The field approximation is constructed by restricting shape functions defined on a structured background grid on the shell surface. As shape functions we use on a background grid the tensor product of cubic splines. This yields C^1-continuous approximation spaces, which are required by the governing equations of fourth order. The parametrization-free formulation allows a natural implementation of the proposed method and manufactured solutions on arbitrary geometries for code verification. Thus, the implementation is verified by a convergence analysis where the error is computed with an exact manufactured solution. Furthermore, benchmark tests are investigated.

Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability.

We deduce a one-dimensional model of elastic planar rods starting from the Föppl-von Kármán model of thin shells. Such model is enhanced by additional kinematical descriptors that keep explicit track of the compatibility condition requested in the two-dimensional parent continuum, that in the standard rods models are identically satisfied after the dimensional reduction. An inextensible model is also proposed, starting from the nonlinear Koiter model of inextensible shells. These enhanced models describe the nonlinear planar bending of rods and allow to account for some phenomena of preeminent importance even in one-dimensional bodies, such as formation of singularities and localization (d-cones), otherwise inaccessible by the classical one-dimensional models. Moreover, the effects of the compatibility translate into the possibility to obtain multiple stable equilibrium configurations.

Closed-Form Saint-Venant Solutions in the Koiter Theory of Shells

In this paper we investigate the deformation of cylindrical linearly elastic shells using the Koiter model. We formulate and solve the relaxed Saint-Venant's problem for thin cylindrical tubes made of isotropic and homogeneous elastic materials. To this aim, we adapt a method established previously in the three-dimensional theory of elasticity. We present a general solution procedure to determine closed-form solutions for the extension, bending, torsion and flexure problems. We remark the analogy and formal resemblance of these solutions to the classical Saint-Venant's solutions for solid cylinders. The special case of circular cylindrical shells is also discussed.

Numerical approximation of the dynamic Koiter's model for the hyperbolic parabolic shell

In this paper, the dynamic Koiter's model is discussed for the hyperbolic parabolic shell, i.e., the saddle surface as its middle surface. The numerical computation for this kind of shell is the most difficult as its complex differential geometry. We provide a numerical algorithm for solving the dynamic problems with finite elements and finite difference methods. At the same time, we do the numerical experiments for the hyperbolic parabolic shell. Concretly, we consider the roof of the Valencia Aquarium as a hyperbolic parabolic shell with a geometric model of the roof deduced. Then, we simulate the deformation of the roof by the dynamic applied force coupled with the dynamic Koiter's model. Finally, a comparison of the displacement distribution on the middle surface is made, which verifies that the dynamic Koiter's model is efficient and that numerical algorithm is stable for the hyperbolic parabolic shell. This result provides a strong theoretical and applicable support for the design and protection of such large-span buildings whose roofs and walls are of one whole object.

A confinement problem for a linearly elastic Koiter's shell

In this Note, we propose a natural two-dimensional model of “Koiter's type” for a general linearly elastic shell confined in a half space. This model is governed by a set of variational inequalities posed over a non-empty closed and convex subset of the function space used for modeling the corresponding “unconstrained” Koiter's model. To study the limit behavior of the proposed model as the thickness of the shell, regarded as a small parameter, approaches zero, we perform a rigorous asymptotic analysis, distinguishing the cases where the shell is either an elliptic membrane shell, a generalized membrane shell of the first kind, or a flexural shell. Moreover, in the case where the shell is an elliptic membrane shell, we show that the limit model obtained via the asymptotic analysis of our proposed two-dimensional Koiter's model coincides with the limit model obtained via a rigorous asymptotic analysis of the corresponding three-dimensional “constrained” model.

Asymptotic justification of the intrinsic equations of Koiter's model of a linearly elastic shell

We show that the intrinsic equations of Koiter's model of a linearly elastic shell can be derived from the intrinsic formulation of the three-dimensional equations of a linearly elastic shell, by using an appropriate a priori assumption regarding the three-dimensional strain tensor fields appearing in these equations. To this end, we recast in particular the Dirichlet boundary conditions satisfied by any admissible displacement field as boundary conditions satisfied by the covariant components of the corresponding strain tensor field expressed in the natural curvilinear coordinates of the shell. Then we show that, when restricted to strain tensor fields satisfying a specific a priori assumption, these new boundary conditions reduce to those of the intrinsic equations of Koiter's model of a linearly elastic shell.

Investigation of imperfect effect on thermal buckling of cylindrical shell with FGM coating

Abstract The shell with functionally graded material (FGM) thermal barrier coating is a novel high temperature resistant structure, which has been increasingly applied in the aerospace, nuclear, turbo machinery and other engineering fields. However, there are some defects for the practical structures due to the limitation of manufacturing technique. But relevant theoretical research on the thermal buckling behavior of the imperfect cylindrical shell is limited in most open literature. Therefore, this work proposed to establish the theoretical solution of the critical temperature of buckling for the cylindrical shell with an axisymmetric imperfect and FGM coating based on the Donnell shell theory, Koiter model and Galerkin method. The theoretical solution deduced in this work agrees well with the existing literature. In addition, the influences of the profile of the axisymmetric imperfection, the volume fraction of the ceramic phase and the types of the thermal loading on the thermal buckling behavior of the coated imperfect cylindrical are further analyzed. The study provides a scientific solution and better understanding for the thermal buckling problem of the coated imperfect cylindrical shells.

The time-dependent Koiter model and its numerical computation

• Existence and uniqueness of the time-dependent Koiter model. • Newmark scheme for the time-dependent Koiter model. • Analyses of existence, uniqueness, stability, convergence and priori error estimate for elastodynamic problems. The theory of elastic shells is one of the most important branches of the theory of elasticity. Among all the shell models, a classical and widely recognized model is the Koiter model. In this paper, we discuss the time-dependent Koiter model, which has not been addressed numerically. We show that the solution of this model exists and is unique. We semi-discretize the space variable and fully discretize the problem using the time discretization by the Newmark scheme. The corresponding analyses of existence, uniqueness, stability, convergence and a priori error estimate are given. Finally, we provide numerical experiments with a portion of spherical shell and a portion of cylindrical shell to demonstrate the efficiency of the model.

A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries

This paper presents a general non-linear computational formulation for rotation-free thin shells based on isogeometric finite elements. It is a displacement-based formulation that admits general material models. The formulation allows for a wide range of constitutive laws, including both shell models that are extracted from existing 3D continua using numerical integration and those that are directly formulated in 2D manifold form, like the Koiter, Canham and Helfrich models. Further, a unified approach to enforce the G1-continuity between patches, fix the angle between surface folds, enforce symmetry conditions and prescribe rotational Dirichlet boundary conditions, is presented using penalty and Lagrange multiplier methods. The formulation is fully described in the natural curvilinear coordinate system of the finite element description, which facilitates an efficient computational implementation. It contains existing isogeometric thin shell formulations as special cases. Several classical numerical benchmark examples are considered to demonstrate the robustness and accuracy of the proposed formulation. The presented constitutive models, in particular the simple mixed Koiter model that does not require any thickness integration, show excellent performance, even for large deformations.

A posteriori analysis of penalized and mixed formulations of Koiter’s shell model

We are interested in finite element approximations of a Koiter model for linearly elastic shells with little regularity. To perform conforming method, we present a penalized and a mixed formulations of the model allowing to approximate and to enforce weakly mechanical constraints, respectively. We establish existence and uniqueness of the solution to the both formulations. Moreover, a posteriori analysis is led yielding an upper bound and a lower bound of the error. Finally, numerical results are presented to illustrate the efficiency of the a posteriori estimators. We therefore, propose a mesh adaptivity strategy relying on these indicators.

On the asymptotic behavior of transmission thin shell problems

In this paper we study the asymptotic behavior of two-dimensional transmission problems for the linear Koiter model of an elastic multi-structure composed of two thin shells with the same thickness e 1. Contrary to the case of membrane shells, the formal limit problem, i.e., for e = 0, fails to give a convenient approximate solution, as the limit problem is ill posed. An appropriate dilation leads to an equivalent problem, for which we prove its strong convergence towards some finite limit.

Singular layers for transmission problems in thin shallow shell theory: Elastic junction case

In this Note we study two-dimensional transmission problems for the linear Koiter's model of an elastic multi-structure composed of two thin shallow shells with the same thickness ε ≪ 1 , in the elastic junction case. We suppose that the loading is singular, that the elastic coefficients are of different order on each part ( O ( ε − 1 ) and O ( 1 ) respectively) and that the elastic stiffness coefficient of the hinge is k ε = O ( ε ) . The formal limit problem fails to give a solution satisfying all boundary and transmission conditions; it gives only the outer solution. We derive the inner limit problem which allows us to describe the transmission layer.

Singular layers for transmission problems in thin shallow shell theory: Rigid junction case

In this Note we study two-dimensional transmission problems for the linear Koiter's model of an elastic multi-structure composed of two thin shallow shells. This work enters in the framework of singular perturbation of problems depending on a small parameter ε. The formal limit problem fails to give a solution satisfying all boundary and transmission conditions; it gives only the outer solution. Both in the case of regular or singular loadings, we derive a limit problem which allows us to determine the inner solution explicitly.

KOITER ESTIMATE REVISITED

We prove a general adimensional energy estimate between the solution of the three-dimensional Lamé system on a thin clamped shell and a displacement reconstructed from the solution of the classical two-dimensional Koiter model. This estimate only involves the thickness parameter ε, constants attached to the mid-surface S, the two-dimensional energy of the solution of the Koiter model and "wavelengths" associated with this latter solution. This bound is in the same spirit as Koiter's heuristic estimate in Ref. 26 and can be viewed as an a posteriori estimation of the modeling error by means of the two-dimensional solution. It is general with respect to the geometry of the mid-surface S which is an arbitrary smooth manifold with boundary. Taking boundary layer terms into account, we prove that our estimates are sharp in the cases of plates and elliptic shells.

Limit behavior of Koiter model for long cylindrical shells and Vlassov model

We propose in this article to consider the limit behavior of the Koiter shell model when one of the characteristic length of the middle surface becomes very large with respect to the other. To do this, we perform a dimensional analysis of Koiter formulation which involves dimensionless numbers characterizing the geometry and the loading. Once reduced to a one-scale problem corresponding to thin-walled beams (long cylindrical shell), using asymptotic expansion technique, we address the limit behavior of Koiter model when the aspect ratio of the shell tends to zero. We prove that at the leading order, Koiter shell model degenerates to a one dimensional thin-walled beam model corresponding to the Vlassov one. Moreover, we obtain a general analytical expression of the geometric constants involved, that improves the empirical expression given by Vlassov.

Adaptive and anisotropic mesh strategy for thin shell problems. Case of inhibited parabolic shells

The aim of this paper is to show the reliability of an adaptive and anisotropic mesh procedure for thin shell problems. We consider singular perturbation problems only for parabolic shells whose behavior is described by the Koiter model. The corresponding system of equations, which depends on the relative thickness ε of the shell, is elliptic except at the limit for ε = 0 where it is parabolic. In a first part of this paper, we study theoretically the phenomena of internal layers appearing during the singular perturbation process, when the loading is somewhat singular. These layers have very different structures either they are along or across the asymptotic lines of the middle surface of the shell. In a second part, numerical computations are performed using a finite element software coupled with an adaptive anisotropic mesh generator. This technique enables to approach accurately the singularities and the layers predicted by the theory especially for very small values of the thickness. The efficiency of such a procedure in comparison with uniform meshes is put in a prominent position.