Fraeijs de Veubeke variational principle-based cylindrical shell finite element model

In this paper a dimensionally reduced cylindrical shell model based on the dual-mixed variational principle of Fraeijs de Veubeke will be presented. The fundamental variables of this variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium. A shell model derived in this way makes the application of the classical kinematical hypotheses unnecessary, and enables us to use unmodified three-dimensional constitutive equations. On the basis of this shell model, a new dual-mixed cylindrical shell finite elements capable of both h- and p-approximation can be derived. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error analysis of mixed finite elements for cylindrical shells

In this paper, based on the Naghdi shell model, we analyze the uniform convergence of mixed finite element methods for cylindrical shell problems using macroelement techniques. We show that Taylor–Hood elements p ₂-P ₁ and P ₁ iso P ₂ are locking free elements for the model problems. Optimal error estimates are presented with these elements. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Priori and a Posteriori Error Analysis of a Wavelet-Based Stabilization for the Mixed Finite Element Method

We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show that the resulting stabilized dual-mixed variational formulation and the associated discrete scheme defined with Raviart–Thomas spaces are well-posed and derive the usual a priori error estimates and the corresponding rate of convergence. Furthermore, a reliable and efficient residual-based a posteriori error estimator and a reliable and quasi-efficient one are provided.     

Dual-mixed finite element analysis of crystalline sub-micron gold

An extended crystal plasticity model is applied to crystalline sub-micron gold in order to study the mechanical response. Numerical results for different crystal sizes are presented and discussed. The governing equations are discretized and, subsequently, solved via a dual-mixed finite element formulation [1, 2]. The evolution equation of the dislocation density is taken as a global field relation additionally to the balance of linear momentum, whereas the flow rule is solved locally at the Gauß point level [3,4]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients

In this work a finite element method for a dual-mixed approximation of Stokes and nonlinear Stokes problems is studied. The dual-mixed structure, which yields a twofold saddle point problem, arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning. The method approximates the velocity, its gradient, and the total stress tensor, but avoids the explicit computation of the pressure, which can be recovered through a simple postprocessing technique. This method improves an existing approach for these problems and uses Raviart-Thomas elements and discontinuous piecewise polynomials for approximating the unknowns. Existence, uniqueness, and error results for the method are given, and numerical experiments that exhibit the reduced computational cost of this approach are presented.     

Axisymmetric Stokes' flow in a spherical shell revisited via the Fokas method. Part I: Irrotational flow

The axisymmetric irrotational Stokes' flow for a spherical shell is analysed by means of the recently developed Fokas method via the use of global relations. Alternative series and new integral representations concerning a system of concentric spheres, yielding, by a limiting procedure, the Dirichlet or Neumann problems for the interior and the exterior of a sphere, are presented. The boundary value problems considered can be classically solved using either the finite Gegenbauer transform or the Mellin transform. Application of the Gegenbauer transform yields a series representation which is uniformly convergent at the boundary, but not convenient for many applications. The Mellin transform, on the other hand, furnishes an integral representation which is not uniformly convergent at the boundary. Here, by algebraic manipulations of the global relation: (i) a Gegenbauer series representation is derived in a simpler manner, instead of solving ODEs and (ii) an alternative integral representation, different from the Mellin transform representation is derived which is uniformly convergent at the boundary. Copyright © 2011 John Wiley & Sons, Ltd.     

The Axisymmetric Deformation of a Thin,or Moderately Thick,Elastic Spherical Cap

A refined shell theory is developed for the elastostatics of a moderately thick spherical cap in axisymmetric deformation. This is a two-term asymptotic theory, valid as the dimensionless shell thickness tends to zero.The theory is more accurate than “thin shell” theory, but is still much more tractable than the full three-dimensional theory. A fundamental difficulty encountered in the formulation of shell (and plate) theories is the determination of correct two-dimensional boundary conditions, applicable to the shell solution, from edge data prescribed for the three-dimensional problem. A major contribution of this article is the derivation of such boundary conditions for our refined theory of the spherical cap. These conditions are more difficult to obtain than those already known for the semi-infinite cylindrical shell, since they depend on the cap angle as well as the dimensionless thickness. For the stress boundary value problem, we find that a Saint-Venant-type principle does not apply in the refined theory, although it does hold in thin shell theory. We also obtain correct boundary conditions for pure displacement and mixed boundary data. In these cases, conventional formulations do not generally provide even the first approximation solution correctly. As an illustration of the refined theory, we obtain two-term asymptotic solutions to two problems, (i) a complete spherical shell subjected to a normally directed equatorial line loading and (ii) an unloaded spherical cap rotating about its axis of symmetry.     

Advanced Shell Element Formulations for Coupled Electromechanical Systems

With the significantly increasing applications of smart structures, piezoelectric material is widely used in branches of engineering sciences. Normally, the Finite Element Method is employed in the numerical analysis of these structures [2]. In this contribution, in order to avoid the locking effects and zero energy modes, the Assumed Natural Strain (ANS) Method [4] is implemented into four‐node piezoelectric shallow shell elements, by using the two‐field variational formulation in which displacements and electric potentials serve as independent variables and the three‐field variational formulation in which the dielectric displacement is taken as an independent variable additionally [3]. Moreover, a quadratic variation of the electric potential through the thickness direction is applied in the two‐field formulation. Numerical examples of piezoelectric sensors and actuators are presented, showing the behaviour of the shell elements by using different hybrid finite element formulations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dual-dual mixed formulation for nonlinear exterior transmission problems

We combine a dual-mixed finite element method with a Dirichlet-to-Neumann mapping (derived by the boundary integral equation method) to study the solvability and Galerkin approximations of a class of exterior nonlinear transmission problems in the plane. As a model problem, we consider a nonlinear elliptic equation in divergence form coupled with the Laplace equation in an unbounded region of the plane. Our combined approach leads to what we call a dual-dual mixed variational formulation since the main operator involved has itself a dual-type structure. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart-Thomas elements. The main tool of our analysis is given by a generalization of the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints.

     

Applications of Rosenbrock-type methods to the p-version of finite elements

In quasistatic solid mechanics the spatial as well as the temporal domain need to be discetized. For the spatial discretization usually elements with linear shape functions are used even though it has been shown that generally the p-version of the finite elemente method yields more effective discretizations, see e.g. [1], [2]. For the temporal discretization diagonal-implicit, see e.g. [4], and especially linear-implicit Runge-Kutta schemes, see e.g. [5], [6], have for smooth problems proven to be superior to the frequently applied Backward-Euler scheme (BE). Thus an approach combining the p-version of the finite element method with linear-implicit Runge-Kutta schemes, so-called Rosenbrock-type methods, is presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupling of FEM and BEM for a nonlinear interface problem: The h–p version

This article presents some numerical examples for coupling the finite element method (FEM) and the boundary element method (BEM) as analyzed in [11]. This coupling procedure combines the advantages of boundary elements (problems in unbounded regions) and of finite elements (nonlinear problems with inhomogeneous data). In [28], experimental rates of convergence for the h version are presented, where the accuracy of the Galerkin approximation is achieved by refining the mesh. In this article we treat the h–p version, combining an increase of the degree of the piecewise polynomials with a certain mesh refinement. In our model examples, we obtain theoretically and numerically exponential convergence, which indicates a great efficiency in particular if singularities appear. © 1995 John Wiley & Sons, Inc.     

Solution of Maxwell equation in axisymmetric geometry by Fourier series decompostion and by use of H (rot) conforming finite element

Summary. This study deals with the mathematical and numerical solution of time-harmonic Maxwell equation in axisymmetric geometry. Using Fourier decomposition, we define weighted Sobolev spaces of solution and we prove expected regularity results. A practical contribution of this paper is the construction of a class of finite element conforming with the H (rot) space equipped with the weighted measure rdrdz. It appears as an extension of the well-known cartesian mixed finite element of Raviart-Thomas-Nédélec [11]–[15]. These elements are built from classical lagrangian and mixed finite element, therefore no special approximations functions are needed. Finally, following works of Mercier and Raugel [10], we perform an interpolation error estimate for the simplest proposed element. Received March 15, 1996 / Revised version received November 30, 1998 / Published online December 6, 1999     

Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

The subject of the paper is the derivation of error estimates for the combined finite volume-finite element method used for the numerical solution of nonstationary nonlinear convection-diffusion problems. Here we analyze the combination of barycentric finite volumes associated with sides of triangulation with the piecewise linear nonconforming Crouzeix-Raviart finite elements. Under some assumptions on the regularity of the exact solution, the L ²(L ²) and L ²(H ¹) error estimates are established. At the end of the paper, some computational results are presented demonstrating the application of the method to the solution of viscous gas flow.     

Coupled multiscale finite element analysis of shell structures

In this paper, a coupled two-scale shell model is presented. A variational formulation and associated linearisation for the coupled global-local boundary value problem is derived. The discretisation of the shell is performed with quadrilaterals, whereas the local boundary value problems at the integration points of the shell are discretised using 8-noded or 27-noded brick elements, or solid shell elements. The coupled boundary value problem is simultaneously solved within a Newton iteration scheme. Solutions for small strain problems are computed within the so-called FE² method. In an important test, the correct material matrix for the stress resultants assuming linear elasticity and a homogeneous continuum is verified. Examples show that the developed two-scale model is able to analyse the global and local mechanical behaviour of heterogeneous shell structures. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An optimal order multigrid method for biharmonic,C 1 finite element equations

Summary In this paper, we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the Hsieh-Clough-Tocher,C ¹ macro finite elements or several otherC ¹ finite elements. Since the multipleC ¹ finite element spaces considered are not nested, the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method. This method converges with the optimal computational order.     

A simplified approach to estimating the stress state of flexible cylindrical shells under nonaxisymmetric load

An approach is proposed to approximate numerical analysis of the stress state of a geometrically nonlinear cylindrical shell under an unsymmetric load. An approximate technique is also proposed to reduce the solution of a two-dimensional nonlinear problem of a cylindrical shell with unsymmetric parameters to the solution of axisymmetric problems. A numerical example is considered. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 61–66, 1986.     

FINITE RINGS WITH COMMUTING NILPOTENT ELEMENTS

As a generalization of Wedderburn's theorem, Herstein [5] proved that a finite ring R is commutative, if all nilpotent elements are contained in the center of R. However a finite ring with commuting nilpotent elements is not necessarily commutative. Recently, in [9] and [10], Simons described the structure of finite rings R with J(R)² = 0 in a variety with definable principal congruences. In this paper, we will consider the difference between the finite commutative rings and the finite rings in which any two nilpotent elements commute with each other. As a consequence, we describe the structure of finite rings R with [J(R), J(R)] = 0 in a variety with definable principal congruences.     

The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity based multigrid theory. In order to apply this theory, we prove regularity results for the axisymmetric Laplace and Maxwell equations in certain weighted Sobolev spaces. These, together with some new finite element error estimates in certain weighted Sobolev norms, are the main ingredients of our analysis.

     

Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow

In this paper we introduce and analyze a new mixed finite element method for the two-dimensional Brinkman model of porous media flow with mixed boundary conditions. We use a dual-mixed formulation in which the main unknown is given by the pseudostress. The original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply the Babu?ka–Brezzi theory to establish sufficient conditions for the well-posedness of the resulting continuous and discrete formulations. In particular, a feasible choice of finite element subspaces is given by Raviart–Thomas elements of order $k \ge 0$ for the pseudostress, and continuous piecewise polynomials of degree $k + 1$ for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Raviart–Thomas and Clément interpolation operators are the main tools for proving the reliability. Then, Helmholtz’s decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are provided.     

Analysis of reinforced concrete structures subjected to dynamic loads with a viscoplastic Drucker–Prager model

The behavior of reinforced concrete structures subjected to dynamic loads is analyzed. The concrete material is modelled by an elasto-viscoplastic law, whose inviscid counterpart is the Drucker–Prager model. A viscous regularization is introduced in order to avoid the mesh dependency effects that usually appear when strain softening occurs. The model is implemented in a general finite element computer code for fast transient analysis of fluid-structure systems, based on an explicit central difference scheme. The model is activated to both continuum elements and layered shell elements. So, realistic numerical analyses of complex 3-D engineering problems are simple and efficient. Three examples, two of which are modelled with layered shell elements, are presented below.