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1.
Alexander Iosilevich Klaus-Jürgen Bathe Franco Brezzi 《International journal for numerical methods in engineering》1997,40(19):3639-3663
This paper addresses the evaluation of the inf–sup condition for Reissner–Mindlin plate bending elements. This fundamental condition for stability and optimality of a mixed finite element scheme is, in general, very difficult to evaluate analytically, considering for example distorted meshes. Therefore, we develop a numerical test methodology. To demonstrate the test methodology and to obtain specific results, we apply it to standard displacement-based elements and elements of the MITC family. Whereas the displacement-based elements fail to satisfy the inf–sup condition, we find that the MITC elements pass our numerical test for uniform meshes and a sequence of distorted meshes. © 1997 John Wiley & Sons, Ltd. 相似文献
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P. Areias K. Matouš 《International journal for numerical methods in engineering》2008,76(8):1185-1201
Non‐linear hyperelastic response of reinforced elastomers is modeled using a novel three‐dimensional mixed finite element method with a nonlocal pressure field. The element is unconditionally convergent and free of spurious pressure modes. Nonlocal pressure is obtained by an implicit gradient technique and obeys the Helmholtz equation. Physical motivation for this nonlocality is shown. An implicit finite element scheme with consistent linearization is presented. Finally, several hyperelastic examples are solved to demonstrate the computational algorithm including the inf–sup and verifications tests. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
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This paper presents the formulation of both the consistent and inconsistent four‐, eight‐ and nine‐noded isoparametric quadrilateral fluid finite elements that are based on Lagrangian frame of reference. The mesh locking phenomenon due to simultaneous enforcement of twin constraints, namely the incompressibility and irrotationality constraints, is studied in detail. The study shows that the characteristic of the locked fluid elements is that it always generates numerous spurious acoustic (volume change) modes upon the enforcement of rotational constraints. That is, the rotational constraints change the character of certain volume change modes. The study further reinforces the necessity of rotational constraints in not only identifying the spurious pressure modes, but also in reducing the computational effort for determining the eigenvalues and eigenvectors. It is found that all fully integrated inconsistent models exhibit locking behaviour. However, the inconsistent eight‐ and nine‐noded elements, integrated with full integration of volumetric stiffness and one point integration of the rotational stiffness matrices, gives excellent performance, although they do not pass the inf–sup test. The four‐ and nine‐noded consistent models are found to give locking free performance while their eight‐noded counterpart exhibited locking behaviour. The study shows that only consistent nine‐noded element models pass the inf–sup test. The utility of these elements in the coupled fluid–structure interaction problem is also demonstrated. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
4.
Olivier Jamond Hachmi Ben Dhia 《International journal for numerical methods in engineering》2013,94(4):374-399
In this paper, we show how one can account for the incompressibility constraint within the multimodel Arlequin framework. The main issue is the treatment of a double constraining of the displacement fields in the gluing model zone. An elastic incompressible structure problem is considered. An incompressible Arlequin formulation of this problem is developed and supported by a stability and consistency result. Its discretization by means of mixed finite elements is detailed. The Inf–Sup condition is numerically discussed for different choices of the Arlequin method parameters. Several numerical tests are conducted and practical recommendations for appropriate choices of these parameters are clearly stated. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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Klaus‐Jürgen Bathe Dena Hendriana Franco Brezzi Giancarlo Sangalli 《International journal for numerical methods in engineering》2000,48(5):745-760
We propose inf–sup testing for finite element methods with upwinding used to solve convection–diffusion problems. The testing evaluates the stability of a method and compactly displays the numerical behaviour as the convection effects increase. Four discretization schemes are considered: the standard Galerkin procedure, the full upwind method, the Galerkin least‐squares scheme and a high‐order derivative artificial diffusion method. The study shows that, as expected, the standard Galerkin method does not pass the inf–sup tests, whereas the other three methods pass the tests. Of these methods, the high‐order derivative artificial diffusion procedure introduces the least amount of artificial diffusion. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
6.
Ferdinando Auricchio Adrien Lefieux Alessandro Reali Alessandro Veneziani 《International journal for numerical methods in engineering》2016,107(2):155-180
An immersed finite element fluid–structure interaction algorithm with an anisotropic remeshing strategy for thin rigid structures is presented in two dimensions. One specific feature of the algorithm consists of remeshing only the fluid elements that are cut by the solid such that they fit the solid geometry. This approach allows to keep the initial (given) fluid mesh during the entire simulation while remeshing is performed locally. Furthermore, constraints between the fluid and the solid may be directly enforced with both an essential treatment and elements allowing the stress to be discontinuous across the structure. Remeshed elements may be strongly anisotropic. Classical interpolation schemes – inf–sup stable on isotropic meshes – may be unstable on anisotropic ones. We specifically focus on a proper finite element pair choice. As for the time advancing of the fluid–structure interaction solver, we perform a geometrical linearization with a sequential solution of fluid and structure in a backward Euler framework. Using the proposed methodology, we extensively address the motion of a hinged rigid leaflet. Numerical tests demonstrate that some finite element pairs are inf–sup unstable with our algorithm, in particular with a discontinuous pressure. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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WaiChing Sun Zhijun Cai Jinhyun Choo 《International journal for numerical methods in engineering》2017,111(7):624-659
An Arlequin poromechanics model is introduced to simulate the hydro‐mechanical coupling effects of fluid‐infiltrated porous media across different spatial scales within a concurrent computational framework. A two‐field poromechanics problem is first recast as the twofold saddle point of an incremental energy functional. We then introduce Lagrange multipliers and compatibility energy functionals to enforce the weak compatibility of hydro‐mechanical responses in the overlapped domain. To examine the numerical stability of this hydro‐mechanical Arlequin model, we derive a necessary condition for stability, the twofold inf–sup condition for multi‐field problems, and establish a modified inf–sup test formulated in the product space of the solution field. We verify the implementation of the Arlequin poromechanics model through benchmark problems covering the entire range of drainage conditions. Through these numerical examples, we demonstrate the performance, robustness, and numerical stability of the Arlequin poromechanics model. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
9.
Nicolas Moës Eric Bchet Matthieu Tourbier 《International journal for numerical methods in engineering》2006,67(12):1641-1669
This paper is devoted to the imposition of Dirichlet‐type conditions within the extended finite element method (X‐FEM). This method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces. Imposing Neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh (observed numerically in earlier papers). On the contrary, much less work has been devoted to Dirichlet boundary conditions for the X‐FEM (or the limiting case of stiff boundary conditions). In this paper, we introduce a strategy to impose Dirichlet boundary conditions while preserving the optimal rate of convergence. The key aspect is the construction of the correct Lagrange multiplier space on the boundary. As an application, we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding (RTM) process while avoiding remeshing. The case of inner conditions is also discussed as well as two important practical cases: material interfaces and phase‐transformation front capturing. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
10.
Suleiman M. BaniHani Suvranu De 《International journal for numerical methods in engineering》2007,70(11):1366-1386
In this paper we use the numerical inf–sup test to evaluate both displacement‐based and mixed discretization schemes for the solution of Reissner–Mindlin plate problems using the meshfree method of finite spheres. While an analytical proof of whether a discretization scheme passes the inf–sup condition is most desirable, such a proof is usually out of reach due to the complexity of the meshfree approximation spaces involved. The numerical inf–sup test (Int. J. Numer. Meth. Engng 1997; 40 :3639–3663), developed to test finite element discretization spaces, has therefore been adopted in this paper. Tests have been performed for both regular and irregular nodal configurations. While, like linear finite elements, pure displacement‐based approximation spaces with linear consistency do not pass the inf–sup test and exhibit shear locking, quadratic discretizations, unlike quadratic finite elements, pass the test. Pure displacement‐based and mixed approximation spaces that pass the numerical inf–sup test exhibit optimal or near optimal convergence behaviour. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
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Alexander Schwarz Karl Steeger Maximilian Igelbüscher Jörg Schröder 《International journal for numerical methods in engineering》2018,115(9):1138-1153
We present geometrically nonlinear formulations based on a mixed least‐squares finite element method. The L2‐norm minimization of the residuals of the given first‐order system of differential equations leads to a functional, which is a two‐field formulation dependent on displacements and stresses. Based thereon, we discuss and investigate two mixed formulations. Both approaches make use of the fact that the stress symmetry condition is not fulfilled a priori due to the row‐wise stress approximation with vector‐valued functions belonging to a Raviart‐Thomas space, which guarantees a conforming discretization of H(div). In general, the advantages of using the least‐squares finite element method lie, for example, in an a posteriori error estimator without additional costs or in the fact that the choice of the polynomial interpolation order is not restricted by the Ladyzhenskaya‐Babu?ka‐Brezzi condition (inf‐sup condition). We apply a hyperelastic material model with logarithmic deformation measures and investigate various benchmark problems, adaptive mesh refinement, computational costs, and accuracy. 相似文献
13.
Takeki Yamamoto Takahiro Yamada 《International journal for numerical methods in engineering》2020,121(9):2001-2032
The authors proposed a quadrilateral shell element enriched with degrees of freedom to represent thickness-stretch. The quadrilateral shell element can be utilized to consider large deformations for nearly incompressible materials, and its performance is demonstrated in small and large deformation analyses of hyperelastic materials in this study. Formulation of the proposed shell element is based on extension of the MITC4 shell element. A displacement variation in the thickness direction is introduced to evaluate the change in thickness. In the proposed approach, the thickness direction is defined using the director vectors at each midsurface node. The thickness-stretch is approximated by the movements of additional nodes, which are placed along the thickness direction from the bottom to the top surface. The transverse normal strain is calculated using these additional nodes without assuming the plane stress condition; hence, a three-dimensional constitutive equation can be employed without any modification. In this work, the authors apply an assumed strain technique to the special shell element to alleviate volumetric locking for nearly incompressible materials. Several numerical examples are presented to examine the effectiveness of the proposed element. 相似文献
14.
Chen Wanji 《International journal for numerical methods in engineering》2004,60(11):1817-1846
A refined discrete degenerated 15‐DOF triangular shell element RDTS15 with high performances is proposed. For constructing the element displacement function, the exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary, and the re‐constitute method for shear strain matrix is adopted. The proposed element can be used in the analysis of both moderate thick and thin plates/shells. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements, and also passed the inf–sup test for free cylindrical shell problems and satisfied both the bending‐ and membrane‐dominated test. Copyright © 2004 John Wiley Sons, Ltd. 相似文献
15.
M. Bruggi P. Venini 《International journal for numerical methods in engineering》2008,73(12):1693-1714
We present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion. The main idea is to adopt a mixed finite‐element discretization scheme wherein not only displacements (as usual) but also stresses are the variables entering the formulation. By doing so, any stress constraint may be handled within the optimization procedure without resorting to post‐processing operation typical of displacement‐based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed. Two dual variational principles of Hellinger–Reissner type are presented in continuous and discrete form that, which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes (Int. J. Numer. Meth. Engng. 1984; 24 (3):359–373). Extensive numerical simulations are performed and ongoing extensions outlined, including the optimization of elastoplastic and incompressible media. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
16.
J. M. A. Csar de S R. M. Natal Jorge 《International journal for numerical methods in engineering》1999,44(2):229-248
A new enhanced strain element, based on the definition of extra compatibles modes of deformation added to the standard four‐node finite element, is initially presented. The element is built with the objective of addressing incompressible problems and avoiding locking effects. By analysing at the element level the deformation modes which form a basis for the incompressible subspace the extra modes of deformation are proposed in order to provide the maximum possible dimension to that subspace. Subsequently another new element with more degrees of freedom is formulated using a mixed method. This is done by including an extra field of variables related to the derivatives of the displacement field of the extra compatible modes defined previously. The performance of the elements proposed is assessed in linear and non‐linear situations. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
17.
Heng Chi Cameron Talischi Oscar Lopez‐Pamies Glaucio H.Paulino 《International journal for numerical methods in engineering》2015,101(4):305-328
Nonlinear elastic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise because nonlinear elastic materials are characterized by non‐convex stored‐energy functions as a result of their ability to undergo large reversible deformations, are incompressible or nearly incompressible, and often times possess complex microstructures. In this work, we propose and explore an alternative approach to model finite elasticity problems in two dimensions by using polygonal discretizations. We present both lower order displacement‐based and mixed polygonal finite element approximations, the latter of which consist of a piecewise constant pressure field and a linearly‐complete displacement field at the element level. Through numerical studies, the mixed polygonal finite elements are shown to be stable and convergent. For demonstration purposes, we deploy the proposed polygonal discretization to study the nonlinear elastic response of rubber filled with random and periodic distributions of rigid particles, as well as the development of cavitation instabilities in elastomers containing vacuous defects. These physically‐based examples illustrate the potential of polygonal finite elements in studying and modeling nonlinear elastic materials with complex microstructures under finite deformations. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
18.
Sudeep K. Lahiri Javier Bonet Jaime Peraire Lluis Casals 《International journal for numerical methods in engineering》2005,63(10):1371-1395
In this paper we present a fractional time‐step method for Lagrangian formulations of solid dynamics problems. The method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly. Energy fluctuations are found to be minimal and stay bounded for long durations. In order to handle incompressibility, a mixed formulation in which the pressure appears explicitly is adopted. The velocity update over a time step is split into deviatoric and volumetric components. The deviatoric component is advanced using explicit time marching, whereas the pressure correction for each time step is computed implicitly by solving a Poisson‐like equation. Once the pressure is known, the volumetric component of the velocity update is calculated. In contrast with standard explicit schemes, where the time‐step size is determined by the speed of the pressure waves, the allowable time step for the proposed scheme is found to depend only on the shear wave speed. This leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
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L.E. Perotti A. Bompadre M. Ortiz 《International journal for numerical methods in engineering》2013,96(7):405-424
We develop a mixed finite‐element approximation scheme for Kirchhoff plate theory based on the reformulation of Kirchhoff plate theory of Ortiz and Morris [1]. In this reformulation the moment‐equilibrium problem for the rotations is in direct analogy to the problem of incompressible two‐dimensional elasticity. This analogy in turn opens the way for the application of diamond approximation schemes (Hauret et al. [2]) to Kirchhoff plate theory. We show that a special class of meshes derived from an arbitrary triangulation of the domain, the diamond meshes, results in the automatic satisfaction of the corresponding inf ? sup condition for Kirchhoff plate theory. The attendant optimal convergence properties of the diamond approximation scheme are demonstrated by means of the several standard benchmark tests. We also provide a reinterpretation of the diamond approximation scheme for Kirchhoff plate theory within the framework of discrete mechanics. In this interpretation, the discrete moment‐equilibrium problem is formally identical to the classical continuous problem, and the two differ only in the choice of differential structures. It also follows that the satisfaction of the inf ? sup condition is a property of the cohomology of a certain discrete transverse differential complex. This close connection between the classical inf ? sup condition and cohomology evinces the important role that the topology of the discretization plays in determining convergence in mixed problems. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献