The Finite Element Method Thomas J R Hughes Pdf Download

The equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established. This enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method. Applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles. These include thin beams and plates, and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics.

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... Approach, concept Application in derivation via principle of virtual work ( ) nodal finite isogeometric or as a mixed method ( ) elements analysis Higher-order basis ( ) [6,7] [ 8,9] Field consistent approach ( ) [10,11,12] [13, 14, 15] Reduced integration Selective reduced integration ( ) [16,17,18,19,20] method and a Hellinger-Reissner approach. For the curved cantilever problem described above, Figure 2 plots the resulting convergence curves in terms of the L 2 errors in the displacements for quadratic, cubic, quartic and quintic B-spline basis functions. ...

... Summing up the two equations in (17) and the two equations in (18), we can write the variational mixed formulation of the circular Euler-Bernoulli ring in curvilinear coordinates in concise format: ...

... In the case of the Euler-Bernoulli beam, selective reduced integration performs numerical integration of the membrane part of the stiffness matrix in (29) with a quadrature rule that cannot 185 accurately integrate all of its polynomials. The idea is closely related to the mechanics-inspired interpretation of locking via parasitic membrane strain components that are then relaxed by using fewer quadrature points than necessary for exact integration [14,16,17,18,27]. On the one hand, reduced selective integration is simple to implement and operates with the same displacementbased standard variational formulation (28). ...

In this paper, we initiate the use of spectral analysis for assessing locking phenomena in finite element formulations. We propose to ``measure'' locking by comparing the difference between eigenvalue and mode error curves computed on coarse meshes with ``asymptotic'' error curves computed on ``overkill'' meshes, both plotted with respect to the normalized mode number. To demonstrate the intimate relation between membrane locking and spectral accuracy, we focus on the example of a circular ring discretized with isogeometric curved Euler-Bernoulli beam elements. We show that the transverse-displacement-dominating modes are locking-prone, while the circumferential-displacement-dominating modes are naturally locking-free. We use eigenvalue and mode errors to assess five isogeometric finite element formulations in terms of their locking-related efficiency: the displacement-based formulation with full and reduced integration and three locking-free formulations based on the B-bar, discrete strain gap and Hellinger-Reissner methods. Our study shows that spectral analysis uncovers locking-related effects across the spectrum of eigenvalues and eigenmodes, rigorously characterizing membrane locking in the displacement-based formulation and unlocking in the locking-free formulations.

... The variational statement of the problem is derived by testing system (47)(48)(49) against ...

... The sub-grid scale is expressed in terms of the residual of the projected (Galerkin) counterpart of Eqs. (47)(48)(49) to obtaiñ ...

In this work an algorithm for topological optimization, based on the topolog-ical derivative concept, is proposed for both nearly and fully incompressible materials. In order to deal with such materials, a new decomposition of the Polarization tensor is proposed in terms of its deviatoric and volumetric components. Mixed formulations applied in the context of linear elasticity do not only allow to deal with incompressible material behavior but also to obtain a higher accuracy in the computation of stresses. The system is stabilized by means of the Variational Multiscale method based on the decomposition of the unknowns into resolvable and subgrid scales in order to prevent fluctuations. Several numerical examples are presented and discussed to assess the robustness of the proposed formulation and its applicability to Topology Optimization problems for incompressible elastic solids.

... By using the standard Bubnov-Galerkin procedure with the test functions δu for displacement field and δρ for density field, the weak forms for equilibrium equation Eq. (3a), mass balance equations Eqs. (22) and (37) are derived. ...

... To treat the hourglass issues, there are many well-known methods available in the literature could be applied, for instance, Fbar [33], Bbar [34], enhanced strain element [35], and Cosserat point approaches [36]. In this study, the selective reduced integration technique in [37] is used instead. The simulation results are obtained from the original model with 100 steps of calculation and compared with the analytical solutions, which are derived from Eq. (19). ...

• Tinh Quoc Bui
• Hung Thanh Tran

In this article, we present a new computational damage approach based on localized mass loss concept and its detailed finite element implementation for brittle fracture under quasi-static loading condition. Formulation of this approach is derived in a general way by means of finite deformation regime and nonlinear isotropic materials. The underlying idea of the theory lies in the fact that cracks are created by massive breakage of atomic bonds, diffusing in a volume of characteristic size, resulting in highly localized mass loss area, and nor geometric description of cracks are required. In degraded domain, the traditional law of conservation of mass is violated locally and that is replaced by the local mass-balance equation, accounting for mass flow in the area of degraded material. A coupled system of equilibrium and local mass-balance equations is thus introduced to govern deformation of the body and evolution of the mass density. In this setting, the failure, in contrast to conventional damage models, is thus driven by localized mass loss, which has physical meaning, nor internal parameters such as phase-field or damage variables are defined. We also introduce new constitutive laws for mass source and mass flux by means of energy decomposition, where only the positive part of strain energy density (SED) function involves in the degradation of mass, accounting for distinction of fracture behavior in tension and compression. The discrete forms of governing equations are solved by a staggered nonlinear algorithm. Importantly, equally linear interpolations for the standard finite elements are used for both displacements and mass density, nor mismatch of approximation exits, convenient in the implementation. In addition, we present a special detection technique to find so-called updated nodes, which aims to prevent numerically unstable issues caused by random distortions of deleted elements. The staggered algorithm is thus modified so that the displacements and mass density are updated separately. This detection technique is then integrated into the staggered algorithm. We also present a procedure for calibration of the required quantities used for the localized mass loss analysis. Numerical experiments for brittle fracture are studied to show the accuracy and performance of the developed damage approach.

...  Reduced integration MALKUS & HUGHES 1978;FLANAGAN & BELYTSCHKO 1981;MURTHY 1994 [PIAN 1964;HERRMANN 1965;PIAN & SUMIHARA 1984;BELYTSCHKO et al 1984YEO & LEE 1997]; ...

• Djamel Boutagouga

Linear and geometrically nonlinear static and dynamic finite element analysis of thin shells using triangular and quadrilateral flat shell elements with in-plane drilling rotational degree of freedom is presented. The flat shell elements are obtained by combining the "DKT" and "DKQ" Discrete Kirchhoff Theory plate bending elements and membrane elements with drilling rotation. The membrane elements developed are a quadrilateral element with drilling rotation based on the modified HUGHES and BREZZI variational formulation and a triangular element with drilling rotation based on the Enhanced Strain formulation. The transient dynamic analysis is carried out using Newmark direct time integration method, while the nonlinear analysis adopts the updated co-rotational Lagrangian description. In this purpose, in-plane co-rotational formulation that considers the in-plane drilling rotation is developed and presented for triangular and quadrilateral membrane elements. Furthermore, a simple and effective in-plane mass matrix that takes into account the in-plane rotational inertia, which permit true representation of in-plane vibrational modes is adopted. Finally, these developments are implemented into three dimensional flat shell finite elements with six degrees of freedom. A finite element analysis program is also developed to check the accuracy of the developed elements. The developed elements are first thoroughly tested for static and dynamic analysis of plane stress problems, then, they are tested for general shell problems. The effectiveness of these elements shown by the selected numerical examples to predict the nonlinear dynamic response of shell structures while remains economic is adequate to ascertain that these elements would perform well in the case of nonlinear static and dynamic analysis of general shells.

... A number of adaptations of the FEM have been developed to overcome these pathologies. These include mixed formulations, in which all variables of interest are approximated explicitly [9][10][11], and discontinuous Galerkin (DG) methods, in which inter-element continuity is abandoned [12][13][14][15]. Similar numerical challenges arise in the nearly inextensible limit with the FEM exhibiting sub-optimal performance and poor accuracy in this context, even in the case of higher-order [16], and mixed formulations [17]. ...

• Daniel van Huyssteen
• B.D. Reddy

This work considers the application of a low-order displacement-based virtual element method (VEM) to plane problems of transversely isotropic hyperelasticity, with a novel approach to the computation of stabilization parameters. The method is applied to a range of numerical examples and a variety of transversely isotropic material models are considered. For each of the material models the performance of the virtual element method is investigated under varying degrees of compressibility and extensibility, including near-incompressibility and near-inextensibility, both separately and in combination. Through the range of examples the convergence behaviour of the method is demonstrated with robust, accurate and locking-free behaviour exhibited for all considered choices of material model and material parameter.

... Within the finite element method various discretization schemes have been developed that solve problems with incompressibility constraints in small and finite deformations, see e.g. [1][2][3][4][5][6][7] and [8]. These are based on different treatments of the incompressibility constraint using Lagrange multiplier, perturbed and augmented Lagrangian as well as Hu-Washizu formulations. ...

Considerable progress has been made during the last decade with respect to the development of discretization techniques that are based on the virtual element method. Here we construct a new scheme for large strain problems that include incompressible material behavior. The idea is to use a formulations analogous to the classical Taylor-Hood element, Taylor and Hood (1972) which is based on a mixed principle where different interpolation functions are used for the deformation and pressure field. In this paper, a quadratic serendipity ansatz for the displacements is combined with a linear pressure field which leads to new virtual element formulations that are discussed and compared in this paper.

... Mathematics 2021, 9, x FOR PEER REVIEW 2 of 16 gration approach was developed by Zienkiewicz et al. [22] and Pugh et al. [23]. The selective integration method was also employed for plate and shell analyses [24,25]. Bathe and Dvorkin [26] proposed the MITC family, and Nguyen et al. [27] developed the MISC element. ...

A Simple three-node Discrete Kirchhoff Triangular (SDKT) plate bending element is proposed in this study to overcome some inherent difficulties and provide efficient and dependable solutions in engineering practice for thin plate structure analyses. Different from the popular DKT (Discrete Kirchhoff Theory) triangular element, using the compatible trial function for the transverse displacement along the element sides, the construction of the present SDKT element is based on a specially designed trial function for the transverse displacement over the element, which satisfies interpolation conditions for the transverse displacements and the rotations at the three corner nodes. Numerical investigations of thin plate structures were conducted, using the proposed SDKT element. The results were compared with those by other prevalent plate elements, including the analytical solutions. It was shown that the present element has the simplest explicit expression of the nine-DOF (Degree of Freedom) triangular plate bending elements currently available that can pass the patch test. The numerical examples indicate that the present element has a good convergence rate and possesses high precision.

In this paper, we take a fresh look at using spectral analysis for assessing locking phenomena in finite element formulations. We propose to "measure" locking by comparing the difference between eigenvalue and mode error curves computed on coarse meshes with "asymptotic" error curves computed on "overkill" meshes, both plotted with respect to the normalized mode number. To demonstrate the intimate relation between membrane locking and spectral accuracy, we focus on the example of a circular ring discretized with isogeometric curved Euler–Bernoulli beam elements. We show that the transverse-displacement-dominating modes are locking-prone, while the circumferential-displacement-dominating modes are naturally locking-free. We use eigenvalue and mode errors to assess five isogeometric finite element formulations in terms of their locking-related efficiency: the displacement-based formulation with full and reduced integration and three locking-free formulations based on the B-bar, discrete strain gap and Hellinger–Reissner methods. Our study shows that spectral analysis uncovers locking-related effects across the spectrum of eigenvalues and eigenmodes, rigorously characterizing membrane locking in the displacement-based formulation and unlocking in the locking-free formulations.

Smoothed finite element method (S-FEM) has attracted lots of attentions in the fields of computational mechanics, especially in solid mechanics and heat transfer problems. In computational fluid dynamics, works on S-FEM were limited to two-dimensional problems. This work aims to extend the S-FEM to three-dimensional (3D) incompressible laminar flows. Wedge element grids and grids with mixed wedge and hexahedral elements are formulated for 3D incompressible laminar flows based on the cell-based S-FEM (CS-FEM). To reduce numerical oscillations, we implemented the streamline-upwind/Petrov-Galerkin method (SUPG) together with the stabilized pressure gradient projection (SPGP). Several examples are presented, including the Beltrami flow, lid-driven cavity flow, backward facing step flow and microchannel flow, to validate and examine the presented method. The results indicate that wedge elements and mixed wedge-hexahedral elements based on the CS-FEM have higher computational efficiency than that of hexahedral elements based on the CS-FEM for the same level of computational accuracy. It is also found that the present CS-FEM performed better than the standard FEM in dealing with pressure stability. The flow characteristics are well captured by the CS-FEM using the mixed wedge-hexahedral elements, and the numerical results are acceptable compared to those of STAR-CCM+.

Highly flexible thin-walled beams with complex open cross-sections are sensitive to torsional and warping effects. The analysis of higher-order vibration modes in these structures needs more accurate and precise methods in order to achieve reliable results and detect the cross-sectional deformations in the structures' free vibration response. This paper analyzes higher vibration modes in a series of thin-walled beams, which were proposed by Chen as benchmark problems. These are all open-section thin-walled beams with complex geometries. Global vibration modes, such as bending and torsion, related to the rigid cross-sectional deformations can be detected via classical and shear refined theories. However, cross-sectional deformations appear at higher frequencies, and these modes are mixed with the global ones. To highlight this fact, this paper compares classical beam theories with refined ones based on the Carrera Unified Formulation (CUF) and the shell results using the commercial finite element (FE) software and the data available from the literature. The CUF FEs based on the power of cross-sectional deformation coordinates (x, z) and those based on the Lagrangian polynomials are implemented and compared using Modal Assurance Criterion. A number of interesting conclusions are drawn about the effectiveness of classical and CUF-based results. The need for models capable of detecting cross-sectional deformations is outlined. In fact, many modes are lost by classical beam theories; on the other hand, they show rigid cross-section modes that do not really exist. This fact is also confirmed by the shell models, which are more expensive in terms of computational costs regarding the efficient CUF ones proposed here.

• Gaetano Fichera

The subject to be developed in this article covers a very large field of existence theory for linear and nonlinear partial differential equations. Indeed, problems of static elasticity, of the propagation of waves in elastic media, and of the thermodynamics of continua require existence theorems for elliptic, hyperbolic and parabolic equations both linear and nonlinear. Even if one restricts oneself to linear elasticity, there are several kinds of partial differential equations to be considered. In static problems we encounter second order systems, either with constant or with variable coefficients (homogeneous and non-homogeneous bodies), scalar second order equations (for instance either in the St. Venant torsion problems or in the membrane theory), fourth order equations (equilibrium of thin plates), eighth order equations (equilibrium of shells). Each case must be considered with several kinds of boundary conditions, corresponding to different physical situations. On the other hand, to every problem of static elasticity corresponds a dynamical one, connected with the study of vibrations in the elastic system under consideration. Moreover, problems of thermodynamics require the study of certain diffusion problems of parabolic type. In addition to that, the study of materials with memory requires existence theorems for certain integro-differential equations, first considered by Volterra.

• Ranbir Sandhu
• Kamar J. Singh

For completeness the finite element bases which are used for approximate solutions of elliptic problems of order 2p by the Ritz method must include the functions corresponding to the constant value of the pth derivative. In actual usage, to ensure a positive definite system of algebraic equations, additional interpolating functions are introduced. This leads to "multiple covering" of some of the system modes and results in overestimation of stiffness. Reduced integration techniques eliminate some of this multiple covering and thereby give improved accuracy. Selective reduced integration has been found useful in the analysis of flexural problems. In this paper we suggest the use of only the minimal covering that is sufficient for convergence. A technique for solution of the discretized system is given. Numerical performance data show remarkable improvement over conventional procedures. The proposed scheme yields good approximation even for very coarse meshes. This indicates the possibility of considerable economy in the cost of obtaining finite element solutions to complex problems, e.g. coupled field problems, three-dimensional problems, stress concentration etc.

• T. J. R. Hughes

The bilinear displacement model, employing one-point Gaussian quadrature on the lambda -term, and the constant pressure variable, bilinear displacement element based upon Herrmann's formulation, have been shown to lead to identical results. A numerical example in support of the analysis has been presented. This result has considerable practical significance since the underintegrated displacement model can be implemented more simply and economically. In particular, programming the element is simpler and the number of equations in practical problems is reduced by approximately 1/3.

• David S. Malkus

In this paper it is shown how the displacement formulation of the theorem of minimum potential energy can be used with the finite element method to approximate both compressible and incompressible equilibria of linearly elastic, isotropic solids. The procedure is shown to be equivalent to the more complicated "mixed principle" technique, due to the use of numerical integration applied to the computation of the element stiffness matrices. Criteria for the choice of integration formulas and elements are discussed, and numerical examples are presented.

Source: https://www.researchgate.net/publication/222621164_Mixed_finite_element_methods-reduced_and_selective_integration_techniques_A_Unification_of_Concepts

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