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Assumed‐deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis
International Journal For Numerical Methods in Engineering
Abstract
An assumed-strain finite element technique for non-linear finite deformation is presented. The weighted-residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed-deformation gradient. Assumed gradient operators are derived via nodal integration from the kinematic-weighted residual. A variety of finite element shapes fits the derived framework: four-node tetrahedra, eight-, 27-, and 64-node hexahedra are presented here. Since the assumed-deformation gradients are expressed entirely in terms of the nodal displacements, the degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials and no volumetric/deviatoric split is required. The consistent tangent operator is inexpensive and symmetric. Furthermore, the material update and the tangent moduli computation are carried out exactly as for classical displacement-based models; the only deviation is the consistent use of the assumed-deformation gradient in place of the displacement-derived deformation gradient. Examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking. A constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Copyright © 2008 John Wiley & Sons, Ltd.
... Beside higher-order elements (either higher-order finite elements or NURBS-based elements [43,44] , various techniques have also been proposed, such as enhanced assumed strain (EAS) [45][46][47][48][49] , stabilization techniques [50][51][52][53] , and smoothing techniques [54] . The aforementioned techniques are not necessarily to be employed separately, but can possibly be combined with each other. ...
... This example is devoted to the performance of CHH8 element in solving a Cook's plane-strain tapered panel with nearly incompressible material. This problem is considered as a standard benchmark in many previous works [49,53,67] . As shown in Fig. 35 , one end of the panel is clamped while a total shear force P = 1600 N is applied on the other end. ...
... The results obtained by the EF, 3D-EM and 3 × 3 × 3 Gaussian scheme are almost the same, though 3D-EM seems to yield slightly better with coarser meshes. Nonetheless, the CHH8 is still not as good as the quadratic element HH20 elements being fused with assumed strain like Tip displacement with respect to mesh densities: a comparison between CHH8 and other element types, including HH8, HH20, QT10MS [53] , and NICE-H8 [49] . QT10MS [53] , and NICE-H8 [49] . ...
The consecutive-interpolation procedure (CIP) has been recently proposed as an enhanced technique for traditional finite element method (FEM) with various desirable properties such as continuous nodal gradients and higher accuracy without increasing the total number of degrees of freedom (DOFs). It is common knowledge that linear finite elements, e.g., four-node quadrilateral (Q4) or eight-node hexahedral (HH8) elements, are not highly suitable for geometrically nonlinear analysis. The elements with quadratic interpolation functions have to be used instead. In this paper, the CIP-enhanced four-node quadrilateral element (CQ4), and the CIP-enhanced eight-node hexahedral element (CHH8), are for the first time extended to investigate geometrically nonlinear problems of two- (2D) and three-dimensional (3D) structures. To further enhance the efficiency of the present approaches, novel numerical integration schemes based on the concept of mid-point rules, namely element mid-points (EM) and element mid-edges (EE) are integrated into the present CQ4 element. For CHH8, the 3D-version of EM (namely 3D-EM) and the element mid-faces (EF) scheme are investigated. The accuracy and computational efficiency of the two novel quadrature schemes in both regular and irregular (distorted) meshes are analyzed. Numerical results indicate that the new integration approaches perform more efficiently than the well-known Gaussian quadrature while gaining equivalent accuracy. The performance of the CIP-enhanced elements, which is examined through numerical experiments, is found to be equivalent to that of quadratic Lagrangian finite element counterparts, while having the same discretization with that by the linear finite elements. In addition, we also apply the present CQ4 and CHH8 elements associated with different numerical integration techniques to nearly incompressible materials.
... An alternative to enrichment consists in the so-called smoothed finite element technology. In node smoothing, stabilization is required [13,16,18,27,36], for edge smoothing [20,35] and face smoothing [33] stabilization is not required. However, in all three smoothing options, Jacobian densification occurs (cf. ...
... This feature will be discussed later. In (13), the flow vector is retained from step n as N n . Omitting the argument of the yield function, the integrated system (12-15) is solved for Δλ using the Newton-Raphson method. ...
... The element ensures continuous displacements, pressures and J 2 . In contrast with current alternatives such as the stabilized nodal integration [13,16,18,27,36] and smoothed edges [20,35] and faces [33], the conventional implementation is retained and no stabilization parameters are required. The present formulation follows the traditional finite element assembling process, while good results are obtained in bending and incompressibility situations. ...
With the goal of improving upon the accuracy of D. Arnold's MINI element for finite strain plasticity, and more precisely calculate the elastic/plastic interface, we extend this element formulation to include, as nodal degrees-of-freedom, a function of the second invariant of the deviatoric stress, J 2. A finite-strain J 2 À u À p mixed formulation of the classical low-order tetrahedron element is introduced. We therefore have continuous displacements , pressures and J 2. This element contains an internal displacement bubble that is not condensed out. For hyperelastic materials, we adopt a relative Green-Lagrange formulation whose conjugate stress approximates the Cauchy stress. For the elasto-plastic case, we combine this formulation with the elastic Mandel stress construction , which is power-consistent with the plastic strain rate. In contrast with nodally integrated and varia-tional multiscale methods, there are no additional parameters. High accuracy is obtained for four-node tetrahedra with three incompressibility and bending benchmarks being solved. Accuracy similar to the F hexahedron are obtained. Although the ad-hoc factor is removed and performance is competitive, computational cost is higher than MINI's, with each tetrahedron containing 23 degrees-of-freedom.
... Bonet et al [14] mentioned, but only theoretically, a stabilization using Stabilized Upwind Petrov-Galerkin (SUPG) method to eliminate spurious modes. Broccardo et al [22] used nodally integrated elements for solving large deformation and buckling problems. The stability was provided with a penalty term based on L 2 -norm of difference between assumed and real deformation gradients in their formulation to avoid spuriousness of the buckling modes. ...
... in terms of the adjugate matrix, where adj(·) is the adjugate operator of a matrix. Using (22), we reformulate (20) as ...
... With division by the Jacobian determinant eliminated, the formulation can efficiently deal with extremely thin elements and slivers as demonstrated in Krysl and Kagey [17]. This nodally integrated finite element formulation is referred to as NICE (Nodally Integrated Continuum Elements) formulation by Krysl and collaborators [15,17,16,19,22]. We mention that the assumed nodal strain-displacement matrix formulation is distantly related to the element-based B-bar technique (Hughes [3]). ...
Nodally integrated elements exhibit spurious modes in dynamic analyses (such as in modal analysis). Previously published methods involved a heuristic stabilization factor, which may not work for a large range of problems, and a uniform amount of stabilization was used over all the finite elements in the mesh. The method proposed here makes use of energy-sampling stabilization. The stabilization factor depends on the shape of the element and appears in the definition of the properties of a stabilization material. The stabilization factor is non-uniform over the mesh, and can be computed to alleviate shear locking, which directly depends on the aspect ratios of the finite elements. The nodal stabilization factor is then computed by volumetric averaging of the element-based stabilization factors. Energy-sampling stabilized nodally integrated elements (ESNICE) tetrahedral and hexahedral are proposed. We demonstrate on examples that the proposed procedure effectively removes spurious (unphysical) modes both at lower and at higher ends of the frequency spectrum. The examples shown demonstrate the reliability of energy-sampling in stabilizing the nodally integrated finite elements in vibration problems, just sufficient to eliminate the spurious modes while imparting minimal excessive stiffness to the structure. We also show by the numerical inf-sup test that the formulation is coercive and locking-free.
... Bonet et al [14] mentioned, but only theoretically, a stabilization using Stabilized Upwind Petrov-Galerkin (SUPG) method to eliminate spurious modes. Broccardo et al [22] used nodally integrated elements for solving large deformation and buckling problems. The stability was provided with a penalty term based on L 2 -norm of difference between assumed and real deformation gradients in their formulation to avoid spuriousness of the buckling modes. ...
... in terms of the adjugate matrix, where adj(·) is the adjugate operator of a matrix. Using (22), we reformulate (20) as ...
... With division by the Jacobian determinant eliminated, the formulation can efficiently deal with extremely thin elements and slivers as demonstrated in Krysl and Kagey [17]. This nodally integrated finite element formulation is referred to as NICE (Nodally Integrated Continuum Elements) formulation by Krysl and collaborators [15,17,16,19,22]. We mention that the assumed nodal strain-displacement matrix formulation is distantly related to the element-based B-bar technique (Hughes [3]). ...
Nodally integrated elements exhibit spurious modes in dynamic analyses (such as in modal analysis). Previously published methods involved a heuristic stabilization factor, which may not work for a large range of problems, and a uniform amount of stabilization was used over all the finite elements in the mesh. The method proposed here makes use of energy-sampling stabilization. The stabilization factor depends on the shape of the element and appears in the definition of the properties of a stabilization material. The stabilization factor is non-uniform over the mesh, and can be computed to alleviate shear locking, which directly depends on the aspect ratios of the finite elements. The nodal stabilization factor is then computed by volumetric averaging of the element-based stabilization factors. Energy-sampling stabilized nodally integrated elements (ESNICE) tetrahedral and hexahedral are proposed. We demonstrate on examples that the proposed procedure effectively removes spurious (unphys-ical) modes both at lower and at higher ends of the frequency spectrum. The examples shown demonstrate the reliability of energy-sampling in stabilizing the nodally integrated finite elements in vibration problems, just sufficient to eliminate the spurious modes while imparting minimal excessive stiffness to the structure. We also show by the numerical inf-sup test that the formulation is coercive and locking-free.
... An alternative to introducing additional degrees of freedom is to superimpose a geometrical structure above the tetrahedral mesh to compute quantities that average gradient information from adjacent elements. This approach covers introducing (i) nodal quantities, as for instance in the averaged nodal pressure approach [3,4,5], or in the uniform strain setting [6,7,8,9,10,11], and (ii) face-based or edge-based smoothed strains [12,13]. These methods can be seen as assumed deformation gradient methods derivable from weighted-residual equations [8,10]. ...
... This approach covers introducing (i) nodal quantities, as for instance in the averaged nodal pressure approach [3,4,5], or in the uniform strain setting [6,7,8,9,10,11], and (ii) face-based or edge-based smoothed strains [12,13]. These methods can be seen as assumed deformation gradient methods derivable from weighted-residual equations [8,10]. ...
... The presented method allows for employing standard routines for constitutive updates, stress calculations, and computations of tangent moduli. The implementation dispenses with extensive modifications: all that is needed is to replace the element kinematics-defined deformation gradient ∇ 0 φ with the assumed deformation gradient F where indicated [10,35]. ...
The concept of energy-sampling stabilization is used to develop a mean-strain quadratic 10-node tetrahedral element for the solution of geometrically nonlinear solid mechanics problems. The development parallels recent developments of a “composite” uniform-strain ten-node tetrahedron for applications to linear elasticity and nonlinear deformation. The technique relies on stabilization energy sampling with a mean-strain quadrature, and proposes to choose the stabilization parameters as a quasi-optimal solution to a set of linear elastic benchmark problems. The accuracy and convergence characteristics of the present formulation are tested on linear and nonlinear benchmarks, and compare favorably with the capabilities of other mean-strain and high-performance tetrahedral and hexahedral elements for solids, thin-walled structures (shells), and nearly incompressible structures.
... Averaging based on a geometrical structure above the tetrahedral mesh to compute quantities that incorporate basis function gradient information from adjacent elements is another option: nodal quantities, as for instance in the averaged nodal pressure approach [6,7,8], or in the uniform strain setting [9,10,11,12,13,14], and face-or edge-based smoothed strains [15,16]. These methods can be seen as assumed deformation gradient methods derivable from weighted-residual equations [11,13]. ...
... Averaging based on a geometrical structure above the tetrahedral mesh to compute quantities that incorporate basis function gradient information from adjacent elements is another option: nodal quantities, as for instance in the averaged nodal pressure approach [6,7,8], or in the uniform strain setting [9,10,11,12,13,14], and face-or edge-based smoothed strains [15,16]. These methods can be seen as assumed deformation gradient methods derivable from weighted-residual equations [11,13]. ...
... We may express the material stiffness matrix of a single element in the current configuration as (refer to [13] for additional details of the implementation of the assumedstrain technique) ...
A mean-strain 10-node tetrahedral element is developed for the solution of geometrically nonlinear solid mechanics problems using the concept of energy-sampling stabilization. A uniform-strain tetrahedron for applications to linear elasticity was recently described. The formulation as extended here is able to solve large-strain hyperelasticity. The present 10-node tetrahedron is composed of several four-node linear tetrahedral elements, four tetrahedra in the corners, and four tetrahedra that tile the central octahedron in three possible sets of four-node tetrahedra, corresponding to three different choices for the internal diagonal. We formulate a mean-strain element with stabilization energy evaluated on the four corner tetrahedra, which is shown to guarantee consistency and stability. The stabilization energy is expressed through a stored-energy function, and contact with input parameters in the small-strain regime is made. The neo-Hookean model is used to formulate the stabilization energy. As for small-strain elasticity, the stabilization parameters are determined by actual material properties and geometry of a tetrahedra without any user input. The numerical tests demonstrate that the present element performs well for solid, shell, and nearly incompressible structures. Copyright © 2016 John Wiley & Sons, Ltd.
... In the literature, the poor performance of simplicial tessellations in large deformation analysis of nearly-incompressible solids has been improved through various techniques such as mixed-enhanced elements [6,7,8], pressure stabilization [9,10,11], composite pressure fields [12,13,14], and average nodal pressure/strains [15,16,17,18,19,20]. The last two approaches are broadly based on the idea of reducing pressure (dilatational) constraints to alleviate volumetric locking. ...
... In (20), the hydrostatic pressure p = dΨ dil dJ and the identity ∂J ∂E = JC −1 have been used (for instance, see Ref. [47]). ...
... The adoption of the name 'projection' reflects the fact that (28) stems from the pressure constraint (21b), which is like an L 2 projection. Note the similarity of the operator (31) with the assumed gradient operator of Ref. [20] (see (16) therein) and ...
A displacement-based Galerkin meshfree method for large deformation analysis of nearly-incompressible elastic solids is presented. Nodal discretization of the domain is defined by a Delaunay tessellation (three-node triangles and four-node tetrahedra), which is used to form the meshfree basis functions and to numerically integrate the weak form integrals. In the proposed approach for nearly-incompressible solids, a volume-averaged nodal projection operator is constructed to average the dilatational constraint at a node from the displacement field of surrounding nodes. The nodal dilatational constraint is then projected onto the linear approximation space. The displacement field is constructed on the linear space and enriched with bubble-like meshfree basis functions for stability. The new procedure leads to a displacement-based formulation that is similar to F-bar methodologies in finite elements and iso-geometric analysis. We adopt maximum-entropy meshfree basis functions, and the performance of the meshfree method is demonstrated on benchmark problems using structured and unstructured background meshes in two and three dimensions. The nonlinear simulations reveal that the proposed methodology provides improved robustness for nearly-incompressible large deformation analysis on Delaunay meshes.
... The insensitivity of the NICE to mesh distortion was significantly enhanced by Krysl and Kagey [19]. The extension of the NICE methodology to large-strain hyperelasticity was discussed by Broccardo et al. [20]. It was discovered early on that the node-based uniform strain approach displayed in spurious modes in vibration problems [6]. ...
... Importantly, it is shown in [19,20] that the low count of integration points makes the NICE methodology very attractive for applications where substantial amount of work needs to be performed at each quadrature point, such as in finite element inelasticity. Given the appealing locking-free response of the nodally integrated elements of the NICE variety and their insensitivity to mesh distortion, the appeal of the technique to computational plasticity is obvious [22,23]. ...
... The described approach could be considered a variation on the B-bar technique: the dilatation strain-displacement matrix is constructed as averaging over a patch of elements instead of an average over a single element. 2. The operator in Equation (18) may be constructed in a cost-effective way from the averages of the basis function derivatives instead of as the average of the strain-displacement matrices [20]. 3. ...
In this work, a linear hexahedral element based on an assumed strain finite element technique is presented for the solution of plasticity problems. The element stems from the Nodally Integrated Continuum Element (NICE) formulation and its extensions. Assumed gradient operators are derived via nodal integration from the kinematic-weighted residual; the degrees of freedom are only the displacements at the nodes. The adopted constitutive model is the classical associative von Mises plasticity model with isotropic and kinematic hardening; in particular, a double-step midpoint integration algorithm is adopted for the integration and solution of the relevant nonlinear evolution equations. Efficiency of the proposed method is assessed through simple benchmark problems and comparison with reference solutions. Copyright © 2014 John Wiley & Sons, Ltd.
... Volumetric locking is eliminated and illustrated by examples. The weighted residual approach was later discussed in detail [Broccardo and Krysl (2009)]. Extensions to higher order hexahedra are presented. ...
... In [Bonet et al. (2001)], it is suggested to stabilize the elements by using information on the deformation gradient from the last time step in explicit simulations. A simple penalty formulation was used in [Broccardo and Krysl (2009)]. The stabilization of spurious modes was discussed in detail [Puso and Solberg (2006)]. ...
... This example is a commonly used problem, see [Broccardo and Krysl (2009);Elguedj et al. (2008)], to test combined bending and shear for compressible and nearly incompressible materials. The geometry is presented in Fig. 9. ...
This paper presents and compares continuous assumed gradient (CAG) methods when applied to structural elasticity. CAG elements are finite elements where the strain, i.e., the deformation gradient, is replaced by a C0-continuous interpolation. Similar approaches are found in nodal integration and SFEM. Recently, interpolation schemes for a continuous assumed deformation gradient were proposed for first order tetrahedral and hexahedral finite elements. These schemes try to balance accuracy and numerical efficiency. At the same time, the stability of the interpolation with respect to hourglassing and spurious low energy modes is ensured. This paper recalls the fundamentals of CAG elements, i.e., the formulation and linearization. Furthermore, it extends the approach to second order finite elements. Examples prove convergence and accuracy of the quadratic elements. Two interpolation schemes, one being supported by finite element nodes and interior points and the other being a higher-order tensor-product polynomial, are identified to be most accurate.
... The authors do not mention instabilities, but for linear tetrahedral elements it corresponds to the method of [3]. In [17], the approach of [10] is discussed in detail. Extensions to higher order hexahedra are presented and the appearance of spurious modes eliminated P e e r R e v i e w O n l y 6 S. WOLFF using a penalty method. ...
... In [17], a penalty of the form ...
... This example is a commonly used problem, see [17,5], to test combined bending and shear for compressible and nearly incompressible materials. The geometry is presented in figure 13. ...
This article presents an alternative approach to assumed gradient methods in FEM applied to three-dimensional elasticity. Starting from nodal integration (NI), a general C0-continuous assumed interpolation of the deformation gradient is formulated. The assumed gradient is incorporated using the principle of Hu-Washizu. By dual Lagrange multiplier spaces, the functional is reduced to the displacements as the only unknowns. An integration scheme is proposed where the integration points coincide with the support points of the interpolation. Requirements for regular finite element meshes are explained. Using this interpretation of NI, instabilities (appearance of spurious modes) can be explained. The article discusses and classifies available strategies to stabilize NI such as penalty methods, SCNI, α-FEM. Related approaches, such as the smoothed finite element method, are presented and discussed. New stabilization techniques for NI are presented being entirely based on the choice of the assumed gradient interpolation, i.e. nodal-bubble support, edge-based support and support using tensor-product interpolations. A strategy is presented on how the interpolation functions can be derived for various element types. Interpolation functions for the first-order hexahedral element, the first-order and the second-order tetrahedral elements are given. Numerous examples illustrate the strengths and limitations of the new schemes. Copyright © 2010 John Wiley & Sons, Ltd.
... The element-based shape function gradients in With division by the Jacobian determinant eliminated, the formulation can efficiently deal with extremely thin elements and slivers as demonstrated in Krysl and Kagey [53]. This nodally integrated finite element formulation is referred to as NICE (Nodally Integrated Continuum Elements) formulation by Krysl and collaborators [51,53,52,55,58]. We mention that the assumed nodal strain-displacement matrix formulation is distantly related to the element-based B-bar technique (Hughes [14]). ...
... Bonet et al [50] mentioned, but only theoretically, a stabilization using Stabilized Upwind Petrov-Galerkin (SUPG) method to eliminate spurious modes. Broccardo et al [58] used nodally integrated elements for solving large deformation and buckling problems. The stability was provided with a penalty term based on L 2 -norm of difference between assumed and real deformation gradients in their formulation to avoid spuriousness of the buckling modes. ...
... This so-called smoothed finite element technology has been explored recently. For node smoothing, stabilization is required [18,34,25,15,20], for edge smoothing [22,33] and face smoothing [28] stabilization is not required. However, in all three smoothing options, Jacobian densification occurs (cf. Figure 8 of [20]) and the finite element implementation is changed. ...
... Compared with alternatives such as the stabilized nodal integration [18,34,25,15,20] and smoothed edges [22,33] and faces [28], the conventional implementation is retained. In contrast with formulations based on bubbles [38,26,7], bending performance is improved. ...
Abstract A finite-strain stress-displacement mixed formulation of the classical low-order tetrahedron element is introduced. The stress vector obtained from the face normals is now a (vector) degree-of-freedom at each face. Stresses conjugate to the relative Green-Lagrange strains are used within the framework of the Hellinger-Reissner variational principle. Symmetry of the stress tensor is weakly enforced. In contrast with variational multiscale methods, there are no additional parameters to fit. When compared with smoothed finite-elements, the formulation is straightforward and sparsity pattern of the classical system retained. High accuracy is obtained for four-node tetrahedra with incompressibility and bending benchmarks being solved. Accuracy similar to the \overline{\bm{F}} hexahedron are obtained. Although the ad-hoc factor is removed and performance is highly competitive, computational cost is comparatively high, with each tetrahedron containing 24 degrees-of-freedom. We introduce a finite strain version of the Raviart-Thomas element within a common hyperelastic/elasto-plastic framework. Three benchmark examples are shown, with good results in bending, tension and compression with finite strains.
... This so-called smoothed finite element technology has been explored recently. For node smoothing, stabilization is required [18,34,25,15,20], for edge smoothing [22,33] and face smoothing [28] stabilization is not required. However, in all three smoothing options, Jacobian densification occurs (cf. Figure 8 of [20]) and the finite element implementation is changed. ...
... Compared with alternatives such as the stabilized nodal integration [18,34,25,15,20] and smoothed edges [22,33] and faces [28], the conventional implementation is retained. In contrast with formulations based on bubbles [38,26,7], bending performance is improved. ...
A finite-strain tetrahedron with continuous stresses is proposed and analyzed. The complete stress tensor is now a nodal tensor degree-of-freedom, in addition to displacement. Specifically, stress conjugate to the relative Green-Lagrange strain is used within the framework of the Hellinger-Reissner variational principle. This is an extension of the Dunham and Pister element to arbitrary constitutive laws and finite strain. To avoid the excessive continuity shortcoming, outer faces can have null stress vectors. The resulting formulation is related to the nonlocal approaches popularized as smoothed finite element formulations. In contrast with smoothed formulations, the interpolation and integration domain is retained. Sparsity is also identical to the classical mixed formulations. When compared with variational multiscale methods, there are no parameters. Very high accuracy is obtained for four-node tetrahedra with incompressibility and bending benchmarks being successfully solved. Although the ad-hoc factor is removed and performance is highly competitive, computational cost is high, as each tetrahedron has 36 degrees-of-freedom. Besides the inf-sup test, four benchmark examples are adopted, with exceptional results in bending and compression with finite strains.
... This smoothed finite element technology has been extensively explored recently. For node smoothing, stabilization is required [19,38,29,14,21], for edge smoothing [23,37] and face smoothing [33] stabilization is not required. However, in all three smoothing options, Jacobian densification occurs (cf. Figure 8 of [21]) and the conventional finite element implementation is disrupted. ...
... Compared with alternatives such as the stabilized nodal integration [19,38,29,14,21] and smoothed edges [23,37] and faces [33], no densitification occurs (cf. Figure 8 of [21]) and the conventional implementation is retained. In contrast with formulations based on bubbles [41,31,4], bending performance is greatly improved. ...
A finite-strain tetrahedron with continuous stresses is proposed and analyzed. The complete stress tensor is
now a nodal tensor degree-of-freedom, in addition to displacement. Specifically, stress conjugate to the relative
Green-Lagrange strain is used within the framework of the Hellinger-Reissner variational principle. This is
an extension of the Dunham and Pister element to arbitrary constitutive laws and finite strain. To avoid the
excessive continuity shortcoming, outer faces can have null stress vectors. The resulting formulation is related
to the nonlocal approaches popularized as smoothed finite element formulations. In contrast with smoothed
formulations, the interpolation and integration domain is retained. Sparsity is also identical to the classical
mixed formulations. When compared with variational multiscale methods, there are no parameters. Very high
accuracy is obtained for four-node tetrahedra with incompressibility and bending benchmarks being successfully
solved. Although the ad-hoc factor is removed and performance is highly competitive, computational cost is
high, as each tetrahedron has 36 degrees-of-freedom. Besides the inf-sup test, four benchmark examples are
adopted, with exceptional results in bending and compression with finite strains.
... An alternative approach is to introduce a geometrical structure above the tetrahedral mesh to compute quantities that average basis function gradient information from adjacent elements: nodal quantities, as for instance in the averaged nodal pressure approach [7,8,9], or in the uniform strain setting [10,11,12,13,14,15], and face-or edge-based smoothed strains [16,17]. These methods can be seen as assumed strain (or, in the nonlinear setting, assumed deformation gradient) methods derivable from weighted-residual equations [12,14]. ...
... An alternative approach is to introduce a geometrical structure above the tetrahedral mesh to compute quantities that average basis function gradient information from adjacent elements: nodal quantities, as for instance in the averaged nodal pressure approach [7,8,9], or in the uniform strain setting [10,11,12,13,14,15], and face-or edge-based smoothed strains [16,17]. These methods can be seen as assumed strain (or, in the nonlinear setting, assumed deformation gradient) methods derivable from weighted-residual equations [12,14]. Additionally, we may mention here recent proposals of polygonal discretizations [18]. ...
In this study, a new mean-strain 10-node tetrahedral element (T10MS) is developed using energy-sampling stabilization. The proposed 10-node tetrahedron is composed of several four-node linear tetrahedral elements, four tetrahedra in the corners and four tetrahedra that tile the central octahedron in three possible sets of four-node linear tetrahedra, corresponding to three different choices for the internal diagonal. The assumed strains are calculated from mean “basis function gradients”. The energy-sampling technique introduced previously for removing zero-energy modes in the mean-strain hexahedron is adapted for the present element: the stabilization energy is evaluated on the four corner tetrahedra. The proposed element naturally leads to a lumped mass matrix, and does not have unphysical low-energy vibration modes. For simplicity, we limit our developments to linear elasticity with compressible and nearly incompressible material. The numerical tests demonstrate that the present element performs well compared to the classical 10-node tetrahedral elements for shell and plate structures, and nearly incompressible materials. This article is protected by copyright. All rights reserved.
... Here an assumed-strain finite element technique for the solution of plasticity models is presented. The element stems from the NICE formulation [13] and its further extensions [14,15]. Assumed gradient operators are derived via nodal integration from the kinematic-weighted residual. ...
... 2 ASSUMED-DEFORMATION GRADIENT In the following we briefly address to the weighted residual formulation used to derive the assumed-deformation gradients. The NICE formulation [13,14] aims at produce assumed-strain nodal matrix in the form: ...
Linear tetrahedral elements are a key tool to employs with a complex finite element discretization. For this kind of elements low computational cost, stability and high accuracy are desirable properties in practical three-dimensional application. These features become critical when dealing with problems with plasticity or nearly incompressible materials, since in these cases standard linear tetrahedral elements usually perform poorly. In this work a linear tetrahedral element based on an assumed-strain finite element technique is presented for the solution of von Mises plasticity with linear hardening. A numerical assessment on accuracy and efficiency of the proposed method is presented by means of extensive testing on classical benchmarks.
... The present paper focuses on an assumed-strain finite element technique for the solution of plasticity problems. The element stems from the Nodally Integrated Continuum Element (NICE) formulation [8] and its further extensions [14,15], in particular from the analogous linear hexahedral NICE-H8 formulation which was presented and validated for plasticity problems in [16]. We introduce a de Veubeke straindisplacement functional by eliminating the Lagrange multipliers from the general Hu-Washizu functional. ...
... This nodal quadrature will now be applied to the kinematic equation to derive the assumed strain-displacement operator. Realizing that the strain (13) is multivalued at node K, meaning that the strain-displacement matrix is independently evaluated in each element ej K connected at node K, the summations in Eq. (14) can be switched and then the nodal quadrature of (13) is expressed as Z ...
Linear tetrahedra perform poorly in problems with plasticity, nearly incompressible materials, and in bending. While higher-order tetrahedra can cure or alleviate some of these weaknesses, in many situations low-order tetrahedral elements would be preferable to quadratic tetrahedral elements: e.g. for contact problems or fluid-structure interaction simulations. Therefore, a low-order tetrahedron that would look on the outside as a regular four-node tetrahedron, but that would possess superior accuracy is desirable. An assumed-strain, nodally integrated, four-node tetrahedral element is presented (NICE-T4). Several numerical benchmarks are provided showing its robust performance in conjunction with material nonlinearity in the form of von Mises plasticity. In addition we compare the computational cost of the nodally integrated NICE-T4 with the isoparametric quadratic tetrahedron. Because of the reduced number of quadrature points, the NICE-T4 element is competitive in nonlinear analyses with complex material models.
... However, low-order triangles/tetrahedra are not appropriate for practical use due to their poor performance in many instances such as bending dominated problems, incompressible media and large deformations. In an effort to cope with their poor performance, various techniques have been developed, which can be classified in four approaches: mixed-enhanced elements [12,13,14], pressure stabilization [15,16,17], composite pressure fields [18,19,20], and average nodal pressure/strains [21,22,23,24,25,26,27,28]. The last two approaches are displacement-based methods and are broadly based on the idea of reducing pressure (dilatational) constraints to alleviate volumetric locking in low-order meshes. ...
... Since the pressure is unique only up to a constant, we define P := p : p ∈ L 2 (Ω), Ω p dΩ = 0 and let p ∈ P and δp ∈ P be the trial and test functions for the pressure variable, respectively. By substituting (25) into (22), the stress tensor can be rewritten as ...
We present a displacement-based Galerkin meshfree method for the analysis of nearly-incompressible linear elastic solids, where low-order simplicial tessellations (i.e., 3-node triangular or 4-node tetrahedral meshes) are used as a background structure for numerical integration of the weak form integrals and to get the nodal information for the computation of the meshfree basis functions. In this approach, a volume-averaged nodal projection operator is constructed to project the dilatational strain into an approximation space of equal- or lower-order than the approximation space for the displacement field resulting in a locking-free method. The stability of the method is provided via bubble-like basis functions. Because the notion of an ‘element’ or ‘cell’ is not present in the computation of the meshfree basis functions such low-order tessellations can be used regardless of the order of the approximation spaces desired. First- and second-order meshfree basis functions are chosen as a particular case in the proposed method. Numerical examples are provided in two and three dimensions to demonstrate the robustness of the method, its ability to avoid volumetric locking in the nearly-incompressible regime, and its improved performance when compared to the MINI finite element scheme on the simplicial mesh.
... In this example, introduced by [27], we investigate linear buckling of an elastic 52 × 16 mm block in plane strain (see Figure 12). The problem is modeled with 3-D elements. ...
... We consider a linear stability analysis of an L-shaped flat frame which is clamped at one end and subjected to an in-plane shear load at the other [28,29,27]. The coarsest and finest mesh used, together with the buckling modes, are shown in Figure 13(a). ...
A method for stabilizing the mean-strain hexahedron for applications to anisotropic elasticity was described by Krysl (in IJNME 2014). The technique relied on a sampling of the stabilization energy using the mean-strain quadrature and the full Gaussian integration rule. This combination was shown to guarantee consistency and stability. The stabilization energy was expressed in terms of input parameters of the real material, and the value of the stabilization parameter was fixed in a quasi-optimal manner by linking the stabilization to the bending behavior of the hexahedral element (Krysl, submitted). Here the formulation is extended to large-strain hyperelasticity (as an example, the formulation allows for inelastic behavior to be modeled). The stabilization energy is expressed through a stored-energy function, and contact with input parameters in the small-strain regime is made. As for small-strain elasticity, the stabilization parameter is determined to optimize bending performance. The accuracy and convergence characteristics of the present formulations for both solid and thin-walled structures (shells) compare favorably with the capabilities of mean-strain and other high-performance hexahedral elements described in the open literature and also with a number of successful shell elements.
... In the following, we briefly address to the nodal integrated finite element approach that uses a weighted residual formulation to derive assumed-deformation gradients. The NICE formulation [4,5,6] aims at producing assumed-strain nodal matrix in the form: ...
... In words, the nodal strain-displacement matrix computed on the i − th node is constructed as averages of the strain-displacement matrices from the connected elements, see Figure 2. It is clear that the assumed-strain matrices have the same nonzero structure as the element-wise strain-displacement matrices. In fact, the assumed-strain matrices B J may be constructed from assumed-strain basis function gradients as discussed in detail in reference [5]. In other words, the averaging operation may be applied to the gradients of the basis functions instead of to the strain-displacement matrices: ...
Sequentially Linear Analysis (SLA) is an alternative method that avoids convergence problems derived from the use of classic nonlinear finite element analysis. Instead of using incremental iterative schemes (arc-length control, Newton-Raphson), SLA is a sequential procedure made by a series of linear analysis, able to capture nonlinear behavior, reducing Young Modulus, according to saw-tooth constitutive relation. In this paper an investigation above all the aspects of the methods will be presented using a new element suitable for the SLA: accuracy of the solutions and computational cost, i.e. the time needed to get to satisfactory conclusions of the analysis. In order to test the efficiency of the proposed element, numerical results hailed from different brittle problems, such as glass beam and an ideal masonry tower, are used.
... Some further developments of these techniques can then be found in [6][7][8]. Recently, a displacement-based assumed strain finite element formulation with nodal integration was proposed in [9,10] and was shown to provide an excellent behavior also for the almost ...
... where the test function ıu is designed to vanish along the parts of the boundary @ u (where essential boundary conditions are prescribed) and the test functions ı N have the meaning of strains and could also be considered in the language of Lagrange multipliers. Following the framework introduced by Krysl et al. in [9,10], the new operator D is introduced instead of the standard D operator so that it enables gradients at the elements interfaces and satisfies a priori the kinematic residual statement. Note that the actual strains and the test strains ı will be derived from the trial and test displacements using special operators, aiming at producing better performance elements. ...
SUMMARYA finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.
... Again, for linear tetrahedral elements it corresponds to the method of [5]. In [10], the approach of [6] is discussed in detail. Extensions to higher order hexahedrals are presented and the appearence of spurious modes eliminated using an artificial potential function. ...
... In order to provide a simple expression for the variations of P , the constitutive law is replaced by a linear elastic material. The scaling parameter is determined from ρ = αlc E with parameter α, characteristic length l c and elastic modulus E. In [10], a penalty of the form ...
Nodal integration of finite elements has been investigated recently. Compared with full in- tegration it shows better convergence when applied to incompressible media, allows easier remeshing and highly reduces the number of material evaluation points thus improving effi- ciency. Furthermore, understanding it may help to create new integration schemes in meshless methods as well. The new integration technique requires a nodally averaged deformation gradient. For the tetrahedral element it is possible to formulate a nodal strain which passes the patch test. On the downside, it introduces non-physical low energy modes. Most of these "spurious modes" are local deformation maps of neighbouring elements. Present stabilization schemes rely on adding a stabilizing potential to the strain energy. The stabilization is discussed within this article. Its drawbacks are easily identified within numerical experiments: Nonlinear material laws are not well represented. Plastic strains may often be underestimated. Geometrically nonlinear stabilization greatly reduces computational efficiency. The article reinterpretes nodal integration in terms of imposing a nonconforming C0-continuous strain field on the structure. By doing so, the origins of the spurious modes are discussed and two methods are presented that solve this problem. First, a geometric constraint is formulated and solved using a mixed formulation of Hu-Washizu type. This assumption leads to a consis- tent representation of the strain energy while eliminating spurious modes. The solution is exact, but only of theoretical interest since it produces global support. Second, an integration scheme is presented that approximates the stabilization criterion. The latter leads to a highly efficient scheme. It can even be extended to other finite element types such as hexahedrals. Numerical efficiency, convergence behaviour and stability of the new method is validated using linear tetrahedral and hexahedral elements.
... The method to select ˛ is at present unclear. Although the assumed deformation gradient may lead to spurious modes for certain single-phase solid mechanics problems as demonstrated in [35], nonzero ˛ is not required in the solutions presented in the example section. ...
... Similar penalty energy-based stabilization approaches have been discussed in the content of meshless, nodal averaging and nodal deformation gradient methods in [35,41,42]. Finally, applying the stabilized formulation in temporal discrete variational equation (46) yields ...
An adaptively stabilized finite element scheme is proposed for a strongly coupled hydro-mechanical problem in fluid-infiltrating porous solids at finite strain. We first present the derivation of the poromechanics model via mixture theory in large deformation. By exploiting assumed deformation gradient techniques, we develop a numerical procedure capable of simultaneously curing the multiple-locking phenomena related to shear failure, incompressibility imposed by pore fluid, and/or incompressible solid skeleton and produce solutions that satisfy the inf-sup condition. The template-based generic programming and automatic differentiation (AD) techniques used to implement the stabilized model are also highlighted. Finally, numerical examples are given to show the versatility and efficiency of this model.
... In FEM, several approaches to deal with locking effects are found in the literature. An exhaustive review of these approaches is out of the scope of this paper, but we mention the most relevant ones: reduced/selective integration [1], B-bar technique [2,3], mixed formulations [1], assumed strain methods [4], and nodal integration [5][6][7][8][9][10][11][12][13][14]. Of particular interest for the method proposed in this paper are nodal integration techniques. ...
A recently proposed node-based uniform strain virtual element method (NVEM) is here extended to small strain elastoplastic solids. In the proposed method, the strain is averaged at the nodes from the strain of surrounding linearly precise virtual elements using a generalization to virtual elements of the node-based uniform strain approach for finite elements. The averaged strain is then used to sample the weak form at the nodes of the mesh leading to a method in which all the field variables, including state and history-dependent variables, are related to the nodes and thus they are tracked only at these locations during the nonlinear computations. Through various elastoplastic benchmark problems, we demonstrate that the NVEM is locking-free while enabling linearly precise virtual elements to solve elastoplastic solids with accuracy.
... In FEM, several approaches to deal with locking effects are found in the literature. An exhaustive review of these approaches is out of the scope of this paper, but we mention the most relevant ones: reduced/selective integration [1], B-bar technique [2,3], mixed formulations [1], assumed strain methods [4], and nodal integration [5][6][7][8][9][10][11][12][13][14]. Of particular interest for the method proposed in this paper are nodal integration techniques. ...
A recently proposed node-based uniform strain virtual element method (NVEM) is here extended to small strain elastoplastic solids. In the proposed method, the strain is averaged at the nodes from the strain of surrounding linearly-precise virtual elements using a generalization to virtual elements of the node-based uniform strain approach for finite elements. The averaged strain is then used to sample the weak form at the nodes of the mesh leading to a method in which all the field variables, including state and history-dependent variables, are related to the nodes and thus they are tracked only at these locations during the nonlinear computations. Through various elastoplastic benchmark problems, we demonstrate that the NVEM is locking-free while enabling linearly-precise virtual elements to solve elastoplastic solids with accuracy.
... Remark 2. The foregoing volume-averaging operation is analogous to the projection scheme utilized for mitigating locking in a different meshfree method in Ortiz-Bernardin et al., 38 where the authors adapted the assumed deformation gradient operation in Broccardo et al. 39 to the meshfree context. It is noted, however, that the formulation of Ortiz-Bernardin et al. 38 involves a modification of the strain-displacement matrix in addition to the volume-averaging projection. ...
The material point method (MPM) is frequently used to simulate large deformations of nearly incompressible materials such as water, rubber, and undrained porous media. However, MPM solutions to nearly incompressible materials are susceptible to volumetric locking, that is, overly stiff behavior with erroneous strain and stress fields. While several approaches have been devised to mitigate volumetric locking in the MPM, they require significant modifications of the existing MPM machinery, often tailored to certain basis functions or material types. In this work, we propose a locking‐mitigation approach featuring an unprecedented combination of simplicity, efficacy, and generality for a family of explicit MPM formulations. The approach combines the assumed deformation gradient (F¯) method with a volume‐averaging operation built on the standard particle–grid transfer scheme in the MPM. Upon explicit time integration, this combination yields a new and simple algorithm for updating the deformation gradient, preserving all other MPM procedures. The proposed approach is thus easy to implement, low‐cost, and compatible with the existing machinery in the MPM. Through various types of nearly incompressible problems in solid and fluid mechanics, we verify that the proposed approach efficiently circumvents volumetric locking in the explicit MPM, regardless of the basis functions and material types.
... Besides, meshfree basis functions generally do not vanish on the boundary, which requires additional procedures to impose Dirichlet boundary conditions. Regarding finite elements, there are various nodal integration approaches that use some form of pressure or strain averaging at the nodes; for instance, the average nodal pressure tetrahedral element [63], node-based uniform strain triangular and tetrahedral elements [64], the averaged nodal deformation gradient linear tetrahedral element [65], and the family of nodally integrated continuum elements (NICE) and its derivative approaches [66][67][68][69][69][70][71][72]. Recently, the nodebased uniform strain approach [64] was adopted in the nodal particle finite element method (N-PFEM) [73,74]. ...
We propose a combined nodal integration and virtual element method for compressible and nearly incompressible elasticity, wherein the strain is averaged at the nodes from the strain of surrounding virtual elements. For the strain averaging procedure, a nodal averaging operator is constructed using a generalization to virtual elements of the node‐based uniform strain approach for finite elements. We refer to the proposed technique as the node‐based uniform strain virtual element method (NVEM). No additional degrees of freedom are introduced in this approach, thus resulting in a displacement‐based formulation. A salient feature of the NVEM is that the stresses and strains become nodal variables just like displacements, which can be exploited in nonlinear simulations. Through several benchmark problems in compressible and nearly incompressible elasticity as well as in elastodynamics, we demonstrate that the NVEM is accurate, optimally convergent and devoid of volumetric locking.
... Remark 2. The foregoing volume-averaging operation is analogous to the projection scheme utilized for mitigating locking in a different meshfree method in Ortiz-Bernardin et al. [34], where the authors adapted the assumed deformation gradient operation in Broccardo et al. [35] to the meshfree context. It is noted, however, that the formulation of Ortiz-Bernardin et al. [34] involves a modification of the strain-displacement matrix in addition to the volume-averaging projection. ...
The material point method (MPM) is frequently used to simulate large deformations of nearly incompressible materials such as water, rubber, and undrained porous media. However, MPM solutions to nearly incompressible materials are susceptible to volumetric locking, that is, overly stiff behavior with erroneous strain and stress fields. While several approaches have been devised to mitigate volumetric locking in the MPM, they require significant modifications of the exist- ing MPM machinery, often tailored to certain basis functions or material types. In this work, we propose a locking-mitigation approach featuring an unprecedented combination of simplicity, efficacy, and generality for a family of explicit MPM formulations. The approach combines the assumed deformation gradient method with a volume-averaging operation built on the standard particle–grid transfer scheme in the MPM. Upon explicit time integration, this combination yields a new and simple algorithm for updating the deformation gradient, preserving all other MPM procedures. The proposed approach is thus easy to implement, low-cost, and compatible with the existing machinery in the MPM. Through various types of nearly incompressible problems in solid and fluid mechanics, we verify that the proposed approach efficiently circumvents volumetric locking in the explicit MPM, regardless of the basis functions and material types.
... Besides, meshfree basis functions generally do not vanish on the boundary, which requires additional procedures to impose Dirichlet boundary conditions. Regarding finite elements, there are various nodal integration approaches that use some form of pressure or strain averaging at the nodes; for instance, the average nodal pressure tetrahedral element [63], node-based uniform strain triangular and tetrahedral elements [64], the averaged nodal deformation gradient linear tetrahedral element [65], and the family of nodally integrated continuum elements (NICE) and its derivative approaches [66][67][68][69][69][70][71][72]. Recently, the nodebased uniform strain approach [64] was adopted in the nodal particle finite element method (N-PFEM) [73,74]. ...
We propose a combined nodal integration and virtual element method for compressible and nearly incompressible elasticity, wherein the strain is averaged at the nodes from the strain of surrounding virtual elements. For the strain averaging procedure, a nodal averaging operator is constructed using a generalization to virtual elements of the node-based uniform strain approach for finite elements. We refer to the proposed technique as the node-based uniform strain virtual element method (NVEM). No additional degrees of freedom are introduced in this approach, thus resulting in a displacement-based formulation. A salient feature of the NVEM is that the stresses and strains become nodal variables just like displacements, which can be exploited in nonlinear simulations. Through several benchmark problems in compressible and nearly incompressible elasticity as well as in elastodynamics, we demonstrate that the NVEM is accurate, optimally convergent and devoid of volumetric locking.
... This smoothed finite element technology has been extensively explored recently. For node smoothing, stabilization is required [14,15,16,17,18,19], for edge smoothing [20,21] and face smoothing [22] stabilization is not required. However, in all three smoothing options, global Jacobian densification occurs (cf. Figure 8 of [18]) and the conventional finite element implementation and assembling are altered. ...
A new gradient-enhanced strain-tensor formulation for finite-strain problems is introduced, based on the Raviart-Thomas face-interpolation scheme and the Hellinger-Reissner variational principle. The screened-Poisson equation is employed to relate the kinematic Green-Lagrange strain with the mixed strain. The strain vector obtained from the face normals is now a (vector) degree-of-freedom at each face. In contrast with variational multiscale methods, there are no parameters to fit and stability in compression is verified. When compared with smoothed finite-elements, the formulation is straightforward and sparsity pattern of the classical system retained, albeit with high computational cost. In contrast with traditional gradient-enhanced formulations, a theoretically sound mixed formulation underlies the algorithm. High accuracy is obtained for four-node tetrahedra with incompressibility and bending benchmarks being solved. Traditional finite-strain benchmarks and a quasi-brittle damage numerical test are performed, with very competitive results.
... Enhanced-assumed strain approach has produced many treatments -see for example Simo and Armero (1992), Simo et al. (1993), Puso (2000), and Areias et al. (2003). An attractive variant was presented by Broccardo et al. (2009) where the assumed strain technique is combined with the nodal average strain formulation but only elastic and hyper-elastic scenarios were under consideration. ...
A nodal averaging technique which was earlier used for plane strain and three-dimensional problems is extended to include the axisymmetric one. Based on the virtual work principle, an expression for nodal force is found. In turn, a nodal force variation yields a stiffness matrix that proves to be non-symmetrical. But, cumbersome non-symmetrical terms can be rejected without the loss of Newton-Raphson iterations convergence. An approximate formula of volume for a ring of triangular profile is exploited in order to simplify program codes and also to accelerate calculations. The proposed finite element is intended primarily for quasistatic problems and large irreversible strain i.e. for metal forming analysis. As a test problem, deep rolling of a steel rod is studied.
... While the relaxation provided by the modification of deformation gradient definition is able to cure the locking issue, the usage of non-standard deformation gradient may lead to numerical instability as exhibited in Broccardo et al. [2009], Castellazzi and Krysl [2012]. Moran et al. [1990] suggested replacing the assumed deformation gradient F with a linear interpolation between the original and the assumed deformation gradient, i.e., F = αF + (1 − α)F . ...
This final report details the research conducted by the Columbia University research group supported by the Earth Materials and Processes Program under the contract number W911NF1410658. During the period 1/1/2015 to 9/1/2015, three graduate research assistants Yang Liu (now postdoctoral associate at MIT), Mr. Kun Kang, and Mr SeonHong Na and the PI have utilized the support to conduct research for fluid-infiltrating granular and geological materials. Dr. Simon Salager and his PhD student Dr. Ghonwa Khaddour from Joseph Fourier University at Grenoble France, have also provided important micro-CT images of a Hostun sand speci- men taken undergoing drained triaxial compression test through a no-cost collaboration. The research presented in this report is a team effort of the aforementioned individuals.
The 9-month support from the STIR grant has directly lead to 10 journal publications published in a number of highly respected peer-review journals, in the field of computational mechanics (e.g. Computer Methods in Applied Mechanics and Engineering, International Journal of Numerical Methods in Engineering) geomechan- ics and geotechnical engineerin (e.g. Journal of Geophysical Research, Geotechnique Letters) and engineering mechanics (e.g.Journal of Applied Mechanics, Journal of Engineering Mechanics) published from January 2015 to April 2016. This report provides a brief account of the major accomplishments, which includes the introduction of the tensorial Bishop’s coefficient, the formulation of a multiscale model that connects DEM model with finite element analysis, the application of microCT images to calibrate material parameters for granular materials idealized as higher-order continua and a stabilized large deformation model that captures the fluid-solid interaction of granular materials under non-isothermal condition. In the next few chapters, we will provide a statement of the problems the research team studied (Chapter 1), a summary of major research accomplishments and the potential impact of the research and the relevant to the Army research (Chapter 2) and future work and extensions (Chapter 3). A selection of published results are highlighted in Chapters 4-6.
The PI and the research team are grateful for receiving the crucial support from the Earth Material and Process Program under the directorship of Dr. Julia Barzyk to start the new research team at Columbia. The accomplishment mentioned above cannot be made possible without this support.
... for the stiffness matrix, and the symmetry of the stiffness matrix could no longer be guaranteed. Therefore, as already proposed in Reference [11], we replace the displacement-derived test strains with the assumed test strains e = Dv, which yields ...
A finite element model for linear elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach a patch averaged strain-displacement matrix is constructed for each node of the mesh, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four node and linear hexahedral eight node elements are derived. Numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit. Furthermore, the numerical inf-sup test is applied to a selection of problems, in order to show the stability of the present formulation.
... These techniques can be classified in four approaches: mixed-enhanced elements [66][67][68], pressure stabilization [69][70][71], composite pressure fields [72][73][74], and average nodal pressure/strains [48,[75][76][77][78][79]. The last two approaches are broadly based on the idea of reducing pressure (dilatational) constraints to alleviate volumetric locking. ...
A Galerkin-based maximum-entropy meshfree method for linear and nonlinear elastic media is developed. The standard displacement-based Galerkin formulation is used to model compressible linear elastic solids, whereas the classical u-p mixed formulation for near-incompressible linear elastic media is adopted to formulate a volume-averaged nodal technique in which the pressure variable is eliminated from the analysis. This results in a single-field formulation that is devoid of volumetric locking. A modified Gauss integration technique that alleviates integration errors in meshfree methods with guaranteed patch test satisfaction to machine precision is devised. The performance of the maximum-entropy meshfree method is assessed for problems in compressible and near-incompressible linear elastic media using three-node triangular and four-node tetrahedral background meshes. Both structured and unstructured meshes are considered to assess the accuracy, performance and stability of the maximum-entropy meshfree method by means of various numerical experiments, which include patch tests, bending dominated problem, combined bending-shear problem, rigid indentation, Stokes flow and numerical stability tests.
An extension of the volume-averaged nodal technique is proposed for the analysis of near-incompressible nonlinear elastic solids in two dimensions. In the nonlinear version, the volume change ratio of the dilatational constraint, namely J, is volume-averaged around nodes leading to a locking-free displacement-based formulation. The excellent performance of the maximum-entropy meshfree method for problems in near-incompressible nonlinear elastic solids is demonstrated via three standard two-dimensional numerical experiments—a combined bending-shear problem, a plane strain compression of a rubber block and a frictionless indentation problem. Three-node structured and unstructured triangular background meshes are employed and the results are compared to two finite element methods that use such meshes, namely, the linear displacement/constant pressure triangle and the linear displacement/linear pressure triangle enriched with a displacement bubble node (MINI element). The two-dimensional nonlinear simulations reveal that the maximum-entropy meshfree method effectively improves the poor performance of linear triangular meshes in the analysis of near-incompressible solids at finite strains.
... While the relaxation provided by the modification of deformation gradient definition is able to cure the locking issue, the usage of non-standard deformation gradient may lead to numerical instability as exhibited in [60,61]. Moran et al. [56] suggested replacing the assumed deformation gradient F with a linear interpolation between the original and the assumed deformation gradient, that is, e F D˛F C .1 ˛/F ; where˛is a stabilization parameter in which˛D 0 leads to the pure F-bar formulation and˛D 1 leads to the standard formulation. ...
An adaptively stabilized monolithic finite element model is proposed to simulate the fully coupled thermo-hydro-mechanical behavior of porous media undergoing large deformation. We first formulate a finite-deformation thermo-hydro-mechanics field theory for non-isothermal porous media. Projection-based stabilization procedure is derived to eliminate spurious pore pressure and temperature modes due to the lack of the two-fold inf-sup condition of the equal-order finite element. To avoid volumetric locking due to the incompressibility of solid skeleton, we introduce a modified assumed deformation gradient in the formulation for non-isothermal porous solids. Finally, numerical examples are given to demonstrate the versatility and efficiency of this thermo-hydro-mechanical model.
... In the compressible problem, ES-FEM also gives relatively good convergence; however when Poisson's ratios are close to 0.5, its convergence becomes slow. Through the problem tested, we believe that the present method can be well applied to some relevant problems [48,49,14,16]. ...
We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bES-FEM and bFS-FEM) for nearly-incompressible elasticity problems. The crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element, while the pressure is a piecewise constant function. The meshes of triangular or tetrahedral elements required by these methods can be generated automatically. The enrichment induces a softening in the bilinear form allowing the weakened weak (W2) procedure to produce a high-quality solution, free from locking and that does not oscillate. We prove theoretically that both methods confirm the uniform inf–sup and convergence conditions. Four numerical examples are given to validate the reliability of the bES-FEM and bFS-FEM.
... An assumed-strain finite element technique for the solution of plasticity models is presented. The element stems from the NICE formulation [1] and its further extensions [2,3] that uses the weighted residual method to enforce weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed-deformation gradient. Assumed gradient operators are derived via nodal integration from the kinematic-weighted residual. ...
... In this work the nodally integrated plate element (NIPE) formulation is presented for the analysis of laminated composite plates based on the first-order shear deformation theory. The nodally integrated approach aims at providing smoothed derivative quantities by constructing nodal strain-displacement operators [23][24][25][26]. In particular the NIPE approach [27] develops assumed strain finite element for shear deformable plates weakly enforcing the balance and the kinematic equations. ...
... The structure of B b J has the same nonzero structure of the element-wise strain-displacement matrix. In particular the assumed-strain matrix B b J may be constructed from assumed-strain basis function gradients as discussed in detail in [9]. In general, it will be beneficial to define the assumed gradient operator as ...
The nodally integrated plate element formulation is an assumed strain finite element technique for shear deformable plates that enforce weakly the balance and the kinematic equations. The strain–displacement operators are derived via nodal integration satisfying a priori the kinematic weighted residual statement. The present work analyzes the NIPE technique, including the new element, by testing thoughtfully the sensitivity of the elements to severe geometry distortions. We propose an improvement that confers robustness to all element shapes developed by the NIPE formulation.We present also a new nine-node NIP element configuration. The new nine-node element uses bi-quadratic interpolations of the transverse displacement and rotations and is computed by means of a nine-node quadrature rule.A brief review of the improved triangular and quadrangular NIPEs is reported for elastic plate analyses. A few challenging benchmarks carried out on extreme distorted meshes illustrate the performance of the introduced NIP-Q9 element. We detail that the new NIPE formulation confers insensitivity to extreme distortions for the quadratic quadrilateral element and allows to solve for severe distortion with the NIPE family.
A finite-strain stress-displacement mixed formulation of the classical low-order tetrahedron element is introduced. The stress vector obtained from the face normals is now a (vector) degree-of-freedom at each face. Stresses conjugate to the relative Green-Lagrange strains are used within the framework of the Hellinger-Reissner variational principle. Symmetry of the stress tensor is weakly enforced. In contrast with variational multiscale methods, there are no additional parameters to fit. When compared with smoothed finite-elements, the formulation is straightforward and sparsity pattern of the classical system retained. High accuracy is obtained for four-node tetrahedra with incompressibility and bending benchmarks being solved. Accuracy similar to the F‾ hexahedron are obtained. Although the ad-hoc factor is removed and performance is highly competitive, computational cost is comparatively high, with each tetrahedron containing 24 degrees-of-freedom. We introduce a finite strain version of the Raviart-Thomas element within a common hyperelastic/elasto-plastic framework. Three benchmark examples are shown, with good results in bending, tension and compression with finite strains.
In this paper, we present novel techniques of using quadratic Bézier triangular and tetrahedral elements for elastostatic and implicit/explicit elastodynamic simulations involving nearly incompressible linear elastic materials. A simple linear mapping is proposed for developing finite element meshes with quadratic Bézier triangular/tetrahedral elements from the corresponding quadratic Lagrange elements that can be easily generated using the existing mesh generators. Numerical issues arising in the case of nearly incompressible materials are addressed using the consistent B‐bar formulation, thus reducing the finite element formulation to one consisting only of displacements. The higher‐order spatial discretization and the nonnegative nature of Bernstein polynomials are shown to yield significant computational benefits. The optimal spatial convergence of the B‐bar formulation for the quadratic triangular and tetrahedral elements is demonstrated by computing error norms in displacement and stresses. The applicability and computational efficiency of the proposed elements for elastodynamic simulations are demonstrated by studying several numerical examples involving real‐world geometries with complex features. Numerical results obtained with the standard linear triangular and tetrahedral elements are also presented for comparison.
We present a framework to efficiently solve large deformation contact problems with nearly incompressible materials by implementing adaptive re-meshing. Specifically, nodally integrated elements are employed to avoid mesh locking when linear triangular or tetrahedral elements are used to facilitate mesh re-generation. Solution variables in the bulk and on contact surfaces are transferred between meshes such that accuracy is maintained and re-equilibration on the new mesh is possible. In particular, the displacement transfer in the bulk is accomplished through a constrained least squares (LS) problem based on nodal integration, while the transfer of contact tractions relies on parallel transport. Finally, a residual-based error indicator is chosen to drive adaptive mesh refinement. The proposed strategies are applicable to both two or three dimensional analysis and are demonstrated to be robust by a number of numerical examples. This article is protected by copyright. All rights reserved.
This article presents some aspects of continuous assumed gradient (CAG) methods applied to explicit structural dynamics. CAG elements are finite elements in which the strain (i.e., the deformation gradient) is replaced by a C-0-continuous interpolation. Examples provide measures for the accuracy and numerical efficiency of CAG methods. First-order hexahedral and tetrahedral elements are tested. The improvements in accuracy are even larger than in static examples, and increased numerical costs can be balanced by a larger critical time step. DOI: 10.1061/(ASCE)EM.1943-7889.0000389. (C) 2012 American Society of Civil Engineers.
The nodally integrated continuum element (NICE) formulation is an assumed-strain finite element technique derived from a weighted residual statement that weakly enforces both the balance equation and the kinematic equation. The original NICE formulation has a number of desirable attributes (e.g., resistance to volumetric locking), but, similar to classical finite elements, it is sensitive to a geometrical distortion of the finite element mesh. The present work analyzes the NICE technique from this viewpoint, the source of the sensitivity to the shape of the element is identified, and an improvement of the NICE formulation is proposed. We illustrate the performance of the revised NICE formulation on extremely distorted meshes. The tetrahedral meshes contain zero-volume or negative-volume elements, including slivers, and the new NICE formulation is shown to have the condition number of the stiffness matrix under control even in the presence of slivers. Furthermore, insensitivity to distortions is demonstrated for quadratic and cubic hexahedral elements. The proposed improvement confers robustness to all element shapes treated by the NICE formulation. The approximation properties of the original NICE formulation are preserved, in particular the improved version is also locking free, and at the same time, the need for stabilization also carries over. Copyright © 2011 John Wiley & Sons, Ltd.
A method which combines the incompatible modes method with the physical stabilization method is developed to provide a highly efficient formulation for the single point eight-node hexahedral element. The resulting element is compared to well-known enhanced elements in standard benchmark type problems. It is seen that this single-point element is nearly as coarse mesh accurate as the fully integrated EAS elements. A key feature is the novel enhanced strain fields which do not require any matrix inversions to solve for the internal element degrees of freedom. This, combined with the reduction of hourglass stresses to four hourglass forces, produces an element that is only 6.5 per cent slower than the perturbation stabilized single-point brick element commonly used in many explicit finite element codes. Copyright © 2000 John Wiley & Sons, Ltd.
An assumed-strain finite element technique is presented for linear, elastic small-deformation models. Weighted residual method (reminiscent of the strain–displacement functional) is used to weakly enforce the balance equation with the natural boundary condition and the kinematic equation (the strain–displacement relationship). A priori satisfaction of the kinematic weighted residual serves as a condition from which strain–displacement operators are derived via nodal integration. A variety of element shapes is treated: linear triangles, quadrilaterals, tetrahedra, hexahedra, and quadratic (six-node) triangles and (27-node) hexahedra. The degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Furthermore, the numerical inf–sup test is applied in select problems and several variants of the proposed formulations (linear triangles, quadrilaterals, tetrahedra, hexahedra, and 27-node hexahedra) pass the test. Examples are used to illustrate the performance with respect to sensitivity to shape distortion and the ability to resist volumetric locking. Copyright © 2008 John Wiley & Sons, Ltd.
This paper considers the solution of problems in three-dimensional solid mechanics using tetrahedral finite elements. A formulation based on a mixed-enhanced treatment involving displacement, pressure and volume effects is presented. The displacement and pressure are used as nodal quantities while volume effects and enhanced modes belong to individual elements. Both small and finite deformation problems are addressed and sample solutions are given to illustrate the performance of the formulation. Copyright (C) 2000 John Wiley & Sons, Ltd.
The present contribution gives an overview of several classes of element technology in large deformation problems. In particular, the nonlinear enhanced strain method, the B-Bar method and a recently developed reduced integration concept with hourglass stabilization are included in the discussion. It is shown in the present paper that the hourglass contribution needed to stabilize the one Gauss point formulation can be chosen such that an equivalence with any of the three formulations mentioned in the above is obtained. The advantage of this observation is evident: stabilized one point formulations are extremely robust and much more efficient from the computational point of view than their fully integrated counterparts. In addition, the reformulation of nonlinear mixed formulations as stabilization technique allows valuable insights into the still open problem of non-physical instabilities at the element level.
This paper presents projection methods to treat the incompressibility constraint in small- and large-deformation elasticity and plasticity within the framework of Isogeometric Analysis. After reviewing some fundamentals of isogeometric analysis, we investigate the use of higher-order Non-Uniform Rational B-Splines (NURBS) within the B¯ projection method. The higher-continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results. A new non-linear F¯ projection method, based on a modified minimum potential energy principle and inspired by the B¯ method is proposed for the large-deformation case. It leads to a symmetric formulation for which the consistent linearized operator for fully non-linear elasticity is derived and used in a Newton–Raphson iterative procedure. The performance of the methods is assessed on several numerical examples, and results obtained are shown to compare favorably with other published techniques.
A simple four-node quadrilateral and an eight-node hexahedron for large strain analysis of nearly incompressible solids are proposed. Based on the concept of deviatoric/volumetric split and the replacement of the compatible deformation gradient with an assumed modified counterpart, the formulation developed is applicable to arbitrary material models. The closed form of the corresponding exact tangent stiffnesses, which have a particularly simple structure, is derived. It ensures asymptotically quadratic rates of convergence of the Newton-Raphson scheme employed in the solution of the implicit finite element equilibrium equations. From a practical point of view, the incorporation of the proposed elements into existing codes is straightforward. It requires only small changes in the routines of the standard displacement based 4-node quadrilateral and 8-node brick. A comprehensive set of numerical examples, involving hyperelasticity as well as multiplicative elasto-plasticity, is provided. It illustrates the performance of the proposed elements over a wide range of applications, including strain localisation problems, metal forming simulation and adaptive analysis.
Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes are presented. The elements use the linear interpolation functions of the original mesh, but each element is associated with a single node. As a result, a favourable constraint ratio for the volumetric response is obtained for problems in solid mechanics. The uniform strain elements do not require the introduction of additional degrees of freedom and their performance is shown to be significantly better than that of three-node triangular or four-node tetrahedral elements. In addition, nodes inside the boundary of the mesh are observed to exhibit superconvergent behaviour for a set of example problems. Published in 2000 by John Wiley & Sons, Ltd.
A stabilized, nodally integrated linear tetrahedral is formulated and analysed. It is well known that linear tetrahedral elements perform poorly in problems with plasticity, nearly incompressible materials, and acute bending. For a variety of reasons, low-order tetrahedral elements are preferable to quadratic tetrahedral elements; particularly for nonlinear problems. But the severe locking problems of tetrahedrals have forced analysts to employ hexahedral formulations for most nonlinear problems. On the other hand, automatic mesh generation is often not feasible for building many 3D hexahedral meshes. A stabilized, nodally integrated linear tetrahedral is developed and shown to perform very well in problems with plasticity, nearly incompressible materials and acute bending. The formulation is analytically and numerically shown to be stable and optimally convergent for the compressible case provided sufficient smoothness of the exact solution u ∈ C2 ∩ (H1)3. Future work may extend the formulation to the incompressible regime and relax the regularity requirements; nonetheless, the results demonstrate that the method is not susceptible to locking and performs quite well in several standard linear and nonlinear benchmarks. Published in 2006 by John Wiley & Sons, Ltd.
This paper provides an assessment of the average nodal volume methodology originally proposed by Bonet and Burton (Commun. Numer. Meth. Engng. 1998; 14:437–449) for the analysis of finitely strained nearly incompressible solids. An implicit version of the average nodal pressure formulation is derived by re-casting the original concept in terms of average nodal volume change ratio within the framework of the F-bar method proposed by de Souza Neto et al. (Int. J. Solids Struct. 1996; 33: 3277–3296). In this context, a linear triangle for implicit plane strain and axisymmetric analysis of nearly incompressible solids under finite strains is obtained. An exact expression for the corresponding element stiffness matrix is presented. This allows the use of the full Newton–Raphson algorithm, ensuring quadratic rates of asymptotic convergence in the global equilibrium iterations. The performance of the procedure is thoroughly assessed by means of numerical examples. The results show that the nodal averaging technique substantially reduces the volumetric locking tendency of the linear triangle and allows an accurate prediction of deformed shapes and reaction forces in situations of practical interest. However, the formulation is found to produce considerable checkerboard-type hydrostatic pressure fluctuations which poses severe limitations on its range of applicability. Copyright © 2004 John Wiley & Sons, Ltd.
This paper presents a simple linear tetrahedron element that can be used in explicit dynamics applications involving nearly incompressible materials or incompressible materials modelled using a penalty formulation. The element prevents volumetric locking by defining nodal volumes and evaluating average nodal pressures in terms of these volumes. Two well-known examples relating to the impact of elasto–plastic bars are used to demonstrate the ability of the element to model large isochoric strains without locking. © 1998 John Wiley & Sons, Ltd.
A class of ‘assumed strain’ mixed finite element methods for fully non-linear problems in solid mechanics is presented which, when restricted to geometrically linear problems, encompasses the classical method of incompatible modes as a particular case. The method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part, formulated in the context of a three-field variational formulation. The resulting class of mixed methods provides a possible extension to the non-linear regime of well-known incompatible mode formulations. In addition, this class of methods includes non-linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements. The good performance of the proposed methodology is illustrated in a number of simulations including 2-D, 3-D and axisymmetric finite deformation problems in elasticity and elastoplasticity. Remarkably, these methods appear to be specially well suited for problems involving localization of the deformation, as illustrated in several numerical examples.
This paper proposes a new technique which allows the use of simplex finite elements (linear triangles in 2D and linear tetrahedra in 3D) in the large strain analysis of nearly incompressible solids. The new technique extends the F-bar method proposed by de Souza Neto et al. (Int. J. Solids and Struct. 1996; 33: 3277–3296) and is conceptually very simple: It relies on the enforcement of (near-) incompressibility over a patch of simplex elements (rather than the point-wise enforcement of conventional displacement-based finite elements). Within the framework of the F-bar method, this is achieved by assuming, for each element of a mesh, a modified (F-bar) deformation gradient whose volumetric component is defined as the volume change ratio of a pre-defined patch of elements. The resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near-incompressibility. As the original F-bar procedure, the present methodology preserves the displacement-based structure of the finite element equations as well as the strain-driven format of standard algorithms for numerical integration of path-dependent constitutive equations and can be used regardless of the constitutive model adopted. The new elements are implemented within an implicit quasi-static environment. In this context, a closed form expression for the exact tangent stiffness of the new elements is derived. This allows the use of the full Newton–Raphson scheme for equilibrium iterations. The performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two- and three-dimensional numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.
Nodal integration can be applied to the Galerkin weak form to yield a particle-type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions. This particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods. To a lesser degree, nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods. In this work, aspects regarding stability, consistency, efficiency and explicit time integration are explored within the context of nodal integration. Both small and large deformation numerical examples are provided. Copyright © 2007 John Wiley & Sons, Ltd.
The treatment of zero energy modes which arise due to one-point integration of first-order isoparametric finite elements is addressed. A method for precisely isolating these modes for arbitrary geometry is developed. Two hourglass control schemes, viscous and elastic, are presented and examined. In addition, a convenient one-point integration scheme which analytically integrates the element volume and uniform strain modes is presented.
A three-field mixed formulation in terms of displacements, stresses and an enhanced strain field is presented which encompasses, as a particular case, the classical method of incompatible modes. Within this frame-work, incompatible elements arise as particular ‘compatible’ mixed approximations of the enhanced strain field. The conditions that the stress interpolation contain piece-wise constant functions and be L2-ortho-gonal to the enhanced strain interpolation, ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation. The preceding conditions are formulated in a form particularly convenient for element design. As an illustration of the methodology three new elements are developed and shown to exhibit good performance: a plane 3D elastic/plastic QUAD, an axisymmetric element and a thick plate bending QUAD. The formulation described herein is suitable for non-linear analysis.
This paper presents a new linear tetrahedral element that overcomes the shortcomings in bending dominated problems of the average nodal pressure element presented in Bonet and Burton (Communications in Numerical Methods in Engineering 1998; 14:437–439) Zienkiewicz et al. (Internatinal Journal for Numerical Methods in Engineering 1998; 43:565–583) and Bonet et al. (Internatinal Journal for Numerical Methods in Engineering 2001; 50(1):119–133). This is achieved by extending some of the ideas proposed by Dohrmann et al. (Internatinal Journal for Numerical Methods in Engineering 2000; 47:1549–1568) to the large strain nonlinear kinematics regime. In essence, a nodal deformation gradient is defined by weighted average of the surrounding element values. The associated stresses and internal forces are then derived by differentiation of the corresponding simplified strain energy term. The resulting element is intended for use in explicit dynamic codes (Goudreau and Hallquist, Computer Methods in Applied Mechanics and Engineering 1982; 33) where the use of quadratic tetrahedral elements can present significant difficulties. Copyright © 2001 John Wiley & Sons, Ltd.
We demonstrate the locking-free properties of the displacement formulation of p-finite elements when applied to nearly incompressible Neo-Hookean material under finite deformations. For an axisymmetric model problem we provide semi-analytical solutions for a nearly incompressible Neo-Hookean material exploited to investigate the robustness of p-FEM with respect to volumetric locking. An analytical solution for the incompressible case is also derived to demonstrate the convergence of the compressible numerical solution towards the incompressible case when the compression modulus is increased. Copyright © 2007 John Wiley & Sons, Ltd.
The now classical enhanced strain technique, employed with success for more than 10 years in solid, both 2D and 3D and shell finite elements, is here explored in a versatile 3D low-order element which is identified as HIS. The quest for accurate results in a wide range of problems, from solid analysis including near-incompressibility to the analysis of locking-prone beam and shell bending problems leads to a general 3D element. This element, put here to test in various contexts, is found to be suitable in the analysis of both linear problems and general non-linear problems including finite strain plasticity. The formulation is based on the enrichment of the deformation gradient and approximations to the shape function material derivatives. Both the equilibrium equations and their variation are completely exposed and deduced, from which internal forces and consistent tangent stiffness follow. A stabilizing term is included, in a simple and natural form. Two sets of examples are detailed: the accuracy tests in the linear elastic regime and several finite strain tests. Some examples involve finite strain plasticity. In both sets the element behaves very well, as is illustrated in numerous examples. Copyright © 2003 John Wiley & Sons, Ltd.
Some constitutive and computational aspects of finite deformation plasticity are discussed. Attention is restricted to multiplicative theories of plasticity, in which the deformation gradients are assumed to be decomposable into elastic and plastic terms. It is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non-symmetric even for associated plastic flow. Both explicit (i.e. no equilibrium iteration) and implicit finite element formulations are considered. An assumed strain formulation is used to accommodate the near-incompressibility associated with fully developed isochoric plastic flow. As an example of explicit integration, the rate tangent modulus method is reviewed in some detail. An implicit scheme is derived for which the consistent tangents, resulting in quadratic convergence of the equilibrium iterations, can be written out in closed form for arbitrary material models. All the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly. Examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method.
In the case of linear elasticity, a direct connection between the concept of reduced integration with hourglass stabilization and a mixed method can usually be established. In the non-linear case, this is in general not possible. To overcome this difficulty we suggest in this paper a new concept based on a Taylor expansion of the constitutively dependent quantities with respect to the centre of the element. The push-forward of the second (linear) term of the Taylor series for the first Piola–Kirchhoff stress tensor to the current configuration determines the so-called hourglass stabilization part of the residual force vector. Due to the fact that the element uses only one Gauss point and the hourglass stabilization part is computed by means of a simple functional evaluation, the present element technology is very efficient from the computational point of view.In contrast to the 2D case the computation of the Jacobi determinant only in the centre of the 3D element does not yield the correct volume, if the element shape deviates from being a parallelipiped. It is shown in the paper that the error becomes negligibly small for a relatively coarse discretization. The formulation is free of volumetric locking and can compete with shell formulations up to an aspect ratio of about hundred. For bending-dominated problems, at least two elements over the thickness are needed in order to compute the onset of plastification correctly. The element behaves very robustly in finite elasticity and inelasticity, also when large element distortions occur.
Computational aspects of a geometrically exact stress resultant model presented in Part I of this work are considered in detail. In particular, by exploiting the underlying geometric structure of the model, a configuration update procedure for the director (rotation) field is developed which is singularity free and exact regardless the magnitude of the director (rotation) increment. Our mixed finite element interpolation for the membrane, shear and bending fields presented in PartII of this work are extended to the finite deformation case. The exact linearization of the discrete form of the equilibrium equations is derived in closed form. The formulation is then illustrated by a comprehensive set of numerical experiments which include bifurcation and post-buckling response, we well as comparisons with closed form solutions and experimental results.
Improved three-dimensional tri-linear elements for finite deformation problems are developed based on an assumed enhanced strain methodology which, in the linear regime, incorporates the classical method of incompatible modes as a particular case. Three crucial modifications of a recently proposed element, which reduces to Wilson's brick in the linear regime, are introduced to prevent locking response in distorted configurations and to maintain proper rank, while preserving excellent performance in bending dominated and localization problems: (i) a modified quadrature rule; (ii) an additional enhancement of the divergence term; and (iii) a modification of the isoparametric shape function derivatives for the three-dimensional problem. Moreover, these modified shape function derivatives are shown to improve the performance of the standard tri-linear brick in distorted configurations. In addition, a strategy is described to circumvent the memory storage requirements in the static condensation procedure of the enhanced strain parameters. The excellent performance of the improved methodology is illustrated in representative numerical simulations.
We introduce a displacement-pressure () finite element formulation for the geometrically and materially nonlinear analysis of compressible and almost incompressible solids. The () formulation features the a priori replacement of the pressure computed from the displacement field by a separately interpolated pressure; this replacement is performed without reference to any specific material description. Considerations for incremental nonlinear analysis (including contact boundary conditions) are discussed and various () elements are studied. Numerical examples show the performance of the () formulation for two- and three-dimensional problems involving isotropic, orthotropic, rubber-like and elasto-plastic materials.
In the present contribution, an innovative brick element formulation for large deformation problems in finite elasticity is discussed. The new formulation can be considered as a reduced integration plus stabilization concept with the stabilization factors being computed on the basis of the enhanced strain method. Such an idea has not been applied yet in the context of large deformation 3D problems and leads to a surprisingly well-behaved locking-free element formulation. Crucial to the method is the notion of the so-called equivalent parallelepiped. The major advantages of this element technology are its simplicity and robustness. Since the element quantities are evaluated only in the center of the element, the approach is also very efficient from the numerical point of view.
This paper focuses on the treatment of volume constraints which in the context of elasto-plasticity typically arise as a result of assuming volume-preserving plastic flow. Projection methods based on the modification of the discrete gradient operator B, often proposed on an ad-hoc basis, are systematically obtained in the variational context furnished by a three-field Hu-Washizu principle. The fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume-preserving and dilatational parts, without relying on rate forms of the weak form of momentum balance. This approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient, and enables one to exactly enforce the condition of volume-preserving plastic flow. Within the framework proposed in this paper, rate forms and incrementally objective algorithms are entirely bypassed.
We develop and analyse a composite 'CT3D' tetrahedral element consisting of an ensemble of 12 four-node linear tetrahedral elements, coupled to a linear assumed deformation defined over the entire domain of the composite element. The element is designed to have well-defined lumped masses and contact tractions in dynamic contact problems while at the same time, minimizing the number of volume constraints per element. The relation between displacements and deformations is enforced weakly by recourse to the Hu-Washizu principle. The element arrays are formulated in accordance with the 'assumed-strain' prescription. The formulation of the element accounts for fully non-linear kinematics. Integrals over the domain of the element are computed by a five-point quadrature rule. The element passes the patch test in arbitrarily distorted configurations. Our numerical tests demonstrate that CT element has been found to possess a convergence rate comparable to those of linear simplicial elements, and that these convergence rates are maintained as the near-incompressible limit is approached. We have also verified that the element satisfies the Babuska-Brezzi condition for a regular mesh configuration. These tests suggest that the CT3D element can indeed be used reliably in calculations involving near-incompressible behaviour which arises, e.g., in the presence of unconfined plastic flow. Copyright (C)2001 John Wiley Sons, Ltd.
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B andF projection methods for nearly incompressible Copyright c
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Curved cantilever, four-element mesh
Figure 3. Curved cantilever, four-element mesh. Undeformed and final deformed shape.
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An averaged nodal deformation gradient linear tetrahedral element for large strain explicit dynamic applications
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