P1‐nonconforming quadrilateral finite element for topology optimization

This investigation focuses on an alternative approach to topology optimization problems involving incompressible materials using the P1‐nonconforming finite element. Instead of using the mixed displacement‐pressure formulation, a pure displacement‐based approach can be employed for finite element formulation owing to the Poisson locking‐free property of the P1‐nonconforming element. Moreover, because the P1‐nonconforming element has linear shape functions that are defined at element vertices, it has considerably fewer degrees of freedom than other quadrilateral nonconforming elements and its implementation is as simple as that of the conforming bilinear element. Various problems dealing with incompressible materials and pressure‐loaded structures found in published works are solved to verify the applicability of the proposed method. The application of the method is extended to the optimal design of fluid channels in the Stokes flow. This is done by expressing pressure in terms of volumetric strain rates and developing a velocity‐field‐only finite element formulation. The optimization results obtained from all the problems considered in this study are in close agreement with those found in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

On finite element formulations for nearly incompressible linear elasticity

In this paper we present a mixed stabilized finite element formulation that does not lock and also does not exhibit unphysical oscillations near the incompressible limit. The new mixed formulation is based on a multiscale variational principle and is presented in two different forms. In the first form the displacement field is decomposed into two scales, coarse-scale and fine-scale, and the fine-scale variables are eliminated at the element level by the static condensation technique. The second form is obtained by simplifying the first form, and eliminating the fine-scale variables analytically yet retaining their effect that results with additional (stabilization) terms. We also derive, in a consistent manner, an expression for the stabilization parameter. This derivation also proves the equivalence between the classical mixed formulation with bubbles and the Galerkin least-squares type formulations for the equations of linear elasticity. We also compare the performance of this new mixed stabilized formulation with other popular finite element formulations by performing numerical simulations on three well known test problems.     

Locking‐free continuum displacement finite elements with nodal integration

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.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

Assumed‐deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis

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.     

A uniform nodal strain tetrahedron with isochoric stabilization

A stabilized node‐based uniform strain tetrahedral element is presented and analyzed for finite deformation elasticity. The element is based on linear interpolation of a classical displacement‐based tetrahedral element formulation but applies nodal averaging of the deformation gradient to improve mechanical behavior, especially in the regime of near‐incompressibility where classical linear tetrahedral elements perform very poorly. This uniform strain approach adopted here exhibits spurious modes as has been previously reported in the literature. We present a new type of stabilization exploiting the circumstance that the instability in the formulation is related to the isochoric strain energy contribution only and we therefore present a stabilization based on an isochoric–volumetric splitting of the stress tensor. We demonstrate that by stabilizing the isochoric energy contributions only, reintroduction of volumetric locking through the stabilization can be avoided. The isochoric–volumetric splitting can be applied for all types of materials with only minor restrictions and leads to a formulation that demonstrates impressive performance in examples provided. Copyright © 2008 John Wiley & Sons, Ltd.     

A three-field formulation for incompressible viscoelastic fluids

This paper presents a new stabilized finite element method for incompressible viscoelastic fluids. A three-field formulation is developed wherein Oldroyd-B model is coupled with the mass and momentum conservation equations for an incompressible viscous fluid. The variational multiscale (VMS) framework is employed to develop a stabilized formulation for the coupled momentum, continuity and stress equations. Based on the new stabilized method a family of linear and higher-order triangle and quadrilateral elements with equal-order velocity-pressure-stress fields is developed. Stability and convergence property of the various elements is studied and optimal rates are attained in the norms considered. The method is applied to some benchmark problems and accuracy and computational economy of the formulation is investigated for various flow conditions.     

Efficient and accurate multilayer solid‐shell element: non‐linear materials at finite strain

We present in this paper an efficient and accurate low‐order solid‐shell element formulation for analyses of large deformable multilayer shell structures with non‐linear materials. The element has only displacement degrees of freedom (dofs), and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out‐of‐plane bending) and to remedy volumetric locking. Based on the mixed Fraeijs de Veubeke‐Hu‐Washizu (FHW) variational principle, the in‐plane and out‐of‐plane bending behaviours are improved and the locking associated with (nearly) incompressible materials is avoided via a new efficient enhancement of strain tensor. Shear locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. Two non‐linear 3‐D constitutive models (Mooney–Rivlin material and hyperelastoplastic material at finite strain) are applied directly without requiring the enforcement of the plane‐stress assumption. In particular, we give a simple derivation for the hyperelastoplastic model using spectral representations. In addition, the present element has a well‐defined lumped mass matrix, and provides double‐side contact surfaces for shell contact problems. With the dynamics referred to a fixed inertial frame, the present element can be used to analyse multilayer shell structures undergoing large overall motion. Numerical examples involving static analyses and implicit/explicit dynamic analyses of multilayer shell structures with both material and geometric non‐linearities are presented, and compared with existing results obtained from other shell elements and from a meshless method. It is shown that elements that did not pass the out‐of‐plane bending patch test could not provide accurate results, as compared to the present element formulation, which passed the out‐of‐plane bending patch test. The present element proves to be versatile and efficient in the modelling and analyses of general non‐linear composite multilayer shell structures. Copyright © 2005 John Wiley & Sons, Ltd.     

On using linear elements in incompressible plane strain problems: a simple edge based approach for triangles

In this paper a simple iterative method is presented for finite element solution of incompressible plane strain problems using linear elements. Instead of using a mixed formulation approach, we use an equivalent displacement/velocity approach in an iterative manner. Control volumes are taken for regions which are to exhibit incompressible behaviour. For triangular elements the control volume is chosen as the area built on the parts of each pair of elements at the sides of an edge. In this case, elements are let to exchange volume. It is shown that the proposed edge based approach removes the deficiency of the linear triangular elements i.e. locking effect. Similar edge based approach is applied to the linear quadrilateral elements. However, if the control volume is chosen as the element volume the formulation gives similar results as the discontinuous mixed formulation using one pressure point without exhibiting instability behaviour. The formulation is based on decomposition of the displacement/velocity field into deviatoric and volumetric parts. The volumetric part is iteratively eliminated without confronting locking or instability phenomenon. The iterative procedure is very cheap and simple to be implemented in any FEM code. Several examples are given to demonstrate the performance of the procedure. Copyright © 2004 John Wiley & Sons, Ltd.     

Patch‐averaged assumed strain finite elements for stress analysis

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, 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.     

Stabilized finite element method for viscoplastic flow: formulation with state variable evolution

A stabilized, mixed finite element formulation for modelling viscoplastic flow, which can be used to model approximately steady‐state metal‐forming processes, is presented. The mixed formulation is expressed in terms of the velocity, pressure and state variable fields, where the state variable is used to describe the evolution of the material's resistance to plastic flow. The resulting system of equations has two sources of well‐known instabilities, one due to the incompressibility constraint and one due to the convection‐type state variable equation. Both of these instabilities are handled by adding mesh‐dependent stabilization terms, which are functions of the Euler–Lagrange equations, to the usual Galerkin method. Linearization of the weak form is derived to enable a Newton–Raphson implementation into an object‐oriented finite element framework. A progressive solution strategy is used for improving convergence for highly non‐linear material behaviour, typical for metals. Numerical experiments using the stabilization method with hierarchic shape functions for the velocity, pressure and state variable fields in viscoplastic flow and metal‐forming problems show that the stabilized finite element method is effective and efficient for non‐linear steady forming problems. Finally, the results are discussed and conclusions are inferred. Copyright © 2002 John Wiley & Sons, Ltd.     

A simple,stable, and accurate linear tetrahedral finite element for transient,nearly, and fully incompressible solid dynamics: a dynamic variational multiscale approach

We propose a new approach for the stabilization of linear tetrahedral finite elements in the case of nearly incompressible transient solid dynamics computations. Our method is based on a mixed formulation, in which the momentum equation is complemented by a rate equation for the evolution of the pressure field, approximated with piecewise linear, continuous finite element functions. The pressure equation is stabilized to prevent spurious pressure oscillations in computations. Incidentally, it is also shown that many stabilized methods previously developed for the static case do not generalize easily to transient dynamics. Extensive tests in the context of linear and nonlinear elasticity are used to corroborate the claim that the proposed method is robust, stable, and accurate. Copyright © 2015 John Wiley & Sons, Ltd.     

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

This paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.     

Stabilization of mixed tetrahedral elements at large deformations

This paper presents stabilized mixed finite element formulations for tetrahedral elements at large deformations using volume and area bubble functions. To this end, the corresponding weak formulations are derived for the standard two‐field method, the method of incompatible modes and the enhanced strain method. Then, the weak formulations will be linearized. Furthermore, the matrix formulations for the weak formulations and its linearizations are summarized. The numerical results for incompressible rubber‐like materials using a Neo‐Hookean material law show the locking‐free performance and the drastic damping of the stresses for the new stabilized tetrahedral elements in finite deformation problems. This paper is an extension of the works published by the authors regarding small deformation problems for linear elasticity and plasticity. Copyright © 2011 John Wiley & Sons, Ltd.     

Displacement/pressure mixed interpolation in the method of finite spheres

The displacement‐based formulation of the method of finite spheres is observed to exhibit volumetric ‘locking’ when incompressible or nearly incompressible deformations are encountered. In this paper, we present a displacement/pressure mixed formulation as a solution to this problem. We analyse the stability and optimality of the formulation for several discretization schemes using numerical inf–sup tests. Issues concerning computational efficiency are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Scaled boundary finite element method for compressible and nearly incompressible elasticity over arbitrary polytopes

In this paper, a purely displacement-based formulation is presented within the framework of the scaled boundary finite element method to model compressible and nearly incompressible materials. A selective reduced integration technique combined with an analytical treatment in the nearly incompressible limit is employed to alleviate volumetric locking. The stiffness matrix is computed by solving the scaled boundary finite element equation. The salient feature of the proposed technique is that it neither requires a stabilization parameter nor adds additional degrees of freedom to handle volumetric locking. The efficiency and the robustness of the proposed approach is demonstrated by solving various numerical examples in two and three dimensions.     

Smooth finite element methods: Convergence,accuracy and properties

A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi‐equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one‐subcell element with a quasi‐equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one‐cell smoothed four‐noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

A super‐element for crack analysis in the time domain

A super‐element for the dynamic analysis of two‐dimensional crack problems is developed based on the scaled boundary finite‐element method. The boundary of the super‐element containing a crack tip is discretized with line elements. The governing partial differential equations formulated in the scaled boundary co‐ordinates are transformed to ordinary differential equations in the frequency domain by applying the Galerkin's weighted residual technique. The displacements in the radial direction from the crack tip to a point on the boundary are solved analytically without any a priori assumption. The scaled boundary finite‐element formulation leads to symmetric static stiffness and mass matrices. The super‐element can be coupled seamlessly with standard finite elements. The transient response is evaluated directly in the time domain using a standard time‐integration scheme. The stress field, including the singularity around the crack tip, is expressed semi‐analytically. The stress intensity factors are evaluated without directly addressing singular functions, as the limit in their definitions is performed analytically. Copyright © 2004 John Wiley & Sons, Ltd.     

A discrete splitting finite element method for numerical simulations of incompressible Navier–Stokes flows

The presence of the pressure and the convection terms in incompressible Navier–Stokes equations makes their numerical simulation a challenging task. The indefinite system as a consequence of the absence of the pressure in continuity equation is ill‐conditioned. This difficulty has been overcome by various splitting techniques, but these techniques incur the ambiguity of numerical boundary conditions for the pressure as well as for the intermediate velocity (whenever introduced). We present a new and straightforward discrete splitting technique which never resorts to numerical boundary conditions. The non‐linear convection term can be treated by four different approaches, and here we present a new linear implicit time scheme. These two new techniques are implemented with a finite element method and numerical verifications are made. Copyright © 2005 John Wiley & Sons, Ltd.