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.     

Modification of the quadratic 10‐node tetrahedron for thin structures and stiff materials under large‐strain hyperelastic deformation

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 10‐node tetrahedron for applications to linear elasticity and nonlinear deformation. The technique relies on stabilization by 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.     

A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations

This paper presents a variational multiscale residual-based stabilized finite element method for the incompressible Navier–Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multiscale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska–Brezzi (inf–sup) condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressure-velocity elements comprising 4-and 10-node tetrahedral elements and 8- and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the lid-driven cavity flow problem.     

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Extension to nearly incompressible implicit and explicit elastodynamics in finite strains

We present a novel unified finite element framework for performing computationally efficient large strain implicit and explicit elastodynamic simulations using triangular and tetrahedral meshes that can be generated using the existing mesh generators. For the development of a unified framework, we use Bézier triangular and tetrahedral elements that are directly amenable for explicit schemes using lumped mass matrices and employ a mixed displacement-pressure formulation for dealing with the numerical issues arising due to volumetric and shear locking. We demonstrate the accuracy of the proposed scheme by studying several challenging benchmark problems in finite strain elastostatics and nonlinear elastodynamics modelled with nearly incompressible hyperelastic and von Mises elastoplastic material models. We show that Bézier elements, in combination with the mixed formulation, help in developing a simple unified finite element formulation that is accurate, robust, and computationally very efficient for performing a wide variety of challenging nonlinear elastostatic and implicit and explicit elastodynamic simulations.     

Remeshing strategies for large deformation problems with frictional contact and nearly incompressible materials

We present a framework to efficiently solve large deformation contact problems with nearly incompressible materials by implementing adaptive remeshing. 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 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‐dimensional or three‐dimensional analysis and are demonstrated to be robust by a number of numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

F‐bar aided edge‐based smoothed finite element method using tetrahedral elements for finite deformation analysisof nearly incompressible solids

A new smoothed finite element method (S‐FEM) with tetrahedral elements for finite strain analysis of nearly incompressible solids is proposed. The proposed method is basically a combination of the F‐bar method and edge‐based S‐FEM with tetrahedral elements (ES‐FEM‐T4) and is named ‘F‐barES‐FEM‐T4’. F‐barES‐FEM‐T4 inherits the accuracy and shear locking‐free property of ES‐FEM‐T4. At the same time, it also inherits the volumetric locking‐free property of the F‐bar method. The isovolumetric part of the deformation gradient ( F ^iso) is derived from the F of ES‐FEM‐T4, whereas the volumetric part ( F ^vol) is derived from the cyclic smoothing of J(=det( F )) between elements and nodes. Some demonstration analyses confirm that F‐barES‐FEM‐T4 with a sufficient number of cyclic smoothings suppresses the pressure oscillation in nearly incompressible materials successfully with no increase in DOF. Moreover, they reveal that our method is capable of relaxing the corner locking issue arising at the corner in the cylinder barreling analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics

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.     

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.     

Topology optimization with incompressible materials under small and finite deformations using mixed u/p elements

This paper focuses on topology optimization utilizing incompressible materials under both small‐ and finite‐deformation kinematics. To avoid the volumetric locking that accompanies incompressibility, linear and nonlinear mixed displacement/pressure (u/p) elements are utilized. A number of material interpolation schemes are compared, and a new scheme interpolating both Young's modulus and Poisson's ratio (E ‐ν interpolation) is proposed. The efficacy of this proposed scheme is demonstrated on a number of examples under both small‐ and finite‐deformation kinematics. Excessive mesh distortions that may occur under finite deformations are dealt with by extending a linear energy interpolation approach to the nonlinear u/p formulation and utilizing an adaptive update strategy. The proposed optimization framework is demonstrated to be effective through a number of representative examples.     

A new finite element approach for solving three‐dimensional problems using trimmed hexahedral elements

In this paper, a novel finite element approach is presented to solve three‐dimensional problems using trimmed hexahedral elements generated by cutting a simple block consisting of regular hexahedral elements with a computer‐aided design (CAD) surface. Trimmed hexahedral elements, which are polyhedral elements with curved faces, are placed at the boundaries of finite element models, and regular hexahedral elements remain in the interior regions. Shape functions for trimmed hexahedral elements are developed by using moving least square approximation with harmonic weight functions based on an extension of Wachspress coordinates to curved faces. A subdivision of polyhedral domains into tetrahedral sub‐domains is performed to construct shape functions for trimmed hexahedral elements, and numerical integration of the weak form can be carried out consistently over the tetrahedral sub‐domains. Trimmed hexahedral elements have similar properties to conventional finite elements regarding the continuity, the completeness, the node–element connectivity, and the inter‐element compatibility. Numerical examples for three‐dimensional linear elastic problems with complex geometries show the efficiency and effectiveness of the present method. Copyright © 2015 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.     

Hybrid hexahedral element for solids,plates, shells and beams by selective scaling

A robust two-field hexahedral element capable of handling plate/shell, beam and nearly incompressible material analyses without locking are presented. Starting with the assumed stress element of Pian and Tong,⁷ parasitic strain components leading to locking in plate, shell and beam analyses are first identified. Locking can be alleviated by scaling down selectively the parasitic strain components in the leverage matrix. Unfortunately, the element then fails the patch test. However, patch test correction and reduction in computation can be achieved by the recently proposed admissible matrix formulation. The resulting element is lock-free and very efficient. All matrices involved in constructing the stiffness matrix can be derived explicitly. The accuracy of the element is tested by popular bench-mark problems.     

Reliable second‐order hexahedral elements for explicit methods in nonlinear solid dynamics

Second‐order hexahedral elements are common in static and implicit dynamic finite element codes for nonlinear solid mechanics. Although probably not as popular as first‐order elements, they can perform better in many circumstances, particularly for modeling curved shapes and bending without artificial hourglass control or incompatible modes. Nevertheless, second‐order brick elements are not contained in typical explicit solid dynamic programs and unsuccessful attempts to develop reliable ones have been reported. In this paper, 27‐node formulations, one for compressible and one for nearly incompressible materials, are presented and evaluated using non‐uniform row summation mass lumping in a wide range of nonlinear example problems. The performance is assessed in standard benchmark problems and in large practical applications using various hyperelastic and inelastic material models and involving very large strains/deformations, severe distortions, and contact‐impact. Comparisons are also made with several first‐order elements and other second‐order hexahedral formulations. The offered elements are the only second‐order ones that performed satisfactorily in all examples, and performed generally at least as well as mass lumped first‐order bricks. It is shown that the row summation lumping is vital for robust performance and selection of Lagrange over serendipity elements and high‐order quadrature rules are more crucial with explicit than with static/implicit methods. Whereas the reliable performance is frequently attained at significant computational expense compared with some first‐order brick types, these elements are shown to be computationally competitive in flexure and with other first‐order elements. These second‐order elements are shown to be viable for large practical applications, especially using today's parallel computers. Published in 2010 by John Wiley & Sons, Ltd.     

Finite element computation of sloshing modes in containers with elastic baffle plates

A finite element method to approximate the vibration modes of a plate in contact with an incompressible fluid is analysed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, discretized by standard piecewise linear tetrahedral finite elements. Gravity waves on the free surface of the liquid are considered in the model. The plate is modelled by Reissner–Mindlin equations discretized by MITC3 locking‐free elements. Implementation issues are discussed and numerical experiments are presented. In particular, the method is compared with analytical approximations and with an experimental study which has been recently reported. Copyright © 2002 John Wiley & Sons, Ltd.     

Efficient analysis of transient heat transfer problems exhibiting sharp thermal gradients

In this paper, heat transfer problems with sharp spatial gradients are analyzed using the Generalized Finite Element Method with global-local enrichment functions (GFEM ^gl). With this approach, scale-bridging enrichment functions are generated on the fly, providing specially-tailored enrichment functions for the problem to be analyzed with no a-priori knowledge of the exact solution. In this work, a decomposition of the linear system of equations is formulated for both steady-state and transient heat transfer problems, allowing for a much more computationally efficient analysis of the problems of interest. With this algorithm, only a small portion of the global system of equations, i.e., the hierarchically added enrichments, need to be re-computed for each loading configuration or time-step. Numerical studies confirm that the condensation scheme does not impact the solution quality, while allowing for more computationally efficient simulations when large problems are considered. We also extend the GFEM ^gl to allow for the use of hexahedral elements in the global domain, while still using tetrahedral elements in the local domain, to allow for automatic localized mesh refinement without the use of constrained approximations. Simulations are run with the use of linear and quadratic hexahedral and tetrahedral elements in the global domain. Convergence studies indicate that the use of a different partition of unity (PoU) in the global (hexahedral elements) and local (tetrahedral elements) domains does not adversely impact the solution quality.     

Triangles and tetrahedra in explicit dynamic codes for solids

Explicit dynamic codes which are used currently for the study of plastic deformations in impact, or with some modification for metal forming, suffer two serious limitations. First, only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing. Second, even with the use of such elements, special devices such as reduced integration must be introduced to avoid locking and reduce costs. These necessitate complex hour glass control, mending-type procedures. The main difficulties are those due to the need of treatment of (almost) incompressible deformation modes. Recently, similar difficulties have been overcome in the context of fluid dynamics soil dynamics and we show here how the processes introduced there can be adopted effectively to the present problem, thus allowing an almost unrestricted choice of element interpolations. © 1998 John Wiley & Sons, Ltd.     

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Many finite elements exhibit the so‐called ‘volumetric locking’ in the analysis of incompressible or quasi‐incompressible problems.In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective–diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi‐implicit and explicit 2D and 3D non‐linear transient dynamic analysis of an impact problem and a bulk forming process are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Non‐linear analysis of structures using high performance hybrid elements

In this work, we describe the formulation and implementation for stress‐based hybrid elements for conducting non‐linear analysis of elastic structures. The motivation behind developing these elements is that they should be as simple to use as standard displacement‐based isoparametric brick elements, but at the same time, be relatively immune to the shortcoming that these elements suffer from, namely, ‘locking’ problems which occur when they are used to model plate/shell geometries, almost incompressible materials or when the elements are distorted, and so on. The formulation is based on a two‐field mixed variational principle. Numerical examples are presented to demonstrate the excellent performance of the proposed elements on a variety of challenging problems involving very large deformations, buckling, mesh distortions, almost incompressible materials, etc. Copyright © 2006 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.     

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.