A new element‐by‐element method for trajectory calculations with tetrahedral finite element meshes

A new method to track massless particles in a three‐dimensional flow field is presented. The method is based on an element‐by‐element approach coupled with a predictor–corrector shooting scheme and does not use any time step. By analogy with time‐dependent schemes, the number of shootings is related to an equivalent number of time steps. The method has been implemented in a finite element framework using unstructured tetrahedral finite element meshes. However, it is general enough so that it can be implemented in finite difference and finite volume frameworks as well. It has been tested on a variety of flow systems namely: the Poiseuille flow in an empty circular pipe, the rotating flow in a stirred tank, the shear flow in a square tank and the flow through a static mixer. Accuracy has been found to depend on the accuracy of the velocity computation, the number of points per element and the level of mesh refinement. Copyright © 2006 John Wiley & Sons, Ltd.     

A three‐dimensional finite element method with arbitrary polyhedral elements

The ‘variable‐element‐topology finite element method’ (VETFEM) is a finite‐element‐like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement‐based formulation is also given, which effectively treats the case of near‐incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight‐node hex elements in a 3D finite‐deformation elastic–plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter. Copyright © 2006 John Wiley & Sons, Ltd.     

Bubble‐enhanced smoothed finite element formulation: a variational multi‐scale approach for volume‐constrained problems in two‐dimensional linear elasticity

This paper presents a bubble‐enhanced smoothed finite element formulation for the analysis of volume‐constrained problems in two‐dimensional linear elasticity. The new formulation is derived based on the variational multi‐scale approach in which unequal order displacement‐pressure pairs are used for the mixed finite element approximation and hierarchical bubble function is selected for the fine‐scale displacement approximation. An area‐weighted averaging scheme is employed for the two‐scale smoothed strain calculation under the framework of edge‐based smoothed FEM. The smoothed fine‐scale solution is shown to naturally contain the stress field jump of the smoothed coarse‐scale solution across the boundary of edge‐based smoothing domain and thus provides the possibility to stabilize the global solution for volume‐constrained problems. A global monolithic solution strategy is employed, and the fine‐scale solution is solved without the consideration of approximating the strong form of the fine‐scale equation. Several numerical examples are analyzed to demonstrate the accuracy of the present formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

Three‐dimensional meshfree‐enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites

A three‐dimensional microstructure‐based finite element framework is presented for modeling the mechanical response of rubber composites in the microscopic level. This framework introduces a novel finite element formulation, the meshfree‐enriched FEM, to overcome the volumetric locking and pressure oscillation problems that normally arise in the numerical simulation of rubber composites using conventional displacement‐based FEM. The three‐dimensional meshfree‐enriched FEM is composed of five‐noded tetrahedral elements with a volume‐weighted smoothing of deformation gradient between neighboring elements. The L₂‐orthogonality property of the smoothing operator enables the employed Hu–Washizu–de Veubeke functional to be degenerated to an assumed strain method, which leads to a displacement‐based formulation that is easily incorporated with the periodic boundary conditions imposed on the unit cell. Two numerical examples are analyzed to demonstrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Hybrid finite element/volume method for shallow water equations

A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.     

A face‐centred finite volume method for second‐order elliptic problems

This work proposes a novel finite volume paradigm, ie, the face‐centred finite volume (FCFV) method. Contrary to the popular vertex and cell‐centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in two dimensions) to construct locally conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin formulation with constant degree of approximation, and thus inheriting the convergence properties of the classical hybridisable discontinuous Galerkin. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element‐by‐element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived, and numerical evidence of optimal convergence in two dimensions and three dimensions is provided. Numerical examples are presented to illustrate the accuracy, efficiency, and robustness of the proposed methodology. The results show that, contrary to other finite volume methods, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy, and robustness using simplicial elements, facilitating its application to problems involving complex geometries in three dimensions.     

Polyhedral elements by means of node/edge‐based smoothed finite element method

The node‐based or edge‐based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces, and so retain a good geometric adaptability. The strain smoothing technique and implicit shape functions based on the linear point interpolation make the element formulation simple and straightforward. The resulting polyhedral elements are free from the excessive zero‐energy modes and yield a robust solution very much insensitive to mesh distortion. Several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior. The smoothed finite element method‐based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.     

The finite element method for the mechanically based model of non‐local continuum

In this paper the finite element method (FEM) for the mechanically based non‐local elastic continuum model is proposed. In such a model each volume element of the domain is considered mutually interacting with the others, beside classical interactions involved by the Cauchy stress field, by means of central body forces that are monotonically decreasing with their inter‐distance and proportional to the product of the interacting volume elements. The constitutive relations of the long‐range interactions involve the product of the relative displacement of the centroids of volume elements by a proper, distance‐decaying function, which accounts for the decrement of the long‐range interactions as long as distance increases. In this study, the elastic problem involving long‐range central interactions for isotropic elastic continuum will be solved with the aid of the FEM. The accuracy of the solution obtained with the proposed FEM code is compared with other solutions obtained with Galerkins' approximation as well as with finite difference method. Moreover, a parametric study regarding the effect of the material length scale in the mechanically based model and in the Kr”oner–Eringen non‐local elasticity has been investigated for a plane elasticity problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element formulation for modelling large deformations in elasto‐viscoplastic polycrystals

Anisotropic, elasto‐viscoplastic behaviour in polycrystalline materials is modelled using a new, updated Lagrangian formulation based on a three‐field form of the Hu‐Washizu variational principle to create a stable finite element method in the context of nearly incompressible behaviour. The meso‐scale is characterized by a representative volume element, which contains grains governed by single crystal behaviour. A new, fully implicit, two‐level, backward Euler integration scheme together with an efficient finite element formulation, including consistent linearization, is presented. The proposed finite element model is capable of predicting non‐homogeneous meso‐fields, which, for example, may impact subsequent recrystallization. Finally, simple deformations involving an aluminium alloy are considered in order to demonstrate the algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

Nonlinear rotation‐free three‐node shell finite element formulation

A new triangular thin‐shell finite element formulation is presented, which employs only translational degrees of freedom. The formulation allows for large deformations, and it is based on the nonlinear Kirchhoff thin‐shell theory. A number of static and dynamic test problems are considered for which analytical or benchmark solutions exist. Comparisons between the predictions of the new model and these solutions show that the new model accurately reproduces complex nonlinear analytical solutions as well as solutions obtained using existing, more complex finite element formulations. Copyright © 2013 John Wiley & Sons, Ltd.     

A Lagrangian cell‐centred finite volume method for metal forming simulation

The current article presents a Lagrangian cell‐centred finite volume solution methodology for simulation of metal forming processes. Details are given of the mathematical model in updated Lagrangian form, where a hyperelastoplastic J₂ constitutive relation has been employed. The cell‐centred finite volume discretisation is described, where a modified discretised is proposed to alleviate erroneous hydrostatic pressure oscillations; an outline of the memory efficient segregated solution procedure is given. The accuracy and order of accuracy of the method are examined on a number of 2‐D and 3‐D elastoplastic benchmark test cases, where good agreement with available analytical and finite element solutions is achieved. Copyright © 2016 John Wiley & Sons, Ltd.     

A vertex‐based finite volume method applied to non‐linear material problems in computational solid mechanics

A vertex‐based finite volume (FV) method is presented for the computational solution of quasi‐static solid mechanics problems involving material non‐linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two‐ and three‐dimensional element types. A detailed comparison between the vertex‐based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. Copyright © 2002 John Wiley & Sons, Ltd.     

Frame‐indifferent beam finite elements based upon the geometrically exact beam theory

This paper describes a new beam finite element formulation based upon the geometrically exact beam theory. In contrast to many previously proposed beam finite element formulations the present discretization approach retains the frame‐indifference (or objectivity) of the underlying beam theory. The space interpolation of rotational degrees‐of‐freedom is circumvented by the introduction of a reparameterization of the weak form corresponding to the equations of motion of the geometrically exact beam theory. Copyright © 2002 John Wiley & Sons, Ltd.     

Energy and momentum conserving elastodynamics of a non‐linear brick‐type mixed finite shell element

The paper presents aspects of the finite element formulation of momentum and energy conserving algorithms for the non‐linear dynamic analysis of shell‐like structures. The key contribution is a detailed analysis of the implementation of a Simó–Tarnow‐type conservation scheme in a recently developed new mixed finite shell element. This continuum‐based shell element provides a well‐defined interface to strain‐driven constitutive stress updates algorithms. It is based on the classic brick‐type trilinear displacement element and is equipped with specific gradient‐type enhanced strain modes and shell‐typical assumed strain modifications. The excellent performance of the proposed dynamic shell formulation with respect to conservation properties and numerical stability behaviour is demonstrated by means of three representative numerical examples of elastodynamics which exhibit complex free motions of flexible structures undergoing large strains and large rigid‐body motions. Copyright © 2001 John Wiley & Sons, Ltd.     

Use of higher‐order shape functions in the scaled boundary finite element method

The scaled boundary finite element method is a novel semi‐analytical technique, whose versatility, accuracy and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using higher‐order polynomial functions for the shape functions. Two techniques for generating the higher‐order shape functions are investigated. In the first, the spectral element approach is used with Lagrange interpolation functions. In the second, hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the p‐version of the finite element method. To check the accuracy of the proposed procedures, a plane strain problem for which an exact solution is available is employed. A more complex example involving three scaled boundary subdomains is also addressed. The rates of convergence of these examples under p‐refinement are compared with the corresponding rates of convergence achieved when uniform h‐refinement is used, allowing direct comparison of the computational cost of the two approaches. The results show that it is advantageous to use higher‐order elements, and that higher rates of convergence can be obtained using p‐refinement instead of h‐refinement. Copyright © 2005 John Wiley & Sons, Ltd.     

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

A new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ?? which encloses a finite computational domain Ω. On ?? a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ??, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C⁰ finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.     

A class of variational strain‐localization finite elements

We present a class of finite elements for capturing sub‐grid localization processes such as shear bands and void sheets. The elements take the form of a double surface and deform in accordance with an arbitrary constitutive law. In particular they allow for the development of displacement and velocity jumps across volume element boundaries. The thickness of the localized zone is set by an additional field variable which is determined variationally. The localization elements are inserted, and become active, only when localized deformations become energetically favourable. The implementation presented in this work is three‐dimensional and allows for finite deformations. The versatility and predictive ability of the method are demonstrated through a simple shear test and the simulation of the dynamic impact of a pre‐notched C300 steel sample by a steel projectile. Copyright © 2004 John Wiley & Sons, Ltd.     

Extended finite element method coupled with face‐based strain smoothing technique for three‐dimensional fracture problems

In this work, the extended finite element method (XFEM) is for the first time coupled with face‐based strain‐smoothing technique to solve three‐dimensional fracture problems. This proposed method, which is called face‐based smoothed XFEM here, is expected to combine both the advantages of XFEM and strain‐smoothing technique. In XFEM, arbitrary crack geometry can be modeled and crack advance can be simulated without remeshing. Strain‐smoothing technique can eliminate the integration of singular term over the volume around the crack front, thanks to the transformation of volume integration into area integration. Special smoothing scheme is implemented in the crack front smoothing domain. Three examples are presented to test the accuracy, efficiency, and convergence rate of the face‐based smoothed XFEM. From the results, it is clear that smoothing technique can improve the performance of XFEM for three‐dimensional fracture problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A finite element semi‐Lagrangian method with L2 interpolation

High‐order accurate methods for convection‐dominated problems have the potential to reduce the computational effort required for a given order of solution accuracy. The state of the art in this field is more advanced for Eulerian methods than for semi‐Lagrangian (SLAG) methods. In this paper, we introduce a new SLAG method that is based on combining the modified method of characteristics with a high‐order interpolating procedure. The method employs the finite element method on triangular meshes for the spatial discretization. An L² interpolation procedure is developed by tracking the feet of the characteristic lines from the integration nodes. Numerical results are illustrated for a linear advection–diffusion equation with known analytical solution and for the viscous Burgers’ equation. The computed results support our expectations for a robust and highly accurate finite element SLAG method. Copyright © 2012 John Wiley & Sons, Ltd.     

Edge‐based finite element techniques for non‐linear solid mechanics problems

Edge‐based data structures are used to improve computational efficiency of inexact Newton methods for solving finite element non‐linear solid mechanics problems on unstructured meshes. Edge‐based data structures are employed to store the stiffness matrix coefficients and to compute sparse matrix–vector products needed in the inner iterative driver of the inexact Newton method. Numerical experiments on three‐dimensional plasticity problems have shown that memory and computer time are reduced, respectively, by factors of 4 and 6, compared with solutions using element‐by‐element storage and matrix–vector products. Copyright © 2001 John Wiley & Sons, Ltd.