A new stabilized finite element method for reaction–diffusion problems: The source‐stabilized Petrov–Galerkin method

This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well‐known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one‐dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi‐dimensional case and for non‐uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi‐dimensional problems. For the one‐dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright © 2008 John Wiley & Sons, Ltd.     

Dispersion analysis of stabilized finite element methods for acoustic fluid interaction with Reissner–Mindlin plates

The application of stabilized finite element methods to model the vibration of elastic plates coupled with an acoustic fluid medium is considered. A complex‐wavenumber dispersion analysis of acoustic fluid interaction with Reissner–Mindlin plates is performed to quantify the accuracy of stabilized finite element methods for fluid‐loaded plates. Results demonstrate the improved accuracy of a recently developed hybrid least‐squares (HLS) plate element based on a modified Hellinger–Reissner functional, consistently combined with residual‐based methods for the acoustic fluid, compared to standard Galerkin and Galerkin gradient least‐squares plate elements. The technique of complex wavenumber dispersion analysis is used to examine the accuracy of the discretized system in the representation of free waves for fluid‐loaded plates. The influence of fluid and coupling matrices resulting from consistent implementation of pressure loading in the residual for the plate equation is examined and clarified for the different finite element approximations. Copyright © 2001 John Wiley & Sons, Ltd.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.     

Homogenized free surface flow in porous media for wet‐out processing

This paper presents a novel porous media model for homogenized free surface flow, representing wet‐out composites processing. The model is derived from concepts of homogenization applied to a compressible two‐phase flow, accounting for capillary effects and the concept of relative permeability. Based on mass balance considerations, we obtain a nonlinear set of equations of convection‐diffusion type involving the mixture (fluid) pressure and the degree of saturation as primary fields. A staggered Galerkin finite element approach is employed to decouple the solution. Moreover, the streamline upwind/Petrov‐Galerkin technique is applied to attenuate the oscillations in the saturation solutions. The model accuracy and convergence of the finite element solutions are demonstrated through 1‐dimensional and 2‐dimensional examples, representing resin transfer molding flow processes.     

Eulerian elasto‐visco‐plastic formulations for residual stress prediction

A stabilized, Galerkin finite element formulation for modeling the elasto‐visco‐plastic response of quasi‐steady‐state processes, such as welding, laser surfacing, rolling and extrusion, is presented in an Eulerian frame. The mixed formulation consists of four field variables, such as velocity, stress, deformation gradient and internal variable, which is used to describe the evolution of the material's resistance to plastic flow. The streamline upwind Petrov–Galerkin method is used to eliminate spurious oscillations, which may be caused by the convection‐type of stress, deformation gradient and internal variable evolution equations. A progressive solution strategy is introduced to improve the convergence of the Newton–Raphson solution procedure. Two two‐dimensional numerical examples are implemented to verify the accuracy of the Eulerian formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

A fully coupled finite element analysis of water‐table fluctuation and land deformation in partially saturated soils due to surface loading

A fully coupled numerical model is presented for the water‐table fluctuation and land deformation in partially saturated soils due to surface loading. This numerical model is developed based on the poroelastic governing equations for groundwater flow in deforming variably saturated porous media and the Galerkin finite element method. The numerical model is verified and validated against a one‐dimensional consolidation problem concerning surface loading on a soil column which has six different initial water‐table elevations. The numerical model is then applied to a two‐dimensional consolidation problem of surface loading on a partially saturated soil at a construction site. Results from the numerical simulations of both problems show that the water table fluctuates in the partially saturated soils, and the unsaturated zone above the water table has significant effects on the consolidation behaviour of the partially saturated soils under surface loading. Such effects are caused by the permanent absorption of a portion of the mechanical loading stress and the weak hydromechanical coupling between the solid skeleton deformation field and the groundwater flow field in the unsaturated zone due to its partial saturation. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element modelling of saturated porous media at finite strains under dynamic conditions with compressible constituents

Two finite element formulations are proposed to analyse the dynamic conditions of saturated porous media at large strains with compressible solid and fluid constituents. Unlike similar works published in the literature, the proposed formulations are based on a recently proposed hyperelastic framework in which the compressibility of the solid and fluid constituents is fully taken into account when geometrical non‐linear effects are relevant on both micro‐ and macroscales. The first formulation leads to a three‐field finite element method (FEM), which is suitable for analysing high‐frequency dynamic problems, whereas the second is a simplification of the first, leading to a two‐field FEM, in which some inertial effects of the pore fluid are disregarded, hence the second formulation is suitable for studying low‐frequency problems. A fully Lagrangian approach is considered, hence all terms are expressed with reference to the material setting; the balance equations for the pore fluid are also expressed in terms of the chemical potential and the mass flux of the pore fluid in order to take the compressibility of the fluid into account. To improve the numerical response in the case of wave propagation, a discontinuous Galerkin FEM in the time domain is applied to the three‐field formulation. The results are compared with analytical and semi‐analytical solutions, highlighting the different effects of the discontinuous Galerkin method on the longitudinal waves of the first and second kind. Copyright © 2010 John Wiley & Sons, Ltd.     

A modified least‐squares mixed finite element with improved momentum balance

The main goal of this contribution is to provide an improved mixed finite element for quasi‐incompressible linear elasticity. Based on a classical least‐squares formulation, a modified weak form with displacements and stresses as process variables is derived. This weak form is the basis for a finite element with an advanced fulfillment of the momentum balance and therefore with a better performance. For the continuous approximation of stresses and displacements on the triangular and tetrahedral elements, lowest‐order Raviart–Thomas and linear standard Lagrange interpolations can be used. It is shown that coercivity and continuity of the resulting asymmetric bilinear form could be established with respect to appropriate norms. Further on, details about the implementation of the least‐squares mixed finite elements are given and some numerical examples are presented in order to demonstrate the performance of the proposed formulation. Copyright © 2009 John Wiley & Sons, Ltd.     

On stability and convergence of finite element approximations of Biot's consolidation problem

Stability and convergence analysis of finite element approximations of Biot's equations governing quasistatic consolidation of saturated porous media are, discussed. A family of decay functions, parametrized by the number of time steps, is derived for the fully discrete backward Euler–Galerkin formulation, showing that the pore-pressure oscillations, arising from an unstable approximation of the incompressibility constraint on the initial condition, decay in time. Error estimates holding over the unbounded time domain for both semidiscrete and fully discrete formulations are presented, and a post-processing technique is employed to improve the pore-pressure accuracy.     

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.     

A discontinuous Galerkin method with plane waves for sound‐absorbing materials

Poro‐elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper, we present a discontinuous Galerkin method (DGM) with plane waves for poro‐elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave‐based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave‐based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material. Copyright © 2015 John Wiley & Sons, Ltd.     

Global 2D digital image correlation for motion estimation in a finite element framework: a variational formulation and a regularized,pyramidal, multi‐grid implementation

In this study, the inverse problem of reconstructing the in‐plane (2D) displacements of a monitored surface through a sequence of two‐dimensional digital images, severely ill‐posed in Hadamard's sense, is deeply investigated. A novel variational formulation is presented for the continuum 2D digital image correlation problem, and critical issues such as semi‐coercivity and solution multiplicity are discussed by functional analysis tools. In the framework of a Galerkin, finite element discretization of the displacement field, a robust implementation for 2D digital image correlation is outlined, aiming to attenuate the spurious oscillations which corrupt the deformation scenario, especially when very fine meshes are adopted. Recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid. Multi‐grid cycles are performed ascending and descending along the pyramid, with only one Newton iteration per level irrespective of the tolerance satisfaction, as if the problem were linear. At each level, the conventional least‐square matching functional is herein enriched by a Tychonoff regularization provision, preserving the solution against an unstable response. The algorithm is assessed on the basis of both synthetic and truly experimental image pairs. Copyright © 2013 John Wiley & Sons, Ltd.     

Poromechanics of saturated media based on the logarithmic finite strain

In this paper, we introduce the mathematical formulation and numerical implementation of a coupled thermo-hydro-mechanical model for saturated poromaterials undergoing logarithmic finite deformation and corotational rates. The model combines (i) the thermodynamics of standard materials, (ii) the frame indifferent hyperelastoplasticty, (iii) the orthogonality condition of maximum dissipation, as well as (iv) the principles of conservations of mass, energy and momenta. This formulation involves new developments based on the logarithmic strain measures and corotational rates which overcome the aberrant oscillations classically encountered in large simple shear. It also takes into accounts recent findings on the thermodynamics of dissipative materials which consist of deriving the yielding conditions and flow rules from suitable free energy and dissipation functions. This framework resulted in the implementation of a new finite element algorithm based on Galerkin’s method. The numerical procedures used in this paper involve the spectral decomposition of the logarithmic strain measures, the gradient split techniques as well as the return mapping method. The formulation is validated using the classical problems of Terzaghi and strip loading consolidation.     

Thermo‐elasto‐plastic finite element analysis of quasi‐state processes in Eulerian reference frames

An Eulerian finite element formulation for quasi‐state one way coupled thermo‐elasto‐plastic systems is presented. The formulation is suitable for modeling material processes such as welding and laser surfacing. In an Eulerian frame, the solution field of a quasi‐state process becomes steady state for the heat transfer problem and static for the stress problem. A mixed small deformation displacement elasto‐plastic formulation is proposed. The formulation accounts for temperature dependent material properties and exhibits a robust convergence. Streamline upwind Petrov–Galerkin (SUPG) is used to remove spurious oscillations. Smoothing functions are introduced to relax the non‐differentiable evolution equations and allow for the use of gradient (stiffness) solution scheme via the Newton–Raphson method. A 3‐dimensional simulation of a laser surfacing process is presented to exemplify the formulation. Results from the Eulerian formulation are in good agreement with results from the conventional Lagrangian formulation. However, the Eulerian formulation is approximately 15 times faster than the Lagrangian. Copyright © 2001 John Wiley & Sons, Ltd.     

A hybrid finite element formulation of the linear biphasic equations for hydrated soft tissue

A hybrid finite element method has been developed for application to the linear biphasic model of soft tissues. The biphasic model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, one solid and one fluid, and employs mixture theory to derive governing equations for its mechanical behaviour. These equations are time dependent, involving both fluid and solid velocities and solid displacement, and will be solved by spatial finite element and temporal finite difference approximation. The first step in the derivation of this hybrid method is application of a finite difference rule to the solid phase, thus obtaining equations with only velocities at discrete times as primary variables. A weighted residual statement of the temporally discretized governing equations, employing C° continuous interpolations of the solid and fluid phase velocities and discontinuous interpolations of the pore pressure and elastic stress, is then derived. The stress and pressure functions are chosen so that the total momentum equation of the mixture is satisfied; they are jointly referred to as an equilibrated stress and pressure field. The corresponding weighting functions are chosen to satisfy a relationship analogous to this equilibrium relation. The resulting matrix equations are symmetric. As an illustration of the hybrid biphasic formulation, six-noded triangular elements with complete linear, several incomplete quadratic, and complete quadratic stress and pressure fields in element local co-ordinates are developed for two dimensional analysis and tested against analytical solutions and a mixed-penalty finite element formulation of the same equations. The hybrid method is found to be robust and produce excellent results; preferred elements are identified on the basis of these results.     

Investigation of diffusion in p-version ‘LSFE’ and ‘STLSFE’ formulations

This paper deals with investigation of diffusion for p-version least squares finite element formulation (LSFEF) and p-version space-time coupled least squares finite element formulation (STLSFEF) for steady-state and transient problems. Convection dominated flows result in hyperbolic system of equations which leads to ill-conditioned matrices when using Galerkin formulation. Various techniques (SUPG, SUPG-with discontinuity capturing operator etc.) have been devised to overcome the difficulties arising primarily due to hyperbolic terms and sharp gradients. In this paper, it is demonstrated that when using p-version STLSFEF or LSFEF, no such difficulties are encountered in formulation as well as in the solution procedure. Almost all numerical processes suffer from numerical diffusion to some extent, however, it is demonstrated in this paper that in p-version STLSFE and LSFE formulations numerical diffusion can be completely eliminated by mesh refinement and p-level increase and the formulations are free of inherent diffusion. Several model problems are considered with dominant convective terms to investigate diffusion in p-version LSFEF and STLSFEF. Two dimensional convection-diffusion problems are used as steady state representative cases. One dimensional transient problems considered in this paper include pure advection, convection-diffusion and Burgers' equation. Numerical results are also compared with exact solutions and those reported in the literature.     

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.     

Conservation properties of a time FE method. Part I: time‐stepping schemes for N‐body problems

In the present paper one‐step implicit integration algorithms for the N‐body problem are developed. The time‐stepping schemes are based on a Petrov–Galerkin finite element method applied to the Hamiltonian formulation of the N‐body problem. The approach furnishes algorithmic energy conservation in a natural way. The proposed time finite element method facilitates a systematic implementation of a family of time‐stepping schemes. A particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals. The influence of various standard as well as non‐standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements. Copyright © 2000 John Wiley & Sons, Ltd.     

Inf–sup testing of upwind methods

We propose inf–sup testing for finite element methods with upwinding used to solve convection–diffusion problems. The testing evaluates the stability of a method and compactly displays the numerical behaviour as the convection effects increase. Four discretization schemes are considered: the standard Galerkin procedure, the full upwind method, the Galerkin least‐squares scheme and a high‐order derivative artificial diffusion method. The study shows that, as expected, the standard Galerkin method does not pass the inf–sup tests, whereas the other three methods pass the tests. Of these methods, the high‐order derivative artificial diffusion procedure introduces the least amount of artificial diffusion. Copyright © 2000 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.