Elastic wave propagation in unsaturated porous media using hybrid‐Trefftz stress elements

The stress model of the hybrid‐Trefftz finite element is formulated for the analysis of elastodynamic problems defined on unsaturated porous media. The supporting mathematical model is the theory of mixtures with interfaces and considers the full coupling between the solid, fluid and gas phases, including the effect of seepage acceleration. Hybrid‐Trefftz stress elements use the free‐field regular solutions of the homogeneous Navier (or Beltrami) equation to construct the approximation of the generalized stresses in the domain of the element. The influence of non‐homogeneous terms in the Navier equation is modelled using solutions of the corresponding static problem. The resulting elements are highly convergent under p‐refinement and robust to both low and high excitation frequencies, as the trial functions embody relevant physical information on the modelled phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.     

Hybrid‐Trefftz stress element for bounded and unbounded poroelastic media

The equations that govern the dynamic response of saturated porous media are first discretized in time to define the boundary value problem that supports the formulation of the hybrid‐Trefftz stress element. The (total) stress and pore pressure fields are directly approximated under the condition of locally satisfying the domain conditions of the problem. The solid displacement and the outward normal component of the seepage displacement are approximated independently on the boundary of the element. Unbounded domains are modelled using either unbounded elements that locally satisfy the Sommerfeld condition or absorbing boundary elements that enforce that condition in weak form. As the finite element equations are derived from first‐principles, the associated energy statements are recovered and the sufficient conditions for the existence and uniqueness of the solutions are stated. The performance of the element is illustrated with the time domain response of a biphasic unbounded domain to show the quality of the modelling that can be attained for the stress, pressure, displacement and seepage fields using a high‐order, wavelet‐based time integration procedure. Copyright © 2010 John Wiley & Sons, Ltd.     

Hybrid‐stress six‐node prismatic elements

This paper presents three novel hybrid‐stress six‐node prismatic elements. Starting from the element displacement interpolation, the equilibrating non‐constant stress modes for the first element are identified and orthogonalized with respect to the constant stress modes for higher computational efficiency. For the second element, the non‐constant stress modes are non‐equilibrating and chosen for the sake of stabilizing the reduced‐integrated element. The first two elements are intended for three‐dimensional continuum analysis with both passing the patch test for three‐dimensional continuum elements. The third element is primarily intended for plate/shell analysis. Shear locking is alleviated by a new assumed strain scheme which preserves the element accuracy with respect to the twisting load. Furthermore, the Poisson's locking along the in‐plane and out‐of‐plane directions is overcome by using the hybrid‐stress modes of the first element. The third element passes the patch test for plate/shell elements. Unless the element assumes the right prismatic geometry, it fails the patch test for three‐dimensional continuum elements. It will be seen that all the proposed elements are markedly more accurate than the conventional fully integrated element. Copyright © 2004 John Wiley & Sons, Ltd.     

Computational electro‐elasticity and magneto‐elasticity for quasi‐incompressible media immersed in free space

In this work, a mixed variational formulation to simulate quasi‐incompressible electro‐active or magneto‐active polymers immersed in the surrounding free space is presented. A novel domain decomposition is used to disconnect the primary coupled problem and the arbitrary free‐space mesh update problem. Exploiting this decomposition, we describe a block‐iterative approach to solving the linearised multiphysics problem, and a physically and geometrically based, three‐parameter method to update the free space mesh. Several application‐driven example problems are implemented to demonstrate the robustness of the mixed formulation for both electro‐elastic and magneto‐elastic problems involving both finite deformations and quasi‐incompressible media. Copyright © 2016 John Wiley & Sons, Ltd.     

A consistent u‐p formulation for porous media with hysteresis

This paper presents a continuum formulation based on the theory of porous media for the mechanics of liquid unsaturated porous media. The hysteresis of the liquid retention model is carefully modelled, including the derivation of the corresponding consistent tangent moduli. The quadratic convergence of Newton's method for solving the highly nonlinear system with an implicit finite element code is demonstrated. A u‐p formulation is proposed where the time discretisation is carried out prior to the space discretisation. In this way, the derivation of all consistent moduli is fairly straightforward. Time integration is approximated with the Theta and Newmark's methods, and hence the fully coupled nonlinear dynamics of porous media is considered. It is shown that the liquid retention model requires also the consistent second‐order derivative for quadratic convergence. Some predictive simulations are presented illustrating the capabilities of the formulation, in particular to the modelling of complex porous media behaviour. Copyright © 2014 John Wiley & Sons, Ltd.     

Hybrid‐stress solid elements for shell structures based upon a modified variational functional

In this paper, we start with a modified generalized laminate stiffness matrix that serves as a remedy to resolve the thickness locking and some abnormalities encountered by solid‐shell elements in laminate analyses. A modified Hellinger–Reissner functional having displacement and a set of generalized stresses as independent fields is devised. Based upon the functional, eight‐node and 18‐node hybrid‐stress solid‐shell elements are proposed. A number of benchmark tests on homogenous and laminated plates/shells are conducted. The accuracy of the elements is promising. Copyright © 2002 John Wiley & Sons, Ltd.     

Object‐oriented finite element analysis of thermo‐hydro‐mechanical (THM) problems in porous media

The design, implementation and application of a concept for object‐oriented in finite element analysis of multi‐field problems is presented in this paper. The basic idea of this concept is that the underlying governing equations of porous media mechanics can be classified into different types of partial differential equations (PDEs). In principle, similar types of PDEs for diverse physical problems differ only in material coefficients. Local element matrices and vectors arising from the finite element discretization of the PDEs are categorized into several types, regardless of which physical problem they belong to (i.e. fluid flow, mass and heat transport or deformation processes). Element (ELE) objects are introduced to carry out the local assembly of the algebraic equations. The object‐orientation includes a strict encapsulation of geometrical (GEO), topological (MSH), process‐related (FEM) data and methods of element objects. Geometric entities of an element such as nodes, edges, faces and neighbours are abstracted into corresponding geometric element objects (ELE–GEO). The relationships among these geometric entities form the topology of element meshes (ELE–MSH). Finite element objects (ELE–FEM) are presented for the local element calculations, in which each classification type of the matrices and vectors is computed by a unique function. These element functions are able to deal with different element types (lines, triangles, quadrilaterals, tetrahedra, prisms, hexahedra) by automatically choosing the related element interpolation functions. For each process of a multi‐field problem, only a single instance of the finite element object is required. The element objects provide a flexible coding environment for multi‐field problems with different element types. Here, the C++ implementations of the objects are given and described in detail. The efficiency of the new element objects is demonstrated by several test cases dealing with thermo‐hydro‐mechanical (THM) coupled problems for geotechnical applications. Copyright © 2006 John Wiley & Sons, Ltd.     

Three dimensional hybrid‐Trefftz stress finite elements for plates and shells

Three‐dimensional hybrid‐Trefftz stress finite elements for plates and shells are proposed. Two independent fields are approximated: stresses within the element and displacement on their boundary. The required stress field derived from the Papkovitch‐Neuber solution of the Navier equation, which a priori satisfies the Trefftz constraint, is generated using homogeneous harmonic polynomials. Restriction on the polynomial degree in the coordinate measured along the thickness direction is imposed to reduce the number of independent terms. Explicit expressions of the corresponding independent polynomials are listed up to the fifth order. Illustrative applications to evaluate displacements and stresses are conducted by hexahedral hybrid‐Trefftz stress element models. The hierarchical p‐ and h‐refinement strategy are exploited in the numerical tests.     

Natural element approximations involving bubbles for treating mechanical models in incompressible media

In this paper, a new approach is proposed to address issues associated with incompressibility in the context of the meshfree natural element method (NEM). The NEM possesses attractive features such as interpolant shape functions or auto‐adaptive domain of influence, which alleviates some of the most common difficulties in meshless methods. Nevertheless, the shape functions can only reproduce linear polynomials, and in contrast to moving least squares methods, it is not easy to define interpolations with arbitrary approximation consistency. In order to treat mechanical models involving incompressible media in the framework of mixed formulations, the associated functional approximations must satisfy the well‐known inf–sup, or LBB condition. In the proposed approach, additional degrees of freedom are associated with some topological entities of the underlying Delaunay tessellation, i.e. edges, triangles and tetrahedrons. The associated shape functions are computed from the product of the NEM shape functions related to the original nodes. Different combinations can be used to construct new families of NEM approximations. As these new approximations functions are not related to any node, as they vanish at the nodes, from now on we refer these shape functions as bubbles. The shape functions can be corrected enforcing different reproducing conditions, when they are used as weights in the moving least square (MLS) framework. In this manner, the effects of the obtained higher approximation consistency can be evaluated. In this work, we restrict our attention to the 2D case, and the following constructions will be considered: (a) bubble functions associated with the Delaunay triangles, called b1‐NEM and (b) bubble functions associated with the Delaunay edges, called b2‐NEM. We prove that all these approximation schemes allow direct enforcement of essential boundary conditions. The bubble‐NEM schemes are then used to approximate the displacements in the linear elasticity mixed formulation, the pressure being approximated by the standard NEM. The numerical LBB test is passed for all the bubble‐NEM approximations, and pressure oscillations are removed in the incompressible limit. Copyright © 2005 John Wiley & Sons, Ltd.     

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

In this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.     

A generalized finite‐strain damage model for quasi‐incompressible hyperelasticity using hybrid formulation

A new generalized damage model for quasi‐incompressible hyperelasticity in a total Lagrangian finite‐strain framework is presented. A Kachanov‐like reduction factor (1 ? D) is applied on the deviatoric part of the hyperelastic constitutive model. Linear and exponential softening are defined as damage evolution laws, both describable in terms of only two material parameters. The model is formulated following continuum damage mechanics theory such that it can be particularized for any hyperelastic model based on the volumetric–isochoric split of the Helmholtz free energy. However, in the present work, it has been implemented in an in‐house finite element code for neo‐Hooke and Ogden hyperelasticity. The details of the hybrid formulation used are also described. A couple of three‐dimensional examples are presented to illustrate the main characteristics of the damage model. The results obtained reproduce a wide range of softening behaviors, highlighting the versatility of the formulation proposed. The damage formulation has been developed to be used in conjunction with mixing theory in order to model the behavior of fibered biological tissues. As an example, the markedly different behaviors of the fundamental components of the rectus sheath were reproduced using the damage model, obtaining excellent correlation with the experimental results from literature. Copyright © 2015 John Wiley & Sons, Ltd.     

On a two‐level finite element method for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions, where the finite‐dimensional space(s) employed consist of piecewise polynomials enriched with residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method (TLFEM) is described and its application to the Navier–Stokes equation is displayed. Numerical solutions employing the TLFEM are presented for three benchmark problems. We compare the numerical solutions using the TLFEM with the numerical solutions using a stabilized method. Copyright © 2001 John Wiley & Sons, Ltd.     

Lower‐dimensional interface elements with local enrichment: application to coupled hydro‐mechanical problems in discretely fractured porous media

In this study, we develop lower‐dimensional interface elements to represent preexisting fractures in rock material, focusing on finite element analysis of coupled hydro‐mechanical problems in discrete fractures–porous media systems. The method adopts local enrichment approximations for a discontinuous displacement and a fracture relative displacement function. Multiple and intersected fractures can be treated with the new scheme. Moreover, the method requires less mesh dependencies for accurate finiteelement approximations compared with the conventional interface element method. In particular, for coupled problems, the method allows for the use of a single mesh for both mechanical and other related processes such as flow and transport. For verification purposes, several numerical examples are examined in detail. Application to a coupled hydro‐mechanical problem is demonstrated with fluid injection into a single fracture. The numerical examples prove that the proposed method produces results in strong agreement with reference solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Free‐slip boundary conditions for simulating free‐surface incompressible flows through the particle finite element method

The possibility of using free‐slip conditions within the context of the particle finite element method (PFEM) is investigated. For high Reynolds number engineering applications in which tangential effects at the fluid–solid boundaries are not of primary interest, the use of free‐slip conditions can alleviate the need for very fine boundary layer meshes. Two novel ways for the imposition of free‐slip conditions in the framework of the PFEM are presented. The proposed approach emphasizes robustness and simplicity, while retaining a sufficient level of generality. These two methods are then tested in the case of dam break and sloshing problems, and their respective advantages and drawbacks are discussed. It is also shown how the use of free‐slip conditions can indirectly improve mass conservation properties of the PFEM, even when coarse meshes are employed. Copyright © 2016 John Wiley & Sons, Ltd.     

A stabilized volume‐averaging finite element method for flow in porous media and binary alloy solidification processes

A stabilized equal‐order velocity–pressure finite element algorithm is presented for the analysis of flow in porous media and in the solidification of binary alloys. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. The analysis is performed in a single domain with a fixed numerical grid. The fluid flow scheme developed includes SUPG (streamline‐upwind/Petrov–Galerkin), PSPG (pressure stabilizing/Petrov–Galerkin) and DSPG (Darcy stabilizing/Petrov–Galerkin) stabilization terms in a variable porosity medium. For the energy and species equations a classical SUPG‐based finite element method is employed. The developed algorithms were tested extensively with bilinear elements and were shown to perform stably and with nearly quadratic convergence in high Rayleigh number flows in varying porosity media. Examples are shown in natural and double diffusive convection in porous media and in the directional solidification of a binary‐alloy. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient finite element procedure for analyzing three‐phase porous media based on the relaxed Picard method

Effective simulation of the solid‐liquid‐gas coupling effect in unsaturated porous media is of great significance in many diverse areas. Because of the strongly nonlinear characteristics of the fully coupled formulations for the three‐phase porous media, an effective numerical solution scheme, such as the finite element method with an efficient iterative algorithm, has to be employed. In this paper, an efficient finite element procedure based on the adaptive relaxed Picard method is developed for analyzing the coupled solid‐liquid‐gas interactions in porous media. The coupled model and the finite element analysis procedure are implemented into a computer code PorousH2M, and the proposed procedure is validated through comparing the numerical simulations with the experimental benchmarks. It is shown that the adaptive relaxed Picard method has salient advantage over the traditional one with respect to both the efficiency and the robustness, especially for the case of relatively large time step sizes. Compared with the Newton‐Raphson scheme, the Picard method successfully avoids the unphysical ‘spurious unloading’ phenomenon under the plastic deformation condition, although the latter shows a better convergence rate. The proposed procedure provides an important reference for analyzing the fully coupled problems related to the multi‐phase, multi‐field coupling in porous media. Copyright © 2014 John Wiley & Sons, Ltd.     

On the a priori and a posteriori error analysis of a two‐fold saddle‐point approach for nonlinear incompressible elasticity

In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two‐fold saddle‐point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well‐known generalization of the classical Babu?ka–Brezzi theory is applied to show the well‐posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi‐efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A mixed finite volume formulation for determining the small strain deformation of incompressible materials

A finite volume formulation for determining small strain deformations in incompressible materials is presented in detail. The formulation includes displacement and hydrostatic pressure variables. The displacement field varies linearly along and across each cell face. The hydrostatic pressure field associated with each face is uniform. The cells that discretize the structure are geometrically unrestricted, each cell can have an arbitrary number of faces. The formulation is tested on a number of linear elastic plane strain benchmark problems. This testing reveals that when meshes of multifaceted cells are employed to represent the structure then locking behaviour is exhibited, but when triangular cells are used then accurate predictions of the displacement and stress fields are produced. Copyright © 1999 John Wiley & Sons, Ltd.     

Hybrid displacement function element method: a simple hybrid‐Trefftz stress element method for analysis of Mindlin–Reissner plate

In order to develop robust finite element models for analysis of thin and moderately thick plates, a simple hybrid displacement function element method is presented. First, the variational functional of complementary energy for Mindlin–Reissner plates is modified to be expressed by a displacement function F, which can be used to derive displacement components satisfying all governing equations. Second, the assumed element resultant force fields, which can satisfy all related governing equations, are derived from the fundamental analytical solutions of F. Third, the displacements and shear strains along each element boundary are determined by the locking‐free formulae based on the Timoshenko's beam theory. Finally, by applying the principle of minimum complementary energy, the element stiffness matrix related to the conventional nodal displacement DOFs is obtained. Because the trial functions of the domain stress approximations a priori satisfy governing equations, this method is consistent with the hybrid‐Trefftz stress element method. As an example, a 4‐node, 12‐DOF quadrilateral plate bending element, HDF‐P4‐11 β, is formulated. Numerical benchmark examples have proved that the new model possesses excellent precision. It is also a shape‐free element that performs very well even when a severely distorted mesh containing concave quadrilateral and degenerated triangular elements is employed. Copyright © 2014 John Wiley & Sons, Ltd.