Localized Jacobian ILU preconditioners for hydraulic fractures

We discuss the properties of a class of sparse localized approximations to the Jacobian operator that arises in modelling the evolution of a hydraulically driven fracture in a multi‐layered elastic medium. The governing equations involve a highly non‐linear coupled system of integro‐partial differential equations along with the fracture front free boundary problem. We demonstrate that an incomplete LU factorization of these localized Jacobians yields an efficient preconditioner for the fully populated, stiff, non‐symmetric system of algebraic equations that need to be solved multiple times for every growth increment of the fracture. The performance characteristics of this class of preconditioners is explored via spectral analysis and numerical experiment. Copyright © 2005 John Wiley & Sons, Ltd.     

Exploiting group symmetry in truss topology optimization

We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in Ohsaki et al. (Comp. Meth. Appl. Mech. Eng. 180:203–217, 1999). We show how one may automatically obtain symmetric designs, by eliminating the ‘redundant’ symmetry in the SDP problem formulation. This has the advantage that the original SDP problem is substantially reduced in size for trusses with large symmetry groups.     

Boundary element based formulations for crack shape sensitivity analysis

The present paper addresses several BIE-based or BIE-oriented formulations for sensitivity analysis of integral functionals with respect to the geometrical shape of a crack. Functionals defined in terms of integrals over the external boundary of a cracked body and involving the solution of a frequency-domain boundary-value elastodynamic problem are considered, but the ideas presented in this paper are applicable, with the appropriate modifications, to other kinds of linear field equations as well. Both direct differentiation and adjoint problem techniques are addressed, with recourse to either collocation or symmetric Galerkin BIE formulations. After a review of some basic concepts about shape sensitivity and material differentiation, the derivative integral equations for the elastodynamic crack problem are discussed in connection with both collocation and symmetric Galerkin BIE formulations. Building upon these results, the direct differentiation and the adjoint solution approaches are then developed. In particular, the adjoint solution approach is presented in three different forms compatible with boundary element method (BEM) analysis of crack problems, based on the discretized collocation BEM equations, the symmetric Galerkin BEM equations and the direct and adjoint stress intensity factors, respectively. The paper closes with a few comments.     

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 dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems

We present a novel multigrid (MG) procedure for the efficient solution of the large non‐symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi‐layered elastic medium. The governing equations involve a highly non‐linear coupled system of integro‐partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N³). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green's functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub‐sets of the finest grid are used to treat mixed variable problems associated with ‘pinch points.’ Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss–Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

Axial symmetric elasticity analysis in non‐homogeneous bodies under gravitational load by triple‐reciprocity boundary element method

In general, internal cells are required to solve elasticity problems by involving a gravitational load in non‐homogeneous bodies with variable mass density when using a conventional boundary element method (BEM). Then, the effect of mesh reduction is not achieved and one of the main merits of the BEM, which is the simplicity of data preparation, is lost. In this study, it is shown that the domain cells can be avoided by using the triple‐reciprocity BEM formulation, where the density of domain integral is expressed in terms of other fields that are represented by boundary densities and/or source densities at isolated interior points. Utilizing the rotational symmetry, the triple‐reciprocity BEM formulation is developed for axially symmetric elasticity problems in non‐homogeneous bodies under gravitational force. A new computer program was developed and applied to solve several test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Two‐point constitutive equations and integration algorithms for isotropic‐hardening rate‐independent elastoplastic materials in large deformation

This paper presents alternative forms of hyperelastic–plastic constitutive equations and their integration algorithms for isotropic‐hardening materials at large strain, which are established in two‐point tensor field, namely between the first Piola–Kirchhoff stress tensor and deformation gradient. The eigenvalue problems for symmetric and non‐symmetric tensors are applied to kinematics of multiplicative plasticity, which imply the transformation relationships of eigenvectors in current, intermediate and initial configurations. Based on the principle of plastic maximum dissipation, the two‐point hyperelastic stress–strain relationships and the evolution equations are achieved, in which it is considered that the plastic spin vanishes for isotropic plasticity. On the computational side, the exponential algorithm is used to integrate the plastic evolution equation. The return‐mapping procedure in principal axes, with respect to logarithmic elastic strain, possesses the same structure as infinitesimal deformation theory. Then, the theory of derivatives of non‐symmetric tensor functions is applied to derive the two‐point closed‐form consistent tangent modulus, which is useful for Newton's iterative solution of boundary value problem. Finally, the numerical simulation illustrates the application of the proposed formulations. Copyright © 2008 John Wiley & Sons, Ltd.     

Stokes flow around cylinders in a bounded two‐dimensional domain using multipole‐accelerated boundary element methods

The multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N²) (iterative solvers) or O(N³) (gaussian elimination) to O(N log N) or O(N), where N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes flow results in Fredholm equations of the second kind, which generally produce a well‐posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the first kind, which, upon discretization, produce ill‐conditioned systems of equations. The model system here is a two‐dimensional wide‐gap couette viscometer, where particles are suspended in the fluid between the cylinders. This is a typical system that is efficiently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is sufficiently well‐conditioned to allow the use of fast multipole methods. The direct approach results in severe ill‐conditioning, to a point where application of the multipole method leads to non‐convergence of the solution iteration. Copyright © 1999 John Wiley & Sons, Ltd.     

Non‐local damage model based on displacement averaging

Continuum damage models describe the changes of material stiffness and strength, caused by the evolution of defects, in the framework of continuum mechanics. In many materials, a fast evolution of defects leads to stress–strain laws with softening, which creates serious mathematical and numerical problems. To regularize the model behaviour, various generalized continuum theories have been proposed. Integral‐type non‐local damage models are often based on weighted spatial averaging of a strain‐like quantity. This paper explores an alternative formulation with averaging of the displacement field. Damage is assumed to be driven by the symmetric gradient of the non‐local displacements. It is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium. Around physical boundaries of the analysed body, both formulations differ and the non‐local displacement model generates spurious damage in the boundary layers. The paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the non‐local weight function. Alternatively, an implicit gradient formulation could be used. Issues of algorithmic implementation, computational efficiency and smoothness of the resolved stress fields are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

The nsBETI method: an extension of the FETI method to non‐symmetrical BEM‐FEM coupled problems

The finite element tearing and interconnecting (FETI) method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second‐order elasticity problems. In the boundary element tearing and interconnecting (BETI) method, a direct extension of the FETI algorithm to the BEM, the symmetric Galerkin BEM formulation, is used to obtain symmetric system matrices, making possible to apply the same FETI conjugate gradient solver. In this work, we propose a new BETI variant labeled nsBETI that allows to couple substructures modeled with the FEM and/or non‐symmetrical BEM formulations. The method connects non‐matching BEM and FEM subdomains using localized Lagrange multipliers and solves the associated non‐symmetrical flexibility equations with a Bi‐CGstab iterative algorithm. Scalability issues of nsBETI in BEM–BEM and combined BEM–FEM coupled problems are also investigated. Copyright © 2012 John Wiley & Sons, Ltd.     

Large deflection analysis of elastic space membranes

In this paper a solution to the problem of elastic space (initially non‐flat) membranes is presented. A new formulation of the governing differential equations is presented in terms of the displacements in the Cartesian co‐ordinates. The reference surface of the membrane is the minimal surface. The problem is solved by direct integration of the differential equations using the analogue equation method (AEM). According to this method the three coupled non‐linear partial differential equations with variable coefficients are replaced with three uncoupled equivalent linear flat membrane equations (Poisson's equations) subjected to unknown sources under the same boundary conditions. Subsequently, the unknown sources are established using a procedure based on the BEM. The displacements as well as the stress resultants are evaluated at any point of the membrane from their integral representations of the solution of the substitute problems, which are used as mathematical formulae. Several membranes are analysed which illustrate the method and demonstrate its efficiency and accuracy as compared with analytical and existing numerical methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Wave propagation in a simplified modelled poroelastic continuum: fundamental solutions and a time domain boundary element formulation

In finite element formulations for poroelastic continua a representation of Biot's theory using the unknowns solid displacement and pore pressure is preferred. Such a formulation is possible either for quasi‐static problems or for dynamic problems if the inertia effects of the fluid are neglected. Contrary to these formulations a boundary element method (BEM) for the general case of Biot's theory in time domain has been published (Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach. Lecture Notes in Applied Mechanics. Springer: Berlin, Heidelberg, New York, 2001.). If the advantages of both methods are required it is common practice to couple both methods. However, for such a coupled FE/BE procedure a BEM for the simplified dynamic Biot theory as used in FEM must be developed. Therefore, here, the fundamental solutions as well as a BE time stepping procedure is presented for the simplified dynamic theory where the inertia effects of the fluid are neglected. Further, a semi‐analytical one‐dimensional solution is presented to check the proposed BE formulation. Finally, wave propagation problems are studied using either the complete Biot theory as well as the simplified theory. These examples show that no significant differences occur for the selected material. Copyright © 2005 John Wiley & Sons, Ltd.     

SGBEM with Lagrange multipliers applied to elastic domain decomposition problems with curved interfaces using non‐matching meshes

An original approach to the solution of linear elastic domain decomposition problems by the symmetric Galerkin boundary element method is developed. The approach is based on searching for the saddle‐point of a new potential energy functional with Lagrange multipliers. The interfaces can be either straight or curved, open or closed. The two coupling conditions, equilibrium and compatibility, along an interface are fulfilled in a weak sense by means of Lagrange multipliers (interface displacements and tractions), which enables non‐matching meshes to be used at both sides of interfaces between subdomains. The accuracy and robustness of the method is tested by several numerical examples, where the numerical results are compared with the analytical solution of the solved problems, and the convergence rates of two error norms are evaluated for h‐refinements of matching and non‐matching boundary element meshes. Copyright © 2010 John Wiley & Sons, Ltd.     

Higher‐order accurate time‐step‐integration algorithms by post‐integration techniques

In this paper, a framework to construct higher‐order‐accurate time‐step‐integration algorithms based on the post‐integration techniques is presented. The prescribed initial conditions are naturally incorporated in the formulations and can be strongly or weakly enforced. The algorithmic parameters are chosen such that unconditionally A‐stable higher‐order‐accurate time‐step‐integration algorithms with controllable numerical dissipation can be constructed for linear problems. Besides, it is shown that the order of accuracy for non‐linear problems is maintained through the relationship between the present formulation and the Runge–Kutta method. The second‐order differential equations are also considered. Numerical examples are given to illustrate the validity of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Planar four node piezoelectric elements with drilling degrees of freedom

Several new planar four node piezoelectric elements with drilling degrees of freedom are presented. We begin by deriving two families of variational formulations accounting for piezoelectricity and in‐plane rotations. The first family retains the skew‐symmetric part of the stress tensor, while in the second, the skew part of stress is eliminated from the functional. The finite elements derived from two of the variational formulations derived in this paper are investigated. The first element is based on an ‘irreducible’ form, while the other is based on a fully mixed functional, with both stress and electric flux density assumed. Our new elements are shown to be accurate and robust in comparison with a number of existing elements, for several benchmark test problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluation of the basic formulations for the cumulative probability of brittle fracture with two different spatial distributions of microcracks

The random distribution of microcracks in terms of their size, shape, orientation and spatial location has direct impact on the cumulative probability of brittle fracture induced failure, with the effect of spatial distribution being rarely explored. Recently, two weakest link theory‐based formulations for the cumulative probability of brittle fracture induced failure have been proposed for the spatial distribution of microcracks obeying the Poisson postulates and the uniform distribution, respectively. This work compares these two new formulations with the currently commonly adopted one built on the Poisson postulates under both the uniform and the non‐uniform uniaxial loading conditions. It is concluded that under general loading conditions involving non‐uniform stress states, the existing formulation is equivalent to or closely approximate to neither of the two new formulations thus should be discarded, because of its inaccurate derivation. The new formulations are featured with unique symmetry or self‐similarity in their expressions. Their capability in revealing the size effect or the scaling law of failure is highlighted and validated by a set of published uniaxial and biaxial flexural strength data of brittle material.     

Symmetric coupling of multi‐zone curved Galerkin boundary elements with finite elements in elasticity

The coupling of Finite Element Method (FEM) with a Boundary Element Method (BEM) is a desirable result that exploits the advantages of each. This paper examines the efficient symmetric coupling of a Symmetric Galerkin Multi‐zone Curved Boundary Element Analysis method with a Finite Element Method for 2‐D elastic problems. Existing collocation based multi‐zone boundary element methods are not symmetric. Thus, when they are coupled with FEM, it is very difficult to achieve symmetry, increasing the computational work to solve the problem. This paper uses a fully Symmetric curved Multi‐zone Galerkin Boundary Element Approach that is coupled to an FEM in a completely symmetric fashion. The symmetry is achieved by symmetrically converting the boundary zones into equivalent ‘macro finite elements’, that are symmetric, so that symmetry in the coupling is retained. This computationally efficient and fast approach can be used to solve a wide range of problems, although only 2‐D elastic problems are shown. Three elasticity problems, including one from the FEM‐BEM literature that explore the efficacy of the approach are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Cell‐based volume integration for boundary integral analysis

The evaluation of volume integrals that arise in boundary integral formulations for non‐homogeneous problems was considered. Using the “Galerkin vector” to represent the Green's function, the volume integral was decomposed into a boundary integral, together with a volume integral wherein the source function was everywhere zero on the boundary. This new volume integral can be evaluated using a regular grid of cells covering the domain, with all cell integrals, including partial cells at the boundary, evaluated by simple linear interpolation of vertex values. For grid vertices that lie close to the boundary, the near‐singular integrals were handled by partial analytic integration. The method employed a Galerkin approximation and was presented in terms of the three‐dimensional Poisson problem. An axisymmetric formulation was also presented, and in this setting, the solution of a nonlinear problem was considered. Copyright © 2012 John Wiley & Sons, Ltd.