Modelling cohesive crack growth using a two-step finite element-scaled boundary finite element coupled method

A two-step method, coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM), is developed in this paper for modelling cohesive crack growth in quasi-brittle normal-sized structures such as concrete beams. In the first step, the crack trajectory is fully automatically predicted by a recently-developed simple remeshing procedure using the SBFEM based on the linear elastic fracture mechanics theory. In the second step, interfacial finite elements with tension-softening constitutive laws are inserted into the crack path to model gradual energy dissipation in the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the proposed method. The numerical results demonstrate that this two-step SBFEM-FEM coupled method can predict both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for crack growth problems with strong snap-back phenomenon. The effects of the tensile strength, the mode-I and mode-II fracture energies on the predicted load-displacement relations are also discussed.     

Fully-automatic modelling of cohesive crack growth using a finite element-scaled boundary finite element coupled method

This study develops a method coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for fully-automatic modelling of cohesive crack growth in quasi-brittle materials. The simple linear elastic fracture mechanics (LEFM)-based remeshing procedure developed previously is augmented by inserting nonlinear interface finite elements automatically. The constitutive law of these elements is modelled by the cohesive/fictitious crack model to simulate the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. The crack is assumed to grow when the mode-I stress intensity factor K_I vanishes in the direction determined by LEFM criteria. Other salient algorithms associated with the SBFEM, such as mapping state variables after remeshing and calculating K_I using a “shadow subdomain”, are also described. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the new method. The results show that this SBFEM-FEM coupled method is capable of fully-automatically predicting both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for problems with strong snap-back. Parametric studies were carried out on the crack incremental length, the concrete tensile strength, and the mode-I and mode-II fracture energies. It is found that the K_I ? 0 criterion is objective with respect to the crack incremental length.     

A coupled cubic hermite finite element/boundary element procedure for electrocardiographic problems

The problem considered is that of trying to determine the potential distribution inside a human torso as a result of the heart's electrical activity. We describe here a high order (cubic Hermite) coupled finite element/boundary element procedure for solving such electrocardiographic potential problems inside an anatomically accurate human torso. Details of the cubic Hermite boundary element procedure and its coupling to the finite element method are described. We then present two and three dimensional test results showing the success, efficiency and accuracy of this high order coupled technique. Some initial results on an anatomically accurate torso are also given.     

A coupled hp‐finite element scheme for the solution of two‐dimensional electrostrictive materials

As part of the ongoing research within the field of computational analysis for the coupled electro‐magneto‐mechanical response of smart materials, the problem of linearised electrostriction is revisited and analysed for the first time using the framework of hp‐finite elements. The governing equations modelling the physics of the dielectric are suitably modified by introducing a new total Cauchy stress tensor (A. Dorfmann and R.W. Ogden. Nonlinear electroelasticity. Acta Mechanica, 174:167–183, 2005), which includes the electrostrictive effect and a staggered partitioned scheme for the numerical solution of the coupling phenomena. With the purpose of benchmarking numerical results, the problem of an infinite electrostrictive plate with a circular/elliptical dielectric insert is revisited. The presented analytical solution is based on the theoretical framework for two‐dimensional electrostriction proposed by Knops (R.J. Knops. Two‐dimensional electrostriction. Quarterly Journal of Mechanics and Applied Mathematics, 16:377–388, 1963) and uses classical techniques of complex variable analysis. Our presentation, to the best of our knowledge, provides the first correct closed form expression for the solution to the infinite electrostrictive plate with a circular/elliptical dielectric insert, correcting the errors made in previous presentations of this problem. We use this analytical solution to assess the accuracy, efficiency and robustness of the hp‐formulation in the case of nearly incompressible electrostrictive materials. Copyright © 2012 John Wiley & Sons, Ltd.     

A finite element solution of coupled electrokinetic and hydrodynamic flow in porous media

The finite element method is applied to the solution f seepage problems where both electrokinetic and hydrodynamic forces occur. The resulting system of coupled equations is used to solve both the steady state and transient conditions in a one-dimensional system and compared with the theoretical result. An analysis is then made of a more complex two-dimensional problem where the application of electro-osmosis may be used successfully to prevent piping.     

Generalized finite element analysis with T-complete boundary solution functions

The use of a complete and nonsingular set of Trefftz functions in the solution of quasi-harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary-type approximation. Procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated.     

An iterative solution scheme for systems of boundary element equations

In this paper an iterative scheme of first order is developed for the purpose of solving linear systems of equations. In particular, systems that are derived from boundary integral equations are investigated. The iterative schemes to be considered are of the form Ex^(k+1) = Dx^(k) + d, where E and D are square matrices. It will be assumed that E is a lower matrix, i.e. the coefficients above the central diagonal are zero. It will be shown that by considering matrix D embedded in a vector space and reducing its size with respect to a chosen metric, that convergence rates can be substantially improved. Equation ordering and parameter matrices are used to reduce the magnitude of D. A number of examples are tested to illustrate the importance of the choice of metric, equation ordering and the parameter matrix. Computation times are determined for both the iterative procedure and Gauss elimination indicating the usefulness of iteration which can be orders of magnitude faster.     

The coupling of the finite element method and boundary solution procedures

The finite element method is now recognized as a general approximation process which is applicable to a variety of engineering problems—structural mechanics being only one of these. Boundary solution procedures have been introduced as an independent alternative which at times is more economical and possesses certain merits. In this survey of the field we show how such procedures can be utilized in conventional FEM context.     

A simple scheme for the finite element solution of a pure advection equation

The present paper deals with the finite element solution of an advection equation. The method of characteristic lines is combined with the finite element method. Three interpolating functions are employed: a cubic, a quadratic and a linear polynomials. The proposed scheme has two key features. One is the simplicity of its algorithm. The other is the combination of the proposed algorithm with the limiting procedure for the purpose of the suppression of numerical oscillations. Three interpolating functions are tested through four numerical examples: the advection of a box-shaped profile, the advection of an elliptic profile, the rotation of a cosine-shaped hill, and the advection of a square-shaped hill. The effectiveness of the proposed scheme is shown by comparing the present numerical results with other popular finite element schemes, i.e., SUPG, Taylor-Galerkin methods and so on. In the present test calculations, the scheme with a quadratic interpolating function has given the best results. The computing time of the proposed scheme is much faster than that of other well-known finite element schemes.     

A posteriori estimates of the solution error caused by discretization in the finite element,finite difference and boundary element methods

A theory is described which guarantees an upper and lower bound estimate of the discretization error in numerical solutions of elliptic boundary value problems. This method gives bounded global estimates of the error in the energy norm. Pointwise estimates of the error in the solution variable or its derivatives can then be obtained if the numerical solution is exhibiting pointwise monotonic convergence. The versatility of this method is illustrated by its application to numerical solutions from finite element, finite difference and boundary element methods.     

An efficient fluid-solid coupled finite element scheme for weapon fragmentation simulations

An efficient finite element (FE) scheme to deal with a class of coupled fluid-solid problems is presented. The main ingredients of such methodology are: an accurate Q1/P0 solid element (trilinear in velocities and constant piecewise-discontinuous pressures); a large deformation plasticity model; an algorithm to deal with material failure, cracking propagation and fragment formation; and a fragment rigidization methodology to avoid the possible numerical instabilities that may produce pieces of material flying away from the main solid body. All the mentioned schemes have been fully parallelized and coupled using a loose-embedded procedure with a well-established and validated computational fluid dynamics (CFD) code (FEFLO). A CSD and a CFD/CSD coupled case are presented and analyzed.     

Experimental analysis of masonry infilled frames using digital image correlation

A measurement technique (digital image correlation) is used for a critical evaluation of simplified models for infilled framed structures. It allows for the assessment of displacement and strain fields in the panel of interest. Several specimens, including infilled and partially infilled frames were subjected to cyclic lateral loads. It was found that two very different deformation mechanisms appear in the masonry panels namely, a first one during the hardening phase and another one, completely different, during the softening stage. For the former, strain concentration bands are observed. After the peak load, horizontal bands appear in the middle of the panels. The Polyakov assumption, i.e. that the panel can be replaced by struts in the analysis, is validated in the hardening stage. However, the orientation of the struts suggested in the literature was not found experimentally. The experimental results demonstrate that the inclination of the bands depends on the brick dimensions and arrangement. Further, the final failure mechanism corresponds to a sliding shear mode.     

Dynamic soil–structure interaction analysis of layered unbounded media via a coupled finite element/boundary element/scaled boundary finite element model

In this paper, a coupled model based on finite element method (FEM), boundary element method (BEM) and scaled boundary FEM (SBFEM) (also referred to as the consistent infinitesimal finite element cell method) for dynamic response of 2D structures resting on layered soil media is presented. The SBFEM proposed by Wolf and Song (Finite‐element Modelling of Unbounded Media. Wiley: England, 1996) and BEM are used for modelling the dynamic response of the unbounded media (far‐field). The standard FEM is used for modelling the finite region (near‐field) and the structure. In SBFEM, which is a semi‐analytical technique, the radiation condition at infinity is satisfied exactly without requiring the fundamental solution. This method, also eliminates the need for the discretization of interfaces between different layers. In both SBFEM and BEM, the spatial dimension is decreased by one. The objective of the development of this coupled model is to combine advantages of above‐mentioned three numerical models to solve various soil–structure interaction (SSI) problems efficiently and effectively. These three methods are coupled (FE–BE–SBFEM) via substructuring method, and a computer programme is developed for the harmonic analyses of SSI systems. The coupled model is established in such a way that, depending upon the problem and far‐field properties, one can choose BEM and/or SBFEM in modelling related far‐field region(s). Thus, BEM and/or SBFEM can be used efficiently in modelling the far‐field. The proposed model is applied to investigate dynamic response of rigid and elastic structures resting on layered soil media. To assess the proposed SSI model, several problems existing in the literature are chosen and analysed. The results of the proposed model agree with the results presented in the literature for the chosen problems. The advantages of the model are demonstrated through these comparisons. Copyright © 2004 John Wiley & Sons, Ltd.     

On the finite element solution of an exterior boundary value problem

This paper compares three methods for dealing with an exterior boundary value problem by the Finite Element Method, one of which involves using an infinite element. The methods are illustrated by application to the problem of ground water flow round a tunnel with permeable invert. The use of a special trial function with a variable parameter in the infinite element gives a particularly efficient method of solution.     

An efficient finite element solution using a large pre-solved regular element

In this paper a finite element algorithm is presented using a large pre-solved hyper element. Utilizing the largest rectangle/cuboid inside an arbitrary domain, a large hyper element is developed that is solved using graph product rules. This pre-solved hyper element is efficiently inserted into the finite element formulation of partial differential equations (PDE) and engineering problems to reduce the computational complexity and execution time of the solution. A general solution of the large pre-solved element for a uniform mesh of triangular and rectangular elements is formulated for second-order PDEs. The efficiency of the algorithm depends on the relative size of the large element and the domain; however, the method remains as efficient as a classic method for even relatively small sizes. The application of the method is demonstrated using different examples.     

A coupled boundary/finite element method for the computation of magnetically and electrostatically levitated droplet shapes

A coupled finite element and boundary element method is developed to predict the magnetic vector and scalar potential distributions in the droplets levitated in an alternating magnetic or electrostatic field. The computational algorithm entails the application of boundary elements in the region of free space and finite elements in the droplet region, the two being coupled along the droplet–air interface. The coupled boundary and finite element scheme is further integrated with a WRM‐based algorithm to predict the free surface deformation of magnetically and electrostatically levitated droplets. Several corner treatments for the boundary and finite element coupling and their implications to free surface calculations are discussed. Detailed formulation and numerical implementation are given. Numerical results are compared with available analytical solutions whenever available. A selection of computed results is presented for mag‐ netically or electrostatically levitated droplets under both terrestrial and microgravity conditions. Copyright © 1999 John Wiley & Sons, Ltd.     

Open boundary treatment of the finite amplitude wave using the boundary element method

In this paper the boundary element method is applied to the analysis of non-linear free-surface waves. A particular concern is the treatment of open boundary at the input flow boundary and output flow boundary, which uses the mass-flux and energy-flux because of the continuity of fluid. By assuming the fluid to be inviscid and incompressible and the flow to be irrotational, the problem is formulated mathematically as a two-dimensional non-linear problem in terms of a velocity potential. The equation (Laplace equation) and the boundary conditions are transformed into two boundary integral equations. Due to the non-linearity of the problem, the incremental method is used in the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the finite element method and also with the experimental values. Good agreements are obtained.     

A scaled boundary finite element formulation for poroelasticity

This paper develops the scaled boundary finite element formulation for applications in coupled field problems, in particular, to poroelasticity. The salient feature of this formulation is that it can be applied over arbitrary polygons and/or quadtree decomposition, which is widely employed to traverse between small and large scales. Moreover, the formulation can treat singularities of any order. Within this framework, 2 sets of semianalytical, scaled boundary shape functions are used to interpolate the displacement and the pore fluid pressure. These shape functions are obtained from the solution of vector and scalar Laplacian, respectively, which are then used to discretise the unknown field variables similar to that of the finite element method. The resulting system of equations are similar in form as that obtained using standard procedures such as the finite element method and, hence, solved using the standard procedures. The formulation is validated using several numerical benchmarks to demonstrate its accuracy and convergence properties.     

A frequency-domain approach for modelling transient elastodynamics using scaled boundary finite element method

This study develops a frequency-domain method for modelling general transient linear-elastic dynamic problems using the semi-analytical scaled boundary finite element method (SBFEM). This approach first uses the newly-developed analytical Frobenius solution to the governing equilibrium equation system in the frequency domain to calculate complex frequency-response functions (CFRFs). This is followed by a fast Fourier transform (FFT) of the transient load and a subsequent inverse FFT of the CFRFs to obtain time histories of structural responses. A set of wave propagation and structural dynamics problems, subjected to various load forms such as Heaviside step load, triangular blast load and ramped wind load, are modelled using the new approach. Due to the semi-analytical nature of the SBFEM, each problem is successfully modelled using a very small number of degrees of freedom. The numerical results agree very well with the analytical solutions and the results from detailed finite element analyses.     

Non‐linear seismic behaviour of concrete gravity dams using coupled finite element–boundary element technique

In this paper, the seismic response of concrete gravity dams is presented using the concept of Continuum Damage Mechanics (CDM) and adopting the hybrid Finite Element–Boundary Element technique (FE–BE). The finite element method is used for discretization of the near field and the boundary element method is employed to model the semi‐infinite far field. Because of the non‐linear nature of the discretizied equations of motion modified Newton–Raphson approach has been used at each time step to linearize them. Damage evolution based on tensile principal strain using mesh‐dependent hardening modulus technique is adopted to ensure the mesh objectivity and to calculate the accumulated damage. The methodology employed is shown to be computationally efficient and consistent in its treatment of both damage growth and damage propagation in gravity dams. Other important features considered in the analysis are: (1) realistic damage modelling for concrete that allows isotropic as well as anisotropic damage state and exhibits stiffness recovery upon load reversals. (2) softening initiation and strain softening criteria for concrete, and (3) proper modelling of semi‐infinite foundation using FE–BE method that allows to consider dam–foundation interaction analysis. As an application of the proposed formulation a gravity dam has been analysed and the results are compared with different foundation stiffnesses. The results of the analysis indicate the importance of including rock foundation in the seismic analysis of dams. Copyright © 1999 John Wiley & Sons, Ltd.