Combined single and smeared crack model in combined finite‐discrete element analysis

Large‐scale discrete element simulations, combined finite‐discrete element simulations as well as a whole range of related problems, involve a large number of separate bodies that interact with each other and in general deform and fracture. In this context there is a need for a robust fracture algorithm applicable to simultaneous multiple fracturing of large numbers of bodies. In this work a fracture model for both initiation and propagation of mode I loaded cracks in concrete in the context of the combined finite‐discrete element method is reported. The algorithm is based on accurate approximation of experimental stress–strain curves for concrete in tension. Copyright © 1999 John Wiley & Sons, Ltd.     

An accurate and robust contact detection algorithm for particle‐solid interaction in combined finite‐discrete element analysis

When applying the combined finite‐discrete element method for analysis of dynamic problems, contact is often encountered between the finite elements and discrete elements, and thus an effective contact treatment is essential. In this paper, an accurate and robust contact detection algorithm is proposed to resolve contact problems between spherical particles, which represent rigid discrete elements, and convex quadrilateral mesh facets, which represent finite element boundaries of structural components. Different contact scenarios between particles and mesh facets, or edges, or vertices have been taken into account. For each potential contact pair, the contact search is performed in an hierarchical way starting from mesh facets, possibly going to edges and even further to vertices. The invalid contact pairs can be removed by means of two reasonable priorities defined in terms of geometric primitives and facet identifications. This hierarchical contact searching scheme is effective, and its implementation is straightforward. Numerical examples demonstrated the accuracy and robustness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

Detonation gas model for combined finite‐discrete element simulation of fracture and fragmentation

The equation of state for expansion of detonation gas together with a model for gas flow through fracturing solid is proposed and implemented into the combined finite‐discrete element code. The equation of state proposed enables gas pressure to be obtained in a closed form for both reversible and irreversible adiabatic expansion, while the gas flow model proposed considers only 1D compressible flow through cracks, hence avoiding full 2D or 3D gas flow through the fracturing solid. When coupled with finite‐discrete element algorithms for solid fracture and fragmentation, the model proposed enables gas pressure to be predicted and energy balance to be preserved. Copyright © 2000 John Wiley & Sons, Ltd.     

A spline wavelet finite‐element method in structural mechanics

The wavelet‐based methods are powerful to analyse the field problems with changes in gradients and singularities due to the excellent multi‐resolution properties of wavelet functions. Wavelet‐based finite elements are often constructed in the wavelet space where field displacements are expressed as a product of wavelet functions and wavelet coefficients. When a complex structural problem is analysed, the interface between different elements and boundary conditions cannot be easily treated as in the case of conventional finite‐element methods (FEMs). A new wavelet‐based FEM in structural mechanics is proposed in the paper by using the spline wavelets, in which the formulation is developed in a similar way of conventional displacement‐based FEM. The spline wavelet functions are used as the element displacement interpolation functions and the shape functions are expressed by wavelets. The detailed formulations of typical spline wavelet elements such as plane beam element, in‐plane triangular element, in‐plane rectangular element, tetrahedral solid element, and hexahedral solid element are derived. The numerical examples have illustrated that the proposed spline wavelet finite‐element formulation achieves a high numerical accuracy and fast convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite strain,finite rotation quadratic tetrahedral element for the combined finite–discrete element method

In the past, the combined finite–discrete element was mostly based on linear tetrahedral finite elements. Locking problems associated with this element can seriously degrade the accuracy of their simulations. In this work an efficient ten‐noded quadratic element is developed in a format suitable for the combined finite–discrete element method (FEMDEM). The so‐called F‐bar approach is used to relax volumetric locking and an explicit finite element analysis is employed. A thorough validation of the numerical method is presented including five static and four dynamic examples with different loading, boundary conditions, and materials. The advantages of the new higher‐order tetrahedral element are illustrated when brought together with contact detection and contact interaction capability within a new fully 3D FEMDEM formulation. An application comparing stresses generated within two drop experiments involving different unit specimens called Vcross and VRcross is shown. The Vcross and VRcross units of ～3.5 × 10⁴kg show very different stress generation implying different survivability upon collision with a deformable floor. The test case shows the FEMDEM method has the capability to tackle the dynamics of complex‐shaped geometries and massive multi‐body granular systems typical of concrete armour and rock armour layers. Copyright © 2009 John Wiley & Sons, Ltd.     

Penalty function method for combined finite–discrete element systems comprising large number of separate bodies

Large‐scale discrete element simulations, the combined finite–discrete element method, DDA as well as a whole range of related methods, involve contact of a large number of separate bodies. In the context of the combined finite–discrete element method, each of these bodies is represented by a single discrete element which is then discretized into finite elements. The combined finite–discrete element method thus also involves algorithms dealing with fracture and fragmentation of individual discrete elements which result in ever changing topology and size of the problem. All these require complex algorithmic procedures and significant computational resources, especially in terms of CPU time. In this context, it is also necessary to have an efficient and robust algorithm for handling mechanical contact. In this work, a contact algorithm based on the penalty function method and incorporating contact kinematics preserving energy balance, is proposed and implemented into the combined finite element code. Copyright © 2000 John Wiley & Sons, Ltd.     

A least‐squares coupling method between a finite element code and a discrete element code

This paper presents a coupling method between a discrete element code CeaMka3D and a finite element code Sem. The coupling is based on a least‐squares method, which adds terms of forces to finite element code and imposes the velocity at coupling particles. For each coupling face, a small linear system with a constant matrix is solved. This method remains conservative in energy and shows good results in applications. Copyright © 2014 John Wiley & Sons, Ltd.     

Fracture and fragmentation of thin shells using the combined finite–discrete element method

A model for fracture and fragmentation of multilayered thin shells has been developed and implemented into the combined finite–discrete element code. The proposed model incorporates an extension of the original combined single and smeared fracture approach to multilayered thin shells; it then combines these with an interaction algorithm that is based on the original distributed potential contact force approach. The developed contact kinematics preserves both energy and momentum balance, whereas the developed fracture model is capable of modelling complex fracture patterns such as fracture of laminated glass under impact. Copyright © 2013 John Wiley & Sons, Ltd.     

A semi‐analytical curved element for linear elasticity based on the scaled boundary finite element method

This work introduces a semi‐analytical formulation for the simulation and modeling of curved structures based on the scaled boundary finite element method (SBFEM). This approach adapts the fundamental idea of the SBFEM concept to scale a boundary to describe a geometry. Until now, scaling in SBFEM has exclusively been performed along a straight coordinate that enlarges, shrinks, or shifts a given boundary. In this novel approach, scaling is based on a polar or cylindrical coordinate system such that a boundary is shifted along a curved scaling direction. The derived formulations are used to compute the static and dynamic stiffness matrices of homogeneous curved structures. The resulting elements can be coupled to general SBFEM or FEM domains. For elastodynamic problems, computations are performed in the frequency domain. Results of this work are validated using the global matrix method and standard finite element analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

A quasi‐implicit characteristic–based penalty finite‐element method for incompressible laminar viscous flows

In this paper, a novel characteristic–based penalty (CBP) scheme for the finite‐element method (FEM) is proposed to solve 2‐dimensional incompressible laminar flow. This new CBP scheme employs the characteristic‐Galerkin method to stabilize the convective oscillation. To mitigate the incompressible constraint, the selective reduced integration (SRI) and the recently proposed selective node–based smoothed FEM (SNS‐FEM) are used for the 4‐node quadrilateral element (CBP‐Q4SRI) and the 3‐node triangular element (CBP‐T3SNS), respectively. Meanwhile, the reduced integration (RI) for Q4 element (CBP‐Q4RI) and NS‐FEM for T3 element (CBP‐T3NS) with CBP scheme are also investigated. The quasi‐implicit CBP scheme is applied to allow a large time step for sufficient large penalty parameters. Due to the absences of pressure degree of freedoms, the quasi‐implicit CBP‐FEM has higher efficiency than quasi‐implicit CBS‐FEM. In this paper, the CBP‐Q4SRI has been verified and validated with high accuracy, stability, and fast convergence. Unexpectedly, CBP‐Q4RI is of no instability, high accuracy, and even slightly faster convergence than CBP‐Q4SRI. For unstructured T3 elements, CBP‐T3SNS also shows high accuracy and good convergence but with pressure oscillation using a large penalty parameter; CBP‐T3NS produces oscillated wrong velocity and pressure results. In addition, the applicable ranges of penalty parameter for different proposed methods have been investigated.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

The maximum principle violations of the mixed‐hybrid finite‐element method applied to diffusion equations

The abundant literature of finite‐element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial–boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time‐dependent solutions for such problems during small time duration obtained by using a non‐conforming mixed‐hybrid finite‐element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite‐difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non‐physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial–boundary value problem will be presented. One of the propositions given to avoid any non‐physical oscillations is to use the mass‐lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd.     

A new finite‐element formulation for electromechanical boundary value problems

In this paper, a new finite‐element formulation for the solution of electromechanical boundary value problems is presented. As opposed to the standard formulation that uses scalar electric potential as nodal variables, this new formulation implements a vector potential from which components of electric displacement are derived. For linear piezoelectric materials with positive definite material moduli, the resulting finite‐element stiffness matrix from the vector potential formulation is also positive definite. If the material is non‐linear in a fashion characteristic of ferroelectric materials, it is demonstrated that a straightforward iterative solution procedure is unstable for the standard scalar potential formulation, but stable for the new vector potential formulation. Finally, the method is used to compute fields around a crack tip in an idealized non‐linear ferroelectric material, and results are compared to an analytical solution. Copyright © 2002 John Wiley & Sons, Ltd.     

An h‐hierarchical adaptive procedure for the scaled boundary finite‐element method

The scaled boundary finite‐element method (a novel semi‐analytical method for solving linear partial differential equations) involves the solution of a quadratic eigenproblem, the computational expense of which rises rapidly as the number of degrees of freedom increases. Consequently, it is desirable to use the minimum number of degrees of freedom necessary to achieve the accuracy desired. Stress recovery and error estimation techniques for the method have recently been developed. This paper describes an h‐hierarchical adaptive procedure for the scaled boundary finite‐element method. To allow full advantage to be taken of the ability of the scaled boundary finite‐element method to model stress singularities at the scaling centre, and to avoid discretization of certain adjacent segments of the boundary, a sub‐structuring technique is used. The effectiveness of the procedure is demonstrated through a set of examples. The procedure is compared with a similar h‐hierarchical finite element procedure. Since the error estimators in both cases evaluate the energy norm of the stress error, the computational cost of solutions of similar overall accuracy can be compared directly. The examples include the first reported direct comparison of the computational efficiency of the scaled boundary finite‐element method and the finite element method. The scaled boundary finite‐element method is found to reduce the computational effort considerably. Copyright © 2002 John Wiley & Sons, Ltd.     

Scaled boundary finite‐element analysis of a non‐homogeneous elastic half‐space

As a result of stresses experienced during and after the deposition phase, a soil strata of uniform material generally exhibits an increase in elastic stiffness with depth. The immediate settlement of foundations on deep soil deposits and the resultant stress state within the soil mass may be most accurately calculated if this increase in stiffness with depth is taken into account. This paper presents an axisymmetric formulation of the scaled boundary finite‐element method and incorporates non‐homogeneous elasticity into the method. The variation of Young's modulus (E) with depth (z) is assumed to take the form E=m_Ez^α, where m_E is a constant and αis the non‐homogeneity parameter. Results are presented and compared to analytical solutions for the settlement profiles of rigid and flexible circular footings on an elastic half‐space, under pure vertical load with αvarying between zero and one, and an example demonstrating the versatility and practicality of the method is also presented. Known analytical solutions are accurately represented and new insight regarding displacement fields in a non‐homogeneous elastic half‐space is gained. Copyright © 2003 John Wiley & Sons, Ltd.     

Unstructured,curved elements for the two‐dimensional high order discontinuous control‐volume/finite‐element method

Quadrilateral and triangular elements with curved edges are developed in the framework of spectral, discontinuous, hybrid control‐volume/finite‐element method for elliptic problems. In order to accommodate hybrid meshes, encompassing both triangular and quadrilateral elements, one single mapping is used. The scheme is applied to two‐dimensional problems with discontinuous, anisotropic diffusion coefficients, and the exponential convergence of the method is verified in the presence of curved geometries. Copyright © 2014 John Wiley & Sons, Ltd.     

A discrete splitting finite element method for numerical simulations of incompressible Navier–Stokes flows

The presence of the pressure and the convection terms in incompressible Navier–Stokes equations makes their numerical simulation a challenging task. The indefinite system as a consequence of the absence of the pressure in continuity equation is ill‐conditioned. This difficulty has been overcome by various splitting techniques, but these techniques incur the ambiguity of numerical boundary conditions for the pressure as well as for the intermediate velocity (whenever introduced). We present a new and straightforward discrete splitting technique which never resorts to numerical boundary conditions. The non‐linear convection term can be treated by four different approaches, and here we present a new linear implicit time scheme. These two new techniques are implemented with a finite element method and numerical verifications are made. Copyright © 2005 John Wiley & Sons, Ltd.