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.     

3D dynamics of discrete element systems comprising irregular discrete elements—integration solution for finite rotations in 3D

An algorithm for transient dynamics of discrete element systems comprising a large number of irregular discrete elements in 3D is presented. The algorithm is a natural extension of contact detection, contact interaction and transient dynamics algorithms developed in recent years in the context of discrete element methods and also the combined finite‐discrete element method. It complements the existing algorithmic procedures enabling transient motion including finite rotations of irregular discrete elements in 3D space to be accurately integrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulation of reinforced concrete structures under impact loading

This study presents the performance of a combined finite‐discrete element method for prediction of the structural response of reinforced concrete beams under impact loading. A combination of finite and discrete element methods enables the modelling of the concrete and the reinforcement before the concrete cracking, as well as a discontinuous nature of the concrete caused by fracture and fragmentation under high impact loading. Discretization of the concrete with triangular finite elements is coupled with one‐dimensional reinforcing bars embedded inside the concrete finite elements. The cracking in the concrete activates the joint elements used to simulate the non‐linear behavior of both concrete and reinforcement. Numerical analysis based on experimental test data has been carried out to simulate the main features of the reinforced concrete beams impacted by free‐falling drop‐weights. A high level of accuracy was demonstrated in various comparisons between the experimental tests and the analysis results, including peak displacement, crack pattern, damage level and failure modes of reinforced concrete beams.     

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.     

A marching cubes based failure surface propagation concept for three‐dimensional finite elements with non‐planar embedded strong discontinuities of higher‐order kinematics

This paper presents an advanced failure surface propagation concept based on the marching cubes algorithm initially proposed in the field of computer graphics and applies it to the embedded finite element method. When modeling three‐dimensional (3D) solids at failure, the propagation of the failure surface representing a crack or shear band should not exhibit a strong sensitivity to the details of the finite element discretization. This results in the need for a propagation of the discrete failure zone through the individual finite elements, which is possible for finite elements with embedded strong discontinuities. Whereas for two‐dimensional calculations the failure zone propagation location is easily predicted by the maximal principal stress direction, more advanced strategies are needed to achieve a smooth failure surface in 3D simulations. An example for such method is the global tracking algorithm, which predicts the crack path by a scalar level set function computed on the basis of the solution of a simplified heat conduction like problem. Its prediction may though lead to various scenarios on how the failure surface may propagate through the individual finite elements. In particular, for a hexahedral eight‐node finite element, 256 such cases exist. To capture all those possibilities, the marching cubes algorithm is combined with the global tracking algorithm and the finite elements with embedded strong discontinuities in this work. In addition, because many of the possible cases result in non‐planar failure surfaces within a single finite element and because the local quantities used to describe the kinematics of the embedded strong discontinuities are physically meaningful in a strict sense only for planar failure surfaces, a remedy for such scenarios is proposed. Various 3D failure propagation simulations outline the performance of the proposed concept. Copyright © 2013 John Wiley & Sons, Ltd.     

Bounds for element size in a variable stiffness cohesive finite element model

The cohesive finite element method (CFEM) allows explicit modelling of fracture processes. One form of CFEM models integrates cohesive surfaces along all finite element boundaries, facilitating the explicit resolution of arbitrary fracture paths and fracture patterns. This framework also permits explicit account of arbitrary microstructures with multiple length scales, allowing the effects of material heterogeneity, phase morphology, phase size and phase distribution to be quantified. However, use of this form of CFEM with cohesive traction–separation laws with finite initial stiffness imposes two competing requirements on the finite element size. On one hand, an upper bound is needed to ensure that fields within crack‐tip cohesive zones are accurately described. On the other hand, a lower bound is also required to ensure that the discrete model closely approximates the physical problem at hand. Both issues are analysed in this paper within the context of fracture in multi‐phase composite microstructures and a variable stiffness bilinear cohesive model. The resulting criterion for solution convergence is given for meshes with uniform, cross‐triangle elements. A series of calculations is carried out to illustrate the issues discussed and to verify the criterion given. These simulations concern dynamic crack growth in an Al₂O₃ ceramic and in an Al₂O₃/TiB₂ ceramic composite whose phases are modelled as being hyperelastic in constitutive behaviour. Copyright © 2004 John Wiley & Sons, Ltd.     

Embedded representation of fracture in concrete with mixed finite elements

As an alternative to the smeared and discrete crack representations, an embedded representation of fracture for finite element analysis of concrete structures is presented. The three-field Hu–Washizu variational statement is extended to bodies with internal discontinuities. The extended variational statement is then utilized for formulating elements with a discontinuous displacement field. The new elements are capable of modelling different deformation modes of an internal discontinuity at the element level. The satisfactory performance of the embedded crack representation is verified by several case studies on concrete fracture.     

NBS contact detection algorithm for bodies of similar size

Large-scale discrete element simulations, as well as a whole range of related problems, involve contact of a large number of separate bodies. In this context an efficient and robust contact detection algorithm is necessary. There has been a number of contact detection algorithms with total detection time (CPU time needed to detect all couples close to each other) proportional to Nln(N) (where N is the total number of separate bodies) reported in recent years. In this work a contact detection algorithm with total detection time proportional to N is reported. The algorithm is termed NBS, which stands for no binary search. In other words, the proposed algorithm involves no binary search at any stage. In addition the performance of the algorithm in terms of total detection time is not influenced by packing density, while memory requirements are insignificant. The only limitation of the algorithm is its applicability to the systems comprising bodies of similar size. © 1998 John Wiley & Sons, Ltd.     

Simulations of fracture and fragmentation of geologic materials using combined FEM/DEM analysis

Results are presented from a study investigating the effect of explosive and impact loading on geological media using the Livermore distinct element code (LDEC). LDEC was initially developed to simulate tunnels and other structures in jointed rock masses with large numbers of intact polyhedral blocks. However, underground structures in jointed rock subjected to explosive loading can fail due to both rock motion along preexisting interfaces and fracture of the intact rock mass itself. Many geophysical applications, such as projectile penetration into rock, concrete targets, and boulder fields, require a combination of continuum and discrete methods in order to predict the formation and interaction of the fragments produced. In an effort to model these types of problems, we have implemented Cosserat point theory and cohesive element formulations into the current version of LDEC, thereby allowing for dynamic fracture and combined finite element/discrete element simulations. Results of a large-scale LLNL simulation of an explosive shock wave impacting an elaborate underground facility are also discussed. It is confirmed that persistent joints lead to an underestimation of the impact energy needed to fill the tunnel systems with rubble. Non-persistent joint patterns, which are typical of real geologies, inhibit shear within the surrounding rock mass and significantly increase the load required to collapse a tunnel.     

MR linear contact detection algorithm

Large‐scale discrete element simulations, as well as a whole range of related problems, involve contact of a large number of separate bodies and an efficient and robust contact detection algorithm is necessary. There has been a number of contact detection algorithms with total detection time proportional to N ln(N) (where N is the total number of separate bodies) reported in the past. In more recent years algorithms with total CPU time proportional to N have been developed. In this work, a novel contact detection algorithm with total detection time proportional to N is proposed. The performance of the algorithm is not influenced by packing density, while memory requirements are insignificant. The algorithm is applicable to systems comprising bodies of a similar size. The algorithm is named MR (Munjiza–Rougier: Munjiza devised the algorithm, Rougier implemented it). In the second part of the paper the algorithm is extended to particles of different sizes. The new algorithm is called MMR (multi‐step MR) algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of fracture/breakup of concrete magazine using cohesive element

To determine the potential hazard zone around an ammunition storage magazine is the research interest of many researchers working on explosive safety. Carrying out an explosion test of an ammunition storage magazine is expensive. As such, numerical modeling offers an economical solution to supplement the field tests, and it has been widely used to simulate the magazine breakup. In the literature and own previous studies, numerical simulations often resorted to the Lagrangian finite‐element method and used the nodal‐split algorithm to model the concrete fracture/breakup upon internal detonation. Pitfalls in that approach have not been fully recognized [ 1 – 3 ]. In it, all elements initially connected to the same node would be spilt in all directions once the nodal strain reached the threshold. This strain‐based splitting criterion lacks of sound physical meaning. To overcome those pitfalls, the present study adopts a kind of interface element, namely a cohesive element, to simulate the fracture/breakup. Strain rate dependent experiments were carried out to obtain the fracture criteria, which were then applied to define the threshold for element‐face split. Results show that it yields good agreement with experiment tests, including Split Hopkinson Pressure Bar (SHPB) on concrete specimen and breakup of concrete magazine upon detonation.     

A novel discrete element method based on the distance potential for arbitrary 2D convex elements

A new 2‐dimensional discrete element method, which is able to simulate a system involving a large number of arbitrary convex elements, is proposed. In this approach, a novel distance potential function is defined using a normalized format of the penetrated distance between contact couples, while a holonomic and precise algorithm for contact interaction is established, accounting for the influence of the tangential contact force. Furthermore, the new contact detection algorithm is well suited for nonuniform blocks unlike the common no binary search method that requires uniform elements. The proposed method retains the merit of the combined finite‐discrete element method and avoids its deficiencies. Compared with the existing finite‐discrete element method, the distance potential function has a clear physical meaning, where the calculation of contact interaction avoids the influence of the element shape. Accordingly, the new method completely gets rid of the restraint of uniform element type and can be applied to arbitrary convex elements. The new method is validated with well‐known benchmark examples, and the results are in very good agreement with existing experimental measurement and analytical solutions. Finally, the proposed method is applied to simulate the Tangjiashan landslide.     

A contact algorithm for 3D discrete and finite element contact problems based on penalty function method

A contact algorithm in the context of the combined discrete element (DE) and finite element (FE) method is proposed. The algorithm, which is based on the node-to-surface method used in finite element method, treats each spherical discrete element as a slave node and the surfaces of the finite element domain as the master surfaces. The contact force on the contact interface is processed by using a penalty function method. Afterward, a modification of the combined DE/FE method is proposed. Following that, the corresponding numerical code is implemented into the in-house developed code. To test the accuracy of the proposed algorithm, the impact between two identical bars and the vibration process of a laminated glass plate under impact of elastic sphere are simulated in elastic range. By comparing the results with the analytical solution and/or that calculated by using LS-DYNA, it is found that they agree with each other very well. The accuracy of the algorithm proposed in this paper is proved.     

An implicit adaptive node‐splitting algorithm to assess the failure mechanism of inelastic elastomeric continua

This contribution presents a mesh adaptive crack propagation scheme for the evaluation of the viscoelastic fracture response of elastomers at large strains and up to high loading rates. The approach accounts for micromechanical based features of both elastic and viscoelastic bulk responses of idealized polymer networks. To this end, the Bergstörm–Boyce model is considered to introduce hyperelastic and nonlinear finite viscoelastic responses. Moreover, the crack driving force and the crack driving direction are predicted by the material force approach. A consistent thermodynamic framework for the combined configurational motion in viscoelastic continua at finite strain regime is discussed. The fracture toughness of non‐strain‐crystallizing elastomers shows strong rate dependency and the energy release rate versus the rate of tearing to be a fundamental material property. Therefore, in this contribution, a dynamic fracture criterion, which is a function of the rate of crack growth, is shown to be adequate in numerical simulations. The use of the presented method enables to study fracture behaviour of any material nonlinearity within the implicit time integration. Main feature of the proposed algorithm is restructuring the overall discrete system by duplication of crack front DOFs based on minimization of the overall energy via the Griffith criterion. Copyright © 2014 John Wiley & Sons, Ltd.     

A combined finite‐discrete element method for simulating pharmaceutical powder tableting

The pharmaceutical powder and tableting process is simulated using a combined finite‐discrete element method and contact dynamics for irregular‐shaped particles. The particle‐scale formulation and two‐stage contact detection algorithm which has been developed for the proposed method enhances the overall calculation efficiency for particle interaction characteristics. The irregular particle shapes and random sizes are represented as a pseudo‐particle assembly having a scaled up geometry but based on the variations of real powder particles. Our simulations show that particle size, shapes and material properties have a significant influence on the behaviour of compaction and deformation. Copyright © 2005 John Wiley & Sons, Ltd.     

Adaptive combined DE/FE algorithm for brittle fracture of plane stress problems

A novel adaptive combined DE/FE algorithm is proposed to simulate the fracture procedure of brittle materials of plane stress problems. The main concept of the approach is that a model is composed of the finite element completely at the initial stage without any discrete element generated until portion of the model grid becoming severely deformed; and then the model is fragmented into two subdomains, the finite element (FE) and the discrete element (DE) subdomains. The interface force between the two subdomains is calculated by using the penalty method. An extrinsic cohesive fracture model is employed to simulate the brittle fracture procedure only in the DE subdomain. The adaptive algorithm may allow for the use of the accurate and efficient FEs in the lower distorted region and the DEs which are automatically generated in the severely deformed FE region . The feasibility of the adaptive algorithm is validated by the impact fracture simulation of a glass beam. The comparison of calculation time consumption shows that the adaptive algorithm has a higher efficiency than the DEM. At last, the impact fracture behavior of a laminated glass beam is simulated, and the cracks propagation is compared with the experimental results showing that the adaptive algorithm can be implemented to capture some fracture characteristics of brittle materials.     

FRP-混凝土三点受弯梁损伤粘结模型有限元分析

该文采用双线形损伤粘结模型研究带切口FRP-混凝土三点受弯梁(3PBB)I型加载下的界面断裂性能。通过有限元参数分析,详细讨论了界面粘结强度、界面粘结能、混凝土抗拉强度、混凝土断裂能对3PBB受力性能的影响。数值模拟表明,FRP-混凝土界面有两种破坏形式,包括FRP-混凝土界面的损伤脱粘和界面混凝土的损伤脱粘破坏,与实验所观察到的现象一致。两种破坏形式尽管在宏观上均表现为界面脱粘,但破坏机制却不同。FRP-混凝土界面的损伤粘结模型与混凝土的拉伸塑性损伤模型相结合,不但再现了3PBB的宏观力学性能,数值分析得到的荷载-位移曲线接近实验结果,而且还能详细展示FRP-混凝土界面的损伤、断裂破坏过程以及损伤在FRP-混凝土界面和界面混凝土之间的转移,能够预测构件的承载力,有助于界面优化设计,这是单纯以能量判据预测裂纹发展的经典断裂力学方法所无法做到的。     

POD‐based model reduction with empirical interpolation applied to nonlinear elasticity

The constantly rising demands on finite element simulations yield numerical models with increasing number of degrees‐of‐freedom. Due to nonlinearity, be it in the material model or of geometrical nature, the computational effort increases even further. For these reasons, it is today still not possible to run such complex simulations in real time parallel to, for example, an experiment or an application. Model reduction techniques such as the proper orthogonal decomposition method have been developed to reduce the computational effort while maintaining high accuracy. Nonetheless, this approach shows a limited reduction in computational time for nonlinear problems. Therefore, the aim of this paper is to overcome this limitation by using an additional empirical interpolation. The concept of the so‐called discrete empirical interpolation method is translated to problems of solid mechanics with soft nonlinear elasticity and large deformations. The key point of the presented method is a further reduction of the nonlinear term by an empirical interpolation based on a small number of interpolation indices. The method is implemented into the finite element method in two different ways, and it is extended by using different solution strategies including a numerical as well as a quasi‐Newton tangent. The new method is successfully applied to two numerical examples concerning hyperelastic as well as viscoelastic material behavior. Using the extended discrete empirical interpolation method combined with a quasi‐Newton tangent enables reductions in computational time of factor 10 with respect to the proper orthogonal decomposition method without empirical interpolation. Negligibly, orders of error can be reached. Copyright © 2015 John Wiley & Sons, Ltd.     

Phase‐field material point method for brittle fracture

The material point method for the analysis of deformable bodies is revisited and originally upgraded to simulate crack propagation in brittle media. In this setting, phase‐field modelling is introduced to resolve the crack path geometry. Following a particle in cell approach, the coupled continuum/phase‐field governing equations are defined at a set of material points and interpolated at the nodal points of an Eulerian, ie, non‐evolving, mesh. The accuracy of the simulated crack path is thus decoupled from the quality of the underlying finite element mesh and relieved from corresponding mesh‐distortion errors. A staggered incremental procedure is implemented for the solution of the discrete coupled governing equations of the phase‐field brittle fracture problem. The proposed method is verified through a series of benchmark tests while comparisons are made between the proposed scheme, the corresponding finite element implementation, and experimental results.     

An efficient algorithm for modelling progressive damage accumulation in disordered materials

This paper presents an efficient algorithm for the simulation of progressive fracture in disordered quasi‐brittle materials using discrete lattice networks. The main computational bottleneck involved in modelling the fracture simulations using large discrete lattice networks stems from the fact that a new large set of linear equations needs to be solved every time a lattice bond is broken. Using the present algorithm, the computational complexity of solving the new set of linear equations after breaking a bond reduces to a simple triangular solves (forward elimination and backward substitution) using the already Cholesky factored matrix. This algorithm using the direct sparse solver is faster than the Fourier accelerated iterative solvers such as the preconditioned conjugate gradient (PCG) solvers, and eliminates the critical slowing down associated with the iterative solvers that is especially severe close to the percolation critical points. Numerical results using random resistor networks for modelling the fracture and damage evolution in disordered materials substantiate the efficiency of the present algorithm. In particular, the proposed algorithm is especially advantageous for fracture simulations wherein ensemble averaging of numerical results is necessary to obtain a realistic lattice system response. Copyright © 2005 John Wiley & Sons, Ltd.