Experimental validation of a computationally efficient beam element for combined finite–discrete element modelling of structures in distress

 The combined finite/discrete element method is adopted to model the pre-failure and post failure transient dynamics of reinforced concrete structures. For this purpose a novel beam element is introduced in order to increase CPU and RAM efficiency. In this paper the accuracy and reliability of this element is assessed when used in dynamic loading conditions. Experiments, which have already been undertaken at the Swiss Federal Institute of Technology, are used for comparison and validation. The results indicate that the element introduced is capable of accurately modelling inertia and contact effects in pre and post failure dynamics, up to collapse. Received: 1 August 2002 / Accepted: 9 January 2002 A debt of gratitude is owed to Prof. Bachman and Mrs. N. Ammann for their kind and sincerely appreciated assistance in the provision of the reports from the Swiss Federal Institute of Technology. The assistance provided by Dr. W. Ammann and Dr. S. Heubbe-Walker in the translation of the reports is also gratefully acknowledged.     

NSCD discrete element method for modelling masonry structures

The aim of this paper is to present a discrete element approach for the study of stonework. In the present work, a masonry structure is considered as a collection of rigid or deformable blocks, interacting together by contact laws. In this paper, we use the non‐smooth contact dynamics (NSCD) resolution method mainly used for the modelling of granular media. In the considered masonry structures we define, on an elementary cell, average local strain and stress tensors. These definitions are valid under dynamical loading of the structure, taking into account rotations. We present their use on academic and on real masonry structures. Copyright © 2005 John Wiley & Sons, Ltd.     

A calibration framework for discrete element model parameters using genetic algorithms

In this research, a universal framework for automated calibration of microscopic properties of modeled granular materials is proposed. The proposed framework aims at industrial scale applications, where optimization of the computational time step is important. It can be generally applied to all types of DEM simulation setups. It consists of three phases: data base generation, parameter optimization, and verification. In the first phase, DEM simulations are carried out on a multi-dimensional grid of sampled input parameter values to generate a database of macroscopic material responses. The database and experimental data are then used to interpolate the objective functions with respect to an arbitrary set of parameters. In the second phase, the Non-dominated Sorting Genetic Algorithm II (NSGA-II) is used to solve the calibration multi-objective optimization problem. In the third phase, the DEM simulations using the results of the calibrated input parameters are carried out to calculate the macroscopic responses that are then compared with experimental measurements for verification and validation.The proposed calibration framework has been successfully demonstrated by a case study with two-objective optimization for the model accuracy and the simulation time. Based on the concept of Pareto dominance, the trade-off between these two conflicting objectives becomes apparent. Through verification and validation steps, the approach has proven to be successful for accurate calibration of material parameters with the optimal simulation time.     

A spherical discrete element model: calibration procedure and incremental response

When using spherical elements within the Discrete Element Method, computational costs can be kept low even for large numbers of elements. However, this oversimplification of the granular geometry has drawbacks when quantitatively assessing the model even for frictional geomaterials. To overcome this limitation, the local constitutive law must at least take into account the transfer of a moment between elements. This moment, which is added to normal and shear local interaction forces, increases the number of local parameters. Moreover, when local plastic thresholds are considered, the calibration of the model becomes tricky. With such a set of local parameters, a calibration procedure is proposed, which attempts to define the respective role of each parameter in the macroscopic behavior. A series of numerical simulations of triaxial compression tests has been performed to check the capability of this model to get good quantitative results and the incremental behavior of the numerical medium is studied by performing a series of axisymmetric stress probes with varying directions. The corresponding strain responses are measured. From different initial stress states, the results indicate that the incremental response is well described by elastoplasticity with a single mechanism, and a non-associative flow rule.     

Two-dimensional discrete element modelling of bender element tests on an idealised granular material

The small-strain (elastic) shear stiffness of soil is an important parameter in geotechnics. It is required as an input parameter to predict deformations and to carry out site response analysis to predict levels of shaking during earthquakes. Bender element testing is often used in experimental soil mechanics to determine the shear (S-) wave velocity in a given soil and hence the shear stiffness. In a bender element test a small perturbation is input at a point source and the propagation of the perturbation through the system is measured at a single measurement point. The mechanics and dynamics of the system response are non-trivial, complicating interpretation of the measured signal. This paper presents the results of a series of discrete element method (DEM) simulations of bender element tests on a simple, idealised granular material. DEM simulations provide the opportunity to study the mechanics of this testing approach in detail. The DEM model is shown to be capable of capturing features of the system response that had previously been identified in continuum-type analyses of the system. The propagation of the wave through the sample can be monitored at the particle-scale in the DEM simulation. In particular, the particle velocity data indicated the migration of a central S-wave accompanied by P-waves moving along the sides of the sample. The elastic stiffness of the system was compared with the stiffness calculated using different approaches to interpreting bender element test data. An approach based upon direct decomposition of the signal using a fast-Fourier transform yielded the most accurate results.     

Benchmark tests for verifying discrete element modelling codes at particle impact level

Over the past 30 years, the Discrete Element Method (DEM) has rapidly gained popularity as a tool for modelling the behaviour of granular assemblies and is being used extensively in both scientific and industrial applications. However, it is far from clear from reviewing the literature whether the large number of DEM codes have been verified and checked against fundamental benchmark problems. DEM simulates the dynamics of each particle in an assembly by calculating the acceleration resulting from all the contact forces and body forces. It is clearly necessary that such a model be validated or verified by comparing with experimental results, analytical solutions or other numerical results (e.g. Finite Element Analysis (FEA) results) at particle impact level. There appears to be no standard benchmark tests against which DEM codes can be verified. It is thus essential and useful to establish a set of standard benchmark tests to confirm that these DEM codes are modelling the particle dynamics as intended. This paper proposes a set of benchmark tests to verify DEM codes at particle impact level for spherical particles. The analytical solutions derived from elasticity theory for elastic normal collision of two spheres or a sphere with a rigid plane are first reviewed. These analytical solutions apply only to the elastic regime for normal impact. Secondly, the analytical solutions of frictional oblique impact between two spheres or a sphere with a rigid plane are scrutinized and derived. These analytical solutions originate from the dynamics principles and should be satisfied for any DEM contact force model with prescribed friction and restitution coefficients. A set of eight benchmark tests are designed and performed using commercial DEM codes. Test 1 and Test 2 consider the elastic normal impact of two spheres or a sphere with a rigid plane, whereas the other tests (Test 3–Test 8) investigate the energy dissipation due to the collision. These benchmark tests also involve different types of material. The DEM results were compared with the analytical solutions, experimental or FEA results found in the literature. All benchmark tests showed good to excellent match, providing a quantitative verification for the codes used in this study. These benchmark tests not only verify DEM codes but also enhance the understanding of fundamental impact phenomena for modelling a large number of particles.     

A modelling and verification approach for soybean seed particles using the discrete element method

To build a discrete element method (DEM) model of soybean seed particles, the shape and size of soybean seed particles were measured and analysed. The results showed that the shape of a soybean seed particle could be approximated to an ellipsoid and that the dispersity in size could be approximated by a normal distribution. Additionally, a certain functional relationship between the primary dimension and secondary dimensions was determined. On this basis, an approach for modelling soybean seed particles based on the multi-sphere (MS) method was proposed. The soybean seed particle was simplified to an ellipsoid with the averaged size of one hundred randomly selected soybean seeds. The model of a single soybean seed particle was built by filling spheres within the ellipsoid. For modelling soybean seed assembly, the primary dimension was generated according to the normal distribution, and the other secondary dimensions were calculated based on their relationships with the primary dimension. In this way, the model of soybean seed assembly with different sizes and distributions was built. In this paper, four varieties of soybean seed were used. By comparing the simulated results and experimental results both in piling tests and “self-flow screening” tests, when the number of filling spheres was five, the simulated results were close to those obtained experimentally. Therefore, the feasibility and validity of the modelling method for soybean seed particles that we proposed were verified. Finally, an application case was employed to show how to use the soybean seed particle model and the discrete element method to analyse the discharging process of a silo.     

The combined plastic and discrete fracture deformation framework for finite-discrete element methods

The combined finite-discrete element method (FDEM) was originally developed for fracture and fragmentation of brittle materials, more specifically for cementitious and rock-like materials. In this work, a combination of a discrete crack and plastic deformation has been combined and applied to FDEM simulation of fracture. The deformation is described using a FDEM-specific mechanistic approach with plastic deformation being formulated in material embedded coordinate systems leading to multiplicative decomposition and plastic flow, that is, resolved in stretch space; this is combined with the FDEM fracture and fragmentation criteria. The result and main novelty of the present work is a robust framework for simulation of large strain solid deformation combined with a multiplicative decomposition-based model that simultaneously involves elasticity, plasticity, and fracture.     

A simplified updated Lagrangian approach for combining discrete and finite element methods

This paper proposes the use of a specific combination of discrete and finite element methods for the simulation of systems of deformable bodies in order to reduce the computational cost, when certain assumptions can be made. In particular, the Updated Lagrangian finite element formulation and the central difference time integration method are employed together with certain simplifying assumptions in order to linearize this highly nonlinear contact problem and obtain solutions with realistic computational cost and sufficiently good accuracy. Furthermore, the paper discusses software implementation issues and the advantages that the Java technologies can offer in the development of such engineering applications.     

Particle shape effects in discrete element modelling of cohesive angular particles

This paper investigates the effect of particle angularity on general granular response concentrating on flow and stress-strain behaviour. A 2D polygon DEM model is developed to a 3D polyhedron model. The effect of particle shape on the response of polygons in simple shear and polyhedra under gravity flow is investigated using regular shapes with rounded vertices. The study concentrates on the angularity rather than aspect ratio by comparing circles, near squares and near equilateral triangles in 2D and spheres, tetrahedra and octahedra in 3D. In summary the more angular the particle the greater the resistance to the forcing load and the flowability is reduced. A mix of spheres and octahedra demonstrates an approximate linear combination of effects.     

On the central difference algorithm in discrete element modelling of impact

The paper provides a comprehensive analysis of the accuracy and stability of the central difference scheme when applied to simulate a simple impact problem in the context of the discrete element approach. It is revealed that the algorithm exhibits some different behaviour due to the inherent non‐linearity/discontinuity of the impact system. Particularly, for an elastic or slightly inelastic impact, the stable/unstable region governing the selection of a maximum time step size is essentially different from that defined by the conventional linear stability criterion, and thus a smaller time step should be employed in order to reduce the possible occurrence of an unexpected numerical instability. Copyright © 2005 John Wiley & Sons, Ltd.     

Probabilistic calibration of discrete element simulations using the sequential quasi-Monte Carlo filter

The calibration of discrete element method (DEM) simulations is typically accomplished in a trial-and-error manner. It generally lacks objectivity and is filled with uncertainties. To deal with these issues, the sequential quasi-Monte Carlo (SQMC) filter is employed as a novel approach to calibrating the DEM models of granular materials. Within the sequential Bayesian framework, the posterior probability density functions (PDFs) of micromechanical parameters, conditioned to the experimentally obtained stress–strain behavior of granular soils, are approximated by independent model trajectories. In this work, two different contact laws are employed in DEM simulations and a granular soil specimen is modeled as polydisperse packing using various numbers of spherical grains. Knowing the evolution of physical states of the material, the proposed probabilistic calibration method can recursively update the posterior PDFs in a five-dimensional parameter space based on the Bayes’ rule. Both the identified parameters and posterior PDFs are analyzed to understand the effect of grain configuration and loading conditions. Numerical predictions using parameter sets with the highest posterior probabilities agree well with the experimental results. The advantage of the SQMC filter lies in the estimation of posterior PDFs, from which the robustness of the selected contact laws, the uncertainties of the micromechanical parameters and their interactions are all analyzed. The micro–macro correlations, which are byproducts of the probabilistic calibration, are extracted to provide insights into the multiscale mechanics of dense granular materials.     

Modelling dilation in an idealised asphalt mixture using discrete element modelling

This paper investigates the use of discrete element modelling (DEM) to simulate the behaviour of a highly idealised bituminous mixture under uniaxial and triaxial compressive creep tests. The idealised mixture comprises single-sized spherical (sand-sized) particles mixed with bitumen and was chosen so that the packing characteristics are known (dense random packing) and the behaviour of the mixture will be dominated by the bitumen and complex aggregate interlock effects will be minimised. In this type of approach the effect of the bitumen is represented as shear and normal contact stiffnesses. A numerical sample preparation procedure has been developed to ensure that the final specimen is isotropic and has the correct volumetrics. Elastic contact properties have been used to investigate the effect of the shear and normal contact stiffnesses on bulk material properties. The bulk modulus was found to be linearly dependent on the normal contact stiffness and independent of the shear contact stiffness. Poisson’s ratio was found to be dependent on only the ratio of the shear contact stiffness to the normal contact stiffness. An elastic contact has been assumed for the compressive normal contact stiffness and a viscoelastic contact for shear and tensile normal contact stiffness to represent the contact behaviour in idealised mixture. The idealised mixture is found to dilate when the ratio of compressive to tensile contact stiffness increases as a function of loading time. Uniaxial and triaxial viscoelastic simulations have been performed to investigate the effect of stress ratio on the rate of dilation with shear strain for the sand asphalt. The numerical results have been validated with experimental data.     

The importance of modelling ballast particle shape in the discrete element method

The discrete element method has been used to model railway ballast. Particles have been modelled using both spheres and clumps of spheres. A simple procedure has been developed to generate clumps which resemble real ballast particles much more so than spheres. The influence of clump shape on the heterogeneous stresses within an aggregate has been investigated, and it has been found that more angular clumps lead to a greater degree of homogeneity. A box test consisting of one cycle of sleeper load after compaction has been performed on an aggregate of spheres and also on an alternative aggregate of clumps. The interlocking provided by the clumps provides a much more realistic load- deformation response than the spheres and the clumps will be the basis for future work on ballast degradation under cyclic loading.     

Deformation localisation modelling of polymer foam microstructure under compression: A new approach by discrete element modelling

A discrete element model (DEM) has been developed to represent the behaviour of the microscopic structure of polymer cellular material, consisting of closed-cells. In DEM, the polymer foam is represented as an assembly of particles which model the closed-cells. The behaviour of the particle is based on the Gibson model and depends on morphologic and mechanical parameters. The present numerical study demonstrates the effect of deformation localisation on the microstructure. It is noted that the cell morphologic parameters and the distribution of various size cells in the specimen have a significant influence of the local deformation. The effect of macroscopic faults is also studied.     

Finite element modelling of multiple cohesive discrete crack propagation in reinforced concrete beams

This paper presents a finite element (FE) model for fully automatic simulation of multiple discrete crack propagation in reinforced concrete (RC) beams. The discrete cracks are modelled based on the cohesive/fictitious crack concept using nonlinear interface elements with a bilinear tensile softening constitutive law. The model comprises an energy-based crack propagation criterion, a simple remeshing procedure to accommodate crack propagations, two state variable mapping methods to transfer structural responses from one FE mesh to another, and a local arc-length algorithm to solve system equations characterised by material softening. The bond-slip behaviour between reinforcing bars and surrounding concrete is modelled by a tension-softening element. An example RC beam with well-documented test data is simulated. The model is found capable of automatically modelling multiple crack propagation. The predicted cracking process and distributed crack pattern are in close agreement with experimental observations. The load-deflection relations are accurately predicted up to a point when compressive cracking becomes dominant. The effects of bond-slip modelling and the efficiency and effectiveness of the numerical algorithms, together with the limitations of the current model, are also discussed.     

Comparative discrete element modelling of a vibratory sieving process with spherical and rounded polyhedron particles

Current spherical particle usage in discrete element modelling (DEM) is not able to accurately reflect the particle shape effect in some specific industrial applications. This study specifically investigated the effect of particle shape in discrete element modelling of a vibratory sieving process, with the focus on comparing results from spherical and non-spherical modelling methods. The particle size distribution of an iron ore material was initially obtained experimentally through vibratory sieving tests. An identical process was replicated in DEM with both spheres and non-spherical particles, and resulting particle size distributions were subsequently compared against the experimental results. A rounded polyhedron shape was utilised to calibrate and generate non-spherical particles based on a 2D particle shape characterisation process. Modelling results suggested that the rounded polyhedron method was able to accurately reflect the particle-sieve contacts without excessive rolling resistance tuning, which was required by the spheres.     

Assessment of mill lifter bar deflection measurements using wavelets and discrete element methods

This paper shows how Partial Least Square Regression (PLS) methods can be used to model sensor data of spectral character. The modelling approach has been applied on a tumbling mill where a strain gauge sensor measures the deflection of a lifter bar when it hits the charge. The deflection of the lifter bar during every mill revolution gives rise to a characteristic signal profile that is shown to contain information on both the charge position and grinding performance. As a signal pre-processing method the discrete wavelet transform is used. It distinctly shows a capability of signal feature extraction where both time and frequency are of interest. Its well-known ability to achieve good data compression without loss of information is also demonstrated, a data reduction ratio of 20:1 is obtained here. Modelling results demonstrate that different operating conditions are well distinguishable from each other and by that the finding of proper operating regimes are highly feasible. Grinding parameters that are normally measured in the laboratory are now readily modelled from the on-line signal. A further objective of this paper is to link the experimentally obtained strain gauge sensor data with computational data from a discrete element mill model (DEM). This enables to visualise the charge motion and helps to interpret the complex phenomena that take place inside a grinding mill measured by the strain gauge sensor. The approach taken is to simulate the behaviour of a rubber lifter when it is exposed to forces from the grinding charge in a two-dimensional DEM mill model using a particle flow code. The deflection profile obtained from the DEM simulation shows a reasonably good correspondence to pilot mill measurements.     

Enhanced continua and discrete lattices for modelling granular assemblies

This article discusses the derivation of continuum models that can be used for modelling the inhomogeneous mechanical behaviour of granular assemblies. These so-called kinematically enhanced models are of the strain-gradient type and of the strain-gradient micro-polar type, and are derived by means of homogenizing the micro-structural interactions between discrete particles. By analysis of the body wave dispersion curves, the enhanced continuum models are compared to corresponding discrete lattice models. Accordingly, it can be examined up to which deformation level the continuum models are able to accurately describe the discrete particle behaviour. Further, the boundary conditions for the enhanced continuum models are formulated, and their stability is considered. It is demonstrated how to use the body wave dispersion relations for the assessment of stability.