A method for representing structural boundaries by mathematical equations and triangle meshes in DEM simulations

The representation of structural boundaries is of significant importance for the applications of the discrete element method in an industry. In recent decades, triangle meshes are extensively used for representing structural boundaries. Structural boundaries of industrial objects are composed by regular shapes and irregular shapes in many occasions. A method for representing structural boundaries with regular shapes and irregular shapes has been developed by combining mathematical equations and triangle meshes for computational efficiency. When structural boundaries are represented by mathematical equations and triangle meshes, gaps or protuberances may exist at the connection boundaries between regular shapes and irregular shapes. For the exactness of representation of structural boundaries, gaps or protuberances are identified, expressed and treated successively, so that the geometrical shapes composed by regular shapes described with mathematical equations and irregular shapes described with triangle meshes can replace the original structural shapes. Two series of numerical tests have been conducted to verify this method. The results showed that this method can effectively represent complicated structural boundaries containing regular shapes and irregular shapes and greatly reduce computational cost in the contact detection between particles and structural boundaries. Copyright © 2016 John Wiley & Sons, Ltd.     

A combined approach for modeling particle behavior in granular impact damper using discrete element method and cellular automata

A particle impact damper is a vibration absorber type that consists of a container attached to a primary vibrating structure. It also contains many particles that are constrained to move inside the container, whereby the damping effect can be obtained by collision between particles and the container. The discrete element method (DEM) has been developed for modeling granular systems, where the kinematics of each particle are calculated numerically using the equations of motion. However, the computational time is significant since the algorithm checks for particle contacts for all possible particle combinations. The use of a cellular automata (CA) modeling technique may provide increased computational efficiency due to the local updating of variables, and the discrete treatment of time and space. In this study, we propose a new approach combining DEM with CA for modeling a granular damper under a forced excitation. We use DEM to describe the particle motion according to the equations of motion, while CA is introduced for the particle contact checks in discrete space. We also investigate the effect of simplification in the contact force model, which allows the unit time step of numerical integration to become larger than that used in the strict model. It is shown that the suggested particle contact scanning method and the force approximation model contribute to the reduction of the computational time, and neither degenerates the calculation accuracy nor causes the numerical instability.     

An energy‐consistent material‐point method for dynamic finite deformation plasticity

Energy consistency for the material‐point method (MPM) is examined for thermodynamically consistent hyperelastic‐plastic materials. It is shown that MPM can be formulated with implicit, three‐ field variational, finite element algorithms which dissipate energy and conserve momentum for that class of material models. With a consistent mass matrix the resulting overall numerical method inherits the energy‐dissipative and momentum‐conserving properties of the mesh solution. Thus, the proposed MPM algorithm satisfies by construction a time‐discrete form of the second law of thermo‐ dynamics. Properties of the method are illustrated in numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Polygon‐based contact resolution for superquadrics

The representation of discrete objects in the discrete element modelling is a fundamental issue, which has a direct impact on the efficiency of discrete element implementation and the dynamic behaviour of particulate systems. Disks and spheres are the most commonly used geometric shapes due to their geometric simplicity and computational efficiency, but they are unable to provide resistance to rolling motion. For this reason, some non‐circular/spherical objects, such as polygons/polyhedrons, superquadrics, or the clustering of disks/spheres to form irregular shapes, are introduced. When superquadrics are used as discrete elements, the bottleneck of contact resolution is associated with the searching for intersections of two non‐linear functions, which is a very expensive operation and may sometimes fail in finding the solution. In this work, an efficient and robust algorithm is proposed for contact resolution of 2D superquadrics, in which any superquadric is approximated with a convex polygon through adaptive sampling; then by clipping two polygons, an efficient linear algorithm is performed to search for intersections and overlap area of the polygons; the contact forces and directions are determined by employing a newly established corner/corner contact model. It is important to highlight that the proposed methodology can also be extended to general non‐circular discrete object cases. The performance of the algorithm is demonstrated via numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical simulation of explosive welding using the material point method

Explosive welding involves detonation of explosive, interactions of fluid-structure and plastic deformations of metal plates at the instant of the explosion. Conventional mesh-based methods such as the finite element method (FEM) and finite difference method (FDM), are limited in simulation of the explosive welding when mesh distortion and interaction of different materials occur. In order to describe process of the explosive welding and accurately predict physical parameters for the explosive welding, numerical simulation of the explosive welding which involves multi-physical phenomenon is performed by using material point method (MPM). Not only can major physical phenomena of explosion impact be well captured, but also some important technical parameters for the explosive welding can be attained based on the MPM simulation. Through the comparison with the experimental results, it is shown that the MPM is a robust tool in simulation of the explosive welding.     

单层网壳结构弹塑性屈曲分析的离散单元法研究

该文提出离散元塑性区法,即将任意2个球元的接触截面划分成若干小面积,通过各小面积的应力状态描述整个截面的塑性发展过程,较离散元塑性铰法更精确。该文推导了杆系离散元截面应变增量计算公式,建立了截面在三维应力-应变状态下的结构弹塑性本构方程、加卸载准则、截面内力积分公式以及计算分析流程。离散元弹塑性屈曲分析的追踪策略与弹性屈曲分析完全相同,即仍采用离散元力控制法或位移控制法。采用Fortran语言自编程序对若干单层网壳结构算例进行弹塑性屈曲分析,验证了离散元塑性区法的正确性和适用性,拓宽了离散单元法在工程领域的应用范围,为结构分析提供了新路径。     

Space–time approach to numerical analysis of a string with a moving mass

Inertial loading of strings, beams and plates by mass travelling with near‐critical velocity has been a long debate. Typically, a moving mass is replaced by an equivalent force or an oscillator (with ‘rigid’ spring) that is in permanent contact with the structure. Such an approach leads to iterative solutions or imposition of artificial constraints. In both cases, rigid constraints result in serious computational problems. A direct mass matrix modification method frequently implemented in the finite element approach gave reasonable results only in the range of relatively low velocities. In this paper we present the space–time approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the space–time finite element domain. The resulting characteristic matrices include inertia, Coriolis and centrifugal forces. A simple modification of matrices in the discrete equations of motion allows us to gain accurate analysis of a wide range of velocities, up to the velocity of the wave speed. Numerical examples prove the simplicity and efficiency of the method. The presented approach can be easily implemented in the classic finite element algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

A DEM-based method for predicting the wear evolution of structural boundary composed of spherical boundary elements

In the discrete element method (DEM) simulation for wear prediction, structural boundary is now represented extensively by triangular meshes with high resolution, which brings a huge computational cost. A DEM-based method for predicting the wear evolution of structural boundary has been developed for computational efficiency. The structural boundary subjected to wear is represented by the spherical boundary elements in the DEM simulation in combination with the inside triangles and fitting curved surface in wear prediction. Wear prediction is performed through a series of evolution steps. In each evolution step, the collision energies by particles at structural boundary are collected via the DEM simulation and assigned to the boundary elements. Then, the volume losses of structural boundary are predicted across each relevant boundary element. Finally, the new geometry of structural boundary in response to wear is described by moving the boundary elements along the depths of wear individually. Through converting the contact detection between structural boundary and particles into between spherical boundary elements and particles, our method greatly reduces the computational cost in the DEM simulation. Through two numerical tests, our method has been verified to be an efficient and accurate method for the wear prediction of structural boundaries with different resolutions.     

A new contact detection method for arbitrary dilated polyhedra with potential function in discrete element method

Contact detection significantly affects the computational efficiency of discrete element simulations, especially for irregularly shaped elements. The dilated polyhedron is constructed by the Minkowski sum of a dilated sphere and a core convex polyhedron. One of the greatest advantages of using the dilated polyhedron in contact detection lies in its ability to be solved by calculating the nearest distance between corresponding core polyhedra. The approximate envelope function (AEF) of a dilated polyhedron is formed by the weighted summation of the second-order dilated function of the polyhedral and spherical functions. The AEF can be used to represent the element in the optimization model for the contact center. Geometric calculations are then employed for the contact points on the core polyhedron, whereupon the contact detection is solved. The accuracy and stability of the proposed method by a 3-D Voronoi tessellation are validated using analytical solutions and previously published simulation results. The efficiency tests show that the speedup of the CPU-based multithread algorithm can reach 14 on a desktop. The direct shear test of the Voronoi shaped ballast is analyzed by this method. The shear stress under different vertical pressure is compared with previously published experimental and simulated results.     

基于FORM 有限元可靠度方法的结构整体概率抗震能力分析

将多级力控制Pushover分析方法与一次可靠度方法(First Order Reliability Method, FORM)相结合,提出了结构整体概率抗震能力分析的FORM 有限元可靠度方法。利用OpenSees 的有限元可靠度分析模块,将该方法应用于钢筋混凝土框架结构,并与蒙特卡洛模拟法分析结果进行对比。结果表明:该方法具有较高的精度和效率,为结构整体概率抗震能力分析提供了另外一种思路。     

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.     

Benchmark solutions of stationary random vibration for rectangular thin plate based on discrete analytical method

This paper aims to accurately and efficiently achieve the benchmark solutions of stationary stochastic responses for rectangular thin plate. Firstly, the exact solutions of free vibration for thin plate with SSSS, SSSC, SCSC, SFSF, SSSF and SCSF boundary conditions are introduced to random vibration analysis. Based on pseudo excitation method (PEM), the analytical power spectral density (PSD) functions of the transverse deflection, velocity, acceleration and stress responses for thin plate under random base acceleration excitation are derived. Subsequently, to enhance computational efficiency, the discrete analytical method (DAM) that realizes the discretization for the modal coordinates and frequency domain is proposed. Finally, the efficiency of DAM and the accuracy of benchmark solutions are scrutinized by comparison with the analytical solutions and finite element solutions.     

Analysis and reduction of quadrature errors in the material point method (MPM)

The material point method (MPM) has demonstrated itself as a computationally effective particle method for solving solid mechanics problems involving large deformations and/or fragmentation of structures, which are sometimes problematic for finite element methods (FEMs). However, similar to most methods that employ mixed Lagrangian (particle) and Eulerian strategies, analysis of the method is not straightforward. The lack of an analysis framework for MPM, as is found in FEMs, makes it challenging to explain anomalies found in its employment and makes it difficult to propose methodology improvements with predictable outcomes. In this paper we present an analysis of the quadrature errors found in the computation of (material) internal force in MPM and use this analysis to direct proposed improvements. In particular, we demonstrate that lack of regularity in the grid functions used for representing the solution to the equations of motion can hamper spatial convergence of the method. We propose the use of a quadratic B‐spline basis for representing solutions on the grid, and we demonstrate computationally and explain theoretically why such a small change can have a significant impact on the reduction in the internal force quadrature error (and corresponding ‘grid crossing error’) often experienced when using MPM. Copyright © 2008 John Wiley & Sons, Ltd.     

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 discontinuous Galerkin method with plane waves for sound‐absorbing materials

Poro‐elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper, we present a discontinuous Galerkin method (DGM) with plane waves for poro‐elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave‐based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave‐based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material. Copyright © 2015 John Wiley & Sons, Ltd.     

耦合有限元物质点法及其在流固耦合问题中的应用

有限元法在求解大变形问题时会遇到网格畸变和时间步长严重减小的问题。物质点法在大变形问题中无网格扭曲问题,且粒子代表了物质流动,无需界面追踪算法。但在小变形问题中,物质点法的精度和效率均低于有限元法。该课题组针对冲击侵彻问题提出的耦合有限元物质点法分别采用有限元法和物质点法模拟小变形和大变形物体,物体间的相互作用通过接触算法实现,既保留了有限元针对小变形问题高精度高效率的特点,又避免了材料大变形给有限元法带来的网格扭曲和时间步长严重减小的问题,还可以自动追踪界面。在流固耦合问题中,固体变形较小而流体变形较大,因此也适合用耦合有限元物质点法求解。该文简要介绍了耦合有限元物质点法的基本原理,并将其应用于流固耦合问题中,取得了较好的效果,表明耦合有限元物质点法是分析流固耦合问题的一种有效的方法。     

Examining the smooth and nonsmooth discrete element approaches to granular matter

The smooth and nonsmooth approaches to the discrete element method (DEM) are examined from a computational perspective. The main difference can be understood as using explicit versus implicit time integration. A formula is obtained for estimating the computational effort depending on error tolerance, system geometric shape and size, and on the dynamic state. For the nonsmooth DEM (NDEM), a regularized version mapping to the Hertz contact law is presented. This method has the conventional nonsmooth and smooth DEM as special cases depending on size of time step and value of regularization. The use of the projected Gauss‐Seidel solver for NDEM simulation is studied on a range of test systems. The following characteristics are found. First, the smooth DEM is computationally more efficient for soft materials, wide and tall systems, and with increasing flow rate. Secondly, the NDEM is more beneficial for stiff materials, shallow systems, static or slow flow, and with increasing error tolerance. Furthermore, it is found that just as pressure saturates with depth in a granular column, due to force arching, also the required number of iterations saturates and become independent of system size. This effect make the projected Gauss‐Seidel solver scale much better than previously thought. Copyright © 2014 John Wiley & Sons, Ltd.     

Polyarc discrete element for efficiently simulating arbitrarily shaped 2D particles

A new two‐dimensional discrete element type, termed the ‘polyarc’ element is presented in this paper. Compared to other discrete element types, the new element is capable of representing any two‐dimensional convex particle shape with arbitrary angularity and elongation using a small number of shape parameters. Contact resolution between polyarc elements, which is the most computation‐extensive task in DEM simulation only involves simple closed‐form solutions. Two undesirable contact scenarios common for polygon elements can be avoided by the polyarc element, so the contact resolution algorithm for polyarc elements is simpler than that for polygon elements. The extra flexibility in particle shape representation induces little or no additional computational cost. The key algorithmic aspects of the new element, including the particle shape representation scheme, the quick neighbor search algorithm, the contact resolution algorithm, and the contact law are presented. The recommended contact law for the polyarc model was formulated on the basis of an evaluation of various contact law schemes for polygon type discrete elements. The capability and efficiency of the new element type were demonstrated through an investigation of strength anisotropy of a virtual sand consisting of a random mix of angular and smooth elongated particles subjected to biaxial compression tests. Copyright © 2011 John Wiley & Sons, Ltd.     

钢筋混凝土框架结构破坏性能的离散单元法模拟

离散单元法是模拟结构破坏的一种有效的分析方法。通过引入节点单元和考虑混凝土的非线性对矩形离散单元模型进行了改进,并给出了改进后离散单元模型的破坏准则和基本方程以及弹簧系数等计算参数的确定;改进后的模型采用了双链表技术,提高了模型的计算效率。对在爆炸荷载作用下的钢筋混凝土框架结构的倒塌破坏过程进行了模拟,结果表明:采用改进后的离散单元法可以有效地模拟钢筋混凝土框架结构的倒塌破坏过程。