440 Science in China Series G: Physics, Mechanics & Astronomy 2006 Vol.49 No.4 440—450

DOI: 10.1007/s11433-006-2011-1

A general finite element model for numerical simulation of structure dynamics WANG Fujun1, LI Yaojun1, Han K.2 & Feng Y.T.2

1. College of Water Conservancy and Civil Engineering, China Agricultural University, Beijing 100083, China;

2. Civil & Computational Engineering Centre, University of Wales Swansea, Swansea SA2 8PP, U.K. Correspondence should be addressed to Wang Fujun (email: wangfj@cau.edu.cn) Received August 21, 2005; accepted March 13, 2006

Abstract A finite element model used to simulate the dynamics with continuum and dis-continuum is presented. This new approach is conducted by constructing the general contact model. The conventional discrete element is treated as a standard finite element with one node in this new method. The one-node element has the same features as other finite elements, such as element stress and strain. Thus, a general finite element model that is consistent with the existed finite element model is set up. This new model is simple in mathematical concept and is straightforward to be combined into the existing standard finite element code. Numerical example demonstrates that this new approach is more effective to perform the dynamic proc-ess analysis in which the interactions among a large number of discrete bodies and continuum objects are included.

Keywords: finite element method, discrete element method, structure dynamics, numerical simulation.

1 Introduction

The finite element method (FEM) is firmly accepted as a powerful general technique for the numerical solution of a variety of continuum problems encountered in engineer-ing[1,2]. However, FEM is difficult to perform the analysis of a number of discrete parti-cles existing, such as the process of brittle fracture under impact, filling of grain bin, classification of apple by size, mining and transporting of ores. The contact among parti-cles dominates the fast change of configuration of these systems, which results in huge computation spending[3]. For these problems, the discrete element method (DEM)[4―6], rather than FEM, is usually adopted to perform the dynamic simulation. Because most engineering problems usually include continuum bodies and discontinuum objects, and sometimes involve the continuum/discontinuum transformation, a combined method FDEM that combines FEM and DEM was proposed by Owen et al. in the 1990s[7,8].

www.scichina.com www.springerlink.com

A general finite element model for numerical simulation of structure dynamics 441

FDEM can be used to simulate the dynamic material separation and progressive failure processes that include fast continuum/discontinuum transformations. However, FDEM is essentially the assembly of two independent systems: FEM and DEM. Because the two different coding and implementing forms coexist in one program, FDEM has the disad-vantages of high programing complexity and low running efficiency, and is difficult to perform parallel computation[9].

To overcome the disadvantages of FDEM, an improved method, the general finite element method (GFEM), is proposed in this paper. The discrete element algorithm is recomposed according to the standard finite element algorithm. A new special finite ele-ment with only one node is introduced into FEM by careful design of element features including stress and strain. The new one-node element can be treated as an ordinary finite element, thus two original different coding and implementing systems FEM and DEM could be unified in GFEM. The mathematics presentation of GFEM is plain and simple. With a little improvement, the existing conventional FEM programs can be used to effec-tively solve the problems including finite elements and discrete elements simultaneously.

2 Formulation of GFEM

2.1 General descriptions

In GFEM, the entire space domain of a problem that may include continuum or dis-crete bodies is discretized by using elements into small objects. The movement and de-formation of the system are described with the motion and the status change of the ele-ments in the system. All elements in GFEM have two basic features in geometry and physics. The geometrical characteristics include shape, node, size and arrangement (original and instantaneous arrangement). The physical properties include element mass, contact status, internal force, external force, motion and deformation under loads, etc.

For the discretization of continuum domain in macro-scale, the ordinary procedure of FEM is yet adopted by using all standard finite elements, such as 3D solid element, shell element, and beam element.

For the discretization of discontinuum objects (or discrete particles), some special fi-nite elements with only one node are introduced to present the objects. The 2D one-node elements usually include disc, ellipse and polygon; the typical 3D one-node elements are sphere and cuboid. In the definition of one-node element, additional control parameters, such as radius vector or boundary vectors, are required to describe the element shape. This is just like the nodal fiber vectors introduced in shell element. For the 3D sphere element, the control parameter is its radius vector. DOF of the node of a 3D sphere is 6, i.e. three displacements and three rotations. To avoid making the method complex, the deformation of one-node element is supposed to be very limited. Thus the 3D sphere element can be represented by only one fixed value of radius. In order to simplify the description of GFEM, only 3D sphere element is discussed in the following sections.

When GFEM is used to simulate quasi-brittle materials such as ceramic, concrete and rock, continuum/discontinuum transformation may happen if the status of local elements reaches the critical conditions during deformation. At this moment, the fracture model is

442 Science in China Series G: Physics, Mechanics & Astronomy

employed, and the small discrete elements coming from large continuum are represented by one-node elements.

The transformation from standard finite element to one-node element, the stress and strain descriptions of one-node element, and the contact forces that represent the interac-tions among elements are the key issues for GFEM. Detailed description of the proce-dures of GFEM including governing equations, contact detection and contact force com-putation, internal force computation, and transformation from standard finite element to one-node element will be given in the next sub-sections.

2.2 Governing equations

Standard finite element approximations for continuum domains yield a nonlinear sys-tem of ordinary differential equations[1,2]: (1) int ext c 0,+ + − − =Mu Cu F F F

where M and C are the mass and damping matrices respectively, intF is the global vector of internal nodal forces, extF is the vector of consistent nodal forces for the applied ex-ternal loads, cF is the vector of consistent nodal contact forces, is the global vector of nodal accelerations and is the global vector of nodal velocities.

uu

For the discontinuum domain, separate objects with individual material properties are possibly connected only along their boundaries. These objects are modeled by one-node elements. These elements are governed by the Newton’s second law[7,9]: ext c+ 0,− − =Mu Cu F F (2) where M is the general mass matrix, i.e. the element mass for displacement computation and rotation inertia for rotation computation; C is the damping matrix; extF and cF are the vectors of the external forces and contact forces, respectively. All types of ele-ments including conventional finite elements and one-node elements are coupled by con-tact forces cF .

It should be mentioned that eq. (2) can be viewed as a special case of eq. (1) after one-node element is merged into a finite element system with new definition of its geo-metrical and physical features. Thus the complexity of algorithm design for solving eqs. (1) and (2) is reduced. We can use one single eq. (1) to describe the GFEM system.

2.3 General contact forces

One of the key issues of GFEM is the description of general contact forces among ad-jacent elements. General contact forces can be expressed as c ,n t= + +F P F F (3) where the terms on the right side are the forces of central potential, normal contact and tangential contact, respectively. It should be mentioned that the three kinds of forces may not exist at the same time, which depends on contact status and physical models. When the contact forces are computed by numerical method, the contact forces are related to not only the current positions and status of two elements in a contact pair, but also the contact history at previous time steps. We refer to contact forces as general contact forces

A general finite element model for numerical simulation of structure dynamics 443

because the contact forces may exist even if two elements are not really in mechanical contact.

To study the interaction mechanism and general contact force computation, we take two 3D sphere elements (could be any kind of element) as an example of contact pair, shown in Fig. 1. , , 1v 2v 1ω and 2ω denote respectively the velocities of the centers, and the angular velocities of two spheres; n denotes the normal vector between two sphere centers.

Fig. 1. Two spheres in contact.

(i) Central potential interaction force. The central potential interaction force is the attractive/repulsive force between two elements, which depends on the micro-structure of molecules of elements. Sometimes, the medium-ranged interaction forces, such as attrac-tion forces in liquid bridges, are also viewed as central potential interaction force. If the element sizes and material parameters are given, central potential interaction force can be determined in terms of Lennard-Jones potential that is widely used in solid physics the-ory . The [5] central potential interaction force is usually very limited for most engineering problems, thus it can be ignored in numerical calculation of GFEM.

(ii) Normal contact force. Normal contact force is the normal component of contact force in a true contact pair, and is the dominant part of contact force. A number of algo-rithms could be used to deal with the normal contact force, e.g. Lagrangian multiplier method and penalty method[10−13]. The major difference between various contact algo-rithms lies in the different ways of enforcing the contact constraint. The penalty based contact interaction laws evaluate contact forces through the gap/penetration between two elements and penalty coefficients, and are consistent with explicit time integration[12]. Therefore, this kind of method is often adopted for contact cases. The typical penalty methods that can be used for GFEM are as follows.

The linear Hooke model[10]: ,n nk δ=F n (4)

where is a model parameter to be chosen, δ is the maximum penetration between two elements. The linear Hooke model is possibly the simplest normal contact model.

nk

444 Science in China Series G: Physics, Mechanics & Astronomy

The improved Hertz model[13]: 3/ 2 1/ 2 ,n nk Rδ=F n (5) where R is the radius of sphere. This model is suitable for the contact between sphere and plate.

Winkler model[13]: ,n nk V=F n (6) where V is the overlap volume between two elements.

In the above three models, the two elements in a contact pair are supposed to be in to-tal contact status. If one vertex or one edge of an element contacts with another element, the contact is a partial contact. For this case, the penalty algorithms need to be modified. (For discussion of the modification, see ref. [13].)

(iii) Tangential contact force. The tangential contact force is the friction in nature between two elements in a contact pair. The tangential direction is identical with the rela-tive tangential motion path, and is perpendicular to the normal direction. The classical Coulomb law is the most typical method for friction calculation. As an alternative to the classical Coulomb law we will develop a modified Coulomb law which will include the effect of contact history and is consistent with GFEM. The modified friction law will be formulated by analogising to the theory of elasto-plasticity.

Consider two spheres in contact, as shown in Fig. 1. By the classical assumptions of the frictional law an element rests until the maximum shear force is reached. In reality, however, elements show tangential micro-displacements within the contact area before the sliding process starts. Here, we sort the contact into elastic phase and plastic phase.

The elastic phase is pre-sliding phase. In this phase, the sliding potential exists, but sliding has not started. The tangential contact force results in the elastic deformation of element, in which the micro-displacement is very limited. The relation between the tan-gential contact force and the micro-displacement is slightly nonlinear. A reasonable ap-proximation of the nonlinear behaviour is given by a linear relation[14]. The tangential contact force tF is defined as

,et t tk=F u (7)

where is the elastic tangential stiffness or penalty coefficient; is the relative tangential micro-displacement between two elements within the contact area.

tk etu

When the elements start to slide within the contact interface, the plastic phase of con-tact comes into existence, and the sliding is referred to as plastic deformation. The tan-gential contact force Ft can be determined by t μ= nF F , (8)

where μ is the dynamic frictional coefficient. For discussion of μ, see ref. [14]. The next step for the tangential contact force computation is to determine the contact

phase. Noting the analogy with elasto-plastic theory, we also need to construct some functions related to element position and motion status in previous and current time steps. At the current time instant 1nt + , the relative velocity vector at the contact point c is

A general finite element model for numerical simulation of structure dynamics 445

computed by (9) 1 2 ,r c c= −v v v

where and are the velocity vectors at the contact point respectively for contact spheres 1 and 2, and evaluated respectively by

1cv 1cv

1 1 1 1,c c= + ×v v ω r 2 2 2c 2c= + ×v v ω r . (10)

rv can be decomposed into two components, in the normal direction and in

the tangential direction of the contact plane: nrv

trv

( ) , (11) nr r= ⋅v v n n

n.

tr r r= −v v v

The current tangential direction is thus defined by .

t tt r r=n v v (12)

The incremental relative tangential displacement is then computed by .

t tt r r tt tΔ = Δ = Δu v v n (13)

It should be noted that this displacement is contributed to by the transverse movement and the relative rotation of the two spheres. Then, the tangential contact force can be computed by the following procedure.

First, suppose that the tangential contact force at the previous time instant is ntn

tF .

Since the contact plane and tangential direction at 1nt + may not coincide with the values

at due to the relative rotation between the two spheres, ntn

tF has to be properly transformed to the current contact plane. The transformation is achieved by the following length-preserving co-rotation: ,n

tR R t= nF T F (14)

where ntRF is the co-rotated n

tF in the current contact plane, is a specially se-lected orthogonal transformation matrix that satisfies the following two conditions:

RT

,n ntR t=F F 0.n

tR ⋅ =F n (15)

Next, evaluate a trial tangential contact force and the slip criterion : trialtF trial( )tΦ F

trial ,nt tR tk t= + ΔF F u (16)

trial trial( )t tΦ = − .nμF F F (17)

Last, the tangential contact force tF at current instant 1nt + is determined by

trial trial

trial

, ( )

( ) 0

t t tt

n t t

if

if

Φ

μ Φ

⎧⎪= ⎨>⎪⎩ ，

F n FF

F n F

≤0,

. (18)

2.4 Damping force

Cu in eqs. (1) and (2) is damping force, denoted by

d =F Cu . (19)

446 Science in China Series G: Physics, Mechanics & Astronomy

Very limited information is available on the element damping matrix C in nonlinear situations. It is therefore customary to assume that the damping matrix is proportional to the mass and stiffness matrix. This is known as Rayleigh damping[1,2]. As an alternative method, the damping matrix is determined by the product of damping coefficient and mass matrix, without the calculation of stiffness matrix.

2.5 Description of stress for one-node element

The concept of stress for one-node element in GFEM is based on continuum mechan-ics. The stress is defined at the center of an element. To discuss the stress for one-node element, we consider a sphere element that has adjacent elements. It is supposed that this sphere is divided into two halves by a cross-section that is through its center and is in the direction perpendicular to coordinate

mi

ix . Ignoring the external forces, and con-sidering the contact forces only, we have the momentum balance equation of the first half along j direction on cross-section i:

1

11

,m

kj ij j

kF A M aσ

=

− =∑ (20)

where kjF is the j component of the force of adjacent element k acting on the current

half-sphere, m1 is the number of forces acting on the current half-sphere, ijσ is the av-

erage stress in direction j on the cross-section i, A is the area of the cross-section, M1 and aj are the mass and acceleration of the first half-sphere respectively.

The motion of the entire sphere is governed by

1

,m

kj j

kMa F

=

= ∑ (21)

where M is the total mass of the sphere. Considering and then

1m m m= + 2 1 2M M= =

/ 2,M

1 2

1 1

1 .2

m mk k

ij j jk k

F FA

σ= =

⎛ ⎞= −⎜⎜

⎝ ⎠∑ ∑ ⎟⎟ (22)

This is the average stress of a sphere at its center under the condition that the moment transmission at the contact point is not considered.

Correspondingly, the strain of one-node element can be defined, and is not given here.

2.6 Continuum to discontinuum transition

When GFEM is used to simulate quasi-brittle materials, the continuum/discontinuum transformation may happen if the status of local elements reaches the critical conditions during deformation. Fracture in quasi-brittle materials is generally an anisotropic phe-nomenon, with the coalescence and growth of micro-cracks occurring in the directions that attempt to maximize the subsequent energy release rate and minimize the strain en-ergy density. For successful modeling the continuum/discontinuum transformation, a suitable fracture model should be used to govern the material failure and fracture propa-gation. On the other hand, a special algorithm should be employed in handling the dis-

A general finite element model for numerical simulation of structure dynamics 447

continuities such as shear bands and cracks. A variety of fracture models have appeared over the years, of which the softening

plasticity and the damage theory are most commonly adopted in the nonlinear finite ele-ment analysis of failure. Details of various fracture models can be found in refs. [15,16].

To deal with the shear bands and cracks, a nodal fracture scheme[3] is employed in this work. The scheme is a three stage procedure: (i) Creation of a failure map for the whole domain; (ii) Assessment of the failure map to identify where, and in what direction, frac-tures should be inserted; (iii) Updating of the mesh, topology and associated data. The failure direction is defined in terms of a weighted average of the maximum failure strain directions of all elements connected to the node and the crack will subsequently propa-gate orthogonal to this failure direction. Associated with the failure direction, a failure plane is defined with the failure direction as its normal and the failed nodal point lying on the plane. A crack is then inserted through the failure plane. If a crack is inserted exactly through the failure plane, some ill-shaped elements may be generated and the local re-meshing is then needed to eliminate them.

After crack inserting and mesh updating, some one-node elements may be employed for representing the discontinuum elements if their sizes reach a criterion.

3 Numerical procedure

The numerical procedure for solving the governing eq. (1) of GFEM is almost the same with conventional FEM[2]. It should be mentioned that eq. (1) should be solved by the explicit time integration scheme because the engineering problem discussed in this study may include heavy dynamic contacts among structures and a large number of parti-cles. The central difference algorithm is the most widely used explicit scheme. In this scheme, the displacements at time instant 1nt + can be obtained explicitly by values at

time instant and nt 1nt −[2].

11 2 int ext c( ) ( ( ) ( ) +( ) ) 2 .

2 2n n n n nt tt

−+ −Δ ⎡ Δ⎛ ⎞ ⎛ ⎞= + Δ − + + − −⎜ ⎟ ⎜ ⎟

1n ⎤⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

u M C F F F Mu M C u (23)

If the mass matrix M and damping matrix C are diagonal then the computation of eq. (23) becomes trivial. If shell elements with large deformation are included in the system, the TL, UL or TUL schemes can be used to perform the nonlinear analysis of shell ele-ments[17].

One important aspect of GFEM analysis is to monitor contact between a large number of discrete objects and also any continuum regions present. In the last decade or so, many contact searching algorithms have been proposed. The existing global contact searching algorithms[18,19] and local searching algorithm[20] can be used to perform the contact de-tect.

4 Numerical example

This example simulates a 3D ore hopper filling process. The ore particles are initially

448 Science in China Series G: Physics, Mechanics & Astronomy

regularly packed at the top of the hopper and then are allowed to fall under the action of gravity, as shown in Fig. 2(a). The ore particles are supposed to be spheres with the same size, and their radii are 10 mm. The total number of ore particles is 8768. They are ar-ranged on 13 layers above a support plate, occupying a 500 mm × 500 mm × 220 mm cuboid domain. The hopper bin is also a cuboid with size of 480 mm × 480 mm × 500 mm. The height of the hopper is 250 mm, the section of the hopper is changed from 966 mm × 966 mm to 100 mm × 100 mm. The material properties for the hopper and the hopper bin are: Young’s Modulus E=2.1×1011 N/m2, Poisson’s ratio ν = 0.29, mass den-sity ρ = 7.86×105 kg/m3. The material properties for ore particles are: E=6.25×1010 N/m2, ν = 0.2, ρ = 2.367×103 kg/m3.

Fig. 2. Hopper filling: configurations at time 0 s (a), 0.5 s (b), 1.0 s (c) and 1.5 s (d).

The ore particles are represented by one-node elements, and the hopper and hopper bin are discretized by quadrilateral finite elements. The normal contact forces and tangential contact forces, i.e. friction, are all considered. To simplify the evaluation of computation performance, the time step is fixed at 1×10−5 s. The configuration change of the system within 1.5 s was computed on a PC Cluster parallel environment in which 4 CPUs are used. Fig. 2 shows the system configurations at time 0, 0.5, 1.0, 1.5 s.

This problem was also analyzed by using combined finite/discrete method (FDEM) by the authors[21]. Compared with the result of FDEM[21], the simulation of GFEM has the

A general finite element model for numerical simulation of structure dynamics 449

same accuracy, but the time consumed decreases from 13528 s to 10023 s, which means that the computational efficiency increases by 24% for this problem. On the other hand, the stresses of some concerned elements are also obtained. Therefore, GFEM is suitable for simulating the dynamic process, in which a large number of discrete bodies are in-cluded.

5 Conclusions and discussion

A general finite element model (GFEM) used to simulate the dynamics with contin-uum and non-continuum is described in this paper. The conventional discrete element is treated as a standard finite element with one node in this new method. By constructing the general contact model and stress expression for one-node element, the discrete ele-ment algorithm is integrated into finite element algorithm. Thus a unified general finite element model that has low programming complexity is set up. The model is consistent with the existing conventional finite element approach. The new one-node element has the same feature as other finite elements, such as element stress and strain, however, the stress is evaluated on the element node rather than Gauss point. Numerical example shows that the new model is fast and the memory consumption is low compared with the conventional finite element model and the discrete element model. This method is suit-able for simulating the dynamic process, in which a large number of discrete bodies are included.

However, GFEM is under development and has some shortages. For example, only 2D disk and 3D sphere one-node elements have been constructed; other kinds of one-node elements (such as triangle, cuboid, ellipsoid) are under construction. We should pay at-tention to the contact model used for treating different kinds of contact pair. Some key coefficients dominating the simulation such as penalty coefficient, damping parameters, and fracture criterion should be further studied.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 10372114 and 90510007), the Program for New Century Excellent Talents in University of China (Grant No. NCET-04-0133), and the Engineering and Physical Sciences Research Council (EPSRC) of UK (GR/R21219).

References

1 Zienkiewicz O C, Taylor R L. The Finite Element Method, 5th ed. Boston: Butterworth-Heinemann, 2000 2 Wang X C. Finite Element Method (in Chinese). Beijing: Tsinghua University Press, 2003 3 Owen D R J, Feng Y T, Souza N E A, et al. The modeling of multi-fracturing solids and particulate media. Int J

Numer Methods Eng, 2004, 60: 317―339 4 Cundall P A, Hart R D. Numerical modeling of discontinua. Eng Comput, 1992, 9: 101―113 5 Tang Z P. Three-dimensional DEM theory and its application to impact mechanics. Sci China Ser E-Eng Mater

Sci, 2003, 33: 989―998 6 Liu K X, Gao L T. A review on the discrete element method. Advances in Mechanics (in Chinese), 2003, 33:

483―490 7 Munjiza A, Owen D R J, Bicanic N. A combined finite/discrete element method in transient dynamics of frac-

turing solid. Eng Comput, 1995, 12: 145―174

450 Science in China Series G: Physics, Mechanics & Astronomy

8 Xu J L, Tang Z P. Combined discrete-finite element multiscale numerical method and its application. Chinese Journal of Computational Physics (in Chinese), 2003, 20: 477―482

9 Wang F J, Feng Y T, Owen D R J. Parallelization for finite-discrete element analysis on distributed-memory environment. Int J Comput Eng Sci, 2004, 5: 1―23

10 Zhong Z H. Finite Element Procedures for Contact-Impact Problems. Oxford: Oxford University Press, 1993 11 Belytschko1 T, Daniel W J T, et al. A monolithic smoothing-gap algorithm for contact-impact based on the

signed distance function. Int J Numer Methods Eng, 2002, 55: 101―125 12 Han K, Peric D, Owen D R J, et al. A combined finite/discrete element simulation of shot peening processes (I):

Studies on 2D interaction laws. Eng Comput, 2000, 17: 593―619 13 Han K, Peric D, Owen D R J, et al. A combined finite/discrete element simulation of shot peening processes

(II): 3D interaction laws. Eng Comput, 2000, 17: 680―702 14 Wriggers P, Van V T, Stein E. Finite element formulation of large deformation impact-contact problems with

friction. Comput Struct, 1990, 37(3): 319―331 15 Yu J. A contact interaction framework for numerical simulation of multi-body problems and aspects of damage

and fracture for brittle materials. Dissertation for the Doctoral Degree. Swansea: University of Wales Swansea, 1999

16 Klerck P A. The finite element modelling of discrete fracture in quasi-brittle materials. Dissertation for the Doctoral Degree. Swansea: University of Wales Swansea, 2000

17 Wang F J, Cheng J G, Yao Z H. TUL deformation formulation for explicit dynamic analysis based sup-per-parametric shell elements. J Tsinghua Univ (Sci & Tech)(in Chinese), 2001, 41(11): 12―16

18 Wang F J, Cheng J G, Yao Z H. A contact searching algorithm for finite element analysis of contact-impact problems. Acta Mechanica Sinica, 2000, 16(4): 374―382

19 Feng Y T, Owen D R J. An augmented spatial digital tree algorithm for contact detection in computational me-chanics. Int J Numer Methods Eng, 2002, 55: 159―176

20 Wang F J, Cheng J G, Yao Z H. FFS contact searching algorithm for dynamic finite element analysis. Int J Nu-mer Methods Eng, 2001, 52: 655―672

21 Wang F J, Feng Y T, Owen D R J, et al. Parallel analysis of combined finite/discrete element systems on PC cluster. Acta Mechanica Sinica, 2004, 20: 534―540