An Enhanced Energy Conserving Time Stepping

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2000; 00:1–6 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02]

An Enhanced Energy Conserving Time Stepping Algorithm for Frictionless Particle Contacts † R. Bravo,

1 ∗

J. L. P´erez-Aparicio

2

and T. A. Laursen

3

1

Department Structural Mechanics and Hydraulic Engineering, Universidad de Granada 18071 Granada, Spain 2 Department of Continuum Mechanics and Theory of Structures, Universidad Polit´ ecnica de Valencia 46022 Valencia, Spain 3 Department of Mechanical Engineering and Materials Science, Comp. Mechanics Lab, Duke University Box 90287, 127 Hudson Hall, Durham, NC 27708-0287, USA

SUMMARY An Enhanced Energy Conserving Algorithm (EECA) formulation for time integration of frictionless contact–impact problems is presented. In it the energy, linear and angular momentum are conserved for every contact using an enhanced Penalty method. Previous formulations for these problems have shown that the total bodies’ energy decreases for contact due to an artificial energy transfer between the penalty spring and the contacting bodies. Consequently, they are not able to reproduce a physical response after a single contact, introducing errors in trajectories and velocities. Through the conserving balance equations, EECA computes a physical response by inserting for every contact an additional amount of linear momentum and contact force. The structure of these equations defines the additional linear momentum to restore the energy and the enhanced Penalty method based on a spring and a dashpot. This method approximately enforces the first and second Kuhn–Tucker conditions. The new algorithm has been applied to several frictionless rigid problems using the Discrete Element Method. The first two problems consist of the simulation and analytical comparison of the Newton’s Cradle and Carom problems (billiard pool problem). The last two are the hopper filling process and the breaking of a pool ball’s triangular arrangement, both of which involve a medium number of contacts. Application of this formulation will be straightforward to elastic and general shaped bodies using the Finite Element Method. c 2000 John Wiley & Sons, Ltd. Copyright key words: Momenta and Body’s Energy Conservation, Contact, Discontinuous Deformation Analysis, Enhanced Penalty

∗ Correspondence to: e-mail: [email protected] Department Structural Mechanics and Hydraulic Engineering University of Granada, 18071 Granada, Spain

Contract/grant sponsor: MFOM I+D grant; contract/grant number: (2004/38), MICIIN grant BIA–2008–00522 and “Ayudas Investigaci´ on” from UPV2010

c 2000 John Wiley & Sons, Ltd. Copyright

Received Dec 2002 Revised 18 September 2002

2

´ R. BRAVO , J. L. PEREZ–APARICIO, T. A. LAURSEN

1. INTRODUCTION The development of time stepping algorithms applied to computational mechanics problems has been an intensive research field during the last few years. All algorithms try to emulate the conserving properties of the corresponding continuous problem (linear, angular momentum and energy) as time evolves. Refs. [1] and [2] developed numerical integration schemes for contact problems that approximately conserve energy in time by developing several kinematic compatibility conditions on the contact surface. Based on another approach, [3] created an algorithm that transferred and dissipated energy between contacting bodies and the Penalty spring, so that the total energy of the bodies decreases, finally achieving stability. This methodology was based on the persistency condition (second Kuhn–Tucker) without the imposition of any geometrical impenetrability constraint. The algorithm was improved by [4], who minimized the energy transfer by imposing an additional penalty based on linear momentum and a new gap formulation. The zero balance body energy is reached after but not during contact. Ref. [5] presented an energy restoring scheme that enforced the impenetrability condition and conserved the total bodies’ energy during contact. This scheme is based on a predictor– corrector energy restoring strategy that mimics the post–impact velocity jump. Contact is computed in the predictor and an energy balance produces a post–contact conserving velocity in the corrector. The present work, Enhanced Energy Conserving Algorithm (EECA), develops an energy, angular and linear momenta (energy–momentum) conserving time stepping scheme for frictionless contact using an enhanced Penalty method. This method considers a spring and a dashpot that approximately enforce the first and second Kuhn–Tucker conditions (no penetration and persistency). For every contact, the energy conservation equation provides an additional amount of linear momentum and contact force that adds or substracts energy equal to the amount transferred to the penalty spring. To compare the performance of published and EECA formulations, Fig. 1 shows the evolution of the total body energy with the Signorini problem before, during, and after contact. Both [6] and EECA keep the energy constant while [4] and [3] cannot. Consequently, the latter are not able to reproduce a physical response after a single contact, introducing errors in trajectories and velocities. The current work has the following structure: Section 2 briefly describes the formulation of the Discontinuous Deformation Analysis (DDA), which is a member of the Discrete Element Method. DDA’s ability to simulate the behavior of systems composed of a high number of bodies continuously interacting through contact makes it a good candidate for demonstrating the conserving properties of EECA (see Fig. 2). Fundamental concepts and equations of EECA’s formulation are presented in Section 3. The conserving time integration scheme of [7] is modified by adding an additional linear momentum and a contact force (a priori unknowns) for every contact, thus making the energy and momentum constant. Conservative properties of the continuous formulation are preserved in the discrete formulation by expressing the contact force by the Discrete Derivative from [8]. Finally, for simplicity and comparison with analytical results, EECA’s formulation will be developed for rigid bodies, although it is possible to extend it to other constitutive laws. Energy–momentum conservation is discussed in Section 4. An analysis of EECA’s conserving properties defines the balance equations that linear momentum and contact force have to c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls

Int. J. Numer. Meth. Engng 2000; 00:1–6

ENERGY CONSERVING ALGORITHM FOR FRICTIONLESS CONTACT

Ebod

1 0 0 1 0 1 0 1 0 1 0Chawla (1998) 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Time t 0 1 0 1 After 0 1 0 1

Love (2000), NECA Petocz (1999)

Before

3

During

Figure 1. Rigid Signorini problem. Artificial energy transfer to penalty spring decreases energy body in [3], [4], while EECA, [6] conserves energy during contact

(q03 , p30 )

K

(q01 , p10 )

ξ

(q04 , p40 ) (q02 , p20 )

Figure 2. DDA numerical contact model, composed of a penalty spring and dashpot that approximately enforce first and second Kuhn–Tucker conditions

fulfill. This analysis proves that linear and angular momenta are always conserved with any momentum and force, but they have to be related in order to conserve energy. Another analysis in Subsection 4.3 develops an enhanced Penalty method and studies the influence of the velocity Penalty parameter on the additional linear momentum, contact force, gap, and penetration velocity. The relationship is parametrically studied with the Signorini contact problem. In Section 5, four representative contact problems have been numerically simulated with EECA and Newmark–β algorithms. The first and second consist of the numerical and analytical comparison of two simple contact problems: Carom and Newton’s Cradle. The third simulation studies a pool breaking balls problem, in which a ball subjected to an initial velocity impacts against a triangular arrangement of another 55 balls under two different hypotheses: a) all balls have the same diameter or, b) there are diameter imperfections in some of them. The c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



4

fourth problem simulates a hopper filling process with a medium number of disks using EECA and Newmark–β.

2. DDA FORMULATION The description of this method will be performed on a single particle, but it is generalizable to a system composed of nbd ones. DDA is based on the laws of classical mechanics, more specifically on Hamilton’s principle. We briefly present here this principle and then apply it to DDA. 2.1. Hamilton’s Principle This is an energy method constructed by means of the Lagrangian function:

˙ = L(Q, Q)

nbd X ˙ i ) − V (Qi ) T (Q

(1)

i=1

i

˙ ) is the kinetic energy, V (Qi ) the potential energy, which depend on velocity where T (Q i Q˙ and position Qi , respectively. Capital letters signify that the variables are defined on the continuum body. The Legendre transformation applied to the Langrangian function implies the ˙ i by the linear momentum coordinate P i , yielding the Hamiltonian function substitution of Q H:

H(Q, P ) =

nbd X i=1

T (P i ) + V (Qi )

(2)

The differential equations of motion (Eqs. 3) are obtained by differentiating the Hamiltonian function respect to P and Q, respectively. ˙i Q

=

i P˙

=

−1 ∂H(Q, P ) = Mi P i i ∂P ∂H(Q, P ) − = −K i Qi + F (Qi , t) ; ∂Qi

i = 1,...,nbd

(3)

where M i and K i represent the mass and stiffness matrices of a nonlinear system subjected to a load F i dependent on time and displacement. 2.2. Application of Hamilton’s Principle to DDA DDA computes the displacements Qi (t) and the linear momenta P i (t) of a designated point x0 , y0 of each body (usually the center of gravity) through Eqs. 3. Displacements q(x, y, t) and linear momenta p(x, y, t) at any point x, y of the body are expressed from a linear combination of the coordinates x0 , y0 using shape functions N (x, y) of a certain order, Eqs. 4, 5. c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



5

Qi = N i (x, y) q i ;

δQi = δN i (x, y) q i + N i (x, y) δq i

(4)

P i = N i (x, y) pi ;

δP i = δN i (x, y) pi + N i (x, y) δpi

(5)

These combinations applied to Eqs. 2 and 3 allow us to obtain the Hamiltonian equations (Eq. 7) at x, y for q(x, y, t) and p(x, y, t).

H(q, p) =

nbd X i=1

q˙ i

=

p˙ i

=

T (N i (x, y) pi ) + V (N i (x, y) q i )

i ∂H(q, p) h i−1 i = M N (x, y) pi ∂pi ∂H(q, p) = − K i N i (x, y) q i + N i (x, y)F (q i , t) − i ∂q

(6)

(7)

with the following initial conditions: q i (0) = q i0 ;

pi (0) = pi0 ;

i = 1,. . . ,nbd

(8)

−1

for simplicity and from now on, M i N i (x, y), K i N i (x, y), and N i (x, y)F (q i , t) will be −1 denoted by M i , K i and F (q i , t).

3. NEW ALGORITHM FORMULATION EECA is developed from the Simo–Tarnow’s [9] conserving energy–momentum time integration scheme, a discrete approximation of a Hamiltonian system (Eqs. 7) in configuration n + 1/2. In this algorithm, the first of Eqs. 9 relates q in , q in+1 and pin , pin+1 . The second is the discrete approximation of the second Newton’s law at n + 1/2.

−1

q˙ i = M i

pi

→

p˙ i = f ic

→

−1 q in+1 − q in = M i pin+1/2 ∆t pin+1 − pin = f ic n+1/2 ∆t

(9)

Given two rigid bodies i, k in contact, Eqs. 10 conserves total momenta and bodies’ energy by adding to Eq. 9 two additional variables: a linear momentum pik c n+1/2 and a contact force ′ik ik f ′ik c n+1/2 (subscript n+1/2 for pc n+1/2 and f c n+1/2 will be omitted in the latter for simplicity). ′ik Zero values for pik c n+1/2 and f c n+1/2 mean that bodies i and k are not in contact.

c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



6

q in+1 − q in ∆t

=

pin+1 − pin ∆t

=

M

i−1

nbd X

pin+1/2

+

nbd X

pik c

k=1 k6=i

−1

= Mi

(pin+1/2 + pic )

′ik i ′i (f ik c + fc ) = fc + fc

(10)

k=1 k6=i

ik Variables f ′ik c and pc are increments between n and n+1 proportional to certain parameters ik ik named ψ1 and ψ2 (Eqs. 11). It will be shown in Section 4 that in order for Eqs. 10 to become a conservative solution, the two constants are related by the conserving balance equations. K ik is the penalty stiffness for contact, gn+1 and gnik the gaps, and Rik n+1/2 the unit normal vector on the contact surface.

pic

=

nbd X

k=1 k6=i

f ′i c

=

nbd X

pik c

=

nbd X 1 k=1 k6=i

f ′ik c =

k=1 k6=i

2

nbd X 1

k=1 k6=i

2

t

ik i i ψ2ik Rik n+1/2 Rn+1/2 (pn+1 − pn ) | {z } local projection ik ψ1ik K (gn+1 − gnik ) Rik n+1/2

(11)

Since frictionless contact geometrically constrains the bodies’ motion and velocity in the normal direction to the contacting surfaces, pik c must be formulated in local contact coordinates and transformed to global ones by the Rik n+1/2 vector in Eq. 11. The contact force f ik c (Eq. 12) is formulated using Gonzalez’s Discrete Derivative [10], which provides a discrete expression that inherits the conserving properties of the continuous function.

f ic

=

nbd X

k=1 k6=i

f ik c

=

nbd ik X V (gn+1 ) − V (gnik )

k=1 k6=i

ik − g ik gn+1 n

Rik n+1/2

=

nbd X

ik ik K Rik n+1/2 (gn+1 − gn )

(12)

k=1 k6=i

ik ik )2 and V (gnik ) = 21 K(gnik )2 are potential contact energies. where V (gn+1 ) = 12 K(gn+1 1 represents the constant The algorithm is graphically explained in Fig. 3, in which the plane 2 is the constant energy surface ∆Ebd = 0 where any conservative solution must lie. Plane body plus penalty contact energy surface, where part of the body’s energy is transferred to 1 At the penalty spring ∆Ebd < 0. Before contact (configuration n), the pair q n , pn lies on . n + 1/2, there is an energy transfer to the contact spring in order to satisfy the impenetrability 3 The pair q n+1/2 , pn+1/2 lies on . 2 In a single time condition, represented by the arrow . 1 at n+1 and, in addition, step, EECA computes pc and f ′c to shift the pair q n+1/2 , pn+1/2 to satisfies the impenetrability condition.




∆Ekin = 0

qn , pn

7

qn+1 , pn+1

1

3 ∆Ekin < 0 4 ∆Epc + ∆Efc′ = 0

⋆ qn+1/2 , p⋆n+1/2

2

1 of constant body energy ∆Ebd = 0. Surface 2 of body+contact constant energy. Figure 3. Surface 3 body–penalty energy transfer ∆Ebd < 0. Arrow , 4 EECA computes pc and f ′c , shifting Arrow 1 q n+1/2 , pn+1/2 to

4. LINEAR AND ANGULAR MOMENTA AND TOTAL BODIES’ ENERGY CONSERVATION The energy and momenta conserving properties of EECA are analyzed in this Section. Transformation of Eqs. 10 permits us to find the conserving balance equations to relate pik c and f ′ik c for every contact. 4.1. Linear momentum balance The second of Eqs. 10 expresses the discrete balance of the linear momentum for a rigid body and the absence of external forces. For a body i, this balance is equal to the resultant of the contact forces f ic plus f ′i c (Eq. 13); that is, the discrete variation of linear momentum is equal to the contact forces pin+1 − pin = f ic + f ′i (13) c ∆t The total linear momentum balance (Eq. 14) is the summation over that of each body and equals the resultant of the contact forces that act on all bodies nbd . n

n

n

n

bd bd X bd bd tot X X X ptot pin+1 − pin n+1 − pn ′ik (f ik f ic + f ′i = = c + fc ) c = ∆t ∆t i=1 i=1 i=1

(14)

k=1 k6=i




8

ki ′ik Every contact satisfies f ik = −f ′ki (action–reaction); the right term of c = −f c and f c c tot Eq. 14 is then null and the total linear momentum is conserved: ptot n+1 = pn .

4.2. Angular momentum balance Under the same hypotheses considered previously, the angular momentum balance (Eq. 15) for a body i is equal to the cross product of the second of Eqs. 13 multiplied by the addition of the displacements at n and n + 1.   J in+1 − J in   ≡   ∆t

(pin+1 − pin ) nbd X × (q in+1 + q in ) (15) ′ik i i  ∆t = (f ic + f ′i ) × (q in+1 + q in ) = (f ik  c c + f c ) × (q n+1 + q n )    k=1 k6=i

where J i is the angular momentum of body i. The total balance of J is the summation over that of each body (Eq. 16). n

n

n

bd bd bd X tot X X (pin+1 − pin ) J tot n+1 − J n ′ik i i (f ik = × (q in+1 − q in ) = c + f c ) × (q n+1 + q n ) ∆t ∆t i=1 i=1

(16)

k=1 k6=i

ik i i k The action–reaction condition and the gap definition gn+1/2 Rik n+1/2 = (q n+1 +q n )−(q n+1 + q kn ) transforms Eq. 16 into Eq. 17.

tot J tot n+1 − J n ∆t

= =

nbd nbd X X

i=1 k=i+1 nbd nbd X X

i ′ik i k k (f ik c + f c ) × (q n+1 + q n ) − (q n+1 + q n ) ′ik ik ik (f ik c + f c ) × (gn+1/2 Rn+1/2 )

(17)

i=1 k=i+1

′ik ik ik For frictionless contact f ik c + f c and gn+1/2 Rn+1/2 are parallel, then their cross product tot is zero and the total angular momentum is conserved: J tot n+1 = J n . Eqs. 14 and 17 show that total linear and angular momenta are conserved for any pik c and f ′ik c , meaning that the algorithm conserves both momenta without enforcing any balance equation.

4.3. Total bodies’ energy conservation From the previous hypotheses, it is possible to obtain the total bodies’ energy balance equation for contact by multiplying the first and second set of Eqs. 10 by (pin+1 −pin )t and −(q in+1 −q in )t and adding the resulting equations for all contacting bodies nbd . The result is the equilibrium of energies: ∆Ekin = −∆Ef c − ∆Epc − ∆Ef ′c . Considering that ∆Ekin is the bodies’ energy balance, En+1 − En between configurations n and n + 1 is c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


9


En+1 − En =

=

nbd nbd X X

i=1 k=1 k6=i

"

(q in+1

q in )t

− {z ∆Ef ik c

|

f ik c

}

nbd X

−1

(pin+1 − pin )t M i {z i=1 | i ∆Ekin

+ (pin+1 |

−

pin )t

M

i−1

pin+1/2 }

pik c

{z ∆Epik (ψ2ik ) c

}

+ (q in+1

q in )t

f ′ik c

− {z } | ik ∆Ef ′ik (ψ ) 1 c

#

(18)

Notice that for rigid bodies and no external forces, ∆Ekin is the only body’s energy. As ψ1ik and ψ2ik may be positive or negative, ∆Epc , ∆Ef ′c may add or subtract energy for every contact; the total bodies’ energy conservation is then enforced by zeroing the right part of Eq. 18, ∆Epc + ∆Ef c + ∆Ef ′c = 0, or nbd nbd X X

i=1 k=1 k6=i

"

(pin+1

|

−

pin )t

i−1

M {z ∆Epik (ψ2ik ) c

pik c

}

+ (qin+1 |

q in )t

− {z ∆Ef ik c

f ik c

}

+ (q in+1 |

q in )t

− {z

f ′ik c

∆Ef ′ik (ψ1ik ) c

}

#

=0

(19)

This equation provides infinite pairs ψ1ik , ψ2ik that satisfy the total bodies’ energy conservation and can be decoupled enforcing the energy conservation for every contact. Using again the action–reaction principle and noticing the reciprocity ψ1ik = ψ1ki , ψ2ik = ψ2ki , Eq. 19 may be rewritten for contact between bodies i and k as: −1 k−1 ki k k t pc + (q in+1 − q in )t − (q kn+1 − q kn )t f ik (pin+1 − pin )t M i pik c c + (pn+1 − pn ) M {z } | {z } | | {z } ik ik ∆E ik ∆Epik (ψ ) ∆E ki (ψ ) f pc c 2 2 c + (q in+1 − q in )t − (q kn+1 − q kn )t f ′ik c =0 {z } | ik (ψ ) ∆Ef ′ik 1 c

(20)

The previous expression implies that for each contact, ∆Ef ik = −∆Epik (ψ2ik )−∆Epki (ψ2ik )− c c c ′ik ki ∆Ef ′ik (ψ1ik ); therefore the energy added or subtracted by pik is equal to the c , pc and f c c ik ik contact potential. The increments ∆Epik (ψ2 ), ∆Epki (ψ2 ) conserve the kinetic body energy c c ik while ∆Ef ′ik (ψ ) adjusts the contact force to the conservative solution. Eq. 20 provides the 1 c ik ik first part to find the relation between ψ1 and ψ2 for each contact. The algorithm will always be stable since the energy balance is enforced to zero. If q n+1 , pn+1 is a solution, the energy conservation equation fulfills

∆Ekin + ∆Epc + ∆Ef c + ∆Ef ′c = 0 | {z } | {z } 0 0 c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls

(21)



10

5. DYNAMIC CONTACT, ENHANCED PENALTY METHOD For every contact, Eq. 20 provides a non–unique relation between ψ1ik and ψ2ik that conserves the total bodies’ energy. A dynamic equation associated with the descriptive algorithm of Eqs. 10 will be developed in this Section. Addtionally, from Eq. 20 we will define an enhanced penalty contact model consisting of a spring and a dashpot. With the new equation and model, the relation between the two ψ’s becomes unique. In the following, superscripts will be omitted. Notice that qn+1 , qn , pn+1 and pn are scalars. The dynamic equation for the Signorini contact problem is developed from Eqs. 22 to 27 by taking into consideration the following steps: i) From Eq. 10 and expressing qn+1 and pn+1 as a function of qn and pn :    1 − ψ2   1 + ψ2 −1 1 1 − qn qn+1   2  2       = (22)       1 − ψ1  p  1 + ψ1 pn+1 2 2 n Ω 1 Ω 1 − 2 {z } | 2 A p where A is the amplification matrix, Ω = ω∆t, ω = K/M (M is the body mass). With notation D = 1 + [Ω2 (1 + ψ1 + ψ2 )2 ]/4 its invariants are i i2 o Ωh 1 n 2h 1 − Ω (1 − (ψ1 + ψ2 )2 ; 1 − (ψ1 + ψ2 )2 + Ω2 L2 = 2 1 − (23) L1 = D D 4 





ii) The characteristic equation of A is: λ2 − L1 λ + L2 = 0. If I is the identity matrix, the Cayley–Hamilton theorem establishes that A2 − L1 A + L2 I = 0

(24)

iii) Defining y n−1 = (qn−1 , pn−1 ∆t)t , y n = (qn , pn ∆t)t , y n+1 = (qn+1 , pn+1 ∆t)t , multiplying (Eq. 24) by y n−1 , and considering that y n+1 = A2 y n−1 , y n = Ay n−1 y n+1 − L1 y n + L2 y n−1 = 0

(25)

iv) Expanding the first row of Eq. 25 in Taylor’s series centered at tn+1/2 ψ1 + ψ2 q˙n+1/2 + ω 2 qn+1/2 2 ∆t ... ψ1 + ψ2 2 2 q n+1/2 + ω ∆t − q¨n+1/2 + ω q˙n+1/2 + O(∆t2 ) 2 2 0 = q¨n+1/2 + ω 2 ∆t

(26)

and the associated dynamic equilibrium equation is c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



2ξω }| { ψ1 + ψ2 2 q˙n+1/2 + ω 2 qn+1/2 0 = q¨n+1/2 + ω ∆t 2{z | {z } | } | {z } Inertia Spring Dashpot z

11

(27)

Eq. 27 represents an enhanced penalty model that is composed of a spring and a dashpot that control the gap and the penetration velocity, respectively. The dashpot is defined by parameter ξ and is considered a penalization for velocities. Eq. 27 now provides a unique relation between ψ1 , ψ2 and ξ:

ψ1 + ψ2 =

4ξ Ω

(28)

This expression can be easily generalized to any contact between rigid bodies i and k:

ψ1ik + ψ2ik =

4ξ Ωik

(29)

p where Ωik = ω ik ∆t, ω ik = K/M , and M is now the largest of the two contacting masses. The combination of Eqs. 20 and 29 provides an explicit expression for ψ1ik and ψ2ik . 4ξ Ωik ; D1 + D2

N1 + N2

ψ1ik

=

N1

≡

N2

≡

D1

≡

D2

≡

ψ2ik =

4ξ − ψ1ik Ωik

i (q n+1 − q in )t − (q kn+1 − q kn )t f ik c i i t k k t ik ik (q n+1 − q n ) − (q n+1 − q n ) K Rik n+1/2 (gn+1 − gn ) 1 i i−1 ikt (pn+1 − pin )t Rik Rn+1/2 (pin+1 − pin ) + n+1/2 M 2 1 k−1 kit Rn+1/2 (pkn+1 − pkn ) + (pkn+1 − pkn )t Rki n+1/2 M 2 1 ik − K(gn+1 − gnik )2 2

(30)

Inserting these expressions into Eq. 10 and taking into account that ξ = c/(2M ω ik ) (c is the dashpot damping coefficient) c 4ξ =2 Ωik K ∆t c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls

(31) Int. J. Numer. Meth. Engng 2000; 00:1–6


12

( ) nbd h X q in+1 − q in N2 N1 i i 4ξ i−1 i i 1− pn+1/2 + − (pn+1 − pn ) =M ∆t Ωik D1 + D2 D1 + D2 k=1 {z } k6=i | Additional momentum pik c pin+1 − pin = ∆t f ic +

nbd h X k=1 k6=i

f ic +

i N2 4ξ N1 ik ik ik ik ik K Rik (g − g ) + K R (g − g ) = (32) n+1/2 n+1 n n+1/2 n+1 n D1 + D2 Ωik D1 + D2 g˙ n+1/2 z }| { ik gn+1 − gnik i 2 c Rik n+1/2 ∆t } | {z

nbd h X

N2 N1 ik ik K Rik n+1/2 (gn+1 − gn ) + D1 + D2 | {z } D1 + D2 k=1 i k6=i ˆ Spring force f c Damping force f id {z | Additional contact force f ′c

}

This decomposition shows that for every contact, EECA includes a damping force that controls the penetration velocity (gap’s vibration). The dashpot and spring terms exactly enforce the Kuhn–Tucker impenetrability conditions as if K, ξ → ∞ in standard algorithms. From the energetic point of view, this impenetrability is (f c + f ′c ) g = 0

→ No energy during contact

(f c + f ′c ) g˙ = 0

→ No power during contact

(33)

The influence of ξ in several parameters is now parametrically studied with the rigid Signorini problem. Figs. 4 and 5 show this influence in a unit mass disk with initial velocity V0 = (0, −1) m/s, K = 106 N/m, and ∆t = 0.002 s. This analysis concludes that as ξ increases • The gap penetration tends to zero (Fig. 4 left) since ξ dampens the penetration velocity and reduces the gap. • The addtional contact force f ′c increases since the penalty pair becomes stiffer (Fig. 5 left). The total contact force is conservative, thus f c + f ′c is constant. • The parameter ψ1 increases (Fig. 4 right) due to its proportionality with f ′c . • ∆Epc and ∆Ef c tend to zero (Fig. 5 right) . The closer the algorithm solution to the Kuhn–Tucker conditions, the smaller the energy transfer between bodies and penalty spring. The main conclusion is that the energy inserted by ∆Epc and ∆Ef ′c decreases (Fig. 5 right). Additionally, ∆Epc → 0 implies that pc , ψ2 → 0 (Fig. 4 left and right). c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


13


0

ψ1 ψ2

0 0

80 Penalty ξ

0

160

0

80 Penalty ξ

Parameter ψ2

-0.001

600 Parameter ψ1

Gap g

gn+1 pc

Lin. mom. pc

1.4 0

-1.4 160

Figure 4. Influence of ξ in g, pc , ψ1 and ψ2 . As ξ increases, the algorithm’s solution approximates the Kunh–Tucker conditions. The artificial energy transfer decreases, pc , gn+1 and ψ2 tend to zero (left and right), and ψ1 increases since f ′c becomes stiffer (right)

0 Energy E

Contact Force f

∆Epc ∆Ef ′c

2

fc f ′c f c + f ′c

0

-2

-2000 0

80 Penalty ξ

160

0

80 Penalty ξ

160

Figure 5. Influence of ξ in f c , f ′c , ∆Epc and ∆Ef ′c . As ξ increases, contact force f ′c increases since the penalty pair becomes stiffer. Conservation implies that f c + f ′c is constant (left), thus f c decreases. ∆Ep c and ∆Ef ′c decreases as the solution is closer to the Kuhn–Tucker conditions (right)

6. NUMERICAL SIMULATIONS Four representative examples have been simulated with EECA based on the Discontinuous Deformation Analysis method applied to rigid disks. The first studies the Carom problem (billiard pool problem), in which the conservation of momenta and total body energy are analyzed using a disk that impacts inside a square, rigid 2D box with an initial incident angle and velocity. A second example analyzes the breaking and rebounding of an arrangement of 55 pool balls for two situations: a) all balls have the same diameter and are initially in contact, and b) several balls have a slightly smaller diameter (imperfections). c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



14

The third consists of the Newton’s Cradle problem, in which the conservative response of three hanged and the aligned disks is studied when one of the extreme impacts the neighbours with a specified velocity. This problem has two analytical solutions corresponding to different hypotheses. Finally, a fourth example simulates the filling of a hopper under the action of gravity. In this process, many more impacts occur and energy conservation is monitored at all times. For the numerical algorithm’s performance comparison, the first, second, and third problems will be additionally simulated with the Newmark–β method using two choices of the β and γ parameters shown in Table I. The Trapezoidal Rule (Trap.) and Maximum Disipation (Max.D.) are chosen since they have zero and largest numerical damping, respectively. M ethod Trapezoidal Rule (Trap.) Max. Dissipation Rule (Max.D.)

β 1/4 0.49

γ 1/2 1

Stability Condition Unconditional Conditional

Table I. Newmark–β parameters for first, second and third problems

6.1. Carom problem The 2D Carom problem, or billiard pool problem, consists of the analysis of a single disk trajectory, velocity and energy inside a one–meter square rigid box. An initial velocity V0 = (4, −4) m/s and a position (0.25, 0.25) m are prescribed. The analysis parameters are: K = 106 N/m, ∆t = 0.0025 s, unit mass, radius 1 cm and two different penalty velocities, low and high ξ = 1 and ξ = 200. Since EECA conserves energy and momenta, the incoming and outgoing angles and velocities are equal, and the trajectories superimpose each other cyclically–results that agree with the analytical solution seen in Figs. 6 left for both ξ’s. The disk impacts in the middle side due the initial conditions choice. Figs. 6 right shows that the horizontal and vertical velocities pre– and post–impact are equal in magnitude and opposite in sign and that the energy is perfectly conserved. If the same problem is simulated using the Newmark–β method for both parameter sets of Table I, the trajectories do not agree with the analytical results; although the Trap. rule (Figs. 7 top) is unconditionally stable for linear problems, it is unstable due to the contact nonlinearities, which produce energy increases at every impact. Both normal (to the boundary) velocity and outgoing angle increase and change sign after each contact (Figs. 7 left and right respectively), yielding a closing trajectories, non–physical result. Max.D. rule (Fig. 7, bottom) is also unstable with impacts, but its intrinsic high numerical damping forces the energy to decrease after each contact. This damping causes the normal velocity to decrease and, consequently, a decrease of the outgoing angle. The tangential velocity is not affected since there is no friction in this simulation. These effects produce an opening trajectory that simulates an inelastic contact but without any physical accuracy. c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


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1 NECA ξ = 1

Vx Vy Ener E

20

4

0.5

10

0 w

-4 0

0

0.5

1

0

0.5

1

0

1

NECA ξ = 200

Vx Vy Ener E

20

0.5

10

0

0.25

Energy E

Velocity x, y

Position y

4

w

-4 0

0

0.25

0.5

Position x

1

0

0.5 Time t

1

0

Figure 6. Trajectory, velocity and energy for EECA. Results agree with the frictionless Carom physics: energy is conserved and normal velocity equal in magnitude and opposite in sign, before and after contact. Initial position depicted by •

6.2. Frictionless breaking of pool balls The frictionless breaking of the billiard game is numerically analyzed in this subsection. The shooting disk a (diameter of 50 mm) is subjected to initial velocity V0 = (0, 10) m/s, impacting against a triangular arrangement of 55 contacting disks. The problem has been analyzed with EECA using parameters K = 106 N/m, ξ = 200, ∆t = 2.5 × 10−4 s, and mass 0.15 kg for each disk. Although in the simulation all disks are rigid, they will not penetrate each other at the same time; the presence of penalty springs propagate a wave that delays the contact transmission. This simulation is focused on the propagation and conservation of the total bodies’ energy and of the linear momenta inside the arrangement. Three cases, with results shown in Figs. c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



16 1

TRAP.

Vx Vy Ener E 120

8

0.5 0

60

w -8

0

0.5

1

0

0.2

1 MAX.D.

Velocity x, y

Position y

4

0.5

0.25

0

0.4

Vx Vy Ener E

16

0

8

Energy E

0

w

-4 0

0

0.25

0.5 Position x

1

0

0.5 Time t

1

0

Figure 7. Trajectory, velocity and energy for Newmark-β. Non–physical behaviour due to boundess energy growth for Trap. and high numerical damping for Max.D. Initial position depicted by •

8 and 9, have been considered: i) all disks have the same diameter, 60 mm; ii) two shaded disks positioned symmetrically (with respect to the impact line) are slightly smaller, diameter 58 mm; and, iii) three of these shaded disks are positioned asymmetrically. For all cases, a impacts b, then a recoils and b contacts the surrounding disks. In the figures, the direction and magnitude of each disk’s linear momentum is indicated by an arrow. Since case i), Fig. 8, is completely homogeneous and symmetric, most of the linear momenta are transmitted along the directions defined by b–c and b–d. This direct propagation explains the high velocity ejection of the disks c and d. Figs. 9 shows the results for cases ii) and iii), where, the linear momenta transmission Lx , Ly and body’s energy E and consequently disk dispersion after impact are symmetric for case ii) and asymmetric for iii). There are no evident trends in the directions of the transmission, c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



Case i)

17

d

c

b a 0s

1.04e−3 s

1.24e−3 s

5.0e−2 s

Figure 8. Frictionless pool balls, case i), with symmetry and equal disks. Most linear momenta are transmitted along sides b–c and b–d. After impact, only vertex balls move

and most disks acquire enough motion to completely disseminate. For ii), the symmetry of the arrangement forces a dispersion, while for iii) the asymmetrical arrangement forces scattering. These results qualitatively agree with those of the pool game, where imperfections in the diameters and positions influence the motion after impact. Although results for the three cases are different, EECA computes and conserves the total linear momentum and bodies’ energy (see Table II).

t (s) Lx (Kgm/s) Ly (Kgm/s) E (J)

Case 0 0.00 8.66 43.3

i) 1.0 0.00 8.66 43.3

Case 0 0.00 8.66 43.3

ii) 1.0 0.00 8.66 43.3

Case 0 0.00 8.66 43.3

iii) 1.0 0.00 8.66 43.3

Table II. Total body energy and linear momenta at the beginning and end of the simulation. Motion is different for the three cases, but linear momenta and bodies’ energy coincide




18

Case ii)

0s

1.3e−3

6.3e−3 s

6.0e−2 s

0s

1.49e−3 s

5.9e−3 s

0.99 s

Case iii)

Figure 9. Frictionless pool balls, case ii), with symmetry and two smaller disks and case iii) with asymmetry and three smaller disks. The disk differences favor linear momenta transmission inside the arrangement. c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


19


6.3. Newton’s Cradle problem The third example simulates the impact of three aligned and adjacent rigid and frictionless disks (Newton’s Cradle problem) using Newmark–β rules and EECA. Figs. 10 show the three initially static disks a, b and c, with a separated to the left. Disk a is subjected to initial velocity Va0 = 1 m/s impacting against b. This problem is solved with two different hypotheses used for both the analytical and the numerical results. In the first, i), disks b and c are initially together, and when a hits b, b and c impact instantaneously. This hypothesis considers that masses b and c behave like a single mass b + c. In the second, ii), all disks are slightly separated and contact b– c occurs after that of a–b. Ref. [11] provides the analytical post–impact velocity solution: Va1 = −1/3 Va0 , Vb1 = Vc1 = 2/3 Va0 for i) and Va1 = Vb1 = 0, Vc1 = Va0 for ii).

Va0 = 1 Va1 = 1/3 Va0 = 1

a

b b

c

a

c

b

Vb1 = 2/3

a

Va0 = 1

b

c

a

b

c

a

c

a

Vc1 = 2/3

c

b Vc1 = 1

gap

Vb = 1

Figure 10. Newton’s Cradle problem. Hypothesis i) (top) with disks b and c in contact, equivalent to a single mass. Hypothesis ii) (bottom), small separation between disks b and c

For both hypotheses, the simulation parameters are K = 107 N/m, ∆t = 0.00025 s, ξ = 200 and unit mass and diameter 10 cm. Table III shows the post–impact velocities for Trap., Max.D. rules, EECA, and analytical solutions. The first two conserve linear momentum but fail to preserve energy since they are unstable in the presence of contact; therefore, their post–impact velocities differ from the analytical ones. The last row of this table shows the error between EECA and the analytical solution. It is clear that post–impact velocities slightly differ for i) and are exact for ii); if more than one contacts occur during the same time step, as in i), results for EECA will be different for each K and ξ choice since the penalty method does not exactly satisfy the impenetrability condition. The analytical formulation is based on this condition; therefore, the numerical velocities tend c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



20

toward the analytical ones only when K, ξ → ∞. For hypothesis ii), only one contact occurs at each time step; then the error is always zero since the numerical conservative solution is unique. Due to the energy balance of Eqs. 32, the linear momentum and the total body energy are conserved by EECA for both hypotheses. Algorithms Trap. Max.D. EECA Analytical %

Va1 -0.82 -1.07 -0.3332 -0.3333 0.0170

Hypothesis i) Vb1 Vc1 Ltot 0.33 1.49 1.0 0.44 1.13 1.0 0.674 0.659 1.0 0.666 0.666 1.0 1.140 1.149 0.0

Etot 1.51 0.90 1.0 1.0 0.0

Va1 -0.33 -0.31 0.0 0.0 0.0

Hypothesis ii) Vb1 Vc1 Ltot 0.08 1.24 1.0 -0.04 1.36 1.0 0.0 1.0 1.0 0.0 1.0 1.0 0.0 0.0 0.0

Etot 0.83 0.97 1.0 1.0 0.0

Table III. Newton’s Cradle problem. Post–impact velocities, linear momentum, and energy. Total linear momentum is conserved for all algorithms, velocities and energy differ from analytical ones due to Newmark–β instability in contact. With EECA, small error for hyp. i) and exact for ii)

Figs. 11 show the velocity errors related to hypothesis i) as a function of time step ∆t (top) and of dashpot penalty ξ (bottom). Each line represents a different value of the spring penalty K. The figures on the left show errors of a, while those on the right show of b and c. Notice that in Table III the error in i) is very small for a but nonnegligible for b and c. The reason can be explained from [12], which proved that if two bodies contact exactly at the beginning of a time step n, the error convergence rate is quadratic and linear otherwise; obviously, a impacts b exactly at n, and the algorithm achieves quadratic convergence rate (Figs. 11 left). Although the current problem studies rigid bodies, the penalty spring propagates an impact wave from a to b and from b to c. This wave delays the second impact b–c for a time α (configuration n + α). Therefore, for b and c the convergence error rate is linear (Figs. 11 right). As expected, the error converges to zero as K, ξ → ∞ and ∆t → 0. 6.4. Hopper filling process This simulation consists of the filling of a 1 m side, 30◦ hopper employing 665 disks of unit mass and radius 0.004 m, under the action of gravity. The disks are initially organized in an alternated array and separated by a small distance, while the hopper is modeled by rigid straight lines. Fig. 12 shows EECA’s evolution at four different times with parameters K = 106 N/m, ξ = 200 and ∆t = 0.0025 s. During the process, the disks located laterally and at the bottom of the array rebound against the inclined rigid sides. Then, contacts propagate through the rest of the disks, changing their trajectories. Since EECA conserves the total bodies’ energy, the disks move at all times and many of them scape the hopper cone. This is a frictionless numerical simulation whose results do not match reality, but the experiment is useful for proving EECA’s conserving properties in the presence of multiple contacts. Since the energy is always conserved, no graphic representation of it has been drawn. The same problem has been simulated using the Newmark–β methods with Trap. and Max.D. rule for large, medium, and small time steps. Figs. 13 and 14 depict the filling process, where it is clear that Trap. is unstable for any ∆t, while Max.D. is so only for small ones. c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


21


100

10

Error %

K = 105 N/m 106 ” 107 ”

K = 105 N/m 106 ” 107 ” 1

0.01

a 1e-05 1e-06

0.0001 Time step log ∆t

Error %

1e-06

0.01

K = 105 N/m 106 ” 107 ”

1

b, c

0.01

0.0001 Time step log ∆t K = 105 N/m 106 ” 107 ”

10

1e-06

0.01

0.01

a

b, c

1e-12 10

10000 Damping log ξ

1e+07

1e-05 10

1000 100000 Damping log ξ

1e+07

Figure 11. Newton’s Cradle Problem. post–impact velocity error analysis as a function of time step ∆t and damping ξ. Convergence is quadratic for disk a and linear for b and c

NECA 0.25s

0.47s

0.65s

0.92s

Figure 12. Hopper filling with EECA. Disks never settle and some escape due to energy conservation




22

TRAP. 0.25s 0.47s ∆t 2.5e−3

0.25s

0.47s 0.65s

∆t 1.0e−3

0.25s

0.47s 0.65s

∆t 5.0e−4

40000

Energy E

∆t = 2.5e−3 s 1.0e−3 5.0e−4

0

0

0.2

0.4

0.6

Time t Figure 13. Hopper filling for three ∆t with Max.D. Unbounded energy growth due to trapezoidal rule instability in the presence of contact with any time step c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



23

MAX.D. 0.25s

0.47s

0.65s

0.92s

0.47s

0.65s

0.92s

∆t 2.5e−3

0.25s

∆t 1.0e−3

0.47s

0.25s

0.65s

∆t 1.0e−5 40000 Body Energy E

∆t = 2.5e−3 s 1.0e−3 1.0e−5

0

0

0.5

1

Time t Figure 14. Hopper filling for three ∆t with Max.D. Although it is unstable for contact, results (top and middle) seem physical for large ∆t due to high numerical dissipation (bottom). While for low, energy increases without bound.




24

Numerical Damping ξN

Instability implies that the motion of the disks increases boundless due to a nonphysical energy blowup, causing many of them to be expelled. Ref. [3] defined an energy norm for Newmark-β methods that decreases when two bodies penetrate and increases when they release. Since the penalty method releases the contact with a positive gap, g > 0, the contact force adds energy to the bodies. On the other hand, it is possible that this additional energy is dissipated by the numerical damping ξN , depending on the type of algorithm and on the time step. Fig. 15 (from [13]) shows the relationship between nondimensional frequency ω∆t and numerical damping ξN for both rules. The Trap. does not introduce damping regardless of the ∆t value; consequently, the bodies always gain energy when they are released. On the contrary, the Max.D. method exhibits the highest damping of the Newmark-β family, increasing with ∆t. Therefore, if large time steps are used, damping can dissipate the energy gain, a situation that is not possible using small time steps. 1 Max. D. Trap. 0.5

0 0

5 Non–dimensional frequency ω∆t

10

Figure 15. Hopper filling. Nondimensional frequency ω∆t vs. numerical damping. Null using Trap. and increasing with ∆t using Max.D.

The bottom graphic of Figs. 13 shows the energy evolution for Trap.; the larger the time step, the sooner the blowup appears, causing the program to stop due to numerical overflow. This is due to the fact that for large time steps the release gap, and therefore the energy gain, are high. The geometrical description of this energy increase is drawn in the top sequence of Figs. 13, where at 0.47 s the simulation is already unstable and with disks dispersing unrealistically. When ∆t decreases (second and third sequences), the release gap is closer to zero and thus the energy gain is small. Although at 0.65 s the result is unstable, for a medium time 0.47 s the simulation is still stable. This behavior can also be studied in the graphic. The time 0.47 s corresponds to a clear instability for the large ∆t but to a stable energy for the other two, while for 0.65 s any time step produces unstable results. In the same graphic, it can be seen that for medium and small ∆t, the body energy decreases at t ≈ 0.3 s since at that time most of the disks are penetrating, while at t ≈ 0.55 s these disks are starting to release. The contact forces introduce then large amounts of energy. In Fig. 14, results using Max.D. are shown that differ from those of Trap., although the same c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls



25

∆t’s are used. For large and medium ∆t, even if the gaps after release are big, the energy gain is dissipated by the numerical damping as seen in the first and second sequences. In the final pictures of both, a fluid–type motion is seen due to the high numerical damping; disks on the top with little damping (due to their low number of contacts) push those at the bottom whose motion is very dampened after many contacts. The values provided by Fig. 15 when ω∆t = 2.5 and ω∆t = 1.0 (large and medium time steps) are ξN = 0.3 and ξN = 0.08, respectively, which are rather high. But by the use of a small ω∆t = 0.01 → ξN ≈ 0, a sudden energy growth that cannot be dissipated appears, causing most of the disks to be expelled from the hopper (third sequence). The graphic at the bottom clearly shows that for the small ∆t and total times t > 0.4 s the behavior is unstable.

7. CONCLUSIONS The new time stepping algorithm EECA, which conserves momenta and total bodies’ energy, has been developed for frictionless rigid body contact problems. This conservation is achieved by modifying the body kinematics for every contact through the addition of a linear momentum and a contact force, thus fulfilling the conserving balance equations for momenta and energy. As a consequence of conservation, the stability of the algorithm is ensured, and the balance equations define an enhanced penalty model composed of a spring K and a dashpot ξ. The role of ξ has been parametrically studied for the Signorini contact problem, concluding that the enhanced penalty model mimics the two Kuhn–Tucker conditions when both penalties tend to infinity. EECA has accurately reproduced the perpetual motion of a disk for the Carom problem. Also, the breaking of pool balls has proven its capacity to represent disk dispersion and internal interactions; penalty springs create an impact wave that delays the contact transmission. For the analytically solved Newton’s Craddle problem, there is a good correspondence in post– impact velocities for the two hypotheses considered; the penalty spring delays the contact transmission so that the second contact converges linearly. Finally, EECA exhibits a good performance for contact problems of medium size, such as the filling of a hopper. These problems (except the second) have also been simulated using the Newmarkβ algorithms: trapezoidal and maximum dissipation rules (zero and maximum numerical damping, respectively). It is shown that the linear momentum is conserved, but these rules are unstable; the trapezoidal exhibits a boundless energy growth regardless of the time step due to its zero damping. For large time steps, the maximum dissipation rule can dissipate the energy blowup through its high damping introduction, but without physical meaning. The formulation and examples of this work have been developed for rigid disks using the Discrete Element Method but may be extended to elastic bodies and the Finite Element Method. The development and implementation of the algorithm to frictional contact is an ongoing work.

ACKNOWLEDGEMENTS

R. Bravo and J.L. P´erez–Aparicio were partially supported by the MFOM I+D (2004/38), MICIIN #BIA-2008-00522 and “Ayudas Investigaci´ on” from UPV2010 grants. c 2000 John Wiley & Sons, Ltd. Copyright Prepared using nmeauth.cls


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