a contact algorithm for explicit dynamic fem analysis e

A contact algorithm for explicit

dynamic FEM analysis

E. Anderheggen", D. Ekchian", K. Heiduschke",

P. Bartelt^

"Swiss Federal Institute of Technology,

Honggerberg, CH-8093 Zurich, Switzerland

&#2%2 AG, Corporal Aesearc^ FI- P

Schaan, Principality of Liechtenstein

ABSTRACT

A numerical contact algorithm which can be used with an explicit timeintegration scheme in a finite element code is introduced. Its effective-ness is demonstrated on low velocity impact and penetration problemscontaining many contact surfaces. The treatment so far is two-dimen-

sional and without friction.

INTRODUCTION

The increasing interest in dynamic penetration and wave propagationphenomena has heightened the need for simulation tools capable ofmodelling multi-body impacts. Engineers who have analysed such prob-lems, have traditionally relied on one-dimensional analytical solutions[1]. These are valid as long as the system is truly one-dimensional andremains elastic. For complex two- or three-dimensional systems in whichplastic deformations and friction effects occur, the finite element method

(FEM) is often used.

In the finite element analysis of impact and wave propagation problems,an explicit time integration scheme is usually employed to solve the sys-tem of equations which governs the dynamic response of the system. Theexplicit algorithm is ideal for this kind of problem since the stable orcritical time step of the algorithm is near the time step required to

Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533

272 Contact Mechanics

accurately follow the propagation of the stress waves. Implicit methodswhich require a much greater computational effort for each time stepwould be unpractical for this application. (For a more detailed explana-tion of the difference between explicit and implicit methods the reader isreferred to a standard finite element text, such as [2]. This will not bediscussed here). The explicit integration scheme has become a popularand standard method available in most commercial finite elementpackages. However, the numerical treatment of the contact or impactphenomena is not as well understood.

The purpose of this paper is to introduce a numerical contact algorithmwhich can be used in conjunction with an explicit integration scheme in afinite element code. The contact algorithm belongs to the class of so-called "nodal projections algorithms" or "simplified Lagrangian methods"[3]. At present the new algorithm is two-dimensional and without fric-tion. The method is general, i.e. it handles node-to-side contact and is notrestricted to node-to-node impact. It requires no mass distribution as in[5] and has been implemented without damping. The method is unsym-metric in that it requires the definition of a "master" and "slave" surface.(However, work is presently underway to symmetrize the algorithm.) Themethod is non iterative which makes it computationally stable.

In the following section, the numerical algorithm will be presented. Af-terwards, several example problems will be discussed. The algorithm willbe applied to a problem in which a nail is driven by a flying piston into asteel substrate. This problem is especially demanding since the analysiscontains eight contacting surfaces. Before discussing such a complex ex-ample, the method will be used in the dynamic stress analysis of a simplebar impact. In this problem, the influence that the algorithm has on thestable time step of the explicit time integration will be demonstrated. Inaddition, the energy (balance) in the system will be carefully observedand analysed.

The paper will not treat topics associated with impact and wave propaga-tion effects such as strain-rate dependent plastic material behaviour underfinite deformations.


Contact Mechanics 273

THE NUMERICAL ALGORITHM

The basic idea behind the algorithm (and all nodal projection algorithms)is to first let the two contacting finite element bodies penetrate into eachother within a single time step. At the end of the time step, additional in-cremental nodal displacements (and velocities and accelerations) are in-troduced into the system such that the non-penetration conditions arestrictly enforced. This can be schematically shown as follows:

Slave Nodes

Master Nodes

Figure 1: Definition of variables. The master node k penetrates the slavesurface between node i and j.

In Figure 1, C% is the contact force at the k-th "master node". It acts per-pendicular to the "slave-side" between nodes i and j. Of course, if thenode does not penetrate, C =0. At node i are the slave forces C^ and C^,which act in the x and y directions, respectively. During contact, themaster and slave forces must be in equilibrium. Hence,

(CJ =[E]{CJ (1)

where {CJ is the vector of Q slave node forces with two entries C^ andCyj per slave node and {C } is the vector of C% master node forces in thedirection perpendicular to the penetrating side. The matrix [E] with com-ponents Eft represents the i-th slave node force in the x- and y-directionsdue to = 1.

The additional incremental displacements {U } and {U j within the timestep are for the master and the slave nodes, respectively



Figure 2: Definition of the length factor a and angle <j).

(3) Assemble the narrowly banded matrix

AtM[MJ-i + [EF [MJ-i [E])

(4) Solve the system of linear equations (6) for all the master contactforces C%:

{C J = (At* ([MJ-i + [ET] [MJ-i [E])-i {P}

(5) Determine the slave forces {CJ from equation (1) and correct thenodal displacements and velocities of the master and slave nodesconsidering the acceleration changes due to the contact forces.

There are several salient features to the algorithm which deserve a fewremarks:

(1) The construction and solution of the contact equations is straightforward if the contact surfaces are "smooth". The search algorithm,finding the penetration lengths {P} and location on the slave sides[E], is also simple. However, in the vicinity of corners or abruptchanges in a contact surface, it is often not clear how to determinethese values. Because of space requirements, this search algorithmwill not be discussed here.

(2) The master contact surface forces C^ found by solving the system ofequations should be all positive. A non-positive C% implies no con-tact, which contradicts the fact that the node was found to be pene-trating a slave surface. We have found that negative C^ forcessometimes do occur. They result when one master node k penetratesdeeply into a slave surface. The adjacent master nodes (k-1, k+1)can have a negative force associated with them. A negative force isusually a signal that the time step is too large.



{UJ= Af[MJ-i{CJ (3)

where At is the time step; [MJ and [MJ are the coiTesponding diagonal

lumped mass matrices.

Let {P} be the vector whose component P% represents the penetrationlength of the k-th master node into the slave side between nodes i and j.The kinematic non-penetration condition within a time-step requires that

{tU + [EF {UJ = {P} (4)

where the same matrix [E] as in equation (1) appears.

Substituting equations (2) and (3) into equation (4) yields

At' [MJ-i (C J + [E]T At* [MJ-i {C,} = {P} (5)

Further substitution of equation (1) into (5) gives

At* ([MJ-i + [EF [MJ-i [E]) (Cm) = {P} (6)

which is a positive-definite narrowly banded equation system with mas-ter-forces {C,J as unknowns and the penetration lengths {P} as the righthand side. The size of the equation is given by the number of masternodes which penetrate the slave surface.

The algorithm can be formulated in a simple five step procedure:

(1) For all master nodes of all possible contact zones determine thepenetrated slave-side (if any) and the penetration lengths P^ of thevector {P} ignoring contact.

For each group of (one or more) master nodes penetrating adjacentslave sides (or the same slave-side) do:

(2) Determine the coefficients E^ which are only functions of the

length factor a which defines the position of the penetrating node kon the side i-j and the angle <j> of side i-j, see Figure 2.



EXAMPLES

In the first example problem a 100 mm cylindrical steel (E = 210'000N/mm*) rod with a velocity of 10 m/s impacts a 200 mm long rod withthe same diameter and material. Poisson's ratio was artificially set to zeroin order to make the impact as one-dimensional as possible. The ends ofthe rods are flat; both rods are free to move. The meshes contained axissymmetric four node elements [6] and were uniformly discretized, i.e. theregions near the impact zone were not more finely modelled. To analysethe effectiveness of the algorithm the time step of the explicit timeintegration scheme varied. Since the smallest element length, 1, was 2mm and the wave propagation speed, c, was 5(10 ) mm/s, the criticaltime step of the time integration is approximately

A T < 1/c = 2 mm / 5 (1.0e6) mm /s = 400 ns.

Figure 3 displays the contact force at the impact surface for various timesteps. The contact force is defined as the sum of all master forces actingon the surface. It can be clearly seen that at a time step of half the criticaltime step (200 ns) the contact force oscillates strongly. At about one-forth the critical time step (100 ns) the results improve dramatically. At atime of 50 ns, the results resemble the one-dimensional analytical solu-tion (a square pulse) with the exception of oscillations at the point of in-itial contact. Interestingly, the oscillations disappear very quickly and donot introduce strong oscillations in the axial stress.

Figure 4 shows the axial stress at a point near the contacting surface ofthe longer bar. The figure shows that a time step of 200 ns (one-half thecritical time step) the stress oscillations are clearly unacceptable. The re-sults improve at a time step of 100 ns (one-forth the critical time step).

A useful method to judge the correctness of a transient dynamic analysisis study the energy in the system [4]. Theoretically, the energy change inthe system should be zero. Moreover,

e = E-I-K = 0

where e is the energy change, E is the external energy is the system,which includes the initial kinetic energy and the energy due to surfacetractions and body forces. K is the kinetic energy in the system; I is theinternal energy. The initial kinetic energy in the system was approxi-



mately 12 J. For the case At = 200 ns, an energy increase of 0.5 J was ob-served. This change (4 %) is judged to be unacceptable. At time stepsunder 100 ns, energy decreases of less than 0.1 J were observed (< 1 %).

In the next example, the numerical simulation of nail driving tool is dem-onstrated. The tool is invaluable on construction sites where quick tem-porary (and permanent) fastenings are required. In this example problema nail is driven by a piston with an initial velocity of 90 m/s into a 20 mmthick steel substrate. The tool is often used to fasten thin metal roofingmaterial - so-called profile sheets - to steel beams. Hence, in this exam-ple, before the nail penetrates the steel substrate, it penetrates two 1 mmthick profile sheets as well. A washer is placed on the nail in order tostabilise it during penetration and to clamp the profile sheets on to thesteel substrate. This example problem is similar to earth penetrationproblems [7] or dynamic rigid punch problems analysed in [8].

The FEM analysis of this example, as shown in Figure 5, contains eightcontact surfaces: (1) piston-nail, (2) nail-washer, (3) nail-first profilesheet, (4) nail-second profile sheet, (5) nail-substrate, (6) washer-firstprofile sheet, (7) first profile sheet - second profile sheet and (8) secondprofile sheet substrate. It is important to point out that such complexproblems are impossible to study analytically or even experimentally.Typically, a FEM analysis is carried out for a variety of reasons such as:

(1) To determine the forces acting on the nail.

(2) To approximate the pull-out force of the nail.

(3) To determine the wave propagation effects in the piston

(4) To study the deformations in the substrate, profile sheets andwasher.

(5) To optimise the clamping forces between the various componentsin the system.

(6) To optimise the energy usage in the system.

(7) To optimise the nail, washer and piston geometries.

Figure 5 shows the deformed mesh plots of the system at several differ-

ent times during the penetration. Of interest is the time t = 100 |ns whenthe washer impacts the profile sheets which are momentarily pressed



against the steel substrate. Afterwards, the washer moves backwards onthe conical nail shaft and since there is no contact between the profilesheets and washer, the profile sheets "relax" and spring back. Obviously,a bad washer design.

The finite element calculation was carried out using a time step of 5 ns oran eighth of the critical time step. Figure 6 contains several interestingnumerical results. Figure 6a contains the contact forces acting against thenail and piston over time. There are approximately in equilibrium witheach other, as expected, and can be viewed as the deceleration andhistory of the nail and piston. Figure 6b shows the radial contact forcebetween the substrate and nail. Finally, Figure 6c shows the rigid-bodyvelocities of all the components in the system. Of interest is the velocityof the washer. It's initial velocity is the same as the nail and piston (90m/s). Afterwards, it impacts the top profile sheet and in the absence offriction changes its direction and moves backward on the nail shaft.

CONCLUSIONS/SUMMARY

A contact algorithm for two-dimensional, frictionless explicit dynamicfinite element analysis has been introduced. The algorithm is general inthat it is not restricted to node-to-node impact, requires no massdistribution or damping. The algorithm has performed satisfactorily. Inorder for the algorithm to work well, i.e. to produce no severe oscilla-tions in stress after impact and to minimise the energy change in thesystem, a smaller time step than the critical time step of the explicit timeintegration must be used. Chatter on the contact surfaces was alsoobserved. However, if the time step was small enough, the chatter disap-pears quickly. Although the algorithm requires the solution of a small,banded system of equations for each time step, the contact algorithm hasused less than five percent of the total solution time of all the problemsthe authors have run to date. Future work will be performed to make thealgorithm independent of the definition of master and slave surfaces andto introduce friction.



0. 10. 20. 30. 40. 50. 60. 70

100.0

80.0-

w 60.0u

40.0

20 0

<Vv-

10. 20. 30. 40.Time

(c)

50. 60. 70.

Figure 3: The contact force (kN) vs time (|is) for (a) At = 200 ns,

(b) At = 100 ns and (c) At = 50 ns.



o.o 10. 20.

Figure 4:

30. 40.Time

50. 60.

30. 40.Time

(C)

The axial stress (N/mnf) vs time (jus) for (a) At = 200 ns,

(b) At =100 ns and (c) At = 50 ns.



(a)

(b)

—r •j~j~t~H

Figure 5: Deformed mesh plots for the nail penetration example (a)

T=0s, (b) T=lOO^s, (c) T=300 (is.



ofa

to0)Pi

2.0

0.00

-2.0

-4.0

-6.0

-8.0

-10.0

-12.0

-14.00.00 50. 100. 150. 200. 250. 300. 350. 400.

Time

Figure 6a: Contact forces (kN) vs time (jus) between the piston and nailand the nail and substrate.

20.0

0.00

-20.0

-40.0

-60.0

-80.0

-100.0

-120.0

Figure

0.00 50. 100. 150. 200. 250. 300. 350. 400.Time

6b: Radial contact force (kN) vs. time (|is) between the nail andsubstrate.

100.0

80.0

60.0

40.0

20.0

0.00

-20.0

•--piston--: Nail:. Wa.sh.0r-.jsheet jlSheet !2

Figure

0.00 50. 100. 150. 200. 250. 300. 350. 400.Time

6c: Rigid body velocities (m/s) vs. time (|is) of several compo-nents in the system.



REFERENCES

[1] Kolsky, H., Stress Waves in Solids, Dover Publications, Inc., New

York, 1963.

[2] Bathe, K., Finite Element Procedures in Engineering Analysis,Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1982.

[3] Hughes, T.J. R. and Belytschko, T., Nonlinear Finite ElementAnalysis, Lecture Notes to a Short Course Taught by , Paris, 1990.

[4] Key, S. and Isay, C, 'On the Use and Implementation of an Ener-gy Balance Equation in Explicit Transient Dynamic Analysis' in Com-putational Aspects of Contact, Impact and Penetration, edited by R. Ku-lak and L. Schwer, Elmepress International, Lausanne, 1991.

[5] Hallquist, J., Goodreau G., and Benson D., 'Sliding Interfaces withContact-Impact in Large-Scale Lagrangian Computations', ComputerMethods in Applied Mechanics and Engineering 51 (1985).

[6] Heiduschke, K., Anderheggen, E., Bartelt, P., 'Axissymmetric Th-ree-Node Triangular and Four-Node Quadrilateral Finite Elements for

Finite Elasto-Plasticity', To appear.

[7] Chen, E., Reedy, E., 'Penetration into Geological Targets: Numeri-cal Studies on Sliding Friction' in Computational Aspects of Contact, Im-pact and Penetration, edited by R. Kulak and L. Schwer, Elmepress In-ternational, Lausanne, 1991.

[8] Hughes, et.al. 'A Finite Element Method for a Class of Contact-Impact Problems', Computer Methods in Applied Mechanics and Engi-

8(1976).


a contact algorithm for explicit dynamic fem analysis e

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