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Int. J. Computer Aided Engineering and Technology, Vol. 1, No. 1, 2008 31

Copyright 2008 Inderscience Enterprises Ltd.

ANSYS and LS-DYNA used for structural analysis

Yucheng Liu

Department of Mechanical Engineering,

University of Louisville,

Louisville, KY 40292, USA

Email: y0liu002@louisville.edu

Abstract: This paper discusses the different features of ANSYS andLS-DYNA in solving structural analysis using finite element approach. ANSYSand LS-DYNA are two popular engineering software tools that are used forfinite element analysis. The difference between ANSYS and LS-DYNA lies indifferent solution procedures and time integration methods they use. ANSYS isan implicit analysis program while LS-DYNA is an explicit analysis program.This paper compares the general properties of the two programs through animpact problem and explains the reason. A simple two degrees of freedomsystem is solved to show how the implicit and explicit methods areimplemented. Conclusively, advantages and limitations of both programs arediscussed as well as their applicable areas.

Keywords: FEA; finite element analysis; implicit analysis; explicit analysis;ANSYS; LS-DYNA.

Reference to this paper should be made as follows: Liu, Y. (2008) ANSYSand LS-DYNA used for structural analysis, Int. J. Computer AidedEngineering and Technology, Vol. 1, No. 1, pp.3144.

Biographical notes: Yucheng Liu is a Post-doctor in Department ofMechanical Engineering, University of Louisville, USA. He earned his PhDfrom University of Louisville in 2005. His research interests includeCAD/CAE, FEA, mechanical design, dynamics and mechanics, vehicle designand analysis and applied mathematics. Until 2008, he has published more than20 journal articles. His other publications include a book, a chapter and 13conference papers. He is a member of SAE and ASME. He is also aProfessional Engineer registered in Ohio.

1 Introduction

Finite element analysis (FEA) algorithm has been integrated with many engineering

software tools for solving most engineering problems. An efficient computer aided

engineering (CAE) tool for static, transient, and dynamic analysis can be extensively

applied in engineering, including model design, optimising, and simulation. ANSYS and

LS-DYNA are the two general-purpose finite element analysis tools that are widely used

for most engineering areas. ANSYS is developed by ANSYS, Inc., and generally used for

numerically solving a wide variety of engineering problems, which include: static and

dynamic structural analysis, heat transfer and fluid problems, and acoustic and

electro-magnetic problems. LS-DYNA is developed by LSTC and mostly applied for

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32 Y. Liu

engineering problems that include non-linear finite element analyses, such as automotive

crashworthiness, occupant safety, and sheet metal forming. Linde et al. (2006) and Rustand Schweizerhof (2003) discussed general properties of ANSYS and LS-DYNA and

demonstrated the benefit of using LS-DYNA to perform quasi-static limit load analyses

in the literatures, the researchers recommended a proper combination of ANSYS and

LS-DYNA, and showed how to apply it to perform dynamic analysis by static analysis

steps. This paper mainly focuses on the capabilities of both software of solving structural

analysis problem, including static and dynamic analyses and exposes the reason by

hand-solving a simple two degrees of freedom problem.

2 ANSYS and LS-DYNA

ANSYS is an implicit FE program which uses a static load step with the defaultincremental iterative solution. In ANSYS, the Newmark time integration method is

employed for the solution of the transient dynamic equilibrium equation [equation (1)].

This method uses finite difference expansions in the time interval t, which will be

demonstrated in following text. ANSYS also uses the Newton-Raphson method to solve

nonlinear problems, in which method, applied load is divided into a series of load

increments and the load increments then are applied over several load steps (Advance

Analysis Technique Guide). Contact analysis is performed with penalty and Lagrange

methods, where only two potential contact surfaces must be specified.

LSDYNA is an explicit FE program and is designed for transient dynamic analysis of

highly nonlinear problems. In LS-DYNA, the transient dynamic equilibrium equation is

solved by the central difference method. Similar to ANSYS, LS-DYNA also uses

Newton-Raphson method to solve the nonlinear problems, including the contact and

impact problems. For the contact problems, three distinct methods are used to handle

such problems, includes the kinematic constraint method, the penalty method and the

distributed parameter method.

Among all the theories and approaches implemented in the two FE programs, the time

integration method is the crucial scheme that decides different properties of both

programs in performing different structural analyses, including static and transient

dynamic analysis. Following example shows how ANSYS and LS-DYNA are used to

prepare static and dynamic analyses and the theoretical parts are discussed later by

exposing different time integration processes.

3 Example

This section presents a simple impact problem to compare general properties of implicit

analysis (ANSYS) and explicit analysis (LS-DYNA). In this example, a thin-walled beam

which is a hollow-box structure enclosed with all surfaces surfers quasi-static load and

dynamic load separately. This beam is made of mild steel, whose Youngs modulus is

200GPa, Poissons ratio is 0.3, yield stress is 250MPa and hardening modulus is 630MPa

(Figure 1). ANSYS and LS-DYNA are used to perform the whole static and dynamic

analyses, from creating the FE model till generating the final results.

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ANSYS and LS-DYNA used for structural analysis 33

Figure 1 Thin-walled steel beam

3.1 Finite element model

The FE model for the square beam is generated in ANSYS and LS-DYNA separately. In

ANSYS model, the beam is modelled with four-node Reissner-Mindlin shell element

SHELL181, which is suitable for analysing thin to moderately-thick shell structures. An

assumed strain formulation of Bathe-Dvorkin is implemented to the SHELL181 to

account for the effects of transverse shear deformation. This element can also be used in

nonlinear analysis as well as linear analysis because it considers the problem of varying

shell thickness, which usually appears in nonlinear problem.

In LS-DYNA, the beam is modelled using full integration shell element: four-node

Belytschko-Tsay shell element with five integration points through the thickness. This

shell element is based on a combined co-rotational and velocity-strain formulation.

Because of its computational efficiency, it is usually the shell element formulation of

choice and has become the default shell element formulation of LS-DYNA (LS-DYNA

Theoretical Manual).

3.2 Quasi-static analysis

The FE model is first used for quasi-static analysis, during the analysis, its axial, bendingand torsional stiffness values are determined as well as its free vibration natural

frequencies. During the analysis, one end of the model is fixed and different types of

static loads are applied on the other end to obtain all the deflections. The axial stiffness is

then given by the quotient of the axial force and the axial deflection. Similarly, the

bending stiffness is determined by the quotient of the downward force and its deflection,

and the torsional stiffness is determined by the quotient of the moment and the angular

rotation of the models. Besides that, modal analyses are also performed to find the

models fundamental natural frequencies and modes.

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34 Y. Liu

Table 1 lists the static analysis results obtained from ANSYS, however, this model

entirely collapsed and yielded huge deformation if running the analysis using LS-DYNA,which could not be correct. To affirm this judgment, the FE models stiffness is evaluated

using popular theories. According to classic mechanics of materials, the axial, bending,

and torsional stiffness of this thin-walled steel beam are:

/axial

K EA L= (1)

33 /bending

K EI L= (2)

/torsion

K GJ L= (3)

The material of beams is mild steel, whose Youngs modulus E is 200GPa and shear

modulus G is 79GPa. For thin-walled square sectional beam with its width b and

thickness t, its area moment of inertia and polar moment of inertia are:

4 4( ( 2 ) ) /12I b b t= (4)

4 4( ( 2 ) ) / 6J b b t= (5)

Therefore, with given beam parameters b = 60mm, t = 1.5mm, and L = 300mm, the axial

stiffness should be around 234000 KN/m; the bending stiffness should be around 4452

KN/m; and the torsional stiffness should be about 106KN-m/rad. Compare with Table 1,

it can be found that both axial and bending stiffness results obtained from computer

analyses fully agree to the theoretical evaluation. Thus, the accuracy of the ANSYS

results is verified through the theoretical evaluation.

Table 1 Static analysis results from ANSYS and LS-DYNA

ANSYS LS-DYNA

Axial stiffness (KN/m) 2.4105

Bending stiffness (KN/m) 4350

Torsional stiffness (KN-m/rad) 112

Fundamental bending frequency (Hz) 267

Fundamental torsional frequency (Hz) 222

Fail to obtain meaning stiffnessand frequencies using LS-DYNA

3.3 Dynamic analysis

The FE model then is used for dynamic analysis. In the dynamic analysis, the square

beam model impacted a rigid wall with an initial velocity V = 15m/s, similar analysis hadbeen performed by Zhong before (Zhong, 1993). ANSYS and LS-DYNA are used to run

the analysis separately; the deformed FE model configurations after the analysis are

plotted in Figure 2 and compared to the results obtained from Zhongs work. From the

Figure 2, it is shown that ANSYS can not successfully solve such impact problem and

failed to simulate the buckling mode of the FE model. However, LS-DYNA is well

capable of running the transient dynamic analysis and the resultant plot is much closed to

the previous one. Figure 3 plot the crushing force, global deformation, and absorbed

energy of the deformed FE model, which are yielded by LS-DYNA. The plotted analysis

results correlate very well to the published results (Zhong, 1993).

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ANSYS and LS-DYNA used for structural analysis 35

Through above analyses, it can be concluded that the ANSYS is a capable

FE-program for static or quasi-static analyses, while the LS-DYNA is specially designed

for transient dynamic analysis of highly nonlinear problems. Next section will thoroughly

discuss the principles used by both FE programs and reveal the reasons that caused

different performances of the ANSYS and LS-DYNA in solving static and dynamic

problems.

Figure 2 Deformed configurations

Note: ANSYS, LS-DYNA, from Zhongs work (Zhong, 1993) (from top to bottom).

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36 Y. Liu

Figure 3 Analysis results from LS-DYNA (a) displacements, (b) crushing forces, (c) absorbed

energies (see online version for colours)

0

20

40

60

80

100

120

140

160

0 0.002 0.004 0.006 0.008 0.01 0.012

Time (sec)

Displacement(mm)

(a)

0

10

20

30

40

50

60

70

80

90

0 0.002 0.004 0.006 0.008 0.01 0.012

Time (sec)

Force(kN)

(b)

0

1

2

3

4

5

6

7

0 0.002 0.004 0.006 0.008 0.01 0.012

Time (sec)

Energ

y(kJ)

(c)

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ANSYS and LS-DYNA used for structural analysis 37

4 Implicit and explicit solution procedures

As mentioned in the text, the primary differences between the ANSYS and the

LS-DYNA are different solution procedures and time integration methods they are using.

The ANSYS is an implicit solver which applies Newmark method whereas the

LS-DYNA is an explicit solver and employs central difference method. This section

thoroughly demonstrates the different integration methods used by ANSYS and

LS-DYNA, and then continues to discuss and compare the attributes of both implicit and

explicit solution procedures.

4.1 Newmark method

For a governing equilibrium equation of a system of finite elements:

[ ] [ ] [ ]M U C U K U R

+ + = (1)

the Newmark integration scheme assumes that the state of displacement, velocity, and

acceleration at time t and t + t is:

1 1[(1 ) ]i i i iU U U U t

+ += + + (2)

211

1[( ) ]

2i i ii iU U U t U U t

++ = + + + (3)

where and are parameters that can be determined to obtain integration accuracy and

stability, in the Newmark method = 1/4 and = 1/2. In solving the displacement,velocity, and acceleration at time t + t, the equilibrium equation (1) can be written as:

1 1 1 1i i i iM U CU KU R

+ + + ++ + = (4)

To solve (4), the acceleration 1iU

+ is first solved from (3) in terms of the displacement

1iU

+ and then substituted into (2). Thus, the acceleration 1iU

+ and velocity 1iU

+ are

solved, both in terms of the unknown displacement 1iU + only. The solved 1iU

+ and

1iU

+ then are substituted into (4) to entirely solve the displacement 1iU + . Afterwards,

1iU

+ and 1iU

+ then can be calculated based on the solved 1iU + using (2) and (3).

4.2 Central difference method

In the central difference method, differently, it is assumed that:

( )1 121

2i i i iU U U U t

+= +

(5)

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38 Y. Liu

1 1

1

( )2i i iU U Ut

+ = (6)

which include the states at time t, t + t, and t t. For the governing equilibriumequation at time t

i i i iM U CU KU R

+ + = (7)

Substitutes (5) and (6) into (7) and we can have:

1 12 2 2

1 1 2 1 1

2 2i i i iM C U R K M U M C U

t tt t t+

+ =

(8)

From (8) it can be seen that 1iU + (displacement at t + t) is determined by both iU

(displacement at t) and 1iU (displacement at t t). Therefore, before starting the

procedure, the displacement at time t = 1, 1U , is determined using (5) and (6), where

i = 0.

2

0 01 02

tU U tU U

= + (9)

After solving 1U , the states of motion 1iU

+ , 1iU

+ and 1iU + then can be fully solved.

When comparing the two solution procedures, it can be found that the implicit

solution which uses the Newmark method requires matrix inversion in calculating 1iU + ,

while the explicit central difference does not. So compared to the implicit integration

method, the explicit integration method can save more computational cost and can handlea dynamic problem faster. However, unlike the implicit integration method, which is

unconditionally stable for large time steps, the explicit integration method is stable only if

the time step size is smaller than the critical time step size for the finite element system

being simulated. The un-damped critical time step size is 2/n(where n is the largestnatural circular frequency and is not constant during the response calculation), which is

usually a very small value.

In order to clearly show the stability of both implicit and explicit schemes, a simple

un-damping structure with the equilibrium equation 2i i ix x r

+ = is solved for stability

analysis and both integration methods are applied separately.

With the Newmark method applied in the implicit analysis we can have:

21 1 1i i ix x r

+ + ++ = (10)

1 1[(1 ) ]i i i ix x x x t

+ += + + (11)

211

1[( ) ]

2i i ii i

x x x t x x t

++ = + + + (12)

Substitutes equations (11) and (12) into (10) and using = 1/4 and = 1/2, followingrelation can be established as:

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ANSYS and LS-DYNA used for structural analysis 39

1

1 1

1

i i

i i i

i i

x x

x A x Lr

x x

+

+ +

+

= +

(13)

where

A = [ ]

[ ] [ ]

2

2

0.25

0.5 0.125 1 0.52

0.25 0.0625 1 0.25 1 0.25

t t

tt

t t

(14)

and

1

2 2

1 1( )

4t

= +

(15)

which will be used to analyse the stability.

If substituting A into the eigenvalue equation Au = u, the eigenvalue then can be

solved. For stability, it is required that 1 . After solving the equation, it can be found

that for any t, the inequality 1 is satisfied. Therefore, the implicit analysis is

unconditionally stable.

With the central difference method applied in the explicit analysis we can have:

2i i i

x x r

+ = (16)

( )1 121

2i i i ix x x xt

+= +

(17)

1 1

1( )

2i i i

x x xt

+ =

(18)

Substitutes (17) and (18) into (16) and solve for1i

x + we obtain that:

( )2 2 21 12i i i ix t x x t r+ = + (19)

Solution (19) can be re-written in the form (13) as:

1

1

i i

i

i i

x xA Lr

x x

+

= +

(20)

where

A =2 1

1 0

(21)

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40 Y. Liu

Substitutes this A into the equation Au = u, and solve for the inequality 1 . From

the mathematical results it can be found that the inequality can be satisfied only when t2/n (where n is the largest natural circular frequency), which shows the explicitanalysis is conditionally stable. The severe time step restriction is one of primary

shortcomings in the use of the explicit integration method.

5 Return mapping algorithm

Besides different time integration methods they used, the explicit algorithm also employs

return mapping method to integrate the differential equations between the times t and t +

t. This is another significant reason why the explicit algorithm is so economic compares

to the implicit one. As concluded in previous literatures (Lee et al., 1999; Plesek and

Korous, 2002; Leclere et al., 2004; Zhang, 1995), the return mapping method is mosteffective, robust, and unconditionally stable compared to other integrate methods. In the

FE analyses, stress updates take place at the Gauss integration points of used finite

elements, where total and incremental strains are calculated. The return mapping

algorithm is to calculate the stress and strain state at time t + t based on the known

stress and strain state at time t. By applying this method, the stress is first evaluated by

solving elastic equations. Afterwards, the obtained stress is taken as an initial condition

for plastic relaxation equations and several plastic corrections are performed in order to

optimise the stress and strain state until the end of the time increment t + t.

From above introduction, it can be seen that the return mapping algorithm is defined

implicitly therefore is unconditionally stable. Meanwhile, the numerical iteration that is

essence of the implicit analysis program is effectively avoided in the return mapping

method. This endows the explicit program solver lots of advantages in solving transientdynamic analysis because the complicated numerical iteration can be avoided.

6 Conclusions

This paper compares two popular FE programs: ANSYS and LS-DYNA, which are

representatives of implicit and explicit analysis solvers, respectively. From the examples

discussed in this article, it is concluded that in running static analysis, ANSYS should be

used because the implicit analysis is unconditionally stable for any time increment t.The static analysis allows a high t during its time integration, whereas the high t maycause unstable solutions if using explicit analysis. On the other side, those dynamic

transient analyses such as high-speed impact problems require small

t in order tocorrectly calculate the varying material properties and geometrics at any time t.

LS-DYNA is best for such analyses, and the explicit analysis can solve such problems

efficiently because it uses central difference method and return mapping algorithm,

therefore the expensive matrix inversions and numerical iteration can be avoided.

Moreover, the small t may even cause failure to convergence during the matrixinversions if using implicit analysis. Summarily, ANSYS and LS-DYNA should be chose

for appropriate FE problems carefully to best utilise their merits.

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ANSYS and LS-DYNA used for structural analysis 41

References

Advanced Analysis Techniques Guide, ANSYS, Inc.

Lee, S.W., Yoon, J.W. and Yang, D.Y. (1999) Comparative investigation into the dynamic explicitand the static implicit method for springback of sheet metal stamping, EngineeringComputations, Vol. 16, No. 3, pp.347373.

Leclere, G., Neme, A., Cognard, J.Y. and Berger, F. (2004) Rupture simulation of 3D elastoplasticstructures under dynamic loading, Computers & Structures, Vol. 82, pp.20492059.

Linde, P., Schulz, A. and Rust, W. (2006) Influence of modelling and solution methods on thepost-buckling behaviour of stiffened aircraft fuselage panels, Composite Structures, Vol. 73,pp.229236.

LS-DYNA Theoretical Manual, Livermore Software Technology Corporation

Plesek, J. and Korous, J (2002) Explicit integration method with time step control forviscoplasticity and creep,Advances in Engineering Software, Vol. 33, pp.621630.

Rust, W. and Schweizerhof, K. (2003) Finite element limit load analysis of thin-walled structuresby ANSYS (implicit), LS-DYNA (explicit) and in combination, Thin-Walled Structures,Vol. 41, pp.227244.

Zhang, Z.L. (1995) Explicit consistent tangent moduli with a return mapping algorithm forpressure-dependent elastoplasticity models, Computer Methods in Applied Mechanics andEngineering, Vol. 121, pp.2944.

Zhong, Z-H. (1993) Finite Element Procedures for Contact-Impact Problems, Oxford SciencePublications.

Appendix

Finally, a simple un-damped, two degrees of freedom system is presented as an example

to show how the implicit and explicit methods solve a real problem.

Example

Given a simple mass-spring system as shown in Figure 4, K1 = 3N/m, K2 = 1.5N/m,

K3 = 1.5N/m, M1 = 1.5kg, M2 = 0.75kg, F1 = 0, and F2 = 7.5N. The governing

equilibrium equation for this system:

1 1 2 21 1 1

2 2 32 2 22

0

0

K K KM U FU

K K KM U FU

+ + = +

which is

1 1

22

1.5 0 4.5 1.5 0

0 0.75 1.5 3 7.5

UU

UU

+ =

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42 Y. Liu

Figure 4 A simple mass-spring system

It is assumed to calculate the response of this system in 12 steps, the Newmark and

central difference methods are used separately.

Before solving the problem, t will be determined. Since both methods will use thesame t to compare their characteristics, an appropriate t has to be calculated to promisethe stability of the central difference method. Generalised eigen problem is firstly solved:

2 2 2

1 2 1 2

4.5 1.5 1.5 02, 5 4.45, 2.8

1.5 3 0 0.75T T

= = = = =

then t can be taken as T2/10 = 0.28 which is far below than the critical value2/n= 0.89.

Solution

a Using Newmark method

The first step is to calculate 0tU

= , we assume that 0tU

= = 0 and 0tU = = 0, and then

use

10 0

2

1.5 0 4.5 1.5 0 0

0 0.75 1.5 3 7.5 7.5t t

UU U

U

= =

+ = =

Substitutes K, M, R and t into equations (2), (3), and (4), we can have:

1 10.14 0.14i i i iU U U U

+ += + +

After performing the iterative calculations, the final results can be obtained.

Time t 2t 3t 4t 5t 6t

U1 0.00673 0.0505 0.189 0.485 0.961 1.58

U2 0.364 1.35 2.68 4.00 4.95 5.34

Time 7t 8t 9t 10t 11t 12t

U1 2.23 2.76 3.00 2.85 2.28 1.40

U2 5.13 4.48 3.64 2.90 2.44 2.31

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ANSYS and LS-DYNA used for structural analysis 43

b Using central difference method

Similarly, substitutes K, M, R, and t into equations (8) and (9), we can have:

1 1

19.1 0 0 33.8 1.5 19.1 0

0 9.6 7.5 1.5 16.1 0 9.6i i i

U U U+

= +

The solution after each time step and the final results are obtained and listed.

Time t 2t 3t 4t 5t 6t

U1 0 0.0307 0.168 0.487 1.02 1.70

U2 0.392 1.45 2.83 4.14 5.02 5.26

Time 7t 8t 9t 10t 11t 12t

U1 2.40 2.91 3.07 2.77 2.04 1.02U2 4.90 4.17 3.37 2.78 2.54 2.60

The two sets of results are compared and plotted in Figures 5 and 6. From the figures, it

can be seen that for this 12-step example, both methods yield very close results and

deliver the same convergent solution.

Figure 5 The displacement for the first degree of freedom (see online version for colours)

0

0.5

1

1.5

2

2.5

3

3.5

1 2 3 4 5 6 7 8 9 10 11 12

time step

displaceme

nt

implicit

explicit

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44 Y. Liu

Figure 6 The displacement for the second degree of freedom (see online version for colours)

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12

time step

displacement

implicit

explicit

Nomenclature

[ ]M Mass matrix

[ ]C Damping matrix

[ ]K Stiffness matrix

U Displacement vector

U

Velocity vector

U

Acceleration vector

R Load vector of system