Finite element analysis of structural instability using an

implicit/explicit switching technique

J.L. Curiel Sosaa,∗, O.A. Begb, J.M Liebana Murilloc

aMechanical Engineering SG, Sheffield Hallam UniversityHoward Street, Sheffield, S1 1WB, UK

bAerospace Engineering, Sheffield Hallam UniversityHoward Street, Sheffield, S1 1WB, UK

cDepartamento de Ingeniera del Diseno, Universidad de Sevilla, Seville, Spain

Abstract

In this paper, we present a study on finite element analysis (FEA) of struc-tural instability by using a switching implicit-explicit algorithm embeddedinto the finite element method. Snap-through or snap-back buckling prob-lems often cause divergence of the finite element method if arc-length meth-ods are not used. The origin of divergence is often associated to criticalpoints. An alternative to the latter is considered herein named the implicit–explicit FEA. The numerical results showed the effectiveness of this switchingtechnique for solving divergence when simulating structural instabilities suchas buckling of an elastic–plastic arch.

Keywords: finite element method, structures, implicit explicit, instability

1. Introduction

The Finite Element Method (FEM) and derivations have proved to be anumerical procedure convenient to solve differential equations systems corre-sponding to several physical models. The solution of the spatially discretisedmomentum equations –strong form– using direct integration –explicit– orthe solution of the weak form through implicit solvers have been performeddepending upon the type of problem. The implicit strategies involve the

∗Jose Luis Curiel Sosa; Tel.: +441142252651; email: j.l.curiel-sosa@shu.ac.ukURL: www.jlcurielsosa.org (J.L. Curiel Sosa)

Preprint submitted to Elsevier November 21, 2012

use of return mapping algorithms [1, 2, 3] for computing stresses. Nonlinearproblems may generate stability problems and, eventually, divergence.

Implicit solvers such as [4, 5] are, in general, robust and theydo not require stability criterion to satisfy convergence as it occurswith the explicit ones. However, in snap-through or snap-backproblems, Newton-Raphson diverges in general when approachingthe buckling point unless Arc-length Methods are used. Explicitsolvers are conditionally stable and need stability condition (interms of the maximum time step that can be performed) to be ful-filled at every time step. For instance, in the circular arch analisedon this study the implicit method diverges after four iterations andthe explicit provides eventually the solution.

In this paper, divergence associated to structural instability is solved bymeans of an implicit/explicit switching technique [6]. The numerical proce-dure starts executing an implicit method until divergence arises and, in thatpoint, switches to an explicit method until the divergence is solved for thatload increment. The execution may return to the implicit scheme (IMP) oncethe divergence source has been passed and the external loading is not totallyapplied. Other strategies involving subcycling are found, e.g. [7] for sub-cycling between distinct material components. A summary of the theoriesused is shown in Figure(1).

Other problems difficult to solve with implicit methods include disconti-nuities or when the total load applied is not divided in small increments inlarge deformation analysis. In order to converge to the solution, explicit FEMmethods, using central finite differences for the discretisation in time may beused subjected to the stability condition. Note that no division of the mesh,e.g. [8, 9], element-partitioning or nodal partitioning for separate treatmentof the solution is perfomed in this study. A connection between full Newton-Raphson Method with implicit backward Euler pseudo-integration is used asthe implicit FEM (IMP). The Euler stepping is very dissipative andothers such as Runge-Kutta would avoid the very small time stepin many occasions. A simplified flowchart of the technique is depicted inFigure(2)

A description of the implicit and explicit schemes coded and the trans-ferring conditions of information between both of them is presented first.Then, numerical test involving buckling analysis –snap-through– has beenconducted for validation as this type of problems does not converge withNewton-Raphson unless arc-length procedures are used. Nevertheless, arc-

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Original initial boundary value problem (IBVP)

Two major numerical approximations:

•Integration algorithm to solve the constitutive equations of the model thatrelate stresses to the history of deformation.

•Finite element discretisation (NFE).

Solve directly the discretisedmomentum equation

Set of incremental (generally non-linear) algebraic equations

Linearization LNIFEDiscretisation (time)

Figure 1: Numerical approximations of the schemes considered. Linearised Non-linearIncremental Finite Element equations (LNIFE) (implicit) and direct time integration byCentral Difference Method (explicit) after FE discretisation. NFE stands for nonlinearsystem of momentum equations after discretisation by finite elements.

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Figure 2: Simplified flowchart of the in-time implicit/explicit algorithm

4

length methods find difficulties of convergence in snap-back analysis if anappropriate path prediction is not provided [11, 12, 13, 14]. Finally, largedeformation analysis of a nonlinear hyperelastic membrane is also presentedto show the robustness of the switching technique.

2. Brief description of concepts in structural instabilities

When geometry, external load and support conditions of thestructure are such that large displacements involve a significantchange in the initial geometry of the structure, the equations ofequilibrium need to be re-formulated for the deformed structure.The external load is thus gradually applied each iteration, and thenew problem that arises as a result of the incremental load ap-plication is conveniently solved. However, in many -even simple-applications, Newton Raphson’s iteration method does not performproperly when handling limit points, which cause some well-knownnumerical singularities to appear. Limit points include both snap-through and snap-back phenomena. Several methods have been de-veloped to cope with these numerical instabilities, being the mostefficient of them those in which the size of each load increment isrelated to the arc-length of the load-strain curve. Structural instabil-ity may be observed by means of either load vs. deflection (scalar) or force vs.displacement (vector) relationships. The response of the structure followingthese curves is usually named equilibrium path. In the case of instabilities,this path is characterized by some special points. A convenient classificationof these special points according to [10] is,

• Turning points, This is generally due to the material of the structure.These failure either may be instantaneous and jump to other state ofequilibrium or may be catastrophic and the structure does not reachstatic equilibrium.

• Critical points, these points can be classified as follows,

Snap-through: the structure behaves in such a manner that a small in-crease in the load brings about a sudden leap in strain.The load–deflection curve shows softening after attained a maxi-mum, i.e. first limit point. Then, the slope of the curve turns pos-itive and the structure hardens (positive stiffness), see Figure(3).

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This type of behaviours may be observed in shallow arches loadedin the mid-span.

Snap-back: on the other hand, snap-back points are characterized bya turning back of the load-strain curve, which involves anoticeable unloading of the structure. In this case the curveturns back, i.e. decreasing deflection, after the first limit point.This causes the appearance of turning points and, later, recoveringincremental deflection with increasing loading, see Figure(3).

Figure 3: Snap-back and snap-through load–deflection responses

3. Elasto-plastic constitutive model

A brief background for two-dimensional elasto-plastic constitutive modelsused in the numerical tests conducted is presented in this section. The basicconstituents of an elasto-plastic model are,

1. Decomposition of strain in elastic and plastic parts.

2. Elastic evolution.

3. Yield criterion which is geometrically expressed by a yield surface.

4. Evolution of the plastic strain: plastic flow rule.

5. Evolution of the yield border: hardening law.

3.1. Yield surface

This surface limits the elastic domain. It has a negative value for elasticdeformations and zero when plastic strains occur, i.e. plastic flow has started.

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It is a surface in the space of stresses. The elastic domain E can be definedas,

E = σ ∥Y (σ,α) < 0 (1)

where α are a set of internal variables to describe hardening of thematerial. And the yield surface is defined as,

P = σ ∥Y (σ,α) = 0 (2)

3.2. Plastic flow rule and hardening law

The evolution of the plastic strains and state variables are chosen by themodeller. Thus, work hardening or accumulated plastic strain are generallyaccepted as internal variables. In a general way the plastic flow rule and thehardening law can be postulated as,

εp = γN(σ,α) = γ∂Ψ

∂σ(3)

where N(σ,α) is the vector of flow. The hardening law is as follows,

α = γH(σ,α) (4)

where H(σ,α) defines the evolution of the hardening variables and is calledthe generalized hardening modulus[15]. N and H can calculated from theexistence of a flow potential. If this potential is the yield function the flowis called associative. Constitutive models for metals are usually chosen asassociative.

3.3. Loading/unloading criterion

Evolution of the plastic flow Eq (3) and hardening Eq (4) is complementedwith the loading/unloading criterion,

Φ ≤ 0 γ ≥ 0 Φ γ = 0 (5)

4. Return mapping algorithm

In a time increment [tn, tn+1], the increment of strain is given by Eq (6).The state variables are the elastic strain εen and the accumulated plastic

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Figure 4: Detail of pseudointegration in perfectly plastic material

Figure 5: Elastic prediction - plastic correction pseudointegration for hardening

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strain (or effective plastic strain) εpn at the beginning of the time interval[tn, tn+1]. The trial state is defined in Eq (7).

∆ε = εn+1 − εn (6)

εe trialn+1 = εen +∆ε (7)

εp trialn+1 = εpn (8)

The trial stress tensor is computed assuming elastic evolution in Eq (9),Figure (5). The trial yield stress depends on the accumulated plastic strainat tn Eq (10).

σtrialn+1 = De : εe trialn+1 (9)

σtrialy n+1 = σy(ε

pn) (10)

If σtrialn+1 lies inside of the trial yield surface Eq (11), the evolution is purely

elastic within the time interval [tn, tn+1] and the trial state is the solution.

Φ(σtrialn+1 , σy n) ≤ 0 (11)

The computational update is simply performed as follows,

εen+1 = εe trialn+1 (12)

εpn+1 = εp trialn+1 = εpn (13)

σn+1 = σtrialn+1 (14)

σy n+1 = σtrialy n+1 = σy n (15)

Otherwise, the evolution is elasto-plastic and the trial state lies outside ofthe elastic domain defined by the yield surface. Therefore, a return map-ping (plastic corrector), Figure (5), is conducted. For instance, the returnmapping equations for the Von-Mises model can be posed as follows,

εen+1 = εe trialn+1 −∆γ

√2

3

sn+1

∥sn+1∥(16)

εpn+1 = εpn +∆γ (17)

0 =√3 J2(sn+1)− σy(ε

pn+1) (18)

This set of algebraic non-linear equations has to be solved for εen+1, εpn+1 and

∆γ. sn+1 is the deviatoric stress tensor Eq (19).

sn+1 = 2Gdev[εen+1] (19)

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The system can be simplified in number of equations with the single-equationreturn mapping having the plastic multiplier ∆γ as variable. This reductionmakes the process more computationally efficient. For more information theinterested reader is referred to [16].

5. Implicit sub-algorithm

The solution of the weak form of the momentum equations by Newton-Raphson Method (NRM) is used. In nonlinear analysis a convenient scheme1 to integrate the rate constitutive equation is required. The algorithm willdepend on the type of material 2 and the consideration of large deformationor not. In this study, the implicit algorithm is devoted to the general case ofpath-dependent materials and large deformation.

The implicit algorithm is based upon a pseudo-time discretisation (Simoand Hughes [2]) considering the transition of deformation between two timepoints. The implicit backward Euler method coupled with the Newton-Raphson iterative scheme is utilized [2, 15]. Thus, if a time increment[tn, tn+1] and set of internal variables αn at tn are given, the deformationtensor ε(tn+1) must determine the stresses σ(tn+1) and internal variablesonly through the integration algorithm, i.e.:

σ(tn+1) = σ(αn, εn+1) (20)

α(tn+1) = α(αn, εn+1) (21)

After discretisation of the domain into finite elements, the problem is com-mitted to find displacements un+1 at time tn+1, so that the incrementalnonlinear FE equation, (22) is satisfied.

R(un+1) = f int(un+1)− f extn+1 = 0 (22)

where internal and external forces vectors are obtained by

f int(un+1) =nelem∧e=1

∫Ω(e)

BT σ(αn, ε(un+1)) dv

(23)

f extn+1 =nelem∧e=1

∫Ω(e)

NT bn+1dv +

∫∂Ω(e)

NT qn+1 ds

(24)

1Integration algorithms like for example the return mapping algorithm2In general, path-dependent materials.

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where N(ξ, η) are bilinear shape functions. N is the tensor containingthe shape functions. The spatial order of error in the approxima-tion is O(hp+1), p is the degree of the polynomial and h is the sizeof the element. As the degree of the polynomial increases, so doesthe order of error and, hence, the convergence accelerates. Below,4-noded quadrilaterals with bilinear shape functions, for both nu-merical tests, have been utilized. bn+1 are the body forces, qn+1 thetraction forces applied over the boundary of the body and B is the linearstrains operator which has the next format (in plane stress/strain analysis)for the generic element (e):

B =

N(e)1,1 0 N

(e)2,1 0 . . . N

(e)nnode,1

0

0 N(e)1,2 0 N

(e)2,2 . . . 0 N

(e)nnode,2

N(e)1,2 N

(e)1,1 N

(e)2,2 N

(e)2,1 . . . N

(e)nnode,2

N(e)nnode,1

Equation (22) needs to be linearized in order to enable a numerical procedure.Details of this may be found in de Souza et al.[15].

5.1. Solution to the implicit incremental problem

As stated above the NRM has been utilized in the solution of equation(22) (note that, in elastic materials, solution is immediate with some algebraicmethod as Frontal [15]) because of its quadratic rate of convergence. Itconsists in solving within each iteration the linearized version of equation(22) for the incremental global displacement δu(k) :

KT δu(k) = −R(k−1)(un+1) (25)

where KT is the global tangent stiffness matrix given as,

KT =∂R

∂ un+1

∣∣∣∣u(k−1)n+1

(26)

which is obtained by assemble of element stiffness matrices:

k(e)T =

∫Ω(e)

BT DB dv (27)

where D is the consistent tangent stiffness matrix ( de Souza et al. [15]):

D =∂σ

∂εn+1

∣∣∣∣ε(k−1)n+1

(28)

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BOX I: IMPLICIT INCREMENTAL PROBLEM

1. Initiate (k=0)

u(0)

n+1 = un

R = f int(un)− f ext

2. For all elements, calculate consistent tangent stiffness matrix.

D =∂σ

∂εn+1

3. Assembly of stiffness matrices

k(e)T =

ngaus∑j=1

ξj BT

j Dj Bj

4. Increment iteration counter (k=k+1), assembly, solve the lin-earized equilibrium equation (31) and update stresses and in-ternal variables:

u(k)n+1 = u

(k−1)n+1 + δu(k)

ε(k)n+1 = Bu

(k)n+1

σ(k)n+1 = σ(αn, ε

(k)n+1)

α(k)n+1 = α(αn, ε

(k)n+1)

5. New internal forces at each element

f int(e) =

ngaus∑j=1

ξj JjBT

j σ(k)n+1, j

6. Gathering of element internal forces vector and updatingresidual.

7. If iterations diverge then go to the EXP scheme,else:(a) If ∥fext−f int∥

∥fext∥ ≤ ϵ then the solution for current external

load is reached and values for this load are from the lastiteration (•)n+1 = (•)(k)n+1

(b) else go to (2).8. If the total load is not completely applied, increment the ex-

ternal load, else exit.

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The implicit scheme in compact form is presented below. Note at point 7 ofBox I that the flow of execution is diverted to explicit if needed.

As indicated below the initial values (displacement and residual) to initial-ize the explicit algorithm are the solution obtained at the last load incrementthat converged.

5.2. Formulation for finite strains

In the case of finite elasticity, the material is independent of the path and,hence, internal variables are not needed in the estimation of stresses. Thus,they can be evaluated without any numerical integration algorithm.The former statement (general case of path-dependent materials) is also ap-plied here to update stresses and other state variables through a numericalintegration algorithm. Moreover, the next formulation in the case of finitestrains are considered in the analysis.The stresses are given by σn+1 = σ(αn,Fn+1 ) (right-hand side is namedalgorithmic incremental constitutive function) with Fn+1 being the deforma-tion gradient at the end of the interval [tn, tn+1].Now, the load vectors are based on the deformed configuration:

f int(un+1) =nelem∧e=1

∫φn+1(Ω(e))

BT σ(αn,F(un+1)) dv

(29)

f extn+1 =nelem∧e=1

∫φn+1(Ω(e))

NT bn+1dv +

∫∂φn+1 (Ω(e))

NT qn+1 ds

(30)

where φn+1(Ω(e)) is the current deformed domain. For details of linearisation

see the work of de Souza et al.A generic iteration of NRM is, as before (see above), applied to solve thestandard linear system.

KT δu(k) = −R(k−1) (31)

where KT is now obtained Eq(32) through G which is the discrete (full)spatial gradient operator. G, in plane stress/strain analysis, has the formatgiven in Eq(33).

KT =nelem∧e=1

∫φn+1(Ω(e))

GT aG dv

(32)

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G =

N

(e)1,1 0 N

(e)2,1 0 . . . N

(e)nnode,1

0

0 N(e)1,1 0 N

(e)2,1 . . . 0 N

(e)nnode,1

N(e)1,2 0 N

(e)2,2 0 . . . N

(e)nnode,2

0

0 N(e)1,2 0 N

(e)2,2 . . . 0 N

(e)nnode,2

(33)

The fourth order tensor a is the consistent spatial tangent modulus and, incartesian components, is defined by eq.(34) at the end of iteration (k − 1).

aijkl =1

J

∂τij∂Fkm

Flm − σilδjk (34)

The main modifications for the case of finite strains are displayed in Box II.

6. Switching between implicit and explicit algorithms

t

t+Δt

t+(n-1)Δt

t+nΔt

I

I

E

MESH DOMAIN

Figure 6: Schematic representation of the evolution of a solution executed by the I/Ealgorithm with detail of the transmission of variables between nodes and time–steps.

The values at the end of the implicit sub–algorithm are transmitted asinitial values for the explicit sub–algorithm and viceversa. Boundary condi-tions are easily managed as these are transferred directly as displacements

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BOX II: IMPLICIT INCREMENTAL PROBLEM FORFINITE STRAINS

1. Computation of consistent spatial tangent moduli

aijkl =1

J

∂τij∂Fkm

Flm − σilδjk

2. Assembly element stiffness matrices

k(e)T =

ngaus∑i=1

ωi jiGT

i ai Gi

3. Updating of deformation gradient

F(k)n+1 = (I−∇x u

(k)n+1)

−1

4. Use of constitutive integration algorithm to update the stressand other state variables

σ(k)n+1 = σ(αn,F

(k)n+1)

α(k)n+1 = α(αn,F

(k)n+1)

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and/or velocities in the nodes of the boundary. The EXP flow is initiatedwith the nodal displacements u and nodal internal forces f int(u) obtained inlast iteration of the IMP flow.

f int(t = 0)|EXP = f int(u)|IMP

u(t = 0)|EXP = u|IMP

Henceforth, subscripts denoting the EXP flow are dropped. The timemarching scheme is denoted by the timestep as normal making clear that itis related to EXP flow. With those nodal values, nodal accelerations ui(0)and nodal velocities ui(0) are initiated in the EXP flow,

ui(0) =f exti − f int

i (t = 0)

Mii

ui(0) = ui(0)∆t(0) + u−i (0)

u−i (0) = 0.0

where ∆t(0) is the initial time step. Flowchart of the I/E algorithm is de-picted in Figure (2). Once the solution is reached the flow is returned to theimplicit sub-algorithm if the external load is not still totally applied (other-wise, the execution is ended). The converged solution values of the explicitsub-algorithm are passed on to the implicit one for the current load incrementand, then, the IMP flow is initiated again for the next load increment. Anupdate of configuration is also necessary in the case of large deformations.

The convergence criterion is based upon the Euclidean norm ofthe residual (3rd column of table 3), which is a measure of relativeerror every iteration.

∥f ext − f int∥∥f ext∥

≤ ϵ (35)

where ϵ is the tolerance provided by the modeller.

7. Explicit sub-algorithm (EXP)

The explicit scheme is based on direct integration –time-steps– the spa-tially discretised –by finite elements– dynamic equilibrium equation Eq (36).Therefore, there is a discretisation of the domain by finite elements and adiscretisation in–time by central finite differences.

Mu(tn) +Cu(tn) + f int(un) = f ext (36)

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where M is the mass matrix, C the damping matrix, f int(un) the internalforces vector, f ext the external forces vector, and u(tn), u(tn),un are, respec-tively, the accelerations, velocity and displacement vectors.

u(tn−1/2) =u(tn) − u(tn−1)

∆tn(37)

7.1. Lumped mass matrix

The mass matrix is evaluated in the undeformed configuration, before thebeginning of the explicit iterations.

M =nelem∧e=1

∫Ω(e)

ρ0 NT

(e) N(e) dω (38)

An approximation needs to be assumed to obtain a diagonal mass matrix. Aspecial lumping technique which ensures mass conservation is desirable. Forinstance, the addition of masses at each node of the mesh must be the totalmass of the body. Thus, one third of the mass of a triangular element isassigned to each node or one fourth of the mass of a quadrilateral is assignedto each node.

7.2. Damping matrix

The explicit in-time integration of the discretised equations of dynamicequilibrium can exhibit spurious oscillations. These can be controlled bydamping values. Furthermore, in highly nonlinear materials it is convenientto use a damping proportional to the stiffness matrix to damp high frequen-cies. Diagonal tensors are desirable in order to get a system of uncoupledequations. A common one is a lumped mass matrix proportional dampingC = αM [20]. There are other options such as the proposed damping rela-tionship by Munjiza [19] which includes stiffness and mass relation to obtaindamping values.

L = M(M−1K)m

(39)

If m = 1 a stiffness proportional damping is introduced, which results in highfrequencies being damped. However, the critical time step decreases withincreasing damping, leading to increasing computational cost. All frequenciesare damped effectively if m = 0.5. Others such as the Rayleigh dampingis widely employed in engineering applications and is a linear combinationof mass and stiffness. This damping can be tuned to damp high or lowfrequencies [20].

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7.2.1. Mass Proportional Damping

The proportionality must be in terms of natural frequencies. The naturalfrequencies associated to degree of freedom (named i by simplicity in theequations) are functions of a stiffness approximation as indicated in Eq (40).Stabilisation is out of the main scope of this paper. A discussion about stabi-lization of numerical computations via the introduction of artificial dampingmay be consulted in [18].

Ci = 2ξωiMi (40)

where ξ is the damping ratio and ωi are the natural frequencies asso-ciated to degree of freedom i,

ω2i = Ki/Mi (41)

Rayleigh D.

Stiffness proportional D.

Mass proportional D.

Figure 7: Summary of possible damping factors. They were used depending upon the typeof problem.

7.2.2. Damping proportionality constant

Numerical damping proportionality constant α can be chosen from thedynamic relaxation method. This is convenient for problems with highnon-linear material and/or geometric behavior. In addition, this damps a

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wide range of frequencies with equivalent efficiency. The dynamic relaxationmethod computes every deformation mode to define an equivalent frequency.This method chooses the smallest frequency in order to achieve a criticalproportionality constant as follows:

ωmin = min

uTKuuTMu

1∆t

Then, the damping is defined under minimum frequency proportionality:

C = 2ωmin ξB (42)

7.3. Time step criteria

The timestep for the Central Differences Method (CDM) must be lessthan a specific value to guarantee the stability of the scheme and therefore itsconvergence. This value is bounded by the natural frequencies and dampingratio relationships as stated in Eq (43).

∆t ≤ min2

ωi

(−ξi +√1 + ξ2i ) (43)

where ωi are the natural frequencies and ξi the fraction of critical dampingat each node i of the mesh. This inequality is satisfied if the maximum fre-quency in the mesh is elected. The maximum frequency can be calculated

knowing the maximum eigenvalue of the mesh as ωmax =

√λ(mesh)max . More-

over, the maximum eigenvalue can be bounded by the maximum elementeigenvalue λ

(mesh)max ≤ λ

(e)max [21]. The eigenvalue computation adds more pro-

cessor time, so an alternative practical way is to use formulas that can boundadequately the timestep for some problems. Thus Eq (44) is obtained for lin-ear problems. When nonlinearities emerge a reduction of this upper-boundmust be considered (typically the range goes from 80 to 98 per cent) takinginto account the type of problem.

∆t ≤ minLe

ce(44)

ce =

√E(e)

ρ(e)(45)

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where Le is a characteristic element length which is frequently adoptedas the minimum length in the smallest element of the mesh and ce the wave-speed Eq (45). Thus, we may see that critical the time step decreases withmesh refinement and increasing stiffness of the material. The ratio of criticaltime step needed is called Courant’s number in the literature. Therefore, thetime cost of execution depends basically on the element size and nonlinearityof material. The frequencies for all nodes and degrees of freedom in the meshare approximately calculated by the code. The maximum frequency boundsthe critical time step Eq(48). A particular expression is usually chosen,depending of the type of analysis. Eq (46) for plane strain problems or Eq(47) for plane stress.

∆t ≤ Le

√ρ(1 + ν)(1− 2ν)

E(1− ν)(46)

∆t ≤ Le

√ρ(1− ν2)

E(47)

These steps of time are very restrictive and not adapted to new conditionsof the next iteration. Timestep adaptivity has been utilized in the program-ming, it attempts to perform a proper step time under the conditions of thenatural frequencies of the system at this time point. Thus the critical timestep is selected as,

∆t (tn+1) =2

maxiωi(tn)(48)

This statement has given place to a faster performance than using a con-stant time step as Eqs (46) and (47). The natural frequencies are determinedfrom homogeneous problem Eq (49). Its analytical solution is in the formu(t) = ue−jωt (j =

√−1), substituting in Eq (49) an eigenvalue problem is

accomplished.

Mu+Ku = 0 (49)

introducing the analytical solution form,

u(t) = u exp(−jωt)

u(t) = −jωu exp(−jωt)

u(t) = −ω2u exp(−jωt)

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which gives after introduction to :

−ω2M(cos(ωt)− jsin(ωt)) +K(cos(ωt)− jsin(ωt)) = 0

which leads to the classical eigenvalue problem

| − ω2M+K| = 0

where ω2 are the eigenvalues of the system that provide the natural frequen-cies for each node and degree of freedom, i.e. each variable.In this explicit code the stiffness matrix is never computed and an approxi-mation is done in order to know the natural frequencies,

Kii(tn) ≃f inti (un)− f int

i (un−1)

un−1/2i ∆tn

(50)

with this approximation the stiffness matrix becomes diagonal and thecomputational cost of the eigenvalue problem is saved because the naturalfrequencies may be easily calculated from Eq(51).

ωi(tn) =

√Ki(tn)

Mi

(51)

and the critical timestep is selected as stated in 48.

8. Numerical tests

In this section, numerical results using the switching implicit–explicittechnique are provided. Details of convergence based in the euclidean norm ofthe residual are shown. A snap-trough buckling problem for an elastic-plasticarch as well as a large deformation of the Cook’s membrane are presented.

8.1. Test 1: Elastic-plastic circular arch

A circular arch loaded with a concentrated force at the centre and clampedat the two ends is analysed herein. No rotation is permitted at the supports.The finite element model consists of 20 quadratic quadrilateral elements withfour nodes. The numerical integration is performed on 4 gauss points withinthe finite element. A plane stress state is considered with the uniform thick-ness of the arch set to 10 cm. The geometry is shown in Figure (8). The

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Figure 8: Circular arch clamped at the edge-supports

Material properties

Density 7, 860 · 10 kgcm3

Young Modulus 210 · 105 Ncm2

Poisson ratio 0.3

Table 1: Material properties of the elastic-plastic material

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material is a Von-Mises perfectly plastic material (see material characteris-tics in Tables (1,2)).

The density is used in the explicit code as this uses the dynamicmomentum equation. Density is needed for computation of themass matrix. Inertia term and damping term will tend to zero inthe static solution.

εp σ( Ncm2 )

0.0 24, 500.00.001167 24, 500.0

Table 2: Material parameters for the plastic evolution.

When the arch is loaded in the central-top node with 6.6 104 N theIMP sub-algorithm diverges in the fourth iteration. At that point EXPsub-algorithm is automatically initiated (see Table (3)). The final verticaldisplacement field, over the undeformed configuration, is represented in Fig-ure (10).

Flow i/tn Error(%) Max. Resid. f inty (kcn) uy(kcn) Stage

IMP 1 47.48 237.70 - - initialIMP 2 25.72 163.08 -7430.40 -0.605 -IMP 3 1.479 7336.16 -29695.92 -1.078 divergingIMP 4 862.37 0.44 · 10+7 -62912.7 -1.119 swapEXP 0.3943256 972567 5493540 -5542582 -1.625 oscillat.EXP 3.33 0.802 2633.40 -63371.6 -1.522 -EXP 6.28 0.45 1885.83 -64114.2 -1.703 -EXP 99.99 0.000188 0.0096 -65998.2 -1.043 convergingEXP 150 0.000008 0.0406 -65999.97 -1.044 convergingEXP 170.56 1.0 · 10−6 0.008 -65999.99 -1.044 solution

Table 3: Relative residual norm(%) (error(%) in the table) and maximum residual. Verticalinternal force (N) and displacement (cm) at central node of the arch(kcn) . In the secondcolumn, the iteration number(i) is shown for the IMP sub-algorithm and the time (tn) forthe EXP sub-algorithm

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1E-10

1E-07

0.0001

0.1

100

100000

iterations

err

or

(%)

F1=1e4 N

F2=5e4 N

F3=6.5e4 N

F4=6.6e4 N

Figure 9: Relative residual norm error (%) on a logarithmic scale (circular elasto-plasticarch). F1, F2 and F3 converge with the implicit strategy. However, F4 needed theswitching from implicit to the explicit one to obtain the solution

Figure 10: Contours of vertical displacement in the elastic-plastic circular arch deformed.Central node loaded with point force 6.6 · 104 N

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8.2. Test 2: Nonlinear large deformation analysis: Cook’s membrane

A non-linear material model such as Ogden model [22] for hyperelastic –rubber-like– materials, is an option to assess a code with these characteristicsas divergence of implicit Newton–Raphson methods suddenly appears if thereis no load control. In the test solved by I/E, the load is totally appliedfrom the first time point. The implicit code showed divergence if the loadis not divided in smaller increments. Thus, I/E switches to the explicitscheme after detecting that the implicit solution is diverging according tothe criterion displayed in Box I. The geometry in Figure (11) corresponds toCook’s membrane problem which is generally used to assess the convergencein elements which are nearly incompressible under shear and bending strains[23] or [24]. In fact, this problem was evident in the top-left corner of themembrane.

Figure 11: Cook’s membrane (dimensions in mm). Geometry and finite element mesh inthe undeformed configuration

The membrane is clamped in the left edge and a shearing force of 100Nis distributed on the right side, Figure (11). Ogden material is assumed withbulk modulus κ = 40.0942x104 and shear modulus µ = 80.1938 (in consistentunits). Two simulations were considered. Firstly, use of IMP sub-algorithmalone allowing incremental loadings (to avoid divergence) until the total loadis applied. And secondly, total load applied from the beginning by I/E. FourIMP iterations (Newton-Raphson Method) were executed (divergence stage)

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Figure 12: Displacement ux (mm) by I/E

and, then, switching to EXP sub-algorithm to reach solution with an errorestimator equal to 0.2% . The solutions obtained by I/E for displacementsand stresses may be visualized in Figures (12, 14, 16, 18, 20). They can becommpared with the solution by IMP with incremental loading, Figures (13,15, 17, 19). Note that explicit iterations were stopped when the residualerror-norm was 0.2%.

Figure 13: Vertical displacement uy (mm) by IMP

The results are in good agreement with the IMP solutions reported in theliterature, see for example [24]. Simo et al. [23] uses an IMP algorithm withcontrol load of ∆F = 1.0 (elastic) ∆F = 0.5 (elastic− plastic) ranging from0 to 100 obtaining a displacement at the reference node (right top corner inFigure (11)) of approximately 6.9mm. Therefore, further refinement of

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Figure 14: Vertical displacement uy (mm) by I/E

Figure 15: Shear stress σxy (N/mm2) by IMP (division of total external loads by incre-ments).

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Figure 16: Shear stress σxy (N/mm2) by I/E

Figure 17: Stress σxx (N/mm2) by implicit method

Figure 18: Stress σxx (N/mm2) by I/E

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Figure 19: Stress σyy (N/mm2) by implicit method

Figure 20: Stress σyy (N/mm2) by I/E

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the mesh provides no significant differences.

9. Conclusion

In this paper, we have presented a finite element analysis investigation us-ing a switching implicit explicit technique. Firstly, details of the methodologywas presented in detail for nonlinear materials, in general, and, in particular,for large deformations. The results showed that the switching technique canbe a promising technique for optimising computational cost and velocity ofconvergence when using the Finite Element Method. The method is versatileand, therefore, future research could point to the development of the tech-nique for other types of problems in Mechanics that are subjected to othernumerical instabilities when using the FEM.

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