on the optimisation of viscoplastic constitutive modelling using a numerical feedback damping...

8/9/2019 On the Optimisation of Viscoplastic Constitutive Modelling Using a Numerical Feedback Damping Algorithm 2005 Co

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On the optimisation of viscoplastic constitutive

modelling using a numerical feedback damping algorithm

J. Pinho-da-Cruz *, F. Teixeira-Dias

Departamento de Engenharia Mecanica, Universidade de Aveiro, Campus Universitario de Santiago, 3810-193 Aveiro, Portugal

Received 16 January 2004; received in revised form 18 June 2004; accepted 29 July 2004

Abstract

The authors propose a mathematical model that, in the presence of a constant time step algorithm and a smooth

evolution of a state variable, increases the performance of the numerical process, forces the convergence of the numer-

ical solution and, consequently, improves the overall quality of the results. This method reduces the total number of

time steps of a simulation process and minimizes the necessary CPU time. The proposed algorithm is based on the

mathematical adjustment of the evolution of a chosen state variable. The resulting numerical signals are analysed

and properly characterised. Several numerical signals, obtained from different non-linear simulation examples and con-

ditions, are studied. Based on the characterisation of the numerical signals and considering that the numerical resultsreflect the behaviour of a vibratory systemthe numerical codewith its own intrinsic mass, springand dashpot ele-

ments, the authors develop a numerical damping algorithm and present its implementation. The algorithm is applied

and tested with a non-linear finite element example, using a viscoplastic constitutive model. The authors also present

a set of numerical validation tests consisting of the simulation of the development of residuals stresses that arise from

the fabrication process of particle reinforced metal matrix composites (MMC). The cooling down stage of an AlSiCp20%vol. MMC is simulated. In order to evaluate the performance of the algorithms, some results, obtained with and

without the application of the optimisation algorithm, are presented and thoroughly compared. The numerical damper

algorithm proves to be very efficient.

2004 Elsevier B.V. All rights reserved.

Keywords:Viscoplastic behaviour; Numerical damper; Optimisation; Incremental process; Finite element

0045-7825/$ - see front matter 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2004.07.013

* Corresponding author. Tel.: +351 234 370830; fax: +351 234 370953.

E-mail address: [email protected](J. Pinho-da-Cruz).

Comput. Methods Appl. Mech. Engrg. 194 (2005) 21912210

www.elsevier.com/locate/cma
mailto:[email protected]:[email protected]


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1. Introduction

On the present day, a large number of numerical simulation processes involve the use of highly non-

linear algorithms, constitutive models and formulations. This is the case, for example, of some fabricationprocesses involving large-scale plasticity or whenever strong stress gradients are induced. Non-linear for-

mulations inevitably lead to the implementation of incremental algorithms with generical time steps

[tn, tn

+ Dt] and, consequently, often complex time integration procedures. Thus, the need for efficient

and low-cost time step optimisation algorithms is ever more needed.

Many algorithms for time step optimisation, adequate for non-linear behaviour models, can be imple-

mented with automatic formulations that anticipate the evolution of the constitutive model parameters

and correct the time step in order to avoid divergence and withdrawal from the constitutive behaviour.

This constitutive evolution is often highly unstable. Consequently, in the present work, the authors pro-

pose a numerical stabilisation method based on an analogy with the dynamic response of a classic vibra-

tory system. The proposed method is based on the mathematical treatment of the evolution of a chosen

internal state variablethe control variable. The evolution of this state variable, also called numerical sig-

nal, is analysed and characterised in terms of its dispersion and amplitude divergence. Based on this char-

acterisation, the authors develop a set of numerical damping algorithms and implementation techniques

that are applied and tested with non-linear finite element examples, using a viscoplastic constitutive model

[1,2].In this paper, only constant time increments Dtwill be considered in the development of the optimi-

sation algorithm.

2. Numerical damping model

Non-linear systems seem to prevail against linear systems whenever real problems are analysed. This is

mostly so because of friction forces, damping elements, resistive elements, etc., that often exhibit non-linear

behaviour. However, linear dynamics often lead to fairly good descriptions of real systems. In these systems

forces increase linearly with parameters such as position and velocity.

2.1. Stability states of the free motion of an SDOF system

The fundamental single degree-of-freedom (SDOF) vibratory system (schematically represented in Fig.

1(a)) can be described by a single mass m, suffering the action of an external force f(t), connected to a spring

of stiffness kand a dashpot c.

Newtons second law of motion applied to the SDOF vibratory system leads to the following well-known

non-homogeneous second-order ordinary differential equation (ODE) of motion:

mxc

_xkx ft 1

with initial conditions

xjt0 x0 and _xjt0 v0: 2

The free vibration of an SDOF vibratory system, represented in Fig. 1(b), corresponds to

mxc _xkx 0: 3

In accordance to the values of the damping ratio,n, the SDOF system may be (i) underdamped (vd.Fig.

2), (ii) critically-damped or (iii) overdamped (vd.Fig. 3). Positive initial displacement (x0) and velocity (v0)

were considered for the representations of the system responses in Figs. 2 and 3.

2192 J. Pinho-da-Cruz, F. Teixeira-Dias / Comput. Methods Appl. Mech. Engrg. 194 (2005) 21912210


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The free motion of an SDOF vibratory system can be statically or dynamically stable/unstable. Consid-

ering a positive inertia, m, the various free motion stability states of a vibratory system can be related to the

algebraic signals ofkandc.

A vibratory system is statically stable if any displacement from equilibrium induces a force that drives

the system back to its equilibrium position. The system is statically unstable if this force tends to drive it

away from its equilibrium position. Therefore, a positive value of spring stiffness kis associated to static

stability, whereas a negative value is associated with static instability. On the other hand, dynamical stabil-

ity of a vibratory system is related to the signal of the damping force. A positive damping force (c> 0) acts

in the opposite direction of the velocity vector, leading to the dissipation of kinetic energy of the system

and, consequently, to dynamic stability. Fig. 2 illustrates the free motion of a statically and dynamically

Physicalr

esponse

Time

Fig. 2. Free vibration of underdamped SDOF vibratory system.

(a)

(b)

Fig. 1. Schematic representation of: (a) a single degree-of-freedom (SDOF) vibratory system and (b) an SDOF with free vibration.

J. Pinho-da-Cruz, F. Teixeira-Dias / Comput. Methods Appl. Mech. Engrg. 194 (2005) 21912210 2193


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stable vibratory system having constant positive values ofkand c. A negative damping force (c< 0) will

provide energy to the system instead of dissipating it, leading to an increase of the free vibration amplitudesand, consequently, to dynamic instability.Fig. 4illustrates the free motion of a statically stable and dynam-

ically unstable vibratory system (k> 0 and c< 0). It should be noted that dynamic stability implies static

stability, but the reverse is not necessarily true: a statically stable system may be dynamically unstable [7].

2.2. Numerical damping algorithm

In the present work, authors assume that, at each integration point, the numerical results of a non-linear

finite element analysis (FEA) can reflect the behaviour of a virtual vibratory systemthe numerical code

with its own intrinsic mass, springand dashpot elements. Thus, when analysing the virtual system, the two

main hypotheses are:

Time

Physicalresponse

Overdamped system

Critically-damped system

Fig. 3. Free vibration of critically-damped and overdamped SDOF vibratory systems.

Phys

icalresponse

Time

Fig. 4. Free vibration of a dynamically unstable vibratory system (k> 0 and c < 0).



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(i) The behaviour of the numerical solution, at each integration point, is similar to the free vibration of a

single degree-of-freedom system.

(ii) Each integration point is associated to constant values of mass 1 mand of spring stiffness k, while the

damping coefficient c depends on the overall process time t.

In this context, the numerical damping algorithm consists on the replacement, for every integration point

(IP), of some transformation functionC of a chosen state variableW(t) by its arithmetic average aCin theinterval [t

n, tn

+ Dt], where n2 N is the number of the increment, i.e.

n2 N 8IP : CWtnDt : CWtnDt Cwtn

2 aC: 4

Note that ifC[W(t)] = W (t) (C is the identity transformation), then

WtnDt : WtnDt Wtn

2 aW: 5

The value of state variable W(tn

+ Dt) is therefore replaced by its arithmetic average along [tn, tn

+Dt], des-

ignated by aW. On the other hand, ifC[W(t)] = ln[W(t)] withW(t) > 0 "t(Cis the logarithmic transforma-tion), then

lnWtnDt : lnWtnDt lnWtn

2 ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiWtnDtWtn

p 6

leading to

WtnDt :

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWtnDtWtnp cW: 7The value of state variable W(tn+ Dt) is therefore replaced by its geometric average along [tn, tn+Dt], des-

ignated by cW.Some features of the presented algorithm can be related to the concept of feedback or to the closed-loop

method of Discrete-Time Control Systems[3]. In fact, the constitutive model (CM) of the finite element

code can be seen as an open-loop system (vd.Fig. 5(a)). With the implementation of the damping algo-

rithm the CM turns into a sort of closed-loop system due to the fact of the previous value of the internal

state variable, at instantt, being fed back, along with its actual value, to the constitutive model at instant

t+ Dt. This is numerically performed replacing the transformation function C by its arithmetic mean (vd.

Fig. 5(b)). InFig. 5(b)Z1 corresponds to the hardware memory where the value of the internal state var-

iable for the previous step is stored, and D is a delay[3].

This damping algorithm can be implemented in constitutive models with more than one internal state

variable that exhibit oscillatory numerical instabilities. When this is the case, the damping algorithm can

be applied simultaneously to all internal state variables.

It should also be noted that the philosophical background of the presented algorithm is substantially

different from those related to energy decaying schemes [46], which attempt to develop robust algorithms

for integrating time semidiscrete equations associated with stiff non-linear finite element problems.

1 It should be noted that, even if the mass varies, i.e. m =m(t) with m(t)5 0 "t, all the terms of Eq.(3) can be divided by m(t),

resulting in an equation that describes the free vibration of an SDOF system with unitary mass and specific values of damping

coefficient,c 0(t) = c(t)/m, and spring stiffness, k0(t) = k(t)/m.



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3. Constitutive modelling

In order to test the optimisation algorithms described in the previous paragraphs, the authors deliberately

chose a numerical example that uses a highly non-linear constitutive model. This example is the numerical

simulation of the development of thermal residual stresses in multiphase materials, namely in particle rein-

forced aluminium matrix composites. To solve this example, a set of two constitutive models is used to de-

scribe the micromechanical behaviour of the two constituent materialsreinforcement and matrix. Thesemodels are thoroughly described, for example, by Anand[8] and Teixeira-Dias and Menezes [9,10].

The constitutive model for the reinforcement is a classical thermoelastic model [11]. For most metallic

matrix composites, the temperature levels reached during some fabrication processes are frequently close

to the melting point of the matrix material. Thus, one should assume that the matrix material exhibits a

rate-dependent behaviour. Consequently, the behaviour model for the matrix material should be based

on the following physical aspects:

(i) plastic strains are highly rate-dependent and

(ii) the internal state of the material determines its mechanical and thermal response.

Additionally, when adopting such a constitutive model, it is often necessary to guarantee that it will be pos-sible to model physical phenomena such as, for example:

(i) the effects of strain-rate and temperature;

(ii) recrystallisation and restoration processes;

(iii) internal damage of the material and

(iv) the evolution of the crystalline structure of the material.

The use of a single state variable to represent all the physical phenomena described above is clearly a

limiting option. However, it is possible to model hardening and strain rate sensitivity with such a law

[12]. In order to do this, the model uses an internal state variable srepresenting strain resistanceand

Fig. 5. Feedback control analogy of the damping algorithm: (a) open-loop system and (b) closed-loop system.



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two other macroscopic external state variablesCauchys stress tensor r and temperature T. Thus, the

most appropriate model that can be used to represent the constitutive behaviour of the metallic matrix

material is a thermoelasticviscoplastic model.

To specify the constitutive relations for the matrix material it is first assumed that the total strain ratetensor, D, is decomposed as[10]

D De Dth Dvp; 8

whereDvp is the viscoplastic strain rate tensor and De andDth are the elastic and thermal strain rate tensors,

respectively. The viscoplastic strain rate tensor is assumed to be isochoric, i.e. tr(Dvp) = 0.

Most temperature- and rate-dependent constitutive models assume that the material is isotropic and that

the plastic strain rate depends on the stress state, the temperature and a set of state variables defining the

materials strain history[8,13,14]. Based on this assumption, the viscoplastic strain rate tensor is defined by

the following flow rule [12]:

Dvp 3 _e

p

2r r

0; 9

where r 0 is Cauchys deviatoric stress tensor and r is von Mises equivalent stress, defined as

r 3

2r

0: r

0

1=2: 10

For the particular constitutive model considered in this work, the equivalent plastic strain rate _ep

is a

function of

(i) the equivalent stress r;

(ii) the internal state variables, representing the strain resistance, and

(iii) the temperature T, i.e.

_ep fr;s; T: 11

The evolution of the internal state variable s is itself defined as a function of r, s and T

os

ot gr;s; T _e

phr;s; T: 12

The definition of functionfr;s; T, necessary for the complete definition of the adopted flow rule (Eq. (9)),is assumed to be the following[1518]:

_ep

fs;r; T A exp Q

RgT

sinh n

r

s

1=m; 13

where A, Q, m and n are material parameters and Rg is the universal gas constant. The evolution of the

internal state variable s, defined in Eq.(12), is[10]

_s _eph0 1

s

sH

asgn 1 ssH

h i; 14

where h0 represents the hardening rate, ais a material parameter and sw is the saturation value ofs, asso-

ciated with a given temperature/strain rate pair of values [14], given by

sH s_e

p

A exp

Q

RgT

n: 15



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In the process chosen as example for the present work, it is reasonable to neglect rotations and consider

only small strains[19]. Thus, a generic small strainsapproach is adopted[20]and the evolution of the Cau-

chy stress tensor can be defined as

_r CeM :De: 16

After some mathematical manipulation one can define the following constitutive law for the metallic matrix

material[9,11]:

_r 2lMDDvp kM trD 3 kM

2

3lM

aM _T

I; 17

wherelM, kM and aMare the coefficients of Lameand the coefficient of thermal expansion for the matrix

material, respectively. _T is the cooling rate and Iis the second-order identity tensor.

4. Numerical modelling

The set of constitutive equations used in the present work have, among others, the big advantage of

using a scalar parameter to represent the internal state of the material. This parameter models the resistance

of the material to plastic flow. This kind of constitutive equations are numerically stable [21,22], which

facilitates their implementation into numerical simulation codes.

4.1. Numerical integration

The numerical integration of the constitutive models is made with a forward gradient algorithm, in the

form initially proposed by Peirce et al. [23]and afterwards developed by Anand et al. [21], Lush et al.[14],

Teodosiu and Menezes[12]and Teixeira-Dias and Menezes[10].With the time integration procedure the correct values of all the state variables in configuration n + 1 are

determined from the known values of the same variables in the previous configuration, n. For this purpose,

the viscoplastic part of the strain rate tensor in the current configuration, Dvpw, is approximated using a

weighting factor between its value in configurations n andn+ 1. This approximation is made as

Dvpwjn1 DvpjnUD

vpjn1Dvpjn; 18

whereU 2[0, 1] is the weighting factor. The value ofDvp at instantn + 1, in the right-hand side of Eq. (18),is determined with a first-order Taylor approximation. The plastic strain increment can be determined from

this approximation as

Dep

Z tn1tn

Dvp

dt 19

Consequently, the increments of stress are calculated.

4.2. Virtual work principle

For the thermomechanical model described in previous sections, the virtual work principle can be writ-

ten as[20]ZV

rdedV

ZS

tdu dS; 20



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whereVis the volume of the solid body andSthe external surface of the volume where stress is restricted as

t t. As external mechanical loads are not considered in the present example, principle (20)simplifies to

ZV

r

dedV 0: 21

The metal matrix composite (MMC) is discretised in hexahedral three-dimensional finite elements with

eight nodes and eight integration points. In order to avoid the development of locking effects the volumetric

part of the displacement gradient is interpolated with a reduced selective scheme [22,24].

The finite element discretisation of Eq.(21), for time increment n, leads to the following standard linear

algebraic system of equations:

KnDun Dfn; 22

where Dun

is the displacement increment vector, Kn

is the corresponding global stiffness matrix and Dfn

is

the incremental nodal force vector, all evaluated at instant tn

and corresponding to time increment Dt. The

solution of the system of equations (22)is then used to update the configuration of the system and all the

state variables.

5. Results and discussion

The case study used in this work is the simulation of the cooling down stage of an AlSiCpmetal matrix

composite. Initial and final process temperatures are Ti= 933K and Tf= 293K, respectively, and the cool-

ing rate is _T 100Ks1. All numerical simulations are performed using a three-dimensional unit cell rep-resentative of a spherical particle reinforced MMC with 20% volume fraction of reinforcement. The finite

element mesh developed for this analysis is shown inFig. 6(a). The boundary conditions specified were such

that coordinate planesOxy, Oxz and Oyz correspond to planes of symmetry.Fig. 6(b) shows the distribu-

tion of the normal stress rxx. Note that, as expected, during the cooling down stage, the gradient of normalstresses tends to be higher near the matrix-reinforcement interface. This is mainly due to the discontinuity

in material properties and the mismatch between the thermal expansion coefficients of the component

materials.

Fig. 6. (a) Finite element mesh and (b) distribution of the normal stressrxx

of the spherical particle reinforced AlSiCp 20%vol. unit

cell.



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5.1. Accurate numerical results

Anaccuratereference solution was obtained, for comparison purposes, using a very small constant time

increment, Dt= 0.001 s, which corresponds to a temperature incrementDT=0.1K, and a forward gradi-ent time integration weighting factor U= 0.7 (see Eq. (18))[10].As stated earlier, this parameter U 2[0,1]controls the estimation of the plastic strain rate increment in the forward gradient integration procedure. It

should be noted that U= 0, U2 ]0,1[ and U= 1 lead, respectively, to explicit, semi-implicit and implicitintegration schemes. The accurate, or optimal, equivalent plastic strain rate _e (EPSR) evolution with the

temperatureTis presented inFig. 7. All numerical results were obtained on an integration point in the ma-

trix material, close to the matrixreinforcement interface.

5.2. Non-damped numerical results

Several numerical simulations were performed consideringU = 0.25 and time increments Dtbetween 0.01

and 0.02s, which correspond to temperature increments 2 DTbetween 1.0 and 2.0K. It should be noted

that, since the cooling rate is considered to be constant, the results can be analysed in terms of temperature,

instead of time, increments. Figs. 810 present the temperature evolution of the equivalent plastic strain

rate _ep

with temperature for DT= 1.0K, 1.4K and 1.8K, respectively. Similar results were observed for

the remainder of the temperature increments studied. For comparison purposes the accurate evolution of_e

pis also presented.

Analysing these results, the following can be observed:

(i) there is dispersion in all but the accurate results;(ii) the results oscillate asymmetrically about the optimal evolution;

(iii) although temperature evolutions of the equivalent plastic strain rate _ep

stabilise in the considered cool-

ing temperature range for temperature increments of 1.0 K and 1.4 K, this is not the case for

DT= 1.8K and

(iv) dispersion seems to increase proportionally with the size of the incrementDT.

Fig. 7. Accurate evolution of the equivalent plastic strain rate (EPSR) with temperature (DT=0.1K and U = 0.7).

2 Although temperature incrementsDTare negative, i.e. are decrements, since _T


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It should also be noted that the disappearance/appearance of numerical oscillations can be explained by

conditional stability of the semi-implicit integration method: if the time step is small enough, stability is

attained, otherwise instabilities (oscillations) will emerge. Therefore, non-damped oscillations vanish as

the time step size is reduced since it becomes sufficiently small to stabilise the semi-implicit integrationmethod and, consequently, lead to a non-oscillatory numerical solution.

In terms of the hypotheses presented in Section 2.2, there seems to be no similarity between the behav-

iour of these numerical results and a mechanical vibration. In fact, the asymmetrical character of the evo-

lution of the numerical results seems to preclude such a fact. Therefore, a qualitative comparison of results

is only possible with symmetry of the numerical results. However, a detailed analysis of these results reveals

that, although the arithmetic averages of _ep

, for each interval [Tn, T

n+ DT], did not follow the general trend

of the accurate results, this was the case when the geometric average was considered. This observation sug-

gests the existence of symmetry in the logarithmic evolution of the equivalent plastic strain rate. This fact is

illustrated in Figs. 1113 for temperature increments DTof 1.0K, 1.4K and 1.8K, respectively. Similar

results were observed for the remainder of the temperature increments studied.

0

2

4

6

8

250350450550650750850950

EPSR

[x103s-1]

Signal 1.0

Temperature [K]

Optimal

Fig. 8. Evolution of the equivalent plastic strain rate (EPSR) with temperature (DT= 1.0K).

0

6

12

18

24

250350450550650750850950

EPSR

[x103s-1]

Optimal

Signal 1.4

Temperature [K]




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Note that a logarithmic transformation of the equivalent plastic strain rate leads to a symmetric evolu-

tion of the temperature about the accurate results. In the presence of symmetry, there seems to be a higherqualitative similarity between the behaviour of these numerical results and a mechanical vibration. The evo-

lution of the transformed results suggests that the system behaves as an underdamped, single degree-of-free-

dom, free vibrating system with variable damping coefficient c. The amplitude variation of the numerical

results corresponds to an alternate dissipation/accumulation of the energy of the system, which, according

to the results shown inFigs. 2 and 4, can be associated to the existence of a time-dependent damping factor

c. On the other hand, the spring stiffness k, hypothetically constant, has a positive value since every dis-

placement from the equilibriumoptimalposition induces a force that drives the systemnumerical

resultback to its equilibrium position, i.e. the system is statically stable. However, since the damping coef-

ficientc may be either positive or negative, the system is not necessarily dynamically stable, i.e. amplitude

may either decrease or increase.

0

5

10

15

20

25

30

250350450550650750850950

EPSR

[x103s-1]

Temperature [K]

Signal 1.8

Optimal


1

10

250350450550650750850950

EPSR

[x103s-1]

Temperature [K]

Signal 1.0

Optimal

Fig. 11. Logarithmic evolution of the equivalent plastic strain rate (EPSR) with temperature (DT= 1.0K).



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5.3. Implemented damping methodologies

As was referred previously, dynamic stability always assumes static stability, but the opposite is not al-

ways true. A statically stable system may be dynamically unstable[7]. Therefore, our system seems to fit inthe second case: it is statically stable, but dynamically unstable. So, if dynamic stability is the main objec-

tive, it is essential to force the positiveness of the damping factor c. In physical terms, a positive damping

factor leads to energy dissipation and a decrease in the amplitude. Therefore, if a decrease of the oscillation

amplitude is forced, it may be possible to guarantee dynamic stability and, consequently, reduce dispersion.

In this context, the main objective of the numerical damping algorithm described in Section 2.2 is to reach

numerical dynamic stability. It forces the replacement of the actual numerical value, or its transformation,

by averaging it with its predecessor. This methodology refrains the unstable tendencies and, for oscillating

numerical signals, leads to the desired approximation to the optimalequilibrium positionresults. In the

present workW(t) andC[W(t)] correspond to the equivalent plastic strain rate _ep

and ln_ep, respectively, i.e.

C is the logarithmic function. The importance ofC resides in the fact that symmetry of a non-damped

1

10

100

2503504505506507508509500.1

Temperature [K]

Signal 1.4

Optimal

EPSR[

x103s

-1]


1

10

100

2503504505506507508509500.1

Temperature [K]

Signal 1.8

Optimal

EPSR[

x103s

-1]




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numerical signal eliminates spurious factors that may distort the analysis of the damped results, since the

signal does not possess average value deviation.

The implemented damping methodologies correspond to the application of the algorithm presented in

Section 2.2 for both the identity and the logarithmic transformations. Thus, for all temperature intervals,the value of state variable W(Tn

+ DT) is replaced by its arithmetic average aW in [Tn, T

n+ DT]. This

approach will be henceforth denominated as a-method. Alternatively, the value of the state variable

W(Tn

+ DT) is replaced by its geometric average cW in [Tn, T

n+ DT]. This approach will be henceforth

denominated as c-method.

5.4. Damped numerical results

Based on the previously referred damping methodologies, several numerical simulations were performed

in order to evaluate the performance and validity of both the a- and c-methods.

5.4.1. Damped numerical resultsthe a-method

Figs. 14 and 15present the evolution of the a-damped equivalent plastic strain rate (EPSR) with tem-

perature, forDT= 1.0K andDT= 1.8K, respectively. The corresponding non-damped results and accurate

results are also shown for comparison purposes. Similar results were observed for the remainder of the tem-

perature increments.

It can be observed that the evolution of the a-damped numerical results have a behaviour similar to an

underdamped vibratory system (seeFig. 2). However, some degree of asymmetry can be observed on the a-

damped results. Moreover, both the underdamped amplitudes and stabilisation time seem to increase with

the temperature increment DT.

5.4.2. Damped numerical resultsthe c-method

Figs. 16 and 17present the logarithmic evolution of the c-damped equivalent plastic strain rate (EPSR)

with temperature, forDT= 1.0K andDT= 1.8K, respectively. The corresponding non-damped results andaccurate results are also presented for comparison purposes. Similar results were observed for the remain-

der of the temperature increments.

It can be seen that c-damped results are almost symmetric. This is related to the fact that a symmetric

oscillating signal does not have average value deviation. c-damped results seem to behave like an under-

damped vibratory system (seeFig. 2). Finally, as occurred for the a-method, both the underdamped ampli-

0

2

4

6

8

250350450550650750850950

EPSR

[x103

s-1]

- method

Signal 1.0

Optimal

Temperature [K]

Fig. 14. Evolution of the a-damped equivalent plastic strain rate (EPSR) with temperature (DT= 1.0K).



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tudes and stabilisation time seem to increase with the temperature increment DT. In comparison to the a-

method, thec-method leads to numerical results with slightly greater underdamped amplitudes, but with a

more uniform decrease and a smaller time to stabilisation.

The first step (n= 1) of the application of the geometric average (c-method) to the logarithmic evolutionof EPSR with temperature for DT= 1.0K is illustrated inFig. 18. It should be noted that point Cfollowed

upwards instead of horizontally. This is in accordance with the inertia hypothesis. In fact, in the absence of

inertial effects there is no reason for the system to keep its previous numerical dynamical state. On the

other hand, the non-damped results clearly illustrate the existence of a positive spring stiffness k, as well as

the alternating motion that it originates.

5.4.3. Dispersion resultsa- andc-methods

In order to evaluate the efficiency of both the a- and the c-methods, relative dispersion of non-damped

and damped signals, for each DT, were determined using the root mean square deviation (RMSD), defined

by:

0

5

10

15

20

25

30

250350450550650750850950

EPSR

[x103s

-1]

- methodSignal 1 .8Optimal

Temperature [K]

Fig. 15. Evolution of the a-damped equivalent plastic strain rate (EPSR) with temperature (DT= 1.8K).

1

10

250350450550650750850950

EPSR

[x103s-1]

Temperature [K]

Optimal

Signal 1.0

- method

Fig. 16. Logarithmic evolution of the c-damped equivalent plastic strain rate (EPSR) with temperature (DT= 1.0K).



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RMSDffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiP

NTS

i1 C

W

T

i C

WH

T

i

2

NTS

vuuut ; 23

where C is the transformation function, W(Ti) is the equivalent plastic strain rate _ep

at temperature Ti,

Ww(Ti) is the accurate value at temperature Ti, and NTS is the number of temperature steps, given by

NTSjTf Tij

jDTj : 24

Fig. 19shows the evolution of the relativea- andc-RMSD factors with the total number of temperature

steps. The relative RMSD factor was defined, for both the methods, by the ratio of the RMSD factors of

the non-damped and damped numerical signals, respectively. It can be observed that:

0

1

10

100

250350450550650750850950

EPSR

[x103s

-1]

OptimalSignal 1.8 - method

0.1

Temperature [K]

Fig. 17. Logarithmic evolution of the c-damped equivalent plastic strain rate (EPSR) with temperature (DT= 1.8K).

Fig. 18. Detail of the application of the dampingc-method to the logarithmic evolution of equivalent plastic strain rate (EPSR) with

temperature (DT= 1.0K).



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(i) botha- and c-methods led to a maximum relative damping effect for around 530 temperature incre-

ments, i.e. for DT= 1.2K;

(ii) for a number of steps smaller than 320, i.e. forDT> 2.0K, the relative a-RMSD tends towards unity,

indicating that, for DT= 2.0K, the dispersion of the non-damped and damped signals are quantita-

tively similar, although qualitatively distinct;

(iii) both relativea- andc-RMSD values decrease forDT< 1.2 K, and tend towards unity, since dispersion

tends to vanish for small values ofDT. Note that, although the a-method seems to be more efficient

than thec-method, this is not true for a number of temperature steps over 470. In fact, the asymmetry

of the linear-scaled non-damped numerical signals, due to mean value deviation (see Figs. 810), leads

to an overestimation of the a-RMSD factors, which are minima for the logarithmic-scaled signals (seeFigs. 1113).

Although a symmetrising transformation leads to better convergence results, in practice, since it is usu-

ally unknown, the damping methodology must be implemented considering the identity transformation, i.e.

the arithmetic mean of the control variable. The main reason for considering the logarithmic transforma-

tion in this work was to investigate the influence of symmetry on damped results. It can be concluded that

this leads to a better performance. Moreover, those differences do not invalidate the general use of the iden-

tity transformation. The symmetrising transformation may be thought as the upper bound of the algorithm

efficiency and not as a conditio sine qua non for its systematic implementation.

5.4.4. Damping numerical resultsnon-initial damper application

Numerical simulations were performed in order to observe the effect of a non-initial damping effect. A

temperatureTd= 710K was chosen for the application of damping. Figs. 20 and 21show, respectively, the

temperature evolution and logarithmic evolution of the a- and c-damped equivalent plastic strain rate

(EPSR) for Td= 710K and DT = 1.8K. Non-damped results and optimal results are also presented for

comparison purposes. The accurate results are also presented. Similar results were observed for the remain-

der of the temperature increments.

It can be observed that the behaviours of the non-initiallya- andc-damped numerical signal are similar

to the initially damped ones. However, the damping effect is more pronounced due to the sudden applica-

tion of damping.

0

1

2

3

4

5

67

8

9

10

300 350 400 450 500 550 600 65 0

RelativeRMSD

method

- method

Number of temperature steps, NTS

Fig. 19. Evolution of the relative a- and c-root mean square deviation (RMSD) factors with the number of temperature steps.



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6. Conclusions

A numerical damper algorithm applicable to the finite element simulation of non-linear behaviour of

materials was presented. It was considered that the evolution of the numerical results of a non-linear finite

element analysis can reflect the behaviour of a virtual vibratory systemthe numerical codewith its own

intrinsic mass, springand dashpot elements. The damping algorithm was applied and tested with a non-

linear finite element example, using a rate dependent constitutive model, in order to evaluate its capabilities.

The set of numerical validation tests consisted on the optimisation of the simulation of the development of

residuals stresses that arise from the fabrication process of particle reinforced metal matrix composites. The

cooling down stage of an AlSiCp MMC with 20% volume fraction of reinforcement was simulated [911].

The numerical damping algorithm forces the replacement of the actual numerical value of a state vari-

able, or its functional transformation, by performing an average with its predecessor numerical value. Two

0

5

10

15

20

25

30

250350450550650750850950

Temperature [K]

EPSR

[x103s-1]

Optimal

Signal 1.8

-method

Fig. 20. Evolution of the non-initiallya-damped equivalent plastic strain rate (EPSR) with temperature (Td= 710K and DT= 1.8K).

Temperature [K]

EPSR

[x103s-1]

1

10

100

250350450550650750850950

Optimal

Signal 1 .8

-method

0.1

Fig. 21. Logarithmic evolution of the non-initiallyc-damped equivalent plastic strain rate (EPSR) with temperature (Td= 710K and

DT= 1.8K).



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distinct transformation functions were used: the identity and the logarithmic functions. The logarithmic

function was used because it symmetrises the evolution of the state variable. In the presence of a constant

time step finite element algorithm, the application of these two methodologies refrained the dispersion of

the oscillating numerical signals. Their application increased the performance of the process and forcedthe convergence of the numerical solutions.

The symmetrising logarithmic transformation leads to the best convergence resultsthe upper bound of

the algorithm efficiency. Moreover, in general implementations the identity transformation should be used.

The presented numerical damper algorithm allows the execution of simulation processes using larger

time steps, thus saving CPU time, and increases the precision of the numerical results.

This numerical damper algorithm proved to be very efficient and appropriate when predicting non-linear

thermomechanical behaviour of composite materials.

Acknowledgement

The authors acknowledge financial support given by FCTFundacao para a Ciencia e a Tecnologia;

Programa Operacional Ciencia, Tecnologia, Inovacao do Quadro Comunitario de Apoio III, FEDER.

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on the optimisation of viscoplastic constitutive modelling using a numerical feedback damping...

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