Computational Mechanics of Composite Materials part 14 ppsx

Wavelet projection of variances for n=6 The expected values and their wavelet projections are greater than the corresponding deterministic values of ft computed for Varl=0.. As is docum

k m k m

(( ) ( ))

) ( 22

k m k m

where X∈(Ω,σ, P), p∈Z and ω t∈ℜ Therefore, the first two probabilistic

moments of Y can be calculated as

2

12

2 0

X Var X

Y Y

Y E

∂

∂+

)()

(

2

X Var X

Y Y

) l p ( X

l k

Therefore, the expected values are determined

[ ]Y E [ ]X cos t p ( p ) X cos tVar ( X )

12

376 Computational Mechanics of Composite Mater ials

and variances as

(pX ) cos tVar ( X ) )

Y (

in the second order perturbation approach The visualisation of all wavelet functions and their approximations are presented below using the symbolic computation package MAPLE [182] The following function is used

( = , t∈[ ,1] where l(ω) belongs to the additional random space

with the expected value E[l]=10 and the variance equal to Var(l)=4; p=-1 The

wavelet projection are shown for n=3,…,6 in case of the expected values – in

Figures 7.54-7.57 and the wavelet approximations for the variance for n=4,5,6 are

shown in Figures 7.58-7.60

Figure 7.54 Wavelet projection of expected values for n=3

Figure 7.55 Wavelet projection of expected values for n=4

Multiresolutional Analysis 377

Figure 7.56 Wavelet projection of expected values for n=5

Figure 7.57 Wavelet projection of expected values for n=6

Figure 7.58 Wavelet projection of variances for n=4

378 Computational Mechanics of Composite Mater ials

Figure 7.59 Wavelet projection of variances for n=5

Figure 7.62 Wavelet projection of variances for n=6

The expected values and their wavelet projections are greater than the

corresponding deterministic values of f(t) computed for Var(l)=0 Since the

expectations and their deterministic origins are very similar, the convergence of

analysed projections is quite the same – for n=6 the approximation error on the

interval [0,1] in practice can be neglected The situation changes in the case of variances where projection of the 6th order is not quite smooth; for n=2 cannot be

accepted at all because of the constant function resulting from the wavelet projection

As is documented in Table 7.3, the total computational cost by means of the consumed time and memory necessary to obtain wavelet projection increases nonlinearly together with this projection order Taking into account that the time of the linear equation system solution shows the same tendency, the very exact solution of (7.120) with 7th and even higher order wavelet projection needs more powerful computers The last column of the computer test shows that the approximation of variances needs essentially more time and memory than the analogous projection of zeroth order moments (deterministic values) and the expectations (first moments) It should be documented by the relevant numerical tests, if the computational symbolic projection cost increases together with the order of the probabilistic moment being projected onto the same wavelet family

f(t) secs/MB

As was demonstrated above, the wavelet-based multiresolutional

computational techniques can be very efficient, considering the capability of

heterogeneity analysis on extremely different geometrical scales in the same time

Such phenomena appear frequently in engineering composites – at the interface

between the components, on microscale connected with the periodicity cell, for a

window on mesoscale for a couple of reinforcing fibres or particles as well as for

the macroscale connected with the global composite structure As can be observed,

the wavelet-based numerical methods (especially the Finite Element Method) can

be successfully used even for the heterogeneous media with random or stochastic

microstructure thanks to implementation of a randomisation method (simple

algebra, PDF integration, Monte Carlo simulation, stochastic perturbation or even

spectral analyses)

The homogenisation method discussed in this chapter enables us to apply an

alternative approach, where the effective material parameters (or its probabilistic

moments) are determined first and then the entire composite is analysed using

traditional computational techniques Wavelet-based multiresolutional approach to

the homogenisation problem should, however, be formulated to introduce the

components characteristics on many scales into the final effective structural

parameters As was demonstrated in the mathematical considerations, homogenised

properties in multiscale analysis and classical macro-micro passage are essentially

different, even in a deterministic formulation, which was observed previously in

three scale Monte Carlo simulation based homogenisation studies for the

fibre-reinforced composites [191,197]

Finally, let us note that due to the character of the homogenised 1D elastostatic

problem, computational studies on effective coefficient probabilistic behaviour can

be applied without any further modifications in the heat conduction problem of a

composite with exactly the same multiscale internal structure as well as for any

linear field problem with random coefficients defined by their first two

380 Computational Mechanics of Composite Mater ials

probabilistic moments The real and imaginary parts of the effective coefficient for the wave equation can be used in acoustic wave propagation in random media It is observed that for wave propagation, homogenised coefficients strongly depend on the same range on angular velocity and the interrelation of material properties of the layered medium components

The most important result of the homogenisation-based Finite Element modelling of the periodic composite beams is that replacing the real composite behaviour is very well approximated by the homogenised model response For a smaller number of bays in the periodic structure, wavelet-based homogenisation gives more accurate results, while the classical approach is more efficient for the increasing number of bays Maximum deflections of the analysed beams are approximated by all the models with the same precision, which increases for increasing number of bays in the whole structure

The wavelet-based multiresolutional homogenisation method introduces new opportunities to calculate effective parameters for the composites with material properties given in various scales by some wavelet functions This method is more attractive from the mathematical point of view However it is characterised by new, closer bounds on the homogenisability of composite structures, but it eliminates all formal problems resulting from the assumption of small parameter existence between macro- and microscales Now, practically any number of various scales can be considered in composite materials and structures, which is important in all these cases, where material properties are obtained through signal detection and its analysis Finally, obtaining satisfactory agreement between the real and homogenised structures enables the application of this method to the forced vibrations of deterministic systems as well as the use of dynamical systems with stochastic parameters

The second order perturbation wavelet projection gives complicated formulae for approximation of the original functions or matrices, which enables fast wavelet-based discretisation of random variables and/or fields It is necessary to recall the algebraic restrictions on the first two probabilistic moments of the input

to achieve the coefficient of variation to be essentially smaller than 0.15

However it is documented by the above numerical examples that the wavelet projection of the expected value and its deterministic origin have almost the same character – the same order of approximation is necessary to achieve the same convergence and error level Wavelet projection of variance (and higher order probabilistic characteristics) needs greater precision, especially for smaller values

of the projection order n Let us note that analogous projection for random

functions or operators defined in two– or three–dimensional spaces can be done by the use of Daubechies wavelets in a similar manner to that presented here

Symbolic computations package MAPLE [61,70] (as well as other numerical tools of this class) is very efficient in wavelet projections of various discrete and/or continuous functions because the efficiency of the projection (and its averaged error) can be recognised graphically in specially adopted plots Otherwise, a special purpose numerical error routine should be implemented and applied

Multiresolutional Analysis 381

The most important result of the homogenisation-based Finite Element modelling of the periodic composite beams is that the real composite behaviour is very well approximated by the homogenised model response The multiresolutional homogenisation technique giving a more accurate approximation of the real structure behaviour is decisively more complicated in numerical implementation because of the necessity of applying the combined symbolic-FEM approach A wavelet-based space-time decomposition should be applied in computational modelling of the transient heat transfer problems in heterogeneous media

Furthermore, mathematical and numerical studies should be conducted to increase nonstationary heat transfer modelling in unidirectional composites by the application of the homogenisation method In the case of small contrast between heat capacities of the constituents, the method proposed was verified as effective; the situation changes when the value of contrast parameter increases dramatically

8 Appendix

8.1 Procedure of MCCEFF Input File Preparation

The instructions described below deal with the preparing of input data file to the MCCEFF analysis in the case there is no need to use the mesh generator

1 Heading line (12a4) general information

2 General information about the problem homogenised (6i5)

Column Variable Description

1 -5 NUMNP Total number of nodal point in the structure discretised 6-10 NELTYP Total number of finite element groups (=1)

11-15 LL Total number of load cases (=3)

36-40 KEQB Total number of non-zero degrees of freedom in the main

matrix 66-70 MK Total number of random trials

71-75 NBN Total number of nodal points of the interfaces

General comments:

A NELTYP variable is provided due to the original POLSAP code to extend in the

next version the MCCEFF code with the analysis of the engineering structures homogenised (e.g fibre-reinforced plates and shells) However due to its constant value it may have been omitted

B LL variable is provided taking into account that in the next versions of the

program the rest of the effective tensor components will be computed (in the 3D homogenisation problem) There are three different components of the elasticity tensor homogenised for the plane strain problems being solved by the program

C KEQB parameter should be modified (default value is equal to 0) if the program

MCCEFF in the process of main stiffness matrix formation or solution of the fundamental algebraic equations system stops running The value of the parameter

is to be taken from the interval [0,NEQB], where NEQB is the total number of the degrees of freedom of the composite cell The probability of the successful

computations increases with decreasing KEQB parameter

Appendix 383

3 Nodal points data (7i5,4d10.0,3i2)

Column Variable Description

16-20 IX(N,3) Displacement boundary conditions codes

21-25 IX(N,4) =0, free degree of freedom

26-30 IX(N,5) =1, fixed degree of freedom

31-35 IX(N,6)

66 -70 K(N) Nodal point generation code

71 -72 M1 Number of the internal region

73 -74 M2 Number of the external region

75 -77 M3 The interface end code (=1)

General comments:

A Nodal point numbering has to be continuous and to start from number 1, which

should denote the centre of the fibre (considering stress boundary conditions computations)

B Interface nodal points numbering has to be provided in the anticlockwise system

and the distances between any two points must be equal

C The structure being discretised should be placed in the YZ plane; the X

coordinate will be used in the next version for the analysis of the 3D composite problems

D The regions of the different materials should have increasing number starting

from the central component (fibre in two-component composites) and continuous

to the external boundary of the cell

E In the case of half or quarter of the periodicity cell analysis the M3 parameter

should be used to underline the ends of the interface being cutted

4 General finite elements data (3i5)

Column Variable Description

6 -10 NUMEL Total number of finite elements

11 -15 NUMMAT Total number of composite components

The total number of the materials used should be greater than 10

5.2 Detailed data two lines for any different temperature (8d10.0/3d10.0)

1-10 Coefficient of thermal expansion an

11 -20 Coefficient of thermal expansion as

21-30 Coefficient of thermal expansion at

Appendix 385

6 Probabilistic parameters (2d10.0/2i5,4d10.0)

Column Description

1 -10 Variance of Young modulus

11 -20 Variance of Poisson ratio

1 -5 Averaged material type:

= 1, material without defects

= 2, material with interface defects

= 3, material with volume defects

6-10 Structural defects type:

= 1, circle

= 2, triangle

= 3, rectangle

= 4, hexagon

11-20 Expected value of the geometrical parameter

21 -30 Variance of the geometrical parameter

31 -40 Expected value of defects total number

41-50 Variance of defects total number

7 Finite elements description (7i5)

56-60 Finite elements generation code:

=0 (default) - the lack of generation

386 Random Composites

enable the user to gain a better understanding of model computed The lines indicated by ‘*’ or without any indication are program execution lines

*heading

corner supported shell

**all material and geometrical parameters are defined in US units

Appendix 387

1,3,2,1,3,20,3

**master element number, number of elements to be defined in the first

**row generated including the master element, increment in node numbers

**of corresponding nodes from element to element in row (default is 1),

**increment in element numbers in row (default is 1), numbers of rows to

**be defined (default is 1), increment in node numbers of corresponding

**nodes from row to row, increment in element numbers of corresponding

**elements from row to row, numbers of layers to be defined (defined is

**1), increment in node numbers of corresponding nodes from layer to

**layer, increment in element numbers of corresponding elements from

**absolute value of compressive stress, absolute value of plastic strain (the

**first stress-strain point must be at zero plastic strain and defines the

**initial yield point)

*failure ratios

1.16 , 0836

**ratio of the ultimate biaxial compressive stress to the uniaxial

**compressive ultimate stress (default is 1.16), absolute value of the ratio

**of uniaxial tensile stress at failure to the uniaxial compressive stress at

**failure (default is 0.09), the ratio of the magnitude of a principal

**component of plastic strain at ultimate stress in biaxial compression to

**the plastic strain at ultimate stress in uniaxial compression, the ratio of

**the tensile principal stress value at cracking, in plane stress, when the

**other non-zero principal stress component is at the ultimate compressive

**stress value, to the tensile cracking stress under uniaxial tension

*tension stiffening

**definition of retained tensile stress normal to a crack is a function of the

**deformation in the direction of the normal to the crack

388 Random Composites

1.,0

0.,2.e-3

**fraction of remaining stress to stress at cracking, absolute value of the

**direct strain minus the direct strain at cracking

*rebar,element=shell,material=slabmt,geometry=isoparametric,name=yy

**definition of the rebars, reinforced element type, material name, rebar

**geometry type (isoparametric or skew), name of the rebars group

slab,.014875,1.,-.435,4

**definition of rebars geometry, cross-sectional area of each rebar, spacing

**of the rebars in the plane of the shell, position of the rebars in the shell

**direction, edge number to which rebar are similar [4]

**option RESTART controls the writing to and reading of the restart file,

**which is used by the postprocessor; the option will create a restart

**file after each increment at which the increment number is exactly

**divisible by N, and at the end of each step of the analysis, regardless of

**the value of N at that time

*step,inc=30

**option STEP must begin each step definition, parameter INC is equal to

**the maximum number of increments in a step (upper bound, the default

**value is 10)

*static,riks

**this option indicates that the step should be analysed as a static load

**step; the Riks method is chosen by the RIKS parameter

Appendix 389

.05,1.,,,,1,3,-1

**initial time increment, time period of the step, minimum time increment

**allowed, maximum increment allowed, maximum value of the load

**proportionality factor for the Riks method, node number at which the

**value is being monitored, degree of freedom being monitored, value of

**the total displacement (or rotation) at the node and degree of freedom

**which, if crossed during an increment, ends the step at that increment

*cload

**concentrated loading definition

1,3,-5000

**node number, number of the corresponding degree of freedom, loading

**magnitude in the orientation given by the user by ordering nodes into

**shell elements

*el print,frequency=10

**option provided tabular printed output of element variables; parameter

**FREQUENCY is equal to the output frequency measured in the

**increments performed (if this option is omitted, very large printed output

**files will be produced by large models in multiple increment analysis!)

s

**all stress components

sinv

**all stress invariants (MISES,TRESC,PRESS-equivalent pressure stress,

**INV3-third stress invariant)

**this option allows nodal variables to be written to the ABAQUS results

**file (no nodal variables will be written to the results file unless this

**option is used!)