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present work is the follow - up on part i of this paper @xcite , discussing a relatively simple model capable of describing the behavior of a massive structure until the point of  localized failure .
the model contains all the ingredients for taking into account both the  diffuse damage mechanism , which leads to the appearance of microcracks , as well as  the  failure process characterized by the propagation of macrocracks .
perhaps the most important advantage of the proposed model is the fact that all its parameters have a clear physical interpretation and can be straightforwardly visualized in terms of the shape of a stress - strain diagram .
in addition , influence of each parameter is dominant only for a specific , easily recognizable , stages of material behavior .
this kind of a priori knowledge has a potential to greatly simplify the model calibration and will be systematically used throughout the  paper .
in part ii of this paper , we discuss the identification of the model parameters from experimental measurements made on a structural level . generally speaking , the complexity of the identification procedure is determined by the choice of experimental setup . solely from the identification point of view , the simplest experiment to execute is the uniaxial tensile test . in this case
, the strain field stays mostly homogeneous during the whole procedure and the global response represented by the load - displacement diagram is very similar to the stress - strain curve for one material point ; see section [ tensile ] for more details .
the  model parameters can be then directly determined from the shape of the load - displacement curve .
such a uniform loading is , however , very difficult if not impossible to impose in a laboratory test , especially for quasi - brittle materials .
therefore , other tests are often used in  experimental practice .    the three - point bending test , in particular ,
is considered to be much simpler to perform and its results are well - reproducible .
therefore , we focus on the identification procedure for the proposed model parameters directly from results of three - point bending test .
main difficulty is in this case imposed by heterogeneity of the stress and the strain fields , which is present since the very start of the experiment .
the macro - scale measurements provide the load - deflection curve that integrates data from different parts of the specimen experiencing different regimes of ( in)elastic behavior .
for that reason , the possibility of a simple determination of model parameters from load - deflection curve is lost and an advanced calibration procedure needs to be applied .    to take advantage of the model specific structure ,
already mentioned above , the identification procedure should be divided into three sequential stages discussed in detail in  section  [ tpb ] . from the algorithmic point of view
, the material calibration can then be understood as  a  sequential optimization problem .
such approach has two main advantages : first , solving three simpler identification steps in a batch form is typically much more efficient then the full - scale problem ; second , it allows to use only a subset of simulations for initial stages of identification process .
a variety of techniques is available to identify material parameters via optimization methods , see e.g.  ( * ? ? ?
* and reference therein ) .
the gradient - based methods are usually considered to be the most computationally efficient optimization algorithms available and  as  such have been successfully used in a variety of identification problems , e.g. @xcite . for the  current model , however , analytic determination of sensitivities is fairly difficult , mainly due the history dependency of the model as well as complex interaction of individual parameters .
the accuracy of numerical approximation to the exact sensitivities , on  the  other hand , is driven by the choice of pseudo - time step used in numerical simulations .
clearly , to reduce the computational time , the pseudo - time step should be used as large as possible .
therefore , the response - based objective function will not be smooth and gradient - based methods are unlikely to be very successful .
as an alternative , techniques of soft - computing can be employed for optimization of complex objective functions .
for example , stochastic evolutionary algorithms have been successfully used for solution of identification problems on a level of material point @xcite or  on  a  level of simple structures @xcite . for the current case ,
however , complexity of  the  optimization can be attributed rather to its non - smooth character than to the appearance of multiple optima ; the family of problems where evolutionary algorithms are the most successful methods .
this opens the way to more specialized tools , which deliver higher efficiency when compared to usually time - consuming evolutionary algorithms .
the approach adopted in the present work is based on an adaptive smoothing of the objective function by artificial neural networks  ( see , e.g. ,  @xcite for alternative ann - based solutions to identification problems ) . in particular , the approximated model is provided by the radial basis function network , described in section  [ rbfn ] , dynamically evolved by  minima located by a real - encoded genetic algorithm , briefly reviewed in section  [ grade ] . the proposed sequential numerical strategy is systematically verified in section  [ results ] with  attention paid to a detailed assessment of the proposed stochastic algorithm reliability .
final remarks and conclusions can be found in section [ conclu ] .
in the present section , we give a brief description of  the model on which the identification procedure is based .
for the readers interested in more details , the complete description of  the  model is given in part i of this paper .
as already mentioned , the proposed model is capable of taking into account two different types of dissipation ( e.g. see @xcite ) :    * a bulk dissipation induced by the appearance of uniformly distributed microcracks .
this bulk dissipation is taken into account by the use of a classical continuum damage model ; * a surface dissipation induced by the development of macrocracks responsible for  the  collapse of the structure . as presented in part
i of this paper , this phase is taken into account by the use of a strong discontinuity model .
the surface dissipation is taken into account by the introduction of a traction / displacement jump relation .
therefore , two different models are involved in the constitutive description : the one associated with the bulk material and the one associated with the displacement discontinuity .
both are built on the same scheme considering the thermodynamics of continuous media and interfaces .
the key points of the construction of each of the two models are summarized in table [ tab : tab_continuum ] and table [ tab : tab_discrete ] .
.main ingredients of the continuum damage model [ cols="<,^",options="header " , ]
we have proposed a sound identification procedure for material parameters of the constitutive model for representing the localized failure of massive structures .
the most pertinent conclusions can be stated as follows :    \i ) the sequential identification approach employed for the uniaxial tensile test can be extended to the three - point bending test .
the resulting algorithm is very straightforward and has a clear link with the structure of the constitutive model .
moreover , each of three stages uses only a part of the test simulation , which leads to substantial computational time savings .
\ii ) due to the physical insight into the model , it was possible to construct simple objective functions @xmath3 , @xmath4 and @xmath5 with a high sensitivity to the relevant parameters
. this led to  non - smooth and non - convex objective functions , which were optimized by robust soft - computing methods .
\iii ) the proposed identification procedure was verified on @xmath6 independent optimization processes executed for each objective function . in the worst case , the reliability of  the  algorithm is @xmath7 due to very small number of objective functions calls set in the termination condition . from our experience with evolutionary algorithms @xcite , such a result is rather satisfactory .
\iv ) as the result of a sequential character of the identification procedure , the errors in  identified parameters accumulate . therefore , the values need to be determined with higher accuracy then usually required in applications ( i.e. @xmath8 ) and achievable by neural network - based inverse analysis @xcite .
\v ) the major difficulty of the proposed methods is to properly identify the three stages of structural behavior . from the point of view of method verification , where the reference load - deflection diagram is not noisy ,
the problem was successfully resolved . to fully accept the procedure
, however , the experimental validation of the method appears to be necessary .
the financial supports for this work by the czech ministry of education ( research project msm  6840770003 ) and the french ministry of research are gratefully acknowledged .
jz acknowledges the support of the ens - cachan invited professor funding program .
d.  brancherie and a.  ibrahimbegovic .
novel anisotropic continuum - discrete damage model capable of representing localized failure .
part i : theoretic formulation and numerical implementation . , accepted for  publication ( 2008 ) .
a.  kuerov and m. lep and j.  zeman .
back analysis of microplane model parameters using soft computing methods .
, * 14*(2):219 - 242 , 2007 ( special issue of the international symposium on neural networks and  soft computing ( nnsc-2005 ) )
in part i of this paper we have presented a simple model capable of describing the localized failure of a massive structure . in this part , we discuss the identification of the model parameters from two kinds of experiments : a uniaxial tensile test and a three - point bending test . 
 the former is used only for illustration of material parameter response dependence , and we focus mostly upon the latter , discussing the inverse optimization problem for which the specimen is subjected to a heterogeneous stress field .    novel anisotropic continuum - discrete damage model capable   +   +   +    a.kuerov@xmath0 , d.brancherie@xmath1 , a.ibrahimbegovic@xmath2 , j.zeman@xmath0 and z.bittnar@xmath0 + @xmath0 department of structural mechanics , faculty of civil engineering , czech technical university in prague , th ' akurova 7 , prague , czech republic + @xmath1 laboratoire roberval , universit de technologie de compigne , + bp 20529 , 60205 compigne cedex , france + @xmath2 ecole normale suprieure de cachan , laboratoire de mcanique et technologie , + 61 , av . 
 du prsident wilson 94235 cachan , france