Grouping detection of uncertain structural process by means of cluster analysis

Transcription

1 Grouping detection of uncertain structural process by eans of cluster analysis Alesandra Piotrow, Martin Liebscher 2, Stephan Pannier, Wolfgang Graf Institute for Structural Analysis, TU Dresden, Gerany 2 DYNAore GbH, Stuttgart, Gerany Suary: Structural analysis under consideration of the uncertainty of input paraeters, such as loads, aterial, and geoetry leads to uncertain tie-dependent results. Such uncertain structural process shows for the uncertain input paraeters all possible behaviours of a structure. Modelling of uncertainty in input paraeters when only incoplete or epert nowledge based inforation is available requires the introduction of the uncertainty odel fuzziness. A fuzzy process is a fuzzy set of real valued processes, whereas each of the possesses an assigned ebership value indicating the degree of possibility. In order to obtain an engineering interpretation of a fuzzy process soe representative crisp processes have to be chosen fro nuerous realisations of this uncertain function. In this paper a cluster analysis based approach for grouping siilar and detecting different tie-dependent structure behaviours is introduced. The siilarity of processes within one cluster is assessed with siilarity etrics: neighbouring location, affinity, and correlation. The uncertain assignent of real valued realizations of fuzzy process to clusters is eecuted with the Fuzzy-c-Means cluster algorith. The capability of this approach is deonstrated within the controlled collapse siulation of a reinforced concrete fraewor structure carried out in LS-DYNA. In this eaple an analysis of a fuzzy process is perfored by eans of cluster ethods and "-level discretization in order to select collapse sequences, significantly differing fro each other and having other degrees of possibility. Keywords: structural processes, cluster analysis, uncertainty

2 Introduction An appropriate structural analysis requires an adequate specification of input paraeters, such as loads, aterial paraeters and geoetrical properties. These paraeters are often affected by significant uncertainty. Data uncertainty is predoinantly caused by the way the inforation in engineering practice is gained: fro ineact observations, iprecise easureents or epert specifications. Thus, the consideration of input paraeters with crisp values only can be isleading and ay yield incorrect conclusions. Generally, the uncertainty can be classified with respect to its source either as aleatory or as episteic. Aleatory uncertainty is associated with objectivity and being principally described with the uncertainty odel randoness. This odel is investigated on the basis of the probability theory ethods. In order to apply randoness soe strict conditions have to be et especially the i.i.d. (identically independently distributed) paradig ust be fulfilled. When describing an input paraeter as a rando variable a large saple size is epected and the available data has to be obtained in constant conditions. In ost engineering applications eeting those requireents is nearly ipossible. The data reproduction conditions are unstable and the nuber of eperients is often insufficient for evaluating the distribution type and distribution function. This lac of inforation can be copensated by soe epert nowledge based evaluations. The inforation gained in such a anner is often affected by subjective/episteic uncertainty. In order to describe inforal or leical uncertainty the odel fuzziness can be utilized. When the inforation content of uncertainty is ainly objective but soe subjective influences are siultaneously taen into account the odel fuzzy randoness is ost suitable. In [4] the application of uncertainty odels to the structural analysis is presented. Considering data uncertainty within structural analysis leads to uncertain tie-dependent results z ( ). Each trajectory z( ) of such uncertain process represents an output of a single deterinistic solution, for eaple a selected force, displaceent or stress. Thereby, a fuzzy process z ( ) contains nuerous real-valued processes z ( ). However, in the engineering practice often copact solutions, allowing a technical interpretation are valuable. To reconcile this need with cople, uncertain structural analysis results a grouping detection approach has been developed. This concept draws upon dividing all realisations of an uncertain process in groups with respect to the criterion of functions siilarity. For each group a representative process, indicating soe characteristic structural behaviour is identified. The approach presented within this paper is based on algoriths of cluster analysis. These ethods have found wide application in any fields, e.g. in data ining, pattern recognition, iage analysis, bioinforatics and also in engineering sciences. In [2] the ipleentation of cluster algoriths for the evaluation of uncertain structural analysis results is shown. Eplorative data analysis has been applied for solving the design proble with the aid of the inverse solution. In this paper a Fuzzy-c-Means cluster algorith is adopted for the classification of fuzzy process trajectories. The presented cluster algorith based approach for structural analysis provides a realistic and applicable solution, because uncertainty of structural paraeters is considered and erely several representative structural behaviours are chosen for the engineering interpretation. The aboveentioned ethod has been adapted to the controlled collapse siulation of a reinforced concrete structure. Reduction of the data base to a certain nuber of groups enables the detection of coplicated interrelationships within a structure collapse.

3 2 Uncertain process within a structural analysis 2. Structural analysis with respect to uncertainty In deterinistic structural analysis crisp vectors of siulation input paraeters including loads, aterial and geoetry paraeters are apped with the aid of a coputational odel M onto vectors of siulation result paraeters z, containing e.g. internal forces, stresses and displaceents. The proble ay be forulated as follows: M : z. () Therefore, the apping odel M, also denoted as fundaental solution, represents a crisp dependency between structural paraeters and structural responses. In the herein presented approach the uncertain siulation input paraeters are odelled with the characteristic fuzziness. In consequence, the siulation result paraeters z are obtained as fuzzy quantities. F : FA z. (2) In Eq. (2) fuzzy input vectors are apped onto fuzzy result vectors z with the aid of the deterinistic fundaental solution, represented in this particular case by the finite eleent ethod. The nuerical realization of a fuzzy structural analysis is shown in Fig.. The application of "-discretization to fuzzy-variables,..., n enables the specification of "-level sets X,,..., X n, at the level ", where [, ]. The "-sets fro several input paraeters for the n- diensional crisp subspace X. Each point fro the subspace X represents a cobination of siulation input paraeters and defines a crisp vector. For a chosen input vector the deterinistic fundaental solution is eecuted and a crisp result vector z is obtained. The ai of the fuzzy structural analysis is to deterine a ebership function (z) for a particular response z. Therefore, on each "-level " all possible result values have to be identified. This ay be realized by deterining the inial and the aial result values z in, z a with the aid of "-level-optiization []. Repeating the "-level-optiization on each " leads to the specification of the ebership function (z). fuzzy input variables,d and 2, d fuzzy result variable z (z) (,d, 2,d) -level set X,,d crisp subspace X apping odel M z,l (in) z,r (a) z -level optiization -level set X 2, 2,d Figure : Fuzzy structural analysis The presented approach ay be etended by introducing siulation input paraeters as fuzzy functions ( t ). In result, the output paraeters z are obtained as fuzzy functions z ( t), as well. If the arguent t represents the tie coordinate a fuzzy function is defined as a fuzzy process and denoted by z ( ).

4 2.2 Nuerical treatent of uncertain processes When introducing fuzzy processes as input and output paraeters Eq. (2) turns to: F FA : ( ) z( ). (3) A fuzzy process z ( ) can be interpreted as a sequence of fuzzy variables evaluated for certain points in tie, see Fig. 2. { z = z ( ) z F( Z) } z ( ) = (4) t t z z (z) Figure 2: Fuzzy process 2 3 In order to allow a nuerical treatent of z ( ) the fuzzy-bunch paraeter representation is applied. A fuzzy process can be described with the aid of fuzzy bunch paraeters s and crisp arguent z ( ) = z( s, ) = { z = (, ) z s z F(Z) }. (5) Each crisp bunch paraeter vector s s with an assigned ebership value (s) describes eactly one crisp process z ( ) = z( s, ) z ( ) with ( z( )) = ( s). Therefore, a fuzzy process can be considered as a fuzzy set of real-valued processes z ( ) z ( ). A gradual ebership of a fuzzy process trajectory to a fuzzy set is epressed with (s). In consequence, a fuzzy bunch paraeter can be interpreted as a representation of z ( ) within the fuzzy analysis. If a fuzzy process with a fuzzy-bunch paraeter s is considered as input paraeter, then the application of "-level-discretization to s yields "-sets S. They for together with "-sets fro other input variables the n-diensional crisp subspaces S. Further steps of fuzzy analysis are then eecuted as described in section 2.. Assuing that one bunch paraeter vector s fro the subspace S deterines one real valued process on the level, then evaluating any vectors s S results in a bunch of processes foring a set on. Such "-function set Z ( ) is forulated as follows: { z( s, ) s s S ; (, ] } Z ( ) =. (6) If the nuber of "-levels is sufficiently high, then z ( ) can be represented by the assigned "-function sets. { ( Z ( ), ( Z ( ))) ( Z ( )) = (,] } z ( ) = z( s, ) = (7) In each "-function set real valued processes are contained, that posses at least the assigned ebership ". In addition, on every "-level bounding functions are specified with respect to " function sets. They indicate the interval boundaries at function discretization points.

5 3 Cluster analysis approach The principal idea of cluster analysis approach is to detect soe data structures within the analysed saple in order to obtain an appropriate data classification. The cluster analysis is carried out under the assuption that soe siilarities eist between the eleents of the available data set. Eleents are available in various fors: as points of the data set or as functions/processes. The classification of points succeeds within the division of the whole data set into saller groups so called clusters. As a classification criterion the siilarity of set eleents is introduced. An appropriate classification is obtained when the requireents of hoogeneity and heterogeneity are et. On the one hand, hoogeneity of a cluster can be assued when the eleents of a cluster indicate high siilarity. On the other hand, heterogeneity of cluster classification is eant when eleents of different clusters show distinct dissiilarity. An accepted classification of cluster ethods distinguishes between deterinistic and uncertain cluster approaches. The essential difference between those two proceedings lies in the type of the assignent of eleents to clusters it can be either deterinistic or uncertain. With deterinistic cluster approach one eleent is assigned to eactly one group. Several types of such ethods have been developed, including especially the hierarchic and partitionate cluster algoriths. A deterinistic algorith that found wide application, inter alia, in engineering sciences is the -Medoid-Cluster approach. The uncertain cluster approaches are considered within the concepts of probabilistics and fuzziness. Fuzzy cluster ethods base on gradual assignent of eleents to a cluster. This gradual assignent is deterined within the ebership function, taing values in [,]. If a ebership of an eleent to a cluster is defined with the value, then the eleent certainly belongs to the entioned cluster. Otherwise, the ebership value indicates that no affiliation of the eleent to a cluster can be stated. The gradual assignent is restricted by a rule, assuing that for a single eleent of the set the su of ebership values to all obtained clusters is equal to. In this paper the uncertain cluster approach: Fuzzy-c-Means is introduced. In order to obtain essential results of cluster analysis it is indispensable to find a nuber of clusters, which optially describes the available data set. Coonly, cluster analysis for different nuber of clusters is perfored and afterwards the obtained data classification is assessed with respect to several quality criterions. These are, inter alia, a partition coefficient, partition entropy and silhouette coefficient. Partition coefficient is used as a easure for assessing the unabiguousness of the classification. n C 2 PC( U ) =, i (8) = i = High values of the partition coefficient indicate that a clear division of eleents into n C clusters was obtained. The partition entropy bases upon the SHANNON entropy. It refers to the inforation content of the investigated data. nc (, i, i, i, i = i = PE U ) = - ln( ) + (- ) ln(- ) (9) Low entropy signifies an unabiguous classification. Silhouette coefficient, see Eq. (), is a easure for assessing the hoogeneity and heterogeneity of cluster configuration. n C bi ai SC, i aai, bi, where () i

6 a i = ( d(, ) h= i h, h ( h=, h i h) i h) () b i in h,...,n C, h j (d(i, h ) h, j i j). (2) ( h, j i j) j The cluster classification quality can be assessed on the basis of silhouette coefficient, as shown in Table. Table : Quality of cluster classification on basis of silhouette coefficient SC Cluster classification quality,7, very good,5,7 good,26,5 bad - an artificial classification is obtained,25 no appropriate classification Presented ethods can be successfully adapted to grouping of functions, in this case to grouping of fuzzy process realizations z ( ) z ( ). It is assued that fuzzy processes are discretized in the tie doain. The application of Fuzzy-c-Means cluster approach should yield a cluster classification, characterized by hoogeneity and heterogeneity. The eecution of Fuzzy-c-Means algorith requires the specification of siilarity easures for processes. Several aspects, lie neighbouring location, affinity, and correlation are taen into account by the siilarity assessent. All these easures are cobined under consideration of user-specified weights. Thereby, the neighbouring location has a high priority in the presented approach. It is defined as a difference between functional values of two processes z ( ) and z ( ), where z ), z ( ) z ( ). 2 ( 2 d( z n t (3) T ( ), z2( )) = ( z( i ) - z2( i )) ( z( i ) - z2( i )) nt i= 4 Grouping detection of structural processes In the approach presented within this paper a special case of Eq. (3) is introduced. Uncertain tie independent input paraeters are apped with the aid of a deterinistic fundaental solution onto tie dependent structural responses z( ). F FA : z( ) (4) The application of "-level-discretization to N siulation input paraeters,..., N yields "-level sets X,,..., X N,, defined as intervals <, l,, r >,..., < N, l, N, r >. The assignent of a trajectory z ( ) z ( ) to a certain "-level is defined within the ebership of z ( ) to an appropriate "-function set Z ( ). Foring "-function sets Z ( ) is essential for the presented approach. According to Eq. (4) an "-function set Z ( ) on the "-level consists of all functions z ( ) obtained for the siulation input paraeter values within the intervals, >,..., <, > <, l, r N, l N, r. As entioned in the section 2.2 each "-function set coprises all real valued functions, that posses at least the assigned ebership ". Each realization of the -th "-function set z( ) Z ( ) is contained in the l-th "-function set Z l ( ), l <, while not all z( ) Zl ( ) are contained in Z ( ). The application of cluster analysis requires the identification of highest possible "-level for each realization z ( ) z ( ).

7 If the trajectory z( ) is obtained through the evaluation of the -diensional input vector, containing variables, i [, ],then: i z( )) = in( ( )). (5) ( i The assignent of z() to the highest "-level ("-function set) is deterined by the inial ebership value of an input variable i. In the proposed approach the cluster analysis, based on the Fuzzy-c-Means algorith is applied to the fuzzy process z ( ). Grouping of trajectories is conducted on the "-function set Z = ( ). Thereby, N cluster centres are obtained, whereas each of the identifies one deterinistic, discrete function p, i ( ), i [, N]. Since the function values at certain points in tie were obtained with the Fuzzy-c- Means cluster approach, the function p, i ( ) doesn t represent a trajectory of a fuzzy process z ( ). However, the Fuzzy-c-Means algorith proposes a gradual assignent of fuzzy process realizations to clusters. Therefore, a real valued process p, i ( ) with the highest ebership to a cluster can be identified and regarded as a representative process for this particular cluster. In general, the grouping detection algorith yields representative processes eeplifying a particular structural behaviour. Since, those processes p, i ( ) are trajectories of a fuzzy process z( ) z( ), they are characterized by a ebership value (z( )) =, indicating the degree of possibility. 5 Application of grouping detection approach to controlled collapse siulation The approach presented in section 4 is applied to the controlled collapse siulation of a three-storied fraewor structure of reinforced concrete. The blasting strategy presued a tilt collapse of the structure, obtained by blasting two front coluns on the first floor. Reoving those supports causes failure zones in the upper part of reaining first floor coluns and leads to a desired ineatic chain. A nuerical odel of the above entioned fraewor structure consists of 233 eight-node heahedral finite eleents. Within the proposed finite eleent discretization soe structural parts of the building e.g. coluns and ceilings are considered as rigid. After reoving of two front coluns a collapse ineatics is initialized and areas of intensified local daage arise. In order to detect those failure zones an erosion odel was applied. The identified local daage zones act lie hinges. Within the erosion odel plastic strain is verified in every tie step and eleents, that satisfy the condition fro Eq. (6), are reoved fro the siulation. (6) pl pl, crit Reinforced concrete was described with the aid of a piecewise linear plasticy odel. Paraeters of this aterial odel were obtained for insufficient data and are thus affected by uncertainty. Due to the predoinantly episteic character of the uncertainty, quantities describing erosion and plastic effects were odelled as fuzzy-variables. Uncertain paraeters of the aterial odel with the uncertain stress-strain curve are shown in Fig. 3. stress ( 2, fail, ) 2 ( fail ) ( ) ( 2 ) strain 2 fail Figure 3: Fuzzy aterial odel; piecewise linear plastic aterial behaviour

8 The application of those uncertain quantities to the controlled collapse siulation yields ultiple collapse scenarios. For each of the several tie-dependent structural responses are obtained. Within the presented eaple a displaceent of a node, located in the failure zone is chosen for further investigation, see Fig. 4. failure zone deterinistic process z-displaceent of node 777 [c] tie [s] Figure 4: Finite eleent odel of the fraewor structure The analysed tie-dependent displaceent is a fuzzy process z ( ). The application of Fuzzy-c- Means cluster approach to z ( ) leads to the classification of all coputed trajectories z ( ) z ( ) into five clusters, see Fig 5. The Fuzzy-c-Means algorith yields an gradual assignent of each trajectory to every cluster, described with a ebership value. In Fig. 5 for a particular z ( ) a crisp assignent to a cluster is obtained taing into account the highest value of. 8 C z-displaceent of node 777 [c] C5 C4 C2 C tie [s] Figure 5: The assignent of fuzzy process realizations to clusters

9 odel 3 p,4 () The appropriateness of obtained classification is assessed with quality criterions as shown in Table 2. A relatively high value of the silhouette coefficient indicates that heterogeneity of cluster configuration and hoogeneity within each cluster has been achieved. The coparison of obtained values with Table suggests a good classification result. The partition coefficient as well indicates an unabigous assignent of fuzzy process realisations to clusters. Value Table 2: Quality criterions values Quality criterion.42 Entropy.86 Partition coefficient.668 Silhouette coefficient Within the fuzzy-cluster approach representative processes p, i ( ) have been identified, which eeplify five different collapse ineatics. The ain differences between those structural behaviours appear in the failure zone and influence the further ineatic chain. Within odels and 2, associated with p, ( ) and p, 2( ) a tilt-vertical collapse can be observed, see Fig. 6. Due to the high plastic strain values a separation within the failure zone proceeds. The isolated upper part of the building falls at reaining two coluns of the first floor. Such a tilt-vertical collapse significantly differs fro the planned strategy and should be concerned as iperissible. The only distinction between those odels can be ade relative to the bucling behaviour of the first floor pillars. Diverse forulation of the aterial law iplies different copliance of the reaining coluns. Within odels 3 and 4 ( p, 3( ) and p, 4( ) ) the division of the failure zone occurs as well, but the separated upper part of the structure falls forward on the prepared fall zone, avoiding the collision with coluns. Soe differences between those two odels appear in the later collapse stage, due to diverse inetic energy of the falling part shortly after the separation. In effect, the successive foration of failure zones proceeds differently. Model 5 ( p, 5( ) ) represents the desired tilt collapse. The separation within the connection of first floor coluns and the ceiling does not occur, which iplies the intended foration of further failure zones. odel p,5 () odel 2 p,2 () odel 4 p, () odel 5 p,3 () t= s t=. s t=.6 s t=.8 s Figure 6: Different collapse ineatics, identified with the cluster approach

10 Since the identified displaceent curves are realisations of a fuzzy-process, they are assessed by ebership values, indicating the degree of possibility. In Fig. 7 representative processes are shown. 8 p, () =.25 6 z-displaceent of node 777 [c] p,5 () = p,2 () =.5 á= p,3 () =.75 p,4 () = tie [s] p, i ( Figure 7: Representative processes ) Hence, soe conclusions can be drawn regarding the degree of possibility of the identified processes. The process p, 5( ), associated with the structural behaviour of odel 5 belongs to the "-set S and is therefore described with the degree of possibility.75. For the fuzzy input paraeters, =.75 specified as entioned above the occurrence of this structural behaviour sees to be quite possible. According to Fig.6 siilarity between processes, characterised with bunch paraeters.75 and can be stated. Since both collapse ineatics correspond to the intended blasting strategy, such a result suggests a successful realization. The ebership values of the reaining representative processes indicate lower degrees of possibility. Even though, they should be taen into consideration during planning of the blasting strategy in order to guarantee a safe realization of structure collapse. 6 Conclusions In this paper a grouping detection approach, based on cluster algoriths has been presented. The application of this ethod to uncertain processes enables an engineering-oriented interpretation of uncertain structural analysis results. As input paraeters are predoinantly specified on the basis of incoplete, dubious or fluctuating inforation, the odel fuzziness has been applied for the uncertainty quantification. Nuerous realizations of fuzzy process have been classified into definite nuber of clusters with the Fuzzy-c-Means cluster approach. Thereby, each cluster has been characterized by a representative process, that eeplifies a particular structural behaviour. Representative processes obtained with presented grouping detection approach are assessed by a ebership value indicating the degree of possibility. Thus, the proposed ethod yields realistic results, because uncertainty is considered and on the other hand provides an applicable solution, because erely several representative structural behaviours are chosen for the engineering interpretation. The capability of the approach was deonstrated by eans of controlled collapse siulation.

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