Engineering Applications of Arti?cial Intelligence 20(2007)709–720
Intelligent support of engineering analysis using ontology and
case-based reasoning $
Peter Wriggers a ,Marina Siplivaya b ,Irina Joukova c ,Roman Slivin c,?
a University of Hannover,Institute of Computational Mechanics and Building Mechanics,Appelstrasse 9a,30-167Hannover,Germany b
Volgograd State University of Architecture and Civil Engineering,IST Department,Akademicheskaya 1,400074Volgograd,Russia
c
Volgograd State Technical University,CAD Department,Lenina 28,400131Volgograd,Russia
Received 3December 2006;accepted 4December 2006
Available online 31January 2007
Abstract
Accuracy and reliability of the FEM analysis results depend heavily on the quality of the decisions made during the analysis process.As there are no industry-level systems for support of non-algorithmic tasks of FEM-based engineering analysis,such tasks are carried out by engineers on the basis of expert knowledge and experience.However,to exploit contemporary potentialities of FEM to solve a complex engineering problem requires high level of expertise;this restricts application of achievements of FE analysis in industry.
In this paper,the concept of intelligent support of engineering analysis using knowledge-based system is presented,which is a promising way to increase quality of complex analysis.r 2007Elsevier Ltd.All rights reserved.
Keywords:Intelligent support;Engineering analysis;Ontology;Case-based reasoning
1.Introduction
Accuracy and reliability of the FEM analysis results depend heavily on the quality of the decisions made during the analysis process.As there are no industry-level systems for support of non-algorithmic tasks of FEM-based engineering analysis,such tasks are carried out by engineers on the basis of expert knowledge and experience.However,to exploit contemporary potentialities of FEM to solve a complex engineering problem requires high level of expertise;this restricts application of achievements of FE analysis in industry.
In this paper,the concept of intelligent support of engineering analysis using knowledge-based system is
presented,which is a promising way to increase quality of complex analysis.2.State-of-the-art
Some of the tasks within the FEM analysis process are of algorithmic nature—meshing (assuming that all the para-meters of the mesh are speci?ed),FEM computation itself,visualization of the results;while other tasks are non-algorithmic and imply knowledge-based approach:pro-blem classi?cation,making decisions about computation model parameters (geometry simpli?cations,?nite elements type and size,material model,etc.)and numeric algorithm type and parameters,evaluation and interpretation of the numeric results.
Algorithmic tasks are effectively automated by analysis subsystems of modern CAD and CAE systems (ABAQUS,I-DEAS,ANSYS,MSC.NASTRAN-based systems,etc.)and stand-alone tools for speci?c tasks (meshing tools,etc.)or domains (Moaveni,2004).While AI technologies are widely applied in the ?elds of conceptual design,diagnostic and fault detection,relatively little work has been done to
https://www.doczj.com/doc/fc12920271.html,/locate/engappai
0952-1976/$-see front matter r 2007Elsevier Ltd.All rights reserved.doi:10.1016/j.engappai.2006.12.002
$
The work is ?nancially supported by the ‘Mikhail Lomonosow’grants of DAAD and Ministry of Education and Science of Russia.?Corresponding author.
E-mail addresses:wriggers@ibnm.uni-hannover.de (P.Wriggers),marina@unix.cad.vstu.ru (M.Siplivaya),kreative@vistcom.ru (I.Joukova),romanmom@yandex.ru (R.Slivin).
automate and/or support non-algorithmic tasks of the engineering analysis,which imply heuristic classi?cation and decision making.In particular,the following re-searches can be mentioned:GENIUS expert system for controlling the iterative FEM computation process (Wriggers and Tarnow,1989);REAP methodology,which aims at FE model construction on the basis of modular reuse of fragments of existing models (Chien,2002).Contemporary AI technologies can be used to repres-ent knowledge and model-reasoning processes used in solving non-algorithmic tasks of FEM-based engineering analysis.A system for intelligent support of FEM-based engineering analysis can be designed on the basis of these technologies.Development of such system is a promising research ?eld.
The contact mechanics problems were selected as a test domain of engineering analysis because of their importance and frequent occurrence in engineering practice,and their signi?cant complexity as well.3.Concept of the intelligent support
The process of engineering analysis has been studied using IDEF methodology.1The following decision-making tasks to be supported were determined at different stages of the process of solving a contact mechanics problem using FEM (Wriggers,2004):
1.Problem identi?cation:
1.1.mechanical device classi?cation,analysis aim de?nition,
1.2.contact pairs identi?cation and their properties de?nition,
1.3.material properties de?nition,
1.4.contact problem properties de?nition.
2.Preprocessing stage:
2.1.geometry simpli?cations for building solid model,2.2.?nite elements type and size selection,2.
3.loading (boundary conditions)de?nition,2.3.loading
2.4.material model de?nition,2.5.analysis type de?nition.
https://www.doczj.com/doc/fc12920271.html,putation:
3.1.algorithm selection,
3.2.interactive control of algorithm parameters.
4.Post-processing stage:4.1.results visualization,
4.2.results interpretation in the domain terms.The input information of the supported engineering analysis process consists of:textual description of the problem,graphical description of the technical object (draft,sketch)and implicit information,which can be
obtained by the automated system in the interactive querying procedure.The reasoning process performed by the automated intelligent support system comprises the following fundamental steps:
1.Engineering problem properties (physical object type and properties,purpose of the analysis)extraction from the problem description.
2.De?nition of the formal contact mechanics problem(s)corresponding to the engineering problem under con-sideration on the basis of the obtained engineering problem properties using:
https://www.doczj.com/doc/fc12920271.html,rmation about the previously solved problem(s)for which the analogy relation can be established with the problem under consideration,
https://www.doczj.com/doc/fc12920271.html,rmation about dependencies between properties of the engineering problem and the corresponding formal problem(s).
3.De?nition of the solution technique(s)corresponding to the formal contact problem under consideration on the basis of the inferred formal contact problem properties using:
https://www.doczj.com/doc/fc12920271.html,rmation about the previously solved problem(s)for which the analogy relation can be established with the problem under consideration,
https://www.doczj.com/doc/fc12920271.html,rmation about dependencies between properties of the formal contact problem and corresponding solution technique(s).The general schema of this process is presented in Fig.1.Thus,the reasoning process has the following properties:reasoning schema ‘‘engineering problem 2formal pro-blem 2solution’’assuming ‘2’denotes ‘‘many-to-many’’relations,use of information about previously solved problems and processing both qualitative and quantitative information about the object under analysis.Knowledge extraction and formalization currently fo-cuses on the stage of passing on from a formal problem to a solution routine in the domain of contact mechanics.The set of the most signi?cant properties describing a contact problem and a solution routine were determined (Feng and Prinja,2002;Konter,2000).Both dependencies between contact problem properties and solution routine properties and inter-dependencies between solution routine properties were formalized.
The set of characteristic properties depends on the domain and engineering problem type.For a contact mechanics problem such properties include:1.dimensionality (2-D,3-D),
2.sliding (small sliding,large sliding,large rotation),
3.friction model (frictionless,stick,slip,stick-slip combination),
4.initial undeformed geometry (?at,curved),
5.initial contact conditions (interference ?t,tied-bonded ?t,gaps),
6.behavior under loading (stationary,advancing,reced-ing contact,self-contact)and
1
IDEF means ICAM de?nition,where ICAM stands for integrated computer-aided manufacturing;now IDEF is a widely accepted as a general process modeling methodology.
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7.rigidity(deformable–deformable or deformable–rigid bodies contact).
The basic components of the solution routine of a contact problem can be speci?ed as the following (Wriggers,1995,2004):
1.Finite-element model:
1.1.meshing type(matching,non-matching),
1.2.?nite-element type:
1.2.1.general?nite elements,
1.2.2.contact?nite elements(node-to-node,node-to-
surface,surface-to-surface,etc.).
2.Numerical solution routine:
2.1.solution method(penalty functions,Lagrangian
multiplier,etc.),
2.2.numerical algorithm(Newton—Raphson,etc.). The use of information about previously solved pro-blems can be facilitated by the case-based reasoning(CBR) technology,which models the process of solving a problem by establishing the analogy relation between the current problem and previously solved problem(s).Within the suggested reasoning schema there are two steps of CBR process,shown in Fig.2.The main tasks of CBR implementation are:to design case description model,to formalize analogy relation,to develop case search,evalua-tion and adaptation procedures.
The process of the engineering analysis has been re-engineered according to the IDEF methodology taking into account participation of the automated knowledge-based decision-support system.The functional structure and general architecture of this system have been determined and the requirements for its reasoning mechan-isms and knowledge base have been set up.
4.Knowledge representation model
According to the results of decision-making process, analysis of the following types of knowledge are to be represented within the knowledge representation model: 1.Mechanical engineering problems including:
1.1.mechanical devices types,
https://www.doczj.com/doc/fc12920271.html,ponent parts of a mechanical device,
1.3.properties of a mechanical device and its parts,
1.4.spatial and functional relations between component
parts within a mechanical device.
2.Contact mechanics problems including:
2.1.contact problems types,
2.2.properties of a contact problem.
3.Knowledge about solution procedures,including:
3.1.analysis types,
3.2.algorithms and their parameters.
Fig.1.Main stages of the engineering analysis process.
Fig.2.The reasoning process schema based on CBR.
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4.Dependencies between engineering problem properties
and formal contact problem type and properties.
5.Dependencies between formal contact problem type and
properties and solution procedure properties.
6.Cases:a case’s description includes de?nitions of:
6.1.physical object(mechanism),
6.2.engineering analysis problem set for this mechan-
ism,
6.3.formulation of a corresponding formal contact
problem,
https://www.doczj.com/doc/fc12920271.html,ed numerical solution routine speci?cation. Upon the comparative analysis formal ontologies has been selected as the knowledge representation model type (Gruber,1993;Ushold and Gruninger,1996).Mechanisms have been designed to represent the required knowledge ?eld elements listed above using the modeling primitives available within the selected model type—concepts,roles and individuals.These mechanisms are based on introduc-tion of concepts and roles with pre-de?ned semantics. Knowledge representation model has been developed which comprises:(1)ontology of mechanisms,(2)ontology of contact problems,(3)ontology of solution techniques, (4)special roles which relate concepts and individuals from the ontologies1–3(‘‘describes’’/‘‘is_described_by’’for mechanism-contact problem relation,‘‘solves’’/‘‘is_ solved_by’’for contact problem—solution relation)and (5)special concepts—subsets(descendants)of the‘rule’concept,which represents rules and constraints re?ecting dependencies between properties of engineering problems, contact problems and solutions.
So,the ontology-based domain knowledge representa-tion model M can be structured according to the used ‘‘engineering problem—formal problem—solution rou-tine’’reasoning schema in the following way:
M?f DO;FP;SM;RUL g,
where DO is the set of elements of description of objects and engineering problems,FP the set of elements of formal problems descriptions,SM the set of elements of solution routines descriptions and RUL the set of dependencies between properties of objects and engineering problems, formal problems and solution routines.
Let us view the ontology branch(sub-ontology),which represents objects(mechanisms)and engineering problems.
ObjectDescription—set of physical objects(mechan-isms),engineering problems and their properties de-scriptions.
ObjectDescription?Object[ObjectProperty[Problem [ProblemProperty,
Object C ObjectDescription—set of objects descrip-tions,
ObjectProperty C ObjectDescription—set of object properties and their values descriptions.
Problem C ObjectDescription—set of engineering problems descriptions.
ProblemProperty C ObjectDescription—set of engi-neering problems properties and their values descrip-tions.
Let us view the ontology branch(sub-ontology),which represents formal problems and their properties.
FormalProblemDescription—set of formal problems and their properties descriptions.
FormalProblemDescription?FormalProblem[
FormalProblemProperty.
FormalProblem C FormalProblemDescription—set of formal problems descriptions.
FormalProblemProperty C FormalProblemDescrip-tion—set of formal problems properties and their values descriptions.
Let us view the ontology branch(sub-ontology),which represents solution routines and their properties.
SolutionDescription?Solution[SolutionProperty.
Solution C FormalProblemDescription—set of solution routines descriptions.
SolutionProperty C ObjectDescription—set of solution routines properties and their values descriptions.
At the same time,
Properties—set of all the properties of model elements (objects,engineering problems,formal problems and solution routines).
Properties?ObjectProperty[ProblemProperty[
FormalProblemProperty[SolutionProperty.
Properties?BasicProperties[ComplexProperties, where
BasicProperties—properties of basic types(number, string),
bp A BasicProperties,bp?{Name,Value}—a property, Name—name of the property,Value—a value of the property,
ComplexProperies—properties with discontinuous values,
cp C ComplexProperties,cp?{Name,Values},Values—set of possible property values cp;cp?[i Val i,Val i C Values,
isGreaterThen C R,isGreaterThen.Domain?Prop-erty,isGreaterThen.Range?Property;isLessThen C R,isLessThen.Domain?Property,isLessThen.Range ?Property—ordering relationships de?ned on the set of properties an their values;these relations are to be used by the qualitative reasoning mechanism.
Let us view the description of a real-world object (mechanism or mechanism part).
P.Wriggers et al./Engineering Applications of Arti?cial Intelligence20(2007)709–720 712
Obj A Object—object description,
Obj?{Name,ObjType,Parts,Props,Relations,FPrs}, where
Name—name of the object,
ObjType—type(class)of the object,a reference to a more general concept describing the object;Obj C ObjType,ObjType C Object,
Parts—set of component parts of the object,Parts C Object,
8part,part A Parts)hasPart(Obj,part)?1,
isPartOf(part,Obj)?1,
8part,part e Parts)hasPart(Obj,part)?0,
isPartOf(part,Obj)?0,
hasPart C R,hasPart.Domain?Object,
hasPart.
Range?Object.
The hasPart/isPartOf roles determine aggregation rela-tions between the object and its component parts: isPartOf C R,isPartOf.Domain?Object,isPartOf. Range?Object.
8a,8b,hasPart(a,b)?isPartOf(b,a).
Props–set of object’s properties,
Props?BasicProps[ComplexProps,
where BasicProps–properties of basic types(number, string),
ComplexProps–properties with discontinuous values. Props C ObjectProperties,BasicProps?Props\ BasicProperties,
ComplexProps?Props\ComplexProperties.
8Obj,8prop,prop A Obj.Props) hasObjectProperty(Obj,prop)?1, IsPropertyOfObject(prop,Obj)?1. hasObjectProperty C R,hasObjectProperty.Domain?Object,hasObjectProperty.Range?ObjectProperty. isPropertyOfObject C R,
isPropertyOfObject.Domain?ObjectProperty,
is isPropertyOfObject.Range?Object.
8a,8b,hasObjectProperty(a,b)?isPropertyOfObject (b,a).
Relations–set of relations,which the object participates in:
Relations?Relations1[Relations2,
8Obj,8rel,rel A Obj.Relations1)(ent,ent A C,
rel(Obj,ent)?1.
8Obj,8rel,rel e Obj.Relations1)8ent,ent A C,
rel(Obj,ent)?0.
8Obj,8rel,rel A Obj.Relations2)(ent,ent A C,rel(ent, Obj)?1.
8Obj,8rel,rel e Obj.Relations2)8ent,ent A C,rel(ent, Obj)?0.
isSubjectToProblem C Relations1—relation indicating that an engineering problem is set for the object;
8Pr,Pr A Problem,8Obj A Object,isSubjectToProblem (Obj,Pr)?1—problem Pr is set for the object Obj; isSubjectToProblem(Obj,Pr)?0—problem Pr is not set for the object Obj.
isDescribedBy C Relations1—relation indicating that an object is described by a formal problem(sub-problem); 8Pr,Pr A FormalProblem,8Obj A Object, isDescribedBy(Obj,Pr)?1—the object Obj is described by the problem Pr;
isDescribedBy(Obj,Pr)?0—the object Obj is not described by the problem Pr.
FPrs—set of formal problems,which can describe the given object;
FPrs C FormalProblems;
8Obj,Obj A Object,8Fpr,FPr A Obj.FPrs) describes(Fpr,Obj)?1,
isDescribedBy(Obj,FPr)?1;
8Obj,Obj A Object,8Fpr,FPr e Obj.FPrs) describes(Fpr,Obj)?0,
isDescribedBy(Obj,FPr)?0.
PR—description of an engineering problem set for a physical object(mechanism)Obj,FP—formal contact problem description,and SL—a solution routine descrip-tion are represented in a similar way,including properties, components and relations.
Let us consider a representation of a complex relation between objects.An example of such relation can be a contact pair formed by solids between which a contact relation(‘contacts’)exists.
ComplexRelation C Object—set of concepts represent-ing complex relations between objects;
cr A ComplexRelation—a complex relation,
cr?{{Object},InO,Cond},
where
InO—set of objects which participate in the relation;
InO C Object.
8cr,cr A ComplexRelation,8Obj,Obj A cr.InO) involves(cr,Obj)?1,
isInvolvedIn(Obj,cr)?1;
8cr,cr A ComplexRelation,8Obj,Obj e cr.InO) involves(cr,Obj)?0,
isInvolvedIn(Obj,cr)?0.
The set of sample contact mechanics problems from NAFEMS Benchmark(Feng and Prinja,2002)was described according to the presented model.Let us consider one of these sample problems.The system consists of the two parallelepipeds(bricks)in contact.The bottom side of the foundation is constrained in the vertical direction,with the center of this side constrained in all directions.The upper brick is subject to the distributed stationary loading.
The system consists of the following elements:
1.Brick1(foundation),
P.Wriggers et al./Engineering Applications of Arti?cial Intelligence20(2007)709–720713
2.Brick 2and
3.Contact pair ‘Brick 1–Brick 2’.
The following properties of the problem are speci?ed:1.linear elastic material of both bodies with equal E,2.stationary distributed loading (pressure q ).
The following properties can be inferred for the corresponding formal contact problem:1.contact type—small sliding,
2.initial undeformed geometry—?at,
3.behavior under loading—stationary contact,
4.rigidity—deformable-deformable bodies contact,
5.
surface friction—frictionless.
The following decisions can be inferred using the current knowledge base for the solution routine for the
considered problem from the viewpoint of the maximal accuracy:1.meshing type—matching,
2.geometrical representation of contact—node-to-node,
3.solution method—Lagrangian multiplier method,
4.
numerical algorithm—Newton–Raphson algorithm.
The sample physical system and the corresponding ontology fragment are presented in Fig.3.
These ontologies are formally described with OWL-DL—the language from the description logic (DL)family,which combine advantages of structured (e.g.frame-based)and logical approaches to knowledge representation,featuring both expressive and reasoning capabilities (Su and Ilebrekke,2004).The knowledge base is developed
using Prote
ge 3.1.1ontology development environment with OWL Plugin.
Fig.3.A sample physical system and the corresponding ontology fragment.
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5.Reasoning mechanism
CBR is successfully applied to the wide range of problems in various domains(Kolodner,1993;Bartsch-Spurl et al.,1999).The key tasks of the CBR process are: (1)query formulation in terms of the knowledge represen-
tation model,
(2)retrieval of cases which are the most similar to the
query,
(3)adaptation of the selected cases to match the query and
(4)cases retention in the knowledge base(learning of the
system).
The performed analysis has shown that existing CBR algorithms(Bogaerts and Leake,2004;Marinilli et al., 2004,etc.)cannot be directly applied to FEM solutions of engineering analysis problems,because this domain fea-tures some speci?c properties:engineering problem(case) description is,as a rule,of complex structure,which can vary depending on the technical object and the problem type;knowledge?eld elements differ signi?cantly in the level of formalization;values of qualitative and quantita-tive parameters are to be evaluated taking into account the context(values of other parameters).So the current problem appeared to be the adaptation of the CBR algorithms for the implementation of the intelligent support of the FEM-based engineering analysis of techni-cal objects.
The following algorithms were developed to operate on the suggested domain model M in the framework of the CBR mechanism implementation:CBR-query formulation support algorithm;case retrieval algorithm which uses class (concept)-based similarity(CBS)computation algorithm and property(slot)-based similarity(SBS)computation algorithm;case adaptation algorithm,which uses produc-tion rules and similarity paths.
CBR-query formulation support algorithm allows redu-cing the amount of the routine work needed to input information and enforcing domain model integrity.It can be viewed as a means for the physical system and engineering problem description auto-completion based on the information provided by the user.It supports inheritance of the components,properties and relations from classes,creation of virtual components corresponding to complex structural relations,model veri?cation,etc. The case retrieval algorithm,which is the key part of the CBR reasoner,is based on the similarity measure S.As the case indexes,which can be physical systems,engineering problems or formal contact problems,are represented in the formal ontology by individuals,the similarity of cases is reduced to the similarity between the individuals i1and i2 of the ontology.It can be represented as
Sei1;i2T?FeCBSei1;i2T;SBSei1;i2TT,
where S is the(global)similarity of the individuals,CBS the concept-based similarity(similarity computed by classes CL),SBS the slot-based similarity(similarity computed by relations which represent parameters V),F a real-valued composition function.
The suggested case retrieval algorithm,unlike known S-based algorithms(Gomez-Albarran et al.,1999),does not calculate the full S value for all the cases in the knowledge base.For most cases only the CBS value is calculated, which is much simpler to compute than the SBS,then the maximal possible value S m of S for the given CBS is estimated,and SBS is computed only when S m is greater than the current maximal S or greater than some given threshold.
The CBS computation algorithm uses the vector space model(Salton and McGill,1983),where each instance is represented by an n-dimensional https://www.doczj.com/doc/fc12920271.html,ponents of this vector correspond to user-de?ned(non-system)ontol-ogy concepts.If the instance is subsumed by a concept c, the corresponding vector component is assigned the appropriate value of the weighting function W(c),if not—zero.Also,in DL ontology the subsumption relation between an instance and a concept can be postulated by the user or inferred by the reasoning mechanism.In the latter case,the in?uence of this concept is moderated by multiplication of the corresponding vector component by the penalty factor p fc,which allows re?ecting of the trust level to the logical conditions in the description of the concept.
CBS is then computed using the well-known cosine measure(the cosine of the angle between the vectors of individuals):
CBSei1;i2T?
v1v2
jj v1jj jj v2jj
,
where v1,v2—vectors,which represent instances i1,i2, respectively.
This approach which uses weight coef?cients W instead of binary values0and1(as in known algorithm described in Gomez-Albarran et al.,1999)as vector components allows adequate evaluation of instance similarity in unbalanced ontologies containing branches with signi?-cantly different number of concept hierarchy levels(which is often the case in real knowledge bases)and also provides some tuning capabilities by adjusting concept weights.The schema of the CBS computation algorithm and an example of the ontology fragment.
The algorithm of the CBS computation and a sample ontology fragment are presented in Fig.4.
The SBS computation algorithm is based on one-to-one comparison of individuals’relations,which can represent their parameters V and/or structural relations RL.The SBS computation process is recursive;it starts on the given individuals i1and i2,compares one-to-one all the indivi-duals related to them by ontology relations(it includes computation of CBS and SBS for those individuals which causes recursion),and stops on individuals with no relations,for which SBS is not computed.
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The developed SBS algorithm features selection of the most appropriate mapping between the relation sets of the compared individuals by testing all the possible mapping against the maximal similarity criterion.This ensures the correctness of the obtained SBS values.Also,use of weighting coef?cients W for relations allows ?exible tuning of the algorithm and use of role similarity functions (tables)adequately handles ontologies with multiple similar relations.
For the two individuals i 1and i 2and a given mapping between their relations SBS is computed as weighted sum of similarities of all the relation tuples within the selected mapping:SBS ei 1;i 2T?
X n 1j 1?0X n 2j 2?0
w j 1w j 2L S eR j 1;R j 2T,
where n 1is the number of relations of the individual i 1,n 2the number of relations of the individual i 2,w j 1the weight of relation j 1of instance i 1,w j 2the weight of relation j 2of the individual i 2,R j 1the j 1th relation of the individual i 1,
R j 2–j 2th relation of the individual i 1,L S the similarity
function of the two given relations (local similarity).
For relations,which represent real-valued parameters modeled in the proposed model M by qualitative variables V the local similarity L S is computed using the developed algorithm for projection of qualitative variables landmarks (see below).For other relations L S eR 1;R 2T?R S eR 1;R 2TS ei R 1;i R 2T,
where R 1,R 2is the relations being compared,i R 1,i R 2the individuals,related to the given individuals i 1,i 2by relations R 1and R 2respectively,R S the given role similarity function (by default—1for roles with the same name,0otherwise).
The general schema of the suggested similarity computa-tion algorithm is presented in Fig.5.
The conceptual difference of the proposed algorithm from the known one (Gomez-Albarran et al.,1999)is the method of one-to-one mapping of the compared relation sets of individuals i 1and i 2.The SBS value obtained can differ signi?cantly depending on the way the relations are
Fig.4.Schema of the CBS computation algorithm and a sample ontology fragment.
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mapped.An example is shown in Fig.2.SBS is calculated for the two individuals representing‘sphere’physical objects,each of them being in contact with two other physical objects—brick(parallelepiped)and cylinder.In case when the‘contacts’relations of spheres are mapped ‘brick–brick’and‘cylinder–cylinder’the SBS value is1, which is intuitively the correct result;but when they are mapped‘brick–cylinder’and‘cylinder–brick’,which for-mally is also a valid mapping,the result can be0.Under some conditions,the latter result can be produced by known SBS computing algorithms(Gomez-Albarran et al., 1999).The problem is even more likely to arise if the numbers of individuals’relations are not equal.The example physical systems and the corresponding ontology fragment are shown in Fig.6.
The case adaptation algorithm performs adaptation by substitution:for a given CBR query Q and the case CS being adapted(the most similar case retrieved by the case retrieval algorithm)it determines the set of parameters VD which have different values for Q and CS(or missing in one of them and present in the other)and for each of them it selects a value which does not violate any of dependen-cies DP(CS).It differs from known algorithms of substitution adaptation of structural cases(Gonzales-Calero et al.,2000)in that it uses both production rules and similarity paths as dependencies(explanations)and produces the quantitative estimation of the adaptation effect.
Adaptation with similarity paths uses dependencies represented in the ontology graph by paths from indivi-duals corresponding to properties of the problem(case index)to individuals corresponding to properties of the case solution.Besides individuals,which represent domain concepts similarity paths can include system pre-de?ned individuals,which represent basis logical operators AND and NOT,which allows expressing arbitrary logical conditions.Examples of similarity paths are given in Fig.7. De?nition1.Path A is quasi-isomorphic to path B if after removal of all the individuals from paths A and B,which are subsumed by a concept‘LogicOperator’and their incoming‘implies’relations the resulting paths contain the same set of elements(individuals,relations)in the same order.
De?nition2.Path A is accomplished for an individual if for any individual i in it subsumed by a concept‘AND’all the paths ending on i contain O and for any individual i in it subsumed by a concept‘NOT’all the paths ending on i do not contain O.
Fig.5.General schema of the similarity computation algorithm.
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Adaptation using the similarity paths consists of the two conceptual steps:
1.Search for all the similarity paths SP i ,beginning on the individuals VD c which represent parameters VD of the case index CS which have values different from those parameters of the CBR query Q or a missing in one of the individuals Q,CS and present in the other one,
2.Search for all the similarity paths SP j ,quasi-isomorphic to paths SP i,beginning on the individuals VD q ,which
Fig.6.Potential ambiguity in SBS
computation.
Fig.7.Similarity paths examples (‘‘ContactPairProblem1—LargeSliding—NodeToSegment contact representation’’,‘‘ContactPairProblem2—SmallSlid-ing AND StationaryContact—NodeToSegment contact representation’’).
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represent parameters VD of the CBR query Q and accomplished for the individuals VD q.
Adaptation with the similarity paths supports use of weighting coef?cients W assigned to elements of paths,but is rather complex and does not use standard inference routines of OWL DL.Algorithm of case adaptation using production rules does not support weighting coef?cients W but effectively uses standard OWL DL reasoner capabil-ities and allows for rules formulation in the description logic language.
A production rule RL is represented by an ontology concept CR subsumed to the pre-de?ned concept‘rule’. The logical expressions,which constitute necessary and suf?cient conditions set of CR represent the left part (antecedent)of the RL,and the logical expressions,which constitute necessary conditions set of CR represent the right part(consequent)of the RL.
Adaptation using rules is then based on generating all the possible values of the parameters VD q being adapted and?ltering them against the production rules by testing the subsumption relation between the adapted case index individual and the concept which represents a rule.An example is shown in Fig.8.
The algorithm for projection of landmarks of qualitative variables is developed which is used by the SBS computa-tion algorithm described above.It allows estimation of a value of a real parameter of a physical system in the context of its other parameters(the qualitative‘meaning’of a real value).An example can be the following:a force applied to a physical body can lead to stress which is greater than its yield stress or less than it,which corresponds to the two different values of the qualitative variable which represents the force.
The algorithm forms the full set of landmarks of a qualitative variable z by adding the landmarks,which correspond to the landmarks of qualitative variables related to z by some qualitative constraints to the explicitly speci?ed landmarks of z.The main steps of the algorithm are:
1.To form the set of variables VR,which are related to the
given variable z by some qualitative constraints,
2.For each of the found variables v A VR:
2.1.to determine the type of monotony of the depen-
dence between z and v;
2.2.to add all the landmarks of the variable v to the
domain of the variable z determining the order
relation between them using the type of monotony
obtained at step2.1.
The proposed algorithms allow effective and correct retrieval of cases from the knowledge base,which are the most similar to the query formulated by a user,and subsequent adaptation of the most similar case to the current problem.
Fig.8.An example of rule representation by an ontology concept.
P.Wriggers et al./Engineering Applications of Arti?cial Intelligence20(2007)709–720719
6.Summary
The concept of intelligent support of engineering analysis is presented which assumes use of integrated knowledge-based CAE system.The reasoning mechanism of the systems implements‘‘engineering problem2formal problem2solution’’schema and is based on the CBR technology.
The model for knowledge representation about physical systems and engineering analysis cases is developed on the basis of ontology formally described in OWL-DL.
CBR algorithms are developed for this model,including case retrieval,adaptation and supplementary algorithms. Currently,an automated knowledge-based system for intelligent support of engineering analysis is being devel-oped using the proposed models and algorithms.The system is being implemented as a web application on the https://www.doczj.com/doc/fc12920271.html, platform.The developed models and algorithms are to be evaluated on the test knowledge base in the domain of contact mechanics.
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