HAL Id: hal-00686188
https://hal-upec-upem.archives-ouvertes.fr/hal-00686188
Submitted on 8 Apr 2012
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Construction of a probabilistic model for the soil impedance matrix using a non-parametric method
R. Cottereau, D. Clouteau, Christian Soize
To cite this version:
R. Cottereau, D. Clouteau, Christian Soize. Construction of a probabilistic model for the soil
impedance matrix using a non-parametric method. 6th International Conference on Structural Dy-
namics, Université de Marne-la-Vallée, Sep 2005, Paris, France. pp.841-846. �hal-00686188�
#ONSTRUCTION OF A PROBABILISTIC MODEL FOR THE SOIL IMPEDANCE MATRIX USING A NONPARAMETRIC METHOD
2 #OTTEREAU $ #LOUTEAU
# 3OIZE
!"342!#4 #ONSTRUCTION CODES DEMAND EVER INCREASING EARTHQUAKERESISTING FEATURES FOR STRATEGIC BUILDINGS SUCH AS DAMS AND NUCLEAR PLANTS AND THE INCORPORATION OF UNCERTAINTY IN THE DESIGN MODELS FOR THESE STRUCTURES PARTICULARLY IN SOIL DOMAINS BECOMES A MAJOR ISSUE 0ARAMETRIC METHODS AND A RECENT NONPARAMETRIC METHOD ARE CONSIDERED FOR THE CONSTRUCTION OF A PROBABILISTIC MODEL OF THE SOIL IMPEDANCE MATRIX THE LATTER SUPPLYING IN TERESTING FEATURES PROVIDED THAT THE MATRICES OF A CERTAIN MEAN MODEL CAN BE IDENTIlED 4HIS DIFlCULTY IS TACKLED USING A HIDDEN STATE VARIABLES MODEL ENSURING CAUSALITY OF THE IMPEDANCE AND NECESSARY POSITIVE DElNITENESS CONDITIONS ON THE GENERATED MATRICES 4HE IDENTIlCATION OF THE UNCERTAIN PARAMETERS IN THE SOIL AND THE DIFlCULT TASK OF QUANTIFYING THEIR VARIABILITY ARE NOT REQUIRED AND COMPUTATIONAL COSTS ARE SIGNIlCANTLY REDUCED
'
STRUCTURE
7
&IGURE UNBOUNDED DOMAIN AND COUPLING BOUNDARY
).42/$5#4)/.
,ET BE A THREEDIMENSIONAL OPEN HALF SPACE OF WITH A SMOOTH BOUNDARY &IG ,ET BE A BOUNDED PART OF COUPLING WITH ANOTHER DOMAIN ,ET BE A STRESS lELD DElNED ONAND THE CORRE SPONDING DISPLACEMENT lELD 4HE PRIMAL FORMULATION OF THE LOCAL PROBLEM ASSOCIATED TO LEADS TO THE FOL LOWING LINEAR OPERATOR EQUATION
WHERE IS THE 3TEKLOV0OINCAR|E OPERATOR CORRE SPONDING TO THE CONDENSATION OF THE DYNAMIC STIFFNESS OPERATOR OF THE DOMAIN 4HE DUAL FORMULATION LEADS
TO WHEREIS THE mEXIBILITY OPERATOR FOR MALLY VERIFYING
0ARTICULARLY IMPORTANT IN EARTHQUAKE ENGINEERING AS WELL AS IN MANY OTHER CIVIL ENGINEERING AND AEROSPACE ENGINEERING APPLICATIONS THE COMPUTATION OF THE IMPEDANCES OF UNBOUNDED DOMAINS HAS BEEN EXTENSIVELY STUDIED IN A DETERMINISTIC FRAMEWORK &OR A BOUNDED THIS OPERATOR CAN BE APPROXIMATED WITH lNITE ERROR BY AN IMPEDANCE MATRIX 7OLF 1UANTIlCATION OF UNCERTAINTY ON THESE OPERATORS HAS BEEN ADDRESSED MORE RECENTLY 3CHUELLER -ANO LIS LEADING TO THE FOLLOWING LINEAR STOCHASTIC OPERATOR EQUATION
WHERE IS A STOCHASTIC lELD AND IS THE STOCHASTIC STIFFNESS OPERATOR #ONSIDERING UNCERTAINTY ONLY IN A BOUNDED VOLUME OF &IG IS THEN A PERTURBATION OF A DETERMINISTIC OPERATOR AND THEREFORE ALL REALISATIONS OF CAN BE APPROXI MATED WITH lNITE ERROR ON A COMMON BASIS LEADING TO
WHERE IS THE DISPLACEMENT VECTOR IS AN APPROX IMATION OF THE STOCHASTIC STRESS VECTOR AND IS THE STOCHASTIC STIFFNESS MATRIX
4HIS PAPER PRESENTS EXISTING METHODS TO COMPUTE THE SOIL IMPEDANCE MATRIX SECTION STRESSING THE
'
STRUCTURE
7 D 7
&IGURE BOUNDED UNCERTAIN DOMAIN IN UNBOUNDED DETERMIN ISTIC DOMAIN AND COUPLING BOUNDARY
APPEAL OF THE NONPARAMETRIC METHOD WHICH REQUIRES THE CONSTRUCTION OF A PROBABILISTIC MODEL SECTION BASED ON REAL POSITIVE DElNITE MATRICES AND FOR WHICH CAUSALITY IS ENFORCED 4HE IDENTIlCATION OF THE MEAN MATRICES FOR THIS MODEL IS THEN PRESENTED SECTION ENABLING APPLICATION OF THIS METHOD IN A SIMPLE CASE SECTION
#/-054!4)/.!, -%4(/$3
)N THIS SECTION TWO CLASSES OF COMPUTATIONAL METHODS TO ACCOUNT FOR THE UNCERTAINTY IN ARE PRESENTED THE CLASSICAL PARAMETRIC METHODS AND A MORE RECENT NONPARAMETRIC METHOD
4HE 3TOCHASTIC &INITE %LEMENT -ETHOD 3&%- IS A CLASSICAL TOOL TO COMPUTE #ORNELL ,IKE ITS DETERMINISTIC EQUIVALENT IT REQUIRES THE DISCRETIZA TION OF DOMAINS AND AND THEREFORE THE CREATION OF AN ARTIlCIAL BOUNDARY BRINGING ALONG PROBLEMS OF UNPHYSICAL WAVE REmECTIONS 5SING A &INITE %LEMENT -ETHOD &%- "OUNDARY %LEMENT -ETHOD "%- COUPLING APPROACH TO MODEL THE PROBLEM 3AVIN AND
#LOUTEAU ALLOWS FOR CORRECT CONSIDERATION OF THE UNBOUNDEDNESS OF DOMAIN AND REDUCES THE SIZE OF THE DISCRETIZED DOMAIN (OWEVER IT STILL IMPLIES IN PRACTICAL SITUATIONS A VERY LARGE NUMBER OF VARIABLES AND HIGH COMPUTATIONAL COSTS
"ESIDES THESE CLASSICAL DRAWBACKS OF THE &%- THESE AND OTHER PARAMETRIC METHODS REQUIRE THE IDEN TIlCATION OF THE UNCERTAIN PARAMETERS AND THE QUAN TIlCATION OF THAT UNCERTAINTY THAT IS TO SAY APPROPRI ATE PROBABILISTIC MODELS OF THESE UNCERTAIN PARAME TERS HAVE TO BE CONSTRUCTED BASED ON GIVEN STATISTICS 3UCH PROBLEMS ARE NOT SIMPLE PARTICULARLY IN THE CASE OF SOILS WHERE SOURCES OF UNCERTAINTY ARE NUMEROUS AND MEASUREMENT DIFlCULTIES HINDER THE RECOLLECTION OF ACCURATE DATA &AVRE
4HE PROPAGATION OF THE UNCERTAINTY FROM THE PARAM ETERS TO THE RESPONSE OF THE SYSTEM IS THEN USUALLY PER FORMED VIA -ONTE#ARLO SIMULATIONS LEADING TO PRO HIBITIVE COSTS PARTICULARLY WHEN UNCERTAINTY ON SEV ERAL PARAMETERS HAS TO BE CONSIDERED !LSO SINCE THE CORRELATION BETWEEN THESE PARAMETERS IS DIFlCULT TO AS SESS PHYSICALLY UNSOUND SYSTEMS CAN BE COMPUTED
2ECENTLY 3OIZE INTRODUCED A NONPARAMETRIC METHOD 3OIZE WHERE THE APPLICATION OF THE MAXIMUM ENTROPY PRINCIPLE *AYNES TO THE REDUCED MATRIX MODEL OF A SYSTEM LEADS TO A PROBABILISTIC MODEL USING ONLY THE INFORMATION AVAILABLE 4HIS METHOD IS BASED ON THE DIRECT CONSTRUCTION OF A PROBABILISTIC MODEL OF THE GENERALIZED MASS DAMPING AND STIFFNESS MATRI CES OBVIATING THE IDENTIlCATION OF THE UNCERTAIN LOCAL PARAMETERS AND THE CONSTRUCTION OF THEIR PROBABILIS TIC MODEL #OARSE STATISTICAL STUDIES ON THE PARAMETERS ARE THEREFORE NOT NEEDED AND PHYSICALLY UNSOUND RE SULTS ARE AVOIDED AS LONG AS THE PHYSICS WERE CORRECTLY INTRODUCED IN THE MODEL 4HE NONPARAMETRIC METHOD ALSO ACCOUNTS FOR MODELLING ERRORS
)N THE CASE OF BOUNDED UNCERTAIN DOMAINS LET IT BE THE STIFFNESS MATRIX CAN BE WRITTEN AS A QUADRATIC FUNCTION OF FREQUENCY IN TERMS OF A POSITIVE DElNITE MATRIX OF MASS AND POSITIVE MATRICES OF DAMPING AND STIFFNESS
$ 4HIS ENSURES CAUSALITY OF THE CORRESPONDING MODEL IN THE TIME DOMAIN SINCE EQUATION WITH THE IMPEDANCE IN THE FORM OF IS RELATED TO A SECOND ORDER DIFFERENTIAL EQUATION IN THE TIME DOMAIN 4HE MEAN MATRICES OF THE PROBABILISTIC MODEL ARE IDENTI lED WITH THE MATRICES OF THE DETERMINISTIC MODEL
)N THE CASE OF A BOUNDED UNCERTAIN DOMAIN IN SIDE AN UNBOUNDED DETERMINISTIC DOMAIN THE SAME
&%-"%- COUPLING APPROACH CAN BE USED INTRO DUCING THE UNCERTAINTY IN BY MEANS OF THE NON PARAMETRIC METHOD 4HIS LEADS TO A mEXIBILITY MATRIX IN THE FORM
WITH
WHERE ANDARE MATRICES FOL LOWING FROM THE DISCRETIZATION OF RESPECTIVELY THE TRACES ON BOUNDARY OF OPERATORS AND AND THE RESTRICTIONS ON DOMAIN OF THE SAME OPERATORS 4HESE OPERATORS ARE DElNED IF IS 'REENS FUNCTION OF THE DETERMINISTIC DOMAIN
A VOLUMIC LOAD FUNCTION DElNED ON AND A SURFACIC LOAD FUNCTION DElNED ON BY
DERIVES FROM THE SAME OPERATOR AS BUT IS PROJECTED ON A DIFFERENT BASIS OF FUNCTIONS BE ING THE IMPEDANCE OF DOMAIN IT CAN THEN BE EX PANDED AS IN AND THE NONPARAMETRIC METHOD CAN BE USED TO GENERATE THE MATRICES OF MASS STIFFNESS AND DAMPING AND ULTIMATELY CONDENSATION ONUSING TO GENERATE AND !LTHOUGH SOME DIFlCULTIES HAVE TO BE ADDRESSED 3OIZE AND #HEBLI SUCH AS THE EXISTENCE OF RIGID BODY MODES WHICH TAKE DOWN THE POSITIVE DElNITENESS OF MATRI CES OF DAMPING AND STIFFNESS THIS APPROACH IS FEASIBLE 5NFORTUNATELY THE COMPUTATIONAL COST IS NOT LOWER THAN THAT OF THE 3&%-"%- METHOD !LL INTER NAL DEGREES OF FREEDOM $/&S OF THE &INITE %LEMENT MODEL ARE CONSIDERED WHEN ALL IS NEEDED IS THEIR TRACE ON BOUNDARY
4HE CONSTRUCTION OF A PROBABILISTIC MODEL DIRECTLY FOR THE SOIL IMPEDANCE WOULD ADD TO THE ADVANTAGES OF THE NONPARAMETRIC METHOD AN IMPORTANT REDUCTION IN COMPUTATIONAL COSTS !S CANNOT BE EX PANDED AS IN IN THE CASE OF AN UNBOUNDED DOMAIN A CAUSAL REDUCED MODEL OF THE SOIL IMPEDANCE HAS TO BE CONSTRUCTED AND THE IDENTIlCATION OF THE MEAN MA TRICES FOR THIS MODEL HAS TO BE PERFORMED
02/"!"),)34)# -/$%, &/2 4(% 3/), )-0%$!.#%
4HE NONPARAMETRIC METHOD IS BASED ON THE POSSIBILITY OF GENERATING THE ANALYTICAL PROBABILITY DENSITY FUNC TION OF A FREQUENCY INDEPENDENT REAL POSITIVE DElNITE OR POSITIVE MATRIX GIVEN ITS MEAN AND A CERTAIN DIS PERSION PARAMETER 4O BE ABLE TO APPLY THIS METHOD THE MODEL FOR THE SOIL IMPEDANCE MATRIX MUST THEN BE COMPOSED ONLY OF SUCH MATRICES AND AS STATED IN SEC TION CAUSALITY HAS TO BE ENFORCED IN ORDER TO BOUND UNPHYSICAL RESULTS 4HREE METHODS ARE DESCRIBED HERE AFTER BEGINNING WITH THE +RAMERS+RONIG RELATIONS WIDELY USED IN EXPERIMENTAL PHYSICS
)NITIALLY DEVELOPPED FOR ELECTROMAGNETIC PROBLEMS TO LINK THE REAL AND IMAGINARY PARTS OF THE COMPLEX SUS CEPTIBILITY +RAMERS AND OF THE COMPLEX REFRAC TION INDEX +RONIG THE +RAMERS+RONIG RELA TIONS HAVE LATER BEEN RECOGNIZED A WIDER RANGE OF AP PLICATION "EING BUILT SOLELY ON CAUSALITY THEY HAVE TO
BE VERIlED BY THE FREQUENCY RESPONSE FUNCTION &2&
OF ANY PHYSICAL SYSTEM IN THE PRESENT CASE AND STATE THAT
OR EQUIVALENTLY
WHERE AND ARE RESPECTIVELY THE
REAL AND IMAGINARY PARTS OF AND REFERS TO
#AUCHYS PRINCIPAL PART
)N MANY APPLICATIONS CAN BE MEASURED EXPERIMENTALLY AND EQUATION CAN THEN BE USED TO RECONSTRUCT BUT NUMERICALLY IS OF NO EASIER ACCESS THAN OR FOR WHICH THE +RAMERS+RONIG RELATIONS ARE OF NO HELP TO CON STRUCT THE PROBABILITY MODEL OF
!NOTHER METHOD TO ENFORCE THE CAUSALITY OF THE IMPEDANCE IS TO EXPAND IT IF POSSIBLE ON A BASIS OF CAUSAL FUNCTIONS LIKE THAT OF THE (ARDY FUNCTIONS 0IERCE IS SAID TO BE A (ARDY FUNCTION ON THE UPPER HALF PLANE IF AND ONLY IF IT IS THE ,APLACE TRANS FORM OF SOME CAUSAL FUNCTION &UNCTIONS FOR"
FORM AN ORTHONORMAL BASIS OF THE SPACE OF (ARDY FUNC TIONS 4HEREFORE ANY CAUSAL MATRIX CAN BE SOUGHT ON A BASIS OF (ARDY FUNCTIONS
5NFORTUNATELY VERY LITTLE IS KNOWN ON THE PROPERTIES OF THE FOR" SUCH AS POS ITIVE DElNITENESS IMPEDING APPLICATION OF THE NON PARAMETRIC METHOD !LSO THE NUMBER OF TERMS RE QUIRED IN THE RIGHTHAND SIDE OF TO OBTAIN A COR RECT APPROXIMATION OF MIGHT BE IMPORTANT
5LTIMATELY IS SOUGHT AS THE CONDENSATION ON
OF A MECHANICAL SYSTEM GOVERNED BY A SECOND OR DER DIFFERENTIAL EQUATION WITH CONSTANT COEFlCIENTS !S THIS IS IN GENERAL UNTRUE IF SUCH A STRUCTURE IS TO BE RE TAINED TO ENSURE CAUSALITY HIDDEN VARIABLES HAVE TO BE INTRODUCED WHICH WILL BE LINKED ONLY INDIRECTLY TO THE INTERNAL $/&S OF THE SYSTEM #HABAS AND 3OIZE
4HE SYSTEM IS THEREFORE DISCRETIZED IN $/&S ON THE BOUNDARYAND HIDDEN STATE VARIABLES 4HIS LAST NUMBER HAS TO BE ACCURATELY CHOSEN SO THAT THE MODEL CAN ACCOUNT FOR THE VARIATIONS OF THE &2& 4HE TOTAL NUMBER OF $/&S OF THE DISCRETIZATION IS 4HE IMPEDANCE OF THIS SYSTEM CAN BE EXPANDED AS IN IN TERMS OF REAL POSITIVE DElNITE MATRICES OF MASS DAMPING AND STIFFNESS 4HE IMPEDANCE CAN BE BLOCK DECOMPOSED IN
WHERE FOR
IN AND ARE
REAL POSITIVE DElNITE MATRICES AND
REAL MATRICES AND AND
REAL POSITIVE DElNITE MATRICES #ONDENSATION ONTHEN LEADS TO
!SSUMING AS USUALLY DONE THAT IS DIAGONAL IZED BY THE EIGENVECTORS SOLUTIONS OF THE GENERALIZED
EIGENVALUE PROBLEM CAN
BE WRITTEN
)N OTHER WORDS THE BOUNDARY IMPEDANCE OF THIS ME CHANICAL SYSTEM HAS THE FORM
WHERE AND ARE TWO POLYNOMI
ALS OF FREQUENCY WITH CONSTANT COEFlCIENTS MATRI CIAL FOR AND SCALAR FOR 4HE ORDERS OF AND
VERIFY AND
4HIS FORMULATION ENSURES THE CAUSALITY OF AS THEN CORRESPONDS IN THE TIME DOMAIN TO A DIF FERENTIAL EQUATION WITH CONSTANT COEFlCIENTS AND THE IMPEDANCE IS COMPUTED USING ONLY THE REAL POSI TIVE DElNITE MATRICES AND 4HE NON PARAMETRIC METHOD CAN THEN BE APPLIED TO THIS MODEL TO OBTAIN A PROBABILISTIC MODEL OF THE SOIL IMPEDANCE MATRIX PROVIDED THAT THE MEAN MATRICES
AND
CAN BE IDENTIlED
)$%.4)&)#!4)/. /& 4(% -%!. -/$%,
!S FOR THE NONPARAMETRIC METHOD APPLIED TO REDUCED
&INITE %LEMENT UNCERTAIN MODELS WHERE THE MEAN MA TRICES ARE IDENTIlED WITH THE MATRICES OF THE DETERMIN ISTIC MODEL THE MEAN MATRIX OF THE SOIL IMPEDANCE
&IGURE 4HE REAL PART OF THE XSWAY ELEMENT OF THE IMPEDANCE MATRIX OF A HOMOGENEOUS HALF SPACE WITHOUT HETEROGENEITY AND WITH BOUNDED HETEROGENEITY ALL -ONTE#ARLO TRIALS AND
MEAN OF THESE TRIALS
&IGURE 4HE IMAGINARY PART OF THE XSWAY ELEMENT OF THE IMPEDANCE MATRIX OF A HOMOGENEOUS HALF SPACE WITHOUT HETERO GENEITY AND WITH BOUNDED HETEROGENEITY ALL -ONTE#ARLO TRIALS ANDMEAN OF THESE TRIALS
WILL BE ASSIMILATED TO THE SOIL IMPEDANCE MATRIX OF THE DOMAIN WITHOUT THE UNCERTAIN DOMAIN !L THOUGH NOT MATHEMATICALLY SOUND THIS HYPOTHESIS IS PHYSICALLY APPEALING AND THE COMPUTATIONS PERFORMED WITH THE PARAMETRIC 3&%-"%- METHOD DESCRIBED IN SECTION SUSTAIN IT &IG
4HE IDENTIlCATION OF THE MEAN MATRIX OF THE SOIL IMPEDANCE WITH THE SOIL IMPEDANCE MATRIX OF DE TERMINISTIC SOIL CONSISTS IN lNDING AND WHICH MINIMIZE
WHERE THE FOR ARE THE FREQUENCIES AT WHICH THE DETERMINISTIC IMPEDANCE HAS BEEN
COMPUTED AND IS THE CONDENSATION ONOF THE
MEAN MATRIX
GIVEN BY 4HIS OPTIMIZATION PROBLEM IS IN GENERAL NONLINEAR IN THE PARAMETERS
"EFORE STUDYING THE GENERAL CASE OF IDENTIFYING THE MEAN IMPEDANCE MATRIX FOR"HIDDEN STATE VARIABLES THE SIMPLE CASE WHERE THE HALF SPACE WITH THE UNCER TAIN DOMAIN IS MODELED BY A MASS SPRING DASH POT SYSTEM IS PRESENTED
"
)N THAT CASE
AND THE MINIMIZATION PROBLEM BECOMES LINEAR IN THE PARAMETERS !N EXACT SOLUTION FOR
AND
CAN THEN BE FOUND
WHERE
AND
" "
7ITH THE INTRODUCTION OF HIDDEN VARIABLES THE MINI MIZATION OF BECOMES A NONLINEAR PROBLEM -ANY DIFFERENT METHODS EXIST FOR THE RESOLUTION OF SUCH PROBLEMS (EYLEN ET AL MOST OF THESE DEPEND ING ON AN INITIAL VALUE THAT HAS TO BE GUESSED TO BE GIN THE OPTIMIZATION PROCESS 4HIS INITIAL VALUE CAN BE SOUGHT USING A LINEARIZED FORM OF
WHERE FOR AND ARE DElNED
AS IN FOR 4HE IDENTIlCATION ON A BASIS OF ORTHOGONAL POLYNOMIALS 0INTELON ET AL
"ULTHEEL AND 6AN "AREL IS PARTICULARLY ADAPTED 4HE MINIMIZATION PROCESS IS THEN COMPLETED BY A CLAS SICAL NONLINEAR OPTIMIZATION PROBLEM STARTING FROM THE VALUE COMPUTED THROUGH MINIMIZATION OF
%8!-0,%
4HE APPLICATION OF THE NONPARAMETRIC METHOD TO COM PUTE THE PROBABILISTIC MODEL OF THE SOIL IMPEDANCE MATRIX IS PERFORMED IN FOUR STEPS
$ETERMINISTIC IS COMPUTED USING CLASSI CAL COMPUTATIONAL TOOLS
4HE MEAN MATRICES
AND
OF THE PROBABILISTIC MODEL ARE IDENTIlED USING THE RE SULTS OF SECTION
'IVEN A DISPERSION PARAMETER THE MAXIMUM ENTROPY PRINCIPLE GIVES THE PROBABILITY DENSITY FUNCTION OF MATRICES AND 5SING -ONTE#ARLO TRIALS THE REALISATIONS OF MA
TRIX ARE COMPUTED AND THE MOMENTS DE RIVED
5SING THIS METHODOLOGY FOR THE CASE OF A SUPERl CIAL FOUNDATION ON A HOMOGENOUS HALF SPACE THE RE SULTS FOR A PARAMETRIC METHOD ARE SHOWN IN &IG AND WITH A DISPERSION FACTOR FOR ALL MATRICES AND CONSIDERING NO HIDDEN VARIABLES THE FOLLOWING MEANS AND TYPICAL DEVIATIONS CAN BE COMPUTED FOR -ONTE#ARLO TRIALS
!
! #
! #
4HESE VALUES HAVE TO BE COMPARED TO THE VALUES OB TAINED WITH THE PARAMETRIC METHOD
!
! #
! #
4HE ONLY SIGNIlCANT DIFFERENCE LIES IN THE MASS TYP ICAL DEVIATION AND IS DUE TO THE SIMPLICITY OF THE MODEL USED NO HIDDEN VARIABLES #ONSIDERING EACH -ONTE#ARLO TRIAL OF THE IMPEDANCE MATRIX INDEPEN DENTLY THE IDENTIlCATION IN TERMS OF MASS DAMPING AND STIFFNESS MATRICES FOR THE PARAMETRIC METHOD LEADS IN SOME CASES TO A NON POSITIVE DElNITE MATRIX OF MASS 3INCE THESE REALISATIONS ARE OUT OF REACH FOR THE NONPARAMETRIC METHOD THE RESULTS ARE NECESSAR ILY CONDENSED CLOSER AROUND THE MEAN VALUE (IDDEN STATE VARIABLES ARE REQUIRED IN ORDER TO TAKE INTO AC COUNT MORE PRECISELY THE PHYSICS OF THE IMPEDANCE MATRIX AND THEREFORE ACCOUNT FOR ITS VARIATIONS USING ONLY POSITIVE DElNITE MATRICES 7HEN CONSIDERING A LAYERED HALF SPACE THIS NEED FOR MORE $/&S WILL BE COME EVEN MORE CRITICAL
#/.#,53)/.3
4HE NONPARAMETRIC METHOD PRESENTED IN THIS PAPER ALLOWS FOR THE CONSTRUCTION OF A PROBABILISTIC MODEL OF THE SOIL IMPEDANCE MATRIX IN AN OBJECTIVE MANNER )T DOES NOT REQUIRE A PREVIOUS IDENTIlCATION OF THE UNCER TAIN PARAMETERS AND THE CONSTRUCTION OF A PROBABILISTIC MODEL FOR EACH OF THEM AS STATISTICAL DATA IS USUALLY SCARCELY AVAILABLE AND IT ACCOUNTS FOR MODELLING ER RORS #OMPARED TO OTHER POSSIBLE IMPLEMENTATIONS OF THE NONPARAMETRIC METHOD IN UNBOUNDED DOMAINS IT ACHIEVES A DRAMATIC REDUCTION IN COMPUTATIONAL TIME
!#+./7,%$'%-%.43
4HIS WORK HAS BEEN SUPPORTED BY |%LECTRICIT|E DE &RANCE 2ESEARCH $EVELOPMENT TO WHICH THE AUTHORS ARE VERY THANKFUL
2%&%2%.#%3
"ULTHEEL ! 6AN "AREL - 6ECTOR ORTHOGONAL POLY NOMIALS AND LEAST SQUARES APPROXIMATION
6OL .O n
#HABAS & 3OIZE #HRISTIAN -ODELING MECHANICAL SUBSYSTEMS BY BOUNDARY IMPEDANCE IN THE lNITE ELE MENT METHOD
n
#ORNELL ! # &IRST ORDER UNCERTAINTY ANALYSIS OF SOILS DEFORMATION AND STABILITY
(ONG +ONG n
&AVRE *, %RRORS IN GEOTECHNICS AND THEIR IMPACT ON
SAFETY 6OL .O n
(EYLEN 7ARD ,AMMENS 3TEFAN 3AS 0AUL ,EUVEN "ELGIUM +ATHOLIEKE 5NIVERSITEIT ,EUVEN
*AYNES % 4 )NFORMATION THEORY AND STATISTICAL MECHAN
ICS 6OL .O n
+RAMERS ( ! ,A DIFFUSION DE LA LUMI{ERE PAR LES
ATOMES 6OL
UME n
+RONIG 2 DE , /N THE THEORY OF DISPERSION OF XRAYS 6OL .O n
-ANOLIS ' $ 3TOCHASTIC SOIL DYNAMICS n
0IERCE , " (ARDY FUNCTIONS *UNIOR PAPER 0RINCETON 5NIVERSITY HTTPWWWPRINCETONEDU LBPIERCE 0INTELON 2 2OLAIN 9 "ULTHEEL ! 6AN "AREL -
&REQUENCY DOMAIN IDENTIlCATION OF MULTIVARIABLE SYS TEMS USING VECTOR ORTHOGONAL POLYNOMIALS
3AVIN % #LOUTEAU $ #OUPLING A BOUNDED DO MAIN AND AN UNBOUNDED HETEROGENEOUS DOMAIN FOR ELAS TIC WAVE PROPAGATION IN THREEDIMENSIONAL RANDOM ME DIA
n
3CHUELLER ' ) ! STATEOFTHEART REPORT ON COMPU TATIONAL STOCHASTIC MECHANICS
6OL .O n
3OIZE #HRISTIAN ! NONPARAMETRIC MODEL OF RANDOM UN CERTAINTIES FOR REDUCED MATRIX MODELS IN STRUCTURAL DY
NAMICS n
3OIZE # #HEBLI ( 2ANDOM UNCERTAINTIES MODEL IN DYNAMICS SUBSTRUCTURING USING A NONPARAMETRIC PROBABILISTIC MODEL
6OL .O n
7OLF * 0 %NGLE
WOOD #LIFFS . * 0RENTICE(ALL )NC
