1、AN AMERICAN NATIONAL STANDARDASME V however, they shouldnot contain proprietary names or information.Requests that are not in this format may be rewritten in the appropriate format by the Committeeprior to being answered, which may inadvertently change the intent of the original request.ASME procedu
2、res provide for reconsideration of any interpretation when or if additionalinformation that might affect an interpretation is available. Further, persons aggrieved by aninterpretation may appeal to the cognizant ASME Committee or Subcommittee. ASME does not“approve,” “certify,” “rate,” or “endorse”
3、any item, construction, proprietary device, or activity.Attending Committee Meetings. The VSQE will not be discussed in this Standard. Calculationverification is defined as the process of determiningthe solution accuracy of a particular calculation. Bothnumerical code verification and calculation ve
4、rificationwill be demonstrated by applying a simple beam ele-ment code to successively more finely meshed modelsof statically loaded beams.4.2 ValidationValidation is defined as the process of determining thedegree to which a computational model is an accuraterepresentation of the real world from th
5、e perspective ofthe intended uses of the model. One of the steps inthe validation process is a comparison of the resultspredicted by the model with corresponding quantitiesASME V failures often teach more thansuccesses.V however, for conciseness only the tip deflectionis considered here.For the plan
6、ned experiments, tapered beams are tobe embedded at their wide endinto a stiff fixture approx-imating a “fixed-end” or cantilevered boundary condi-tion. The beams are to be loaded continuously along theouter half of their lengths.During the experiment planning, it was acknowl-edged that the “fixed-e
7、nd” boundary condition can onlybe approximated in the laboratory. Thus in the modeldevelopment the translational constraint at the bound-ary will be assumed fixed, but the rotational constraintwill be assumed to vary linearly with the magnitude ofthe moment reaction.Additionally, the beam model to b
8、e developed isassumed to have negligible shear deformation, and thusshear deformation is ignoredin the mathematical model.For the prescribed magnitude of the loading, the deflec-tion of the beam will be small relative to beam depth,so a small-displacement theory will be used, and thebeam material is
9、 assumed to be linear elastic.These assumptions feed directly into the conceptualmodel of the physical structure, which will be definedprecisely in para. 6.1, and which guides both the devel-opment of the validation experiments and the definitionof the mathematical model.5.2 Verification Requirement
10、sIn this example, both code and calculation verificationwill be performed. The requirements for code verifica-tion are as follows:(a) It is conducted using the same system responsequantities as will be measured and used for validation.(b) It demonstrates that the numerical algorithm con-verges to th
11、e correct solution of a problem closely relatedto the reality of interest as the grid is refined. This canbe difficult or impractical in many cases, but without it,the code is not verified.6(c) It demonstrates that the algorithm converges atthe expected rate.The requirement for calculation verificat
12、ion in generalis to demonstrate that the numerical error (due to incom-plete spatial or iterative convergence) in the SRQs ofinterest be a small fraction of the validation requirement.In this example, the validation requirement will be 10%,and the numerical error is required to be no greater than2%
13、of that (i.e., 0.2%).5.3 Validation Approaches, Metrics, andRequirementsTwo different validation approaches are demonstratedin this Standard. They differ mainly in the source ofinformation used to quantify the uncertainties in thecomputed and measured values of the SRQ. The V the CDF is the integral
14、 of the PDF). This metric,sometimes referred to as the “area” metric, is illustratedin Fig. 6, and more detail about it is given below.The area metric MSRQis the area between the experi-ment and model CDF 3, normalized by the absolutemean of the experimental outcomes. Thus, if FSRQ(y)isthe CDF of ei
15、ther the model-predicted or measured SRQvalues, thenMSRQp1H20904SRQexpH20904H20885H11557H11557H20904FSRQmod(y)FSRQexp(y)H20904dy (1)whereSRQexpp the mean of the experimental outcomesThis metric is nonnegative and vanishes only if thetwo CDFs are identical. To help understand what themetric represent
16、s, it can be shown that in the specialcase where the two CDFs do not cross, the integral ineq. (1) is the absolute value of the difference betweenthe means, and in general, it is a lower bound on themean of the absolute value of the difference betweenSRQmodand SRQexp4. In the deterministic case, whe
17、reboth CDFs are step functions, the area is simply theabsolute value of the difference between the two uniquevalues.For both validation approaches in this example, thevalidation requirement is taken asMSRQ 0.1 (2)Obviously, satisfaction of a particular validationrequirement is the desired outcome of
18、 the validationassessment. However, the V therefore, beam deflectionsoccur in a plane.(e) The beam boundary constraint is fixed in transla-tion and constrained against rotation by a linear rota-tional spring.6.2 Mathematical ModelThe mathematical model uses the information fromthe conceptual model,
19、including idealizing assumptionsconcerning the behavior of the beam, to derive equationsgoverning the structures behavior. For the beam consid-ered here, the assumptions listed when defining theconceptual model in para. 6.1 combine to yield the equa-tions of static BernoulliEuler beam theory:d2dx2H2
20、0898EI(x)d2dx2w(x)H20899p q(x), 0 x L,w(0) pdwdxH20904xp0p frEI(0)d2wdx2H20904xp0,H20900EI(x)d2dx2w(x)H20901H20904xpLp 0,(3)ddxH20900EI(x)d2dx2w(x)H20901H20904xpLp 0,I(x) p112H20902b0H208981H9251xLH20899h3H20900b0H208981H9251xLH208992tH20901H20851h 2tH208523H20903whereb0p width at the supportASME V
21、rather, thedifferential equations are documented in the softwaresuser or theory manual (e.g., in descriptions of variousbeam element types and options). Inthat case, it is highlyrecommended that the analyst review those models,equations, and assumptions (given the options chosenin the code) to ensur
22、e they are consistent with theintended use of the model. Without carefully consider-ing these equations, an error or inconsistency in themathematical modeling can easily occur.6.3 Computational ModelThe computational model provides the numericalsolution of the mathematical model, and normally doesso
23、 in the framework of a computer program. The rangeof discretization approaches (e.g., finite element, finitedifference) and options within each approach in com-mercial software is often extensive. The analyst needsto find a balance between representing the physicsrequiredbytheconceptualmodelandtheco
24、mputationalresources required by the resulting computationalmodel. For example, finite element type options for themathematical/computational model for the airfoil alu-minum skin would include the following:(a) continuum elements: use solid elements throughthe thickness of the aluminum skin(b) shell
25、 elements: plane stress assumption throughthe skin thickness10(c) plate elements: same as shell element but omitsurface curvatures(d) hollow-section beam elements: strains areassumed to be primarily axial and torsional(e) closed-section beam elements: equivalent constantcross-sectional properties of
26、 a solid section(f) uniform closed-section beam elements: averagecross section properties along lengthIn this example, the computer code used to solve forthe beam deflections and rotations was specially writtenfor this application example. It can be used only to ana-lyze beam structures. It is a fin
27、ite element programemploying BernoulliEuler beam elements with con-stant cross section. If the element itself is loaded by anycombination of uniform distributed load, transverse endforces, and couples at the ends, then the relative dis-placements and rotations at the ends are exact in thecontext of
28、BernoulliEuler beam theory. On the otherhand, when used to model a tapered beam, these relativedeformations are only approximations.The beam considered here is shown schematically inFig. 3. The length of the beam is 2 m, the depth is 0.05 m,the width varies linearly from 0.20 m at the supportedend t
29、o 0.10 m at the free end, and the wall thickness is0.005 m. The material is aluminum, with a modulus ofelasticity of 69.1 GPa. A uniform distributed load of500 Nm is applied vertically in the downward directionon the outer half of the beam.7 VERIFICATIONCode verification seeks to ensure that there a
30、re noprogramming errors and that the code yields the accu-racy expected of the numerical algorithms used toapproximate the solutions of the underlying differentialequations. This is in contrast with calculation verification,which is concerned with estimating the discretizationerror in the numerical
31、solution of the specific problem ofinterest. The distinction is subtle but important, becausecode verification requires an independent, highly accu-ratereferencesolutionandcan (andusuallywill)operateon a problem that is different from the problem ofinterest.What is important in code verification is
32、that all por-tions of the code relevant to the problem at hand befully exercised to ensure that they are mistake free. Thisis done by comparing numerical results with analyticalsolutions, and in the process, confirming that the numer-ical solution converges to the exact one at the expectedrate as th
33、e mesh is refined.At the root of both code and calculation verificationis the concept of the order of accuracy of a numericalalgorithm. Under h-refinement (variation of elementsize, as contrasted with p-refinement or variation of alge-braic order of interpolation functions), it is defined asthe expo
34、nent p in the power series expansionASME V further, it can be shownanalytically that the theoretical order of accuracy of thenumerical algorithm when applied to this specific prob-lem is in fact 2. First consider the results of the initialcoding. The error plot strongly implies that the resultsare s
35、ystematically converging to the exact solution, buteven without performing further calculations, the slopeof the line indicates an order of accuracy much closerto 1 than 2. During the early stages of the developmentof this example, these were the computed results. Theunexpectedly low observed order
36、of accuracy prompteda detailed review of the coding, and an error was found.Upon correction of that error, the second set of resultswas obtained. Now the observed order of accuracy asindicated by the slope of the error plot is seen to bemuch closer to the theoretical value of 2. The error wasuninten
37、tional, and could not possibly have provided abetter illustration of the value of numerical codeverification.It now remains to extract a precise numerical estimatefor the observed order of accuracy of the correctednumerical solution. This is accomplished by computingthe logarithmic slope between the
38、 last two points on theASME Vthe second uses only two but requires an assumed valuefor the order of convergence. Both are based onRichardson extrapolation 5, which will now be explained.As in the prior section, it is assumed that the numericalvalue of the quantity of interest is related to the exact
39、ASME V H9268expp0.753p 0.25 mm(14)H9262mod(pwmod) p 14.2 mm; H9268modp0.713p 0.24 mmThese inputs are now used to compute the validationmetrics. As shown in Fig. 8, the area between the CDFswas computed to be 0.8 mm, so according to eq. (1)the area metric is MASRQp 5.3%. (In keeping with thepropertie
40、s listed in para. 5.3 and that the two CDFs crossonly near one extreme where they are practically equal,this value is almost exactly the absolute relative differ-ence of the means.) Because the validation metric fallswithin the 10% requirement, the model is assessed asvalid.9 VALIDATION APPROACH 2Va
41、lidation Approach 2 considers the case whereuncertainty data are available, and employs a straight-forward probabilistic analysis to relate model inputuncertainties to the model output uncertainty. The areametric is then used to assess the validity of the model.9.1 Validation ExperimentsFor validati
42、ng a model, multiple replications of thevalidation experiment are highly recommended. Thiswill account for inevitable variations that exist in thetest article fabrication, experimental setup, and mea-surement system. For any application, the questions that15must be addressed and answered in the V&V
43、Planinclude the following:(a) How many experiments should be conducted?(b) Should different technicians or subcontractorsassemble the system to be tested?(c) Should different experimental facilities conductthe experiment?These types of questions must be answered on a case-by-case basis.In this examp
44、le the data used to characterize theresponse of the system come from validation experi-mentson 10nominally identicalbeams, constructedwiththe nominal dimensions listed in para. 6.3.After constructing each beam, the same proceduredescribed in para. 8.2 concerning measurements relativeto the gravitati
45、onal equilibrium conditions of the beamis used. The uncertainty in the response of the beam isdue to random and systematic uncertainty in the multi-ple experimental measurements as well as variabilityin the properties of the test article. In this example,properties variations are assumed to be confi
46、ned to themodulus of elasticity in the material used to constructeach beam, and the support flexibility. These indepen-dentsources ofuncertaintycan beseparated usingstatis-tical design of experiment techniques 6, 7.Uncertainty in the experimental measurements willbe due to a number of random and sys
47、tematic uncertain-ties. Some examples of random uncertainty are trans-ducer noise, attachment of individual transducers, andsetup and calibration of all of the instrumentation foreach test. Examples of systematic uncertainties are cali-bration of the transducers, unknown bias errors in theexperiment
48、al procedures, and unknown bias errors inthe experimental equipment. The measured displace-ments in the validation experiment are denoted aswiexp,ip1, ., 10. The measurements are given in Table 3.These data can be used to compute the sample meanand standard deviation of the experimental tipdeflectio
49、ns:w expp110H2085810ip1wexpip 15.4 mm(15)H9268exppH209061101H2085810ip1H20849wexpi w expH208502p 0.57 mmFor use in the area metric, an empirical CDF can beconstructed from the validation experiment data. Thedata listed in Table 3 are sorted in ascending order anda probability value of i/N is assigned to each of the datapoints. The empirical CDF is shown in Fig. 9. This CDFis “stair-stepped” because of the finite number of datapoints, each with a single associated probability.9.2 Model Uncertainty QuantificationUncertainty quantification provides the basis forqua
