1、AN AMERICAN NATIONAL STANDARDASME V however, each section of this Standard may also be viewed as a standalone presentation on each of the relevant topics. The intent of this document is validation in which uncertainty is determined for both the experimental data and the simulation of the experiment.
2、 However, the material in Sections 2, 3, and 4 can be studied independently of the remain-der of the document as they are important in their own right. A readers guide follows:Section 1 presents an introduction to the concepts of verifi cation and validation, the defi nitions of error and uncer-tain
3、ty, and the introduction of the overall validation methodology and approach as defi ned in this Standard. The key concepts of this Section are the validation comparison error and the validation standard uncertainty. It is shown that validation standard uncertainty is a function of three standard unc
4、ertainties associated with errors due to numerical solution of the equations, due to simulation inputs, and due to experimental data. Section 2 presents two key topics:(a) the details of a method for code verifi cation based on the technique of the method of manufactured solutions(b) the details of
5、a method for solution verifi cation based on the technique of the Grid Convergence Index (an exten-sion of Richardson Extrapolation).The outcome of Section 2 is a method for estimating the standard uncertainty associated with numerical errors.Section 3 presents two different approaches for estimatin
6、g the standard uncertainty associated with errors in simu-lation input parameters. One approach evaluates response of the simulation or system in a local neighborhood of the input vector, while the other approach evaluates response in a larger global neighborhood. The fi rst approach is com-monly re
7、ferred to, for example, as the sensitivity coeffi cient method, and the second approach is generally referred to as the sampling or Monte Carlo method. Section 4 presents a brief overview of the method presented in the ASME PTC 19.1-2005 Test Uncertainty standard for estimating uncertainty in an exp
8、erimental result. At the conclusion of this Section, the reader will have methods for estimating the key uncertainties required to complete a validation assessment.Section 5 presents two approaches for estimating the validation standard uncertainty given the estimates of uncer-tainty associated with
9、 numerical, input, and experimental data errors as developed in the three previous sections. At the conclusion of this Section, the reader will have the necessary tools to estimate validation standard uncertainty and the error associated with the mathematical model.Section 6 presents a discussion of
10、 the interpretation of the key validation metrics of validation comparison error and validation uncertainty. It is shown that the validation comparison error is an estimate of the mathematical model error and that the validation uncertainty is the standard uncertainty of the estimate of the model er
11、ror.Section 7 summarizes the methods presented in the previous sections by implementing them in a comprehensive example problem working through each element of the overall procedure and results in a complete validation assess-ment of a candidate mathematical model. Finally, several appendices are in
12、cluded in this Standard. Some are considered as part of the Standard and are iden-tifi ed as mandatory appendices. Other included appendices are considered as nonmandatory or supplementary and are identifi ed as such.ASME V however, they should not contain proprietary names or information.Requests t
13、hat are not in this format will be rewritten in this format by the Committee prior to being answered, which may inadvertently change the intent of the original request.ASME procedures provide for reconsideration of any interpretation when or if additional information that might affect an interpretat
14、ion is available. Further, persons aggrieved by an interpretation may appeal to the cognizant ASME Committee or Subcommittee. ASME does not approve, certify, rate, or endorse any item, construction, proprietary device, or activity.Attending Committee Meetings. The V estimates must be made of the sta
15、ndard uncertainties in all input parameters that contribute to uinputand of the stan-dard uncertainties in the experiment that contribute to uD. Code verifi cation and solution verifi cation processes, discussed in Section 2, result in estimation of unum. Code verifi cation is the process of determi
16、ning that a code is mathematically correct for the simulations of interest (i.e., it can converge to a correct continuum solution as the discretization is refi ned). Code verifi cation involves error evaluation from a known benchmark solution. Solution verifi cation is the process of estimating nume
17、rical uncer-tainty for a particular solution of a problem of interest. Solution verifi cation involves error estimation rather than evaluation from a known benchmark solution.Techniques for estimation of uinput, the standard uncer-tainty in the solution S due to the standard uncertainties in the sim
18、ulation input parameters, are presented in Sec-tion 3. Obviously, estimates of the standard uncertainties of all of the input parameters are required. Then uinputis determined from propagation by either of the following:(a) using a sensitivity coeffi cient (local) method that requires estimates of s
19、imulation solution sensitivity coeffi cients(b) using a Monte Carlo (sampling, global) method that makes direct use of the input parameter standard uncertainties as standard deviations in assumed parent population error distributionsThe standard uncertainty in the experimental result uDis determined
20、 using well-accepted techniques 24, 9 de-veloped by the international community over a period of decades and is discussed in Section 4 of this document. The estimate uDis the standard uncertainty appropriate for D.It includes all effects of averaging, includes all random and systematic uncertainty c
21、omponents, and includes effects of any correlated experimental errors and any other factors that infl uence D and uD. As explained previously, when D and uDare used in the validation comparison any random uncertainty components have been fossilized and uDis a systematic standard uncertainty.The esti
22、mation of uvalfor a range of practical V that is the concern of validation. Note, however, that the solution and its error estimation from a solution verifi cation will be used in the validation process. In this way, code veri-fi cation, solution verifi cation, and validation are cou-pled into an ov
23、erall process for assessing the accuracy of the computed solution.The verifi cation methods discussed in this Section are specifi c to grid-based simulations. These include primarily fi nite difference, fi nite volume, and fi nite el-ement methods in which discrete grid intervals are de-fi ned betwe
24、en computational nodes. The grids may be unstructured or structured (including nonorthogonal 5The term “solution verifi cation” is used in this Standard; in other references the term “calculation verifi cation” is also used inter-changeably with “solution verifi cation” and is the equivalent term us
25、ed by Freitas 2 and in the ASME V it is also necessary that the solution structure be suffi ciently complex that all terms in the governing equation(s) of the code being tested are ex-ercised. A perception may exist, and has often been stated in research journal articles, that general accuracy verif
26、i ca-tion of codes for diffi cult problems (e.g., the full Navier-Stokes equations of fl uid dynamics) is not possible because exact solutions exist only for relatively simple problems that do not fully exercise a code. This percep-tion has resulted in a haphazard and often piecemeal approach to cod
27、e verifi cation. In actuality, there exists a systematic approach based on grid convergence tests that is both tractable and effective (subsection 2-3.3). Some modeling approaches such as large eddy simula-tion (LES) and direct numerical simulation (DNS) may pose some challenges to the use of grid c
28、onvergence for assessing code accuracy, but fundamentally the ap-proach discussed in this standard may be applied (see subsection 2-5 for an additional discussion). 6Dynamic grid methods include adaptive, Lagrangian, or arbitrary Lagrangian Eulerian. Free Lagrangian methods such as discrete vortex a
29、nd discrete element methods may also use the approach defi ned in this Section, where the Lagrangian markers and initial distribution can be viewed as analogous to a grid dis-tribution. Based on the initial distribution of Lagrangian markers, a refi nement strategy may be deployed to determine “grid
30、” conver-gence order and an assessment of uncertainty.ASME V detailed examples of the implementation of the method are given in Nonmandatory Appendix A for a heat conduction problem.As noted previously, Code Verifi cation requires an exact, analytical solution to a nontrivial problem that covers the
31、 same options as the problem to be eventually addressed with the verifi ed code. The formulation of an exact, ana-lytical solution may seem diffi cult for nonlinear systems of PDEs, but in fact it is relatively easy. MMS starts at the end, with a suffi ciently complex solution form (e.g., hy-perboli
32、c tangents or other transcendental functions). A linear solution, however, would not exercise the terms in our PDEs. Also, tanh is easily evaluated and differen-tiated, and contains all orders of derivatives (other func-tional forms also possess this attribute). One can use tanh, or another nonphysi
33、cal analytical solution, or a physically realistic solution (an approximate solution to a physical problem) in the MMS method as long as suffi cient com-plexity is embedded in the functional form.2-3.3.1 Simple 1-D Example of MMS. To emphasize the generality of the MMS concept, as in references 4, 6
34、, 7 the example solution is selected before the governing equa-tions are specifi ed. Then the same solution may be used for different problems, where the problem consists of a set of ASME V other examples are given in references 4, 8.2-3.3.2 General Operator Formulation of MMS. In the general MMS ap
35、proach, the problem is written symbolically as a nonlinear (system) operator L.L f(x, y, z, t) 5 0 (2-3-15)Choose a manufactured solution and denote it by M. f 5 M(x, y, z, t) (2-3-16)The problem is now changed to a new operator, L9, such that the solution to L9 f(x, y, z, t) 5 0 (2-3-17)is exactly
36、the manufactured solution M. The most gen-eral and straightforward approach is to determine L9 by adding a source term to the original problem.L9 f 5 L f 2 Q (2-3-18)The required source term is evaluated by passing the manufactured solution M through the operator, L.Q 5 LM (2-3-19)So instead of solv
37、ing the original problem L(f ) 5 0 with an unknown solution, L(f ) 5 Q or equivalently, L9(f ) 5 0, which has the known solution, M, is solved. Bound-ary values, for any boundary condition to be tested, are governing PDEs and boundary conditions. The chosen solution V(t,x) in this example is the fol
38、lowing:V(t, x) 5 A 1 sin(B), B 5 x 1 Ct (2-3-1)This 1-D transient solution is applied to the nonlinear Burgers equation, often taken as a model problem for CFD algorithm development 4. vy t 5 2v vy x 1 a 2vy x 2(2-3-2)or, using the more compact subscript notation to indicate partial derivatives,v t5
39、 2 vv x1 a v xx(2-3-3)Incidentally, this specifi ed solution V(t,x) is the exact solution for the constant velocity advection equation with boundary condition of v(t,0) 5 A 1 sin(Ct). However, the physical realism of the solution selected for MMS is irrelevant to the code verifi cation process. All
40、that is re-quired of the solution is that it be nontrivial, and that it exercise the computational algorithm appropriately. The source term Q(t,x) is determined that, when added to the Burgers equation for v(t,x), produces the solution v(t,x) 5 V(t,x). The Burgers equation is written as an op-erator
41、 (nonlinear) of v,L(v) ; v t1 vv x2 a v xx5 0 (2-3-4)Then the source function Q that produces V by operat-ing on V with L is evaluated.Q (t, x) 5 LV(t, x) 5 Vy t 1 V Vy x 2 a 2Vy x 2(2-3-5)By elementary operations on the manufactured solution V(t,x) stated in eq. (2-3-1), Q (t, x) 5 C cos(B) 1 A 1 s
42、in(B) cos(B) 1 a sin(B) (2-3-6)If the modifi ed equation is now solvedL(v) ; v t1 vv x2 a v xx5 Q(t, x) (2-3-7)orv t5 2 vv x1 a v xx1 Q(t, x) (2-3-8)with compatible initial and boundary conditions, the exact solution of the modifi ed problem will be V(t,x) given by eq. (2-3-1).The initial conditions
43、 are obviously just v(0,x) 5 V(0,x) everywhere. The boundary conditions are determined from the manufactured solution V(t,x) given by eq. (2-3-1). Note that the domain of the solution is not even specifi ed as yet. To consider the usual model 0 # x # 1 or something like 210 # x # 100, the same solut
44、ion eq. (2-3-1) applies, but of course, the boundary values are determined at the corresponding locations in x. Note also that the type of boundary condi-tion as yet has not been specifi ed. This aspect of the meth-odology has often caused confusion. It is widely known that different boundary condit
45、ions on a PDE produce different answers, but not everyone recognizes immediately that the same solution V(t,x) can be produced by more than one set ASME V 68. 2-3.3.3 Application of MMS to Verifi cation of Codes. Once a nontrivial exact analytic solution has been generated, by this method of manufac
46、tured solutions or perhaps another method, the solution is now used to verify a code by performing systematic discretization convergence tests (usually, grid convergence tests) and monitoring the convergence as h 0, where h is a measure of discretization e.g., Dx (in space), Dt (in time) in a fi nit
47、e difference or fi -nite volume code, and element size in a fi nite element code, number of vortices in a discrete vortex method, number of surface facets in a radiation problem, etc.The principal defi nition of “order of convergence” is based on the behavior of the error of the discrete solu-tion.
48、There are various measures of discretization error Eh, but in some sense this discussion is always referring to the difference between the discrete solution f(h) (or a functional of the solution, such as lift coeffi cient) and the exact (continuum) solution, E h5 f(h) 2 f exact(2-3-20)For an order p
49、 method and a well-behaved problem, the error in the solution E hasymptotically will be proportional to hp. This terminology applies to the “consistent” method-ologies of fi nite difference methods (FDM), fi nite volume methods (FVM), fi nite element methods (FEM), vortex-in-cell, etc., regardless of solution smoothness.7Thus, E h5 f(h) 2 f exact5 C h p1 H.O.T (2-3-21)7This order of convergence description will not apply to global spectral methods or to p-refi nement FEM, but the exact solutions of MMS will still be useful for code verifi
