Predictivity-I! Verification! Computational Fluid Dynamics. Computational Fluid Dynamics

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1 Predictivity-I! To accurately predict the behavior of a system computationally, we need:! A correct and accurate code: Verification! An model that accurately describes the process under study: Validation! Grétar Tryggvason! Spring 2011! A set of input parameters whose value and range is known: Uncertainty Quantification! Verification! From: Schlesinger, S. Terminology for Model Credibility, Simulation, Vol. 32, No. 3, 1979; Cited in W. L. Oberkampf and T.G. Trucano. Verification and Validation in. SANDIA REPORT SAND (2002)! Verification: Show that the code solves the equations that it is intended to solve with the expected accuracy. Verification consists of Code Verification (which can be done once and for all) and Solution Verification which must be done for all new problems.! Code Verification is independent of any physical reality. The goal is simply to show that it correctly solves the equations that it is intended to solve. If an exact solution is available, then it can be used. If not, the Method of Manufactured Solutions (MMS) can be used.! The key tool to verify a code and solution correctness is grid refinement.!

2 Method of Manufactured Solutions! The basic idea is to add a source term to the equations that force the solution to take a given value. Suppose we have:! L(u) = 0 Taking! u = q obviously does not satisfy this equation. However if we add a source to the RHS, given by! u = q g = L(q) Then is a solution to! L(u) = g Thus, the Method of Manufactured Solutions consists of! 1. Picking a function q 2. Computing the source term! 3. Solving the original equation with the new source term! Notice that there is no requirement that the function q satisfies the equations or that it has any physical meaning.! Trivial Example! d 2 u = 0; 0 < x < 1; u(0) = 1; u(1) = 0; 2 dx The solution is given by:! u = 1 x Take:! q = 1 x 2 ; Solve:! d 2 u dx 2 = 2; dq dx = 2x; d 2 q 2 dx = 2; g = 2 du dx = 2x + C 1 ; u = x2 + C 1 x + C 2 C 1 = 0; C 2 = 1; u = 1 x 2 As we intended!! For a more real example consider! f x = D 2 f x 2 With the manufactured solution:! q = A + sin(x + Ct) First we rewrite our equation as:! L( f ) = f x D 2 f x = 0 2 So our source is! g = q t + q q x D 2 q x 2 Our manufactured solution is! q = A + sin(x + Ct) We have! q q = C cos(x + Ct); t So the source term is! = cos(x + Ct); x 2 q = sin(x + Ct); 2 x g = C cos(x + Ct) + ( A + sin(x + Ct) )cos(x + Ct) + sin(x + Ct); From: P. J. Roache. Fundamental of Verification and Validation. Hermosa, 2009!

3 Therefore,! f = A + sin(x + Ct) Is an EXACT solution to! f x = D 2 f x + g 2 g = C cos(x + Ct) + ( A + sin(x + Ct) )cos(x + Ct) + Dsin(x + Ct); % Method of Manufactured Solution for the 1D Burgers equation! % ! n=61; nstep=2000; length=2*pi;h=length/(n-1);diff=0.05;dt=1.0/nstep! f=zeros(n,1); y=zeros(n,1); ex=zeros(n,1); time=0.0; for i=1:n; x(i)=h*(i-1);end! A=1.0; C=1.0; for i=1:n, f(i)=a+sin(x(i)); end; %initial conditions! for m=1:nstep+1,m! for i=1:n, ex(i)=a+sin(x(i)+c*time); end; %exact solution! hold off;plot(f,'linewidt',6); axis([1 n -1.0, 3.0]); % plot solution! hold on;plot(ex,'r','linewidt',2); %pause; % plot exact solution! err=0.0;for i=1:n, err=err+h*(ex(i)-f(i))^2; end;err=sqrt(err)! y=f; % store the solution! for i=2:n-1,! g=c*cos(x(i)+c*time)+(a+sin(x(i)+c*time))*cos(x(i)+c*time)+diff*sin(x(i)+c*time);! f(i)=y(i)-0.5*(dt/h)*y(i)*(y(i+1)-y(i-1))+...!! diff*(dt/h^2)*(y(i+1)-2*y(i)+y(i-1))+dt*g; % advect by centered differences! end;! g=c*cos(x(n)+c*time)+(a+sin(x(n)+c*time))*cos(x(n)+c*time)+diff*sin(x(n)+c*time);! f(n)=y(n)-0.5*(dt/h)*y(n)*(y(2)-y(n-1))+diff*(dt/h^2)*(y(2)-2*y(n)+y(n-1))+dt*g; % do endpoints! f(1)=f(n); % for periodic! time=time+dt % boundaries! end;! The error versus grid spacing! The manufactured solution is used in code verification exactly as we would use an exact solution to the original equations. If we do not, for example, get the expected convergence rate, then there must be an error somewhere! Error in boundary conditions destroys second order convergence!! Solution at time 1! 1! 2! Correct second order convergence! 1! 2! h! The manufactured solution can be selected in many different ways. We could, for example, pick:! q = e x sin( t) Which gives us the source! g = e x cos t ( ) + ( e x sin( t) ) 2 De x sin( t) For the Burgerʼs equation! f x = D 2 f x 2 From: P. J. Roache. Fundamental of Verification and Validation. Hermosa, 2009! Guidelines for the MMS! 1. Manufactured solutions should be smooth analytic functions like polynomials, trigonometric, or exponential functions so that the solution is conveniently computed.! 2. The solution should exercise every term in the equations! 3. The solution should have sufficient number of derivatives! 4. The derivatives should be bounded by small constants! 5. The solution should not prevent the code from running to completion! 6. The solution should be defined on a connected subset of twoor three-dimensional space! 7. The solution should be constructed in a manner such that the differential operators in the PDEʼs make sense.! Adopted from: K. Salari and P. Knupp. Code Verification by the Method of Manufactured Solutions. SAND (2000)!

4 Generating the source terms can be complicated for complex operators. This can, however, easily be done using symbolic manipulation software such as MAPLE or Mathematics! Verification involves two different steps:! Code verification! Generally done once to ensure that the code is correct, using for example the method of manufactured solutions! Solution verification! Done every time the code is used to produce a solution to ensure that the solution errors are acceptable (that the solution is converged)! Suppose we have a supposedly pth-order solution on two grids where h 2 =h 1 /r. The error is then! E grid1 = Ch p E grid 2 = C h r The ratio of the errors is! E grid1 = Ch p E grid 2 Ch p r p = r p p = log E grid1 E grid 2 p Which allows us to compute the actual order:! / log r ( ) The actual order p, for the Burgerʼs equation example, computed as on the previous slide! Theoretical convergence rate! h! Euler Equations for 2D Flow! Roy, Nelson, and Smith tested two codes using:! The method of manufactured solutions does not address all issues of code verification, such as domain size and boundary conditions. Nevertheless, it is emerging as one of the major tool in ensuring that a given set of equations is correctly solved.! From: C.J. Roy, C.C. Nelson, T.M. Smith, C.C. Ober, Verification of Euler/Navier Stokes codes using the method of manufacturedsolutions, Int. J. Numer. Meth. Fluids 44 (6) (2004) ! Cited in: C.J. Roy. Review of code and solution verification procedures for computational simulation. Journal of Computational Physics 205 (2005) !

5 Solution Verification (Correct code, wrong solution)! While code verification is usually done once, solution verification needs in principle to be done every time the code is used to generate a solution. In practice an experienced user will have a good idea about the necessary resolution.! A correct code but insufficient resolution or other numerical parameters (iteration errors, for example) can lead to inaccurate and even wrong solutions. For new problems the accuracy must be verified.! The approximate solution can be written as a Taylor series around the exact solution f(0):! f ( h) = f ( 0) + Ch 2 + HOT Here, f(0) is the (unknown) exact solution and C is a constant determining the magnitude of the error. Given f(h) and f(2h), estimate C. Once C is known, we can use the above formula to find a better estimate for the solution. This procedure is called Richardson Extrapolation and is widely used in practice.! We have:! f ( h) = f ( 0) + Ch 2 + HOT similarly, the solution on twice as coarse grid is:! f ( 2h) = f ( 0) + C4h 2 + HOT subtracting to eliminate the h 2 term:! 4 f ( h)! f ( 2h) = 4 f ( 0)! f ( 0) + HOT Solving for the exact solution:! f ( 0) = 4 f ( h)! f ( 2h) + HOT 3 Since the Higher Order Terms (HOT) are at least O(h 3 ), f(0) is a better estimate than either f(h) or f(h 2 ). Similar formulas can be derived for schemes of different orders.! For an p-th order scheme we have:! ( ) = f ( 0) + Ch p + HOT f h similarly, the solution on a finer grid is:! ( ) = f ( 0) + C( h / r) p + HOT f h / r subtracting to eliminate the h n term:! f ( h) f ( h / r)r p = f ( 0) 1 r p ( ) + HOT Solving for the exact solution:! f ( 0) = f ( h) f ( 2h)r p + HOT 1 r p Uncertainty Quantification!

6 In most cases uncertainties do not only come from the numerical solution but also from the problem specification. Those uncertainties include:! Material properties (density, viscosity, etc)! Domain geometry! Boundary conditions! Model assumption! In principle these uncertainties can be treated in the same way as experimental uncertainties.! In practice, the quantification of uncertainties requires us to assume that the error follows a particular distribution and it is easiest to deal with uncorrelated errors.! The identification of the uncertainty in all model and input parameters is challenging and there are considerable questions whether standard uncertainty quantification will ever be able to deal with one-off events or Black Swans. The role of incorrect use of uncertainty models (Black-Scholes, etc) in the recent financial crisis suggests caution!! Validation! Validating a theory consists of comparing its predictions with experimental results. It is thus at the core of science and as such not a computational issue.! However, scientific computing has greatly increased our abilities to solve complex models and this is leading to more and more complex models, with new issues and challenges for validation.! At the present time there is no real theory covering how to validate a computational model, except compare the predictions with experiments or observations for selected cases. For complex models, build by assembling sub-models, the sub-models are usually validated independently.! C.J. Roy. Review of code and solution verification procedures for computational simulation. Journal of Computational Physics 205 (2005) ! AIAA AIAA Guide for the Verification and Validation of Computational Fluid Dynamics Simulations. American Institute of Aeronautics and Astronautics.! Standard for Verification and Validation in and Heat TransferV V !