Verification, Validation and Variational Crimes in FE Computations

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1 Verification, Validation and Variational Crimes in FE Computations R Muralikrishna 1, S Mukherjee 2 and G Prathap 3 1 Engineering Analysis Centre of Excellence, Bangalore , India 2 National Aerospace Laboratories, Bangalore , India 3 CSIR Centre for Mathematical Modelling and Computer Simulation, Bangalore , India

2 Validation and verification in computational engineering science have attracted some attention recently. I Babuska and J T Oden Verification and validation in computational engineering science: basic concepts Comput. Methods Appl. Mech. Engg. 2004, to appear

3 Computational engineers are clever at inventing artifices and tricks which take liberties with the canonical formulation. In such cases, where variational crimes are deliberately or accidentally introduced, verification and validation must be interpreted carefully.

4 In this lecture, the problem of finite element computations from such extra-variational considerations is examined critically. The use of non-conforming formulations or the use of lumped mass approaches, which are extra-variational steps, introduces additional errors. The errors due to these extra-variational steps fortuitously compensate for each other, leading one to the erroneous conclusion that a very accurate computational model has been achieved, and that the model has been verified and validated! This has important implications, especially where the aim is to derive error estimators for adaptive mesh refinement using non-conforming elements and/or lumped mass approaches.

5 Hierarchy of models Reality Conceptual model Mathematical Model Numerical Model Computational Model Error Model: Model of the Model

6 V & V Verification: Computational Model Numerical Model (is your code correct?) Validation: Conceptual Model Reality (is your model correct?)

7 Virtual work for elastostatics. The weak form in terms of the energy inner product for the exact solution u to the problem. a(u,u) = (f,u) (1) a(u,u h ) = (f,u h ) (2) (1) and (2) refer to the exact solution of the elastostatic problem. In (1), the trial function and test function are taken as u and the virtual work argument establishes that (1) is truly satisfied only when u is the exact solution at the point of equilibrium. In (2), we take note of the fact that the test function u h (the Ritz or finite element solution) need not be the exact displacement function for the virtual work principle to be true. For convenience, we take this to be the discrete finite element displacement field, as long as it is admissible (that is, satisfies all the geometric boundary conditions).

8 Virtual work for elastostatics (contd.) (3) takes the step towards discretisation. By using u h for both the trial and test function, we get the actual finite element equations, with the right hand side leading to the consistent load vector and the left hand side representing the stiffness matrix. a(u h,u h ) = (f,u h ) (3)

9 Virtual work for elastostatics (contd.) We are now in a position to see how the error e = u - u h can be assessed. Comparing (2) and (3) and noting that the energy inner product is bi-linear, we can arrive at a(u,u h ) = a(u h,u h ) and from this we obtain the projection theorem a(u - u h, u h ) = 0 (4)

10 Virtual work for elastostatics (contd.) The finite element solution is therefore seen to be a best-fit or best approximation solution. In most simple linear elastostatics cases, this would imply that the strains or stresses are obtained in a best-fit sense and that there would be points in the element domain where these stresses or strains are very accurately computed (superconvergence).

11 Virtual work for elastostatics (contd.) From the fact that the energy inner product is bi-linear, we can argue that a(u u h,u - u h ) = a(u,u) + a(u h, u h ) -2a(u, u h ) = a(u,u) - a(u h, u h ) - 2[ a(u, u h ) - a(u h, u h )] = a(u,u) - a(u h, u h ) - 2[ a(u - u h, u h )]

12 Virtual work for elastostatics (contd.) Introducing the result from (4), we get an energy error theorem which can be expressed as a(u u h,u - u h ) = a(u,u) - a(u h, u h ) (5) i.e., Energy of the error = Error of the energy

13 Virtual work for elastostatics (contd.) This leads to a useful statement that as the left hand side of (5) is always positive definite, a(u h, u h ) T a(u,u) (6)

14 Virtual work for elastodynamics Now, the kinetic energy of motion enters into the picture, and a classical variational basis for this is already available through the Lagrangian or Hamiltonian statements, where both potential energy and kinetic energy enter into the functional. We shall write the weak form in terms of the energy inner product for the exact solution u to the problem and replace the loading term f with the inertial force term ÆŁ 2 u. The earlier equations now become

15 Virtual work for elastodynamics (contd.) a(u,u) = ÆŁ 2 (u,u) (7) a(u,u h ) = ÆŁ 2 (u,u h ) (8) (7) and (8) refer to the exact solution of the elastodynamic problem.

16 Virtual work for elastodynamics (contd.) a(u,u) = ÆŁ 2 (u,u) (7) In (7), the trial function and test function are taken as u and the virtual work argument establishes that (7) is truly satisfied only when u is the exact eigenfunction and Ł 2 is the exact eigenvalue.

17 Virtual work for elastodynamics (contd.) a(u,u h ) = ÆŁ 2 (u,u h ) (8) In (8), we take note of the fact that the test function u h (the Ritz or finite element solution) need not be the exact displacement function for the virtual work principle to be true, while Ł 2 remains the exact eigenvalue. Again, we take u h to be the discrete finite element displacement field, as long as it is admissible (that is, satisfies all the geometric boundary conditions).

18 Virtual work for elastodynamics (contd.) We now take the step towards discretisation. By using u h for both the trial and test function, we get the actual finite element equations, with the right hand side leading to the consistent load vector and the left hand side representing the stiffness matrix. a(u h, u h ) = Æ(Ł h ) 2 (u h, u h ) (9) This equation will now reflect the error due to the finite element discretisation, appearing both in the eigenfunction, and in the eigenvalue.

19 Virtual work for elastodynamics (contd.) This makes the assessment of the errors a trifle more complicated than in the elastostatics case earlier, as there is the error in the eigenfunction, u - u h as well as the error in the eigenvalue, (Ł h ) 2 - (Ł) 2 to be assessed.

20 Virtual work for elastodynamics (contd.) Comparing (8) and (9) and also noting that the energy inner product is bi-linear, we can arrive at [a(u,u h ) - [a(u h, u h )] - Æ[Ł 2 (u, u h ) (Ł h ) 2 (u h, u h )] = 0 or a(u - u h,u h ) - Æ(Ł 2 u - (Ł h ) 2 u h, u h ) = 0 (10) becomes the projection theorem for elastodynamics.

21 Virtual work for elastodynamics (contd.) The finite element solution is still seen to be a best-fit or best approximation solution. However, unlike the simple linear elastostatics cases, this would imply that the strains or stresses are only nearly obtained in a best-fit sense. It is no longer possible now to relate the error of the energy to the energy of the error, as was possible for the elastostatics case as in equation (5) earlier. We shall first take up the error of the energies.

22 Virtual work for elastodynamics (contd.) From equation (7) and (9), we have, [a(u,u) - a(u h, u h )] - Æ[Ł 2 (u,u) - (Ł h ) 2 (u h, u h )] = 0 (11)

23 Virtual work for elastodynamics (contd.) We now take up the case of the energy of the errors. For this, we have a(u u h,u - u h ) = a(u,u) + a(u h,u h ) - 2 a(u,u h ) = Æ[Ł 2 (u,u) - 2Ł 2 (u, u h ) + (Ł h ) 2 (u h, u h ) = Æ[Ł 2 (u,u) - 2Ł 2 (u, u h ) + (Ł) 2 (u h, u h ) + [(Ł h ) 2 - Ł 2 ] (u h, u h )] = Æ[Ł 2 (u u h,u - u h ) + [ (Ł h ) 2 - Ł 2 ](u h, u h )] (12)

24 Virtual work for elastodynamics (contd.) This is indeed Lemma 6.3 of Strang and Fix1. We started with three virtual work equations (equations (7) to (9)) and this led very easily to a projection theorem (equation (10)), and separate energy-error theorem (equation (11)) and error-energy theorem (equation (12)). Unlike the case for elastostatics, we cannot derive a simple relationship between the energy of the error and the error of the energy. Also, it is not easy to show that for a conforming and variationally correct formulation, the discretised eigenvalue would always be higher than the exact eigenvalue, whereas for the elastostatics problem, we did have an elegant lower bound result (equation (6)). This is because an eigenvalue problem does not give a unique displacement field u or u h.

25 Virtual work for elastodynamics (contd.) One can side-step this issue by introducing into equation (11), the idea of a normalized generalized mass, where ( ρu, u) = ( ρu h, u h ) = 1. This gives us, (ω h ) 2 - ω 2 = - [a(u,u) - a(u h,u h )] What is important to us now is that the error is still governed by the error in the strain energy, and therefore, as long as no variational crimes are committed, the orders of convergence for the elastostatic case should apply.

26 Numerical Experiments In this section, we indicate the thrust of the work done, but refer the author to the web location: rep/rrcm0306.pdf for a full account of numerical experiments by Muralikrishna and Prathap (2003). An in-house package has been developed comprising the ACM, BFS and QUAD4 elements. Numerical experiments are first carried out using the consistent mass matrices for all three element types. This is followed by experiments with lumped mass matrices.

27 Numerical Experiments It is important that we clearly define the measure of error to be used. For elastodynamics, where, a variationally correct formulation would have produced higher frequencies, one can define error as Error = (ω h ) 2 /ω 2-1 If the problem is not formulated in a variationally correct way, boundedness is lost, and it may be possible that frequencies are lower than the analytical ones, in which case it may be necessary to use Error in the logarithmic plots.

28 Numerical benchmarks the free vibration response of a simply supported plate A plate of dimensions 40mm x 40mm, 0.4 mm thick, is chosen and with elastic properties, ν=0.3, E= N/mm2, and mass density ρ =7850 kg/m3. We start with an accurate high density mesh of uniformly sized elements (NxN mesh) and then compute the error in the frequencies with increasing mode number (for a plate, m or n, varying one and keeping the other fixed).

29 Numerical benchmarks the free vibration response of a simply supported plate (contd.) For the simply supported plate, the exact solution involves simple trigonometric waves; for example, the (m,n) mode will be the eigenfunction sin(mpx/a)sin(npy/b). The ratio of the element length, h = a/n to the wave-length of the m th wave? = a/m, is now r = h/? = m/n, when the other mode number (say n) is kept fixed. The errors will now be a function of r, and this will mean that the order of convergence with one of the mode numbers of the frequency (say m) will follow the exact trend as with h, if N and n are fixed for the problem. Thus, with one single eigenvalue computation, we can sweep for the errors for varying mode number m.

30 Numerical benchmarks the free vibration response of a simply supported plate sweep test For this, we choose again, the simply supported plate but use only a 2x2 mesh. Also, we sweep the central nodal point of the mesh of the plate. y x

31 Natural Frequencies of a Simply Supported Plate BFS Consistent = O(h 4 ) Variation of Log(Error) with Log(mode number) for a SS Plate Log(Error) n=1 n=2 n=3 n=4 n=5 O(h^4) -6 Log(m) BFS, Consistent Mass

32 Fundamental Frequency of a Simply Supported Plate BFS Element Consistent Mass Matrix

33 Fundamental Frequency of a Simply Supported Plate BFS Element Consistent Mass Matrix Variation of Fundamental Frequency of a SS plate 2x2 Mesh - Full plate Frequency (Hz) BFS-Consistent Exact Solution Distance along diagonal of plate

34 Natural Frequencies of a Simply Supported Plate ACM Consistent = O(h 2 ) Variation of Log( Error ) with Log(mode number) for a SS Plate Log( Error ) n=1 n=2 n=3 n=4 n=5 O(h^2) m=n modes -3 Log(m) ACM Element, Consistent Mass

35 Fundamental Frequency of a Simply Supported Plate ACM Element Consistent Mass Matrix

36 Fundamental Frequency of a Simply Supported Plate ACM Element Consistent Mass Matrix Variation of Fundamental Frequency of a SS Plate- Full plate 2x2 Mesh 240 Fundamental Frequency ACM-Consistent Exact Solution Distance along diagonal

37 Natural Frequencies of a Simply Supported Plate BFS Lumped = O(h 2 ) Variation of Log( Error ) with Log(mode number) for a SS Plate Log( Error ) n=1 n=2 n=3 n=4 n=5 O(h^2) m=n modes -3.5 Log(m) Lumped Mass,BFS

38 Fundamental Frequency of a Simply Supported Plate BFS Element Lumped Mass Matrix

39 Fundamental Frequency of a Simply Supported Plate BFS Element Lumped Mass Matrix Variation of Fundamental Frequency along Diagonal Frequency (Hz) BFS-Lumped Exact Solution Distance along diagonal

40 Natural Frequencies of a Simply Supported Plate ACM Lumped = O(h 2 ) Variation of Log( Error ) with Log(mode number) for a SS Plate Log( Error ) n=1 n=2 n=3 n=4 n=5 O(h) m=n modes O(h^2) -3 Log(m) ACM Element,Lumped

41 Fundamental Frequency of a Simply Supported Plate ACM Element Lumped Mass Matrix

42 Fundamental Frequency of a Simply Supported Plate ACM Element Lumped Mass Matrix Variation of Fundamental Frequency of a SS Plate- Full plate 2x2 Mesh 350 Fundamental Frequency ACM-Lumped Exact Solution Distance along diagonal

43 Numerical Experiments Concluding Remarks A detailed study of the errors produced due to the use of extra-variational processes in the plate bending problems such as the loss of conformity in the ACM element, and the use of lumping for the mass matrix, are available in: Muralikrishna, R. and Prathap, G. (2003), Studies on the variational correctness of finite element elastodynamics of some plate elements, Research Report CM 0306, CSIR Centre for Mathematical Modelling and Computer Simulation, Bangalore, August. It is shown that whenever the variational principles are not adhered to, there is no guaranteed boundedness of the results, or an assurance on the variationally correct rate of convergence. This has important implications where error measures are used in adaptive mesh refinement.

44 Concluding remarks A detailed study of the errors produced due to the use of extra-variational processes in the plate bending problems has been done. It is shown that whenever the variational principles are not adhered to, there is no guaranteed boundedness of the results. The principles of validation and verification rest on very shaky foundations when such extravariational steps are routinely introduced.

Advanced Vibrations. Distributed-Parameter Systems: Approximate Methods Lecture 20. By: H. Ahmadian

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