Bifurcation Analysis and Reduced Order Modelling of a non-adiabatic tubular reactor with axial mixing

Transcription

1 Bifurcation Analysis and Reduced Order Modelling of a non-adiabatic tubular reactor with axial mixing M.A Woodgate, K.J Badcock, and B.E Richards Department of Aerospace Engineering, University of Glasgow, Glasgow, G12 8QQ, United Kingdom Aerospace Engineering Report an update of 0013 Abstract This report explores the practicality of reduced order modelling for coupled computational fluid dynamics and computational structural dynamics simulations used in flutter predition. The system of ordinary different equations can have a size of the order 10 6 and the derivation of the reduced order model must be accurate, robust and fast for this size of problem. These issues are examined for a non-adiabatic tubular reactor which serves as a model problem. A direct calculation based on a Hopf Bifurcation formulation is used to calculate the bifurcation parameter and the corresponding critical eigenvalue and eigenvector is obtained as part of this, which are then exploited as part of a model reduction method using the centre manifold theorem. The reduced model, which is of dimension two, can be used to explore the behaviour of the solution, and in particular either a limit cycle amplitude or damping characteristics, to the full (large dimensional) system in a computationally efficient manner. The reduction method has been formulated to be extendable to aeroelastic systems of large dimension. 1 Introduction The use of coordinate transformations to simplify nonlinear equations in the vicinity of a bifurcation is a well known technique for systems of low order. The original variables are transformed so that a small number of critical variables are isolated which can describe qualitatively the behaviour of the full system for changes in the parameter near the bifurcation point. An overview of methods used for investigating nonlinear flutter of an aerofoil is given by Lee et al [1]. A number of reduction techiques can be applied to nonlinear systems of equations. These include Volterra methods, Proper Orthogonal Decomposition (POD), and system identification techniques. 1

2 2 The application of Reduced Order Modelling (ROM) techiques to aeroelastic systems is driven by the desire for extremely fast calculations, requiring a few seconds CPU time on a cheap computer, that are well suited for applications to design, clearance and simulators. The idea behind reduction is to maintain the large scale features of the aerodynamics described by Euler or Navier-Stokes modelling, which are critical in transonic aeroelastic predictions, whilst meeting the efficiency requirements. In recent years the application of POD has been successful in predicting limit cycle oscillations (LCOs) in the transonic regime [2]. A technique called subspace projection was used for time marching of the full system with the full system residual and unknowns projected onto a set of basis vectors derived from a small number of full system time response calculations. For this method however, the number of degrees of freedom was reduced by 4 orders of magnitude while the time marching with subspace projection only reduced the required CPU time by between one and two orders. A hybrid Volterra-POD method is being developed to remove this drawback [3] and obtain the same order deduction in computational cost as degrees of freedom. Another possibility is the use of the centre manifold to form the ROM. If the asymptotic solution of the equations of motion only is of interest it is appealing to consider the low dimensional centre manifold instead of the whole system. In the case of limit cycle oscillations after a Hopf bifurcation the corresponding centre manifold is two dimensional. A possible drawback of this approach lies in the use of series expansions. These expansions are truncated and are only good approximations if the higher order terms are convergent and small [4]. The use of coordinate transformations to simplify nonlinear equations in the vicinity of a bifurcation is a well known technique for systems of low order (eg order 10 in [5]). However, the methods are generally only applicable for systems of a size which is much smaller than the systems obtained from CFD based aeroelastic discretisations as transformations to Jordan canonical form in general destroy the sparsity of the aeroelastic operator. The current work is aimed at developing a method which can be applied for systems of realistic size. 2 Hopf Bifurcations of Dynamical Systems Consider a continuous time system depending on a parameter µ ẋ = f(x, µ), x R n, µ R 1, (1)

3 3 where f is smooth with respect to both x and µ. The eigenvalues of the Jacobian matrix A = f/ x, referred to as the eigenvalues, are important for determining the stability characteristics of the equilibria of the system. Let x = x 0 be a hyperbolic equilibrium 1 of the system for µ = µ 0. There are only two ways in which the hyperbolicity condition can be violated. Either a simple real eigenvalue approaches zero and we have λ 1 = 0, or a pair of simple complex eigenvalues reach the imaginary axis and we have λ 1,2 = ±iω 0, ω 0 > 0 for some value of the parameter. It can be shown that we need more than one parameter to allocate extra eigenvalues on the imaginary axis. There are 3 topological classes of hyperbolic equilibrium, namely stable nodes, saddles and unstable nodes. These are distinguished by the positive and negative real parts of the eigenvalues. For one parameter bifurcations two of these types are possible. The first is called a fold and is associated with the appearance of a zero eigenvalue. This is also referred to as a limit point or a turning point. The second type is the Hopf bifurcation which is associated with the appearance of a purely imaginary eigenvalue. It should be noted that a fold can exist for a system of dimension one whereas a Hopf requires at least two dimensions. The number of possible types of bifurcation increase with the number of free parameters. For a two parameter problem there are additionally cusp, generalized Hopf (Bautin), Bogdanov-Takens, Fold-Hopf and Hopf-Hopf bifurcations. However, in the current work it is the one parameter Hopf bifurcations that are of interest. An invariant manifold of a nonlinear system of equations near an equilibrium point or a limit cycle is determined by the structure of its vector field. Two methods can be used to simplify the original nonlinear system; the centre manifold or the normal form theory. The normal form theory is a method for transforming the original nonlinear differential equation to a simpler standard form by an appropriate change of coordinates so that the main features of the manifold become clearer. To illustrate first consider the nonlinear system of equations without the parameter ẋ = f(x), x R n (2) where f is sufficiently smooth. We assume that the linear system has 2 and only 2 critical eigenvalues with zero real part and that the remaining m = n 2 eigenvalues have negative real parts since we 1 i.e. there are no eigenvalues of f/ x on the imaginary axis

4 4 are at a Hopf bifurcation. Then the system (2) can be transformed to { u = Bu + g(u, v) v = Cv + h(u, v) (3) where u R 2 and v R m. B is a 2 2 matrix with its eigenvalues on the imaginary axis and C is a m m matrix with no eigenvalues on the imaginary axis. The functions g and h have at least quadratic terms. The centre manifold W c of system (3) can be locally represented as a graph of a smooth function, W c = {(u, v) : v = V (u)} V : R 2 R m and due to the tangent property of W c, V (u) = O( u 2 ). The Reduction Principle says system (3) is locally topologically equivalent near the origin to { u = Bu + g(u, V (u)) v = Cv (4) The important thing to notice is that the equations in (4) are decoupled for u and v. The first equation is the restriction of (3) to its centre manifold. The dynamics of the structurally unstable system (3) are essentially determined by this restriction, since the second equation in (4) is linear. Since bifurcations are determined by the normal form coefficients of these restricted systems at the critical parameter µ = 0 we have to be able to calculate centre manifolds and the equations or maps that restrict to this manifold up to sufficiently high order terms. Coefficients of the Taylor expansion of the function v = V (u) representing the centre manifold W c can be computed in a recursive manner. Avoiding the system transformation into its eigenbasis is an essential feature when the system has millions of equations. Indeed full eigenvalue-eigenvector decomposition of such systems is impossible. The centre manifold can be transformed by using only the vectors corresponding to the critical eigenvalues of A and its transpose A T. The method projects the system into its critical eigenspace and complement. Suppose we have a Taylor expansion of the system ẋ = Ax + F (x), x R n (5) where F (x) has at least quadratic terms. The matrix A has a pair of complex eigenvalues on the imaginary axis λ 1,2 = iω, ω > 0. Let q be the right eigenvector corresponding to λ 1. Then q is the right eigenvector corresponding to λ 2 and Aq = iωq, A q = iω q.

5 5 The left eigenvector p also has the same property A T p = iωp, A T p = iω p. These can be normalised such that p, q = 1 where p, q = n i=1 p iq i. The eigenspace S corresponding to ±ω is two dimensional and is spanned by {Rq, Iq}. The eigenspace T corresponds to all the other eigenvalues of A and is n 2 dimensional. Then y T if and only if p, y = 0. Since y R n while p is complex then two real constraints on y exist and hence it is possible to decompose any x R n as x = zq + z q + y where z C 1, zq + z q S, and y T. The complex variable z is a coordinate of S so since p, q = 0. So the equation (5) has the form { z = p, x y = x p, x q p, x q { ż = iωz + p, F (zq + z q + y) ẏ = Ay + F (zq + z q + y) p, F (zq + z q + y) q p, F (zq + z q + y) q. This system is (n + 2) dimensional but we have two constraints on y. This system is expanded in a Taylor series in z, z and y to give the following approximation { ż = iωz z 2 + G 11 z z + 1G 2 02 z 2 + 1G 2 21z 2 z + G 10, y z + G 01, y z +... ẏ = Ay + 1H 2 20z 2 + H 11 z z + H 02 z (6) where G 20, G 11, G 02, G 21 C 1 ; G 01, G 10, H ij C n. This approach leads to the scalars and vectors all being functions of the F or inner products of p and F (i.e. the original function). The manipulation of this system is feasible, even for systems of large dimension. The centre manifold can be represented by y = k(z, z) = 1 2 k 20z 2 + k 11 z z + k 02 z 2 + O z 3 with the constraint p, k ij = 0. The vectors w ij C n can be found from the linear equations (2iωI A)k 20 = H 20 Ak 11 = H 11 (7) ( 2iωI A)k 02 = H 02

6 6 These equations are invertable since 0 and ±2iω are not eigenvalues of A. We can now write the restricted equation as ż = iωz + 1G 2 20z 2 + G 11 z z + 1G 2 02 z 2 + 1(G G 10, A 1 H 11 + G 01, (2iωI A) 1 H 20 )z 2 z +... If we write F (x) in terms of functions B(x, y) and C(x, y, z) then Then we can express and hence the restricted equation is in the form F (x) = 1 2 B(x, x) C(x, x, x) + O x 4 (8) G 10, y = p, B(q, y), G 01, y = p, B( q, y) z = iωz + 1G 2 20z 2 + G 11 z z + 1G 2 02 z 2 + 1(G p, B(q, A 1 H 11 ) + p, B( q, (2iωI A) 1 H 20 ) )z 2 z +... (9) where and G 20 = p, B(q, q) G 11 = p, B(q, q) G 02 = p, B( q, q) G 21 = p, C(q, q, q) (10) Using these equations and the identities { H20 = B(q, q) p, B(q, q) q p, B(q, q) q H 11 = B(q, q) p, B(q, q) q p, B(q, q) q. (11) A 1 q = 1 iω q, A 1 q = 1 iω q, (2iωI A) 1 q = 1 iω q, (2iωI A) 1 q = 1 3iω q, the restricted equation can be rewritten as where z = iωz g 20z 2 + g 11 z z g 02 z g 21z 2 z g 20 = p, b(q, q) g 11 = p, b(q, q) and g 21 = p, C(q, q, q) 2 p, B(q, A 1 B(q, q)) + p, B( q, (2iωI A) 1 B(q, q)) + 1 p, B(q, q) p, B(q, q) iω 2 iω p, B(q, q) 2 1 p, B( q, q) 2 3iω

7 7 The first Lyapunov coefficient to be written as l 1 (0) = 1 2ω 2 Re(ig 20g 11 + ωg 21 ) = 1 2ω 2 Re[ p, C(q, q, q) 2 p, B(q, A 1 B(q, q)) + p, B( q, (2iωI A) 1 B(q, q)) ] (12) The real beauty of this formula is that it does not require a preliminary transformation of the system into its eigenbasis and l 1 (0) is expressed using original terms assuming that only the critical left and right eigenvectors of the matrix are known. From the definition of the bifurcation of a stable equilibrium to a limit cycle we can define a system which directly gives the parameter value at which the stability changes, and the critical eigenvectors required for the reduction, as F = f (f y iωi)q q T Q i = 0 X = [x, q, µ, ω] T where q = q 1 + iq 2. Taking real and imaginary parts of q f f y q 1 + ωq 2 F = f y q 2 ωq 1 r T q 1 = 0 X = [x, q 1, q 2, µ, ω] T. r T q 2 1 Here r is a scaling vector and ω is the imaginary part of the eigenvalue with zero real part. This system will be referred to as the augmented system throughout this paper. The augmented system can be solved by applying Newton s method with a Newton update given by F n X X = F n where F X = f y 0 0 f µ 0 (f y q 1 ) y f y Iω (f y q 1 ) µ q 2 (f y q 2 ) y Iω f y (f y q 2 ) µ q 1 0 r T r T 0 0 This formulation was put forward for two dimensional aeroelastic systems in reference [6] and has been used to solve three dimensional systems in [7]..

8 8 Finally, consider the parameterized equation ẋ = f(x, µ) where x R n and µ R 1. Suppose that at µ = 0 the system has a non-hyperbolic equilibrium x = 0 with n 0 eigenvalues on the imaginary axis. It is then possible to write an extended system as the system { ẋ = f(x, µ) µ = 0 (13) The Jacobian of (13) at the equilibrium point (x, µ) = (0, 0) is the (n + 1) (n + 1) matrix J = having n eigenvalues on the imaginary axis. ( fx (0, 0) f µ (0, 0) 0 0 ) 3 Results 3.1 Model Problem To test the solution methodology for the augmented system and model reduction, a model problem is considered which describes the unsteady behaviour of a non-adiabatic tubular reactor with axial mixing [8] [9] y t Θ t = = 1 2 y P e m x y 2 x ( µy exp Γ Γ ) Θ 1 2 Θ P e h x Θ 2 x β(θ Θ) + µαy exp ( Γ Γ ) Θ (14) where P e m, P e h, β, α, Γ, and Θ are fixed constants and µ is the bifurcation parameter. The boundary conditions (t > 0) are given by y x = P e m(y 1) Θ x = P e m(θ 1) (x = 0) y x = Θ x = 0 (x = 1). For the results presented here the constants are set to P e m = 5, P e h = 5, β = 2.5, α = 0.5, Γ = 25, and Θ = 1.0.

9 9 The system is discretised using a cell centred finite difference scheme so that the first and second differences are approximated by 2 y x 2 = y i+1 2y i + y i 1 i h 2 y x = y i+1 y i 1. i 2h Here a uniform mesh of spacing h is used with the i-th point at x i = ih for (i = 0,..., n). The boundary conditions for x = 1 are applied by setting halo cell values to be identical to the values in the adjacent interior cell. There are three possibilities for applying the condition at x = 0. First there is the first order approximation y b = y 0 which leads to y 1 = y 0 hp e m (y 0 1), y 1 y 0 = 1 hp e m, y 1 y 1 = 0. (15) The first of the two second order approximations is y b = (y 0 + y 1 )/2 which leads to y 1 = y 0(2 hp e m ) + 2hP e m 2 + hp e m y 1 y 0 = 2 hp e m 2 + hp e m y 1 y 1 = 0. (16) The alternative second order approximation is y b = (3y 0 y 1 )/2 which leads to an extra term being added into the Jacobian Matrix y 1 = y 0 hp e m [3y 0 y 1 2] 2 y 1 y 0 = 1 3hP e m 2 y 1 y 1 = hp e m 2 (17) For this problem having the first order boundary condition greatly affects the accuracy of the results so that even a grid with 512 cells does not give a grid converged answer, as shown in figures 1 and 2. The solution for the equilibrium is by the full Newton method with the use of the exact Jacobian on the left hand side. For the continuation problem this is solved using a banded LU decomposition. For the solution of the augmented system, since the bandwidth has grown to nearly the width of the full matrix, a full LU decomposition is used. It is possible to use a direct solver for the linear system since the dimension is small in the current problem. To check the direct results, unsteady time stepping was used. An explicit method is used which results in a large number of time steps ( t = 1/500 is required for stability). The bifurcation point is bracketed between a steady solution at one parameter value and an unsteady solution at a second value. Each new calculation halves the length of the region bracketing the bifurcation value. This method however does not give the eigenvalue and eigenvector causing the instability as part of the solution. This information is found as part of the solution of the augmented system.

10 Full Solution Behaviour The rich solution space for this model problem is shown in figure 3 where the solution is characterised by the maximum value of Θ within the domain. This includes stable and unstable equilibria, limit points and Hopf bifurcation points. There is also a hysteresis loop for increasing and decreasing µ. The equilibrium solutions for varying µ are shown in figure 3. For values of µ < and µ > this equilibrium is stable and the solution to equation (14) is steady. For values of µ in between these extremes the equilibrium is unstable and a limit cycle oscillation is formed. Depending on whether the parameter µ is increased (solid line) or decreased (solid and dashed lines) a different equilibrium is obtained, indicating hysteresis. The equilibria were mapped out using the continuation method with Newton s method for the corrector stage. In addition, time marching calculations were done to map out the stability of these equilibria. The time history for Θ at x = 1 is shown in figures 5 and 6 for µ = and µ = respectively. It is clearly seen that the solution is steady in the first case and oscillates in the second. Next, the augmented system was solved to find the bifurcation points. If the initial guess is poor then the solution diverges. For the current calculations the following initial guess was used: µ = 0.16, x 2i = 1.0, x 2i+1 = 0.0, P 1i = n1, P 22i = n1, P 22i+1 = n1, q = P 2 and the eigenvalue i. By changing the initial conditions the Newton iterations can be made to converge to the second Hopf point at µ = Starting from this guess the iterations had to be under-relaxed by a factor 0.5 until the domain of quadratic convergence was reached (the criteria used was based on the initial residual being reduced by half). A sequence of grids was used to show mesh independence and a second method of initialisation was used by taking the final solution from the previous grid in the sequence as the starting solution on the next grid. No relaxation was required using this technique. The convergence of the bifurcation parameter is shown in table 1. The number of Newton iterations required with and without the grid sequencing to initialise the iteration is given in the fourth and sixth columns. The sequenced start-up is obviously very beneficial in reducing the cost of the calculation. From the convergence plot shown in figures 4 for the residual it is clear that some iterations are required before the domain of quadratic convergence is reached when not using the sequenced start-up. However, once the quadratic region is reached the convergence is rapid. The CPU times

11 11 shown in the fifth column of the table scale with N 3 since a full Gaussian elimination was used on the whole matrix for this test problem. The exact Jacobian matrix of the augmented system has a large bandwidth. 3.3 Model Reduction For all reduced models the prediction is only expected to represent well the original model in the neighbourhood of the bifurcation point. The size of this neighbourhood is vitally important if the reduced model is going to be of practical use. The time history for y at x = 1 is shown in figures 7 and 8 for initial defections of 0.01 and respectively in Θ at x = 1. For the larger deflection the reduced model overpredicts the size of the initial oscillation but quickly recovers to obtain the correct amplitude and damping. This overprediction causes a phase shift in the solution with the reduced model response slightly underpredicting the frequency. Figure 9 shows the comparison of the amplitudes for the full and reduced models with varying µ. The straight line shows perfect agreement. As the bifurcation parameter is increased both the size of the amplitude of the oscillation increases as well as the discrepancy between the two models. The time history for y at x = 1 is shown in figures 10 and 11. Close to the bifurcation parameter there is very little difference between the two models while far from the bifuration point the reduced order model overpredicts the amplitude of the oscillation and underpredicts the frequency. Similar behaviour has been obtained on the range of meshes used for the time marching calculations above and can be seen in figures 12 and 13 with a mesh 32 times finer. 4 Extension to Large Systems As the primary objective is to allow the construction of reduced models for 3D configurations when a reasonable resolution for the Euler equations requires at least half a million degrees of freedom, forming the reduced model is a non trivial computational exercise. In this section we consider the feasibility of the extension of the projection method to this type of problem. Demonstration of the bifurcation predicitions for this size problem have already been obtained [7], exploiting sparse matrix solvers.

12 12 The higher order derivatives of equation (8) can be approximated using the finite differences, B(v, v) = 1 h 2 [f(x 0 + hv, α 0 ) + f(x 0 hv, α 0 )] + O(h 3 ) and C(v, v, v) = [f(x 0 + 3hv, α 0 ) 3f(x 0 + hv, α 0 ) + 3f(x 0 hv, α 0 ) f(x 0 3hv, α 0 )] /8h 3 + O(h 3 ) where h is small. Denoting the real and imaginary parts of the eigenvector q by q 1 and q 2 respectively q = q 1 + iq 2, q C n q 1, q 2 R n. The following relations are then true B(q, q) = B(q 1, q 1 ) B(q 2, q 2 ) + 2iB(q 1, q 2 ) B(q, q) = B(q 1, q 1 ) + b(q 2, q 2 ) C(q, q, q) = C(q 1, q 1, q 1 ) + C(q 1, q 2, q 2 ) + ic(q 1, q 1, q 2 ) + ic(q 2, q 2, q 2 ). The terms B(q 1, q 2 ), C(q 1, q 2, q 2 ), and C(q 1, q 1, q 2 ) are in the correct form to use equation (10) but it is enough to be able to calculate the multi-linear functions B(v, w) and C(v, v, w) for v, w R n. Using the identities and hence B(v, w) can be expressed as B(v + w, v + w) = B(v, v) + 2B(v, w) + B(w, w) B(v w, v w) = B(v, v) 2B(v, w) + B(w, w) B(v, w) = 1 [B(v + w, v + w) B(v w, v w)] 4 a similar set of identities holds for C C(v + w, v + w, v + w) = C(v, v, v) + 3C(v, v, w) + 3C(v, w, w) + C(w, w, w) C(v w, v w, v w) = C(v, v, v) 3C(v, v, w) + 3C(v, w, w) + C(w, w, w) and hence C(v, v, w) can be expressed as C(v, v, w) = 1 [C(v + w, v + w, v + w) C(v w, v w, v w) 2C(w, w, w)] 6 With these identities it is possible to calculate all the terms required for both the transformed system (6) and the projected system (9). The direct calculation of the bifurcation point provides q, q, x 0, and ω 0 so only the adjoint eigenvector p must be calculated in addition. This can be done easily and quickly with the inverse power method since we know the value of the eigenvalue and hence

13 13 an excellent shift. This method is already employed in the direct bifurcation solver to obtain initial estimates for q and is cheap compared to the direct bifurcation solution itself. The values of G 20, G 11, and G 02 are calculated using the identities above and require just eight function evaluations and a few inner products. The same applies for G 21. All these terms are fixed and only need to be calculated once. To avoid having to compute B(q, y) and B( q, y) at each iteration, since y is not fixed, two more complex linear systems are required see equations(7). This is due to the reduction onto the centre manifold and again they are fixed at the start and so only need to be calculated once. Once the above information is calculated the use of the reduced model is independent of the number of unknowns in the original system. 5 Conclusions We have shown that the use of the direct bifurcation method can provide extra useful information that can be incorporated into a reduced order model. The method of projection was chosen since it avoids the transformation of the system into its eigenbasis which is inconceivable for aeroelastic systems of realistic size. The tubular reactor is probably a hard test for the reduction method since the solution changes rapidly for very small increases in the bifuration parameter, (eg see the steep gradient in figure 3 around µ = ). A possible approach to practical aeroelastic analysis is to calculate a steady state using the coupled solver, then to calculate the bifurcation (flutter) point using a direct solve which in turn yields the information required for the model reduction. The reduced system can then be used to calculate the damping values below the flutter point which can then be used to compare with flight test data. The only part of this scheme not yet demonstrated for 3d problems is the model reduction, which will be done in the next stage of this work. 6 Acknowledgements This work was supported by EPSRC, MoD, and BAE SYSTEMS.

14 14 References [1] Lee, B. H. K., Price, S.J. and Wong, Y.S., Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos, Progess in Aerospace Sciences, Vol 35, 1999, pp [2] Lucia, D. J., Beran, P. S. and King, P. I., Reduced Order Modeling of an Elastic Panel in Transonic Flow, AIAA , 43rd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, Denver, CO, April [3] Lucia, D. J., Beran, P. S. and Silva, W. A., Aeroelastic System Development Using Proper Orthogonal Decomposition and Volterra Theory, AIAA , 2003 [4] Grzedzinski, J. Practical aspects of center manifold in nonlinear flutter calculations, International Forum on Aeroelasticity and Structural Dynamics, Amsterdam, June [5] Leung, A.Y.T. and Ge, T., An algorithm for Higher Order Hopf Normal forms, Shock and Vibration, Vol 2, No 4, pp , [6] Morton, S. A. and Beran, P.S.,Hopf-Bifurcation Analysis of Airfoil Flutter at Transonic Speeds, J Aircraft, 36, pp , [7] Badcock, K. J., Woodgate, M. and Richards, B. E., Direct aeroelastic bifurcation analysis of a symmetric wing based on the Euler equations, Glasgow University Aerospace Engineering Report 0315, 2003 [8] Ortiz,E.L., Step by step Tau method, Comput. Math Appl, 1, pp [9] Beran, P.S., A Domain-Decomposition Method for Airfoil Flutter Analysis, AIAA Paper , 1998.

15 15 Table 1: Grid convergence for the solution of the Augmented System No. of Cells Bifurcation Eigenvalue Newton CPU Time Nested parameter Iters Iters N/A cells 1st Order BC 32 cells 1st Order BC 64 cells 1st Order BC 128 cells 1st Order BC 256 cells 1st Order BC 0.7 y Figure 1: The grid convergence of the Y solution with a first order treatment of the boundary condition at x = 0 X cells 2nd Order BC 32 cells 2nd Order BC 64 cells 2nd Order BC 128 cells 2nd Order BC 256 cells 2nd Order BC 0.7 y Figure 2: The grid convergence of the Y solution with a second order treatment of the boundary condition at x = 0 X

16 Maximum value of Theta in [0,1] forwards backwards Bifurcation Parameter Figure 3: The equilibrium solution as mapped out by a continuation method varing the bifurcation parameter µ Log of the Residual Iteration Number Figure 4: Convergence of the Log of the residual against iteration number

17 Theta at x= Time Figure 5: The time history of Θ at x = 1 with µ = Theta at x= Time Figure 6: The time history of Θ at x = 1 with µ =

18 Full Model Reduced Model y(1) Time Figure 7: Comparison of the time history computed with full and reduced models of y at x = 1 with µ = and an initial deflection of δθ = Full Model Reduced Model y(1) Time Figure 8: Comparison of the time history computed with full and reduced models of y at x = 1 with µ = and an initial deflection of δθ = 0.001

19 B Reduced Model A Full Model Figure 9: The correspondence of amplitudes for the full and reduced models. The comparison of time histories at point A is shown in Figure 10 and in Figure 11 for point B Full Model Reduced Model y(1) Time Figure 10: Comparison of time histories close to the bifurcation point µ

20 Full Model Reduced Model y(1) Time Figure 11: Comparison of time histories far from the bifurcation point µ The full model was used to compute the solid line and the dot dashed line for the reduced model Full Model Reduced Model y(1) Time Figure 12: Comparison of time histories close to the bifurcation point µ

21 Full Model Reduced Model y(1) Time Figure 13: Comparison of time histories far from the bifurcation point µ The full model was used to compute the solid line and the dot dashed line for the reduced model.