EXTENSIONS OF THE HHT-α METHOD TO DIFFERENTIAL-ALGEBRAIC EQUATIONS IN MECHANICS

Transcription

1 EXTENSIONS OF THE HHT-α METHOD TO DIFFERENTIAL-ALGEBRAIC EQUATIONS IN MECHANICS LAURENT O. JAY AND DAN NEGRUT Abstract. We present second-order extensions of the Hilber-Hughes-Taylor-α HHT-α) method for systems of overdetermined differential-algebraic equations ODAEs) arising for example in mechanics. A detailed analysis of extensions of the HHT-α method is given. In particular a local and global error analysis is presented. Second order of convergence is theoretically demonstrated and practically illustrated by numerical experiments. A new variable stepsizes formula is proposed which preserves the second order of the method. Key words. Differential-algebraic equations, HHT-α method, variable stepsizes.. Introduction. The Hilber-Hughes-Taylor-α HHT-α) method [6, 7] and its generalizations such as the generalized-α method [3, 4] are widely used in structural and flexible multibody dynamics. This paper is concerned with extending the HHT-α method to systems of overdetermined differential-algebraic equations ODAEs) with index 3 constraints and their underlying index 2 constraints, e.g., to systems in mechanics having holonomic constraints. An extension of the HHT-α method to index 2 DAEs, e.g., to systems in mechanics with nonholonomic constraints, is briefly discussed as well. We have found extensions of the HHT-α method preserving its second order of convergence. Our extensions are indirect in the sense that we make use of the partitioned and additive structures of the ODAEs. Detailed mathematical proofs of second order of convergence of extensions of the HHT-α method to the systems of ODAEs considered are given. A new variable stepsizes formula is proposed which preserves the second order of the method. Second order of convergence of these extensions is numerically illustrated on two test problems. For DAEs global error estimates generally do not follow directly from local error estimates. The error propagation mechanism of a method for DAEs is usually more complicated than for ordinary differential equations ODEs). In particular, for DAEs one cannot generally infer a global order of convergence directly from its local error estimates, as an order reduction may occur due to error propagation []. Analysis of the direct extension of the HHT-α method to linear DAEs was performed in [2]. It was shown that for semi-explicit index 3 linear DAEs the direct application of the HHT-α method is inconsistent and suffers from instabilities, but that it may still converge when applied with constant stepsizes, similarly to BDF methods []. A first order extension of the HHT-α method to holonomically constrained mechanical systems was proposed in [2] and is based on projecting the solution of the underlying ODEs onto the constraints after each step. In [] the direct application of the HHT-α method to index 3 holonomically constrained mechanical systems is considered, but no convergence result is given. The extensions of the HHT-α method that we present in this paper have second order of convergence without relying on underlying ODEs and they also directly preserve the underlying index 2 constraints. This paper is organized as follows. In section 2 we describe the original HHT-α method for second order systems of ODEs and we propose a new variable stepsizes Department of Mathematics, 4 MacLean Hall, The University of Iowa, Iowa City, IA , USA. ljay@math.uiowa.edu and na.ljay@na-net.ornl.gov. This material is based upon work supported by the National Science Foundation under Grant No Department of Mechanical Engineering, University of Wisconsin-Madison, 342 Engineering Centers Bldg, 550 Engineering Dr., Madison, WI , USA. negrut@engr.wisc.edu.

2 2 L. O. Jay and D. Negrut formula preserving the second order of the method. In section 3 we present extensions of the HHT-α method to systems of ODAEs with index 3 constraints and underlying index 2 constraints. In section 4 we give a detailed analysis of extensions of the HHT-α method to these systems of ODAEs. In particular, we show existence and uniqueness of the numerical solution, we analyze the local error of the method, its stability with respect to consistent perturbations in the initial values, and prove its global second order of convergence. In section 5 we illustrate the second order of convergence of an extended HHT-α method on two test problems. In section 6 we propose an extension of the HHT-α method for index 2 DAEs. A short conclusion is given in section The HHT-α method for second order systems of ODEs. Second order systems of ODEs y = ft, y, y ) are equivalent to 2.) y = z, z = a, a = ft, y, z). In mechanics, y represents generalized coordinates, z represents the corresponding velocities, a represents the corresponding accelerations, ft, y, z) = M F t, y, z) where M is the mass matrix and F t, y, z) represents external forces. The HHT-α method for the system of equations 2.) can be expressed as an implicit non-standard one-step method as follows [6, 7] y, z, a ) = Φ h y 0, z 0, a 0 ) 2.2a) 2.2b) 2.2c) y =y 0 + hz 0 + h2 2 2β)a 0 + 2βa ), z =z 0 + h γ)a 0 + γa ), a = + α)ft, y, z ) αft 0, y 0, z 0 ), where h is the stepsize and t := t 0 +h. For the HHT-α method the coefficients α, β, γ are chosen according to α [ 3 ], 0 α)2, β =, γ = 4 2 α. The free coefficient α is a damping parameter. Notice that the notation for a 0 and a may be misleading. These values are not really approximations of at 0 ) and at ) respectively, but of at 0 +αh) and at +αh). The coefficient γ = 2 α is determined such that the method is of local order 2 in z when a 0 at 0 + αh) = Oh 2 ). When a 0 at 0 + αh) = Oh), e.g., when a 0 = at 0 ) = ft 0, y 0, z 0 ) the method is only of local order in z for the first step, i.e., z zt 0 + h) = Oh 2 ). However, in this situation since a at + αh) = Oh 2 ) the next step y 2, z 2, a 2 ) has nevertheless an error estimate in z of the form z 2 zt + h) = Oh 3 ). The HHT-α method is thus self-correcting, explaining in part its global second order of convergence even when a 0 is taken as a 0 = ft 0, y 0, z 0 ). The HHT-α method is generally applied with constant stepsizes in order to keep its second order of accuracy. For a 0 h) coming from the previous step with stepsize h, by changing the stepsize from h to h h, a is no more an approximation of local order to at + αh), i.e., it does not satisfy a at + αh) = Oh 2 ). Hence, without any modification the HHT-α method reduces to a first order method for all variables. To reestablish global second order of

3 Extensions of the HHT-α method to DAEs in mechanics 3 convergence while still allowing stepsize changes using a 0 h) from the previous step taken with stepsize h, one can replace the definition of a 0 for the current step by 2.3) a 0 := ft 0, y 0, z 0 ) + h ha 0 h) ft 0, y 0, z 0 )). Simply taking a 0 := αft, y, z ) + α)ft 0, y 0, z 0 ) analogously to 2.2c) leads to the expression z = z 0 + h 2 ft 0, y 0, z 0 ) + 2 ft, y, z ) ) which corresponds to the trapezoidal rule which has no damping parameter α and is thus not recommended except for the first step [0]). 3. Extensions of the HHT-α method to ODAEs. We consider semi-explicit index 3 DAEs of the form 3.a) 3.b) 3.c) 3.d) y =z, z =a + rt, y, λ), a=ft, y, z), 0=gt, y), where we assume that g y t, y)r λ t, y, λ) is invertible in the region of interest. In mechanics 3.d) represents holonomic constraints, λ represents Lagrange multipliers, and rt, y, λ) = M g T y t, y)λ where M is the mass matrix and g T y t, y)λ represents reaction forces coming from the constraints []. Differentiating 3.d) once with respect to t, we obtain additional constraints 3.e) 0 = g t t, y) + g y t, y)z. The whole system of ODAEs 3.) is of index 2. One more differentiation of 3.e) with respect to t leads to 3.2) 0 = g tt t, y) + 2g ty t, y)z + g yy t, y)z, z) + g y t, y)ft, y, z) + rt, y, λ)). We will not make a direct use of these additional constraints 3.2) in the numerical scheme 3.4) below. Nevertheless, it will be useful to consider them in the analysis of the method. From the constraints 3.2), one more differentiation gives an expression for λ, 3.3) λ = g y r λ ) g ttt + 3g tty z + 3g tyy z, z) + g yyy z, z, z) + 3g ty f + r) ) + 3g yy z, f + r) + g y f t + f y z + f z f + r) + r t + r y z) where we have not written explicitly the arguments t, y, z, λ) for f, r, g, f y, r y, g y, etc. We propose a new generalization of the HHT-α method for the system 3.). Although different in essence our approach is reminiscent of the GGL/stabilized index 2 formulation [5]. Here, instead of artificially introducing additional new algebraic variables in 3.a), we consider directly the systems of ODAEs 3.). Given y 0, z 0, a 0 ) we define the extended HHT-α method for 3.) as follows 3.4a) 3.4b) 3.4c) y =y 0 + hz 0 + h2 2 2β)a 0 + 2βa ) + h2 2 b)r 0 + br ), z =z 0 + h γ)a 0 + γa ) + h 2 R 0 + R ), a = + α)ft, y, z ) αft 0, y 0, z 0 ),

4 4 L. O. Jay and D. Negrut where b /2 is a free coefficient, 3.4d) R 0 := rt 0, y 0, Λ 0 ), R := rt, y, Λ ), and Λ 0 is not a value λ 0 coming from the previous step, but Λ 0 and Λ are locally determined by the two sets of constraints 3.4e) 0 = gt, y ), 0 = g t t, y ) + g y t, y )z. This determination of Λ 0 and Λ is an important point. The numerical solution y, z ) thus satisfies both constraints 3.d)-3.e) at each timestep. We propose the simple choice b = 0. For b = 0 and α = 0 the method is an additive combination of the 2-stage Lobatto IIIA and Lobatto IIIB implicit Runge-Kutta coefficients, and is known to be of second order for all variables [9] note that it is not the combination of Lobatto IIIA and Lobatto IIIB coefficients given in [8] since for unconstrained problems the HHT-α method is simply equivalent to the trapezoidal rule, the 2-stage Lobatto IIIA method). To make the method less implicit, one can replace R in 3.4d) by R := rt, ỹ, Λ ) where 3.5) ỹ := y 0 + hz 0. Another possibility is to take b = 0 and to replace the expression R 0 + R )/2 in 3.4b) by the midpoint approximation R /2 = r t 0 + h 2, y ) 0 + y, Λ /2 2 and also with y replaced by 3.5). The results given in this paper remain valid with these simplifications under some minor modifications. In particular, second order of global convergence as shown in Theorem 4.5 also holds. 4. Analysis of the extended HHT-α method for ODAEs. First we show existence and uniqueness of the numerical solution of the extended HHT-α method 3.4). Theorem 4.. Consider the overdetermined system of DAEs 3.) with initial conditions y 0, z 0, a 0 ) = y 0 h), z 0 h), a 0 h)) depending on h and satisfying gt 0, y 0 ) = Oh 3 ), g t t 0, y 0 ) + g y t 0, y 0 )z 0 = Oh 2 ), a 0 at 0 + αh) = Oh). Then for 0 h h 0 there exists a unique solution y, z, a, Λ 0, Λ ) depending on h to the system of equations 3.4) in a neighborhood of y 0, z 0, a 0, λ 0, λ 0 ) where λ 0 = λ 0 h) satisfies g tt t 0, y 0 ) + 2g ty t 0, y 0 )z 0 + g yy t 0, y 0 )z 0, z 0 ) Moreover, we have the estimates + g y t 0, y 0 )ft 0, y 0, z 0 ) + rt 0, y 0, λ 0 )) = Oh). 4.a) 4.b) y y 0 = Oh), z z 0 = Oh), a a 0 = Oh), Λ 0 λ 0 = Oh), Λ λ 0 = Oh).

5 Extensions of the HHT-α method to DAEs in mechanics 5 Remark 4.2. Note that the numerical solution y, z, a, Λ 0, Λ ) is functionnally independent of λ 0. The value λ 0 only indicates a solution branch to which the numerical solution is close. Varying slightly λ 0 to λ 0 + δ with a small perturbation δ = o) does not change the numerical solution y, z, a, Λ 0, Λ ). Proof. The proof of this theorem can be made by application of the implicit function theorem. We first introduce directly the definition 3.4d) of R 0, R into 3.4a) and 3.4b). Then we also replace partially some expressions for y and z explicitly in 3.4e). Multiplying the two equations of 3.4e) by 2/h 2 and /h respectively, we obtain the equivalent system of equations 0=y y 0 + hz 0 + h2 2β)a0 + 2βa + b)rt 0, y 0, Λ 0 ) + brt, y, Λ ) )), 2 0=z z 0 + h γ)a 0 + γa + 2 rt 0, y 0, Λ 0 ) + 2 rt, y, Λ ) )), ) 0=a + α)ft, y, z ) αft 0, y 0, z 0 ), 0= 2 h 2 g t, y 0 + hz 0 + h2 2β)a0 + 2βa + b)rt 0, y 0, Λ 0 ) + brt, y, Λ ) )), 2 0= h g tt, y ) + h g yt, y ) z 0 + h γ)a 0 + γa + 2 rt 0, y 0, Λ 0 ) + 2 rt, y, Λ ) )). Replacing a by its expression 3.4c) in the last two equations and then expanding in h around t 0, y 0, z 0 ) we obtain 0= 2 h 2 gt 0, y 0 ) + 2 h g tt 0, y 0 ) + g y t 0, y 0 )z 0 ) ) +g y t 0, y 0 ) 2β)a 0 + 2βft 0, y 0, z 0 ) + b)rt 0, y 0, Λ 0 ) + brt 0, y 0, Λ ) +g tt t 0, y 0 ) + 2g ty t 0, y 0 )z 0 + g yy t 0, y 0 )z 0, z 0 ) + Oh) 0= h g tt 0, y 0 ) + g y t 0, y 0 )z 0 ) + g tt t 0, y 0 ) + 2g ty t 0, y 0 )z 0 + g yy t 0, y 0 )z 0, z 0 ) +g y t 0, y 0 ) γ)a 0 + γft 0, y 0, z 0 ) + 2 rt 0, y 0, Λ 0 ) + ) 2 rt 0, y 0, Λ ) + Oh). By using the hypotheses of the theorem all equations are satisfied at h = 0 by y 0), z 0), a 0), Λ 0 0), Λ 0)) := y 0 0), z 0 0), a 0 0), λ 0 0), λ 0 0)). The Jacobian at h = 0 of the above equations with respect to y, z, a, Λ 0, Λ ) is given by I O O O O O I O O O I O O b)m 0 bm 0 2 M 0 2 M 0 where M 0 := g y t 0, y 0 )r λ t 0, y 0, λ 0 ) is invertible. Since [ ] [ b)m0 bm 0 b) b = 2 M 0 2 M ] M 0.

6 6 L. O. Jay and D. Negrut the Jacobian at h = 0 is invertible provided b /2. The conclusion and the estimates 4.) now follow by application of the implicit function theorem. We now consider local error estimates: Theorem 4.3. Consider the overdetermined system of DAEs 3.) with initial conditions y 0, z 0, a 0 ) at t 0 satisfying gt 0, y 0 ) = 0, g t t 0, y 0 ) + g y t 0, y 0 )z 0 = 0, a 0 at 0 + αh) = Oh), and let λ 0 be such that g tt t 0, y 0 )+2g ty t 0, y 0 )z 0 +g yy t 0, y 0 )z 0, z 0 )+g y t 0, y 0 )ft 0, y 0, z 0 )+rt 0, y 0, λ 0 )) = 0. Then for 0 h h 0 the numerical solution y, z, a ) at t = t 0 + h to the system of equations 3.4) satisfies 4.2) y yt ) = Oh 3 ), z zt ) = Oh 2 ), a at + αh) = Oh 2 ), where yt), zt)) is the exact solution to 3.) at t passing through y 0, z 0 ) at t 0. If in addition we assume that 4.3) a 0 at 0 + αh) = Oh 2 ) then 4.4) z zt ) = Oh 3 ). Proof. The Taylor series of the exact solution yt), zt)) at t = t 0 + h satisfies yt ) = y 0 + hz 0 + h2 2 f 0 + r 0 ) + Oh 3 ), zt ) = z 0 + hf 0 + r 0 ) + Oh 2 ), where f 0 := ft 0, y 0, z 0 ) and r 0 := rt 0, y 0, λ 0 ). We know from Theorem 4. that Λ 0 h) λ 0 = Oh) and Λ h) λ 0 = Oh). For the numerical solution y, z ) we have by direct application of the estimates 4.) in the definition 3.4a)-3.4b) y = y 0 + hz 0 + h2 2 f 0 + r 0 ) + Oh 3 ), z = z 0 + hf 0 + r 0 ) + Oh 2 ). Hence, we obtain y yt ) = Oh 3 ) and z zt ) = Oh 2 ). From 2.) and 3.4c) for a we have and at + αh) = f α)hf t0 + f y0 z 0 + f z0 f 0 + r 0 )) + Oh 2 ) a = f α)hf t 0 + f y 0 z 0 + f z 0 f 0 + r 0 )) + Oh 2 ). A direct consequence is the estimate a at + αh) = Oh 2 ). It remains to show 4.4) when 4.3) holds. The condition a 0 at 0 + αh) = Oh 2 ) is equivalent to a 0 = f 0 + hαf t0 + f y0 z 0 + f z0 f 0 + r 0 )) + Oh 2 ). The Taylor series of zt ) at t 0 satisfies zt ) = z 0 +hf 0 +r 0 )+ h2 2 ft0 + f y0 z 0 + f z0 f 0 + r 0 ) + r t0 + r y0 z 0 + r λ0 λ ) 0 +Oh 3 )

7 Extensions of the HHT-α method to DAEs in mechanics 7 where λ 0 corresponds to the expression 3.3) evaluated at t 0, y 0, z 0, λ 0 ). Since γ = /2 α we get ) ) 2 + α a α a = f 0 + h 2 f t0 + f y0 z 0 + f z0 f 0 + r 0 )) + Oh 2 ). We also have R 0 = r 0 + hr λ0 Λ 0 0) + Oh2 ), R = r 0 + hr t0 + r y0 z 0 + r λ0 Λ 0)) + Oh2 ). Putting all previous estimates together we obtain z =z 0 + hf 0 + r 0 ) + h2 f t f y 0 z 0 + f z 0 f 0 + r 0 ) + r t0 + r y0 z 0 + r λ0 Λ 0 0) + Λ 0)) ) + Oh 3 ). Thus, it remains to show that Λ 0 0) + Λ 0) = λ 0. This expression can be obtained by expanding 4.5) 0 = h 2 g tt, y ) + g y t, y )z ) around t 0, y 0, z 0, λ 0 ) into h-powers and then letting h 0. First, we write y = y 0 +δ with δ := hz 0 + h 2 /2f 0 + r 0 ) + Oh 3 ) = Oh) and we expand in h and δ g t t 0 + h, y 0 + δ)=g t0 + g tt0 h + g ty0 δ + 2 g y t 0 + h, y 0 + δ)=g y0 + g ty0 h + g yy0 δ + 2 gttt0 h 2 + 2g tty0 hδ + g tyy0 δ, δ) ) + Oh 3 ), gtty0 h 2 + 2g tyy0 hδ + g yyy0 δ, δ) ) + Oh 3 ). Expanding 4.5) in h-powers, grouping the terms, and letting h 0 we finally obtain 0=g ttt0 + 3g tty0 z 0 + 3g tyy0 z 0, z 0 ) + g yyy0 z 0, z 0, z 0 ) + 3g ty0 f 0 + r 0 ) +3g yy0 z 0, f 0 + r 0 ) + g y0 f t 0 + f y 0 z 0 + f z 0 f 0 + r 0 ) + r t0 + r y0 z 0 ) +g y0 r λ0 Λ 0 0) + Λ 0)). From 3.3) this leads to the desired result Λ 0 0)+Λ 0) = λ 0 and therefore z zt ) = Oh 3 ). To analyze the error propagation we introduce the projectors 4.6) Qt, y, λ) := r λ t, y, λ)g y t, y)r λ t, y, λ)) g y t, y), P t, y, λ) := I Qt, y, λ). They have the properties Qt, y, λ)r λ t, y, λ) = r λ t, y, λ), g y t, y)qt, y, λ) = g y t, y), P t, y, λ)r λ t, y, λ) = 0, g y t, y)p t, y, λ) = 0. Before proving global convergence, we need to study changes in the numerical solution with respect to perturbations in consistent initial conditions: Theorem 4.4. Consider ỹ k, z k, ã k ) ŷ k, ẑ k, â k ) at t k satisfying the constraints 3.d) and 3.e). Let y k := ŷ k ỹ k, z k := ẑ k z k, a k := â k ã k, satisfying y k = Oh), z k = Oh), a k = Oh),

8 8 L. O. Jay and D. Negrut and let ỹ k+, z k+, ã k+ ) and ŷ k+, ẑ k+, â k+ ) be the corresponding HHT-α solutions 3.4). Then we have P k+ y k+ =P k y k + hp k z k + /2 β) h 2 P k a k +Oh P k y k + h 2 P k z k + h 3 a k ), hp k+ z k+ =hp k z k + γ) h 2 P k a k +Oh 2 P k y k + h 2 P k z k + h 3 a k ), h 2 a k+ =Oh 2 P k y k + h 2 P k z k + h 3 a k ), Q k+ y k+ =O P k+ y k+ ), Q k+ z k+ =O P k+ y k+ + P k+ z k+ ), where P k := P t k, ỹ k, λ k ), P k+ := P t k+, ỹ k+, λ k+ ), and λ k, λ k+ are such that the constraints 3.2) are satisfied for ỹ k, z k, λ k ) and ỹ k+, z k+, λ k+ ) respectively. Proof. Let λ k be such that the constraints 3.2) are satisfied for ŷ k, ẑ k, λ k ) and satisfying λ k λ k = Oh). By Theorem 4. we have Hence, we also have ỹ k+ ỹ k = Oh), z k+ z k = Oh), ã k+ ã k = Oh), Λ k0 λ k = Oh), Λk λ k = Oh), ŷ k+ ŷ k = Oh), ẑ k+ ẑ k = Oh), â k+ â k = Oh), Λ k0 λ k = Oh), Λk λ k = Oh). y k+ = Oh), z k+ = Oh), a k+ = Oh), Λ k0 = Oh), Λ k = Oh), where Λ k0 := Λ k0 Λ k0, and Λ k := Λ k Λ k. Substracting 3.4abc) for ŷ k+, ẑ k+, â k+ ) from 3.4abc) for ỹ k+, z k+, ã k+ ) and linearizing around t k, ỹ k, z k, ã k ) we obtain 4.8a) 4.8b) y k+ = y k + h z k + h2 2 2β) a k + 2β a k+ ) + h2 2 b)r y,k y k + r λ,k Λ k0 ) + br y,k+ y k+ + r λ,k+ Λ k )) +Oh 2 y k 2 + h 2 y k+ 2 + h 2 Λ k0 2 + h 2 Λ k 2 ), z k+ = z k + h γ) a k + γ a k+ ) + h 2 r y,k y k + r λ,k Λ k0 + r y,k+ y k+ + r λ,k+ Λ k ) 4.8c) +Oh y k 2 + h y k+ 2 + h Λ k0 2 + h Λ k 2 ), a k+ = + α) f y,k+ y k+ + f z,k+ z k+ ) α f y,k y k + f z,k z k ) +O y k 2 + y k+ 2 + z k 2 + z k+ 2 ). =O y k + y k+ + z k + z k+ ). From 0 = gt k+, ŷ k+ ), 0 = gt k+, ỹ k+ ), we obtain by linearization around t k+, ỹ k+ ) 4.9) 0 = g y,k+ y k+ + O y k+ 2 ).

9 Extensions of the HHT-α method to DAEs in mechanics 9 Introducing the expression 4.8a) for y k+ we get 4.0) 2 b)gy,k+ r λ,k h 2 Λ k0 + bg y,k+ r λ,k+ h 2 Λ k ) = g y,k+ y k + hg y,k+ z k + h2 2 2β)g y,k+ a k + 2βg y,k+ a k+ ) + h2 2 b)g y,k+r y,k y k + bg y,k+ r y,k+ y k+ ) + Oh 2 y k 2 + y k+ 2 + h 2 Λ k0 2 + h 2 Λ k 2 ). From 0 = g t t k+, ŷ k+ ) + g y t k+, ŷ k+ )ẑ k+, 0 = g t t k+, ỹ k+ ) + g y t k+, ỹ k+ ) z k+, we obtain by linearization around t k+, ỹ k+, z k+ ) 4.) 0=g ty,k+ y k+ + g yy,k+ z k+, y k+ ) + g y,k+ z k+ +O y k+ 2 + z k+ 2 ). Introducing the expression for h z k+ from 4.8b) we get 4.2) 2 gy,k+ r λ,k h 2 Λ k0 + g y,k+ r λ,k+ h 2 Λ k ) = hg ty,k+ y k+ + hg yy,k+ z k+, y k+ ) + hg y,k+ z k + h 2 γ)g y,k+ a k + γg y,k+ a k+ ) + h2 2 g y,k+r y,k y k + g y,k+ r y,k+ y k+ ) + Oh 2 y k 2 + h y k+ 2 + h z k+ 2 + h 2 Λ k0 2 + h 2 Λ k 2 ), Since by assumption 0 = gt k, ŷ k ) and 0 = gt k, ỹ k ) we obtain by linearization around t k, ỹ k ) 4.3) 0 = g y,k y k + O y k 2 ). Therefore, we can estimate the term g y,k+ y k in 4.0) by g y,k+ y k = g y,k y k + Oh y k ) = Oh y k + y k 2 ). We can also estimate the term g y,k+ z k in 4.0) and 4.2) by We also have in 4.0) and 4.2) g y,k+ z k = g y,k Q k z k + Oh z k ). g y,k+ a k = g y,k Q k a k + Oh a k ), g y,k+ a k+ = g y,k+ Q k+ a k+. For b /2 and h sufficiently small the matrix [ ] b)gy,k+ r λ,k bg y,k+ r λ,k+ g y,k+ r λ,k g y,k+ r λ,k+

10 0 L. O. Jay and D. Negrut is invertible and has a bounded inverse. Therefore, from 4.0) and 4.2) we obtain the following estimates for h 2 Λ k0 and h 2 Λ k 4.4a) h 2 Λ k0 =O h y k + y k 2 + h y k+ + y k+ 2 + h Q k z k + h 2 z k + h z k+ 2 + h 2 Q k a k + h 3 P k a k + h 2 Q k+ a k+ + h 2 Λ k0 2 + h 2 Λ k 2), 4.4b) h 2 Λ k =O h y k + y k 2 + h y k+ + y k+ 2 + h Q k z k + h 2 z k + h z k+ 2 + h 2 Q k a k + h 3 P k a k + h 2 Q k+ a k+ + h 2 Λ k0 2 + h 2 Λ k 2). Multiplying 4.8a) and 4.8b) from the left by P k+ and hp k+ respectively, using P k+ r λ,k = Oh), P k+ r λ,k+ = 0, we get 4.5a) P k+ y k+ = P k y k + hp k z k + h2 2 2β)P k a k + 2βP k+ a k+ ) +Oh y k + h 2 y k+ + h 2 z k + h 3 a k + h 3 Λ k0 4.5b) + h 2 y k 2 + h 2 y k+ 2 + h 2 Λ k0 2 + h 2 Λ k 2 ), hp k+ z k+ = hp k z k + h 2 γ)p k a k + γp k+ a k+ ) +Oh 2 y k + h 2 y k+ + h 2 z k + h 3 a k + h 3 Λ k0 + h 2 y k 2 + h 2 y k+ 2 + h 2 Λ k0 2 + h 2 Λ k 2 ). Inserting the estimates 4.4) and 4.8c) into 4.5) we obtain 4.6a) 4.6b) P k+ y k+ = P k y k + hp k z k + /2 β)h 2 P k a k +Oh y k + h 2 y k+ + h 2 z k + h 2 z k+ + h 3 a k ), hp k+ z k+ = hp k z k + h 2 γ)p k a k +Oh 2 y k + h 2 y k+ + h 2 z k + h 2 z k+ + h 3 a k ). From 4.9) we have Q k+ y k+ = O y k+ 2 ). Thus, y k+ =P k+ y k+ + Q k+ y k+ = P k+ y k+ + O y k+ 2 ) 4.7) =P k+ y k+ + O P k+ y k+ 2 ) = P k+ y k+ + Oh P k+ y k+ ). Similarly, from 4.3) we have 4.8) y k = P k y k + Oh P k y k ). From 4.) we have Q k+ z k+ = O y k+ + h z k+ ) = O P k+ y k+ + h z k+ ). Therefore, z k+ =P k+ z k+ + Q k z k+ = P k+ z k+ + O y k+ + h z k+ ) 4.9) =P k+ z k+ + O P k+ y k+ + h P k+ z k+ ).

11 Extensions of the HHT-α method to DAEs in mechanics Similarly, from 0 = g t t k, ŷ k ) + g y t k, ŷ k )ẑ k and 0 = g t t k, ỹ k ) + g y t k, ỹ k ) z k we have 4.20) z k = P k z k + O P k y k + h P k z k ). Taking into account the above estimates 4.7)-4.8)-4.9)-4.20) into 4.6) finally leads to the desired result. Global convergence of the HHT-α method 3.4) can now be proved: Theorem 4.5. Consider the overdetermined system of DAEs 3.) with initial conditions y 0, z 0 ) at t 0 and a 0 satisfying gt 0, y 0 ) = 0, g t t 0, y 0 ) + g y t 0, y 0 )z 0 = 0, a 0 at 0 + αh) = Oh). Then the HHT-α solution y n, z n, a n ) to the system of equations 3.4) satisfies for 0 h h 0 and t n t 0 = nh Const y n yt n ) = Oh 2 ), z n zt n ) = Oh 2 ), a n at n + αh) = Oh 2 ) n ), where yt), zt)) is the exact solution to 3.) at t passing through y 0, z 0 ) at t 0 and at) is given by 3.c). y k k y k Proof. We consider two neighboring HHT-α approximations y k k, zk k, ak, z k k, a k k ) n k=k with k =,..., n and we denote their difference by y k := k )n k=k, k yk k, z k := z k k zk k, a k := a k k ak k. We assume that y k = Oh), z k = Oh), a k = Oh). These assumptions can be justified by induction, see below. For k = k, y k, z k, a k are just the local error 4.2) of the HHT-α method 3.4) with y k k, z k k ) being the exact solution passing through y k k, zk k ) and y k, z k, a k ) being the HHT-α numerical approximation from the same k k k point. The HHT-α approximations satisfy the constraints 3.d)-3.e) and we get by application of Theorem 4.4 for k = k,..., n P k+ y k+ =P k y k + hp k z k + /2 β) h 2 P k a k +Oh P k y k + h 2 P k z k + h 3 P k a k + h 3 Q k a k ), P k+ z k+ =P k z k + γ) hp k a k +Oh P k y k + h P k z k + h 2 P k a k + h 2 Q k a k ), hp k+ a k+ =Oh P k y k + h P k z k + h 2 P k a k + h 2 Q k a k ), hq k+ a k+ =Oh P k y k + h P k z k + h 2 P k a k + h 2 Q k a k ). Taking a norm of these expressions leads to the estimates with matrix M := P k+ y k+ P k+ z k+ h P k+ a k+ h Q k+ a k+ M P k y k P k z k h P k a k h Q k a k + Oh) h + Oh 2 ) h 2 β + Oh2 ) Oh 2 ) Oh) + Oh) γ + Oh) Oh) Oh) Oh) Oh) Oh) Oh) Oh) Oh) Oh).

12 2 L. O. Jay and D. Negrut Defining the matrix T := γ we can first transform the matrix M by a similarity transformation to the form + Oh) h + Oh 2 ) h 2 β γ ) + Oh2 ) Oh 2 ) N := T MT = Oh) + Oh) Oh) Oh) Oh) Oh) Oh) Oh) Oh) Oh) Oh) Oh) For the matrix N there is a first linear invariant subspace V 2 R 4 associated to the two eigenvalues µ = + Oh) and µ 2 = + Oh). This subspace V 2 is of the form V 2 = spane +Oh), e 2 +Oh)) where e :=, 0, 0, 0) T R 4 and e 2 := 0,, 0, 0) T R 4. For the matrix N there is also a second linear invariant subspace V 34 R 4 associated to the two eigenvalues µ 3 = 0 + Oh) and µ 4 = 0 + Oh). This subspace V 34 is of the form V 34 = spane 3 + Oh), e 4 + Oh)) where e 3 := 0, 0,, 0) T R 4 and e 4 := 0, 0, 0, ) T R 4. Therefore, there is a transformation V = I + Oh) with inverse V = I + Oh) such that NV = V B with B block-diagonal, i.e., B := V NV =, + Oh) Oh) 0 0 Oh) + Oh) Oh) Oh) 0 0 Oh) Oh) For 2 m n, from mh nh Const we obtain giving P n y n P n z n h P n a n h Q n a n M m = T V B m V T = By Theorem 4.4 we have Q k y k = O P k y k ), O) O) O) Oh) O) O) O) Oh) Oh) Oh) Oh) Oh 2 ) Oh) Oh) Oh) Oh 2 ) O) O) O) Oh) O) O) O) Oh) Oh) Oh) Oh) Oh 2 ) Oh) Oh) Oh) Oh 2 )., P n m y n m P n m z n m h P n m a n m h Q n m a n m Q k z k = O P k y k + P k z k ). Hence since y k = P k y k + Q k y k and z k = P k z k + Q k z k, we get y n z n C y n m + z n m + h a n m 4.2) y n m + z n m + h a n m. a n y n m + z n m + h a n m First, we consider the HHT-α solution using the exact value at 0 + αh). We denote it by y k, z k, a k ) n k=0. For k = 0 we have y 0 = y 0, z 0 = z 0, and a 0 = at 0 + αh). Taking m := k in 4.2) leads to y n c y h 3, z n c z h 3, a n c a h 3...

13 Extensions of the HHT-α method to DAEs in mechanics 3 The assumptions y k = Oh), z k = Oh), a k = Oh) are thus justified by induction on k. Summing up these estimates we obtain yt n ) y n zt n ) z n n k = n k = at n + αh) a n y k n z k n n k = y k n c y nh 3 C y h 2, z k n c z nh 3 C z h 2, a k n ak n c a nh 3 C a h 2. Now, suppose that a 0 satisfies a 0 = at 0 + αh) + Oh). We denote the corresponding HHT-α solution using this approximate value of a 0 by y k, z k, a k ) n k=0. We want to estimate y n y n, z n z n, and a n a n. Using 4.2) for m = n, since y 0 = y 0 and z 0 = z 0, we simply obtain y n y n z n z n C h a 0 a 0 h a 0 a 0 = Oh 2 ) a n a n h a 0 a 0 since a 0 a 0 = a 0 at 0 + αh) = Oh). The assumptions y k y k = Oh), z k z k = Oh), a k a k = Oh) are also justified by induction on k. By combining the above estimates we finally we get the desired result y n yt n ) = y n y n + y n yt n ) = Oh 2 ), z n zt n ) = z n z n + z n zt n ) = Oh 2 ), a n at n + αh) = a n a n + a n at n + αh) = Oh 2 ). Remark that the proof of this Theorem remains valid with variable stepsizes provided the values a k are corrected for example by 2.3). 5. Numerical experiments. 5.. A first test problem. We consider the following mathematical test problem [ ] [ ] [ ] [ ] y z z y 2 =, y z 2 z 2 = z 2 + 2y 2 z + e t y λ 2 y 2z 2 2y z y 2 z 2 + y 2 λ 2, [ ] [ 0 = y 2 y 2 ] [ ] [, 0 = 2y y 2 z + y 2z ] 2. Notice that these equations are nonlinear in λ. Consistent initial conditions at t 0 = 0 are given by [ ] [ ] [ ] [ ] y 0) z 0) [ =, =, λ 0) ] = [ ]. y 2 0) z 2 0) 2 The exact solution to this system of DAEs is given by [ ] [ ] [ ] [ ] y t) e t z t) e = y 2 t) e 2t, = t z 2 t) 2e 2t, [ λ t) ] = [ e t ]. We have applied the extended HHT-α method 3.4) with parameters α = 0.5 and b = 0.3 for various stepsizes h. We observe global convergence of order 2 at t =

14 4 L. O. Jay and D. Negrut 0 4 error of extended HHT 0 5 errors in y,z, and a h Fig. 5.. Global errors y n yt n) 2 ), z n zt n) 2 ), a n at n + αh) 2 ) for first test problem at t n = and the extended HHT-α method α = 0.5, b = 0.3). One observes global convergence of order 2 in h. in Fig. 5. as expected from Theorem 4.5. In Fig. 5.2 we have repeated the same numerical experiment by simply replacing y in R with 3.5). We still observe global convergence of order 2. In Fig. 5.3 we have applied the HHT-α method with variable stepsizes alternating between h/3 and 2h/3. We have plotted in Fig. 5.3 the error versus the average stepsize h/2. We observe a reduction to convergence of order one as expected from the remarks in section 2. To reestablish second order of convergence for variable stepsizes we have made use of the modification 2.3) for a n, i.e., a n := ft n, y n, z n ) + h n h n a n h n ) ft n, y n, z n )). We have applied again the HHT-α method with variable stepsizes alternating between h/3 and 2h/3 using this modification. This time we observe second order of global convergence in Fig A pendulum model. The pendulum model in Fig. 5.5 was used to carry out a second set of numerical experiments. Using the notation z i = y i i =, 2, 3), the constrained equations of motion associated with this model are mz 0 0 [ ] mz 2 = mg ml 2 3 z 3 cz 3 k y ) 0 λ, 3 3π λ 2 L siny 3 ) L cosy 3 ) 2 while the constraint equations at the position and velocity levels are [ ] [ ] [ ] [ ] 0 y L cosy = 3 ) 0 z + L siny, = 3 )z 3. 0 y 2 L siny 3 ) 0 z 2 L cosy 3 )z 3

15 Extensions of the HHT-α method to DAEs in mechanics error of extended HHT with modification 0 5 errors in y,z, and a h Fig Global errors y n yt n) 2 ), z n zt n) 2 ), a n at n + αh) 2 ) for first test problem at t n = and the extended HHT-α method α = 0.5, b = 0.3) with modification 3.5). One observes global convergence of order 2 in h. The parameters associated with this model are as follows: mass m = 5, length L = 2, spring stiffness k = 3000, damping coefficient c = 00, gravitational acceleration g = 9.8. All units used herein are SI units. The evolution of the pendulum angle y 3 = θ on the time interval [0, 4] is shown in Fig In the numerical experiments, the pendulum is started from an initial position that corresponds to y 3 = 3π/2, and z 3 = 0. A reference solution was generated by applying an explicit Runge-Kutta method of order 4 RK4) with a small constant stepsize h = The explicit integrator RK4 was used in conjunction with an equivalent underlying ODE problem that provided directly the time evolution of y 3 y 3 = z 4mL 2 3, z cz 3 + k y 3 3π ) + mgl cosy 3 ) = 0. 2 Figs. 5.7 and 5.8 support the convergence results obtained in Theorem 4.5. The global error in y 3 and z 3 at time t = 2 is plotted in these figures against a series of stepsizes used for integration. The plots confirm that the global errors y 3,n y 3 t n ) and z 3,n z 3 t n ) associated with the extended HHT-α method 3.4) are of order two. Note that Figs. 5.7 and 5.8 report results for α = 0, b = 0 and α = 0.3, b = 0 respectively. 6. Extension of the HHT-α method to DAEs with index 2 constraints. Consider semi-explicit index 2 DAEs of the form 6.a) 6.b) y =z, z =a + rt, y, z, ψ),

16 6 L. O. Jay and D. Negrut 0 2 error of extended HHT with variable stepsizes 0 3 errors in y,z, and a h Fig Global errors y n yt n) 2 ), z n zt n) 2 ), a nh n ) at n + αh n ) 2 ) for first test problem at t n = and the extended HHT-α method α = 0.5, b = 0.3) with variables stepsizes and unmodified a n. One observes global convergence of order in h. 6.c) 6.d) a=ft, y, z), 0=kt, y, z), where we assume k z t, y, z)r ψ t, y, z, ψ) is invertible in the region of interest. We can consider for example kt, y, z) = g t t, y) + g y t, y)z from 3.e) and rt, y, z, ψ) = rt, y, ψ) from 3.b). We propose a generalization of HHT-α methods for the system 6.) similar to 3.4), y =y 0 + hz 0 + h2 2 2β)a 0 + 2βa ) + h2 2 R /2, z =z 0 + h γ)a 0 + γa ) + hr /2, a = + α)ft, y, z ) αft 0, y 0, z 0 ), where R /2 = r t 0 + h2, y 0 + h2 z 0, 2 ) z 0 + z ), Ψ /2. The algebraic variable Ψ /2 is determined by the constraint 0 = kt, y, z ). Hence, the numerical solution satisfies the constraint 6.d) at each timestep. replacing the expression for z explicitly in this equation, we obtain equivalently By 0 = h kt, y, z 0 + h γ)a 0 + γa ) + hr /2 ).

17 Extensions of the HHT-α method to DAEs in mechanics error of extended HHT with variable stepsizes 0 5 errors in y,z, and a h Fig Global errors y n yt n) 2 ), z n zt n) 2 ), a nh n ) at n + αh n ) 2 ) for first test problem at t n = and the extended HHT-α method α = 0.5, b = 0.3) with variables stepsizes and modified a n 2.3). One observes global convergence of order 2 in h. Existence and uniqueness of the numerical solution y, z, a, Ψ /2 ) is ensured and can be shown by application of the implicit function theorem. When kt, y, z) is linear in z and rt, y, z, ψ) is linear in ψ we obtain a linear equation for Ψ /2. However, since generally kt, y, z) or ft, y, z) are nonlinear in y, we generally have a system of nonlinear equations to solve in terms of y. If kt, y, z) and ft, y, z) are linear in y and z, and if rt, y, z, ψ) is linear in ψ, we obtain a system of linear equations for y, z, a, Ψ /2 ). Global second order of convergence of the new extended HHT-α method can be proved in a similar way as for 3.). 7. Conclusion. In this paper we have presented second order extensions of the HHT-α method for systems of ODAEs with index 3 and index 2 constraints arising for example in mechanics. We have given detailed mathematical proofs of convergence of extensions of the HHT-α method for semi-explicit ODAEs with index 3 constraints and underlying index 2 constraints. We have taken into account the structure of the equations to extend the HHT-α method to DAEs in order to keep its second order of accuracy. We have also proposed an elementary way to preserve the second order of the HHT-α method when using variable stepsizes, a technique which is also relevant for the HHT-α method when applied to ODEs. The HHT-α method and its extensions to DAEs is relatively simple to express and to implement. However, its analysis in the context of DAEs was found to be surprisingly difficult. After the submission of this manuscript, we learned about a similar extension of the generalized-α method found independently by Lunk and Simeon [0]. They consider problems of the form 3.) with rt, y, λ) linear in λ, whereas in this paper rt, y, λ) may be nonlinear. Their extension is slightly different, when rt, y, λ) = rt, y)λ they replace the term b)r 0 + br = b)rt 0, y 0 )Λ 0 + brt, y )Λ in

18 8 L. O. Jay and D. Negrut Fig Pendulum model. 3.4a) by b)rt 0, y 0 ) + brt, y ))Λ 0. REFERENCES [] K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical solution of initial-value problems in differential-algebraic equations, SIAM Classics in Appl. Math., SIAM, Philadelphia, Second Edition, 996. [2] A. Cardona and M. Géradin, Time integration of the equations of motion in mechanism analysis, Comput. & Structures, ), pp [3] J. Chung and G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, J. Appl. Mech., ), pp [4] S. Erlicher, L. Bonaventura, and O. S. Bursi, The analysis of the generalized-α method for nonlinear dynamics problems, Comput. Mech., ), pp [5] C. W. Gear, G. K. Gupta, and B. J. Leimkuhler, Automatic integration of the Euler- Lagrange equations with constraints, J. Comput. Appl. Math., ), pp [6] H. M. Hilber, T. J. R. Hughes, and R. L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engng Struct. Dyn., 5 977), pp [7] T. J. R. Hughes, Finite element method - Linear static and dynamic finite element analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 987. [8] L. O. Jay, Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, SIAM J. Numer. Anal., ), pp [9], Structure preservation for constrained dynamics with super partitioned additive Runge- Kutta methods, SIAM J. Sci. Comput., ), pp [0] C. Lunk and B. Simeon, Solving constrained mechanical systems by the family of Newmark and α-methods, ZAMM, ), pp [] D. Negrut, R. Rampalli, G. Ottarsson, and A. Sajdak, On an implementation of the HHT method in the context of index 3 differential algebraic equations of multibody dynamics,

19 Extensions of the HHT-α method to DAEs in mechanics 9 time variation of pendulum angle pendulum angle [rad] time [s] Fig Time evolution of pendulum angle y 3. ASME J. Comp. Nonlin. Dyn., ). To appear. [2] J. Yen, L. Petzold, and S. Raha, A time integration algorithm for flexible mechanism dynamics: The DAE-alpha method, Comput. Methods Appl. Mech. Engrg., ), pp

20 20 L. O. Jay and D. Negrut 0 5 error of extended HHT: Pendulum, α=0 0 6 errors in angle and angular velocity h Fig Global errors y 3,n y 3 t n) ), z 3,n z 3 t n) ) at t n = 2 for a simple pendulum and the extended HHT-α method α = 0, b = 0). One observes global convergence of order 2 in h. 0 5 error of extended HHT: Pendulum, α= errors in angle and angular velocity h Fig Global errors y 3,n y 3 t n) ), z 3,n z 3 t n) ) at t n = 2 for a simple pendulum and the extended HHT-α method α = 0.3, b = 0) One observes global convergence of order 2 in h.