Differential-Algebraic Equations (DAEs) Differential-algebraische Gleichungen

Transcription

1 Differential-Algebraic Equations (DAEs) Differential-algebraische Gleichungen Bernd Simeon Vorlesung Wintersemester 2012/13 TU Kaiserslautern, Fachbereich Mathematik 1. Examples of DAEs 2. Theory of DAEs 3. Numerical Methods References / Literatur: Brenan / Campbell / Petzold: Numerical Solution of Initial Value Problems in Differential-Algebraic Equations. SIAM, 1996 Hairer / Wanner: Solving ODEs II. Springer, 1996

2 Vorwort These lecture notes have been compiled as basic reference for the course Differential-Algebraic Equations at the Technische Universität Kaiserslautern. The notes contain most of the relevant formulas and some basic terminology. Students are encouraged to add their own notes and sketches during the lecture. Kaiserslautern, October 2012 Bernd Simeon

3 Chapter 1 Examples 1 This chapter deals with typical examples of differential-algebraic equations (DAEs). We start with a singularly perturbed ordinary differential equation (ODE) and turn then to constrained mechanical systems as typical problem class. In passing we also look at electric circuits. 1.1 Van der Pol s Equation Consider the second order ODE Solution behavior: Case α > 0: Case α < 0: Case α = 0: z(t) + αż(t) + z(t) = 0. The nonlinear term α(z) := µ(z 2 1) with constant parameter µ > 0 acts as a controller (Steuerung) and leads to Van der Pol s equation z + µ(z 2 1)ż + z = 0. (1.1) 1

4 y y1 There is no closed-form solution! Figure 1: Van der Pol s equation in phase space Transforming (1.1) to the new time x = t/µ results in an oscillation whose period does not depend on µ. From u(x) = z(µx) = z(t(x)), u = µż we get 1 µ 2u + (u 2 1)u + u = 0. The case µ 1 is of particular interest, and we define then ε := 1/µ 2 1. Moreover, we introduce Liénhard s coordinates y 2 (x) := u(x), y 1 (x) := εy 2 + (y 3 2/3 y 2 ) and obtain the singularly perturbed system (singulär gestörtes System) y 1 = y 2, εy 2 = y 1 y2/3 3 + y 2. (1.2) Give a sketch of the solution of (1.2) in phase space! What happens in the limit case ε 0? 2

5 1.2 Mechanical Multibody Systems We start with an informal definition. A multibody system is defined to be a set of rigid bodies and interconnection elements. The latter are joints that constrain the relative motion of pairs of bodies and springs and dampers that act as compliant elements. Moreover, the bodies possess a specific mass and geometry whereas the interconnections are treated as massless, Fig. 2. Variational principles dating back to Euler and Lagrange characterize the motion of a multibody system. These principles constitute a methodology to generate the governing equations and are the basis of today s simulation software. In this context, one speaks also of a multibody formalism. damper sliding (translational) joint revolute joint rigid body springs actuator ground Figure 2: Sketch of a multibody system with rigid bodies and typical interconnections Kinematics Let q(t) R n q denote a vector that comprises the coordinates for position and orientation of all bodies in the system depending on time t. We leave the specifics of the chosen coordinates open at this point but will come back to this issue when discussing standard approaches such as absolute and relative 3

6 coordinates below. As in the preceeding chapter, differentiation with respect to time is expressed by a dot and thus we write q(t) and q(t) for the corresponding velocity and acceleration vectors. The term kinematics describes the geometry of the motion q without taking the forces into account that produce it. Revolute, translational, universal, and spherical joints are examples for bonds in a multibody system. They may constrain the motion q and hence determine its kinematics. If constraints are present, we express the resulting conditions on q in terms of n λ constraint equations 0 = g(q). (1.3) Obviously, a meaningful model requires n λ < n q. The equations (1.3) that restrict the motion q are called holonomic constraints, and the rectangular matrix G(q) := g(q) R n λ n q q is called the constraint Jacobian. We remark that there exist constraints, e.g., driving constraints, that may explicitly depend on time t and that are written as 0 = g(q, t). For notational simplicity, however, we omit this dependence in (1.3). A standard assumption on the constraint Jacobian is the full rank condition rank G(q) = n λ, (1.4) which means that the constraint equations are linearly independent. In this case, the difference n s := n q n λ is the number of degrees of freedom (DOF) in the system. One can also say that the degrees of freedom represent the independent kinematical possibilities of the motion. Two routes branch off at this point. The first modeling approach expresses the governing equations in terms of the redundant variables q and uses additional Lagrange multipliers λ(t) R n λ to take the constraint equations into account. 4

7 Γγ Figure 3: Simple pendulum in the plane. Alternatively, the second approach introduces minimal coordinates s(t) R n s such that the redundant variables q can be written as a function q(s) and that the constraints are satisfied for all choices of s, g(q(s)) 0. (1.5) As a consequence, by differentiation of the identity (1.5) with respect to s we get the orthogonality relation G(q(s)) N(s) = 0 (1.6) with the null space matrix N(s) := q(s)/ s R n q n s. Simple Planar Pendulum. An instructive example for the two modeling approaches is given by the simple pendulum of Fig. 3. The pendulum is pivoted at the origin of the inertial reference frame and performs a circular motion. One choice for q are absolute or Cartesian coordinates r 1 (t) q(t) = r 2 (t) (1.7) α(t) where r 1, r 2 are the coordinates of the centroid and α is the angle between the inertial reference frame and a body-fixed frame placed in the centroid. 5

8 The n q = 3 coordinates completely specify the position and orientation of the pendulum in the plane. Due to the revolute joint at the tip, the kinematic constraint is prescribed by n λ = 2 equations ( r1 l 2 0 = cos α ) r 2 l 2 sin α =: g(q) (1.8) where l denotes the length of the pendulum. The constraint Jacobian reads ( ) l 1 0 G(q) = 2 sin α 0 1 l 2 cos α and is obviously of full rank 2. On the other hand, we can directly eliminate r 1 = l 2 cos α and r 2 = l 2 sin α from the constraints and select the angle α as minimal coordinate for the constrained motion. The number of DOF is hence n s = 1, and the redundant coordinates can be written as ( l q(α) = 2 cos α, l ) T sin α, α (1.9) 2 while ( N(α) = l 2 sin α, l ) T 2 cos α, 1 (1.10) is the corresponding null space matrix. The pendulum example might suggest that an adept choice of variables in terms of minimal coordinates is clearly preferable. We postpone the discussion of this issue for the moment and study next the dynamics of a multibody system Dynamics Above we have sketched two approaches to model the kinematics of a multibody system. These two approaches are also fundamental for the dynamics of the system, which is the study of forces and how the motion changes under their influence. 6

9 Lagrange Equations of Type One. Using both the redundant position variables q and additional Lagrange multipliers λ to describe the dynamics leads to the equations of constrained mechanical motion, also called the Lagrange equations of type one M(q) q = f(q, q, t) G(q) T λ, (1.11a) 0 = g(q) (1.11b) where M(q) R n q n q stands for the mass matrix and f(q, q, t) R n q vector of applied and internal forces. for the Hamilton s principle of least action holds for conservative multibody systems, i.e., systems where the applied forces can be written as the gradient of a potential U. The equations of motion (1.11) then follow from t1 t 0 ( ) T U g(q) T λ dt stationary! (1.12) where the kinetic energy possesses a representation as quadratic form T (q, q) = 1 2 qt M(q) q. In the least action principle (1.12), we observe the fundamental Lagrange multiplier technique for coupling constraints and dynamics. Extensions of the multiplier technique exist in various more general settings such as dissipative systems or even inequality constraints. The underlying idea is always the same, namely to add an expression g(q) T λ or G(q) T λ to the variational principle or the resulting system of dynamic equations, respectively. More precisely, in the non-conservative case the Lagrange equations of type one read ( ) d dt q T (q, q) q T (q, q) = f a(q, q, t) G(q) T λ, (1.13) 0 = g(q) with applied forces f a. Carrying out the differentiations and defining the force 7

10 vector f as f(q, q, t) := f a (q, q, t) + q ( ) ( ) 1 d 2 qt M(q) q dt M(q) which includes Coriolis and centrifugal terms, results in the equations of motion (1.11). In the conservative case, it holds f a = U, and the principle (1.12) is equivalent to (1.13). It should be remarked that for ease of presentation, we omit momentarily the treatment of generalized velocities resulting from 3-dimensional rotation matrices. For that case, an additional kinematic equation q = S(q)v with transformation matrix S and velocity vector v needs to be taken into account. q, Lagrange Equations of Type Two. The Lagrange equations of type one lead to a system of second order differential equations with additional constraints (1.11), which is a special form of a differential-algebraic equation. Applying minimal coordinates s, on the other hand, eliminates the constraints and allows generating a system of ordinary differential equations. If we insert the coordinate transformation q = q(s(t)) into the principle (1.12) or apply it directly to the equations (1.11) of type one, the constraints and Lagrange multipliers cancel due to the property (1.5). The equations of motion, the Lagrange equations of type two, then take the form C(s) s = h(s, ṡ, t). (1.14) This system of second order ordinary differential equations bears also the name state space form. For a closer look at the equations (1.14) of type two, we recall the null space matrix N from (1.6) and derive the relations d dt q(s) = N(s)ṡ, d 2 N(s) dt2q(s) = N(s) s + (ṡ, ṡ) s 8

11 for the velocity and acceleration vectors. Inserting these relations into the dynamic equations (1.11a), we conclude that the mass matrix C in (1.14) is given by C(s) = N(s) T M(q(s))N(s) R n s n s, which can be viewed as the projection of M onto the space of minimal coordinates. Furthermore, the force vector h(s, ṡ, t) R n s is given by the expression h(s, ṡ, t) = N(s) T f(q(s), N(s)ṡ, t) N(s) T M(q(s)) N(s) (ṡ, ṡ). s If the mapping from minimal coordinates s to redundant coordinates q is globally defined, then it is obviously possible to transform the equations of type two (1.14) back to the differential-algebraic equation (1.11), and the two modeling approaches turn out to be equivalent. However, the existence of a globally valid set of minimal coordinates is a crucial point that can not be guaranteed in general. Dynamics of Simple Pendulum. To illustrate the two models for the dynamics, we take up the simple pendulum of Fig. 3 and state the resulting equations of motion. The Cartesian coordinates (1.7) lead to the differential-algebraic equation m 0 0 r ( ) λ 1 0 m 0 r 2 = mγ 0 1 (1.15a), 0 0 m l2 12 α l 0 2 sin α l 2 cos α λ 2 ( r1 l 2 0 = cos α ) r 2 l 2 sin α, (1.15b) where m stands for the pendulum s mass and γ for the gravitation constant. Expressing the motion in the minimal coordinate α, however, results in the state space form with moment of inertia J = l 2 m/3. J α = mγ l cos α (1.16) 2 9

12 1.3 Circuit Simulation Figure 4: Differentiator circuit. DAEs arise also frequently in electric circuit simulation. We study here a simple example, the differentiator circuit. Modelling with Kirchhoff s laws: Network equations: L 0 u 1 u 2 u 3 I A I L I V = G G G G u 1 u 2 u 3 I A I L I V V (t)

13 Chapter 2 Theory of DAEs 2 This chapter presents a brief summary of the most important facts on DAEs. For an in-depth exposition on differential-algebraic equations, the reader is referred to the monographs of Brenan, Campbell & Petzold [7], Griepentrog & März [18], Hairer & Wanner [21], Kunkel & Mehrmann [24] and to the survey of Rabier & Rheinboldt [27]. 2.1 Basic Types of DAEs The most general type of a time-dependent differential equation is the fully implicit system F (ẋ, x, t) = 0 (2.1) with state variables x(t) R n x and a nonlinear, vector-valued function F of corresponding dimension. If the n x n x Jacobian F / ẋ is invertible, then by the implicit function theorem, it is, at least formally, possible to transform (2.1) to a system of ordinary differential equations. If F / ẋ is singular, however, (2.1) constitutes a fully implicit system of differential-algebraic equations. In most applications one actually has substantially more structural information available than in (2.1). An important class are linear-implicit systems of the 11

14 form E ẋ = ϕ(x, t) (2.2) with singular matrix E R n x n x and right hand side function ϕ. In some cases, e.g., in electric circuit simulation, the matrix E depends on the unknown states x, but here we will assume that E is constant. By applying Gaussian elimination with total pivoting or the singular value decomposition, it is then possible to transform E into UEV = ( I ) (2.3) with invertible matrices U, V. The block matrix on the right has the same rank as E and features an identity matrix I and a zero block on the diagonal, which can be exploited to introduce new variables V 1 x =: of appropriate dimensions. Multiplying (2.2) from the left by U, we obtain ( ) ( ( ) ) ( ) ẏ y a(y, z, t) = Uϕ V, t =:. 0 z b(y, z, t) ( y z ) (2.4) It is convenient to further convert this system to autonomous form by adding the equation ṫ = 1 and appending time t as variable to the vector y. Keeping the notation unchanged, this yields finally the semi-explicit equation ẏ = a(y, z), (2.5a) 0 = b(y, z). (2.5b) The differential-algebraic system (2.5) shows a clear separation into n y differential equations (2.5a) for the differential variable y(t) R n y and n z constraints (2.5b), which define the algebraic variables z R n z. For the convergence analysis of numerical time integration methods, the system (2.5) is usually the easiest starting point. If the method is invariant under a 12

15 transformation from the linear-implicit system (2.2) to (2.5), the results then also hold for (2.2). Van der Pol s Equation. A classical example for a semi-explicit system is ẏ = z, (2.6a) 0 = y z3 3 + z (2.6b) with scalar variables y and z. The differential-algebraic equation (2.6) originates from the ordinary differential equation ẏ = z, (2.7a) ϵż = y z3 3 + z (2.7b) with parameter ϵ 1. Van der Pol s equation (2.7), here formulated in Liénhard s coordinates [20], is a singularly perturbed system. In the limit ϵ 0, the perturbed equation (2.7b) turns into the constraint (2.6b). Such a close relation between a singularly perturbed system and a differentialalgebraic equation is quite common and can be found in various application fields. Often, the parameter ϵ stands for an almost negligible physical quantity or the presence of strongly different time scales. Analyzing the reduced system, in this case (2.6), usually proves successful to gain a better understanding of the original perturbed equation [25]. In the context of regularization methods, this relation is also exploited, but in reverse direction [22]. One starts with a DAE such as (2.6) and replaces it by a singularly perturbed ODE, in this case (2.7). We will take up this connection in Chapter 3 when discussing stiff mechanical systems. 2.2 The Index The index of a differential-algebraic equation measures its singularity when compared to an ordinary differential equation. This key concept has evolved over 13

16 several decades, and today a number of definitions with different emphasis exist. In this short survey, we focus first on linear constant coefficient systems and the nilpotency index, continue with the differentiation index and finally include also the perturbation index. Linear Constant Coefficient DAEs. The analysis of a linear constant coefficient DAE reveals already a number of fundamental properties. Consider the initial value problem Eẋ = Hx + c, x(t 0 ) = x 0, (2.8) with constant matrices E, H R n x n x and some time-dependent excitation c(t) R n x. While the matrix E is regular in the ODE case and can be brought to the right hand side by formal inversion, it is singular in the DAE case. The singularity, however, may not be arbitrary. We assume that the matrix pencil µe H is regular, i.e., that there exists µ C such that the matrix µe H is regular. Otherwise, the pencil is singular, and (2.8) has either no or infinitely many solutions [8]. If µe H is a regular pencil, there exist nonsingular matrices U and V such that ( ) ( ) I 0 C 0 UEV =, UHV = (2.9) 0 N 0 I where N is a nilpotent matrix, I an identity matrix, and C a matrix that can be assumed to be in Jordan canonical form. The transformation (2.9) is called the Kronecker canonical form [23]. It is a generalization of the Jordan canonical form and contains the essential structure of the linear system (2.8). The construction of (2.9) is particularly easy for µ = 0, i.e., when the right hand side matrix H is regular. Then, the Jordan canonical form of H 1 E yields ( ) H 1 E = T JT 1 R 0, J =, 0 N 14

17 where R corresponds to the Jordan blocks with nonzero eigenvalues and N to the zero eigenvalues. Thus, setting C := R 1 and ( ) U := T 1 H 1 R 1 0, V := T 0 I results in the decomposition (2.9). For the derivation in the general case see Gantmacher [12]. In the Kronecker canonical form (2.9), the singularity of the DAE is represented by the nilpotent matrix N. Its degree of nilpotency, i.e., the smallest positive integer k such that N k = 0, plays a key role when studying closed-form solutions of the linear system (2.8). In passing we note that the transformation matrices U and V in the Kronecker form (2.9) are, in general, not equivalent to the matrices in the decomposition (2.3). An exception is the case k = 1 where N 1 is the zero matrix. Moreover, while standard numerical algorithms compute the decomposition (2.3) in a stable way, the Jordan canonical form and consequently also the Kronecker form are notoriously ill-conditioned problems. To construct a solution of (2.8), we introduce new variables and right hand side vectors ( ) ( ) V 1 y δ x =:, Uc =:. (2.10) z θ Premultiplying (2.8) by U then leads to the decoupled system ẏ = Cy + δ, Nż = z + θ. (2.11a) (2.11b) While the solution of the ODE (2.11a) follows by integrating and results in an expression based on the matrix exponential exp C(t t 0 ), the equation (2.11b) for z can be solved recursively by differentiating. More precisely, it holds N z = ż + θ N 2 z = Nż + N θ = z + θ + N θ. 15

18 Repeating the differentiation and multiplication by N, we can eventually exploit the nilpotency and get This implies the explicit representation k 1 0 = N k z (k) = z + N l θ (l). l=0 k 1 z = N l θ (l). (2.12) l=0 The above solution procedure illustrates several crucial points about differentialalgebraic equations and how they differ from ordinary differential equations. We highlight two of these: (i) The solution of (2.8) rests on k 1 differentiation steps. This requires that the derivatives of certain components of θ exist up to l = k 1. Furthermore, some components of z may only be continuous but not differentiable depending on the smoothness of θ. (ii) The components of z are directly given in terms of the right hand side data θ and its derivatives. Accordingly, the initial value z(t 0 ) = z 0 is fully determined by (2.12) and, in contrast to y 0, cannot be chosen arbitrarily. Initial values (y 0, z 0 ) where z 0 satisfies (2.12) are called consistent. The same terminology applies to the initial value x 0, which is consistent if, after the transformation (2.10), z 0 satisfies (2.12). Nilpotency Index. As seen above, constructing a closed-form solution of a linear DAE is based on repeated differentiation until the nilpotency of the singular part comes into play. Since differentiation is an unstable numerical process, the degree of nilpotency k measures the numerical difficulty associated with the system (2.8). The integer k is therefore called the (nilpotency) index of equation (2.8). 16

19 Obviously, the nilpotency index is restricted to linear systems with a regular matrix pencil. Its generalization to the nonlinear case will be discussed next. Differentiation Index. Both the index concept and the definition of consistent initial values can be generalized for the fully implicit system (2.1). Following Gear [13, 14], we define the index k by k = 0: If F / ẋ is non-singular, the index is 0. k > 0: Otherwise, consider the system of equations F (ẋ, x, t) = 0, d F (ẋ, x, t) = dt ẋ F (ẋ, x, t) x(2) +... = 0, (2.13). d s dtsf (ẋ, x, t) = ẋ F (ẋ, x, t) x(s+1) +... = 0 as a system in the separate dependent variables ẋ, x (2),..., x (s+1), with x and t as independent variables. Then the index k is the smallest s for which it is possible, using algebraic manipulations only, to extract an ordinary differential equation ẋ = ψ(x, t) from (2.13). In the case of linear constant coefficient systems (2.8), the differential index is equivalent to the nilpotency index. As example, we consider the semi-explicit system (2.5) where the unknown variables are a priori partitioned into differential variables y and algebraic variables z. This allows one to determine the index by simply differentiating the constraint equation 0 = b(y, z). We assume that the Jacobian matrix b z (y, z) Rn z n z is invertible (2.14) in a neighborhood of the solution. Differentiating (2.5b) then leads to 0 = d dt b b b(y, z) = (y, z)ẏ + (y, z)ż. y z 17

20 This implies ( b ż = (y, z) z ) 1 b (y, z) a(y, z), (2.15) y which is the desired ordinary differential equation for the variable z. In other words, (2.5) is of index k = 1 if the assumption (2.14) holds. The combination of (2.15) with (2.5a) is called the underlying ODE of the differential-algebraic system. An initial value (y 0, z 0 ) of (2.5) is consistent if the constraint is satisfied, 0 = b(y 0, z 0 ). As second example, we analyze the semi-explicit equation ẏ = a(y, z), (2.16a) 0 = b(y) (2.16b) under the assumption b a (y) y z (y, z) Rn z n z is invertible (2.17) in a neighborhood of the solution. Now, differentiating twice and setting B(y) := b(y)/ y, we get and 0 = B(y)ẏ = B(y)a(y, z) (2.18) 0 = B(y) a (y, z)ż + B(y) a z y (y, z)a(y, z) + d B(y) a(y, z). (2.19) dt Due to assumption (2.17), the index of (2.16) is thus k = 2. As a rule of thumb, differential-algebraic equations of index 2 or higher are generally more difficult to analyze and to solve numerically than ordinary differential equations or DAEs of index 1. One reason for this is the presence of hidden constraints such as (2.18). Note that a consistent initial value for the system (2.16) must satisfy both the original and the hidden constraint, i.e., 0 = b(y 0 ), 0 = B(y 0 )a(y 0, z 0 ). (2.20) In practice, finding such consistent initial values may constitute a challenging problem of its own [7, 30]. 18

21 Perturbation Index. The index is the standard approach to classify differentialalgebraic equations. While the differential index is based on successively differentiating the original DAE until the obtained system can be solved for ẋ, the perturbation index introduced by Hairer, Lubich & Roche [19] measures the sensitivity of the solutions to perturbations in the equation: The system F (ẋ, x, t) = 0 has perturbation index k 1 along a solution x(t) on [t 0, t 1 ] if k is the smallest integer such that, for all functions ˆx having a defect F ( ˆx, ˆx, t) = δ(t), there exists on [t 0, t 1 ] an estimate ( ˆx(t) x(t) c ˆx(t 0 ) x(t 0 ) + max t 0 ξ t ) δ(ξ) max t 0 ξ t δ(k 1) (ξ) whenever the expression on the right hand side is sufficiently small. Note that the constant c depends only on F and on the length of the interval. The perturbation index is k = 0 if ( ˆx(t) x(t) c ˆx(t 0 ) x(t 0 ) + max t 0 ξ t which is satisfied for ordinary differential equations. ξ t 0 ) δ(τ) dτ, If the perturbation index exceeds k = 1, derivatives of the perturbation show up in the estimate and indicate a certain degree of ill-posedness. E.g., if δ contains a small high frequency term ϵ sin ωt with ϵ 1 and ω 1, the resulting derivatives will induce a severe amplification in the bound for ˆx(t) x(t). Unfortunately, the differential and the perturbation index are not equivalent in general and may even differ substantially [9]. As an example, consider the linear-implicit system 0 y y ẏ 1 ẏ 2 ẏ y 1 y 2 y 3 = 0. (2.21)

22 T y M 0 E y(t) (t) M Figure 5: Manifold M, tangent space T y M, and local parametrization. The last equation is y 3 = 0, which immediately implies y 1 = 0 and y 2 = 0. Differentiating these equations once yields the underlying ordinary differential equation, and accordingly the differential index is 1. If the right hand side, on the other hand, is perturbed by δ = (δ 1, δ 2, δ 3 ) T, we can compute the perturbed solution in a way similar to the derivation of (2.12), obtaining eventually an expression for ŷ 1 that involves the second derivative δ (2) 3. The perturbation index is hence 3. The example (2.21) extends easily to arbitrary dimension n y. While the perturbation index equals n y and grows with the dimension, the differential index stays 1 [9]. In case of semi-explicit systems (2.5), however, such an inconsistence does not arise, and both indices can be shown to be equivalent. 2.3 Constraint Manifold and Local State Space Form So far, our discussion of differential-algebraic equations has been mainly inspired by differential calculus and algebraic considerations. A fundamentally different aspect comes into play by adopting the viewpoint of differential equations on manifolds, as introduced by Rheinboldt [28]. To illustrate this approach, we consider again the semi-explicit system (2.16) of index 2 where the constraint 0 = b(y), assuming sufficient differentiability, defines the manifold M := {y R n y : b(y) = 0}. (2.22) The full rank condition (2.17) for the matrix product b/ y a/ z implies 20

23 that the Jacobian B(y) = b(y)/ y R n z n y possesses also full rank n z. Hence, for fixed y M, the tangent space T y M := {v R n y : B(y)v = 0} (2.23) spans the kernel of B and has the same dimension n y n z as the manifold M. Fig. 5 depicts M, T y M, and a solution of the DAE (2.16), which, starting from a consistent initial value, is required to proceed on the manifold. The differential equation on the manifold M that is equivalent to the DAE (2.16) is obtained as follows: The hidden constraint 0 = B(y)a(y, z) can be solved for z(y) according to the rank condition (2.17) and the implicit function theorem. Moreover, for y M it holds a(y, z(y)) T y M, which defines a vector field on the manifold M [1]. Overall, ẏ = a(y, z(y)) for y M (2.24) represents then a differential equation on the manifold [3, 28]. In theory, and also computationally [29], it is possible to map the differential equation (2.24) from the manifold to an ordinary differential equation in a linear space of dimension n y n z. For this purpose, one introduces a local parametrization ψ : E U (2.25) where E is an open subset of R n y n z and U M, see Fig. 5. Such a parametrization is not unique and holds only locally in general. It is, however, possible to extend it to a family of parametrizations such that the whole manifold is covered. For y U and local coordinates ξ E we thus get the relations y = ψ(ξ), ẏ = Ψ(ξ) ξ, Ψ(ξ) := ψ ξ (ξ) Rn y (n y n z ). 21

24 Premultiplying (2.24) by the transpose of the Jacobian Ψ(ξ) of the parametrization and substituting y by ψ(ξ), we arrive at Ψ(ξ) T Ψ(ξ) ξ = Ψ(ξ) T a(ψ(ξ), z(ψ(ξ))). (2.26) Since the Jacobian Ψ has full rank for a valid parametrization, the matrix Ψ T Ψ is invertible, and (2.26) constitutes the desired ordinary differential equation in the local coordinates ξ. In analogy to the mechanical system (1.14) in minimal coordinates, we call (2.26) a local state space form. The process of transforming a differential equation on a manifold to a local state space form constitutes a push forward operator, while the reverse mapping is called a pull back operator [1]. It is important to realize that the previously defined concept of an index does not appear in the theory of differential equations on manifolds. Finding hidden constraints by differentiation, however, is also crucial for the classification of DAEs from a geometric point of view. 2.4 Solution Invariants versus Constraints Differential equations with invariants are sometimes confused with differentialalgebraic equations. In fact, there exists a close relation between both concepts, and this connection plays a major role for index reduction and stabilization techniques. Assume the solution of the ordinary differential equation satisfies the invariant ẏ = ϕ(y), (2.27) 0 = c(y). (2.28) If (2.28) is even valid in a neighborhood of the solution, the invariant is called a first integral. Since (2.28) holds for all t, we may differentiate the invariant and obtain the relation 0 = C(y)ẏ = C(y)ϕ(y) (2.29) 22

25 with C(y) := c(y)/ y. Linear invariants are preserved by standard time integration methods, but other types of invariants require either specific integrators, e.g., symplectic methods in case of Hamiltonian systems, or additional measures to enforce (2.28). In this context, reformulating the equations as a differential-algebraic system ẏ = ϕ(y) C(y) T z, (2.30a) 0 = c(y) (2.30b) is a tempting option. The DAE (2.30) is semi-explicit, with the algebraic variables z playing the role of Lagrange multipliers. Moreover, the invariant (2.28) has turned into the constraint (2.30b). If the Jacobian C(y) has full rank, the condition (2.17) is satisfied, and the DAE (2.30) is thus of index 2. Even more, assume y solves (2.27) and satisfies (2.28) as well as (2.29). Inserting y into the DAE (2.30), we have by differentiating the constraint 0 = C(y)ϕ(y) C(y)C(y) T z = C(y)C(y) T z. Since the square matrix C(y)C(y) T is invertible if C(y) has full rank, we conclude that z = 0. In other words, the extra multipliers vanish, and the pair (y, z) where z 0 solves the DAE (2.30). Appending an invariant as a constraint by means of suitable Lagrange multipliers is not always a good remedy. Properties such as symplecticity require a specific discretization and cannot be simply enforced in this way [21, VII.2]. For higher index problems, however, this approach can be used to lower the index without loosing the information of the original constraint. In Sect below, we will explain this in more detail when studying alternative formulations of the equations of constrained mechanical motion. 23

26 2.5 Analysis of the Equations of Constrained Mechanical Motion The state space form (1.14) is a system of second order ordinary differential equations. Consequently, all results from standard theory on the existence and uniqueness of solutions carry over. In contrast, the equations of constrained mechanical motion (1.11) constitute a differential-algebraic system of index 3, as we will see in the following. For this purpose, it is convenient to rewrite the equations as a system of first order q = v, M(q) v = f(q, v, t) G(q) T λ, with additional velocity variables v(t) R n q. (2.31a) (2.31b) 0 = g(q) (2.31c) Index and Existence of Solutions In order to analyze the differential-algebraic equation (2.31), we apply a stepby-step procedure that identifies the hidden constraints and allows to solve for the Lagrange multiplier λ, which is the algebraic variable. By differentiating the constraints (2.31c) with respect to time, we obtain the constraints at velocity level 0 = d g(q) = G(q) q = G(q) v. (2.32) dt A second differentiation step yields the constraints at acceleration level 0 = d2 g(q) = G(q) v + κ(q, v), dt2 where the two-form κ comprises additional derivative terms. G(q) κ(q, v) := (v, v), (2.33) q Combining the dynamic equation (2.31b) and the acceleration constraints (2.33), 24

27 we finally arrive at the linear system ( ) ( M(q) G(q) T G(q) 0 v λ ) = ( f(q, v, t) κ(q, v) ). (2.34) The matrix on the left hand side has a saddle point structure. We presuppose that ( ) M(q) G(q) T is invertible (2.35) G(q) 0 in a neighborhood of the solution. A necessary but not sufficient condition for (2.35) is the full rank of the constraint Jacobian G as stated in (1.4). If in addition the mass matrix M is symmetric positive definite, (2.35) obviously holds 1. Assuming (2.35) and a symmetric positive definite mass matrix, we can solve the linear system (2.34) for the acceleration v and the Lagrange multiplier λ by block Gaussian elimination. This leads to the explicit expressions v = M(q) 1 ( f(q, v, t) G T (q)λ ), (2.36) λ = ( G(q)M(q) 1 G T (q) ) 1 ( G(q)M(q) 1 f(q, v, t) + κ(q, v) ).(2.37) The representation λ = λ(q, v, t) from (2.37) is now inserted into (2.36), which leads to an ordinary differential equation for the velocity variables v. Under the usual assumption of Lipschitz continuity of the corresponding right hand side, the unique solution (q, v) of (2.31a) and (2.36) is guaranteed and in turn, the multiplier λ is uniquely determined as well. At the same time, the above differentiation steps also define the index of the DAE (2.31). Two differentiation steps result in the linear system (2.34) that allows to solve for the Lagrange multiplier as a function λ(q, v, t). A final third differentiation step yields an ordinary differential equation for this algebraic variable, which implies that the differentiation index is 3. 1 We remark that there are applications where the mass matrix is singular but the prerequisite (2.35) nevertheless is satisfied. 25

28 Note that the solution of the ODE (2.31a) and (2.36) does not necessarily fulfill the differential-algebraic equation (2.31) since the differentiation steps involve a loss of integration constants. However, if the initial values (q 0, v 0 ) are consistent, i.e., if they satisfy the original constraints and the velocity constraints, 0 = g(q 0 ), 0 = G(q 0 ) v 0, (2.38) the solution of (2.31a) and (2.36) also fulfills the original system (2.31). Above all, depending on q 0 and v 0, the initial value λ 0 for the Lagrange multiplier is completely determined by (2.37). Hidden Constraints of Planar Pendulum. As example for the hidden constraints of the equations of motion, we consider the planar pendulum (1.15) with velocity variables v 1 (t) ṙ 1 (t) v(t) = v 2 (t) := ṙ 2 (t). v 3 (t) α(t) Differentiating the constraints (1.15b) at position level, we obtain the constraints at velocity level ( 0 = d r1 l 2 cos α ) ( ) l 1 0 dt r 2 l 2 sin α = 2 sin α v l 2 cos α v 2. v 3 The constraints at acceleration level, finally, are given by ( 0 = d2 r1 l 2 cos α ) ( ) l 1 0 dt 2 r 2 l 2 sin α = 2 sin α v l 2 cos α v 2 + v3 2 v 3 ( l 2 cos α l 2 sin α For completeness, we furthermore state the equation (2.37) for the two arising Lagrange multipliers. It reads ( ) ( ) ( ) λ 1 = mγ 3 sin α cos α l λ sin 2 + mv3 2 2 cos α l α 2 sin α. 26 ).

29 We point out that, in practice, the complexity of the equations of motion prohibits such explicit expressions as given here for the planar pendulum example. Instead, the multibody formalisms provide automatic procedures for numerically evaluating constraints and differentiated constraints Minimax Characterization of Constraints We reconsider next the full rank criterion (1.4). The above analysis steps demonstrate that the full rank of the constraint Jacobian G is a fundamental property. We can reformulate this criterion in terms of the singular values of G, which are given by the factorization [17, 2.5] U T G(q)V = diag (σ 1,..., σ nλ ) R n λ n q with orthogonal matrices U R n λ n λ and V R n q n q. The singular values are ordered as σ 1 σ 2... σ nλ 0, and for the full rank of G we require σ min := σ nλ > 0. Omitting the argument q of G for brevity, we can reformulate this criterion by observing that λ T GG T λ λ T λ = τ T diag (σ 2 1,..., σ 2 min)τ τ T τ σ 2 min for λ 0 and τ := U T λ. In case of τ = (0,..., 0, 1) T R n λ, this inequality is sharp and we conclude σ 2 min = min λ λ T GG T λ λ T λ σ min = min λ G T λ 2 λ 2. Moreover, using the definition of the operator norm we get the identity G T λ 2 = max v v T G T λ 2 v 2 = max v λ T Gv v 2 27

30 since v T G T λ 2 = λ T Gv. Overall, we have thus derived the minimax characterization λ T Gv σ min (G) = min max > 0. (2.39) λ v v 2 λ 2 In mixed finite elements, the essential minimax condition (2.39) for the regularity of rigid body constraints is generalized to an inf-sup condition for constraints on elastic bodies. Constraint Jacobian of Planar Pendulum. To illustrate the minimax criterion (2.39), we analyze the constraint Jacobian ( ) l 1 0 G(q) = 2 sin α 0 1 l 2 cos α of the planar pendulum example (1.15). Computing the eigenvalues of the symmetric matrix ( ) G(q)G(q) T 1 + l2 = 4 sin2 α l2 4 sin α cos α sin α cos α 1 + l2 4 cos2 α l2 4 yields the squared singular values of G(q), and in this way we find that σ 1 = 2, σ 2 = 1 for all configurations q of the system. The minimax condition (2.39) is hence guaranteed Influence of Perturbations The previous analysis has shown that the equations of motion (2.31) possess a unique solution provided that the full rank or minimax criterion (2.39) holds and the right hand side of (2.36) is Lipschitz continuous. For well-posedness of the equations, however, we need to look at small perturbations. 28

31 As discussed above, the perturbation index measures the sensitivity of the solution of a DAE with respect to perturbations in the equations. The perturbed version of the equations of constrained mechanical motion (2.31) takes the form ˆq = ˆv + γ(t), (2.40a) M(ˆq) ˆv = f(ˆq, ˆv, t) G(ˆq) T ˆλ + δ(t), (2.40b) 0 = g(ˆq) + θ(t). (2.40c) Arnold [2] proved that the position variables q and the velocity variables v are not very sensitive to such perturbations but the Lagrange multipliers λ indeed are. It holds the sharp estimate ( ˆλ(t) λ(t) c 0 + max δ(τ) + max θ(τ) (2.41) τ [t 0,t] τ [t 0,t] ) + max θ(τ) + max θ(τ). τ [t 0,t] τ [t 0,t] In this estimate, 0 := ˆq(t 0 ) q(t 0 ) + S(ˆq(t 0 ))ˆv(t 0 ) S(q(t 0 ))v(t 0 ) denotes the difference in the initial values. Moreover, S(q) := I (M ( ) ) 1 G T GM 1 G T 1 G (q) is a projector onto the kernel of the velocity constraints (2.32), and c is a constant that may depend on the size of max τ δ(τ). What are the consequences of this result? Perturbations in the constraints strongly influence the Lagrange multipliers. In particular, the second time derivative θ is typical for DAEs of index 3. If θ is a small but high frequency signal, we get an amplification of the perturbation and the problem is not wellposed in this formulation. The amplification of perturbations is not the only obstacle to the numerical treatment of the equations of motion (2.31). In a numerical time integration 29

32 scheme, a differentiation step is replaced by a difference quotient, i.e., by a division by the stepsize. Therefore, the approximation properties of the numerical scheme deteriorate and we observe phenomena like order reduction, ill-conditioning, or even loss of convergence. Most severely affected are typically the Lagrange multipliers. In Chapter 3, this issue will be studied in more detail Alternative Formulations In viev of the above discussion, it is mostly not advisable to tackle DAEs of index 3 directly. Instead, it has become standard in multibody dynamics to lower the index first by introducing alternative formulations. Formulation of Index 1. The differentiation process for determining the index revealed the hidden constraints at velocity and at acceleration level. It is a straightforward idea to replace now the original position constraint (2.31c) by one of the hidden constraints. Selecting the acceleration equation (2.33) for this purpose, one obtains the formulation of index 1 q = v, M(q) v = f(q, v, t) G(q) T λ, 0 = G(q) v + κ(q, v). (2.42) This system is obviously of index 1, and at first sight one could expect much less difficulties here. But a closer view shows that (2.42) lacks the information of the original position and velocity constraints, which have become invariants of the system. In general, these invariants are not preserved under discretization, and the numerical solution may thus turn unstable, which is called the drift off phenomenon. To give a first idea of the drift off, we write the position constraint as w(t) := g(q(t)) 30

33 and differentiate twice. The differential equation ẅ = 0 corresponds then to the acceleration constraint (2.33), but we assume now small errors in the right hand side and the initial values and consider instead the initial value problem ẅ = ζ a, ẇ(t 0 ) = ζ v, w(t 0 ) = ζ p (2.43) with constants ζ a, ζ v, ζ p R n λ. By integrating twice, we obtain w(t) = 1 2 (t t 0) 2 ζ a + (t t 0 )ζ v + ζ p, (2.44) We thus observe a violation of the position constraint that grows quadratically with time. In Chapter 3, this intuitive reasoning will be made more precise by studying the discrete error propagation in the constraints when numerically solving the formulation of index 1. For the moment, we conclude that there is an inherent quadratic instability in this formulation. Baumgarte Stabilization. A very early cure for the drift off goes back to Baumgarte [6]. The idea is to combine original and differentiated constraints and form the new constraint 0 = G(q) v + κ(q, v) + 2αG(q)v + β 2 g(q) (2.45) with scalar parameters α and β. In (2.42), the acceleration constraint is then replaced by (2.45), which leaves the index unchanged. The free parameters α and β should be chosen in such a way that becomes an asymptotically stable equation. E.g., if α = β, the solution of the perturbed system is given by 0 = ẅ + 2αẇ + β 2 w (2.46) ẅ + 2αẇ + α 2 w = ζ a, ẇ(t 0 ) = ζ v, w(t 0 ) = ζ p, w(t) = ( ζ p + (t t 0 )(ζ p + αζ v ) ) exp( α(t t 0 )) + ζ a α 2. 31

34 For α > 0, the exponential function decays and damps out the initial deviations ζ p and ζ v in the position and velocity constraints, respectively. The crucial point in Baumgarte s approach is the choice of the parameters. Also, extraneous eigenvalues are introduced in this way. For a detailed analysis of this stabilization and related techniques we refer to Ascher et al. [4, 5]. Formulation of Index 2. Instead of the acceleration constraints, one can also use the velocity constraints (2.32) to replace (2.31c). This leads to the formulation q = v, M(q) v = f(q, v, t) G(q) T λ, 0 = G(q) v. (2.47) Now the index is 2, but similar to the index 1 case, the information of the position constraint is lost. The resulting drift off is noticeable but stays linear, which means a significant improvement compared to (2.42). Nevertheless, additional measures such as stabilization by projection are often applied when discretizing (2.47). The evaluation of the acceleration constraint (2.33) requires expressions with second derivatives of the constraints and is thus computationally rather expensive. For the formulation (2.47), however, there is almost no extra effort when evaluating the velocity constraints since the constraint Jacobian G(q) needs to be evaluated anyway to form the product G(q) T λ. GGL Formulation. On the one hand, we have seen that it is desirable for the governing equations to have an index as small as possible. On the other hand, though simple differentiation lowers the index, it may lead to drift off. An elegant way out of this dilemma is due to Gear, Gupta & Leimkuhler [16]. This formulation starts with the kinematic and dynamic equations (2.31a-b) combined with the constraints at velocity level (2.32). The position constraints 32

35 (1.11b) are interpreted as invariants and appended by means of extra Lagrange multipliers, cf. Section 2.4. In this way, one obtains an enlarged system q = v G(q) T τ, M(q) v = f(q, v, t) G(q) T λ, (2.48) 0 = G(q) v, 0 = g(q) with additional multipliers τ (t) R n λ. A straightforward calculation shows 0 = d dt g(q) = G(q) q = G(q) v G(q)GT (q)τ = G(q)G T (q)τ and one concludes τ = 0 since G(q) is of full rank and hence G(q)G T (q) invertible. With the additional multipliers τ vanishing, (2.48) and the original equations of motion (1.11) coincide along any solution. Yet, the index of the GGL formulation (2.48) is 2 instead of 3. Some authors refer to (2.48) also as stabilized index 2 system. A scaled variant of the GGL formulation (2.48) is also widespread where the kinematic equation is replaced by S q = Sv G(q) T τ. (2.49) The scaling matrix S R n q n q should be symmetric positive definite so that G(q)S 1 G T (q) is invertible and the above conclusion τ = 0 remains valid. Specific choices are S = M(q) or, better with respect to efficiency, S = M d (q) where M d is a diagonal mass matrix obtained from mass lumping. Overdetermined Formulation. From an analytical point of view, one could drop the extra multiplier τ in (2.48) and consider instead the overdetermined 33

36 α α M T M r 2 r 1 r 2 r 1 Figure 6: Constraint manifold (left) and tangent bundle (right) for the planar pendulum (1.15). The constraint manifold is the helix defined by (1.9), and the disjoint union of all tangents to the helix forms the tangent bundle. system q = v, M(q) v = f(q, v, t) G(q) T λ, (2.50) 0 = G(q) v, 0 = g(q). Though there are more equations than unknowns in (2.50), the solution is unique and, given consistent initial values, coincides with the solution of the original system (2.31). Even more, one could add the acceleration constraint (2.33) to (2.50) so that all hidden constraints are explicitly stated. Under discretization, however, an overdetermined system such as (2.50) can only be solved in a least squares sense. As investigated in Führer [10] and Führer & Leimkuhler [11] for the BDF methods, it is possible to construct a least squares objective function that inherits certain properties of the state space form (1.14) and that defines an integration scheme equivalent to the discretization of (2.48). Local State Space Form. In Section 1.2.2, we have contrasted the constrained equations of motion (1.11) with the state space form (1.14), which is a system 34

37 of second order ordinary differential equations. The reasoning about the restrictions of the state space form contained a loose end that we now take up. It is very helpful in this context to view the equations of constrained mechanical motion as differential equations on a manifold, compare Sect The position constraint (2.31c) defines the constraint manifold M := {q R n q : 0 = g(q)}. (2.51) Furthermore, combining (2.31c) and the velocity constraint (2.32) leads to the tangent bundle T M := {(q, v) R n q R n q : 0 = g(q), 0 = G(q)v}. (2.52) Fig. 6 illustrates the constraint manifold and the tangent bundle for the planar pendulum example (1.15). Given (q 0, v 0 ) T M, our task is now to construct a parametrization such that the resulting analogue of the state space form (1.14) can be evaluated. As discussed in Sect. 2.3, such a parametrization in general holds only locally, and this restriction will become clearer when we discuss two established approaches in the following. We start with the method of coordinate partitioning by Wehage & Haug [31]. The full rank of the constraint Jacobian G(q 0 ) implies that, after an appropriate permutation of the columns of G(q 0 ) which we omit here for simplicity, the coordinate vector q can be partitioned into independent coordinates q I (t) R n q n λ and dependent coordinates q D (t) R n λ such that q = ( q I q D ) and G D (q 0 ) := g(q 0) q D R n λ n λ is invertible. (2.53) Recall that n s = n q n λ is the number of DOF, and these degrees of freedom are momentarily identified with q I. By the implicit function theorem, the constraint 0 = g(q I, q D ) defines formally a function q D = η(q I ) in a neighborhood of q 0, and by computing the derivative 35

38 and observing the chain rule, we obtain 0 = q I g(q I, q D ) = G I (q I, q D ) + G D (q I, q D ) η(q I) q I where G I (q) := g(q)/ q I R n λ n s. Overall, it holds η(q I ) q I = (G 1 D G I)(q I, η(q I )) (2.54) for the relation between dependent and independent coordinates. In the language of differential geometry, the mapping ( ) q ψ : E U, q I I η(q I ) (2.55) constitutes a local parametrization of the manifold M, with E being an open subset of R n s and U M. The same partitioning into dependent and independent coordinates can be applied to the velocity vector v = (v I, v D ), and the pair (ψ, Ψ) with Ψ(q I ) = ψ(q I) q I parametrizes the tangent bundle (2.52). Looking back to the state space form (1.14), it is now evident that the independent coordinates q I are a special choice of minimal coordinates, and furthermore, the null space matrix N is given by N(q I ) = ψ(q I) q I = Ψ(q I ) = ( ) I (G 1 D G. (2.56) I)(q I, η(q I )) Thus, all ingredients for evaluating the equations of motion (1.14) are defined. One should be aware, however, that the construction of the local parametrization outlined so far is not suitable for a computer implementation. One reason lies in the second fundamental form N(q I )/ q I (v I, v I ) that is required in (1.14) and that is not given explicitly. 36

39 To this end, we use the coordinate partitioning to express the acceleration constraint (2.33) as which implies that 0 = G I (q) v I + G D (q) v D + κ(q, v), v D = (G 1 D G I)(q) v I G 1 D (q)κ(q, v). (2.57) Due to the parametrization, it holds on the other hand v = = d ( ) N(qI )v I = N(qI ) v I + N(q I) (v I, v I ) dt q I ( ) I 0 G 1 D G v I + I (G 1 D G I). (2.58) (v q I, v I ) I Comparing the second row of (2.58) with (2.57), we conclude that the evaluation of the second fundamental form N(q I )/ q I (v I, v I ) can be accomplished by forming the acceleration term G 1 D (q)κ(q, v) where (q, v) T M. In practice, this procedure is implicitly performed. Given (q I, v I ), the evaluation of the local state space form starts by computing q = ψ(q I ), v = N(q I )v I, which involves the solution of the nonlinear system 0 = g(q I, q D ) for q D. Next, one solves the linear system ( ) ( ) ( ) M(q) G(q) T w f(q, v, t) = (2.59) G(q) 0 λ κ(q, v) for the acceleration w and the Lagrange multiplier λ. Finally, the local state space form, written as a system of first order, is given by q I = v I, v I = w I (2.60) where w I denotes the entries of the acceleration vector that correspond to the independent coordinates. 37