ME451 Kinematics and Dynamics of Machine Systems Numerical Solution of DAE IVP Newmark Method November 1, 2013 Radu Serban University of Wisconsin-Madison

2 Before we get started… Last Time: Stiff differential equations: Implicit numerical integration formulas Accuracy and Stability properties Implicit methods have much larger stability regions However, they require solution of nonlinear algebraic problems Today: Numerical integration of DAE initial value problems in multibody dynamics Newmark integration method Assignments: Homework 9 – 6.3.3, – due November 4 (12:00pm) Project 1 – due Wednesday, November 6, (11:59pm) Midterm 2: Review session – Monday 6:30pm in ME1143 Everything covered under Dynamics, not including today’s lecture Lecture 16 (October 9) – Lecture 23 (October 30)

3 Sample Problem Find the time evolution of the pendulum Steps for Dynamic Analysis: 1. Derive constrained equations of motion 2. Specify initial conditions (ICs) 3. Apply numerical integration algorithm to discretize DAE problem and turn into algebraic problem RSDA

4 Lagrange Multiplier Form of the Constrained Equations of Motion Equations of Motion Position Constraint Equations Velocity Constraint Equations Acceleration Constraint Equations

5 What’s Special About the EOM of Constrained Planar Systems? Linda R. Petzold – “Differential/Algebraic Equations Are Not ODEs” SIAM J. of Scientific and Statistical Computing (1982)

6 Numerical Solution of the DAEs in Constrained Multibody Dynamics This is a research topic in itself We cover one of the simplest algorithm possible We will use Newmark’s formulas to discretize the index-3 DAEs of constrained multibody dynamics Note that the textbook does not discuss this method Nathan M. Newmark (1910 – 1981)

7 Solution Strategy The numerical solution; i.e., an approximation of the actual solution of the dynamics problem, is produced in the following three stages: Stage 1: two integration (discretization) formulas, Newmark in our case, are used to express the positions and velocities as functions of accelerations Stage 2: everywhere in the constrained EOM, the positions and velocities are replaced using the discretization formulas and expressed in terms of the acceleration This is the most important step, since through this “discretization” the differential problem is transformed into an algebraic problem Stage 3: the acceleration and Lagrange multipliers are obtained by solving a nonlinear system

Newmark Integration Formulas (1/2)

9 Newmark Integration Formulas (2/2)

The rigid multibody dynamics problem is more complicated than the Linear Finite Element problem used to introduce Newmark’s formulas Additional algebraic equations: constraints that the solution must satisfy Additional algebraic variables: the Lagrange multipliers that come along with these constraints Newmark’s method can be applied for the DAE problem, with slightly more complexity in the resulting algebraic problem. DAEs of Constrained Multibody Dynamics Linear Finite Element Dynamics Problem Nonlinear Multibody Dynamics Problem

11 Variables in the DAE Problem All these quantities are functions of time (they change in time)

Stage 3: Discretization of the Constrained EOM (1/3) Recall, these are Newmark’s formulas that express the generalized positions and velocities as functions of the generalized accelerations

13 Stage 3: Discretization of the Constrained EOM (2/3) The unknowns are the accelerations and the Lagrange multipliers The number of unknowns is equal to the number of equations The equations that must be solved now are algebraic and nonlinear Differential problem has been transformed into an algebraic one The new problem: find acceleration and Lagrange multipliers that satisfy We have to use Newton’s method We need the Jacobian of the nonlinear system of equations (chain rule will be used to simplify calculations) This looks exactly like what we had to do when for Kinematics analysis of a mechanism (there we solved  (q,t)=0 to get the positions q)

14 Stage 3: Discretization of the Constrained EOM (3/3)

15 Chain Rule for Computing the Jacobian (1/3) Newton’s method for the solution of the nonlinear system relies on the Jacobian Use the chain rule to calculate the above partial derivatives. Note that, from the Newmark formulas we get

16 Consider Apply the chain rule of differentiation to obtain and Chain Rule for Computing the Jacobian (2/3)

17 Consider Apply the chain rule of differentiation to obtain and Chain Rule for Computing the Jacobian (3/3)

18 Newton’s method applied to the system Jacobian obtained as Corrections computed as Solving the Nonlinear System

19 At each integration time step Newton Method for Dynamics Find consistent initial conditions for generalized positions and velocities Calculate the generalized accelerations and Lagrange multipliers Compute the correction vector by solving a linear system with the Jacobian as the system matrix and the residual as the RHS vector. Compute the infinity norm of the correction vector (the largest entry in absolute value) which will be used in the convergence test Is error less than tolerance? NO Need to further improve accelerations and Lagrange multipliers YES

20 Newton-Type Methods Geometric Interpretation Newton method At each iterate, use the direction given by the current derivative Modified Newton method At all iterates, use the direction given by the derivative at the initial guess Quasi Newton method At each iterate, use a direction that only approximates the derivative

21 Quasi Newton Method for the Dynamics Problem (1/3)

22 Approximate the Jacobian by ignoring these terms Nonlinear equations: Exact Jacobian: Approximate Jacobian: Therefore, we modify the solution procedure to use a Quasi Newton method Quasi Newton Method for the Dynamics Problem (2/3)

23 Quasi Newton Method for the Dynamics Problem (3/3)

24 At each integration time step Quasi Newton Method for Dynamics Find consistent initial conditions for generalized positions and velocities. Calculate the generalized accelerations and Lagrange multipliers. Calculate the approximate Jacobian matrix. Only evaluate this matrix at the first iteration and reuse it at subsequent iterations. Compute the correction vector by solving a linear system. Note that the system matrix is constant during the iterative process. Compute the infinity norm of the correction vector (the largest entry in absolute value) which will be used in the convergence test. Is error less than tolerance? NO Need to further improve accelerations and Lagrange multipliers YES

25 Sample Problem Find the time evolution of the pendulum Steps for Dynamic Analysis: 1. Derive constrained equations of motion 2. Specify initial conditions (ICs) 3. Apply numerical integration algorithm to discretize DAE problem and turn into algebraic problem RSDA