Numerical methods for physics simulations.
|
|
- Briana Ryan
- 5 years ago
- Views:
Transcription
1 Numerical methods for physics simulations. Claude Lacoursière HPC2N/VRlab, Umeå Universiy, Umeå, Sweden, and CMLabs Simulations, Montréal, Canada May 8, 26 Typeset by FoilTEX May 5th 26 Outline Typeset by FoilTEX May 5th 26 1
2 Integrating Newton s second law Newton s law states dp dt = f, p = Mẋ for point masses. M is the mass matrix, x is the position vector, p is the momentum vector, and f is the force vector. Integrating these equations of motion means solving the following system of ordinary differential equations: ẋ = v v = M 1 f(x, v) (1) In general, forces depend on both position and velocities but keep it simple and assume f(x) for now. Typeset by FoilTEX May 5th 26 2 Simple Harmonic Oscillator Simple harmonic oscillator in 1D: f(x) = kx, M = m, (a scalar) f(x) m = k m x = ω2 x ω = k/m is the natural frequency, so: ẋ = v v = ω 2 x, (2) solution: x(t) = x() cos(ωt) + v() ω sin(ωt). Because any physical force is smooth, we can always expand: f(x) = a + b(x x ) +... Linear case is easy to study and gives good insight. Typeset by FoilTEX May 5th 26 3
3 Integration Balance: accuracy stability speed Accuracy can also impliy preservation of energy and other physical invariants (symmetry). We want to integrate indefinitely, without having to change the time step all the time so we need: fixed time step unconditional stability ease of implementation Investigate simple methods (first order) on spring systems without damping to get some good idea of what to expect. Typeset by FoilTEX May 5th 26 4 Integrating the Simple Harmonic Oscillator Ignore the literature and follow basic intuition. Discretize time : t = nh where h is small and n = 1, 2,..., x(t) = x(nh) = x n, and similarly for v(t) = v(nh) = v n. Simple formulae for derivatives: ẋ n+1 v n+1 x n+1 x n h v n+1 v n h so using the most naive integration formula, we have: and therefore: ẋ n+1 v n+1 x n+1 x n+1 x n h v n+1 v n h v n = x n + hv n ω 2 x n v n+1 = hω 2 x n + v n (if you stay tuned, we ll explain that this formula is bad for your health; it is called the Explicit Euler integration formula, applied here to the harmonic oscillator) Typeset by FoilTEX May 5th 26 5
4 Simple Harmonic Oscillator cont In matrix form, this is: x n+1 v n+1 = 1 h hω 2 1 which we can also write as z n+1 = Az n with: z n = x n v n ; A = x n v n 1 h hω 2 1 and so: z n = Az n 1 =... = A n z. This is a form of matrix iteration procedure (more on this later). We have stability if z n stays bounded for all n. So, our first interesting matrix problem of the day: what is lim n An? Note that for this case, the energy is E = 1 2 [mv2 + kx 2 ] α z 2 for some scalar α.. Typeset by FoilTEX May 5th 26 6 Intermission: the spectral radius Think of a matrix A and its eigenvalues and eigenvectors: λ i, v i. For any of these, we have: z n = A n v i = λ n i v i so that z n is bounded iff λ i < 1. The spectral radius states that for any consistent matrix norm, the following holds: ρ(a) = max( λ i ) = lim A n 1/n. i n Using that theorem, we can state the following: Theorem 1. Given a matrix A over the complex numbers, the iterations z n = A n z are bounded if and only if ρ(a) 1. Of course, an interesting limit case is when ρ(a) = 1 which leads to oscillations which are bounded forever. For the case where ρ(a) is strictly less than one, the iterates just decay to. To build good integrators, you need cases which have ρ(a) = 1. Typeset by FoilTEX May 5th 26 7
5 Back to Integration What are the eigenvalues of our iteration matrix? det(a λi) = λ h 1 hω 2 1 λ which means that λ ± = 1 ihω and so, = (1 λ) 2 + h 2 ω 2 = ρ(a) = 1 + h2 ω2 > 1. This is always unstable! It is easy to show that the energy increases at each step by the factor 1 + h 2 ω 2. A naïve analysis would suggest that keeping hω < 1 would keep things stable but that is wrong. Adding some damping force of the form γẋ can make the system stable but too much damping makes it unstable again (it is easy to do the analysis). Typeset by FoilTEX May 5th 26 8 Energy Energy as a function of time, hω =.15 γ = 2.5m/h γ =.1mω discretized time (units of hω) Figure 1: Energy of simple harmonic oscillator integrated explicitly. We either bet big oscillations when the oscillator is damped or a blowup. But the energy should strictly decrease at the rate γ. Typeset by FoilTEX May 5th 26 9
6 Intermission: standard stability analysis (simplified) Given ODE ż = f(z), integrator produces z n+1 = φ(h, f, z n, z n 1, ). For linear stability, look at the Dahlqvist (another famous Swede) test equation: ż = λz, z(t) = exp[λt]z(), where λ, z C. For this system, any integration method leads to a linear recurrence relation of the form: w n+1 = Aw n where A is a matrix function which depends on τ = λh and w is a generalized state vector of the form: w n = [z n, z n 1,...] T. Typeset by FoilTEX May 5th 26 1 Intermission: standard stability analysis (simplified) To use this analysis for our case, we need to redifine: and so: s = ωt, dt ds = 1 ω y(s) = x(t), w(s) = ω 1 v(t) y = dy ds = dx(t) dt dt ds = 1 ω v = w(s) dv w = dw ds = 1 ω dt dt ds = ω2 x(t) = x(t) = y(s) (3) If we then set z(s) = y(s) + iw(s) we find that this corresponds to: z = iz where i is the imaginary unit. The analysis is straight forward but it does not apply easily to all the integration cases we want to cover. A matrix format is more convenient. Typeset by FoilTEX May 5th 26 11
7 Discretizing Differently Actually, we had some choice in discretizing. Using Taylor expansions: x n+1 = x n + ẋ n h + O(h 2 ) (4) x n = x n+1 ẋ n+1 h + O(h 2 ) (5) (x n+1 x n )/h = ẋ n + O(h 2 ) = ẋ n+1 + O(h 2 ) (6) x n+1 xn h v n = αv n + (1 α)v n+1 v n+1 vn h ω 2 x n = ω 2 (βx n + (1 β)x n+1 ) and we can chose α 1, β 1 independently. Typeset by FoilTEX May 5th Discretizing Differently The general stepping equation is then: 1 h(1 α) hω 2 (1 β) 1 x n+1 v n+1 = 1 hα hω 2 β 1 The most interesting cases are the ones where α, β are either, 1 or 1/2. Three new stepping formulae: 1 h hω h 1 1 h 2 hω x n+1 v n+1 x n+1 v n+1 x n+1 v n+1 = = = 1 1 x n v n 1 hω 2 1 h 1 2 hω2 2 1, fully implicit x n v n x n v n x n v n, v implicit: the Verlet method, implicit midpoint rule (7) Typeset by FoilTEX May 5th 26 13
8 Discretizing Differently All of these are of the general form: Bz n+1 = Cz n and with a little algebra, we convert them to the form z n+1 = Az n. Computing these matrices is nothing complicated (but tedious). Once you get matrix A, you compute the eigenvalues, carefully check all cases to find out what values of h, ω produce ρ(a) 1. This is the stability regime. Some simple dimensional analysis shows that ρ(a) will always be a function of τ = hω. This is the natural time scale of the system. The following cases will occur: ρ(a) > 1 for all values of τ: unstable, no good (explicit Euler) ρ(a) = 1 for some values of < τ τ: conditionally satble ρ(a) < 1 for all values of τ: unconditionally stable but everything is damped: this is L-stability (implicit Euler) ρ(a) = 1 for all values of τ: unconditionally stable but nothing is damped: this is A-stability (implicit midpoint) Typeset by FoilTEX May 5th Discretizing Differently: Special Cases We cover the different cases and we use: τ = hω. Implicit stepper: Verlet stepper: 1 A ie = 1 + h 2 ω 2 A vv = 1 h hω h 2 ω 2 h hω 2 1, λ ± = 1, ρ(a) < τ 2, λ ± = (1 τ 2 ) ± τ τ In this case, as long as hω < 2, we have λ ± = 1 and ρ(a vv ) = 1. Otherwise, ρ(a v v) > 1. The implicit stepper is even more interesting: A mid = τ2 4 1 τ2 hω 2 4 h 1 τ2 4, λ ± = τ2 2 and in this case, λ ± = ρ(a mid ) = 1 for any value of τ = hω. 1 τ 2 2 iτ Typeset by FoilTEX May 5th 26 15
9 Some Results amplitude Damped Simple Harmonic Oscillator, 1D, c = 1E-1, d = time in units of natural period Variational Implicit midpoint time Vortex Implicit Euler Exact Figure 2: Simple harmonic oscillator in one dimension using different integration methods. At low frequency, they all perform reasonably well as expected but both the Vortex and the implicit Euler method, which produce identical output in this case, dissipate energy. Note that both the variational and the midpoint method produce identical output in this case. Typeset by FoilTEX May 5th Damped Simple Harmonic Oscillator, 1D, c = 1E-4, d = time in units of natural period amplitude Variational Implicit midpoint time Vortex Implicit Euler.6.7 Exact.8.9 Figure 3: Simple harmonic oscillator in one dimension using different integration methods. At higher frequencies, the Vortex and implicit methods dissipate energy very quickly. The midpoint and variational methods are producing the correct output qualitatively but of course, the phase of the signal is erratic. Typeset by FoilTEX May 5th 26 17
10 amplitude Damped Simple Harmonic Oscillator, 1D, c = 1E-6, d = time in units of natural period Variational Implicit midpoint.1.15 time.2 Vortex Implicit Euler.25.3 Exact Figure 4: Simple harmonic oscillator in one dimension using different integration methods. At even higher frequencies, the variational and midpoint methods show signs of instability. A simple computation shows that both integration methods produce an oscillatory signal which diverges linearly to infinity. Vortex and implicit Euler are intrinsically damped and just decay to Typeset by FoilTEX May 5th Summary of reasonably good methods Derivations can be found elsewhere but here is the summary of the one and only explicit method you should bother with at first order: Velocity Verlet: x n+1 = x n + hv n + h2 Verlet or Leap Frog (don t bother with v n ) 2 M 1 f(x n ) (8) v n+1 = v n + h 2 (f(x n) + f(x n+1 )) (9) x n+1 = x n + hv n+1/2 (1) v n+1/2 = v n 1/2 + hf(x n ) (11) Typeset by FoilTEX May 5th 26 19
11 Summary of reasonably good methods Original Verlet: don t even track velocity: x n+1 = 2x n x n 1 + M 1 h 2 f(x n ) Position Verlet, partially implicit: x n+1 = 2x n x n 1 + M 1 h 2 ((1 α)f(x n+1 ) + αf(x n )) Note that the half implicit Verlet method with α = 1/2 is of second order. Typeset by FoilTEX May 5th 26 2 Summary of reasonably good methods Two implicit methods that come in handy: Implicit Euler (dissipative): x n+1 = x n + hv n+1 v n+1 = v n + hf(x n+1 ) = v n + hf(x n + hv n+1 ) (12) Implicit midpoint: x n+1 = x n + h 2 [v n+1 + v n ] v n+1 = v n + hf( x n+1 + x n ) = v n + hf(x n + h 2 4 [v n+1 + v n ]) (13) Must do root finding on the velocity equation: more about this later. Typeset by FoilTEX May 5th 26 21
12 Conclusion of part I It is possible to find simple methods to integrate physical systems in a stable way but careful analysis is necessary. There is a multitude of integration method to choose from and there is no real clear cut method that will work in all cases, satisfying all requirements. It is possible to integrate high frequency problems without resolving them at the time scale of the simulation, and at least capturing some of the overall physics (the midpoint rule). However, there are problems and this is discussed later. Typeset by FoilTEX May 5th Constraints Introduction Simplify the machinery of Erleblen: only holonomic case for now: no velocity or time dependence, no friction constraints ignore rigid rotation details and think of point particles Typeset by FoilTEX May 5th 26 23
13 Constraints Introduction Physical constraints Φ(x) are limits of strong forces. Think of a spring f = kx. As k becomes large, we get the constraint Φ(x) = x = k m mg Figure 5: A stretchable pendulum Typeset by FoilTEX May 5th Constraints Introduction Geometric constraints produce forces. How much? Just enough to keep Φ(x) =. We are going to have to compute the constraint forces somehow. Since motion is tangential to Φ(x) =, a force in the direction normal to Φ does not work. This means that f c = J T λ. Φ/ x T = J T f e (x) External force Resulting force is tangent to the curve Φ(x) = and motion satisfies constraints Φ(x) = Constraint force f c = J T λ Cancel component of f that would violate the constraint Figure 6: A bead on a wire Typeset by FoilTEX May 5th 26 25
14 Constraints Introduction The equation of motion is now: Mẍ = f e + f c = f e + J T λ Φ(x) = (14) This is called a differential algebraic equation (DAE). The variable λ is implicitly defined now as we can rewrite these equations in the form F (ẍ, ẋ, x, λ) =, i.e., as a nonlinear equations in the varaibles ẍ, ẋ, x, λ. Since there is no differential equation for λ, we cannot use the stepping methods that work for ODEs. Typeset by FoilTEX May 5th How can we solve for λ? Let us try to find if there is a linear expression hiding here. Since Φ(x) =, then, Φ = const so that Φ = Φ =. Φ = d dt Φ Φ(x) = ẋ = Jv =. x Φ = d2 dt 2Φ(x) = d dt [Jẋ] = Jẍ + Jẋ = Jẍ + c =. Formally, the equation Φ(x) = is mathematically equivalent to Φ = Jẍ =. So: Mẍ J T λ = f e Jẍ = c (15) This is linear in ẍ and λ: M J T J ẍ λ = f e c Typeset by FoilTEX May 5th 26 27
15 How can we solve for λ? Solve first equation for ẍ and sub in second equation: ẍ = M 1 J T λ + M 1 f e Jẍ = JM 1 J T λ + JM 1 f e = c JM 1 J T λ = JM 1 f e c Put S = JM 1 J T, the Schur complement, and substitute back in: ẍ = M 1 J T S 1 Then use an ODE method, right? JM 1 fe + c + M 1 f e No. Numerically, enforcing Φ(x) = is not the same as Φ(x) =. Though we do follow the tangent of the constraint, we take finite steps and move off the surface. Hard problem but, fortunately, there are good simple methods (and many, many simple looking BAD methods: be careful!) Typeset by FoilTEX May 5th Solve for λ numerically the Rattle way. We can approximate ẍ numerically directly with: ẍ x n+1 2x n + x n 1 h 2 and put this in the equation in the equation of motion so we can write: Mx n+1 h 2 J T λ = M [2x n x n 1 ] + h 2 f e Φ(x n+1 ) = Φ(x n ) + J n (x n+1 x n ) Now, this is linear in x n+1. Remains to chose at which step to evaluate J in the first equation and we can at least try J n. Putting λ = h 2 λ, we have: M J n J T n x n+1 λ = M [2x n x n 1 ] + h 2 f e (x n ) Φ(x n ) + J n x n Typeset by FoilTEX May 5th 26 29
16 Solve for λ numerically the Rattle way. This will still drift though because we are not solving exactly for Φ(x n+1 = but it is still better than discretizing the ODE reduction directly. Curiously, Erleben insists on calling this a velocity based formulation as opposed to the previous format which he (and others) call an acceleration based formulation. To make this stable, we need to solve the nonlinear equations Φ(x n+1 ) =. To see how this is done, first remember that: x n+1 = z n + h 2 M 1 J T n λ where z n is the sum of all the terms that are evaluated at time n. Then: Φ(x n+1 ) = Φ(z n + h 2 M 1 J T n λ) = This is m nonlinear equations in m unknowns. Solve this with Newton iterations say: J ν M 1 J T n λ (ν+1) λ (ν) = 1 h 2Φ(z n + h 2 M 1 J T n λ(ν) ) Start with λ () =, say, and iterate over ν a few times. You could update J ν so that it is reevaluated at the new coordinate x n+1 = z n + h 2 M 1 J T λ (ν) but you can also keep it at J n. This last choice corresponds to a modified Newton method. (I m not going into details because you will not implement this). Typeset by FoilTEX May 5th 26 3 Is there another way? Constraints are the limit of strong forces. instead of constraints? Can we use strong forces numerically Potential: V (x) = k 2 ΦT Φ Force: f c (x) = V x = kj T Φ Maybe λ kφ? Remember that for implicit integration, we need to evaluate f (αx n + (1 α)x n+1 ). Approximate this linearly in x n+1 so that: f (αx n + (1 α)x n+1 ) = f(x n ) + (1 α) f x (x n x n+1 ) and then, using f = kj T Φ: K = 1 k f c x = J T Φ x + J T x Φ = J T J + 2 Φ x x T Typeset by FoilTEX May 5th 26 31
17 Is there another way? Substituting this in the implicit Verlet stepper: x n+1 = 2x n x n 1 + M 1 h 2 f (αx n + (1 α)x n+1 ) and accounting for all the terms, this becomes: I + h 2 k(1 α)m 1 Kn x n+1 = (16) 2x n x n 1 + M 1 h 2 f n + h 2 k(1 α)m 1 K n x n (17) This is essentially the Baraff [1] cloth method when α =. In general, the case with α = 1/2 is best as it corresponds to the midpont rule but one must explictly add some damping. The matrix multiplying x n+1 is symmetric but not necessarily positive definite. In addition, the factor of k is problematic as the problem is ill-posed when this gets too large. Matrix K contains second derivative terms and not well behaved. 2 Φ x x T which are expensive to compute Typeset by FoilTEX May 5th To do relax Rattle using λ = kφ show the difference between the two matrices show what happens in the limit k. Next time: methods. spring-damper systems, cable example, Gauss-Seidel iterations, direct References [1] David Baraff and Andrew Witkin. Large steps in cloth simulation. In Michael F. Cohen, editor, Computer graphics: proceedings: SIGGRAPH 98 Conference proceedings, July 19 24, 1998, Computer Graphics -proceedings- 1998, pages 43 54, New York, NY 136, USA and Reading, MA, USA, ACM Press and Addison-Wesley. ACM order number: Typeset by FoilTEX May 5th 26 33
Physically Based Modeling Differential Equation Basics
Physically Based Modeling Differential Equation Basics Andrew Witkin and David Baraff Pixar Animation Studios Please note: This document is 2001 by Andrew Witkin and David Baraff. This chapter may be freely
More informationPhysically Based Modeling: Principles and Practice Differential Equation Basics
Physically Based Modeling: Principles and Practice Differential Equation Basics Andrew Witkin and David Baraff Robotics Institute Carnegie Mellon University Please note: This document is 1997 by Andrew
More informationModeling and Solving Constraints. Erin Catto Blizzard Entertainment
Modeling and Solving Constraints Erin Catto Blizzard Entertainment Basic Idea Constraints are used to simulate joints, contact, and collision. We need to solve the constraints to stack boxes and to keep
More informationReview Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester
HIGHER ORDER METHODS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE
More informationDifferential Equations
Differential Equations Overview of differential equation! Initial value problem! Explicit numeric methods! Implicit numeric methods! Modular implementation Physics-based simulation An algorithm that
More informationTime scales. Notes. Mixed Implicit/Explicit. Newmark Methods [ ] [ ] "!t = O 1 %
Notes Time scales! For using Pixie (the renderer) make sure you type use pixie first! Assignment 1 questions?! [work out]! For position dependence, characteristic time interval is "!t = O 1 % $ ' # K &!
More informationOrdinary Differential Equations
Chapter 13 Ordinary Differential Equations We motivated the problem of interpolation in Chapter 11 by transitioning from analzying to finding functions. That is, in problems like interpolation and regression,
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 33 Almost Done! Last
More informationCS-184: Computer Graphics
CS-184: Computer Graphics Lecture #25: Rigid Body Simulations Tobias Pfaff 537 Soda (Visual Computing Lab) tpfaff@berkeley.edu Reminder Final project presentations next week! Game Physics Types of Materials
More informationStability of Mass-Point Systems
Simulation in Computer Graphics Stability of Mass-Point Systems Matthias Teschner Computer Science Department University of Freiburg Demos surface tension vs. volume preservation distance preservation
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More information4. Complex Oscillations
4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic
More informationSolution via Laplace transform and matrix exponential
EE263 Autumn 2015 S. Boyd and S. Lall Solution via Laplace transform and matrix exponential Laplace transform solving ẋ = Ax via Laplace transform state transition matrix matrix exponential qualitative
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 29 Almost Done! No
More informationDamped harmonic motion
Damped harmonic motion March 3, 016 Harmonic motion is studied in the presence of a damping force proportional to the velocity. The complex method is introduced, and the different cases of under-damping,
More informationLinear and Nonlinear Oscillators (Lecture 2)
Linear and Nonlinear Oscillators (Lecture 2) January 25, 2016 7/441 Lecture outline A simple model of a linear oscillator lies in the foundation of many physical phenomena in accelerator dynamics. A typical
More informationODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class
ODE Homework Due Wed. 9 August 2009; At the beginning of the class. (a) Solve Lẏ + Ry = E sin(ωt) with y(0) = k () L, R, E, ω are positive constants. (b) What is the limit of the solution as ω 0? (c) Is
More informationPSE Game Physics. Session (6) Angular momentum, microcollisions, damping. Oliver Meister, Roland Wittmann
PSE Game Physics Session (6) Angular momentum, microcollisions, damping Oliver Meister, Roland Wittmann 23.05.2014 Session (6)Angular momentum, microcollisions, damping, 23.05.2014 1 Outline Angular momentum
More informationOrdinary Differential Equations
Ordinary Differential Equations We call Ordinary Differential Equation (ODE) of nth order in the variable x, a relation of the kind: where L is an operator. If it is a linear operator, we call the equation
More informationProblem Set Number 01, MIT (Winter-Spring 2018)
Problem Set Number 01, 18.377 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Thursday, March 8, 2018. Turn it in (by 3PM) at the Math.
More informationPHYSICS 110A : CLASSICAL MECHANICS
PHYSICS 110A : CLASSICAL MECHANICS 1. Introduction to Dynamics motion of a mechanical system equations of motion : Newton s second law ordinary differential equations (ODEs) dynamical systems simple 2.
More informationProblem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension
105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1
More informationConsider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be.
Chapter 4 Energy and Stability 4.1 Energy in 1D Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be T = 1 2 mẋ2 and the potential energy
More informationNumerical solution of ODEs
Numerical solution of ODEs Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology November 5 2007 Problem and solution strategy We want to find an approximation
More informationEN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015
EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 9
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 9 Initial Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationHarmonic oscillator. U(x) = 1 2 bx2
Harmonic oscillator The harmonic oscillator is a familiar problem from classical mechanics. The situation is described by a force which depends linearly on distance as happens with the restoring force
More informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More informationReview: control, feedback, etc. Today s topic: state-space models of systems; linearization
Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered
More informationAutonomous Systems and Stability
LECTURE 8 Autonomous Systems and Stability An autonomous system is a system of ordinary differential equations of the form 1 1 ( 1 ) 2 2 ( 1 ). ( 1 ) or, in vector notation, x 0 F (x) That is to say, an
More informationUnderstand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.
Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics
More informationLinearization of Differential Equation Models
Linearization of Differential Equation Models 1 Motivation We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking
More informationRobotics. Dynamics. Marc Toussaint U Stuttgart
Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, Euler-Lagrange equation, Newton-Euler recursion, general robot dynamics, joint space control, reference trajectory
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear
More informationHarmonic Oscillator I
Physics 34 Lecture 7 Harmonic Oscillator I Lecture 7 Physics 34 Quantum Mechanics I Monday, February th, 008 We can manipulate operators, to a certain extent, as we would algebraic expressions. By considering
More informationFirst-Order Differential Equations
CHAPTER 1 First-Order Differential Equations 1. Diff Eqns and Math Models Know what it means for a function to be a solution to a differential equation. In order to figure out if y = y(x) is a solution
More informationMATH 320, WEEK 11: Eigenvalues and Eigenvectors
MATH 30, WEEK : Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors We have learned about several vector spaces which naturally arise from matrix operations In particular, we have learned about the
More informationSymmetries 2 - Rotations in Space
Symmetries 2 - Rotations in Space This symmetry is about the isotropy of space, i.e. space is the same in all orientations. Thus, if we continuously rotated an entire system in space, we expect the system
More informationSimple Car Dynamics. Outline. Claude Lacoursière HPC2N/VRlab, Umeå Universitet, Sweden, May 18, 2005
Simple Car Dynamics Claude Lacoursière HPC2N/VRlab, Umeå Universitet, Sweden, and CMLabs Simulations, Montréal, Canada May 18, 2005 Typeset by FoilTEX May 16th 2005 Outline basics of vehicle dynamics different
More informationCDS 101 Precourse Phase Plane Analysis and Stability
CDS 101 Precourse Phase Plane Analysis and Stability Melvin Leok Control and Dynamical Systems California Institute of Technology Pasadena, CA, 26 September, 2002. mleok@cds.caltech.edu http://www.cds.caltech.edu/
More informationChapter 15 Periodic Motion
Chapter 15 Periodic Motion Slide 1-1 Chapter 15 Periodic Motion Concepts Slide 1-2 Section 15.1: Periodic motion and energy Section Goals You will learn to Define the concepts of periodic motion, vibration,
More information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationCS 450 Numerical Analysis. Chapter 9: Initial Value Problems for Ordinary Differential Equations
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationWoods Hole Methods of Computational Neuroscience. Differential Equations and Linear Algebra. Lecture Notes
Woods Hole Methods of Computational Neuroscience Differential Equations and Linear Algebra Lecture Notes c 004, 005 William L. Kath MCN 005 ODE & Linear Algebra Notes 1. Classification of differential
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture.1 Dynamic Behavior Richard M. Murray 6 October 8 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More information2.3 Damping, phases and all that
2.3. DAMPING, PHASES AND ALL THAT 107 2.3 Damping, phases and all that If we imagine taking our idealized mass on a spring and dunking it in water or, more dramatically, in molasses), then there will be
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 9 Initial Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign
More informationPhysics 6010, Fall Relevant Sections in Text: Introduction
Physics 6010, Fall 2016 Introduction. Configuration space. Equations of Motion. Velocity Phase Space. Relevant Sections in Text: 1.1 1.4 Introduction This course principally deals with the variational
More informationDamped Oscillation Solution
Lecture 19 (Chapter 7): Energy Damping, s 1 OverDamped Oscillation Solution Damped Oscillation Solution The last case has β 2 ω 2 0 > 0. In this case we define another real frequency ω 2 = β 2 ω 2 0. In
More information2t t dt.. So the distance is (t2 +6) 3/2
Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the
More informationIntroduction to standard and non-standard Numerical Methods
Introduction to standard and non-standard Numerical Methods Dr. Mountaga LAM AMS : African Mathematic School 2018 May 23, 2018 One-step methods Runge-Kutta Methods Nonstandard Finite Difference Scheme
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture 2.1 Dynamic Behavior Richard M. Murray 6 October 28 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationNumerical Methods for Differential Equations Mathematical and Computational Tools
Numerical Methods for Differential Equations Mathematical and Computational Tools Gustaf Söderlind Numerical Analysis, Lund University Contents V4.16 Part 1. Vector norms, matrix norms and logarithmic
More informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More informationMath 216 Final Exam 24 April, 2017
Math 216 Final Exam 24 April, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationIn the presence of viscous damping, a more generalized form of the Lagrange s equation of motion can be written as
2 MODELING Once the control target is identified, which includes the state variable to be controlled (ex. speed, position, temperature, flow rate, etc), and once the system drives are identified (ex. force,
More informationBIBO STABILITY AND ASYMPTOTIC STABILITY
BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to
More informationChapter 15 - Oscillations
The pendulum of the mind oscillates between sense and nonsense, not between right and wrong. -Carl Gustav Jung David J. Starling Penn State Hazleton PHYS 211 Oscillatory motion is motion that is periodic
More informationfor changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df
Physics 106a, Caltech 1 November, 2018 Lecture 10: Hamiltonian Mechanics I The Hamiltonian In the Hamiltonian formulation of dynamics each second order ODE given by the Euler- Lagrange equation in terms
More informationOrdinary Differential Equations (ODEs)
Ordinary Differential Equations (ODEs) 1 Computer Simulations Why is computation becoming so important in physics? One reason is that most of our analytical tools such as differential calculus are best
More informationCHAPTER 7. An Introduction to Numerical Methods for. Linear and Nonlinear ODE s
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 1 A COLLECTION OF HANDOUTS ON FIRST ORDER ORDINARY DIFFERENTIAL
More informationExamination paper for TMA4195 Mathematical Modeling
Department of Mathematical Sciences Examination paper for TMA4195 Mathematical Modeling Academic contact during examination: Elena Celledoni Phone: 48238584, 73593541 Examination date: 11th of December
More informationODEs. September 7, Consider the following system of two coupled first-order ordinary differential equations (ODEs): A =
ODEs September 7, 2017 In [1]: using Interact, PyPlot 1 Exponential growth and decay Consider the following system of two coupled first-order ordinary differential equations (ODEs): d x/dt = A x for the
More informationSimple Harmonic Motion
Simple Harmonic Motion (FIZ 101E - Summer 2018) July 29, 2018 Contents 1 Introduction 2 2 The Spring-Mass System 2 3 The Energy in SHM 5 4 The Simple Pendulum 6 5 The Physical Pendulum 8 6 The Damped Oscillations
More information03. Simple Dynamical Systems
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 03. Simple Dynamical Systems Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationModeling and Experimentation: Compound Pendulum
Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014 Overview This lab focuses on developing a mathematical
More informationOscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is
Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring
More informationIntroduction to Path Integrals
Introduction to Path Integrals Consider ordinary quantum mechanics of a single particle in one space dimension. Let s work in the coordinate space and study the evolution kernel Ut B, x B ; T A, x A )
More informationMATH 1231 MATHEMATICS 1B Calculus Section 2: - ODEs.
MATH 1231 MATHEMATICS 1B 2007. For use in Dr Chris Tisdell s lectures: Tues 11 + Thur 10 in KBT Calculus Section 2: - ODEs. 1. Motivation 2. What you should already know 3. Types and orders of ODEs 4.
More informationPhysics 115/242 Comparison of methods for integrating the simple harmonic oscillator.
Physics 115/4 Comparison of methods for integrating the simple harmonic oscillator. Peter Young I. THE SIMPLE HARMONIC OSCILLATOR The energy (sometimes called the Hamiltonian ) of the simple harmonic oscillator
More informationAssignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class
Assignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class Homeworks VIII and IX both center on Lagrangian mechanics and involve many of the same skills. Therefore,
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions Overview Our next goal is to extend LMI s and optimization to nonlinear
More informationKasetsart University Workshop. Multigrid methods: An introduction
Kasetsart University Workshop Multigrid methods: An introduction Dr. Anand Pardhanani Mathematics Department Earlham College Richmond, Indiana USA pardhan@earlham.edu A copy of these slides is available
More informationAppendix A Equations of Motion in the Configuration and State Spaces
Appendix A Equations of Motion in the Configuration and State Spaces A.1 Discrete Linear Systems A.1.1 Configuration Space Consider a system with a single degree of freedom and assume that the equation
More informationDifferential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm
Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is
More informationDifferential Equations
Differential Equations A differential equation (DE) is an equation which involves an unknown function f (x) as well as some of its derivatives. To solve a differential equation means to find the unknown
More informationNumerical Algorithms as Dynamical Systems
A Study on Numerical Algorithms as Dynamical Systems Moody Chu North Carolina State University What This Study Is About? To recast many numerical algorithms as special dynamical systems, whence to derive
More informationCHAPTER 10: Numerical Methods for DAEs
CHAPTER 10: Numerical Methods for DAEs Numerical approaches for the solution of DAEs divide roughly into two classes: 1. direct discretization 2. reformulation (index reduction) plus discretization Direct
More informationNumerical Methods - Numerical Linear Algebra
Numerical Methods - Numerical Linear Algebra Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Numerical Linear Algebra I 2013 1 / 62 Outline 1 Motivation 2 Solving Linear
More informationMultistep Methods for IVPs. t 0 < t < T
Multistep Methods for IVPs We are still considering the IVP dy dt = f(t,y) t 0 < t < T y(t 0 ) = y 0 So far we have looked at Euler s method, which was a first order method and Runge Kutta (RK) methods
More informationTopic 5 Notes Jeremy Orloff. 5 Homogeneous, linear, constant coefficient differential equations
Topic 5 Notes Jeremy Orloff 5 Homogeneous, linear, constant coefficient differential equations 5.1 Goals 1. Be able to solve homogeneous constant coefficient linear differential equations using the method
More informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationCircular motion. Aug. 22, 2017
Circular motion Aug. 22, 2017 Until now, we have been observers to Newtonian physics through inertial reference frames. From our discussion of Newton s laws, these are frames which obey Newton s first
More informationTwo-Body Problem. Central Potential. 1D Motion
Two-Body Problem. Central Potential. D Motion The simplest non-trivial dynamical problem is the problem of two particles. The equations of motion read. m r = F 2, () We already know that the center of
More informationCS520: numerical ODEs (Ch.2)
.. CS520: numerical ODEs (Ch.2) Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca people.cs.ubc.ca/ ascher/520.html Uri Ascher (UBC) CPSC 520: ODEs (Ch. 2) Fall
More informationIntroductory Physics. Week 2015/05/29
2015/05/29 Part I Summary of week 6 Summary of week 6 We studied the motion of a projectile under uniform gravity, and constrained rectilinear motion, introducing the concept of constraint force. Then
More informationMeaning of the Hessian of a function in a critical point
Meaning of the Hessian of a function in a critical point Mircea Petrache February 1, 2012 We consider a function f : R n R and assume for it to be differentiable with continuity at least two times (that
More informationDue 1/24/12: Section and 6 and the problems below.
Math 221 - Homework assignments General instructions: No effort will be made to read illegible homework. In this connection illegible means: unintelligible grammar, handwriting too small or otherwise unreadable,
More informationENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations
ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential
More informationMATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES
MATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES J. WONG (FALL 2017) What did we cover this week? Basic definitions: DEs, linear operators, homogeneous (linear) ODEs. Solution techniques for some classes
More information18. Linearization: the phugoid equation as example
79 18. Linearization: the phugoid equation as example Linearization is one of the most important and widely used mathematical terms in applications to Science and Engineering. In the context of Differential
More informationL = 1 2 a(q) q2 V (q).
Physics 3550, Fall 2011 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion
More informationFinal 09/14/2017. Notes and electronic aids are not allowed. You must be seated in your assigned row for your exam to be valid.
Final 09/4/207 Name: Problems -5 are each worth 8 points. Problem 6 is a bonus for up to 4 points. So a full score is 40 points and the max score is 44 points. The exam has 6 pages; make sure you have
More informationResonance and response
Chapter 2 Resonance and response Last updated September 20, 2008 In this section of the course we begin with a very simple system a mass hanging from a spring and see how some remarkable ideas emerge.
More informationNumerical Methods for Initial Value Problems; Harmonic Oscillators
Lab 1 Numerical Methods for Initial Value Problems; Harmonic Oscillators Lab Objective: Implement several basic numerical methods for initial value problems (IVPs), and use them to study harmonic oscillators.
More informationOrdinary differential equations. Phys 420/580 Lecture 8
Ordinary differential equations Phys 420/580 Lecture 8 Most physical laws are expressed as differential equations These come in three flavours: initial-value problems boundary-value problems eigenvalue
More information2 Discrete Dynamical Systems (DDS)
2 Discrete Dynamical Systems (DDS) 2.1 Basics A Discrete Dynamical System (DDS) models the change (or dynamics) of single or multiple populations or quantities in which the change occurs deterministically
More information1 Simple Harmonic Oscillator
Physics 1a Waves Lecture 3 Caltech, 10/09/18 1 Simple Harmonic Oscillator 1.4 General properties of Simple Harmonic Oscillator 1.4.4 Superposition of two independent SHO Suppose we have two SHOs described
More information