numeric-geometric techniques for diﬀerential equations … · numeric-geometric techniques for...

Numeric-Geometric Techniques for Differential Equations II. Applications

Numeric-Geometric Techniques for DifferentialEquations II. Applications

Niloofar Mani and Greg Reid

Applied Mathematics, University of Western Ontario

June 25 2009, Conference ACA 2009, Montreal

Table of contents

1 MapleSimWhat is MapleSim and What we can do with it?MapleSim and Maple 13

2 From Part 1 - the previous talkFast Prolongation Method by Reid and Wu

3 Homotopy and Bertini SoftwareHomotopyBertini

4 Example:DAE for a Crane in Maple by fast prolongation method

5 From MapleSimExample from MapleSim done by our method

6 Summery and Future worksWhat is next?

7 Acknowledgements

8 References

MapleSim

What is MapleSim and What we can do with it?

MapleSim

What is MapleSim?[5]

A high-performance multi-domain modeling and simulationtool.

Powered by Maple 13.

MapleSim

A high-performance multi-domain modeling and simulationtool.Powered by Maple 13.

MapleSim

A high-performance multi-domain modeling and simulationtool.Powered by Maple 13.

MapleSim

What can we do with MapleSim?

Use MapleSim to build models that integrate components fromvarious engineering fields into a complete system.

Build component diagrams that represent such systems in agraphical form.Simulated behavior of a system.Use both Symbolic Computation and numeric approach.

MapleSim

Use MapleSim to build models that integrate components fromvarious engineering fields into a complete system.Build component diagrams that represent such systems in agraphical form.

Simulated behavior of a system.Use both Symbolic Computation and numeric approach.

MapleSim

Use MapleSim to build models that integrate components fromvarious engineering fields into a complete system.Build component diagrams that represent such systems in agraphical form.Simulated behavior of a system.

Use both Symbolic Computation and numeric approach.

MapleSim

Use MapleSim to build models that integrate components fromvarious engineering fields into a complete system.Build component diagrams that represent such systems in agraphical form.Simulated behavior of a system.Use both Symbolic Computation and numeric approach.

MapleSim

Use MapleSim to build models that integrate components fromvarious engineering fields into a complete system.Build component diagrams that represent such systems in agraphical form.Simulated behavior of a system.Use both Symbolic Computation and numeric approach.

MapleSim

What are key advantages of MapleSim?

Create system diagram on the screen using icons.

It has connection to Maple so system equations areautomatically generated for us.Fast, it cuts project time and cost.It has 3D visualizer that gives immediate insight into behaviorof our model.

MapleSim

Create system diagram on the screen using icons.It has connection to Maple so system equations areautomatically generated for us.

Fast, it cuts project time and cost.It has 3D visualizer that gives immediate insight into behaviorof our model.

MapleSim

Create system diagram on the screen using icons.It has connection to Maple so system equations areautomatically generated for us.Fast, it cuts project time and cost.

It has 3D visualizer that gives immediate insight into behaviorof our model.

MapleSim

Create system diagram on the screen using icons.It has connection to Maple so system equations areautomatically generated for us.Fast, it cuts project time and cost.It has 3D visualizer that gives immediate insight into behaviorof our model.

MapleSim

Create system diagram on the screen using icons.It has connection to Maple so system equations areautomatically generated for us.Fast, it cuts project time and cost.It has 3D visualizer that gives immediate insight into behaviorof our model.

MapleSim

How to make a model in MapleSim:

MapleSim

Modeling of Single Pendulum:

MapleSim

Modeling of Single Pendulum after simulation:

MapleSim

MapleSim and Maple 13

What does Maple 13 do for MapleSim?

Use Maple commands to analyse the dynamic behavior of aMapleSim model.

View model equations worksheet, test input and output values.Use plotting tools to visualize possible simulation results.

MapleSim

Use Maple commands to analyse the dynamic behavior of aMapleSim model.View model equations worksheet, test input and output values.

Use plotting tools to visualize possible simulation results.

MapleSim

Use Maple commands to analyse the dynamic behavior of aMapleSim model.View model equations worksheet, test input and output values.Use plotting tools to visualize possible simulation results.

MapleSim

Use Maple commands to analyse the dynamic behavior of aMapleSim model.View model equations worksheet, test input and output values.Use plotting tools to visualize possible simulation results.

MapleSim

Our model in Maple 13:

DescriptionUse this template as a starting point for performing advanced analysis on MapleSim physical models.This template allows you to retrieve equations to gain insight into the behavior of your model.

Note: The ability to retrieve equations is currently limited to continuous subsystems.

Model Diagram

MapleSim

Our model in Maple 13:

Model EquationsTo start, click System Update to populate the Subsystem menu with a list of the subsystems in your physical model. From the list, select the subsystem for which you want to retrieve model equations orproperties. Alternatively, select Main to retrieve model equations or properties for the entire system. Next, click Get Equations to retrieve equations from the specified system or subsystem. ClickAssign to variable to manipulate these equations further in Maple. You can also retrieve properties and assign them to variables.

Model Main

System Update

Subsystem: Main

Get Equations

Get Discrete Equations

Get Boolean Equations

Get I/O Variables

Get Initial Equations

Get Parameters

Get Probes

Get Constraints

Get States

Assign to variable:

MapleSim

How can we derive equations of our model?Use Get Equation from that table.

Use Maple commands:mysys := GetProperty(”Simulation0”, ’system’);GetEquations(mysys);To have unsimplified one GetEquations(mysys, simplify =false);bug (see below)!

Use modelica to work around this.

MapleSim

How can we derive equations of our model?Use Get Equation from that table.Use Maple commands:

mysys := GetProperty(”Simulation0”, ’system’);GetEquations(mysys);To have unsimplified one GetEquations(mysys, simplify =false);bug (see below)!

MapleSim

How can we derive equations of our model?Use Get Equation from that table.Use Maple commands:

mysys := GetProperty(”Simulation0”, ’system’);GetEquations(mysys);To have unsimplified one GetEquations(mysys, simplify =false);bug (see below)!

From Part 1 - the previous talk

Fast Prolongation Method by Reid and Wu

From last talk

Fast Prolongation

We often need differentiation to cover all of a DE system’sconstraints – prolongation; and simplify to check if they are“new” – differential elimination. By Reid and Wu in[1, 2]

But a major problem is the exploding expression size.

Fast Prolongation

We often need differentiation to cover all of a DE system’sconstraints – prolongation; and simplify to check if they are“new” – differential elimination. By Reid and Wu in[1, 2]

But a major problem is the exploding expression size.

Fast prolongation method

How to find numerically stable methods to identify all hiddenconstraints without prolongation explosion and complicateddiff. elim.?[1]

Only prolongations w.r.t one variable, # new eqns will notchange after prolongation!

No elimination is needed, only check some simple criteria fortermination and failure of the algorithm.

Homotopy and Bertini Software

Homotopy

When we want to find roots of p(z) = 0.

We can start from roots of q(z) = 0 which we know anddeform these into the p(z) = 0, how? By Sommese andWampler in[3]

H(z , t) := te iθ(zd −1)+(1− t)p(z) = 0 where θ ∈ [0, 2π].

t = tinit ⇒ H(z , tinit) = q(z) roots of q(z).t = tfin ⇒ H(z , tfin) = p(z) roots of p(z)something we wish to solve.

How can we numerically follow solution paths?

H(zi (t), t) ≡ 0 ⇒ dzi (t)dt = −Hz(zi (t), t)

−1Ht(zi (t), t)

Homotopy

−1Ht(zi (t), t)

Homotopy

−1Ht(zi (t), t)

Homotopy

−1Ht(zi (t), t)

Homotopy

−1Ht(zi (t), t)

Homotopy

−1Ht(zi (t), t)

Homotopy

First root by numerical discretisation:

p(z) := (z − 5)(z − (1

2+ 3i))

Homotopy

Second root by numerical discretisation:

p(z) := (z − 5)(z − (1

2+ 3i))

Homotopy

Locating both roots:

p(z) := (z − 5)(z − (1

2+ 3i))

Bertini

Bertini by Bates, Hauenstein, et al.

Purpose: The numerical solution of systems of polynomialequations. See Bertini Manual[4]

Approach: Homotopy continuation implemented in C.

Key Features: Finds isolated solutions using total-degreestart systems.

Bertini

How Bertini works?

Bertini can produce all isolated solutions of a system.

Write function equation and its variables in an input file.

It produces different files.

Bertini

How Bertini works?

Bertini

How Bertini works?

Example:

Example, DAE for a Crane in Maple 13

Example:

DAE for a Crane in Maple by fast prolongation method

Example:

DAE for a Crane

The problem is to determine the horizontal velocity u1(t) of awinch of mass M1, and the angular velocity u2(t) of the winchso that the attached load M2 moves along a prescribedpath.[2]

Belongs to a test set of 27 DAE from diverse applications dueto Visconti. The method succeeded for 23 of these.

Example:

DAE for a Crane

The problem is to determine the horizontal velocity u1(t) of awinch of mass M1, and the angular velocity u2(t) of the winchso that the attached load M2 moves along a prescribedpath.[2]

Belongs to a test set of 27 DAE from diverse applications dueto Visconti. The method succeeded for 23 of these.

Example:

DAE for a Crane

The equations of motion are given with unknowns{x , x ′, z , z ′, d , d ′, r , r ′, θ, τ, u1, u2}:[2]

xt − x ′ = 0, zt − z ′ = 0, dt − d ′ = 0, rt − r ′ = 0

M2 x ′t + τ sin(θ) = 0,M1 d ′

t + C1 dt − u1 − τ sin(θ) = 0

M2 z ′t + τ cos(θ)−mg = 0, J r ′

t + C2 rt + C3 u2 − C32τ = 0

r sin(θ) + d − x = 0, r cos(θ)− z = 0

H1(x , z , t) = 0, H2(x , z , t) = 0.

Example:

DAE for a Crane

To convert the DAE to an algebraic differential system wesubstituted sin(θ) and cos(θ) by n1 and n2 respectively.[2]

Substitute constants C1 = 1 C2 = 1 C3 = 1 J =1 M1 = 1 M2 = 2 g = 10 m = 1.

Example:

DAE for a Crane

To convert the DAE to an algebraic differential system wesubstituted sin(θ) and cos(θ) by n1 and n2 respectively.[2]

Substitute constants C1 = 1 C2 = 1 C3 = 1 J =1 M1 = 1 M2 = 2 g = 10 m = 1.

Example:

DAE for a Crane

Fast prolongation method from our package gives us those ci

and di for i = 1 . . . 13 in 0.062 seconds.[1, 2]

The result does not depend on the coefficients and degree ofH1, H2, since the signature matrix only requires thedifferential or orders of these functions, which are both zero.

Higher degree(≥ 4) leads to uncontrollable expressionexplosion for classical diff-elim methods.

Using that package we have made block structures in zerosecond.

Example:

DAE for a Crane

Example:

DAE for a Crane

Example:

DAE for a Crane

Example:

DAE for a Crane

Set random initial time.

Make Jet variables for sending to bertini software for examplextttt = xtttt.[1]

To solve system I need to calculate initial conditions bybottom up block which I did with Bertini.

After I made initial condition, I used ODE solver such asdsolve to numerically solve the system

Example:

DAE for a Crane

Example:

DAE for a Crane

Example:

DAE for a Crane

Example:

Bertini Part

INPUTvariable_group, x, zfunction, f1, f2f1 = x-.8434065834f2 = z-.6042811395END

Example:

Bertini Part

INPUTvariable_group, dx, dz, xt, ztfunction, f1, f2, f3, f4f1 = xt-dxf2 = zt-dzf3 = xt+.7914377210f4 = zt+1END

From MapleSim

Example from MapleSim done by our method

Example from MapleSim, Gimbal

From MapleSim

Gimbal after simulation

From MapleSim

Equations from Gimbal

The system has 27 eqns and 27 unknowns:

X8(t) = 0, X12(t) = 0, X1(t) − X5(t) = 0, . . .

X2(t) −d

dtX1(t) = 0, X4 −

dtX3(t) = 0, . . .

4 cos(X20(t))X23(t) cos(X21(t))2X25(t) − 8 cos(X20(t))X23(t)X25(t)

−2 cos(X21(t))X25(t)2 sin(X21(t)) − 6X22(t) sin(X20(t)) + . . .

There were many 2-terms eqns of form u − v = 0 orw − dz

dt = 0

From MapleSim

Gimbal

Classical symbolic simplification of the 2-term eqns w.r.tranking so that u < v , w < dz

Yields a system of 3 eqns and 3 unknowns.

From MapleSim

Gimbal

Classical symbolic simplification of the 2-term eqns w.r.tranking so that u < v , w < dz

Yields a system of 3 eqns and 3 unknowns.

From MapleSim

Gimbal

Application:

Fast prolongation is executed in 0.016 seconds and gives usc1 = 0,c2 = 0,c3 = 0,d1 = 2,d2 = 2,d3 = 2.[1, 2]We found Initial Condition randomly since we do not havemissing constraints.

We solve the system by dsolve.

From MapleSim

Gimbal

Application:

Fast prolongation is executed in 0.016 seconds and gives usc1 = 0,c2 = 0,c3 = 0,d1 = 2,d2 = 2,d3 = 2.[1, 2]We found Initial Condition randomly since we do not havemissing constraints.We solve the system by dsolve.

From MapleSim

Gimbal

Application:

Fast prolongation is executed in 0.016 seconds and gives usc1 = 0,c2 = 0,c3 = 0,d1 = 2,d2 = 2,d3 = 2.[1, 2]We found Initial Condition randomly since we do not havemissing constraints.We solve the system by dsolve.

From MapleSim

Result of Gimbal:Confirmation

From MapleSim

Summery and Future Works

Summery and Future works

What is next?

Summery:

Initial interface between MapleSim and Fast prolongation,Bertini software.

Future works:

MapleSim yields many examples to improve our algorithm.Make that initial condition by Bertini automatically.Make our package to work for those systems with unequalnumber of equations and variables.

What is next?

Summery:

Future works:

MapleSim yields many examples to improve our algorithm.

Make that initial condition by Bertini automatically.Make our package to work for those systems with unequalnumber of equations and variables.

What is next?

Summery:

Future works:

MapleSim yields many examples to improve our algorithm.Make that initial condition by Bertini automatically.

Make our package to work for those systems with unequalnumber of equations and variables.

What is next?

Summery:

Future works:

What is next?

Summery:

Future works:

Acknowledgements

Many thanks to Juergen Gerhard, Gilbert Lai, Greg Reid, AustinRoche, Allan Wittkopf and Wenyuan Wu for all their help.

References

Reference

1-G. Reid, W. Wu, S. Ilie.Implicit Riquier Bases for PDAE and their Semi-Discretizations.Journal of Symbolic Computaion. Vol 44. Issue 7. 923–941, 2009.

2-G. Reid, W. Wu.Symbolic-Numeric Computation of Implicit Riquier Bases for PDE.Proc. of ISSAC’07. 377–385, ACM 2007.

3-A.J. Sommese and C.W. Wampler.The Numerical Solution of Systems of Polynomials Arising in Engineeringand Science.World Scientific Press, Singapore, 2005.

4-D. Bates with contributions from J Hauenstein.Bertini User’s Manual.2008.

5-MapleSoft CompanyWWW.MapleSoft.Com,Webinars.2009

References

Thank you!

numeric-geometric techniques for diﬀerential equations … · numeric-geometric techniques for...

Documents