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Efficient Simulation of Physical System Models Using Inlined Implicit Runge-Kutta Algorithms. Vicha Treeaporn Department of Electrical & Computer Engineering The University of Arizona Tucson, Arizona 85721 U.S.A. Topics. Introduction Techniques for Simulation Results An Application.

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efficient simulation of physical system models using inlined implicit runge kutta algorithms

Efficient Simulation of Physical System Models Using Inlined Implicit Runge-Kutta Algorithms

Vicha Treeaporn

Department of Electrical & Computer Engineering

The University of Arizona

Tucson, Arizona 85721 U.S.A

topics
Topics
  • Introduction
  • Techniques for Simulation
  • Results
  • An Application
introduction
Introduction
  • Stiffness
    • Widely varying eigenvalues
  • Explicit algorithms
    • Straightforward to implement
    • Step size limited by numerical stability
  • Implicit algorithms
    • More difficult to implement
    • Additional computational load
    • Needed to simulate stiff systems
      • May use larger step sizes
inline integration
Inline-Integration
  • Merges the integration algorithm with the model
    • Eliminates differential equations
    • Results in difference equations (∆Es)
    • Easily implement implicit algorithms
  • Circuit example inlining Rad3
inlined with rad3
Inlined with Rad3

Evaluate at

Rad3 time instants

Eliminate

derivatives

Integrator equations

sorting3
Sorting
  • 10 equations immediately causalized
  • Need to perform tearing
    • Make assumptions about variables being ‘known’
tearing
Tearing

Tearing

variable

Residual Eq.

tearing1
Tearing

Residual Eq. #2

Tearing variable #2

tearing2
Tearing
  • Completely causalized equations
  • 2 iteration variables, vc and i1
  • Could use this set of equations for simulation
    • Want step-size control
step size control
Step-Size Control
  • Want larger step sizes
    • Reduce the overall computational cost
    • Maintain desired accuracy
      • Compute error estimate
      • Embedding method
        • Shares computations with original method
step size control1
Step-Size Control
  • Explicit RKs
    • Embedding methods have been found
  • Implicit RKs
    • Difficult problem
      • Algorithms are compact
    • Can find embedding methods using two steps
      • Linear polynomial approximation
hw sdirk embedding
HW-SDIRK Embedding
  • 3rd-order accurate
  • Behaves like an explicit method
    • May unnecessarily restrict step size for stiff systems
  • Search for an alternate embedding method
alt hw sdirk embedding
Alt. HW-SDIRK Embedding
  • 3rd-order accurate
  • Implicit method
alt hw sdirk embedding1
Alt. HW-SDIRK Embedding

Stability Domain

Damping Plots

lobatto iiic 6
Lobatto IIIC(6)
  • No embedding method exists
    • Expensive to perform step size control
  • Can search for an embedding method
lobatto iiic 6 embedding method
Lobatto IIIC(6) Embedding Method
  • 5th-order accurate
  • A-Stable
  • Large asymptotic region
lobatto iiic 6 embedding method1
Lobatto IIIC(6) Embedding Method

Stability Domain

Damping Plots

numerical experiments1
Numerical Experiments
  • Tested various algorithms with selected benchmark ODEs
  • Implemented in Dymola/Modelica
ode set b
ODE Set B

Inlined with HWSDIRK and

alternate error method

ode15s

ode set b1
ODE Set B

Error estimate stays near 10-3

Step size grows and

shrinks appropriately

ode set d
ODE Set D

Inlined with Lobatto IIIC(6)

ode15s

an application1
An Application
  • Real-Time, Limited Resources
    • Embedded control systems
      • Model Predictive
        • Add additional system dynamics
        • Simulate missile dynamics in flight for trajectory shaping
  • First solution is faster computer
    • Model may still be too complex
      • Try inlining