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4. Metamodeling

For CFD, FEM, and Look-up Tables.


Benefits of Local Model Network Architecture


  • Distinction between rule premises (if…) and rule consequents (then…)
  • Easy to interpret
  • Well suited for high-dimensional input spaces
  • Gained knowledge about premises and consequents can be used for DoE designs

Artificial Example


AiF Project in cooperation with the Institute of Fluid- and Thermodynamics

(AiF: Arbeitsgemeinschaft industrieller Forschungsvereinigungen)


  • Problem: New EU regulation requires higher degree of efficiency for ventilators
    => Most existing ventilators will be prohibited
    => New efficient dimensioning and optimization strategies required
  • Computational Fluid Dynamic (CFD) models fulfill accuracy requirements, but:
    - Many evaluations necessary for optimization
    - Computationally very expensive
    => Economically unviable!
  • Approach to solving the problem:
    - Metamodel: Model of the CFD model
    - CFD model generates data set
    - Local model network (LMN) is used

Local Model Network Advantages for the AiF Project Tasks

System Identification

  • Automatic model complexity determination saves data for the training
    - Akaike’s corrected information criterion (AICc) instead of validation data
  • Deterministic training procedure (no initialization required)

Design of Experiments

  • Active learning strategy
    - Structure of LMN is utilized to determine future CFD simulations
  • Exploitation of separation between linear and nonlinear effects


  • Local quadratic models can be used
  • Very likely to find global optimum of the LMN
    - M local models => M reasonable initialization points




Construction of Centrifugal Fan

Computational fluid dynamics (CFD) simulations utilized (~4 hours per simulation)
  • Target values
    - Pressure curve and efficiency curve
  • Design parameters
    - Number of blades, inner and outer diameter, several angles, etc.

Two Metamodeling Approaches

1. Direct Metamodeling


  • 10 inputs: Geometric parameters and volume flow.
  • 2 outputs: Efficiency and pressure.

2. Indirect Metamodeling via Characteristic Curves


  • 9 inputs: Geometric parameters.
  • 3 or 4 outputs: Characteristic parameters of efficiency and pressure curves.
Characteristic Curve Model
  • 3 or 4 inputs: Characteristic parameters (metamodel outputs).
  • 2 outputs: Efficiency and pressure.

CFD (black line), Indirect (MIMO, blue line), Indirect (multiple MISO, green line), Direct (red line)


Next Chapter: 5. Nonlinear Dynamic Models     Back to Overview