Shape Optimization Problem for Incompressible Viscous Flow

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Shape Optimization Problem for Incompressible
Viscous Flow Masaki Tashiro, Mutsuto Kawahara
Department of Civil Engineering, Chuo
University, Japan Department of Civil
Engineering, Chuo University, Japan 5th
ICHE2002 September,2002
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• Introduction
• Lagrange multiplier method
• Gradient method
• Numerical example
• Conclusion and Future work

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What is the shape which has minimum drag force?
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Incompressible Stokes equation
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APPROXIMATION
PSPG formulation
The weighiting residiual equation of the PSPG
formulation can be denoted by adding the
stability term of the Petrov-Galerkin formulation
to the Galerkin formulation.
The stability parameter is defined in each
element.
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APPROXIMATION
Finite element equation
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FORMULATION
Performance function
Fluid forces are directly applied to the
performance function.
F1 Drag forces F2 Lift forces
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The volume is being kept in each minimization
iteration step. To keep the volume of the body is
equal to keep the volume of the whole analytical
domain.
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FORMULATION
Lagrange multiplier method
Shape optimization problem with constraint
condition can be transformed to the minimization
problem of the extended performance function.
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FORMULATION
The stationary condition of the extended
performance function can be derived from the
first variation of that.
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FORMULATION
Adjoint equation
When the state and adjoint values are solved, the
optimal condition of this problem is given as
follows.
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FORMULATION
Gradient of the extended performance function
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MINIMIZATION
Gradient algorithm
New shape is found by deforming the surface
according to
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THE PROCESS OF OPTIMALSHAPE
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NUMERICAL EXAMPLE
Objective Drag minimization problem Constraint
conditionArea of body is constant Initial
shape Circle
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NUMERICL EXAMPLE
Finite element mesh of optimal shape
Pressure distribution of optimal shape
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NUMERICL EXAMPLE
Vorticity
Normal derivative
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NUMERICL EXAMPLE
Iteration history
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• Numerical determination of minimum drag shape
• in an incompressible flow was attempted.
• The body shape that satisfied the necessary
  condition was obtained.
• The present method can be applied to other flow..