A Shape Optimization Method in Viscous Flow Using Acoustic Velocity and Four-step Explicit Scheme
The purpose of this study is to derive optimal shapes of
a body located in viscous flows by the finite element method using the
acoustic velocity and the four-step explicit scheme. The formulation
is based on an optimal control theory in which a performance function
of the fluid force is introduced. The performance function should be
minimized satisfying the state equation. This problem can be transformed
into the minimization problem without constraint conditions
by using the adjoint equation with adjoint variables corresponding to
the state equation. The performance function is defined by the drag
and lift forces acting on the body. The weighted gradient method
is applied as a minimization technique, the Galerkin finite element
method is used as a spatial discretization and the four-step explicit
scheme is used as a temporal discretization to solve the state equation
and the adjoint equation. As the interpolation, the orthogonal basis
bubble function for velocity and the linear function for pressure
are employed. In case that the orthogonal basis bubble function is
used, the mass matrix can be diagonalized without any artificial
centralization. The shape optimization is performed by the presented
method.
[1] T.Doyle, M.Gerritsen and G.Iaccarino, Towards sail-shape optimization of
a modern clipper ship, Center for Turblence Research, Annual Research
Briefs 2002, pp.215-224, 2002.
[2] A.Jameson, Aerodynamic Design via Control Theory, Journal of Scientific
Computing, 3:233-260, 1988.
[3] A.Jameson and L.Martinelli, Optimum Aerodynamic Design Using
the Navier-Stokes Equations, Theoret. Comput. Fluid Dynamics(1998)
10:213-237.
[4] E.Katamine, H.Azegami, T.Tsubata and S.Itoh, Solution to shape optimization
problems of viscous flow fields, International Journal of
Computational Fluid Dynamics, Vol. 19, No.1, 45-51, January 2005.
[5] S.Kim, K.Hosseini, K.Leoviriyakit and A.Jameson, Enhancement of Adjoint
Design Methods via Optimization of Adjoint Parameters, AIAA
Paper 2005-0448, 43rd AIAA Aerospace Sciences Meeting and Exhibit,
Reno, NV, January 10-13, 2005.
[6] J.Matsumoto and M.Kawahara, Stable shape identification
for fluid-structure interaction problem using MINI element.
Journal of Applied Mechanics, JSCE, Vol. 3, 2000.
[7] J.Matsumoto, A Relationship between Stabilized FEM and Bubble Function
Element Stabilization Method with Orthogonal Basis for Incompressible
Flows, Journal of Applied Mechanics, JSCE, Vol. 8 August
2005.
[8] S.Nakajima and M.Kawahara, Shape optimization of a body in compressible
inviscid flows, Comp.Meth.Appl.Meth.Engrg., Vol.197.pp.4521-
4530, 2008.
[9] O.Pironneau, On optimal Design in Fluid Mechanics, J.Fluidmech, Vol.
64, PART 1, 1974, 64(1):97-110.
[10] H.Yagi and M.Kawahara, Shape Optimization of a Body Located
in Incompressible Viscous Flow Using Adjoint Method, in
Proc.ECCOMAS2004.
[1] T.Doyle, M.Gerritsen and G.Iaccarino, Towards sail-shape optimization of
a modern clipper ship, Center for Turblence Research, Annual Research
Briefs 2002, pp.215-224, 2002.
[2] A.Jameson, Aerodynamic Design via Control Theory, Journal of Scientific
Computing, 3:233-260, 1988.
[3] A.Jameson and L.Martinelli, Optimum Aerodynamic Design Using
the Navier-Stokes Equations, Theoret. Comput. Fluid Dynamics(1998)
10:213-237.
[4] E.Katamine, H.Azegami, T.Tsubata and S.Itoh, Solution to shape optimization
problems of viscous flow fields, International Journal of
Computational Fluid Dynamics, Vol. 19, No.1, 45-51, January 2005.
[5] S.Kim, K.Hosseini, K.Leoviriyakit and A.Jameson, Enhancement of Adjoint
Design Methods via Optimization of Adjoint Parameters, AIAA
Paper 2005-0448, 43rd AIAA Aerospace Sciences Meeting and Exhibit,
Reno, NV, January 10-13, 2005.
[6] J.Matsumoto and M.Kawahara, Stable shape identification
for fluid-structure interaction problem using MINI element.
Journal of Applied Mechanics, JSCE, Vol. 3, 2000.
[7] J.Matsumoto, A Relationship between Stabilized FEM and Bubble Function
Element Stabilization Method with Orthogonal Basis for Incompressible
Flows, Journal of Applied Mechanics, JSCE, Vol. 8 August
2005.
[8] S.Nakajima and M.Kawahara, Shape optimization of a body in compressible
inviscid flows, Comp.Meth.Appl.Meth.Engrg., Vol.197.pp.4521-
4530, 2008.
[9] O.Pironneau, On optimal Design in Fluid Mechanics, J.Fluidmech, Vol.
64, PART 1, 1974, 64(1):97-110.
[10] H.Yagi and M.Kawahara, Shape Optimization of a Body Located
in Incompressible Viscous Flow Using Adjoint Method, in
Proc.ECCOMAS2004.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:61251", author = "Yoichi Hikino and Mutsuto Kawahara", title = "A Shape Optimization Method in Viscous Flow Using Acoustic Velocity and Four-step Explicit Scheme", abstract = "The purpose of this study is to derive optimal shapes of
a body located in viscous flows by the finite element method using the
acoustic velocity and the four-step explicit scheme. The formulation
is based on an optimal control theory in which a performance function
of the fluid force is introduced. The performance function should be
minimized satisfying the state equation. This problem can be transformed
into the minimization problem without constraint conditions
by using the adjoint equation with adjoint variables corresponding to
the state equation. The performance function is defined by the drag
and lift forces acting on the body. The weighted gradient method
is applied as a minimization technique, the Galerkin finite element
method is used as a spatial discretization and the four-step explicit
scheme is used as a temporal discretization to solve the state equation
and the adjoint equation. As the interpolation, the orthogonal basis
bubble function for velocity and the linear function for pressure
are employed. In case that the orthogonal basis bubble function is
used, the mass matrix can be diagonalized without any artificial
centralization. The shape optimization is performed by the presented
method.", keywords = "Shape Optimization, Optimal Control Theory, Finite
Element Method, Weighted Gradient Method, Fluid Force, Orthogonal
Basis Bubble Function, Four-step Explicit Scheme, Acoustic
Velocity.", volume = "4", number = "11", pages = "1452-6", }