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Gradient-based methods

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As its name means, gradient-based methods need the gradient of objective functions to design variables. The evaluation of gradient can be achieved by [[finite difference method, linearized method or adjoint method]]. Both finite difference method and linearized method has a time-cost proportional to the number of design variables and not suitable for design optimization with a large number of design variables. Apart from that, finite difference method has a notorious disadvantage of subtraction cancellation and is not recommended for practical design application.
 
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Suppose a cost function <math>J</math> is defined as follows,
 
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<math>J=J(U(\alpha),\alpha)</math>
 
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where <math>U</math> and <math>\alpha</math> are the flow variable vector and the design variable vector respectively. <math>U</math> and <math>\alpha</math> are implicitly related through the flow equation, which is represented by a residual function driven to zero.
 
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<math>R(U\alpha,\alpha)=0</math>
 
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Finite difference method:
 
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Linearized method:
 
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Adjoint method:
 

Latest revision as of 01:33, 6 January 2012

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