In many industrial application
In many industrial application ,Optimization of fluid mechanics is an important task. It has been proven to be a challenging task since a large amount of design variables and parameters need to be considered for effective optimized solution. This thesis focuses on the development and application of the adjoint method for flows. Adjoint method has be applied to shape as well as topology optimization problems. Shape optimizing deals with modification of wall surface where as in topology, the geometry is not described by surface parameters but with the volume elements in the entire domain. This feature makes it more robust and flexible as several variables can be controlled and optimized.
In several engineering fields, Adjoint method are favored in optimal control theory 1, 2 3, shape optimization 5, 6, uncertainty or sensitivity analysis. In gradient based optimization, adjoint methods are widely utilized for the computing gradient when the problem related to computing a large number of design variables and parameters. For case of linear governing equations, only the solution of final data from the forward solver is required, whereas, for the cases of nonlinear governing equations the solution data must be stored at every time step which can become restrictive for large problems. The treatment of boundary conditions is also an issue with the continuous method 10, as it’s presenting few difficulties to obtain these conditions for the adjoint solver in the discrete version. This allows the adjoint solution to be solved at the same time as the original problem without the need for storing the solution values at each time step.
The cost function defined to measure the performance of the shape is firstly determined through solving the governing flow equations in each design cycle. The shape is then redesigned by using the optimization method attempting to finally obtain the optimal shape with maximum aerodynamic performance and under required constraints. However, the number of the needed flow calculations depends strongly on their numbers of parameters for design. For the design of complex configuration with a large number of design parameters by using either RSM or GA, the optimization efficiency is low and consequently it may usually be achieved only on a supercomputer.
The continuous adjoint method has been used in the design optimization of both internal and external flows. More efforts are still required on the implementation and application of this optimization method in other engineering application. The current study will introduce firstly the fundamentals related to the continuous adjoint method, the generalized formulation of the Adjoint Navier Stoke Equations and the corresponding boundary conditions and later introduces Adjoint scalar transport equation. Secondly issues is the crucial that is design optimization, such as cost function and parameterization are briefly introduced. Concluded with a series of examples for design optimization.
2.1 Continuous Adjoint method
In the continuous adjoint method, a constrained optimization problem is transformed into an unconstrained type by forming a Lagrangian, with the Lagrange multipliers posing as the adjoint variables. The adjoint problem can be determined through either a variation formulation or by finding shape derivatives. For an optimization problem, an objective/cost function J(?,?), which is to be minimized is defined. The cost function depends on the geometry, ?, as well as flow field, ?. The total variation of J can be expressed as sum of variation with respect to flow variables, and design variables,
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The flow/constraint equation which are the Navier-Stokes equation written in residual form as :
The total variation of R, can be written in the same way as objective function. i.e. sum of variation with respect to flow variables, w, and design variables, ?, which gives
Multiplying above function with an arbitrary Lagrange multiplier, ?, yields
The total variation of the augmented objective function can be written as
Adding Eq.(iv) and (i) to above equation results in
Now, we are free to choose the value of Lagrange multiplier so that the variation with respect to flow variables, ? vanishes,
Integrating by parts the right side of Eq.(vi) result in system of equation called Adjoint equations. Solving that system gives the value of adjoint variables, the Lagrange multipliers.
The total variation of augmented objective function becomes
where the variation is calculated using variation of objective function and inner product between variation of flow equation with respect to the design variables and Lagrange multipliers, the solution to adjoint equation. This are calculated without the need of extra CFD solution of state equation and can be easily applied for the computation of volumetric sensitivities as well as surface sensitivities. Therefore, the quantity of main interest is the total variation ?J of original cost function with respect to design change including alteration in flow filed arising from this design changes. The propagation of information is reversed, compared to primal equation. Comparing the primal and adjoint equation systems, several similarities are observed including pressure and diffusion terms. The adjoint convection term differs from the primal standard convection by decomposing it into a backward convection term and a transpose convection term. The backward convection term indicates how information about the how is transported from downstream to upstream. The adjoint transpose convection(ATC) introduces a high degree of cross-coupling which makes it difficult to be implemented implicitly in a segregated algorithm.
2.2 Discrete Adjoint Formulation
In the discrete adjoint method, we solve a set of governing equations forward and then solve the adjoint problem backwards in time in order to acquire the adjoint variables. considering the discrete NSE with a weak imposition of boundary conditions. If the far filed boundary conditions are also imposed, then the discrete system of equations that is solved is of the form, which are the Navier-Stokes equation written in residual form as