Adjoint method for the optimum planning of industrial pollutant sources

The optimum planning of industrial pollutant sources, which optimize the economic object without violating environmental constraints, is an important and hard task to be conquered. In this paper, an adjoint method is developed to solve the problem. The penalty function is introduced to deal with the environmental inequality constraints, and Lagrange function is constructed to derive the adjoint equation and the gradient of the object function. In this means, the gradient of the object function can be calculated by solving the adjoint equation, and the information from the gradient is used to make the object function descend and approach to an optimal solution after some iterations. A two dimensional, simplified model is used for numerical experiments. The theoretical derivations are verified by the results of the experiments. Furthermore, the adjoint method is showed to be of excellent convergence and efficiency, which is adaptive to the fast development of air quality numerical models and super computers.