To be sure that the domain is simple in orthogonal coordinates, it is divided into a fixed and a variable curves. Then by idea of approximating a curve with broken lines and fixing the y-components, each admissible domain can be represented with finite number of variables (domain variables). Therefore in the first step of shape-measure method, for a given domain, we will determine the optimal control and the optimal value of objective function by applying the obtained results from Rubio. Then a vector function on the set of admissible domains will be set up in which it gives the optimal valve of objective function for a given domain and its related optimal control function. Afterwards, in the second step, by use of a standard minimization algorithm, the nearly optimal domain and its related optimal control function will be determined at the same time. Numerical examples will be also given.
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To get a better approximation, for the first time in optimal shape design problems, we may apply the Glashoff-Gustafson approach while Rubio used the Rudolph's one.