Objectives  Both the mechanical impedance of the foundation and the homogenization of foundation impedance among loading points can affect the radiation noise of underwater structures.

  Methods  The mechanical impedance of a typical long underwater foundation was studied. A mathematical topology model in which the density of web plate was set to be the design variable was established to improve the homogenization of foundation impedance. An effective form of web perforation was put forward according to the result of topology optimization. On this basis, a multi-objective size optimization mathematical model based on tolerance ranking method was proposed. The thicknesses of plates constituting the foundation were used as design variables while the objective was to maximize the mechanical impedance and impedance uniformity simultaneously. Finally an optimum proposal of thicknesses of face plates, webs and knee plates was found.

  Results  The result of topology optimization indicates that hole formed on foundation webs are not the same, and the centers of holes do not coincide with web centers. The size optimization result shows that the maximum dispersion of foundation impedance can be reduced by 9.37% with 33.31% weight saving, while the minimum mechanical impedance remains nearly unchanged by redistributing the material in plates of the foundation by matching thicknesses of face plates, webs and knee plates.

  Conclusions  In order to improve the impedance homogenization among loading points, forms of perforations on foundation webs should not be the same. And foundation components in different areas can be different in order to take into account both the mechanical impedance and impedance homogenization. The study in this paper can provide reference for the optimal design of underwater structure foundation.