Adaptive Finite Element Calculations For Fluid-based Shape Optimization

The shape optimization of an incompressible fluid is widely used in many field-s,and is usually described as a class of optimization problems,with the Navier-Stokes equation and the area volume as the constraint,the minimum total potential energy of the fluid as the objective function.The control variable is the geometry of the region,so that the optimal shape of the region is obtained through the relat-ed optimization method.Traditional methods solve the sensitivity of the problem,however,there are some difficulties for numerical simulations.Firstly,the object in-terface must be smooth enough;secondly,remeshing is computationally expensive.Since the target structure is usually represented by a finite element mesh,both the objective function and its sensitivity are numerically calculated by the finite element method.In this sense,the mesh must be updated as the shape changes to maintain the accuracy of the analysis.To resolve these problems,a phase field method is used to solve the optimal shape of the fluid and the corresponding sensitivity functional is derived.In addition,we propose a novel finite element method on the multi-mesh with different resolution to solve the phase-field model arising in the fluid shape optimization.Using a coarse mesh to solve the state equation,a fine mesh to solve the phase field equation,and a mesh between with size the previous two's to solve the adjoint equation,so that we can solve the finite element variables on the different meshes is by simple linear interpolation.Consequatly,the computational cost is greatly reduced.