Topological sensitivity analysis of a time-dependent nonlinear problem

Mohamed Abdelwahed,Azhar Al Salem,Maatoug Hassine,Nejmeddine Chorfi

doi:10.1186/s13661-019-1138-8

Mohamed Abdelwahed, Azhar Al Salem + Show 2 more

Open Access

https://doi.org/10.1186/s13661-019-1138-8

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Abstract

We are interested in the optimization of the pipe shape allowing minimization of the dissipated energy in time-dependent Navier–Stokes Darcy flow. The used technique is based on the topological gradient method. In the theoretical part, we present an analysis of the topological sensitivity for the dissipated energy function. Some numerical tests are presented to illustrate the developed approach.

Highlights

Let O be a bounded cavity of R2 occupied by a viscous and incompressible fluid modeled by the time-dependent nonlinear Navier–Stokes equations
To optimize the obstacles’ location, we developed in Sect. 2 a topological asymptotic expansion of the dissipation energy function relative to the introduction of an obstacle of small size within the domain O of the fluid flow
4 Conclusion We developed in this work an efficient topological optimization algorithm for determining the optimal shape design of unsteady flow described by the coupled Navier–Stokes and

Summary

Introduction

Let O be a bounded cavity of R2 occupied by a viscous and incompressible fluid modeled by the time-dependent nonlinear Navier–Stokes equations. The topological gradient method consists in finding the asymptotic expansion of the cost function J with respect to a small perturbation of the initial domain. Corollary 2.1 Summing over time, the topological gradient of JT (v) is given by DJT (v) = –(1 – c)κ(y) v(y) v(y) – vadj(y) dt Following this result, we propose the following numerical algorithm: We begin first by choosing O0 = O. DJT vk = –(1 – c)κ vkn vkn – vkadj n , n=0 where vkn and (vkadj)n are, respectively, the numerical solution of the Navier–Stokes Darcy problem and its adjoint at time tn. We use the presented algorithm in order to find the pipe optimal domain connecting the inlet of the cavity and its outlet with minimum dissipated energy.

Example 2

Conclusion