Numerical simulation and optimal shape for viscous flow by a fictitious domain method

Abstract

SUMMARYIn this article we discuss the fictitious domain solution of the Navier-Stokes equations modelling unsteady incompressible viscous flow. The method is based on a Lagrange multiplier treatment of the boundary conditions to be satisfied and is particularly well suited to the treatment of no-slip boundary conditions. This approach allows the use of structured meshes and fast specialized solvers for problems on complicated geometries. Another interesting feature of the fictitious domain approach is that it allows the solution of optimal shape problems without regriding. The resulting methodology is applied to the solution of flow problems including external incompressible viscous flow modelled by the Navier-Stokes equations and then to an optimal shape problem for Stokes and Navier-Stokes flow.

A moving mesh fictitious domain approach for shape optimization problems

“…where the matrix block D is the same one as in (18). The matrix block S for the Lagrange multipliers has the form…”

“…Another possibility is to add some constraints to the extended problem. This can be done using boundary Lagrange multipliers [18,19,44] or using distributed Lagrange multipliers [16,17,27,44].…”

“…It may lead to nonsmooth objective functions, locking effects and low approximation accuracy; see, for example, the review paper [22] by Haslinger. In the shape optimization problems considered in [18,25,29], the fictitious domain method enforces the Dirichlet boundary value in the state equation using Lagrange multipliers on the boundary. In [13,21,23,24], a distributed control is introduced to the right-hand side and the Dirichlet boundary value is added to the objective function as a penalty.…”

A three-dimensional fictitious domain method for incompressible fluid flow problems

“…Glowinski et al 19 also demonstrated the ef®ciency of this type of method for predicting the three-dimensional¯uid¯ow around a sphere imbedded in a rectangular domain and for solving an optimal shape problem involving two-dimensional Navier±Stokes¯ows. 20 The objective of this paper is to introduce a new three-dimensional ®ctitious domain method for the solution of the Navier±Stokes equations in enclosures containing internal parts which may be moving.…”

Fictitious domain/mixed finite element approach for a class of optimal shape design problems

“…We refer to the text books [2,3,4] for a description of this classical approach to shape optimization. In [5] (see also [9]) a new approach based on a combination of the fictitious domain approach and optimal control was proposed in the case when the state is given by a homogeneous Dirichlet boundary value problem. The original optimal shape design problem was formally rewritten as a new one which uses as state problem again a homogeneous Dirichlet problem but posed on a a fixed domain and with the control entering only in the right hand side.…”