Stokes problem with a solution dependent slip bound: Stability of solutions with respect to domains (original) (raw)
Related papers
Shape optimization for Stokes problem with threshold slip
Applications of Mathematics, 2014
We study the Stokes problems in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning smoothness of Ω, solutions to the Stokes system with the slip boundary conditions depend continuously on variations of Ω. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero.
Efficient methods for solving the Stokes problem with slip boundary conditions
Mathematics and Computers in Simulation, 2016
The paper deals with the Stokes flow with the threshold slip boundary conditions. A finite element approximation of the problem leads to the minimization of a non-differentiable energy functional subject to two linear equality constraints: the impermeability condition on the slip part of the boundary and the incompressibility of the fluid. Eliminating the velocity components, one gets the smooth dual functional in terms of three Lagrange multipliers. The first Lagrange multiplier regularizes the problem. Its components are subject to simple bounds. The other two Lagrange multipliers treat the impermeability and the incompressibility conditions. The last Lagrange multiplier represents the pressure in the whole domain. The solution to the dual problem is computed by an active set strategy and a path-following variant of the interior-point method. Numerical experiments illustrate computational efficiency.
Shape optimization in problems governed by generalised Navier-Stokes equations: existence analysis
We study a shape optimization problem for a paper machine headbox which distributes a mixture of water and wood fibers in the paper manufacturing process. The aim is to find a shape which a priori ensures the given velocity profile on the outlet part. The state problem is represented by the generalised Navier-Stokes system with nontrivial boundary conditions. The objective of this paper is to prove the existence of an optimal shape.
Shape optimization for Stokes flows using sensitivity analysis and finite element method
Applied Numerical Mathematics, 2018
In the context of structural optimization in fluid mechanics we propose a numerical method based on a combination of the classical shape derivative and Hadamard's boundary variation method. Our approach regards the viscous flows governed by Stokes equations with the objective function of energy dissipation and a constrained volume. The shape derivative is computed by Lagrange's approach via the solutions of Stokes and adjoint systems. The programs are written in FreeFem++ using the Finite Element method.
Computational Optimization and Applications, 2011
This paper deals with a control-constrained linear-quadratic optimal control problem governed by the Stokes equations. It is concerned with situations where the gradient of the velocity field is not bounded. The control is discretized by piecewise constant functions. The state and the adjoint state are discretized by finite element schemes that are not necessarily conforming. The approximate control is constructed as projection of the discrete adjoint velocity in the set of admissible controls. It is proved that under certain assumptions on the discretization of state and adjoint state this approximation is of order 2 in L 2 (Ω). As first example a prismatic domain with a reentrant edge is considered where the impact of the edge singularity is counteracted by anisotropic mesh grading and where the state and the adjoint state are approximated in the lower order Crouzeix-Raviart finite element space. The second example concerns a nonconvex, plane domain, where the corner singularity is treated by isotropic mesh grading and state and adjoint state can be approximated by a couple of standard element pairs.
Penalty finite element method for Stokes problem with nonlinear slip boundary conditions
Applied Mathematics and Computation, 2008
The penalty finite element method for Stokes problem with nonlinear slip boundary conditions, based on the finite element subspace ðV h ; M h Þ which satisfies the discrete infsup condition, is investigated in this paper. Since this class of nonlinear slip boundary conditions include the subdifferential property, the weak variational formulation associated with Stokes problem is variational inequality. Under some regularity assumptions, we obtain the optimal H 1 and L 2 error estimates between u and u h , and between u and u e h , where the error orders are e þ h for H 1 error and e þ h 2 for L 2 error.
Domain Optimization Problem Governed by a State Inequality with a „Flux” Cost Functional
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1986
Shape optimization with a state unilateral boundary value probleni nnd th.e flux cost functional is analyzed. Uaing the pennlty method the existence of n solution is proved. U iiacrom~e8 p a 6 o~e paccMaTpusaeTcH aanasa cyuccTnosaHMR onTmaJibHot4 06nac.r~ B npobnenrax, ICOTOplJe J'lIpaBJlfllOTCH CUCTt !MaMM OIIMChlBaeMb1%l1.I OnHOCTOPOHHl rl MM rPaHllYllhlMM 3aAallaMU.