On a shape control problem for the stationary Navier-Stokes equations

Gunzburger, Max; Kim, Hong-Chul; Manservisi, Sandro

doi:10.1051/m2an:2000125

Cited by 31 publications

(21 citation statements)

References 36 publications

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“…, and I D (·) the indicator function of the domain D. With this choice of the cost functional and the flow model, existence of solutions of problem (1)- (2) is in general stated [17,19] for…”

Section: Geometry Description State Equations and System Observationmentioning

confidence: 99%

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Manzoni

Quarteroni

Rozza

2011

Numerical Methods in Fluids

108

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SUMMARYIn this paper we further develop an approach previosuly introduced in [23] for shape optimization that combines a suitable low-dimensional parametrization of the geometry (yielding a geometrical reduction) with reduced basis methods (yielding a reduction of computational complexity). More precisely, freeform deformation techniques are considered for the geometry description and its parametrization, while reduced basis methods are used upon a finite element discretization to solve systems of parametrized partial differential equations. This allows an efficient flow field computation and cost functional evaluation during the iterative optimization procedure, resulting in effective computational savings with respect to usual shape optimization strategies. This approach is very general and can be applied to a broad variety of problems. In this paper we apply it to find the optimal shape of aorto-coronaric bypass anastomoses based on vorticity minimization in the down-field region. Blood flows in the coronary arteries are modelled using Stokes equations; afterwards, results have been verified in feedback using Navier-Stokes equations.

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Section: Geometry Description State Equations and System Observationmentioning

confidence: 99%

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Manzoni

Quarteroni

Rozza

2011

Numerical Methods in Fluids

108

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“…The three-step procedure in § 2.3 yields expression (20) for the gradient of the objective function (10). A discrete version of this procedure, as presented in § 4 below, is all that is needed to apply a gradient-based optimization algorithm such as the steepest-descent, conjugate gradient, or a quasi-Newton method with secant approximations of the model Hessian.…”

Section: The Gradient ∇ α Jmentioning

confidence: 99%

“…It has been used in the context of aerodynamic shape optimization by Gunzburger et al [10] and Mohammadi and Pironneu [18], for instance.…”

Section: Regularizationmentioning

confidence: 99%

Shape optimization of an acoustic horn

Bängtsson

Noreland

Berggren

2003

Computer Methods in Applied Mechanics and Engineering

112

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Shape optimization of an acoustic horn is performed with the goal to minimize the portion of the wave that is reflected. The analysis of the acoustical properties of the horn is performed using a finite element method for the Helmholtz equation.The optimization is performed employing a BFGS Quasi-Newton algorithm, where the gradients are provided by solving the associated adjoint equations. To avoid local solutions to the optimization problem corresponding to irregular shapes of the horn, a filtering technique is used that applies smoothing to the design updates and the gradient. This smoothing technique can be combined with Tikhonov regularization. However, experiments indicate that regularization is redundant for the optimization problems we consider here. However, the use of smoothing is crucial to obtain sensible solutions. The smoothing technique we use is equivalent to choosing a representation of the gradient of the objective function in an inner product involving second derivatives along the design boundary.Optimization is performed for a number of single frequencies as well as for a band of frequencies. For single frequency optimization, the method shows particularly fast convergence with indications of super-linear convergence close to optimum. For optimization on a range of frequencies, a design was achieved providing a low and even reflection throughout the entire frequency band of interest.

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“…In the past few years, shape optimization problems have received considerable attentions. On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis to the problems; see e.g., [19,20,21,27,39,46] and references therein. On the practical side, optimal shape design has played an important role in many industrial applications, for example, aerodynamic shape design [25,33,34,35,44], artery bypass design [2,3,6,7,40,43], microfluidic biochip design [4,24,38] and so on.…”

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confidence: 99%

“…The last two equations indicate that the optimized boundary should be connected to the rest of the boundary and z 1 and z 2 are two given constants [21]. For PDE constrained optimization problems, there are two basic approaches: optimize-then-discretize (OTD) and discretize-then-optimize (DTO).…”

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confidence: 99%

Parallel One-Shot Lagrange--Newton--Krylov--Schwarz Algorithms for Shape Optimization of Steady Incompressible Flows

Chen¹,

Cai²

2012

SIAM J. Sci. Comput.

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Abstract. We propose and study a new parallel one-shot Lagrange-Newton-Krylov-Schwarz (LNKSz) algorithm for shape optimization problems constrained by steady incompressible NavierStokes equations discretized by finite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems solve iteratively the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relatively easy to implement, but generally not easy to converge as they are basically nonlinear Gauss-Seidel algorithms with three large blocks. In this paper, we introduce a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously. First, we introduce a moving mesh finite element method for the shape optimization problems in which the mesh equations are implicitly coupled with the optimization problems. Second, we introduce a LNKSz framework based on an overlapping domain decomposition method for solving the fully coupled problem. Such an approach doesn't involve any sequential steps that are necessary for the Gauss-Seidel type reduced space methods. The main challenges in full space approaches are that the corresponding nonlinear system is much harder to solve because it is two to three times larger and its indefinite Jacobian problems are also much more ill-conditioned. Effective preconditioning becomes the most important component of the method. Numerically, we show that LNKSz deals with these challenges quite well. As an application, we consider the shape optimization of an artery bypass problem in 2D. Numerical experiments show that our algorithms perform well on supercomputers with hundreds of processors.Key words. Shape optimization, one-level additive Schwarz preconditioner, parallel computing, one-shot method, finite element, inexact Newton.AMS subject classifications. 49K20, 49Q10, 65F08, 65F10, 65N30, 65N55, 65Y05, 76D05, 90C06, 90C30, 93C201. Introduction. Shape optimization, or optimal shape design, aims to optimize an objective function by changing the shape of the computational domain. In the past few years, shape optimization problems have received considerable attentions. On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis to the problems; see e.g., [19,20,21,27,39,46] and references therein. On the practical side, optimal shape design has played an important role in many industrial applications, for example, aerodynamic shape design [25,33,34,35,44], artery bypass design [2,3,6,7,40,43], microfluidic biochip design [4,24,38] and so on. Most of these optimization problems have constraints imposed by partial differential equations (PDE) or other physical and geometrical conditions. They are considerably more difficult and expensive to solve than the corresponding simulation problems, and often require large scale parallel computers for their ...

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On a shape control problem for the stationary Navier-Stokes equations

Cited by 31 publications

References 36 publications

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Shape optimization of an acoustic horn

Parallel One-Shot Lagrange--Newton--Krylov--Schwarz Algorithms for Shape Optimization of Steady Incompressible Flows

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