Shape optimization for a fluid-elasticity system
Jan Sokolowski2018
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
In this paper, we are interested in a shape optimization problem for a fluid-structure interaction system composed by an elastic structure immersed in a viscous incompressible fluid. The cost functional to minimize is an energy functional involving together the fluid and the elastic parts of the structure. The shape optimization problem is introduced in the 2-dimensional case. However the results in this paper are obtained for a simplified free-boundary 1-dimensional problem. We prove that the shape optimization problem is wellposed. We study the shape differentiability of the free-boundary 1-dimensional model. The full characterization of the associated material derivatives is given together with the shape derivative of the energy functional. A special case is explicitly solved, showing the relevancy of this shape optimization approach for a simplified free boundary 1-dimensional problem. The full model in two spatial dimensions is under studies now.
Related papers
On a numerical shape optimization approach for a class of free boundary problems
Computational Optimization and Applications, 2020
This paper is devoted to a numerical method for the approximation of a class of free boundary problems of Bernoulli's type, reformulated as optimal shape design problems with appropriate shape functionals. We show the existence of the shape derivative of the cost functional on a class of admissible domains and compute its shape derivative by using the formula proposed in [6, 7], that is, by means of support functions. On the numerical level, this allows us to avoid the tedious computations of the method based on vector fields. A gradient method combined with boundary element method are performed for the approximation of this problem, in order to overcome the re-meshing task required by the finite element method. Finally, we present some numerical results and simulations concerning practical applications, showing the effectiveness of the proposed approach.
Boundary integral equation method for shape optimization of elastic structures
International Journal for Numerical Methods in Engineering, 1988
A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.
No . B . 12 / 2007 Shape optimization of systems governed by Bernoulli free boundary problems
In this work we consider shape optimization of systems, which are governed by external Bernoulli free boundary problems. A pseudo-solid approach for solving discrete free boundary problems is introduced. The solution strategy readily allows us to obtain geometrical sensitivities of the system, which can then be used to solve e.g. inverse design problems. Numerical examples show that the location of the free boundary can, to some extent, be controlled by changing the shape of the other component of the boundary.
A shape optimization approach to free boundary value problems
This contribution combines a shape optimization approach to free boundary value problems of Bernoulli type with an embedding domain technique. A theoretical framework is developed which allows to prove continuous dependence of the primal and dual variables in the resulting saddle point problems with respect to the domain. This ensures the existence of a solution of a related shape optimization problem in a sufficiently large class of admissible domains.
Shape Optimization with Symmetric Galerkin Boundary Element Method
In this work a numerical implementation of shape optimization in two-dimensional linear elasticity problems is proposed. The main goal is to propose a robust and efficient methodology for the solution of shape optimization problems regarding the minimization of stress concentration effects. In the proposed implementation, the structural analysis is performed by the Symmetric Galerkin Boundary Element Method (SGBEM), thus disposing of the mesh generation burden. The boundary stress evaluation is carried out by an accurate approach which is ideally suited for problems with stress concentrations. Another relevant feature of the proposed implementation is a suitable partition of the SGBEM equations which aims at reducing the computational effort associated with the structural analysis stage. The solution for the optimization problem is obtained by means of a modern numerical optimization method, the so-called Second Order Conic Programming (SOCP). Specifically, the solution for the non-linear optimization is sought by solving a sequence of SOCP subproblems.
A Simple Boundary Element Formulation for Shape Optimization of 2D Continuous Structures
Proceedings of 18th …, 2005
For the design of nuclear equipment like pressure vessels, steam generators, and pipelines, among others, it is very important to optimize the shape of the structural systems to withstand prescribed loads such as internal pressures and prescribed or ...
Chapter 16: An introduction to shape optimization problems
2014
In this section we give a quick introduction to shape optimization problems in a rather general framework, and we discuss some of their features, especially in relation to the existence of an optimal solution. Our goal is not to give here a detailed presentation of the many problems and results in this very wide field, but only to show that several optimization problems, often very important for applications in mechanics and engineering, cannot be formulated by taking a Banach function space of the competing admissible choices: a more appropriate framework consists in taking as admissible controls the elements of a class of domains. We refer the reader interested in a deeper knowledge and analysis of this fascinating field to one of the several books on the subject [239], [11], [322], [337], to the notes by Tartar [347], or to the collection of lecture notes by Bucur and Buttazzo [144], [145]. A shape optimization problem is a minimization problem where the unknown variable runs ove...
Shape Optimization Approach to the Bernoulli Problem : A Lagrangian Formulation
2017
The exterior Bernoulli free boundary problem is reformulated into a shape optimization setting by tracking the Dirichlet data. The shape derivative of the corresponding cost functional is established through a Lagrangian formulation coupled with the velocity method. A numerical example using the traction method or H gradient method is also provided.
A shape optimization approach for a class of free boundary problems of Bernoulli type
Applications of Mathematics, 2013
We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincaré inequality for variable domains.
Related topics
Related papers
Multiparameter structural shape optimization by the finite element method
International Journal for Numerical Methods …, 1978
The problem of optimal design of the shape of a free or internal boundary of a body is formulated by assuming the boundary shape is described by a set of prescribed shape functions and a set of shape parameters. The Optimization procedure is reduced to determination of these parameters. For constant volume or material cost constraint, the optimality conditions are derived for the case of mean compliance design of elastic structures of a non-linear material. Some additional conditions for the global minimum of the mean compliance are proved. The most typical cases of boundary variations are discussed. The optimal shape problem is next formulated by means of the finite element method and the iterative solution algorithm is discussed by using the optimality criteria. Several simple numerical examples are included.
Optimization of elastic structures using boundary elements and a topological-shape sensitivity formulation
The objective of this work is to present the implementation of topological derivative concepts in a standard BEM formulation. The topological derivative is evaluated at internal points, and those showing the lowest values are used to remove material by opening a circular cavity. Hence, as the iterative processes evolutes, the original domain has holes progressively punched out, until a given stop criteria is achieved. At this point, the optimal topology is expected. Several benchmarks of two-dimensional elasticity are presented and analyzed. Because the BEM does not employ domain meshes in linear cases, the resulting topologies are completely devoid of intermediary material densities. The results obtained showed good agreement with previous available solutions, and demanded comparatively low computational cost. The results prove that the formulation generates optimal topologies, eliminates some typical drawbacks of homogenization methods, and has potential to be extended to other classes of problems. More important, it opens an interesting field of investigation for integral equation methods, so far accomplished only within the finite element methods context.
Geometric Aspects of Shape Optimization
The Journal of Geometric Analysis
We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model p...
Analysis of Dual Boundary Element in Shape Optimization
IOSR Journal of Mechanical and Civil Engineering, 2016
The numerical analysis with dual boundary element method for shape optimal design in twodimensional linear elastic structures is proposed. The design objective is to minimize the structural compliance, subject to an area constraint. Sensitivities of objective and constraint functions, derived by means of Lagrangean approach and the material derivative concept with an adjoint variable technique, are computed through analytical expressions that arise from optimality conditions. The dual boundary element method, used for the discretization of the state problem, applies the stress equation for collocation on the design boundary and the displacement equation for collocation on other boundaries. The perturbation field is described with linear continuous elements, in which the position of each node is defined by a design variable. Continuity along the design boundary is assured by forcing the end points of each discontinuous boundary element to be coincident with a design node. The optimization problem is solved by the modified method of feasible directions available in the PYOPT program and the accuracy and efficiency of the analysis is assessed through two examples of a plate with a hole, making this formulation ideal for the study of shape optimal design of structures.
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.
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 derivatives for general objective functions and the incompressible Navier-Stokes equations
The aim of this paper is to present the shape derivative for a wide array of objective functions using the incompressible Navier-Stokes equations as a state constraint. Most real world applications of computational fluid dynamics are shape optimization problems in nature, yet special shape optimization techniques are seldom used outside the field of elliptic partial differential equations and linear elasticity. This article tries to be self contained, also presenting many useful results from the literature. We conclude with a comparison of different objective functions for the shape optimization of an obstacle in a channel, which can be done quite conveniently when one knows the general form of the shape gradient.
On Shape Optimization for an Evolution Coupled System
Applied Mathematics & Optimization, 2011
A shape optimization problem in three spatial dimensions for an elasto-dynamic piezoelectric body coupled to an acoustic chamber is introduced. Well-posedness of the problem is established and first order necessary optimality conditions are derived in the framework of the boundary variation technique. In particular, the existence of the shape gradient for an integral shape functional is obtained, as well as its regularity, sufficient for applications e.g. in modern loudspeaker technologies. The shape gradients are given by functions supported on the moving boundaries. The paper extends results obtained by the authors in [9] where a similar problem was treated without acoustic coupling.
Some problems of shape optimization arising in stationary fluid motion
ADVANCES IN MATHEMATICAL SCIENCES AND APPLICATIONS, 2004
Abstract. We investigate the existence of a drag-minimizing shape for two classes of optimal-design problem of fluid mechanics, namely the vector Burgers equations and the Navier-Stokes equations. It is known that the two-dimensional Navier-Stokes problem of shape optimization has a solution in any class of domains with at most l holes. We show, for the Burgers equation in three dimensions, that the existence of a minimizer still holds in the classes Oc, r (C) and Ww (C) introduced by Bucur and Zolésio. These classes are defined ...
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.