On coupling wavelets with fictitious domain approaches

We define and analyse a numerical algorithm for the approximation of parabolic equations on a general 2D domain with Dirichlet boundary conditions. It couples wavelet approximations with fictitious domain surface Lagrange multiplier approaches. This algorithm turns out to be precise, fast and numerically efficient.     

Compact gradient tracking in shape optimization

In the present paper we consider the minimization of gradient tracking functionals defined on a compact and fixed subdomain of the domain of interest. The underlying state is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that, in contrast to the situation of gradient tracking on the whole domain, the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the shape problem under consideration. Shape functional and gradient require only knowledge of the Cauchy data of the state and its adjoint on the boundaries of the domain and the subdomain. These data can be computed by means of boundary integral equations when reformulating the underlying differential equations as transmission problems. Thanks to fast boundary element techniques, we derive an efficient algorithm to solve the problem under consideration.     

On a fictitious domain method with distributed Lagrange multiplier for interface problems

In this paper we propose a new variational formulation for an elliptic interface problem and discuss its finite element approximation. Our formulation fits within the framework of fictitious domain methods with distributed Lagrange multipliers. For the underlying mixed scheme we prove stability and convergence. Some preliminary numerical tests confirm the theoretical investigations.     

An embedding domain approach for a class of 2-d shape optimization problems: mathematical analysis

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.     

A stabilized finite element method for a fictitious domain problem allowing small inclusions

The purpose of this work is to approximate numerically an elliptic partial differential equation posed on domains with small perforations (or inclusions). The approach is based on the fictitious domain method, and as the method's interest lies in the case in which the geometrical features are not resolved by the mesh, we propose a stabilized finite element method. The stabilization term is a simple, non‐consistent penalization that can be linked to the Barbosa‐Hughes approach. Stability and convergence are proved, and numerical results confirm the theory.     

On approximation in the fictitious domain method for a bidimensional transport equation

In this article a numerical method for solving a two‐dimensional transport equation in the stationary case is presented. Using the techniques of the variational calculus, we find the approximate solution for a homogeneous boundary‐value problem that corresponds to a square domain D₂. Then, using the method of the fictitious domain, we extend our algorithm to a boundary value problem for a set D that has an arbitrary shape. In this approach, the initial computation domain D (called physical domain) is immersed in a square domain D₂. We prove that the solution obtained by this method is a good approximation of the exact solution. The theoretical results are verified with the help of a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Second-order numerical method for domain optimization problems

This paper is concerned with a second-order numerical method for shape optimization problems. The first variation and the second variation of the objective functional are derived. These variations are discretized by introducing a set of boundary-value problems in order to derive the second-order numerical method. The boundary-value problems are solved by the conventional finite-element method.The authors would like to express their thanks to Mr. T. Masanao, who was an undergraduate student, for his cooperation and comments. They also thank Professor Y. Sakawa of Osaka University for his encouragement.A part of this paper was presented at the IFIP Conference on Control of Boundaries and Stabilization, Clermont-Ferrand, France, 1988.     

Genetic and Random Search Methods in Optimal Shape Design Problems

We describe the application of two global optimization methods, namely of genetic and random search type algorithms in shape optimization. When the so-called fictitious domain approaches are used for the numerical realization of state problems, the resulting minimized function is non-differentiable and stair-wise, in general. Such complicated behaviour excludes the use of classical local methods. Specific modifications of the above-mentioned global methods for our class of problems are described. Numerical results of several model examples computed by different variants of genetic and random search type algorithms are discussed.     

Flow simulation and shape optimization for aircraft design

Within the framework of the German aerospace research program, the CFD project MEGADESIGN was initiated. The main goal of the project is the development of efficient numerical methods for shape design and optimization. In order to meet the requirements of industrial implementations a co-operative effort has been set up which involves the German aircraft industry, the DLR, several universities and some small enterprises specialized in numerical optimization. This paper outlines the planned activities within MEGADESIGN, the status at the beginning of the project and it presents some early results achieved in the project.     

A smoothness preserving fictitious domain method for elliptic boundary-value problems

^** Email: mommer{at}math.uu.nl We introduce a new fictitious domain method for the solutionof second-order elliptic boundary-value problems with Dirichletor Neumann boundary conditions on domains with C² boundary.The main advantage of this method is that it extends the solutionssmoothly, which leads to better performance by achieving higheraccuracy with fewer degrees of freedom. The method is basedon a least-squares interpretation of the fundamental requirementsthat the solution produced by a fictitious domain method shouldsatisfy. Careful choice of discretization techniques, togetherwith a special solution strategy, leads then to smooth solutionsof the resulting underdetermined problem. Numerical experimentsare provided which illustrate the performance and flexibilityof the approach.     

Advances in concrete arch dams shape optimization

This paper presents an efficient methodology to find the optimum shape of arch dams. In order to create the geometry of arch dams a new algorithm based on Hermit Splines is proposed. A finite element based shape sensitivity analysis for design-dependent loadings involving body force, hydrostatic pressure and earthquake loadings is implemented. The sensitivity analysis is performed using the concept of mesh design velocity. In order to consider the practical requirements in the optimization model such as construction stages, many geometrical and behavioral constrains are included in the model in comparison with previous researches. The optimization problem is solved via the sequential quadratic programming (SQP) method. The proposed methods are applied successfully to an Iranian arch dam, and good results are achieved. By using such methodology, efficient software for shape optimization of concrete arch dams for practical and reliable design now is available.     

Analytical and numerical methods in shape optimization

This paper is intended to overview on analytical and numerical methods in shape optimization. We compute and analyse the shape Hessian in order to distinguish well‐posed and ill‐posed shape optimization problems. We introduce different discretization techniques of the shape and present existence and convergence results of approximate solutions in case of well posedness. Finally, we survey on the efficient numerical solution of the state equation, including finite and boundary element methods as well as fictitious domain methods. Copyright © 2008 John Wiley & Sons, Ltd.     

A projection based multiscale optimization method for eigenvalue problems

We present a projection based multiscale optimization method for eigenvalue problems. In multiscale optimization, optimization steps using approximations at a coarse scale alternate with corrections by occasional calculations at a finer scale. We study an example in the context of electronic structure optimization. Theoretical analysis and numerical experiments provide estimates of the expected efficiency and guidelines for parameter selection.     

Spectral domain embedding for elliptic PDEs in complex domains

Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such as rectangles or disks. Domain decomposition methods/spectral element methods extend the applicability of spectral methods to more complex geometries. An alternative is to embed the irregular domain into a regular one. This paper uses the spectral method with domain embedding to solve PDEs on complex geometry. The running time of the new algorithm has the same order as that for the usual spectral collocation method for PDEs on regular geometry. The algorithm is extremely simple and can handle Dirichlet, Neumann boundary conditions as well as nonlinear equations.     

A Smooth Embedding Domain Method Based On the Penalty Approach

It is well known that convergence of the fictitious domain formulation with boundary Lagrange multipliers is slow due to the lower global regularity of its solution. This article presents a smoothed variant of this approach which is based on a formulation in the form of a state constraint optimal control problem. The convergence rate is increased as seen from a model example.     

Optimality criteria coupled adaptive mesh method for optimal shape design of Stokes flow

An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.     

A shape optimization formulation of weld pool determination

In this paper, we propose a shape optimization formulation for a problem modeling a process of welding. We show the existence of an optimal solution. The finite element method is used for the discretization of the problem. The discrete problem is solved by an identification technique using a parameterization of the weld pool by Bézier curves and Genetic algorithms.     

The fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L²‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order in H¹‐norm for the velocity and in L²‐norm for the pressure, where is the penalty parameter. The L²‐norm error estimate for the velocity is upgraded to . Moreover, we derive the a priori estimates depending on for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate in H¹‐norm for the velocity and in L²‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L²‐norm, the convergence rate is improved to . The theoretical results are verified by the numerical experiments.     

Boundary regularity for Maxwell's equations with applications to shape optimization

The dynamic Maxwell equations with a strictly dissipative boundary condition is considered. Sharp trace regularity for the electric and the magnetic field are established for both: weak and differentiable solutions. As an application a shape optimization problem for Maxwell's equations is considered. In order to characterize the shape derivative as a solution to a boundary value problem, the aforementioned sharp regularity of the boundary traces is critical.     

A new hybrid method for solving global optimization problem

In this paper we present a new hybrid method, called the SASP method. The purpose of this method is the hybridization of the simulated annealing (SA) with the descent method, where we estimate the gradient using simultaneous perturbation. Firstly, the new hybrid method finds a local minimum using the descent method, then SA is executed in order to escape from the currently discovered local minimum to a better one, from which the descent method restarts a new local search, and so on until convergence.The new hybrid method can be widely applied to a class of global optimization problems for continuous functions with constraints. Experiments on 30 benchmark functions, including high dimensional functions, show that the new method is able to find near optimal solutions efficiently. In addition, its performance as a viable optimization method is demonstrated by comparing it with other existing algorithms. Numerical results improve the robustness and efficiency of the method presented.