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This paper presents a method to reduce the computational time required to solve shape optimization problems. A volume minimization problem under the mean compliance constraint is chosen as an example of the shape optimization problem. To solve this problem, an iterative algorithm based on the H^1 gradient method is considered as a conventional approach. In this study, we attempt to use a method of model order reduction for solving the linear elasticity problem based on the idea by Karhunen-Loève expansion (KLE). We consider the displacements obtained by the conventional method to be sampling data of a random variable; the orthonormal bases of KLE are defined as eigenfunctions of the eigenvalue problem obtained as the optimality condition of the variance maximization problem for the random variable. The feasibility of the proposed method is illustrated by testing the numerical scheme to a linear elastic body of the connecting rod type.

The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the L^2-distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule approach. Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed. The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem. The feasibility of the proposed method is illustrated through various numerical examples., ファイル公開：2021/09/01

The present paper describes a shape optimization procedure for designing running shoes, focusing on two mechanical properties, namely, the shock absorption and the stability keeping the right posture. These properties are evaluated from two deformations of a sole at characteristic timings during running motion. We define approximate planes for the deformations of sole’s upper boundary by least squares method. Using the planes, we choose the tilt angle in the shoe width direction at the mid stance phase of running motion as an objective function representing the stability, and the sunk amount at the contact phase of running motion as a constraint function representing the shock absorption. We assume that the sole is a bonded structure of soft and hard hyper-elastic materials, and the bonding and side boundaries are variable. In this study, we apply the formulation of nonparametric shape optimization to the sole considering finite deformation and contact condition of the bottom of the sole with the ground. Shape derivatives of the cost (objective and constraint) functions are obtained using the adjoint method. The H^1 gradient method using these shape derivatives is applied as an iterative algorithm. To solve this optimization problem, we developed a computer program combined with some commercial softwares. The validity of the optimization method is confirmed by numerical examples.

We propose a new shape optimization formulation of the Bernoulli problem by tracking the Neumann data. The associated state problem is an equivalent formulation of the Bernoulli problem with a Robin condition. We devise an iterative procedure based on a Lagrangian-like approach to numerically solve the minimization problem. The proposed scheme involves the knowledge of the shape gradient which is established through the minimax formulation. We illustrate the feasibility of the proposed method and highlight its advantage over the classical setting of tracking the Neumann data through several numerical examples.

The solution to a free boundary problem of Bernoulli type, also known as Alt-Caffarelli problem, is studied via shape optimization techniques. In particular, a novel energy-gap cost functional approach with a state constraint consisting of a Robin condition is proposed as a shape optimization reformulation of the problem. Accordingly, the shape derivative of the cost is explicitly determined, and using the gradient information, a Lagrangian-like method is used to formulate an efficient boundary variation algorithm to numerically solve the minimization problem. The second order shape derivative of the cost is also computed, and through its characterization at the solution of the Bernoulli problem, the ill-posedness of the shape optimization formulation is proved. The analysis of the proposed formulation is completed by addressing the existence of optimal solution of the shape optimization problem and is accomplished by proving the continuity of the solution of the state problems with respect to the domain. The feasibility of the newly proposed method and its comparison with the classical energy-gap type cost functional approach is then presented through various numerical results. The numerical exploration issued in the study also includes results from a second-order optimization procedure based on a Newton-type method for resolving such minimization problem. This computational scheme put forward in the paper utilizes the Hessian information at the optimal solution and thus offers a state-of-the-art numerical approach for solving such free boundary problem via shape optimization setting., Online Published: 2019-06-28 ファイル公開：2020-06-28

The solution to a free boundary problem of Bernoulli type, also known as Alt-Caffarelli problem, is studied via shape optimization techniques. In particular, a novel energy-gap cost functional approach with a state constraint consisting of a Robin condition is proposed as a shape optimization reformulation of the problem. Accordingly, the shape derivative of the cost is explicitly determined, and using the gradient information, a Lagrangian-like method is used to formulate an efficient boundary variation algorithm to numerically solve the minimization problem. The second order shape derivative of the cost is also computed, and through its characterization at the solution of the Bernoulli problem, the ill-posedness of the shape optimization formulation is proved. The analysis of the proposed formulation is completed by addressing the existence of optimal solution of the shape optimization problem and is accomplished by proving the continuity of the solution of the state problems with respect to the domain. The feasibility of the newly proposed method and its comparison with the classical energy-gap type cost functional approach is then presented through various numerical results. The numerical exploration issued in the study also includes results from a second-order optimization procedure based on a Newton-type method for resolving such minimization problem. This computational scheme put forward in the paper utilizes the Hessian information at the optimal solution and thus offers a state-of-the-art numerical approach for solving such free boundary problem via shape optimization setting.

We aim to identify the geometry (i.e., the shape and location) of a cavity inside an object through the concept of thermal imaging. More precisely, we present an identification procedure to determine the geometric shape of a cavity with convective boundary condition in a heat-conducting medium using the measured temperature on a part of the surface of the object. The inverse problem of identifying the cavity is resolved by shape optimization techniques, specifically by minimizing a least-squares type cost functional over a set of admissible geometries. The computation of the first-order shape derivative or shape gradient of the cost is carried out through minimax formulation, which is then justified by the Correa–Seeger theorem coupled with function space parametrization technique. We further characterize its boundary integral form using some identities from tangential calculus. Then, we utilize the computed expression for the shape gradient to implement an effective boundary variation algorithm for the numerical resolution of the shape optimization problem. To avoid boundary oscillations or irregular shapes in our approximations, we execute the gradient-based scheme using the H^1 gradient method with perimeter regularization. Also, we propose a novel application of the said method in computing the mean curvature of the free boundary appearing in the shape gradient of the cost functional. We illustrate the feasibility of the proposed method by testing the numerical scheme to several cavity identification problems. Additionally, we also give some numerical examples for the case of corrosion detection since its inverse problem interpreted in the framework of electrostatic imaging is closely related to the focused problem., ファイル公開：2020/01/01

The main topic of this book is optimal design. In order to understand the mathematical structures involved in our study, we will begin by examining two simple problems. Upon finishing this book, the reader should be able to understand that even shape optimization problems of continuum structures possess the same formulation as the problems dealt with in this chapter. Moreover, even if the target continuum is changed from linear elastic body or flow field dealt in this book, the reader will recognize that their corresponding shape optimization problems maintain the fundamental structures introduced in this book.

In Chap. 8 we looked at problems for obtaining the optimal topologies of continua with the densities of continua set to be the design variable. In this chapter, we shall look at the type of shape optimization problems in which the boundary of a continuum varies. The key theory of numerical solution shown in this chapter is published in the paper (Azegami, Trans. Jpn. Soc. Industrial Appl. Math. 23(2), 83–138 (2014)). In this book, we shall look at the theory used there by comparing it to the contents shown in Chaps. 1 to 7.

From this chapter, we finally examine shape optimization problems in continua. Firstly, let us think about a problem seeking the appropriate arrangement of holes in a domain where the boundary value problem of a partial differential equation is defined. Such a problem is known as the topology optimization problem. Here, the term topology refers to the study of geometrical properties and spatial relation of objects unaffected by the continuous change of their shape or size. In mathematics, two mathematical objects are said to belong to the same topology if they are images of two homotopic maps; that is, if one can be continuously deformed into the other. Therefore, letting n be a natural number, a set of n-connected domains are regarded as belonging to the same homotopy groups. Here, the term “topology optimization” in the topology optimization problem refers to the determination of the connectivity of the design domain that optimizes an object’s material distribution through insertion and arrangement of holes in its structure. However, as will be explained in detail later, the shape of the holes actually becomes the target. Therefore, the problems dealt with in this chapter also become included in shape optimization problems in a wider sense. In this book, it will be referred to as the topology optimization problem in the sense that topology is also in the scope of the design.

This paper presents a solution of a shape optimization problem of a flow field for delaying transition from a laminar flow to a turbulent flow. Mapping from an initial domain to a new domain is chosen as the design variable. Main problems are defined by the stationary Navier–Stokes problem and an eigenvalue problem assuming a linear disturbance on the solution of the stationary Navier–Stokes problem. The maximum value of the real part of the eigenvalue is used as an objective cost function. The shape derivative of the cost function is defined as the Frechet derivative of the cost function with respect to arbitrary variation of the design variable, which denotes the domain variation, and is evaluated using the Lagrange multiplier method. To obtain a numerical solution, we use an iterative algorithm based on the $$H^{1}$$ gradient method using the finite element method. To confirm the validity of the solution, a numerical example for two-dimensional Poiseuille flow with a sudden expansion is presented. Results reveal that a critical Reynolds number increases by the iteration of reshaping.

This paper presents numerical results obtained, using a numerical method for a flow field shape optimization, for the design of a room from which people can evacuate smoothly. The domain of interest is a two-dimensional Poiseuille flow with sudden contraction in which a disk is located initially. In fact, the compressible fluid model has been used to describe the dynamics of people for the study of evacuation. However, we set a model involving the incompressible Navier-Stokes equation for the first trial. And the shape optimization to minimize the dissipation energy on the disk is demonstrated under the volume constraint. For reshaping numerically, the traction method is used. Numerical results reveal that the shape in the wake of the disk becomes an acute angle to decrease the dissipation energy monotonically, thereby satisfying the volume constraint. Such a shape has never been inferred from results of earlier studies of the evacuation problem in jamology., "Topology optimization theory and applications toward wide fields of natural sciences". May 7～9, 2014. edited by Takashi Nakazawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

In this article, we report on-going projects about developments of less-invasive artificial blood tubes (Graft) for Hemodyalysis. Blood is flowing over 2m/s because of pressure gradient between artery and vein which are connected by the Graft. It causes a stenosis at the junction between the Graft and the vein, and finally the stenosis raises low efficiency of the Hemodyalysis and more infection disease. For solving this problem, we develop the graft with a sudden contraction (FCG: Flow Control Graft), and some animal experiments provide enough evidence of effectiveness. Now, by operating shape optimization, more efficient and less invasive FCG is developed., "Topology optimization theory and applications toward wide fields of natural sciences". May 7～9, 2014. edited by Takashi Nakazawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

We derive the second-order shape derivatives (shape Hessians) of cost functions for shape optimization problems of domains in which boundary value problems of partial differential equations are defined, and propose an \(H^1\) Newton method to solve the problems using the shape Hessians. In this paper, we formulate an abstract shape optimization problem and show the computations of the first- and second-order shape derivatives of cost functions under the abstract framework. Then, using the shape gradients and Hessians, we propose an \(H^1\) Newton method to solve the given problem. As an illustration, the shape Hessians of a mean compliance and a domain measure are derived and then used for a numerical example.

The present paper describes a solution to a non-parametric shape optimization problem of a brake model to suppress squeal noise. The brake model consists of a rotor and a pad, between which Coulomb friction occurs. The main problem is defined as a complex eigenvalue problem of the brake model obtained from the equation of motion. As an objective cost function, we use the positive real part of the complex eigenvalue generating the brake squeal. The volume of the pad is used as a constraint cost function. The Frechet derivative of the objective cost function with respect to the domain variation, which we refer to as the shape derivative of the objective cost function, is evaluated using the solution of the main problem and the adjoint problem. A scheme by which to solve the shape optimization problem using an iterative algorithm based on the H 1 gradient method (the traction method) for reshaping is presented. Numerical results obtained using a simple rotor-pad model reveal that the real part of the target complex eigenvalue decreases monotonically, thus satisfying the volume constraint.

The present paper introduces a numerical solution to shape optimization problems of domains in which boundary value problems of partial differential equations are defined. In the present paper, the finite element method using NURBS as basis functions in the Galerkin method is applied to solve the boundary value problems and to solve a reshaping problem generated by the H1 gradient method for shape optimization, which has been developed as a general solution to shape optimization problems. Numerical examples of linear elastic continua illustrate that this solution works as well as using the conventional finite element method., This paper was presented in WCSMO-9, Shizuoka.

We present an error estimation for the H1 gradient method, which provides numerical solutions to the shape-optimization problem of the domain in which a boundary value problem is defined. The main result is that if second-order elements are used for the solutions of the main and adjoint boundary value problems to evaluate the shape derivative, and the first-order elements are used for the solution of domain variation in the boundary value problem of the H1 gradient method, then we obtain first-order convergence of the solution of the domain variation with respect to the size of the finite elements.

The present paper describes a method by which to formulate a shape optimization problem of a linear elastic continuum for minimizing the maximum value of a strength measure, such as the von Mises stress. In order to avoid the irregularity of the shape derivative of the maximum value, the Kreisselmeier-Steinhauser function of the strength measure is used as the cost function. In the cost function, a parameter is used to control the regularity of the shape derivative. In the present paper, we propose a rule by which to appropriately determine the parameter. The effectiveness of the proposed rule is confirmed through a numerical example of a cantilever problem.

In this paper, we define shape optimization problems as problems of finding the shapes of domains in which boundary value problems of partial differential equations are defined. A domain mapping from an initial domain to a new domain is chosen as the design variable. Functionals of the design variable and the solution to the boundary value problem are used as cost functions. In this paper, the formulation of the shape optimization problem and a numerical method of solving the problem are presented. In addition, our subsequent works applying this method to product design are introduced: (1) shape optimization of a link mechanism, (2) shape optimization for suppressing brake squeal, (3) a method of designing beads in a shell structure, (4) shape optimization of a flow field to improve hydrodynamic stability, and (5) shape optimization of an electrostatic capacitive sensor.

The importance of the optimal shape design has been increasing in the present industrial design due to the request to make their production more efficient.

This paper presents a numerical analysis method for shape optimization in order to achieve stiffness maximization in thermoelastic fields. Mean compliance is used as an objective functional for the shape optimization problem. The mean compliance minimization problem on the thermoelastic fields is formulated on volume constraint condition. The shape gradient of the shape optimization problems is derived theoretically using the adjoint variable method, the Lagrange multiplier method and the formulae of the material derivative. Reshaping is accomplished using a traction method that was proposed as a solution to shape optimization problems. In addition, a new numerical procedure for the shape optimization is proposed. The validity of the proposed method is confirmed based on the results of 2D numerical analysis.

This paper presents a numerical analysis method for shape optimization of domains with steady-state heat-conduction fields considering the temperature dependence of the thermal conductivity coefficient. In this paper, we formulate two shape optimization problems, namely, maximization of thermal dissipation on heat transfer boundaries and minimization of heat-conduction fields. The shape gradient functions for these shape optimization problems are derived theoretically using the Lagrange multiplier method and formulae of the material derivative. Reshaping is accomplished using the traction method proposed as a solution to the shape optimization problems. The proposed method is validated from the results of two-dimensional numerical analysis. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.20374

This paper presents a numerical solution to a non-parametric shape optimization problem for design of suspension arm in which strength of suspension arm is evaluated by reaction force to plastic buckling load due to compulsory displacement. To deal with buckling phenomena, the geometrical non-linearity and material non-linearity are considered. Hyper-elastic theory is applied to calculate the deformation of suspension arm, under assumption of monotonous loading. Mass and the reaction force integral to the buckling phenomena are chosen as an objective function and a constraint function, respectively. The shape derivatives of these functions are evaluated by the shape optimization theory. A numerical scheme based on a sequential quadratic approximation method is applied to reshape by using the shape gradients. In this scheme, the traction method is used to find the decent directions of the cost functions. The scheme is implemented by using a commercial shape optimization program. In this program, the shape gradients are calculated by a user sub-program which is developed by using the result of non-linear FEM analysis of a commercial solver. The numerical example for a suspension arm model shows 12% of mass reduction while keeping the reaction force integral constant.

The present paper describes a numerical solution to shape optimization problems of contacting elastic bodies for controlling contact pressure. The contacting elastic problem is formulated as the minimization of potential energy with a constraint for penetration based on the large deformation theory. The contact pressure is defined as a Lagrange multiplier for the constraint of penetration in the minimization problem. An error norm of the contact pressure to a desired distribution is chosen as an objective functional. The shape derivative of the functional is theoretically evaluated. Numerical solutions are constructed by the traction method.

The present paper describes a numerical solution of shape optimization problems for non-stationary Navier-Stokes problems. As a concrete example, we consider the problem of finding the shape of an obstacle in a flow field in order to minimize the energy loss integral for an assigned time interval. The primary goal of the present paper is to demonstrate the evaluation of the shape derivative of the energy loss. The traction method is used for the reshaping algorithm. Numerical results show that the shapes of the circle obstacle converge to wedge shapes for the cases of Reynolds numbers of 100 and 250.