Hideyuki Azegami

A SMOOTHING METHOD FOR SHAPE OPTIMIZATION: TRACTION METHOD USING THE ROBIN CONDITION

This paper presents an improved version of the traction method that was proposed as a solution to shape optimization problems of domain boundaries in which boundary value problems of partial differential equations are defined. The principle of the traction method is presented based on the theory of the gradient method in Hilbert space. Based on this principle, a new method is proposed by selecting another bounded coercive bilinear form from the previous method. The proposed method obtains domain variation with a solution to a boundary value problem with the Robin condition by using the shape gradient.

Irregularity Of Shape Optimization Problems AndAn Improvement Technique

Shape optimization problems of linear elastic bodies, flow fields, magnetic fields, etc. for equilibrium types can be generalized as optimization problems of domains in which elliptic boundary value problems are defined. This paper shows that ordinary domain optimization problems do not have sufficient regularity and proposes a technique to overcome this irregularity. It briefly describes the derivation of the shape gradient functions for a self-adjoint shape optimization problem, and shape identification problems of the Dirichlet type, Neumann type and subdomain gradient assigned type. Using these shape gradient functions, the irregularity of ordinary domain optimization problems is shown through a discussion of the ill-posedness that occurs when the gradient method in Hilbert space is applied directly. To overcome this irregularity, the idea of a smoothing gradient method in Hilbert space is proposed. It is conclusively shown that a numerical method based on this idea coincides with the traction method previously proposed by one of the authors and this conclusion is verified by numerical experiments.

Solution to shape optimization problems of viscous flow fields

This paper presents a numerical solution to the shape optimization problems of steady-state viscous flow fields. The minimization problem of total dissipation energy was formulated in the domain of viscous flow fields. The shape gradient of the shape optimization problem was derived theoretically using the adjoint variable method, the Lagrange multiplier method and the formulae of the material derivative. Reshaping was carried out by the traction method proposed by one of the authors as an approach to solving domain optimization problems. The validity of the proposed method was confirmed by results of 2D and 3D numerical analyses.

Etiology of idiopathic scoliosis : Computational study

A review of the literature on the mechanical aspects of the etiology for idiopathic scoliosis reveals that the buckling hypothesis has been presented as a purely mechanical phenomenon. In an attempt to confirm the buckling hypothesis, a numerical simulation of growth and the resulting buckling phenomena was done by means of finite element analysis. It previously was observed that growth was induced in the T4 to T10 vertebrae. Only the sacrum was assumed to be stationary. From the growth analysis, a deformation process that mitigated thoracic kyphosis was obtained as observed in healthy children during early adolescence. From the buckling analysis, the first to the fourth buckling modes that correspond to the first side bending, first forward bending, first rotation, and second side bending modes were obtained. The shape of the fourth buckling mode (second side bending mode) was in good agreement with the clinical shape. Considering the potential for controlling these modes by posture change, it is concluded that the second bending mode in the coronal plane is one of the most likely etiologic candidates in the mechanics of thoracic idiopathic scoliosis.

Buckling and bone modeling as factors in the development of idiopathic scoliosis.

STUDY DESIGN Computational analysis using the finite-element method was used to examine a possible etiology of idiopathic scoliosis. OBJECTIVES To compare changes in the coronal and the transverse planes of idiopathic thoracic scoliosis with changes produced in a finite-element buckling model, and to investigate the influence of bone modeling on the buckling spine. SUMMARY OF BACKGROUND DATA Although it is now widely accepted that growth is related strongly to the onset and progression of scoliosis, the pathomechanism or etiology of idiopathic scoliosis still is not clear. A previous study showed that a buckling phenomenon caused by anterior spinal overgrowth can produce scoliosis, and that the fourth buckling mode matched the clinical characteristics associated with the thoracic type of idiopathic scoliosis. The fourth buckling mode occurs when the first, second, and third buckling modes are prevented. METHODS The spinal finite-element model used in this study consisted of 68,582 elements and 84,603 nodes. The transverse changes seen in the computed tomography images of 41 patients with idiopathic thoracic scoliosis (apex, T8; average Cobb angle, 52.5 degrees) were compared with those produced in the fourth buckling mode. Bone modeling (bone formation and resorption) was simulated as heat deformation caused by changes in temperature. The bone formation and resorption were simulated, respectively, by positive and negative volume changes in proportion to the stress that occurred in the buckling spine. RESULTS Computed tomography images of scoliosis show that as the scoliosis becomes more severe, the thoracic cage decreases on the convex side of the curve and increases on the concave side. The opposite thoracic cage deformation was obtained in the fourth buckling mode. In patients with scoliosis, the sternum essentially remains in its original position with respect to the vertebrae, but in the linear buckling model, it shifted in the direction of vertebral body rotation. In contrast to clinical data, the incremental deformation resulting from bone formation corrected the original curve, and the thoracic cage distorted. On the other hand, incremental deformation resulting from bone resorption worsened the original curve, and the thoracic cage distorted in a manner similar to that described by the clinical data. CONCLUSIONS This computational investigation suggests that scoliotic changes in the spinal column triggered by the buckling phenomenon are counteracted by bone formation, but worsened by bone resorption. The authors hypothesized that scoliosis progressed with resorption of loaded bone. However, it is unclear whether this hypothesis applies to a living body in practice because of the effects from additional factors.

A Domain Optimization Technique For EllipticBoundary Value Problems

A numerical analysis technique is presented for solving optimization problems of geometrical domains in which elliptic boundary value problems are defined. Domain variation is formulated with a one-to-one mapping and its infinitesimal variation with a speed field as advocated by Zolesio. * The sensitivity functions, which we call the shape gradient functions, of domain variation are derived using the Lagrange multiplier method or the adjoint method. 3 By applying the gradient method in functional space* with the shape gradient functions, we propose to analyze an optimal speed field as a displacement of a pseudo-elastic problem that is defined on the design domain and loaded with pseudo-distributed external force, or traction, in proportion to the shape gradient function. This technique is called the traction method because of the procedure used. Numerical results for several linear elastic problems and flow field problems are given.

Non-parametric Shape Optimization Method ForThin-walled Structures Under Strength Criterion

This paper presents a numerical optimization method for shape design to improve the strength of thin-walled structures. A solution to maximum stress minimization problems subject to a volume constraint is proposed. With this solution, the optimal shape is obtained without any parameterization of the design variables for shape definition. It is assumed that the design domain is varied in the in-plane direction to maintain the curvatures of the initial shape. The problem is formulated as a non-parametric shape optimization problem. The shape gradient function is theoretically derived using the Lagrange multiplier method and the adjoint variable method. The traction method, which was proposed as a gradient method in Hilbert space, is applied to determine the smooth domain variation that minimizes the objective functional. The calculated results show the effectiveness and practical utility of the proposed solution in solving minmax shape optimization problems for the design of thin-walled structures under a strength criterion.

Shape optimization of 3D viscous flow fields

This article presents a numerical solution technique for shape optimization problems of steady-state, 3D viscous flow fields. In a previous study, the authors formulated shape optimization problems by considering the minimization of total dissipation energy in the domain of a viscous flow field, proposing a solution technique in which the traction method is applied by making use of a shape gradient. This approach was found to be effective for 2D problems for low Reynolds number flows. In the present study, the validity of the proposed solution technique is confirmed by extending its application to 3D problems.

Solution to boundary shape identification problems in elliptic boundary value problems using shape derivatives

Publisher Summary This chapter briefly defines the identification problems of geometrical boundary shapes of domains in which elliptic boundary value problems are treated. These problems are formulated as minimization problems of squared error integrals between the actual solutions of the elliptic boundary value problems and its reference data with respect to perturbation of the uncertain boundary. The fundamental theory concerning with the shape derivatives of functional with respect to domain perturbation and the gradient method in Hilbert space are also presented by mathematicians. Based on the theories, this chapter provides a concrete solution to the geometrical domain identification problems. It briefly describes the derivation of the shape gradient functions for the shape identification problems of two types referring to boundary value on sub boundary and referring to gradient in subdomain, and introduces the definition of the gradient method in Hilbert space.

Solution To Boundary Shape OptimizationProblems

This paper presents a numericalanalysis method of nonparametricboundaryshape optimizationproblemswith respectto boundaryvalueproblemsof partialdifferential equations. The nonparametricboundary variation can be formulated by selecting a one parameter family of continuous one-to-one mappings from an original domain to variable domains. The shape gradient with respect to domain variation can be evaluated by the adjoint variable method. However, the direct application of the gradient method often results in oscillating shapes. It has been known that the oscillating phenomenon is caused by a lack of smoothness of the shape gradient. To make up the irregularity, a smoothing gradient method and its concrete numerical procedure called the traction method have been presented by the author and coworkers. However, in the previous papers, the numerical procedure of the traction method was not illustrated. This paper presents a generalized description of the traction method and gives a precise algorithm of the traction method.