Parameter-free Shape Optimization Research Articles

Yet another parameter-free shape optimization method

Yet another parameter-free shape optimization method

Parameter-Free Shape Optimization: Various Shape Updates for Engineering Applications

In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update in gradient-based optimization strategies. Instead, an auxiliary problem has to be solved to obtain a gradient from the sensitivity. While several choices for these auxiliary problems were investigated mathematically, the complexity of the concepts behind their derivation has often prevented their application in engineering. This work aims to explain several approaches to compute shape updates from an engineering perspective. We introduce the corresponding auxiliary problems in a formal way and compare the choices by means of numerical examples. To this end, a test case and exemplary applications from computational fluid dynamics are considered.

Parameter-free shape optimization for minimizing plastic work of 3D rigid-plastic solids

Parameter-free shape optimization for minimizing plastic work of 3D rigid-plastic solids

Node-based free-form optimization method for vibration problems of shell structures

We have previously proposed a numerical node-based parameter-free shape optimization method for designing the optimal free-form surface of shell structures. In this paper, this method is extended to deal with two vibration problems including a vibration eigenvalue maximization problem and a frequency response minimization problem. To avoid the repeated eigenvalue problem when a specified vibration eigenvalue is maximized, we provide two optional approaches, i.e., tracking the specified natural mode or increasing all the repeated eigenvalues. Each vibration problem is formulated as a distributed-parameter shape optimization problem, and the derived shape gradient function is applied to the H1 gradient method for the shells proposed by the authors, where the shape gradient function is used as a distributed force function to vary the surface. With this method, the optimal and smooth free-form shape including a natural bead pattern can be obtained. Several calculated examples are presented to demonstrate the effectiveness of the proposed method for the free-form design of shell structures involving vibration problems.

Parameter-free method for the shape optimization of stiffeners on thin-walled structures to minimize stress concentration

This paper presents a parameter-free shape optimization method for the strength design of stiffeners on thin-walled structures. The maximum von Mises stress is minimized and subjected to the volume constraint. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that a stiffener is varied in the in-plane direction and that the thickness is constant. The issue of nondifferentiability, which is inherent in this min-max problem, is avoided by transforming the local measure to a smooth differentiable integral functional by using the Kreisselmeier-Steinhauser function. The shape gradient functions are derived by using the material derivative method and adjoint variable method and are applied to the H1 gradient method for shells to determine the optimal free-boundary shapes. By using this method, the smooth optimal stiffener shape can be obtained without any shape design parameterization while minimizing the maximum stress. The validity of this method is verified through two practical design examples.

Manufacturing constraints in parameter–free shape optimization

AbstractStructural shape optimization has become an important tool for engineers when it comes to improving components with respect to a given goal function. During this process the designer has to ensure that the optimized part stays manufacturable. Depending on the manufacturing process several requirements could be relevant such as demolding or different kinds of symmetry. This work introduces two approaches on how to handle manufacturing constraints in parameter‐free shape optimization. In the so–called explicit approach equality and inequality equations are formulated using the coordinates of the FE‐nodes. These equations can be used to extend the optimization problem. Since the number of the additional constraint equations may be very large we apply aggregation formulations, e.g. the Kreisselmeier‐Steinhauser function, if necessary. In the second approach, the so–called implicit method, the set of design nodes is split in two groups called optimization nodes and dependent nodes. The optimization nodes are now handled as design nodes but the dependent nodes are coupled to the optimization nodes in such a way that the manufacturing constraint is fulfilled. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Non-parametric shape optimization method for natural vibration design of stiffened shells

In this paper, we newly present an effective shape optimization method for natural vibration design of stiffened thin-walled or shell structures. Both the stiffeners and their basic structures are optimized by solving two kinds of optimization problems. The first is a specified eigenvalue maximization problem subject to a volume constraint, and the second is its reciprocal volume minimization problem subject to a specified eigenvalue constraint. The boundary shapes of a thin-walled structure are determined under the condition where the stiffeners and the basic structure are movable in the in-plane direction to their surface. Both problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using the material derivative method and the adjoint variable method. The optimal free-boundary shapes are determined by applying the derived shape gradient function to the H1 gradient method for shells, which is a parameter-free shape optimization method proposed by one of the authors. Several design examples are presented to validate the proposed method and demonstrate its practical usages.

A non-parametric solution to shape identification problem of free-form shells for desired deformation mode

A shape identification method of free-form shells is presented for controlling the static deformation mode to the desired one. This problem is formulated as a parameter-free shape optimization problem, in which a squared displacements error norm on the prescribed region is employed as an objective functional. The shape sensitivity, called shape gradient function, is theoretically derived using the adjoint variable method and the formula of the material derivative, and then applied to a gradient method with Laplacian smoother for shells to determine the smooth optimal shape. Several calculated examples are presented to verify the validity and practical utility of the proposed method.

On the discrete variant of the traction method in parameter-free shape optimization

One of the major challenges in the parameter-free approach to computational shape optimization is the avoidance of oscillating (i.e. non-smooth) boundaries in the optimal design trials. In order to achieve this, we investigate a method for regularization that corresponds to the discrete variant of the so-called traction method. In this approach, the design updates are generated in terms of a displacement field, which is obtained as the solution to an auxiliary boundary value problem that is defined on the actual design domain. The main idea herein is to apply fictitious nodal forces corresponding to the discrete sensitivity of the objective function. We propose an algorithm in which constraint functions will be taken into account by using an augmented Lagrangian formulation and a step-length control ensures a sufficient decrease condition in terms of the objective function within each iteration. We examine the benefits of the proposed regularization method on the basis of some numerical examples in comparison to an unregularized steepest-descent algorithm.

Non-parametric free-form optimization method for frame structures

In this paper, we propose a parameter-free shape optimization method based on the variational method for designing the smooth optimal free-form of a spatial frame structure. A stiffness design problem where the compliance is minimized under a volume constraint is solved as an example of shape design problems of frame structures. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that each member is varied in the out-of-plane direction to the centroidal axis and that the cross section is prismatic. The shape gradient function and the optimality conditions are then theoretically derived. The optimal curvature distribution is determined by applying the derived shape gradient function to each member as a fictitious distributed force both to vary the member in the optimum direction and to minimize the objective functional without shape parametrization, while maintaining the members' smoothness. The validity and practical utility of this method were verified through several design examples. It was confirmed that axial-force-carrying structures were obtained by this method.

2205 骨組構造のノンパラメトリック形状最適化(材料・構造)

In this paper, a parameter-free shape optimization method is presented for designing the optimal free-form of a spatial framed structure. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that each member is varied in the out-of-plane direction to the centroidal axis and the cross section is prismatic. Compliance minimization, vibration eigenvalue maximization and displacements control problems are formulated under a volume constraint, and solved as examples of optimal shape design of spatial framed structures. The shape gradient function and the optimality conditions for the problems are then theoretically derived. The optimal free-form is determined by applying the derived shape gradient function to each member as a fictitious distributed force to vary the frame, while minimizing the objective functional. We call this method the H^1 gradient method for frame structures, a gradient method in the Hilbert space. The validity and practical utility of this method are verified through design examples.

Free-Boundary Shape Optimization Method for Natural Vibration Problem of Thin-Walled Structure

In this paper, we present a shape optimization method for the natural vibration problem of thin-walled or shell structures with or without sitiffeners. The boundary shapes of a shell or stiffeners are determined under the condition where the boundary is movable in the in-plane direction to the surface. The design problems deal with eigenvalue maximization problem and volume minimization problem, which are subject to a volume constraint and an eigenvalue constraint respectively. The both optimization problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using the material derivative method and the adjoint variable method. The optimal free-boundary shapes are obtained by applying the derived shape gradient functions to the H1 gradient method for shells, which is a parameter-free shape optimization method proposed by one of the authors. Several design examples are presented to validate the proposed method and demonstrate its practical utility.

Free-Form Optimization Method for Shape Design of Framed Structure

In this paper, a parameter-free shape optimization method is proposed for designing the smooth optimal free-form of a 3D framed structure. A stiffness design problem where the compliance is minimized under a volume constraint is solved as an example of a shape design problem. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that each member is varied in the out-of-plane direction to the centroidal axis and the cross section is prismatic. The shape gradient function and the optimality conditions for this problem are then theoretically derived. The optimal curvature distribution is determined by applying the derived shape gradient function to each member as a pseudo distributed force to vary the frame, while minimizing the objective functional. We call this method the free-form optimization method for frame, a gradient method in the Hilbert space. The validity and practical utility of this method were verified through several design examples. It was confirmed that axial-force-carrying structures was obtained by this method.

2408 固有振動問題に対する補強リブ付きシェル構造の形状最適化手法

This paper presents a shape optimization method for the natural vibration problem of stiffened thin-walled or shell structures. The boundary shapes of stiffeners are determined under the condition where the boundary is movable in the inplane direction to the surface. The design problems deal with eigenvalue maximization problem and volume minimization problem, which are subject to a volume constraint and an eigenvalue constraint respectively. The both optimization problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using the material derivative method and the adjoint variable method. The optimal free-boundary shapes are obtained by applying the derived shape gradient functions to the H^1 gradient method for shells, which is a parameterfree shape optimization method proposed by one of the authors. Several design examples are presented to validate the proposed method and demonstrate its practical utility.