Non-parametric Shape Optimization Research Articles

Design for the Vent Holes of Gas Turbine Flow Guide Disks Based on the Shape Optimization Method

Two shape optimization methods, based on non-parametric and geometric parameters, were developed to address stress concentrations in the vent holes of gas turbine flow guide disks. The design optimization focused on reducing the maximum equivalent stress at the hole edge in an aero-engine gas turbine flow guide disk. The effectiveness of both methods in achieving this objective was studied. The results indicated that the non-parametric-based optimization method reduced the maximum equivalent stress at the hole edge by 24.5% compared to the initial design, while the geometric parameter-based optimization method achieved a reduction of 20.2%. Both shape optimization methods proved effective in reducing stress concentrations and improving fatigue life. However, the non-parametric shape optimization method resulted in a better design for the vent holes based on the study’s findings.

Lagrangian approach for optimizing fiber bundle placements of CFRP shell structures

We present a non-parametric shape and topology optimization method for fiber bundle placement and orientation design aimed at controlling the rigidity of CFRP shell structures. Since the fiber bundles are modeled by the beam element, the placement and the orientation of the fiber bundles can be explicitly obtained while moving the positions. In order to achieve the target displacement at an arbitrarily specified position, we considered the problem of minimizing the squared error under the length constraint of the fiber bundles, and formulated it as a distributed-parameter optimization problem. Using the derived shape gradient function and the density gradient function, the optimum placement and the orientation of the fiber bundles are determined simultaneously by the H1 gradient method for vector and scalar variables. The effectiveness of the proposed method is confirmed by numerical examples.

Shape optimization of running shoes with desired deformation properties

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 H1 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.

Shape optimization of SMA structures with respect to fatigue

This paper presents an efficient structural optimization approach for shape memory alloys (SMAs) with respect to fatigue. In the proposed method, a nonparametric shape optimization approach based on optimality criteria is applied to change the structural configuration. The design optimization framework is accomplished by relying on commercially available software and tools. In addition, both low- and high-cycle fatigue criteria are used to compute the fatigue factor at each material point, indicating its degree of safeness with respect to fatigue. Meanwhile, a 3D constitutive model is utilized to predict, with good accuracy, the stabilized thermomechanical stress state of a SMA structure subjected to multiaxial nonproportional cyclic loading. Finally, numerical examples are tested to demonstrate the effectiveness of the proposed method.

Non-parametric optimization of railway wheel web shape based on fatigue design criteria

Railway wheels are safety-critical components with complex stress states that make parametric design difficult. Accordingly, this study evaluated a non-parametric shape optimization technique using the traction method for the fatigue design of railway wheel webs, considering design specifications or factors such as the position and size of the rim or hub as constraints. Reduced models were then employed to consider the influence of wheel wear on the stress state of the web to further inform design. The findings of this study indicate that the traction method can be successfully used to optimize the geometry of railway wheel webs according to stress, geometric, or mass constraints.

Non-parametric shape design optimization of elastic-plastic shear panel dampers under cyclic loading

In this work, we develop a non-parametric shape optimization method for designing low yield steel shear panel dampers (SPDs) under cyclic loading to enhance their energy dissipation capacity and verify it by experiments. Considering material nonlinearity (elastic-plastic) and geometric nonlinearity (large deformation), we formulate a shape optimization problem based on the variational method for 3D solid continua under cyclic loading, in which the plastic work in terms of von Mises equivalent stress is used as the objective function under a volume constraint. Considering the optimality conditions for the plastic work maximization problem, we theoretically derive the shape gradient function based on the Lagrange multiplier method, the material derivative formula, and the adjoint variable method. The derived shape gradient function is used to the H1 gradient method for designing the optimal shapes of 3D solid continua. We apply the developed shape optimization method to design SPDs made of low yield steel, which is an elastic-plastic material owning high-energy dissipation capacity. Cyclic enforced displacements acting on SPDs are used for simulating cyclic loading and evaluating the capacity of energy absorption of SPDs. To evaluate the nonlinear finite element analysis, we also perform cyclic loading test on SPDs. Two design examples of SPDs without and with holes are optimized to verify the validity of the developed shape optimization method. The results show that the developed optimization method works effectively to enhance the plastic work of SPDs. As a feature, this method can design SPDs more efficiently when considering the variation in the thickness direction.

Non-parametric shape optimization method for robust design of solid, shell, and frame structures considering loading uncertainty

The robust design of structures is essential to improve their stabilities in structural design optimization and has been studied based on a variety of optimization methods. In this study, we propose a non-parametric optimization method for the robust shape design of solid, shell, and frame structures subjected to uncertainty loadings. We adopt the concept of principal compliance to perform the robust shape design considering loading uncertainty and transform the principal compliance minimization problem into the fundamental eigenvalue maximization problem associated with the weighting coefficients of the unknown loadings. The proposed non-parametric shape optimization method for robust design consists of four main procedures: the eigenvalue analysis of structures, derivation of shape gradient functions considering repeated eigenvalues, velocity analysis based on the H1 gradient method, and shape updating. We perform several design examples to confirm the validity of the proposed non-parametric shape optimization method. The optimal results show that the proposed optimization method works efficiently to reduce the principal compliance and enhance the robust behavior of each design example. As a feature, by setting the weighting coefficients, we can enhance the robust of the structures subjected to the unknown loadings at different loading positions and with different magnitudes of the directions of the admissible loading space.

SNS mercury target design optimization

The mercury target at the Spallation Neutron Source (SNS) in Oak Ridge National Laboratory, Tennessee, works under a number of severe conditions: high temperature, radiation, mercury flow, and proton beam pulse pressures. Insufficient strength and durability of the target stainless steel vessel from beam pulse and thermal cyclic loads can lead to mercury leaks and premature target life termination. The parametric and non-parametric shape optimization tools (Isight, Tosca) from Dassault Systemes Extended Abaqus Package have been adopted at SNS to develop robust and reliable target designs. The durability evaluation with recognized leading Fe-Safe (structural fatigue) and Verity (weld fatigue) software is also included in optimization process. Design of Experiment (DOE) was performed to create the response surface for steel vessel structure. The response surfaces are used for design exploration on a basis of Abaqus and Fe-Safe different analyses results. The shape optimization of steel vessel filled with mercury was performed first time using Tosca optimization tool. Obtained complex results enable the consideration of possible trade-offs for the most feasible design.

A consistent methodology for optimal shape design of graphene sheets to maximize their fundamental frequencies considering topological defects

In recent years, shape design of graphene sheets (GSs) by introducing topological defects for enhancing their mechanical behaviors has attracted the attention of scholars. In the present work, we propose a consistent methodology for optimal shape design of GSs using a combination of the molecular mechanics (MM) method, the non-parametric shape optimization method, the phase field crystal (PFC) method, Voronoi tessellation, and molecular dynamics (MD) simulation to maximize their fundamental frequencies. At first, we model GSs as continuum frame models using a link between the MM method and continuum mechanics. Then, we carry out optimal shape design of GSs in fundamental frequency maximization problem based on a developed shape optimization method for frames. However, the obtained optimal shapes of GSs only consisting of hexagonal carbon rings are unstable that do not satisfy the principle of least action, so we relocate carbon atoms on the optimal shapes by introducing topological defects using the PFC method and Voronoi tessellation. At last, we perform the structural relaxation through MD simulation to determine the final optimal shapes of GSs. We design two examples of GSs and the optimal results show that the fundamental frequencies of GSs can be significantly enhanced according to the optimal shape design methodology.

Non-parametric shape optimization method for designing cable net structures in form finding and stiffness maximization problems

Cable net structures, belonging to the tension structures, have been widely utilized in structural engineering. In this study, we develop a non-parametric shape optimization method based on the H1 gradient method for designing cable net structures in form finding and stiffness maximization problems. We divide the design velocity field of cable net structures into the off-axis and the in-axis components for deriving and applying the shape gradient function in each design problem conveniently. In the form finding problem, we use the total length of cables as the objective functional and minimize it without or with considering the constraint condition of perimeter. In the stiffness maximization problem, the compliance is used as the objective functional and minimized under the constraint condition of perimeter and total length of cables considering large deformation and small strain. Moreover, two-step shape optimization of cable net structures from the form finding problem to the stiffness maximization problem is also implemented. The validity of the developed shape optimization method is confirmed by design examples.

Non-parametric free-form optimal design of frame structures in natural frequency problem

Optimal design of structures with respect to their mechanical behavior is essential and basically required in structural engineering. In this study, we propose a non-parametric free-form optimization method based on the variational method to design frame structures composed of arbitrarily curved linear elastic members. The natural frequency maximization problem of frame structures is formulated as a non-parametric shape optimization problem under the volume constraint. Under the assumption that each member varies in the out-of-plane direction to its centroidal axis, the shape gradient functions and the optimality conditions are theoretically derived by the Lagrange multiplier method and the formulae of the material derivative. Then, the derived shape gradient functions are applied to a gradient method in the Hilbert space with a P.D.E (Partial Differential Equation) smoother, which is referred as the H1 gradient method for frame structures. Moreover, a simple switching technique of the objective functional is presented for overcoming the discontinuity problem of repeated eigenvalues, which often appears in natural frequency maximization problem. With this combination of the three techniques, the optimal free-form frame structures owning smoothly curved members can be obtained without any preliminary shape parameterization, and the effectiveness and validity of the proposed method are verified through three design examples.

Free-Form Optimization of Thin-Walled Structure for Frequency Response Problem

We present a node-based free-form optimization method for designing forms of thin-walled structures in order to control vibration displacements or mode at a prescribed frequency. A squared displacement error norm is introduced at the prescribed surface as the objective functional to control the vibration displacements to target values in a frequency response problem. It is assumed that the thin-walled structure is varied in the normal direction to the surface and the thickness is constant. A nonparametric shape optimization problem is formulated, and the shape gradient function is theoretically derived using the material derivative method and the adjoint variable method. The shape gradient function obtained is applied to the surface of the thin-walled structure as a fictitious traction force to vary the form. With this free-form optimization method, an optimum thin-walled structure with a smooth free-form surface can be obtained without any shape parameterization. The calculated results show the effectiveness of the proposed method for the optimal free-form design of thin-walled structures with vibration mode control.

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.

Shape optimization for suppressing brake squeal

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...

J122034 最大主ひずみに基づく破断強度に関する構造部材の形状最適化解析手法の検討

This paper presents an approach to minimize weight of structural material while keeping breaking strength base on a non-parametric shape optimization problem. As an index for the breaking strength evaluation, the maximum value of principle strain is used including large deformation range. Under assumption of monotonous loading, the hyper-elastic theory is applied in calculating the deformation of structural material in order to reduce computational time. To avoid difficulty of taking derivative, local functional are transposed to global integral functional using Kreisselmeier-Steinhauser (KS) functional. To improve convergence of the calculation, a new method to determine the parameter in KS functional which controls extraction ability of maximum principle strain value is introduced. The mass and the maximum principle strain evaluated by KS functional are chosen as an objective function and a constraint function, respectively. The shape derivatives of these functions are evaluated by the shape optimization theory. The traction method is employed to keep the smoothness of original boundary. An example of numerical calculation for a structural material model shows 10% of mass reduction while keeping the maximum principle strain constant.

Examination of Shape Optimization for the Nonlinear Buckling Phenomenon of Suspension Parts

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.

J1202-2-1 シェルの形状最適化手法の板バネ設計への適用([J1202-2]解析・設計の高度化・最適化II)

In this paper, we present a shape optimization method for plate springs, which is based on the free-form optimization method for the shape design of shell structures we have developed. With the free-form optimization method, an optimum shell with smooth free-form surface can be obtained without any shape parameterization, In order to apply this method to the design of plate springs, we introduce a squared displacements error norm at the loaded points as the objective functional to control the displacements to the desired ones. It is assumed that the shell was varied in the normal direction to the surface and the thickness is constant. A non-parametric shape optimization problem without a volume constraint is formulated, and the shape gradient function is theoretically derived using the material derivative method. The shape gradient function is applied to the shell surface as the pseudo traction force to change the form. The calculated results show the effectiveness of the proposed method for the optimal free-form design of the plate springs.

803 周波数応答問題におけるモードコントロールを目的とするシェル構造のフリーフォルム最適化(OS8. 計算力学と最適化(1),オーガナイズドセッション講演)

In this paper, we present a free-form optimization method for designing shell's form in order to control the vibration mode at a prescribed frequency to a target mode. With the free-form optimization method, an optimum shell with smooth free-form surface can be obtained without any shape parameterization. We introduce a squared displacements error norm at the prescribed surface as the objective functional to control the vibration mode to a desired one in a frequency response problem. It is assumed that the shell was varied in the normal direction to the surface and the thickness is constant. A non-parametric shape optimization problem is formulated, and the shape gradient function is theoretically derived using the material derivative method. The shape gradient function is applied to the shell surface as the pseudo traction force to vary the form. The calculated results show the effectiveness of the proposed method for the optimal free-form design of shell structures aiming at mode control.

1302 最適な曲率分布を有するシェルの形状設計(OS7-1 設計と最適化I-構造最適化-,OS7 設計と最適化)

This paper presents a numerical optimization method for the optimal free form design of shell structures. Using the compliance as an index of stiffness, the stiffness design problem is formulated as a non-parametric shape optimization problem under the assumptions that the shell is varied in the normal direction to the surface and the thickness is constant. The shape gradient function derived for this problem is applied to the Robin-type traction method to determine the optimal and smooth free form, or the optimal curvature distribution. Several shape design problems are solved in order to verify the practical utility of this method. It is also confirmed that the strain energy of the optimal shape is mainly dominated by the membrane component. The calculated results show the proposed method is effective for the shape design of shell structures with the optimal curvature distribution.

252 固有振動数最大化を目的とするシェル構造の形状最適化

This paper presents a numerical optimization method for optimal configuration design of shell structures. It is assumed that the shell surface is varied in the out-of-plane direction to the surface in order to obtain the optimal free-form shape, and the thickness is not changed with respect to the shape variation. A solution to the maximization problem of vibration eigen value subject to a volume constraint is proposed. With this solution, the optimal shape is obtained without shape design parameterization. The problem is formulated as a non-parametric shape optimization problem. The shape gradient function is theoretically derived using the material derivative formulas, Lagrange multiplier method and the adjoint variable method. The Robin type traction method is applied to determine the free-form shell while minimizing the objective functional. The calculated results show the effectiveness of the proposed method for optimal free-form design of shell structures.