Material Derivative Method Research Articles

Shape and topology optimization method for fiber placement design of CFRP plate and shell structures

In this study, we present a novel non-parametric shape-topology optimization method for fiber placement and orientation design, aiming at controlling the stiffness of CFRP (Carbon Fiber Reinforced Plastics) shell structures. The sum of squared error norm for achieving the target displacements on the arbitrarily specified position is minimized under the length constraints as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the density gradient for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The solid isotropic material with penalization (SIMP) method is employed for topology optimization. The gradient functions derived are applied to the H1 gradient method for fibers to determine the optimal shape and topology. With the present method, the optimal fiber placement and orientation can be obtained without any parameterization. Numerical examples are solved to show the validity of the proposed method and the results are discussed.

An abstract Lagrangian framework for computing shape derivatives

In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints in Banach spaces. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of this method to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We also compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method.

Shape optimization method for strength design problem of microstructures in a multiscale structure

AbstractIn this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed: minimization of maximum microstructural stress and minimization of maximum macrostructural stress. The homogenization method is used to bridge the macrostructure and the microstructure and to calculate local microstructural stress. By replacing the maximum stress value with a Kreisselmeier‐Steinhauser function, the difficulty of nondifferentiability of maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem subject to an area constraint including the whole microstructure. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructure, while reducing the objective function and maintaining smooth design boundaries. In the numerical examples, the optimal shapes obtained for minimization of the maximum local stress of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.

Shape Sensitivity of Eigenvalue Functionals for Scalar Problems: Computing the Semi-derivative of a Minimum

This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators involving a variety of boundary conditions. A uniform treatment of these problems is possible by considering them as a problem of calculating the semi-derivative of a minimum with respect to a parameter. The applicability of this approach, which can be traced back to the works of Danskin [8, 9] and Zolésio [28], to the treatment of eigenvalue problems (where the full shape derivative may not exist, due to multiplicity issues), has been illustrated by Zolésio in [29] (see also [10, Chap. 10] and included references). Despite this, some of the recent literature (see, for example, [1] or [7]) on the shape sensitivity of eigenvalue problems still continue to employ methods such as the material derivative method or Lagrangian methods which seem less adapted to this class of problems. The Delfour–Zolésio approach does not seem to be fully exploited in the existing literature: we aim to recall the importance and the simplicity of the ideas from [8, 28], by applying it to the analysis of the shape sensitivity for eigenvalue functionals for a class of elliptic operators in the scalar setting (Laplacian or diffusion in heterogeneous media), thus recovering known results in the case of Dirichlet or Neumann boundary conditions and obtaining new results in the case of Steklov or Wentzell boundary conditions.

Shape optimization method for strength design problem of microstructures in a multiscale structure

In this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed; minimization of the maximum stress of the microstructure and minimization of the maximum stress of the macrostructure. The homogenization method is used to bridge the macrostructure and the microstructures, and also to calculate the local stress in the microstructures. By replacing the maximum value of the stress with a Kreisselmeier-Steinhauser function, the difficulty of non-differentiability on the maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem with the area constraint including the whole microstructures. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructures, while reducing the objective function and maintaining the smooth design boundaries. In the numerical examples, the optimal shapes obtained for the maximum local stress minimization of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.

Topology optimization of thermoelastic structures using MMV method

Thermoelastic problems refer to a coupled phenomenon existing in a wide range of practical applications where elastic responses are influenced by temperature variation. Meanwhile, Moving Morphable Void (MMV) approach receives extensive attentions owing to explicit topological geometric information. In MMV method, a closed B-spline curve defined by several design variables is adopted to describe void material in design domain. In present study, we intend to use the MMV method to topologically optimize the thermoelastic structures under uniform temperature increment and varying temperature field. The weak formulations of heat conduction and elasticity are combined to describe the thermoelastic problems. The optimization mathematical model with the aim of compliance minimization and volume constraint is constructed. The sensitivity of design variables is derived by material derivative method and adjoint sensitivity analysis method. The method of moving asymptote (MMA) algorithm is implemented to update the design variables for finding the best material distribution. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Simultaneous shape and topology optimization method for frame structures with multi-materials

In this study, we present a novel non-parametric shape-topology optimization method for frame structures with multi-materials, aiming at designing more light and stiff frame structures. The sum of squared error norm for achieving the target displacements on the specified members is minimized under the volume constraints of multi-materials as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the material gradient functions for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The generalized solid isotropic material with penalization (GSIMP) method is employed to classify into the multi-materials in topology optimization. The gradient functions derived are applied to the unified H1 gradient method for frame structures to determine the optimal shape and material variations. With this method, the optimal free-form and topology for arbitrary large-degree of design freedom frame structures with multi-materials can be obtained without any shape and topology parameterization. Numerical examples with various materials and different boundary conditions are demonstrated and the results are discussed.

Free-form optimization method for frame structures aiming at controlling time-dependent responses

In this paper, a parameter-free optimization method for controlling the time-dependent response of frame structures is proposed. The proposing method deals with three optimization problems, including a vibration displacement minimization problem, a vibration displacement control problem, and a dynamic compliance minimization problem, which is the special case of the vibration minimization problems. These time-dependent optimum design problems are formulated as distributed-parameter optimization problems and the sensitivity functions for the shape variation to obtain the optimal shape are derived based on the material derivative method, Lagrange multiplier method, and the adjoint method. The derived sensitivity function is applied to the H1 gradient method for frames, which is a gradient method in the function space to determine the optimal shape variation. With the proposed method, the optimal shape for time-dependent response problems such as a forced vibration, a free vibration, or a transient response is obtained while minimizing the objective functional and maintaining the smoothness of a frame structure. Several numerical design examples including a continuous dynamic force or an impulse force as an input force are shown to verify and validate the effectiveness of the proposed method and the results are discussed.

A post-processing method to remove stress singularity and minimize local stress concentration for topology optimized designs

Topology optimization is becoming a popular choice in designing components as it can reduce a component's weight while mostly maintaining its performance. In the designs from a discrete topology optimization, the boundaries are usually jagged and post-processing is required to smooth such boundaries. A post-processing method is presented in the paper to smooth the jagged boundaries. The post-processing method includes a boundary identification algorithm that identifies the external boundaries and internal boundaries (voids). The external boundaries can be further separated into the optimized jagged boundaries to be smoothed and the originally existing boundaries to be kept. A spline fitting method was then applied to smooth the jagged boundaries. For internal voids, a void replacement method was developed to replace the polygonal voids to eliminate the sharp corners and stress singularity around the voids. A novel location optimization method was developed to optimize the void locations with the objective of minimizing the stress concentration around voids. The sensitivity of the maximum stress around all voids was calculated. The p-norm function that approximates the maximum function and the material derivative method that calculates the domain derivative were used to obtain the sensitivity. A gradient-based solver was applied to solve the optimization problem and all voids were moved simultaneously during the line search. Case studies demonstrated that the developed approach can effectively smooth the jagged boundaries and minimize the stress concentration around the internal voids.

Optimum shape design of thin-walled cross sections using a parameter-free optimization method

In this paper, a parameter-free method is presented for optimizing the cross-sectional shape of thin-walled structures, which are often demanded at the early stage in structural designs. The thin-walled cross section is minimized subject to constraints of sectional properties including torsion constant, moment of inertia of area, centroid and shear center of the cross section. The problem is formulated as a distributed shape optimization problem, and the shape gradient function is derived using the Lagrange multipliers and the material derivative method. The H1 gradient method, which was proposed as a gradient method in a Hilbert space, is applied to determine the smooth optimal shape. The constraint conditions are satisfied using a linearised constraint equation. The validity of this parameter-free method is verified through several design examples for obtaining the optimal shape of a thin-walled cross section under the constraints of sectional properties.

Sensitivity analysis of a shape control problem for the Navier--Stokes equations

We deal with a sensitivity analysis of an optimal shape control problem for the stationary Navier-Stokes system. A two-dimensional channel flow of an incompressible, viscous fluid is examined to determine the shape of a bump on a part of the boundary that minimizes the viscous drag. By using the material derivative method and adjoint variables for a shape sensitivity analysis, we derive the shape gradient of the design functional for the model problem.

Reaction forces control design of shell structures in plastic range based on free-form optimization method

In this study, we propose an optimization method for the reaction forces control problem of shell structures in plastic range. The sum of squared error norms between reaction forces in the plastic range and the target forces are minimized under a volume constraint. The total strain theory is employed for simplicity under the assumption that the considered force or enforced displacement increases monotonically, so that the final deformation is path-independent. The optimum shape design problem is formulated as a distributed-parameter system, and we assume that a shell is varied in the out-of-plane direction to the surface whereas the thickness is not varied during the shape change. The shape gradient function and the optimality conditions for this problem are derived theoretically using the material derivative method and the adjoint variable method. The shape gradient function calculated from a non-linear finite-element analysis is applied to the H1 gradient method for shells proposed by one of the authors to determine optimum shape variation. The optimal shape of a shell structure can be obtained without shape parameterization while maintaining surface smoothness. Numerical examples are presented to demonstrate the validity and practical utility of the proposed free-form optimization method.

Simulation of the Continuous Casting and Cooling Behavior of Metallic Glasses.

The development of melt spinning technique for preparation of metallic glasses was summarized. The limitations as well as restrictions of the melt spinning embodiments were also analyzed. As an improvement and variation of the melt spinning method, the vertical-type twin-roll casting (VTRC) process was discussed. As the thermal history experienced by the casting metals to a great extent determines the qualities of final products, cooling rate in the quenching process is believed to have a significant effect on glass formation. In order to estimate the ability to produce metallic glasses by VTRC method, temperature and flow phenomena of the melt in molten pool were computed, and cooling rates under different casting conditions were calculated with the simulation results. Considering the fluid character during casting process, the material derivative method based on continuum theory was adopted in the cooling rate calculation. Results show that the VTRC process has a good ability in continuous casting metallic glassy ribbons.

Tailoring static deformation of frame structures based on a non-parametric shape–size optimization method

In this study, a non-parametric shape–size optimization method is developed for tailoring the static deformation of large-scale frame structures. This deformation control design is one of the important problems in the stiffness design of frame structures, and enables us to create a smart or a high performance structure for a specific ability of deformation. As the objective functional, we introduce the sum of squared error norms for achieving the desired displacements on specified members, and assume that each frame member varies in the off-axis direction with changing cross sections. The shape gradient function, the size gradient function, and the optimality conditions for this problem are theoretically derived with the Lagrange multiplier method, the material derivative method, and the adjoint variable method. The optimal shape–size variations that minimize the objective functional are determined by using the H1 gradient method for frame structures. With the proposed method, the optimal arbitrarily formed frame structures with the optimal cross sections can be obtained without any shape and size parameterization while maintaining their smoothness. The validity and practical utility of this method for tailoring the static deformation of frame structures are verified through design examples.

Free-form optimization of sandwich structures for controlling thermal displacement

Considering the different thermal expansion coefficients of skins and cores, sandwich structures whose design optimization is performed have the potential for application in thermal actuators. In the present work, we propose a free-form optimization method for the design optimization of sandwich structures composed of two thermoelastic materials (A and B) for controlling their thermal displacements. We use the square displacements error norm between the thermal displacement and target displacement at the designated boundary as the objective function, and minimize it by optimizing the interface and the surface shapes of the sandwich structures under thermal load. We derive the shape gradient function by using the material derivative method and apply it to the free-form optimization method for determining the optimal shape. Two volume constraints, the total volume and volume of material A, are considered in the problem formulation. The numerical results show that the thermal deformation of the optimal sandwich structure can be well identified; this is helpful for applying sandwich structures in thermal actuators.

異種材料からなる複合構造体の形状最適設計

Designing composite structures to satisfy specific requirements has been an important topic for a long time. In this study, we review a free-form optimization method for the optimal shape design of composite structures composed of dissimilar materials in terms of the stiffness problem, the shape identification problem, and the vibration control problem. The free-form optimization method is based on the H1 gradient method, in which the shape gradient functions are theoretically derived using the material derivative method, and applied to the design boundaries as traction forces. We set the interface and surface of composite structures as the design boundaries, and consider the volume constraints of the dissimilar materials in each design problem. At last, we give some numerical examples to verify the validity of the proposed free-form optimization method.

固有振動問題における３次元複合構造体の形状最適設計

In this study, we propose a non-parametric optimization method for designing the interface shapes of three-dimensional composite structures composed of dissimilar materials in a natural vibration problem. The problem is defined as that the objective functional consists of a set of specified vibration eigenvalues with weighting coefficients is maximized for specific requirements. The shape gradient function is derived using the material derivative method, and is applied in shape update based on the H1 gradient method. Two design problems of a clamped-clamped beam composed of dissimilar materials in order to maximize the fundamental eigenvalue and maximize the difference between the fundamental and second eigenvalues are carried out. The optimal results show that the proposed interface shape optimization method can successfully maximize the objective functional in each design problem.

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.

Interface shape optimization of designing functionally graded sandwich structures

Functionally graded materials (FGMs) have been proposed to be potential structural materials applied in high-speed spacecraft and power generation industries. In this study, an interface shape optimization method for designing FG sandwich structures with two different materials is proposed to minimize the compliance of FG sandwich structures under the volume constraint. Using the material derivative and adjoint methods, the shape gradient function is derived to determine the optimized interface shapes between different materials in FG sandwich structures without requiring shape design parameterization. FG sandwich structures with two sets of metal–ceramic materials, aluminum–alumina and aluminum–zirconia, are given as examples to verify the validity of the proposed optimization method, respectively. The results show that the compliance of FG sandwich structures with large difference between Young’s modulus of the component materials can be significantly reduced after optimized by the proposed method.

J1240204 H^1勾配法を利用したスポット溶接位置の最適化

Spot welding is widely used for manufacturing sheet metal structures such as automobiles. This paper presents an optimization method for determining the welding positions. The spot welding is assumed to be movable in the in-plane direction of the sheet metal structure, or the tangential direction to the plate surface. The squared error norm of maximum principal stress around the spot welding is defined as the objective functional. The optimization problem is formulated as a distributed-parameter shape optimization problem, and the shape gradient function is derived for this problem using the material derivative method. The optimal position is determined by applying the shape gradient function derived to the H^1 gradient method. We confirm the proposed method is effective for designing the welding position of sheet metal structures through design examples.