Velocity field level‐set method for topological shape optimization using freely distributed design variables

Structural topology optimization.- On basic properties of Michell's structures.- Validation of numerical method by analytical benchmarks and verification of exact solutions by numerical methods.- Introduction to shape and topology optimization.- Homogenization method for shape and topology optimization.- Level set method for shape and topology optimization.- Compliance minimization of two-material elastic structures.- Some notes on topology optimization of vibrating continuum structures.- Topology optimization of vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps.- On optimum shape design and periodicity of band-gape beam structures.- Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration.- Topological design for minimum sound emission from bi-material structures subjected to forced vibration.- Discrete material optimization of vibrating laminated composite plates for minimum sound emission.- Optimization of diffusive transport problems.- Fluid topology optimization: stokes and Navier-stokes models.- Topology optimization of coupled multi-physics problems.- Topology optimization based on the extended finite element method.- Topology optimization of meso- and nano-scale problems.- Topology optimization under uncertainty.- A brief review of numerical methods of structural topology optimization.- The free material design in planar elasticity.

SUMMARYA method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated. The original discrete‐valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain. With the use of the Shepard family of interpolants, this density field is mapped onto the design space defined by a finite number of nodal density variables. The employed interpolation scheme has an explicit form and satisfies range‐restricted properties that makes it applicable for physically meaningful density interpolation. Its ability to resolve more complex spatial distribution of the material density within an individual element, as compared with the conventional elementwise design variable approach, actually provides certain regularization to the topology optimization problem. Numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques. Copyright © 2011 John Wiley & Sons, Ltd.

Free-mounted cooling plate multi-objective topology optimization method towards precision machine tool heat dissipation: An experimental and numerical study

Topology optimization with closed B-splines and Boolean operations

The advances of manufacturing techniques, such as additive manufacturing, have provided unprecedented opportunities for producing multiscale structures with intricate latticed/cellular material microstructures to meet the increasing demands for parts with customized functionalities. However, there are still difficulties for the state-of-the-art multiscale topology optimization (TO) methods to achieve manufacturable multiscale designs with cellular materials, partially due to the disconnectivity issue when tiling material microstructures. This paper attempts to address the disconnectivity issue by extending component-based TO methodology to multiscale structural design. An effective linkage scheme to guarantee smooth transitions between neighboring material microstructures (unit cells) is devised and investigated. Associated with the advantages of components-based TO, the number of design variables is greatly reduced in multiscale TO design. Homogenization is employed to calculate the effective material properties of the porous materials and to correlate the macro/structural scale with the micro/material scale. Sensitivities of the objective function with respect to the geometrical parameters of each component in each material microstructure have been derived using the adjoint method. Numerical examples demonstrate that multiscale structures with well-connected material microstructures or graded/layered material microstructures are realized.

Topological shape optimization of geometrically nonlinear structures using level set method

Random variable of thermo-mechanical coupling (TMC) structure is analyzed first, then the topology optimization model of TMC compliant mechanism is established. According to the uncertainty factors such as temperature, load, volume ratio and thermal expansion coefficient in the process of topology optimization, the robust reliability optimal design method is introduced into the topology optimization process. The various factors on the sensitivity of topology optimization process is analyzed, the best match between various factors is find out, which make the product has robust reliability for the change of the uncertainty. The residual analysis for the most sensitive factors is carried out. Finally the correctness and validity of the method is verified with an example. Introduction When we study the compliant mechanisms by adopting topological optimization method, we only need to give design domain and allocate input and output position, no need from the perspective of a known rigidity. The mechanism come from this method have a input and output relationship of optimized “force—displacement”, thus it attracts people’s attention. Topology optimization research is developing from certain topology optimization direction to reliable topology optimization direction. Jinqing Zhan proposed a reliable topology optimization method of compliant mechanism based on infrastructure method. KHARMANDA proposed a reliable topology optimization method. Zhaokun Li proposed a geometric nonlinear topology optimization research method based on reliable compliant mechanism. Their research turned out that the mechanism based on reliable topology optimization is more reasonable than the mechanism based on certain topology optimization. Yongcong Kuang build a level set function to design variable, stiffness minimization, and proposed corresponding optimization algorithm. Mingtao Cui established a monolithic compliant mechanism multi-target reliable topology optimization design dual mathematical model. This paper combines topology optimization process of compliant mechanism and robust reliability optimal design, analyzes the sensitivity of uncertainty random variables such as design domain’s size, external load and temperature in topology optimization process. Then get the best combination among these random variables. Random Variable Coupling Field Analysis Hot objects will produce thermal expansion, thus caused by the thermal deformation and displacement of combined action of mechanical load and temperature load structure called a hot solid coupling structure, the finite element balance equation :

The objective of this investigation is to carry out the topology design optimization of the pickup suspension of high-speed optical pickups in DVD’s or CD ROM’s. The specific design target here is to increase the torsional eigenfrequency of the pickup as high as possible while keeping the focusing eigenfrequency between 25 Hz and 35 Hz, and the tracking eigenfrequency between 43 Hz and 57 Hz. These frequency ranges are set to meet other design constraints such as servo-controllability. A typical pickup consists of a bobbin having an optical lens and suspension plates to support it. Since the eigenfrequencies of the bobbin itself are in the order of 10 kHz, it can be modeled as a point mass when dealing with the low-frequency dynamic characteristics in consideration. These modeling techniques are verified in the previous researchers. In modeling the bobbin, we also consider its moments of inertia. In designing the suspension system, damping bond is usually applied to control the quality factors at the focusing and tracking frequencies. The location of the damping treatment on the suspension system may be determined by an optimization method, but this is not pursued in this work since the result by the earlier work can be applied to the present suspension system. Unlike static problems such as compliance minimization problems with single and multiple loads or compliant mechanism problems with single and multiple input/outputs, optimization problems involving eigenvalues suffer from mode-switching, which often causes serious difficulties in solution convergence. Mode-switching represents the phenomenon that eigenmodes are changed during the update of design variables. If the design target were simply to maximize a few lowest eigenfrequencies, the mode-switching problem would be handled rather easily. However, the present optimization problem is not the case because we must follow the eigenfrequencies of the specific modes such as tracking, focusing and twisting modes during the optimization process. To handle this situation satisfactorily, we should employ a technique to track the desired modes. In the present investigation, the desired modes are traced by means of the modal assurance criterion (MAC), which has been widely used in experimental modal analysis. For the pickup consisting of a bobbin and suspension plates, the present design goal can be achieved by optimizing the configuration of the suspension plates. To find an optimal plate configuration, we employ the topology optimization method, but the specific method employed here is the multiscale topology optimization method. In the multiscale optimization method, the density design variables are transformed into multiscaled design variables by a series of nonlinear and linear transformations. With the multiscale topology optimization method, the optimization can be completed more efficiently than the optimization by 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization 4-6 September 2002, Atlanta, Georgia AIAA 2002-5483 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. the standard single-scale direct density method. As an optimizer, the feasible direction method is employed to handle effectively multiple constraints on eigenfrequencies. By the present application of the topology optimization method, the twisting eigenfrequency has been maximized while the focusing and tracking eigenfrequencies are kept in the prescribed frequency range. The optimized suspension plate in the present work outperforms the designs improved from the existing rectangular bend-type plates.

This work presents the framework to optimally design a cantilever for torsion mode frequency maximization. A cantilever design problem is formulated by topology and shape optimization methods. The torsion mode frequency is selected as an objective function, and the volume of the cantilever and the first bending mode frequency are constrained. Two optimization problems are defined and sequentially solved for the specific values. A new idea in this work is using a final topology obtained in the topology optimization problem as an initial shape in the shape optimization problem. The torsional mode frequency of the optimized cantilever is well improved in comparison with a nominal cantilever.

Piecewise length scale control for topology optimization with an irregular design domain

In this paper, an explicit and implicit hybrid topology optimization method is proposed for the design of multiple materials structure. The explicit topology optimization employs the moving morphable component (MMC) method to determine where the solid material is present within the design domain. The implicit topology optimization employs the solid isotropic material with penalization (SIMP) method to identify material type within the solid material region. The explicit and implicit topology optimization methods are combined through a surrogate material model, resulting in a new hybrid topology optimization framework known as the MMC–SIMP hybrid topology optimization method. The proposed method retains the advantages of both individual optimization methods, allowing for explicit boundary representation and high design freedom in material selection. The element density function and sensitivity analysis are conducted based on two-phase materials topology optimization. Finally, some numerical examples demonstrate the effectiveness of the proposed method.

Structural topology optimization is a crucial approach for achieving lightweight design. An effective topology optimization algorithm must strike a balance between the objective functions, constraints, and design variables, which essentially reflects the symmetry and tradeoff between the objective and constraints. In this study, a topology optimization method grounded in load path theory is proposed. Element bearing capacity is quantified using the element birth and death method, with an explicit formulation derived via finite element theory. The effectiveness in evaluating structural performance is assessed through comparisons with stress distributions and topology optimization density maps. In addition, a novel evaluation index for element bearing capacity is proposed as the objective function in the topology optimization model, which is validated through thin plate optimization. Subsequently, sensitivity redistribution mitigates checkerboard patterns, while mesh filtering suppresses multi-branch structures and prevents local optima. The method is applied for the lightweight design of a triangular arm, with results benchmarked against the variable density method, demonstrating the feasibility and effectiveness of the proposed method. The element bearing capacity seeks to homogenize the load distribution of each element; the technique in this study can be extended to the optimization of symmetric structures.

A desirable feature in multidisciplinary design optimization (MDO) is to achieve an acceptable preliminary or initial design in an expeditiously fashion. One way to pursue this goal is to describe the structural boundary using a small number of design variables. In this paper we propose an approach to structural optimization that enhances the performance of multidisciplinary design tools because it greatly decreases the number of design variables assigned to structural deflnition. This approach is based on the Level Set Methods and uses the coe‐cients of the Fourier series expansion of the level set function as design variables. The global character of these coe‐cients and the fact that the level set function is real-valued provide the means to reduce the design space dimension. Within the level set framework, the structure is conflned to the region in which the level set function is negative. In this way, the zero level set of this function implicitly describes the structural boundary, which can undergo topological changes during the optimization process as the function evolves. The proposed methodology also averts successive mesh generation. We show that an interface can be described to any desirable accuracy using flnitely many Fourier modes of the signed distance to the interface. The total error is inversely proportional to the number of Fourier modes. We present two validation examples in topology optimization, and use these cases to study the efiects of the methodology parameters on the results. Finally, we propose two examples of application to the enhancement of aircraft roll maneuvers. In the flrst, the goal is to determine the layout of reinforcements in the front spar of a wing-aileron that maximizes the aileron reversal speed, subject to a constraint in the amount of reinforcing material. We also propose the design of a morphing airfoil-aileron conflguration capable of changing its mean camber to maximize the aileron efiectiveness through the activation of a system of actuators, given a restriction on power consumption.

An incremental form interpolation model together with the Smolyak method for multi-material topology optimization

This paper presents an approach for solving the multidisciplinary topology optimization (MTO). To simplifying the description, a three-dimensional (3D) “heat transfer-thermal stress” coupling topology design problem is used as an instance to interpret the solving scheme. Unlike the common multiphysics topology optimization problem which usually modeled in a 3D domain or a 2D domain alternatively, the topology optimization problem mentioned in this paper has a 3D design domain (the design variable is referred as ρ1) and two 2D design domains (the design variable is referred as ρ2and ρ3) together in one mathematical model. Although all the model and solving method are based on a certain design instance, the solving scheme presented in this paper can be used as an efficient method for solving the boundary coupling MTO.