Zero density lower bounds in topology optimization

A domain evolution method to suppress the eigenfrequency and eigenmode errors caused by low-density elements in dynamic topology optimization

This paper focuses on modeling of axial symmetrical bodies which is formalized and generalized by the approach of topological optimization. Both constant velocities of their striking the water surface and arbitrary angles of water-entry are utilized to topological design within non-regular design domain and on the fixed loading surface. The topological optimization is based on the adoption of the solid isotropic microstructure with penalty (SIMP) approach and minimum compliance as the design objective subjective to volume constrain. The initialization of variables is the ratio of material volume limited to the total volume; and the values of variables out of the design domain always evaluate zeros. Numerical examples of water-entry bodies are investigated based on the topology optimization method. The optimum models with actual loads presented and the SIMP models are proved to be reliable and practical at the conceptual design phase of axisymmetric bodies on water entries by applying the topological optimization to a hemispheroid, wedge, and plate.

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 :

Stress-based topology optimization and nonlinear structural topology optimization is gaining increasing attention in order to make topology optimization more realistic. Thus, this paper extends current concepts of topology optimization to the design of structures made of nonlinear materials. An extended bi-directional evolutionary structural optimization (BESO) method for stress minimization topology optimization of material nonlinear structures is proposed in this work. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low-density elements. The maximum von Mises stress is approximated by the p-norm global stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topological variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several 2D benchmark design problems.

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.

Compared with traditional metal-materials, fiber-reinforced composite materials have better performance in many aspects such as strength, stiffness, and fracture resistance. At present, fiber-reinforced composite materials have been widely used in automotive, aerospace, and other industrial fields. This paper proposes a topology optimization method for solving the fundamental frequency maximization of undamped free vibration of continuous fiber-reinforced composite structures. To achieve the simultaneous optimization of the structural topological configuration and the fiber angle. A dynamic topological optimization model is established with the permitted material usage as the constraint and the structure's first-order eigenvalue as the objective function. The model includes density design variables that characterizes the topological configuration of the structure and angular design variables that characterizes the fiber orientation. The analytical sensitivity formulas of the objective function of eigenvalue with respect to density design variables and angle design variables are derived in detail, and the method of moving asymptotes (MMA) is used to solve the optimization problem. Finally, three numerical examples are performed to verify the effectiveness of the proposed method, which includes a static optimization example with the stiffness maximization as the goal and two dynamic optimization examples with the first-order eigenvalue maximization as the goal. The results show that the proposed method can achieve a stable iterative history and fast convergence, and can effectively improve the structural frequency while achieving the integrated optimization of the structural topological configuration and the fiber angle.

Exploring the influence of initial design domain dependencies in concurrent multiscale topology optimization for heat conductivity maximization

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

Topology optimization is the procedure of determining the connectivity, shape and location of voids inside a given design domain. This chapter proposes a quantum-inspired evolutionary algorithm (QEA) for structural topology optimization. It reviews the formulation of the continuum structural topology optimization model and the characteristics of quantum computing. The chapter also proposes the detailed implementation of a QEA-based topology optimization. It discusses two issues: the density-based continuum structural topology optimization formulation and the characteristics of quantum computing, which may be conducive to reducing the computational expense of topology optimization. The entire theoretical procedure of quantum-inspired topology optimization, which mainly combines FEM analysis, design sensitivity solution and quantum-inspired search, is accomplished for continuum structures. As the experimental implementation of quantum computation needs initialization, coherent manipulation, control and read of the fragile quantum system, practically building quantum computers has proved extremely difficult.

Isogeometric analysis (IGA) connects CAE with standard CAD design tools and has been utilized for a decade in structural optimization. FEA has a number of drawbacks, including a lack of linkage between analytic and geometric models, a shortage of continuity between components, and a lack of accuracy for high-order elements. Isogeometric analysis effectively overcomes these obstacles. This paper proposes the isogeometric topological optimization mathematical model depending on the asymmetric constitutive relationship of tension and compression microscopic materials. We provide a new density-based topology optimization formulation in which the design domain is limited to the B-spline space. A rectangular domain with a tensor-product is embedded in a design domain that is arbitrarily shaped. The density field is represented using B-splines. The intrinsic filter provided by the B-spline representation for topology optimization is governed by the degrees of B-spline and the knot spans. This B-spline effectively removes numerical artifacts and controls the minimum feature-length in; when the B-spline basis functions span many analysis elements. Benchmark problems in structural topology optimization are computed through Evolutionary Structural Optimization (ESO) algorithm. Comparisons of the optimal structure topologies using finite element analysis and isogeometric collocation methods are analyzed. The influence of different tension-compression stiffness ratios on the final structure topologies is listed and analyzed. The numerical results show the reliability and performance of the proposed mathematical formulation and asymmetry of natural structure.

With the rapid advancement of computing technology, mathematical modeling and computer simulations have become common and standard methods for design of complex systems. To exercise these models intelligently and eliminate the burden of manual iteration, manipulating inputs and reviewing outputs, optimization strategies are applied in a simulation-based design environment. In reality, the knowledge that engineers have about the design problem is imperfect and incomplete, uncertainty always exists in simulation-based design. The conventional deterministic optimization techniques do not consider the impact of such uncertainties. If the system responses driving the design decisions are very sensitive to these uncertainties, serious problems might confront the designer. Performance of the actual system may differ greatly from expected optimum performance. Again, if the design solution is near one or more constraint boundaries, then even slight uncertainties or changes in the operating environment could produce failed, unsafe designs, and/or result in substantial performance degradation. In recent years, the concept of reliability and robust design has become very popular. Probabilistic design analysis and statistical modeling allow the identification of designs that qualify as not only feasible, but as consistently feasible in the face of uncertainty. Reliability based topology optimization (RBTO) extends the reliability notion to the area of structural topology optimization. Although reliability based design optimization (RBDO) is a well established research area, very few work has been presented in the area of RBTO. In this study, we give an overview of the various works done in the field of RBTO. Ever since Bendsoe and Kikuchi introduced the topology optimization using a homogenization method, many topology optimization methods have been developed for the both linear and nonlinear structures. Traditional topology optimization methods drive the topology of a structure to an optimum design based on the constraint on mass. Kharmanda et al. introduced reliability constraints in deterministic topology optimization problem. They proposed a heuristic strategy that aims to reduce mass while improving the reliability level of the structure without greatly increasing its weight. But the limit state function used by them was not based on failure criteria for the structure. Their formulation considered uncertainty with respect to ∗Design Automation Laboratory †Advisor geometrical dimension and applied load only. Also their reliability analysis seems to be independent of the boundary and loading condition, so their results showed similar values for the uncertain variables for different structures. Tovar et al. have developed the Hybrid Cellular Automaton (HCA) method for structural synthesis of continuum material where the state of each cell is defined by both density and strain energy. In Agarwal, a decoupled RBDO approach is employed such that the topology optimization is separate from the reliability analysis. Patel et al. showed the use of RBTO using the gradient free Hybrid Cellular Automata (HCA) method. Their formulation incorporates uncertainty with respect to material property also. They considered limit state function based on failure modes on the output displacements. This study aims at giving a comparative study on these methods. It points out the advantages and disadvantages of the methods, and their significance in the field of RBTO. As large numbers of design variables are associated with continuum topology optimization problems, RBTO methods are inherently computationally expensive because of additional required system analysis associate with RBDO. Based on the observations, this study presents an efficient method for incorporating reliability analysis in topology optimization.

Topology optimization of bounded acoustic problems using the hybrid finite element-wave based method

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.

A method is developed to systematically remove and reintroduce low density elements from and into the finite element mesh on which the structural topology optimization problem is defined. The material density field which defines the topology and the local ‘stiffness’ of the structure is optimally distributed via non‐linear programming techniques. To prevent elements from having zero stiffness, an arbitrarily small lower bound on the material density is typically imposed to ensure that the global stiffness matrix does not become singular. While this approach works well for most minimum compliance problems, the presence of low density elements can cause computational problems, particularly in structures that exhibit geometric non‐linearities, e.g. in compliant mechanisms. To resolve this problem, a systematic approach for removing and reintroducing low density elements is presented, and the substantial performance improvements both in design and computational efficiency of the method over current methods are discussed. Several structures and compliant mechanisms are designed to demonstrate the method. Copyright © 2003 John Wiley & Sons, Ltd.