자기장 내의 위상최적화 방법에 대한 연구 - 밀도법과 균질화법의 비교 -

The homogenization design method has been successfully applied to obtain the optimal topology of a structure under electrical fields as well as various mechanical environments. This paper introduces the expansion of the application of the homogenization design method into the structure in magnetic fields. The topology optimization on a device in magnetic fields is focused on the maximizing the magnetic flux in an air-gap. This is accomplished by maximizing the magnetic mean compliance defined by the magnetic energy in a given region of the device. The homogenization design method is applied to three-dimensional cases with and without considering the saturation effect of the material. Results show that the homogenization design method is valid to maximize the vector potential in linear cases and the magnetic flux in non-linear cases.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

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.

An approach for the concurrent homogenization-based microstructure type and topology optimization problem

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.

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.

A robust topological derivative-based multi-material optimization approach: Optimality condition and computational algorithm

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.

A globally convergent method to accelerate topology optimization using on-the-fly model reduction

<div class="htmlview paragraph">In various practical structural design problems, decision of a layout of material or stiffeners in the given design space satisfying some design constraints is the most difficult task for design engineers. Also, they must consider many possible cases for safety, performance and some other requirements. But it is very difficult to solve this type of layout (topology) problem by any conventional structural optimization techniques because its mathematical modeling of topology is so complex. To solve this problem, we employed the new topology optimization technique using the homogenization method that was recently introduced by Bensøe and Kikuchi. The method gives the minimum compliance design under certain material constraint. This technique was also extended to three dimensional shells and three dimensional solids by Suzuki and Kikuchi.</div> <div class="htmlview paragraph">In this paper, shape and topology optimization problem under multi-loading conditions is discussed. Various car body design problems are solved to justify the validity and the strength of the present approach for engineering applications.</div>

3R25. Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, Vol 146. - G Allaire (Center of Appl Math, Ecole Polytechnic, Paliseau Cedex, 91128, France). Springer-Verlag, New York. 2002. 456 pp. ISBN 0-387-95298-5. $79.95.Reviewed by MS Qatu (Ford Motor Co, EVB Bldg, MD X8, 20800 Oakwood Blvd, Dearborn MI 48124).This is mainly a research monograph on the theory and application of the homogenization method in shape optimization problems. Applications given in the book are in the fields of composite materials, conductivity, and elasticity with emphasis on the minimization of compliance. The audience for this book includes researchers and advanced graduate students in applied mathematics and engineers interested in the design of shapes of structures for conductive and elastic purposes. It also fits very well for material scientists with inclination towards theoretical aspects of design and analysis of composite materials. Based on the theoretical content and examples in the book, the word shape in the title should rather be understood as topology. In some engineering fields, shape is more related to boundary perturbation, however, the methods presented in this book are designed to define the boundary, rather than to perturb it. This concept of defining the boundary is closer to the idea of topology optimization as a material distribution problem. The book is divided into five chapters. The first chapter covers the subject of homogenization in quite details. Periodicity is first assumed and used to derive homogenized operators in elasticity by means of the two-scale asymptotic expansion. Definitions and proofs of H- and G-convergence are then presented in a rigorous way. The second chapter is titled mathematical modeling of composite materials. In this chapter, Allaire explains carefully the G-closure problem and its interpretations in the mechanical setting. Finding bounds on elastic properties of bi-phase materials is an old problem in mechanics that can be explained and treated mathematically with the help of the G-closure theory. Rank laminated materials are used to demonstrate the bounds derived in the book. Optimal design of materials under electric conductivity considerations is addressed in the third chapter. With an extensive number of proofs, this chapter explains the theory behind the computation of optimal topology (shape) using homogenization theory as a mean to relax the optimization problem. Composite materials are used to relax the problem, and homogenization of conductivity tensors is then used to go from an ill-posed problem to a well-posed one. Structural optimization has been an important subject of research since the days of Lagrange and Saint Venant. This chapter is the counterpart of the previous one, this time, in the field of elastic structures. Composite materials are again introduced to relax the optimization problem and homogenized elastic properties are computed in order to attain a solution to the problem. The minimization of compliance (maximization of stiffness) is addressed intensively in this chapter. The fifth and last chapter is titled Numerical Algorithms. Academic two-dimensional rectangular domains using quadrilateral elements in regular meshes are used to exemplify the theory presented in the previous chapters. Examples include minimization of compliance and maximization of eigenvalues. Multiple loadings and optimum topologies (shapes) of complaint mechanisms were considered. In summary, Shape Optimization by the Homogenization Method is an excellent book that complements very well the extensive literature in the field, including Sanchez-Palencia and Zaoui [1], Bendsøe [2], Cherkaev [3], and Bendsøe and Sigmund [4].

Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

Frequency- and temperature-dependent properties of viscoelastic materials have a significant effect on vibration characteristics of constrained layer damping (CLD) structures. In this article, parametrized level set method (PLSM) is extended to solve the topology optimization problem of CLD plates, in which the viscoelastic material with frequency- and temperature-dependent characteristics is employed as damping core. The single/weighted modal loss factors, which are solved by an iterative method, are defined as objective functions, and topology optimization problem is formulated based on the finite element model for CLD plates which are described by combining parametric level set (PLS) functions. The sensitivities of objective functions with respect to expansion coefficients in PLS are calculated by the use of adjoint vector method. As the optimized results with complex constant shear modulus for viscoelastic core indicates, the proposed method is effective for topology optimization of CLD structures and insensitive to initial design. Numerical examples with the consideration of frequency- and temperature-dependent characteristics for “soft” and “hard” viscoelastic core are implemented. Through comparing the optimal results with different models for viscoelastic core, the influences of frequency- and temperature-dependent characteristics for viscoelastic core on natural frequencies, modal loss factors and optimized layouts are analyzed. Similarity indices are defined for distinguishing the difference among optimized layouts and initial design induced by modal strain energy contour. The optimal design dependence on layer thicknesses is further discussed with additional mass of CLD materials unchanged.

In this paper we consider the classical minimum compliance topology optimization in linear elasticity. In our approach, the objective is to minimize a cost functional that comprises the work of applied body and traction forces obtaining simultaneously the optimal topology and the optimal localization for proportional actuators on the concerning structural domain. A proportional actuator is considered as a spring and one of the objectives is to obtain the optimal local to couple it on the structure. The problem is treated as a topology optimization problem where a given cost functional must be minimized under suitable constraints for external forces, control energy, volume restriction and a set of project variables. The topology optimization procedure here performed uses the solid isotropic material with penalization approach, based on the concept of optimizing the material distribution. The solution of the optimization problem is obtained by using a sequential linear programming and the finite element method is used to discretize the design domain. Several numerical examples are presented showing the effectiveness of the proposed approach.

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 :