Topology optimization with multiple materials via moving morphable component (MMC) method

Zhang, Weisheng; Song, Junfu; Zhou, Jianhua; Du, Zongliang; Zhu, Yichao; Sun, Zhi; Guo, Xu

doi:10.1002/nme.5714

Cited by 128 publications

(54 citation statements)

References 29 publications

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“…Also recently, the work in Ref. [34] extends the moving morphable components method to accommodate multiple materials by seeding the initial design with geometric components made of different materials (i.e., the material of a component does not change during the optimization). The material interpolation scheme in regions where two or more geometric components intersect chooses the stiffer material of any intersecting component.…”

Section: Introductionmentioning

confidence: 99%

Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

Kazemi

Vaziri

Norato

2018

Journal of Mechanical Design

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We present a new method for the simultaneous topology optimization and material selection of structures made by the union of discrete geometric components, where each component is made of one of multiple available materials. Our approach is based on the geometry projection method, whereby an analytical description of the geometric components is smoothly mapped onto a density field on a fixed analysis grid. In addition to the parameters that dictate the dimensions, position, and orientation of the component, a size variable per available material is ascribed to each component. A size variable value of unity indicates that the component is made of the corresponding material. Moreover, all size variables can be zero, signifying the component is entirely removed from the design. We penalize intermediate values of the size variables via an aggregate constraint in the optimization. We also introduce a mutual material exclusion constraint that ensures that at most one material has a unity size variable in each geometric component. In addition to these constraints, we propose a novel aggregation scheme to perform the union of geometric components with dissimilar materials. These ingredients facilitate treatment of the multi-material case. Our formulation can be readily extended to any number of materials. We demonstrate our method with several numerical examples.

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Section: Introductionmentioning

confidence: 99%

Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

Kazemi

Vaziri

Norato

2018

Journal of Mechanical Design

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“…When level set description with extended finite element model is used, the jump term must be calculated exactly. However, this term can be smeared out when the ersatz material is adopted for finite element analysis …”

Section: Sensitivity Analysismentioning

confidence: 99%

“…However, this term can be smeared out when the ersatz material is adopted for finite element analysis. 44 Since the ersatz material is adopted in this work, which will be described in Section 5.3 the jump term is temporarily ignored in the following sensitivity analysis. To obtain the shape derivatives, the adjoint method are employed and the Lagrangian is defined as…”

Section: Sensitivity Analysismentioning

confidence: 99%

A level set–based method for stress‐constrained multimaterial topology optimization of minimizing a global measure of stress

Chu

Xiao

Gao

et al. 2018

Numerical Meth Engineering

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Summary In multimaterial topology optimization of minimizing a global measure of stress, the maximum stresses in different materials may not satisfy the strength design requirements simultaneously if stress constraints for different materials are not considered. In this paper, a level set–based method is presented to handle the stress‐constrained multimaterial topology optimization of minimizing a global stress measure. Specifically, a multimaterial level set model is adopted to describe the structural topology, and a stress interpolation scheme is introduced for stress evaluation. Then, a stress penalty‐based topology optimization model is presented. Meanwhile, an adaptive adjusting scheme of the stress penalty factor is employed to improve the control of the local stress level. To solve the stress‐constrained multimaterial topology optimization problem minimizing the global measure of stress, the parametric level set method is employed, and the sensitivity analysis is carried out. Numerical examples are provided to demonstrate the effectiveness of the presented method. Results indicate that multimaterial structures with optimized global stress can be gained, and stress constraints for different materials can be satisfied simultaneously.

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“…However, unlike DMO, this method employs optimization constraints to ensure these size variables are 0 or 1, and to guarantee that each component has at most one material with a size variable of 1. The moving morphable component method has been applied to the design of multi-material structures [41] with geometric components made of different materials; however, unlike all of the aforementioned multi-material methods, the choice of material for each component is fixed and therefore not part of the optimization. This work focuses on the topology optimization of multi-material lattice structures using the geometry projection method.…”

Section: Introductionmentioning

confidence: 99%

Multi-material topology optimization of lattice structures using geometry projection

Kazemi

Vaziri

Norato

2020

Computer Methods in Applied Mechanics and Engineering

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This work presents a computational method for the design of architected truss lattice materials where each strut can be made of one of a set of available materials. We design the lattices to extremize effective properties. As customary in topology optimization, we design a periodic unit cell of the lattice and obtain the effective properties via numerical homogenization. Each bar is represented as a cylindrical offset surface of a medial axis parameterized by the positions of the endpoints of the medial axis. These parameters are smoothly mapped onto a continuous density field for the primal and sensitivity analysis via the geometry projection method. A size variable per material is ascribed to each bar and penalized as in density-based topology optimization to facilitate the entire removal of bars from the design. During the optimization, we allow bars to be made of a mixture of the available materials. However, to ensure each bar is either exclusively made of one material or removed altogether from the optimal design, we impose optimization constraints that ensure each size variable is 0 or 1, and that at most one material size variable is 1. The proposed material interpolation scheme readily accommodates any number of materials. To obtain lattices with desired material symmetries, we design only a reference region of the unit cell and reflect its geometry projection with respect to the appropriate planes of symmetry. Also, to ensure bars remain whole upon reflection inside the unit cell or with respect to the periodic boundaries, we impose a no-cut constraint on the bars. We demonstrate the efficacy of our method via numerical examples of bulk and shear moduli maximization and Poisson's ratio minimization for two-and three-material lattices with cubic symmetry.

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Topology optimization with multiple materials via moving morphable component (MMC) method

Cited by 128 publications

References 29 publications

Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

A level set–based method for stress‐constrained multimaterial topology optimization of minimizing a global measure of stress

Multi-material topology optimization of lattice structures using geometry projection

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