Conformal Topology Optimization of Heat Conduction Problems on Manifolds Using an Extended Level Set Method (X-LSM)

2021 ◽  
Author(s):  
Xiaoqiang Xu ◽  
Shikui Chen ◽  
Xianfeng David Gu ◽  
Michael Yu Wang
Keyword(s):  
Topology Optimization ◽  
Heat Conduction ◽  
Level Set ◽  
Rectangular Domain ◽  
Free Form ◽  
Topology Optimization Problem ◽  
Parameter Domain ◽  
Covariant Derivatives ◽  
3D Topology ◽  
Free Form Surfaces

Abstract In this paper, the authors propose a new dimension reduction method for level-set-based topology optimization of conforming thermal structures on free-form surfaces. Both the Hamilton-Jacobi equation and the Laplace equation, which are the two governing PDEs for boundary evolution and thermal conduction, are transformed from the 3D manifold to the 2D rectangular domain using conformal parameterization. The new method can significantly simplify the computation of topology optimization on a manifold without loss of accuracy. This is achieved due to the fact that the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar with the conformal mapping. The original governing equations defined on the 3D manifold can now be properly modified and solved on a 2D domain. The objective function, constraint, and velocity field are also equivalently computed with the FEA on the 2D parameter domain with the properly modified form. In this sense, we are solving a 3D topology optimization problem equivalently on the 2D parameter domain. This reduction in dimension can greatly reduce the computing cost and complexity of the algorithm. The proposed concept is proved through two examples of heat conduction on manifolds.

Topology Optimization of Conformal Structures Using Extended Level Set Methods and Conformal Geometry Theory

2018 ◽  
Author(s):  
Qian Ye ◽  
Yang Guo ◽  
Shikui Chen ◽  
Xianfeng David Gu ◽  
Na Lei
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Differential Operators ◽  
Conformal Geometry ◽  
Level Set Methods ◽  
Free Form ◽  
Computational Framework ◽  
Covariant Derivatives ◽  
Free Form Surface ◽  
Free Form Surfaces

In this paper, we propose a new method to approach the problem of structural shape and topology optimization on manifold (or free-form surfaces). A manifold is conformally mapped onto a 2D rectangle domain, where the level set functions are defined. With conformal mapping, the corresponding covariant derivatives on a manifold can be represented by the Euclidean differential operators multiplied by a scalar. Therefore, the topology optimization problem on a free-form surface can be formulated as a 2D problem in the Euclidean space. To evolve the boundaries on a free-form surface, we propose a modified Hamilton-Jacobi equation and solve it on a 2D plane following the conformal geometry theory. In this way, we can fully utilize the conventional level-set-based computational framework. Compared with other established approaches which need to project the Euclidean differential operators to the manifold, the computational difficulty of our method is highly reduced while all the advantages of conventional level set methods are well preserved. We hope the proposed computational framework can provide a timely solution to increasing applications involving innovative structural designs on free-form surfaces in different engineering fields.

Computational Design and Additive Manufacturing of Conformal Metasurfaces by Combining Topology Optimization With Riemann Mapping Theorem

2017 ◽  
Author(s):  
Panagiotis Vogiatzis ◽  
Ming Ma ◽  
Shikui Chen ◽  
Xianfeng David Gu
Keyword(s):  
Topology Optimization ◽  
Additive Manufacturing ◽  
Conformal Mapping ◽  
Level Set ◽  
Computational Design ◽  
Unit Cell ◽  
Free Form ◽  
Computational Framework ◽  
Riemann Mapping Theorem ◽  
Free Form Surfaces

In this paper, we present a computational framework for computational design and additive manufacturing of spatial free-form periodic metasurfaces. The proposed scheme rests on the level-set based topology approach and the conformal mapping theory. A 2D unit cell of metamaterial with tailored effective properties is created using the level-set based topology optimization method. The achieved unit cell is further mapped to the 3D quad meshes on a free-form surface by applying the conformal mapping method which can preserve the local shape and angle when mapping the 2D design to a 3D surface. The proposed level-set based optimization methods not only can act as a motivator for design synthesis, but also can be seamlessly hooked with additive manufacturing with no need of CAD reconstructions. The proposed computational framework provides a solution to increasing applications involving innovative metamaterial designs on free-form surfaces in different fields of interest. The performance of the proposed scheme is illustrated through a benchmark example where a negative-Poisson’s-ratio unit cell pattern is mapped to a 3D human face and fabricated through additive manufacturing.

scholarly journals Stiffness optimization of geometrically nonlinear structures and the level set based solution

2016 ◽  
Vol 7 ◽  
pp. A3 ◽  
Author(s):  
Qi Xia ◽  
Tielin Shi
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Optimization Problem ◽  
Two Dimensions ◽  
Geometrically Nonlinear ◽  
Nonlinear Structures ◽  
Element Analysis ◽  
Topology Optimization Problem ◽  
Stiffness Optimization ◽  
Ks Function

Load-normalized strain energy increments between consecutive load steps are aggregated through the Kreisselmeier-Steinhauser (KS) function, and the KS function is proposed as a stiffness criterion of geometrically nonlinear structures. A topology optimization problem is defined to minimize the KS function together with the perimeter of structure and a volume constraint. The finite element analysis is done by remeshing, and artificial weak material is not used. The topology optimization problem is solved by using the level set method. Several numerical examples in two dimensions are provided. Other criteria of stiffness, i.e., the end compliance and the complementary work, are compared.

2009 A level-set based topology optimization for 2D heat conduction problems using BEM

2013 ◽  
Vol 2013.26 (0) ◽  
pp. _2009-1_-_2009-3_
Author(s):  
Guoxian JING ◽  
Toshiro MATSUMOTO ◽  
Toru TAKAHASHI ◽  
Hiroshi ISAKARI ◽  
Takayuki YAMADA
Keyword(s):  
Topology Optimization ◽  
Heat Conduction ◽  
Level Set

scholarly journals Parametric Level Set-Based Multimaterial Topology Optimization of Heat Conduction Structures

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Yadong Shen ◽  
Jianhu Feng
Keyword(s):  
Topology Optimization ◽  
Heat Conduction ◽  
Objective Function ◽  
Level Set ◽  
Heat Dissipation ◽  
Adjoint Method ◽  
Basis Functions ◽  
Expansion Coefficients ◽  
Quadratic Temperature ◽  
Moving Asymptotes

This paper presents a parametric level set-based method (PLSM) for multimaterial topology optimization of heat conduction structures with volume constraints. A parametric level set-based optimization model of heat conduction structures is built with multimaterial level set (MM-LS) model, which describes the boundaries of different materials by the combination of all level set functions. The heat dissipation efficiency which means the quadratic temperature gradient is conducted as the objective function. The adjoint method is utilized to calculate the sensitivities of the objective function with respect to expansion coefficients of the compactly supported radial basis functions (CSRBFs). The optimal configuration is achieved by updating the expansion coefficients gradually with the method of moving asymptotes (MMA). Several numerical examples are discussed to demonstrate effectiveness of the proposed method for multimaterial topology optimization of heat conduction structures.

Design of Mechanical Structures Considering Harmonic Loads Using Level Set-Based Topology Optimization

2012 ◽  
Author(s):  
Takayuki Yamada ◽  
Toshiro Matsumoto ◽  
Shinji Nishiwaki
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Design Method ◽  
Optimization Method ◽  
Adjoint Equations ◽  
Topology Optimization Problem ◽  
Level Set Function ◽  
Elastic Design ◽  
Objective Functional ◽  
Topology Optimization Method

This paper presents an optimum design method for mechanical structures considering harmonic loads using a level set-based topology optimization method and the Finite Element Method (FEM). First, we briefly discuss the level set-based topology optimization method. Second, a topology optimization problem is formulated for a dynamic elastic design problem using level set boundary expressions. The objective functional is set to minimize the displacement at specific boundaries. Based on this formulation, the topological sensitivities of the objective functional are derived. Next, a topology optimization algorithm is proposed that uses the FEM to solve the equilibrium and adjoint equations, and when updating the level set function. Finally, several numerical examples are provided to confirm the validity and utility of the proposed method.

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