IUTAM Symposium on When topology optimization meets additive manufacturing – theory and methods

Dalian, October. 8-12, 2018, China

GENERATIVE DESIGN OF MULTIFUNCTIONAL CONFORMAL STRUCTURES USING EXTENDED LEVEL SET METHODS (X-LSM) AND

CONFORMAL GEOMETRY THEORY (Pre-Print)

Qian Ye1, Long Jiang1, Shikui Chen1,*, Xianfeng David Gu2,3

1Dept. of Mechanical Engineering, the State University of New York at Stony Brook, 11790, Stony Brook, USA

* Corresponding author: shikui.chen@stonybrook.edu

2Dept. of Computer Science, the State University of New York at Stony Brook, 11790, Stony Brook, USA

3Dept. of Applied Mathematics and Statistics, the State University of New York at Stony Brook, 11790, Stony Brook, USA

Abstract

In this paper, a framework for generatively designing multifunctional conformal structures on free-form surfaces is

proposed. This framework rests on the extended level set methods (X-LSM) and conformal geometry theory. By applying

the X-LSM, the structural topology optimization can be directly performed on the free-form surface to form a load-

carrying structure. Helped by the angle-preserving conformal mapping theory, extra mesoscale functional cellular

structures, such as thermal insulating lining cellular structures can be conformally mapped to the targeted surface to form

a functional structure. By combining the two structures at two scales, a multiscale and multifunctional structure with

desired load-carrying and additional functions can be achieved. The proposed framework is utilized to design a spaceship

model which contains the outer reinforcement frame and inner thermal insulating cellular lining. The proposed method

reveals the intrinsic relationship between topology optimization performed on the surface and in Euclidean space. It

provides a unified computational framework for generatively designing multifunctional conformal structures on surfaces

with increasing applications in different fields of interests.

Keywords: Structural Optimization, Topology Optimization, Multifunction Design, Conformal Structure, Level Set

Method

1. Introduction

Topology optimization can generate innovative designs without requiring a priori knowledge of the design. Paired with

the rapidly developing additive manufacturing (AM) technologies, topology optimization can be a powerful tool to create

high-performance structures with complex geometries [1]. Recently, topology optimization of conformal structures on

surfaces has become a hot research topic due to the wide range of applications in both academia and industry, for example,

the aircraft and aerospace structure design [2], architecture design [3], and conformal flexible electronics design [4].

The research on the topology optimization of conformal structures can be classified into two categories. The first one is

to optimize the overall structural topology directly on the surface. The SIMP method is a commonly used approach [5-7],

but the numerical instability issue [8] can be often observed during the optimization process. An alternative approach is

recently presented by Ye et al.[9, 10] which extended the conventional level set method from the Euclidian space to the

free-form surfaces. At the same time, the X-LSM can inherit the advantages of the level set method, such as the clear

design boundaries and high topological handling capability. The details of X-LSM will be discussed in Section 2.2.

The second one is to perform the optimization at the mesoscale, i.e., designing the conformal cellular structure on the

surface. In engineering applications, the cellular structure is assembled by periodically combining individual unit cells.

This kind of metamaterial has wide applications in energy absorption [11], and thermal isolation at a low self-weight.

Conventionally, topology optimization of cellular structure on regular Cartesian space has been well studied [12-14]. The

typical way is to design a planar unit cell and assemble it into a 2D or 3D array. Designing cellular structures on the

surface or in the irregular domain can be more challenging, since a mapping must be identified between the planar unit

cell and the target surface during the optimization process. Nguyen et al. [15, 16] did pioneering work, where the mapping

process via augmented size matching and scaling (SMS) is used to create a conformal lattice design. During the mapping

process, the surface is meshed by quadrilateral elements, and the unit cell is filled into each of the mesh elements. However,

this mapping procedure depends highly on the mesh quality. Vogiatzis et al. [17] employed the conventional level set

method to design cellular metamaterials on the surface. The unit cell is first optimized in Euclidean space and then mapped

to the surface via conformal mapping [18, 19]. This work is extended to design multi-material structures with conformal

microstructure infills recently [20], with the preservation of structural connectivity and integrity simultaneously. Jiang et

al. employed a concurrent-topology optimization with a parametric level set method to design a multiscale shell-infill

structure [21, 22]. A multi-control-point conformal mapping was utilized to ensure the conformality of the infill

metastructure to the outer shell. In engineering applications, a multiscale structure will possess multiple functions. A

typical example would be a multiscale sandwich structures showing unique mechanical and thermal properties at a lower

density [23, 24].

This paper aims to devise a systematic framework for the generative design of multifunctional conformal structures on

the free-form surface. By applying the X-LSM, the macroscale conformal structure can be directly optimized on the

surface. Meanwhile, the functional lining formed up by the 2d mesoscale planar optimal unit cell can be mapped to the

surface by following the conformal mapping theory. By applying different design objectives for each scale, the multiscale

multifunctional final conformal design can be achieved. The rest of the contents are organized as follows: Sections 2

introduces the general workflow and gives the background information regarding X-LSM and conformal geometry theory.

The proposed framework is verified by three numerical examples in section 3. In Section 4, a brief conclusion is drawn.

2. Method

2.1 Conformal Mapping

Conformal mapping gives the mapping relationship between two Riemannian surfaces, which can preserve the angles.

Suppose given two Riemannian surfaces 𝑔1 and 𝑔2, where 𝑔1 and 𝑔2 are Riemannian metric tensors, a smooth mapping

φ: S1 to S2 is called conformal if the pull-back metric induced by φ and the original metric on the source differ by a scalar

function. Specifically, there exists a real function λ: S → 𝐑, such that

φ∗𝐠𝟐 = e2λ𝐠𝟏

Intuitively, the derivative map dφ: TS1(p) → TS2(φ(p)) is a scaling transformation. The conformal mapping can be

visualized by mapping a checkerboard pattern onto the conformal parametrized surface. As shown in Figure 1, the angles

of the checkerboard pattern are still orthogonal after being mapped onto the human face surface.

In summary, the conformal mapping can be considered as a local scaling process governed by the scalar function 𝜆. It has

been proven that by using conformal mapping, the covariant derivatives on the surface are equivalent to the differential

operators on Euclidean space apart from the scalar function [26, 27]. With conformal mapping, PDEs on the surface can

be formulated to 2D with a modified variation operator which greatly reduces the computational cost.

Figure 1. Visualization of conformal mapping on a human face surface

2.2 Extended Level Set Method

The key idea of X-LSM is to recast the topology optimization problem on the surface as a 2D topology optimization

problem in the Euclidean space [9]. Figure 2 gives an overview of the X-LSM process. With conformal parameterization,

a surface can be mapped to a 2D rectangle domain, where the level set function is defined. The covariant derivatives on

the manifold can be represented by the Euclidean gradient operators multiplied by a scalar function. Keeping this intrinsic

relation in mind, the Euclidean form for the Riemannian Hamilton-Jacobi equation governing the boundary evolution on

the manifold is derived, which can be solved on a 2D plane using classical level-set methods. The Riemannian Hamilton-

Jacobi equation can be written as:

𝜕𝜙

𝜕𝑡+ 𝑒−2𝜆𝑣𝑛|∇𝜙| = 0

where 𝜆 is the conformal factor, 𝑣𝑛 is the normal velocity field and 𝜙 is the level set function. The detailed derivation is

provided in [9, 10].

Figure 2. Overview of the X-LSM

2.3 Design of Multifunctional Conformal Structures

The multifunctional conformal structure can be achieved through topology optimization at two scales, that is, the overall

structure optimization at macroscale, and metamaterial optimization at mesoscale. As shown in Figure 3, the overall

structure is optimized using X-LSM. Meanwhile, after getting the optimal functioning unit cell design, the conformal

mapping is employed to map the 2D metamaterial onto the surface. The method is eligible for multi-scale and multi-

physics structural optimization problem on freeform surfaces. The efficiency of the proposed framework can be ensured

since the X-LSM reduces the computational difficulty by reformulating the 3D problem onto 2D. Unlike the conventional

methods where the unit cells along the boundary have to be cut to fit into the enclosed surface, the proposed framework

can guarantee the conformality, integrity of the mapped metamaterial with limited area distortion.

Figure 3. A computational framework for the generative design of the multifunctional conformal structure

3. Numerical Examples

In this section, two benchmark examples of topology optimization of mean compliance on the free-form surface are

provided first to illustrate the proposed X-LSM scheme. Then, a multifunctional conformal structure design for a

spaceship is provided to demonstrate the performance of the proposed computational framework.

3.1 Torsion Cylindrical Shell

In the first benchmark example, we aim to minimize the mean compliance on a cylindrical surface subject to a pure

torsion [28] using X-LSM. As shown Figure 4.a, the design domain is a cylinder with a radius of 0.5, length 3 and

thickness 0.05 respectively. A clockwise torsion is applied on one end of the surface, and the other end is fixed. The

design domain is meshed by 8832 triangles. The Young’s modulus and density are set as 1 for the solid and 1e-3 for the

void. The Poisson’s ratio is 0.3. The conformal mapped 2D domain is meshed by a 126 × 120 structure grid. The

optimized design with a volume fraction of 0.3 is shown in Figure 4.b. It can be noticed that the result matches well the

optimal analytical Michel frame structures on a torsion rod proposed by Sigmund et al. [28].

(a)Design domain & boundary conditions (b) Topology optimization results

Figure 4. Torsion cylindrical shell design

3.2 Topology Optimization of Conformal Reinforcement on a Chair Surface Using X-LSM

In this example, the X-LSM is applied to design the conformal reinforcement structure of a chair surface with an oval

shape. Please be reminded that the chair surface is an undevelopable surface, which means it cannot be flattened to a 2D

plane without stretching, tearing or squeezing it. The span of the armchair is approximately 0.6m × 0.5m, and the

thickness is 0.04m. The design domain is discretized into 59685 triangular elements. The design domain and boundary

conditions of the armchair surface are shown in Figure 5.a. Three concentrated loads along the z-direction are applied at

the arms and the seat area. Another force is applied on the back along the y-axis. A small portion of the bottom is fixed.

The target volume ratio is 0.4. The linear elastic material is assumed with a Poisson's ratio 0.3 and Young's modulus

E=1 GPa, and Young's modulus 𝐸 = 10−6 GPa is set for the void. The corresponding 2D rectangular plane is discretized

with a 120 × 167 quad mesh. Figure 5.b shows the result of the chair surface with the optimized conformal

reinforcement design.

(a)Design domain & boundary conditions (b) Topology optimization results

Figure 6. Oval chair surface design

3.3 A Conceptual Design of a Multifunctional Spaceship Model

The previous examples consider the optimization on macroscale with one single objective. In this example, a

multifunctional conceptual design of a spaceship is demonstrated. The landing scenario for a cone shape spaceship is

considered, as shown in Figure 6.a. Generally, the spaceship design should consider that the overall spaceship would be

strong enough to resist the mechanical force as well as able to withstand the intense heat generated by the air friction.

Based on the aforementioned design criteria, a multifunctional spaceship design with conformal reinforcement frame and

thermal insulation lining is provided by using the proposed framework detailed in Section 2.3.

a) Schematic figure of a landing spaceship b) Boundary conditions of spaceship model

Figure 7. Spaceship model

3.3.1 Topology optimization of conformal reinforcement on a spaceship

A minimum compliance problem on the spaceship surface is solved by X-LSM. The boundary conditions are set as a

distributed drag force applied on the top side as well as a counterclockwise torque in Figure 6.b. Only 1/8 of the spaceship

surface is treated as design domain considering the symmetry. The surface is meshed by 7983 triangles. The corresponding

2D plane is constructed by an 89 × 501 structural grid. The linear elastic material has the Passions ration 𝜈 = 0.3 and

Young’s modulus 𝐸 = 10−6 GPa . To avoid singularity, a weak material with Young’s modulus 𝐸 = 10−6GPa is set for

the void. Figure 7 is the optimized conformal reinforcement structure on the spaceship surface.

Figure 8. The final design of the conformal reinforcement structure on the spaceship surface

3.3.2 Design of the conformal lining with thermal insulating unit cell

The design domain of the thermal insulating unit cell is a 1 × 1 × 0.1 plate as shown in Figure 8.a. The upper surface and

bottom surface are set as the hot side and the cold side, respectively. Meanwhile, 4 point loads are added on the corners

to ensure structural connectivity. A multi-physics topology optimization problem considering minimum mean compliance

and thermal resistance [29] is solved. The thermal insulating unit cell is optimized by SIMP method with a volume fraction

of 0.2. The optimized unit cell design is shown in Figure 8.b. By using conformal mapping, the thermal insulation unit

cell can be mapped onto the spaceship surface in Figure 9. The final assembled multi-functional spaceship design

including the conformal reinforcement frame and the thermal insulation lining is shown in Figure 10.

(a) Design domain & boundary conditions (b) Topology optimization result

Figure 9. Design of the load-withstanding thermal insulation unit cell

Figure 10. Thermal insulating conformal structures on the spaceship surface

Figure 11. The final multifunctional conformal spaceship

3. Conclusions

In this paper, a new computational framework for the generative design of the multifunctional conformal structure is

proposed. By employing X-LSM, the macroscale structure can be directly optimized on the freeform surfaces. By utilizing

the conformal mapping theory, the functioning metamaterial can be conformally mapped to the surface. Besides, the

conventional finite element analysis can be directly coupled with the proposed framework. Moreover, via conformal

parameterization, the original PDE defined on the 3D undevelopable surface can be solved on the 2D domain, which

boosts the numerical efficiency. Also, the mapped metamaterial shape and the unit cell integrity can always be preserved

by the conformal mapping. Following the proposed scheme, a large variety of topology optimization problems have the

potential to be solved to design multiscale, multifunctional conformal structures.

Acknowledgments

This work is supported by the National Science Foundation (CMMI-1462270 and CMMI-1762287), the Ford University

Research Program (URP), and the start-up fund from the State University of New York at Stony Brook.

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