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Abstract

Lightweight is one of the most important research subjects in modern manufacturing. However, the research on lightweight of shell models is rare, and most limited in topological changes. This article proposes a local heat sensor–based lightweight framework of shell models that consists of model analysis, lightweight modeling and analysis, and three-dimensional printing and practical validation in an optimum iterative procedure. Specifically, first, both geometric features and empirical features are introduced to construct a frame structure. Second, a local diffusion–based heat sensor network is exploited to simulate the stress distribution due to two reasons: one is that they have the similar physical transmissibility and the other is that the heat diffusion is smooth, and it guarantees that the thickness variation is smooth and natural without restriction on the degrees of freedom. Then, a local iteration consists of heat simulation and stress analysis is utilized to further improve the efficiency. Finally, we use three-dimensional printer to manufacture testing models and apply them to practical verification and feedback. Our extensive experiments have exhibited many attractive properties, including the flexibility and freedom of the thickness variation, the effectiveness, and credibility of the lightweight.

Keywords

Lightweight shell model heat sensor diffusion three-dimensional printing

Introduction

Lightweight of models is one of the most important and challenging topics in energy conservation and environmental protection. It can be widely used in automobile, aviation, medicine, and many other fields. As a branch of lightweight modeling, the lightweight problem of shell models is also extensive used in numerous areas. However, because of the deficiencies in conventional manufacturing, the lightweight of shell models has received little attention and most limited in the aspect of topological changes. Specifically, some applications may require to remain the topology and shape, such as the car shell in automobile manufacturing; therefore, it is essential to accomplish the topology and shape preserved lightweight of shell models. In addition, three-dimensional (3D) printing technology has attracted increasing attention from scholars and the modern industrial communities, and the advantages in the aspects of personalization and diversification also bring new energy and opportunity to the lightweight.

In this approach, our efforts are dedicated to reduce the weight by changing the thicknesses of shell models while preserving the topologies and shapes. We devise a lightweight framework that consists of model analysis, lightweight modeling and analysis, and 3D printing and practical validation, as shown in Figure 1. Specifically, both geometric features and empirical features are utilized to constitute a support structure in model analysis. Then, a local diffusion–based heat sensor network is exploited to simulate the stress distribution, for the reason that they have the similar physical transmissibility, and the heat diffusion could promote the smoothness of the thickness transformation, as shown in Figure 2. Furthermore, a local iteration consists of local heat simulation and stress analysis is used to optimize the thicknesses in local regions efficiently. Finally, 3D printed models and practical validation by testing machine are used to make the lightweight results more reliably. The primary contributions of this article can be summarized as follows: (1) We propose a topology and shape preserved lightweight framework of shell models that forms an efficient and credible solution. (2) We exploit a local diffusion–based heat sensor network to simulate the stress distribution, which inherits the robustness and smoothness of the heat diffusion that makes the thickness variation smooth and natural. (3) We utilize 3D printing to execute the practical validation and feedback, which could further promote the reasonable and reliable of the proposed lightweight framework.

Figure 1.

The pipeline of our lightweight framework.

Figure 2.

The heat diffusion and the stress distribution have similar physical transmissibility near the heat source and stress source, respectively: (a) the von Mises stress distribution and (b) the heat diffusion.

Background

Lightweight and shape optimization

The lightweight design can be roughly divided into two categories: lightweight of materials^1,2 and optimization of shapes and structures.^3–5 In this approach, we focus on the later one, which attempts to reduce the weight by modifying the shapes or structures utilizing heat sensor.^6–9 There is plenty of work on lightweight of shapes and structures, especially for automobile lightweight.^10–12 However, most of them concerned mainly on the optimization of frame structures, such as the design of automobile body frame and chassis, seldom enthuse about the lightweight of shell shapes, especially the one with topology and shape preserved. On the other hand, shell models and structures are widely used; however, they have not been studied much due to the cost and limitation of traditional manufacture technology. Recently, shell model optimization has attracted increasing attentions with the development of applicable fields and manufacturing techniques. Such as, Fauche et al.¹³ proposed a topology optimization of the thin-shell bridge structure. Xia et al.¹⁴ applied a level-set method to obtain a shape and topology optimization. Martínez et al.¹⁵ optimized the geometry of an object using an exemplar shape to capture its target appearance. However, the above methods concentrated more on topology optimization than shape optimization. For the shell shape optimization, Zhao et al.¹⁶ developed a patch-based shell optimization method, but the results were highly determined by the segmentations. Zhou et al.¹⁷ proposed a general optimization method that allowed shape deformations, but the results could be influenced by the shapes and model sampling. In contrast to the optimization of general shape and topology, there are still few works on the lightweight of shell models. With the above consideration, the lightweight of shell models is urgently needed.

3D printing

3D printing technology is also called additive manufacturing or rapid prototyping, is an important symbol of the third industrial revolution.¹⁸ It has brought changes in manufacturing processes and production patterns and contains the following characteristics: (1) it is unrestricted to the complexity and diversity of products, making the manufacturing cost irrelevant; (2) it is easy to use and suit for customization; (3) it lowers material and manufacturing costs, and making manufacturing more environmentally friendly. Therefore, 3D printing has attracted broad attention in many fields.^19,20 Recently, research on lightweight of solid models,^21,22 structure support of models,^23,24 and optimization of gravity center²⁵ have been proposed, and these research activities provided solutions from the views of printing cost, the feasibility of printing process, and the stability of printing models, respectively. Previous studies of 3D printing offer available experiences and valuable reference to help us perfect the lightweight of shell models.

The lightweight framework

Given a shell model $M$ , let $S$ be the deformable surface (inside or outside surface), which is usually the inside surface by default on account of the requirements of vision and applications. The users should also specify the materials and applied external force according to the applications. The proposed lightweight framework consists of three main steps, including model analysis, lightweight modeling and analysis, and 3D printing and practical verification.

Model analysis

We first analyze the given models to extract features. For most shell models, there are primary structure characters, which consist of geometric features and empirical features. These features are treated as a braced structure that corresponds to the thicker frame that withstands greater force.

Geometric features

For the geometric features, we concentrate on geometric characters that contribute to lightweight, including the highlight point features and line features. In this approach, we utilize the modified normal voting tensor to extract these features.^26,27

A normal voting tensor $NT (v_{i})$ of a vertex $v_{i}$ can be computed by the sum of the weighted covariance matrices

NT (v_{i}) = \sum_{t_{j} \in N_{t} (v_{i})} μ_{j} n_{t_{j}} n_{t_{j}}^{T}

(1)

where $t_{j}$ is a triangle, $N_{t} (v_{i})$ denotes the set of neighboring triangles of $v_{i}$ , $n_{t_{j}}$ is the normal of triangle $t_{j}$ , and $μ_{j}$ is the weight coefficient (set as 1 in this work). The tensor can be diagonalized by eigenvalues ( $λ_{1} > λ_{2} > λ_{3} \geq 0$ ) and reformulated by a spectral representation

NT (v_{i}) = λ_{1} e_{1} e_{1}^{T} + λ_{2} e_{2} e_{2}^{T} + λ_{3} e_{3} e_{3}^{T}

(2)

where $e_{i}$ is the corresponding eigenvector of $λ_{i}$ , $i = 1, 2, 3$ . Similar to the method proposed in Wang et al.,²⁷ we extract the point features and line features by considering the eigenvalues and the neighboring vertex coincidence, as shown in Figure 3(a).

Figure 3.

The features extraction: (a) the geometric features and (b) the empirical features, including the important pillars and the border lines.

Empirical features

Most often, the geometric features are not sufficient for shell models to constitute an adequate support structure in practical applications. Extra significant features should also be added to supplement the support structure, and we call them empirical features because they depend a great deal on the applications and the user experience. For the empirical features, the user could specify important regions interactively, such as automobile B-pillar and C-pillar and border lines of a car model, as shown in Figure 3(b). Moreover, more empirical features could be added locally in an iterative process according to the later stress analysis.

Lightweight modeling and analysis

Condition setting

Before lightweight modeling, an initial condition should be set. The initial condition includes external force requirements, material categories, and the minimum/maximum thicknesses. We will briefly state them in the following parts, respectively.

The external force requirements are closely associated with the applications. The force setting could be either a distribution over the entire model or a sampling of specified regions, and the later form is used in our experiments for clarity and practicality.

Different materials have diverse material attributes, such as elasticity modulus (EM) and Poisson’s ratio (PR). These attributes are important arguments in stress analysis using finite element method. The common materials and their attributes are selectively listed.²⁸ Since materials are sensitive and polytropic, we repeat the experiment in Tymrak et al.²⁸ using the standard components produced by our 3D printer and recalculate the EM of Poly lactic acid (PLA) that is 3600 MPa. Therefore, the default material in our experiments is the PLA with default EM (3600 MPa) and PR (0.3).

The minimum/maximum thicknesses should also be specified by the users according to the applied external force and materials. In our experiments, the minimum thickness is set to 1 mm and the maximum thickness is set to 2.5 mm, empirically (the maximum lengths of all the models are 200 mm; the material is PLA). However, some applications bring up the thickness terms that might be unknown to the users. In this case, the minimum/maximum thicknesses could be obtained by testing the uniform thickness models with a specified force (beyond the scope of our approach).

Heat diffusion–based stress distribution

Then we use local diffusion–based heat sensor network to simulate the stress distribution for two reasons. One is that they have the similar physical transmissibility. The other is that the smoothness and robustness of heat diffusion could enable the thicknesses of shell models to vary smoothly and naturally.

It is well-known that the heat diffusion over surface $S$ is governed by the heat equation

{\begin{matrix} \frac{\partial u (x, t)}{\partial t} = - Δ u (x, t), & x \in S, t > 0 \\ u (x, 0) = f (x), & x \in S, t = 0 \\ u (x, t) = g (x), & x \in C, t > 0 \end{matrix}

(3)

where $Δ$ is the Laplace–Beltrami operator, $f (x)$ is a function of initial heat value defined on $S$ , $g (x)$ is a function of constraint value $(f (x) = g (x), x \in C, t = 0)$ , and $C$ is a set of constraint source. The results of equation (3) can be obtained by

{\begin{matrix} u (x, t) = \int_{S} h_{t} (x, y) f (y) dy, \\ u (x, t) = g (x), & x \in C \end{matrix}

(4)

where $h_{t} (x, y)$ is the heat kernel between $x$ and $y$ at time $t$ . The multi-scale property of heat kernels implies that for small value of time $t$ , heat kernels can be well approximated linearly within a small geodesic bandwidth using the Gaussian kernel

K_{σ} (x, y) = \frac{1}{\sqrt{4 π σ}} \exp (- \frac{d^{2} (x, y)}{4 σ})

(5)

where $d (x, y)$ is the geodesic distance between x and y, and $σ$ is the bandwidth.

Therefore, a local heat diffusion over $S$ can be efficiently approximated using convolution of Gaussian kernel within a small bandwidth

f (v_{i}) = \sum_{v_{j} \in Ω_{v_{i}}} {\tilde{K}}_{σ} (v_{i}, v_{j}) f_{0} (v_{j})

(6)

where

${\tilde{K}}_{σ} (v_{i}, v_{j}) = K_{σ} (v_{i}, v_{j}) / \sum_{v_{j} \in Ω_{v_{i}}} K_{σ} (v_{i}, v_{j})$ , $Ω_{v_{i}}$ is the small bandwidth region at $v_{i}$ , and $f_{0} (v_{j})$ is the value of initial field at $v_{j}$ . The heat diffusion can be written in matrix form

F = A_{σ} F_{0}

(7)

where $F = [f (v_{1}), \dots, f (v_{n})]^{T}, F_{0} = [f_{0} (v_{1}), \dots, f_{0} (v_{n})]^{T}$ , and $A_{σ}$ is a sparse matrix with elements

A_{σ} (i, i) = {\begin{matrix} 1, & if v_{i} \in C \\ {\tilde{K}}_{σ} (v_{i}, v_{i}), & otherwise \end{matrix}

(8)

A_{σ} (i, j) = {\begin{matrix} 0, & if v_{i} \in C \\ {\tilde{K}}_{σ} (v_{j}, v_{i}), & else if v_{j} \in Ω_{v_{i}} \\ 0, & otherwise \end{matrix}

(9)

The small diffusion region $Ω_{*}$ is fixed for one-step diffusion proportion during the process of the diffusion. According to the semigroup identity of heat kernels, the heat diffusion can be obtained using the local Gaussian kernel convolution iteratively. The heat diffusion field after $k$ steps is

F^{k} = A_{σ} F^{k - 1} = \dots = A_{σ}^{k} F_{0}, k = 0, 1, \dots

(10)

When $k = 0$ , the heat diffusion field is the initial field. As $k$ increases, the heat diffuses to a larger region, and more global behavior results from the convolution. The diffusion turns into matrix-vector multiplication. Since $A_{σ}$ is a sparse matrix, the computation is highly efficient. Other methodologies^29,30 can also be used to solve this problem.

In this approach, $C$ is the set of features, including the geometric features, the empirical features, and the stress source points, that are treated as constant-temperature heat sources (the values are set to 1 by default). $f_{0} (v)$ is the initial heat function, which is corresponding to the initial thicknesses of the shell model ( $f_{0} (v_{\max}) = 1$ , $f_{0} (v_{\min}) = 0$ , where $v_{\max} \in C$ and $v_{\min} \in other$ are the vertices that corresponding to the maximum and minimum thicknesses, respectively). An initial diffusion simulation is first executed (default $k = 5$ , as shown in Figure 4(a)). Then a stress analysis is utilized to determine whether the diffusion will continue in the next phase.

Figure 4.

The heat diffusion simulation in local regions: (a) the initial diffusion $(k = 5)$ in a given local region and (b–d) the diffusion processing (k = 10, 20, 25), respectively.

Finite element–based stress analysis

After the initial diffusion simulation, an initial shell model with corresponding thicknesses can be obtained, where the thicknesses are calculated according to the values of the heat sensor. Specifically, a linear interpolation method is exploited to calculate the thicknesses by corresponding the maximum and minimum thicknesses to the maximum and minimum heat values, respectively.

Then, a finite element–based stress analysis is utilized to obtain the stress distribution of the current shell model. Specifically, take the “top-force and bottom-constraint” for example, a static analysis is first built by setting the main parameters “Material = PLA,”“E = 3600,”“NU = 0.3,” and “ $T = Thickness .^{*}$ ,” where $Thickness .^{*}$ is a file of the thickness value of each vertex.

In this stage, all the local regions that do not meet the requirements will be checked out, where the requirements refer to the given stress threshold or deformation. It means that these selected local regions of the shell model cannot afford the applied force, and they should continue to be thickened by diffusing the heat in these regions locally using equation (10) (the diffusion control parameter k turns to k + 5 in one iterative step). The more heat diffuses, the thicker the local regions of the shell model become.

Repeat the two steps between stress analysis and the heat diffusion iteratively, until there are no unacceptable regions (the max-stress in the region is larger than the given stress threshold). Then, a deformation of deformable surface (internal) is implemented according to the thicknesses values, as shown in Figure 5. Finally, we obtain a shell model with optimized thickness that can tolerate the applied external forces.

Figure 5.

The deformation of shell models: (a) the original deformable surface, (b) the deformed internal surface, (c) the deformed shell model, and (d) the internal view of deformed shell model.

3D printing and practical verification

Since the lightweight shell models are designed to manufacture by 3D printing technology, practical verification is necessary to eliminate the performance impact of production processing. In this approach, we use economical 3D printers to manufacture the lightweight models and exploit the stress testing machine to accomplish the practical verification. Then, a feedback correction is implemented by adjusting the diffusion parameter according to the testing data. Our lightweight shell models could also be manufactured using traditional methods, such as casting and cutting (it is out the scope of our approach). Moreover, the practical verification is optional according to the applications.

Considering the cost of manufacturing and model quality, it is a cost efficient solution to choose fused deposition modelling (FDM)-based 3D printers to print the lightweight models (the 5th-Gen Makerbot 3D Printer). To reduce the impact of 3D printing, we utilize the highest printing accuracy (0.1 mm), and three testing samples are printed for each model. In the verification process, for three testing samples, if there are two or three samples that do not meet the criteria in the conditions, then the lightweight model is considered to be a candidate that needs feedback correction. Specifically, we exploit WDW-50 microcomputer control electronic universal testing machine to execute the verification, as shown in Figure 6. In the feedback, the testing error (the difference between the theory model and the actual testing) and the error regions need to be detected. A diffusion will be implemented in the local error regions using the diffusion simulation in equation (10) if the detected error is larger than the given threshold.

Figure 6.

The verification using WDW-50 microcomputer control electronic universal testing machine.

Experiment and discussion

We now demonstrate the performance in this section. To be convenient for explanation, all the models in our experiments are uniform (the maximum length is 200 mm). The external force applied on the shell models are also consistent: top-force is set to 100 N and front/back-force is set to 40 N.

There are several parameters in our lightweight framework; however, most of them are fixed or set empirically. The maximum and minimum thicknesses are related to the applied force and materials (PLA is used in this approach), and they are set to constant 2.5 and 1 mm, respectively. The bandwidth $σ$ is set to the $2^{*} L_{m}$ , where $L_{m}$ is the mean edge length of a given shell model. The stress threshold used to detect the local unsatisfied regions can be set to a specified value under the maximum tolerated stress according to the materials and applications. In our experiment, the threshold is set to 95% of the maximum tolerated stress of the PLA material. Moreover, the deformation displacement could also be used to detect the local unsatisfied regions, such as the criterion is set to $105 %$ deformation displacement of the original model in the same regions. Figure 7 shows the curves of force and deformation displacement, and we can see that the lightweight model performs well under the applied external force. In order to alleviate the complexity, we only release the diffusion step $k$ and set the step-size to 5 in one iteration in our experiments. Users can adjust $k$ to obtain different diffusion simulation in local regions, as shown in Figure 4.

Figure 7.

The analysis of the force and displacement between the original model (Audi: top and back) and the lightweight one, respectively.

Figure 8 illustrates the comparison results of well-known finite element analysis (FEA) method “solid isotropic material with penalization” (SIMP),³¹ patch-based shell optimization method,¹⁶ and our method. We can see from the cross profiles of the lightweight models, the thicknesses are thinner in less significant regions (tolerance less force), and thickness variation of our method is more smooth and natural than the results using SIMP (Figure 8(b)) and patch-based method (Figure 8(c)). Moreover, different from the previous methods, the freedom of thickness in the proposed approach is not limited, and it can be concluded automatically from the diffusion with fewer parameters. Figure 9 shows the weight measurement of Audi model (material: PLA) under the applied external force (top-force: 100 N; front/back-force: 40 N). The weight of original shell model is 64.2 g, and the lightweight shell model is 46.0 g ( $28.3 %$ weight is lost). Figure 10 exhibits more results of shell models with different topologies, and we can see that the proposed method is robust for complex topology by inheriting the robustness of the heat diffusion. Table 1 lists the comparison of weight between the original models and the lightweight ones under the applied external force while the applied external force (top, front and back) is satisfied, and almost $30 %$ weight can be reduced (models are uniform with the same maximum length 200 mm).

Figure 8.

The comparison of lightweight results with top-force and front/back-force: (a) the original model; (b) the result using the well-known SIMP,²⁸ and the dotted lines mark the discontinuous variation; (c) the result of patch-based method³¹; and (d) our result that is smooth and natural.

Figure 9.

The weight measurement: (a) the weight of original shell model is 64.2 g and (b) the weight of lightweight shell model is 46.0 g.

Figure 10.

Lightweight results of other shell models with different topologies. The proposed method is robust for complex topology: (a) CC, (b) Camry, (c) Tesla, and (d) Beetle.

Table 1.

The comparison between the original models (O) and the lightweight ones (L).

Models	Max-width (mm)	O(g)	L(g)
Audi	84.8	64.2	46.0
CC	79.1	68.5	48.4
Camry	79.3	70.5	51.3
Tesla	83.4	69.2	50.1
Beetle	81.3	61.7	41.5

We would like to note that the proposed lightweight of shell models is a general framework, and it can be applied to most shell models with different materials and external force conditions. However, there are still a few limitations that have room for improvement. The lightweight results are not optimal solutions in theory, and they are existential approximations of the optimal solutions. Besides that, we will perform the stress analysis using nonconforming finite element³² in future work.

Conclusion

In this article, we have presented a heat sensor network–based lightweight framework of shell models. Both theoretical method and practical verification are inclusive, such as voting tensor–based model analysis, diffusion simulation–based lightweight modeling, finite element–based stress analysis, and practical validation using 3D printed models, and these constitute a complete lightweight framework of shell models. The lightweight results of shell models are smooth and natural, and comprehensive experiments have demonstrated the advantages of our method in terms of efficiency, robustness, and reliability.

Footnotes

Handling Editor: Songhua Xu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported in part by the National Natural Science Foundation of China grants (61772104, 61432003, 61720106005, 61300083, 11472073), the Fundamental Research Funds for the Central Universities (DUT17JC27), and National Key Basic Research Program (2017YFB1103700).

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Lightweight framework of shell models utilizing local heat sensors

Abstract

Keywords

Introduction

Background

Lightweight and shape optimization

3D printing

The lightweight framework

Model analysis

Geometric features

Empirical features

Lightweight modeling and analysis

Condition setting

Heat diffusion–based stress distribution

Finite element–based stress analysis

3D printing and practical verification

Experiment and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References