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INDUSTRIAL APPLICATION Automated design of thin-walled packaging structures Ching-Jui Chang therefore, it is normally not used for industrial packaging. There are many different fixed format methods for various industrial packaging applications. Among all fixed format methods, the thin-walled protective packaging has been evolving as the most important one because it features space saving and lightweight while delivering conspicuous property in shock absorption, moisture resistance, and most noticeable, environmental friendly when it is constructed by molded pulp. Thin-walled packaging structures are normally strength- ened by adding rib-shaped features on the thin wall to increase its stiffness. However, when designing a thin- walled packaging structure, the spaces for holding products should be determined before the rib-shaped reinforcement can be placed. Generally, the packaging company receives orders for products to be packaged in the form of computer- aided design (CAD) models or physical products. The CAD models sometimes will not be given in its native CAD formats but only in their surface representation formats due to the product confidentiality. If physical models are provided, they can also be digitalized to generate 3D data for surface representation. For both approaches, Stereo Lithography (STL) file (Jacobs and Reid 1992)isa commonly used data format to describe surface information because triangle is the most basic primitive for computer graphics, and it is easy to be manipulated. The arrangement of product, such as orientation and distribution, and the overall size packaged box is pre-determined based on its display requirement and/or weight distribution. The task of packaging company is to obtain the supporting space between the bounding box and the products first, and then a thin-walled structure can be generated from the surface of the supporting space. Finally, rib-shaped reinforcement are Struct Multidisc Optim (2008) 35:601608 DOI 10.1007/s00158-007-0170-y C.-J. Chang : B. Zheng : H. C. Gea (*) Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA e-mail: gearci.rutgers.edu C.-J. Chang e-mail: chingjuirutgers.edu placed to enhance its stiffness. Currently, the entire process is accomplished by repeated trial-and-errors. Although there are some existing methods for identifying the supporting space between the product and its bounding box, e.g., Boolean-based operations for creating a negative image model and parting line detection method (Fu et al. 1999, 2001; Wong et al. 1998) or injection mold generation method (Priyadarshi and Gupta 2004;Li2002)for generating molded foam, these methods may produce faulty designs for concave shapes. To remedy this deficiency, additional steps before applying these methods, such as the orientation of objects, should be recalculated to obtain an undercut free orientation (Majhi et al. 1999), or concave objects should be convexified first. Furthermore, packaging designers also need to consider offsets as well as parting lines and draft angle for packaging molds when create packaging molds (Qu and Stucker 2003; Tokuyama and Bae 1999). As these procedures often require experienced engineers to intervene, it is very desirable to have an automated design method for thin-walled packaging structures. To address these challenges, an automated process for designing thin-walled packaging structures with the optimal placement of reinforcements is presented in this paper. In this automated process, CAD models of the products are placed in a pre-specified fashion first. Then, a design space for supporting structures is generated between the products and the overall package by an automated computer procedure. The design space is further reduced to a one- layered finite element mesh for modeling the thin-walled structure. Then, a topology optimization method is incor- porated to generate the optimal reinforced structure. The optimal criterion is based on the mean compliance formulation to produce the stiffest structure. The final result from the topology optimization can be used to suggest the optimal reinforcements in the thin-walled structures. In the following sections, detailed discussion on the proposed method will be given. Then, some practical design examples are included to demonstrate the effectiveness of the proposed method. 2 Design space identification To create thin-walled packaging structures with the optimal placement of reinforcements, a thin-walled design space should be determined from a pre-specified packaging bounding box and product arrangement. There are three steps in the design space identification: (1) facet reduction, (2) height detection, and (3) design space generation. The facet reduction eliminates unnecessary calculation by reducing the number of facets of the CAD model. The height detection creates an evenly distributed supports for products and prevents undercuts. The design space gener- ation forms a thin-wall mesh surrounding the products. The details of these steps will be presented in this section. Once the thin-walled design space is identified, topology optimi- zation can be applied to determine the optimal placement of reinforcement, which will be discussed in the next section. 2.1 Facet reduction As the products are often given in the form of surface representation, a commonly used data format, STL format, is implemented in our current study. The STL file format simply describes the surface model of solid parts using triangles, which is one of the basic graphic primitives. Each triangle is specified with its three vertices, arranged in counterclockwise order, along with its surface normal calculated using the right hand rule from its vertices as shown in Fig. 1, which indicates the direction leaving the model. The surface normal is normalized to have unity length. A typical ASCII data of a facet in a STL file consists of its normal vector (u, v, w) and three sets of coordinates of vertices, (x i , y i , z i ). To construct the supporting space between the bounding box and the products, a set of evenly spaced supporting lines is created from the bottom of the bounding box towards the products. Although there is no special requirement on the spacing between supporting lines, it should be slightly smaller than the size of the smallest downward facing triangle such that all features of the product model can be captured. From our numerical experiments, using the spacing as an 80% nominal length of the smallest feature can always produce satisfactory results. Once the spacing is determined, a set of 2D mesh grids can be generated on the bottom plane of bounding box. These mesh grids are positioned as the starting points of the vertical supporting lines, and the ending points are the intersections of supporting lines to the products as shown in Fig. 2a. The intersections between each sup- porting line and products will be calculated as shown in Fig. 2b. Fig. 1 Triangle primitive of a 3D model 602 Chang et al. As there are a lot of facets involved in the intersection calculation, we need to reduce the computational effort by reducing the number of facets in the STL files first. Considering an object as shown in Fig. 3, it is obvious that all facets with green normal vectors should not be included in the intersection test. To remove all facets with non-downward normal vector, we need to transform all entities in the model to have the same coordinate system and then rotate the z-axis to be the direction pointing upwards as the supporting lines. Once these transformations are completed, all facets with a positive z component in the normal vectors should be removed from the comparison pool. This selection process can reduce a great number of facets before the intersection calculation. 2.2 Height detection For each supporting line, the height from the base to the product must be calculated. Although the number of facets has been greatly reduced after the first step, there still has no obvious pattern to locate the corresponding facet for each mesh grid. Because it is computationally expensive to check every grid on the mesh for corresponding points, a pre-test using a blue bounding rectangle enclosed the projected triangle, as shown in Fig. 4, is applied first to limit the grids to be checked. This pre-test is very straight- forward because the mesh grids are pre-defined and spaced regularly. To further improve the efficiency of the calcula- tion, each facet is projected into the mesh grid plan to trace back the mesh grids for the height detection. The red mesh grids enclosed by the projected triangle as shown in Fig. 4 are the corresponding mesh grids which need to be processed further for height detection. To verify whether the mesh grids are the corresponding ones for a given facet, the following procedure is used. Because all the projected facets have their surface normal pointing downwards, if one looks from the top of the bounding rectangle, the edge vectors of the projected triangle must run clockwise; consequently, the inner grids must be always on the right side of every edge. Hence, the easiest way to select the inner points is to compare the z value of the cross product between the edge vector and the vector formed by the end vertex and the targeting grid. When the z value of the outer product is less than or equal to zero, then the grid is inside the triangle as shown in Fig. 5. This test is performed along each edge of the facet. Fig. 3 All facets with green normal vectors should not be included in the intersection test Fig. 2 a Supporting lines from the bottom of the bounding box; b intersections between supporting lines and product Fig. 4 Grids enclosed by a projected facet Fig. 5 Inner test for a grid and a facet Automated design of thin-walled packaging structures 603 If any of the tests confirms that the grid is outside the projected triangle, the point will be removed immediately. After collecting all inner grids for each facet, it is very simple to calculate the intersection point. The plane that embraces a given triangle with facet normal (u k , v k , w k ) can be defined as: u k x v k y w k z u k x a v k y a w k z a where (x a , y a , z a ) is one of the vertex coordinates of the facet. The z coordinate at the product end of the supporting line with respect to a mesh grid (x i , y j ) can be calculated easily as: z i;j z a u k x a C0 x i v k y a C0 y j C0C1 w k In some cases, a supporting line from one mesh grid may intersect with more than one facet. In this case, the shortest height will be assigned to be the actual height on that grid to avoid penetration. 2.3 Design space generation Although the heights of all supporting lines from the mesh grids are obtained after the height detection, the packaging supporting surface still cannot be formed by simply connecting these points because there are some portions of the models may have interference. To overcome this problem, heights of supporting lines at the location where surface normal changes from down facing to up facing need to be relaxed as follows. For a given supporting line, we compare all possible eight neighboring supporting lines for their heights, and the lowest value among all neighboring heights is assigned to be the new height. This method applies to every supporting line to adjust its height to a new one to avoid interferences. After the height relaxation method, all new heights can be connected to form a supporting surface as shown in the blue lines of Fig. 6a. One interesting by-product of this method is that a packaging foam model is obtained immediately before generating a thin-walled design space and topology optimization. As the foam type of packaging method is also very popular in some industrial applications, the current implementation can also be used to create the foam model very efficiently. To generate thin-walled design space, a continuously connected mesh is formed using the heights on the grids because a continuous connected layer of mesh is required to prevent pivot effect from hinge-like meshes. Therefore, a set of four neighboring height grids is used to create continuously connected mesh. For grids with the same Fig. 6 An example of forming a FEA mesh layer. a supporting surface, b first pass mesh, c final mesh a bc Fig. 7 a A ball model with 1,596 facets, b supporting space for a bounding box with height of 45% of the diameter of the ball, and c supporting space for a bounding box with height of 70% of the diameter of the ball 604 Chang et al. height, simply connected mesh can be formed. However, for grids with different heights, the lowest height value of the set of grids is chosen as the base height and an array of connecting hexahedron is generated from the base to the top as shown in Fig. 6b and c. 3 Topology optimization Once the thin-walled design space is generated, designer can assign proper boundary conditions including supports and loadings to simulate various packaging conditions. Because thin-walled structures have very low stiffness against off-plane loading and vibration, they are normally reinforced by proper placement of stiffeners. There are two research directions related to the optimal stiffener design: the composite structure optimization problem and the reinforcement problem. The former models the thin-walled structures using orthotropic material and calculate the optimal thickness and orientation of orthotropic materials (Gea and Luo 2004; Luo and Gea 1998a Pedersen 1989, 1990, 1991), and the latter utilizes the topology optimiza- tion to identify the optimal locations and orientation under various static and vibration applications (Luo and Gea 1997, 1998b, c, 2003; Gea and Luo 1999; Gea and Fu 1995, 1997). The validity of topology optimization result is currently under investigation. In common cases, bead rib is a suitable candidate to be applied on the thin-wall structure to increase stiffness. Some extra considerations need to be taken when bead ribs intersecting each other because stiffness may be decreased in this case. As the main purpose of this paper is to demonstrate an automated design methodology of thin-walled packaging structure using topology optimization, only the minimizing mean compliance formulation is used to demonstrate the applicability of the method. In this paper, a microstructure- based design domain method is employed to formulate this problem due to its simplicity (Gea 1996), and the optimization problem is solved iteratively by generalized convex approximation (Chickermane and Gea 1996). This method has been implemented using C#.NET on a Pentium IV 2.66 GHz personal computer with 1.5 GB RAM. In the next section, examples of automated packaging design using this method are presented and discussed. Fig. 8 Efficient test result on ball models with different number of supporting base grids a b c Fig. 9 a A suspension model, b suspension model and supporting space with 70% height, and c supporting space Automated design of thin-walled packaging structures 605 4 Design examples Four examples are presented in this section. The first two examples are foam packaging designs: example 1 is a simple convex object and example 2 consists of an automobile suspension model with multiple components. These two examples are used to demonstrate the efficiency of the proposed method. The last two examples are thin- walled packaging structures: example 3 is a simple box and example 4 is a toy car model with multiple components. Designs of optimal reinforcement of both models using topology optimization under mean compliance formulation are presented. 4.1 Example 1: a ball model A simple ball model is presented in the first example because it is a strictly convex model as shown in Fig. 7a. The ball contains 1,596 facets, and the bottom of sup- porting space is divided into 200200 mesh grids. First, the height of bounding box is chosen as 45% of the diameter of the ball, which is slightly below the center of ball. A concave foam packaging structure is generated in 0.15 s in a personal computer as shown in Fig. 7b. When the height of bounding box elevates to 70% of the diameter of the ball, the foam packaging structure cannot be obtained by a simple negative model from Boolean operation. However, our method can still produce a correct form packaging space automatically by creating a cylindrical cavity after the supporting space passes the half ball as shown in Fig. 7c. The entire operation is completed within 0.16 s. To further demonstrate the efficiency of the algorithm of generating the supporting surface, the ball model is generated and saved to have a different number of facets, and the supporting space is divided into different amount of grids to compare the time required to generate the supporting surface. Fifty percent of the height is designed to be covered. The result is shown in Fig. 8. The result shows that the amount of facets has minor effect on the time needed to obtain the supporting surface, while the number of base grids plays a major role on that. The complexity of the algorithm is verified to be linear, which can be observed from Fig. 8. 4.2 Example 2: automobile suspension model In the second example, an automobile suspension model is used. This model consists of 55 individual components: wheels, motor, transmission gears, differential gears, frames, etc., totaling 140,640 facets. A 300524 mesh grids are created in the bottom plane of supporting space. The height of the bounding box is chosen as the 75% of the height of the model. It is obvious that models with multi- component will pose great challenge when the parting line Fig. 10 Detection of design space. a bounding box and cut- ting plane, b supporting space, c final design space Fig. 11 Topology optimization result for a box model. a design space with loading and supports, b final design from topology optimization, c location of reinforcements 606 Chang et al. algorithm is use. However, the method presented here can still generate the correct foam packaging structure automat- ically. The final supporting space is generated in 0.83 s. The results are shown in Fig. 9. 4.3 Example 3: a box model A simple square box model is presented in the third example for thin-walled packaging structure design because it is a simple convex model. First, the box is placed in a bounding space that is 30% larger along all its edges. A bounding plane is set at the height equals 50% of the total height as shown in Fig. 10a. Then, the height detection grid is defined to have 3030 points, and the heights are accordingly obtained as shown in Fig. 10b. Followed by the design space generation, a thin-walled design space is obtained as shown in Fig. 10c. The loadings and constraints are assigned by designer according to its applications. As an example to demonstrate this method, a simply supported four corners are modeled with a concentrated load at the center of the bottom plane as shown in Fig. 11a. The allowable volume constraint is 8% of total volume in addition to the base material that have a minimum stiffness bound. A set of symmetric constraints for x- and y-directions is specified. The result of reinforce- ment is shown in Fig. 11b. The optimization process is converged at 38 iterations, and the final optimal reinforce- ment is shown in Fig. 11c with the base material removed to show the locations of the reinforcement. 4.4 Example 4: toy car model The last example is a more complicated toy car model because it consists of many small components such as a car body, four wheels, and two fog lights. Similar to the procedures described in the first example, the height is also set to 50% of the total height of the bounding box and the mesh grids are defined to be 3072. A foam packaging space can be obtained after height detection as shown in Fig. 12. Note that the foam packaging space is an under-cut free struct