Lattice structures with excellent physical properties have attracted great research interest. In this paper, a novel volume parametric modeling method based on the skeleton model is proposed for the construction of three-dimensional lattice structures. The skeleton model is divided into three types of nodes. And the corresponding algorithms are utilized to construct diverse types of volume parametric nodes. The unit-cell is assembled with distinct nodes according to the geometric features. The final lattice structure is created by the periodic arrangement of unit-cells. Several different types of volume parametric lattice structures are constructed to prove the stability and applicability of the proposed method. The quality is assessed in terms of the value of the Jacobian matrix. Moreover, the volume parametric lattice structures are tested with the isogeometric analysis to verify the feasibility of integration of modeling and simulation.

In recent years, the extensive application of additive manufacturing (AM) has rendered possibilities for fabricating complex components, such as lattice structures [

In this paper, a volume parametric modeling method of three-dimensional (3D) lattice structures is proposed based on the skeleton. The models constructed by our method are suitable for IGA without any transformation. In the entire modeling process, the construction of the volume parametric nodes is the key step. The unit-cell is assembled with different nodes by permutation and combination. And the lattice structure is created via the periodic arrangement of unit-cells. In addition, the toric surface is used as the transition surface to make the shape of the node smooth, which is beneficial to reduce stress concentration in the analysis. The main contributions are as follows:

A volume parametric modeling method of complex nodes is achieved.

Different types of volume parametric unit-cells and lattice structures are created.

The framework of integration of modeling and simulation is completed.

The remainder of the paper is organized as follows: In

The lattice structures with excellent properties have been widely applied in many industries. Researchers have proposed numerous modeling methods to construct lattice structures. Fan et al. [

Manufacturing lattice structures is not easy via traditional processing technology due to the complex structure. The development of AM solves this problem. Dong et al. [

Compared with the B-rep models mentioned above, the volume parametric models utilize high-order non-uniform rational B-spline (NURBS) to express the physical domains precisely [

In this paper, a volume parametric modeling method of lattice structures based on the skeleton model is proposed. The method solves the problem of hexahedron segmentation of the complex nodes. And the modeling process is simplified by utilizing volume parametric nodes to assemble lattice structures. In addition, the toric surface is used as the transition surface to make the shape of the node smooth. Most importantly, the model constructed by the proposed method is suitable for isogeometric analysis. The proposed method realizes the integration of modeling and simulation.

The physical domain is defined based on the NURBS [

The skeleton is an abstract description of the geometric model. It is widely used in segmentation, reconstruction, and geometric design. Many extraction algorithms are used to construct the skeleton model [

The unit-cell is the basic element to generate a uniform lattice structure. Several representative unit-cells presented by the skeleton models are shown in

In this paper, we defined three types of nodes according to the number of branches. One branch indicates the end nodes

The connection part between nodes is composed of solid cylinders. It is divided into five primitive volumes to satisfy the parameterization requirement, as shown in

The input parameters to the algorithm include the skeleton model

Based on the research of the previous work [

The posture redirection operation ensures the branch curves penetrate from different surfaces of

This is a nonlinear optimization problem. The length of the orthogonal basis

After the posture redirection operation, there may still exist more one branch curves intersect with the same surface

Each surface of

The outer hexahedrons of the node are constructed based on the inner hexahedron group modeling after the above steps. Firstly, construct the transition surfaces of the node. Then, the inner hexahedron group converts into multiple B-spline surfaces. Finally, the volume interpolation is used to generate the outer hexahedron group.

A toric surface is adopted here to create a closed surface of the node mostly because of the advantages of good smoothness [

Here,

It is necessary to segment the outer surface to construct a closed transition surface. The intersection points of the sampling vector

Supposing there are

When the boundary surface in

The vector of each corner

When the corner point

The sampling vector on the edge

The Newton iteration method is used to find the intersection points on the transition surface. The sampling points and the Coons interpolation algorithm are used to fit the B-spline curves

According to the mapping between

The local patches of the outer hexahedron group and the final volume parametric node are shown in

After completing the modeling of the branch node, the cross-section is obtained, as shown in

The end node

The stretch volumes are constructed through affine transformations of the section group along the skeleton curve as shown in

In addition, sweeping is another method to construct nodes. Both

The other control points

Here,

The joint node is constructed with the two section groups

The unit-cell of the lattice structure describes the local characteristics. The geometric parameters that control the specific shape of the unit-cell mainly include the size (i.e., length, width, and height), section radius, and horizontal angles. According to the geometry and topology of the skeleton model, a volume parametric unit-cell is constructed through translation, rotation, and reflection of the nodes. The relationship between the unit-cell

For example, the skeleton of the simple cubic unit-cell shown in

The unit-cell of the uniform lattice structure is a parallelepiped. Three translation basis vectors intersect at the vertex. The skeleton with three orthogonal translation basis vectors is called the cubic lattice structure. Due to the unique symmetry properties, the cubic lattice structure is the most widely used type. The modeling process mainly includes two steps. Firstly, the unit-cell is constructed as the basic element. Secondly, duplicate the unit-cells in space to form the lattice structure. The center of the unit-cell is used to indicate the location. Finally, unit-cells are arranged along the translation basis vectors to generate the volume parametric lattice structures.

A simple cubic lattice structure that satisfies geometric connectivity is shown in

We constructed several models of node and lattice structure to prove the effectiveness and applicability of the proposed method. In the meantime, the Jacobian values of the models are calculated and visualized to evaluate the quality. And the IGA is utilized to verify the feasibility of integration of modeling and simulation of our method.

Several nodes are constructed with the modeling method presented in

Toric surface | ||||
---|---|---|---|---|

Ort-3-valence | 1 | 4 | 8 | 19 |

Ort-4-valence | 1 | 5 | 8 | 23 |

Ort-5-valence | 1 | 6 | 8 | 27 |

Ort-6-valence | 1 | 7 | 8 | 31 |

4-valence | 1 | 5 | 10 | 23 |

6-valence | 1 | 7 | 8 | 31 |

8-valence | 2 | 10 | 10 | 44 |

10-valence | 2 | 12 | 12 | 52 |

12-valence | 4 | 20 | 14 | 76 |

According to

Name | Node number | Node type | Volume patches | Control vertice number | Joint node length |
---|---|---|---|---|---|

Cell-1 | 20 | 212 | 5724 | 16 | |

Cell-2 | 29 | 328 | 8856 | 16 | |

Cell-3 | 35 | 387 | 10449 | 16 | |

Cell-4 | 71 | 61 | 1647 | 8 | |

Cell-5 | 9 | 84 | 2268 | 8 | |

Cell-6 | 11 | 102 | 1647 | 8 |

Based on the abovementioned unit-cells, several different types of volume parametric lattice structures are constructed as shown in

Name | Volume patches | Control vertice number | Size |
---|---|---|---|

Using cell-1 | 4096 | 110592 | 104 * 64 * 64 |

Using cell-2 | 14760 | 398520 | 66 * 66 * 66 |

Using cell-3 | 17425 | 470205 | 66 * 66 * 66 |

Using cell-4 | 3664 | 98928 | 144 * 80 * 46 |

Using cell-5 | 5044 | 136188 | 144 * 80 * 68 |

Using cell-6 | 6124 | 165348 | 144 * 80 * 68 |

To verify the volume parametric model is suitable for IGA without intersections, overlaps, or large-angle distortions of hexahedrons, the value of the Jacobian matrix is the main index for quality assessment [

The Jacobian values of the nodes and the unit cells are shown in

There are some singularities in the models constructed by the proposed method, which are reflected in the minimum values of the Jacobian. Singularities damage the quality of the model. How to control the singularity is a major issue of geometric modeling. Researchers have proposed some approaches, such as utilizing sparse distributed directional constraints to determine the appropriate singularities [

We applicated IGA to the volume parametric models to verify the applicability. As shown in

The boundary conditions are illustrated in

With the above examples, we have proved the volume parametric models constructed by our method are suitable for IGA, and the feasibility of integration of modeling and simulation is verified too.

In this paper, a volume parametric modeling method of 3D lattice structures based on the skeleton model is proposed. The unit-cell is combined with different volume parametric nodes. And the volume parametric lattice structure is assembled with the periodic arrangements of the unit-cells. The effectiveness and stability of the proposed method are proved with numerous examples. The quality of the volume parametric models is evaluated by Jacobian values. And finally, the feasibility of integration of modeling and simulation is proved with isogeometric analysis.

However, the quality of complex nodes is still not perfect enough, and how to control the singularity is also an important issue. In addition, the manufacturing of volume parametric lattice structures is worth to be studied in the future, and applying these models to crystal dynamics is also significant.