WO2023035132A1 - 一种基于薄壳结构的参数化雕刻设计方法 - Google Patents
一种基于薄壳结构的参数化雕刻设计方法 Download PDFInfo
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- 238000013461 design Methods 0.000 title claims abstract description 64
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- 238000010586 diagram Methods 0.000 abstract description 3
- 238000009826 distribution Methods 0.000 abstract description 3
- 238000004519 manufacturing process Methods 0.000 description 6
- 238000007639 printing Methods 0.000 description 3
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- 238000012938 design process Methods 0.000 description 2
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/10—Geometric CAD
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- G—PHYSICS
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- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
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- G—PHYSICS
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- G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
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- the invention belongs to the field of computer-aided design and mechanical manufacturing, and the main content is to propose a parametric modeling method based on a thin shell structure to carry out engraved and hollowed-out design and manufacture of the structure surface, which can be applied to the fields of architecture, medicine, engineering design and the like.
- the invention utilizes an implicit modeling method to carry out parametric engraving design on the thin shell structure, and combines the structural flexibility minimization optimization model in the field of mechanics to propose a complete set of design and optimization modeling framework.
- the optimized structure can have the maximum structural stiffness under the volume specified by the user.
- the developed inventive algorithm utilizes function representation for analysis, optimization and storage, which has high efficiency, and can be easily extended to different applications, is universal, and the algorithm is also highly robust.
- the purpose of the present invention is to propose a complete set of design and optimization modeling framework.
- a parametric engraving design method based on thin shell structure the steps are as follows:
- the present invention expresses and optimizes the structure based on the method of implicit modeling, and uses the form of function to represent the thin shell structure and carved parts, so that the fusion (Boolean operation) between them becomes simple and easy, and has great advantages. High efficiency and wide versatility, and compatible with mechanical analysis and optimization frameworks.
- a thin shell structure is a layered structure with a certain thickness, which is much smaller than other dimensions of the overall model.
- the input is a manifold surface S 0 of arbitrary shape, which is generally represented by a triangular mesh. Then calculate the vertex normal vector of the surface, and offset the surface equidistantly along the positive and negative directions of the normal vector In this way, a thin shell structure S T with thickness h 0 can be obtained.
- the directed distance field SDF of the thin shell structure in the entire design domain ⁇ is calculated as its function description. That is to say:
- the sculpted part is modeled independently of the input shell structure, and the sculpted holes are implemented by simple Boolean operations with the shell structure.
- the carving part should have the following two characteristics: 1) it can be expressed implicitly in the form of a function; 2) it has controllable parameters, and the parameters can adjust the shape, size, placement and direction of the carving part .
- the function of the hyperellipsoid is expressed as:
- p is an even number, which can control the shape of the hyperellipsoid;
- (x, y, z) ⁇ is any point in the design domain,
- (x 0 , y 0 , z 0 ) is the coordinates of the center of the hyperellipsoid, Constrain it to the center plane of the shell structure, and adjust the position of the hyperellipsoid on the shell structure through it.
- L 1 , L 2 and L 3 can control the lengths of the three axes of the hyperellipsoid, which are used to determine the size of the engraved part. When the lengths of the three axes are equal, the engraved part degenerates into a hypersphere.
- the combination of multiple hyper ellipsoids can be achieved by rotating the super ellipsoid at an equal angle on the xy-plane of the local coordinate system of the hyper ellipsoid. Specifically, if there are n hyperellipsoids combined at one position, the angles to be rotated by the hyperellipsoids are respectively For the kth ellipsoid, compute the rotation matrix
- the new hyperellipsoid ⁇ 0 can be obtained by compounding the previous original rotation matrix R k R 0 , and then the fusion of multiple hyperellipsoids can be obtained through Boolean operations, because the implicit method of function representation is used here, So boolean operations can be transformed into simple function operations:
- the controllable parameters of the hyperellipsoid of the engraving part are ⁇ x 0 , y 0 , z 0 , L 1 , L 2 , L 3 , R 0 , k ⁇ , by adjusting these parameters, the hyperellipsoid can be controlled in The position, size, and placement direction of the thin shell structure. Different parameters can be selected to obtain different engraving and digging effects. Users can apply parameter optimization designs according to different needs to obtain structures that meet individual requirements.
- the engraving and digging design on the thin shell structure can be completed only through simple Boolean operations. And because the representation of the structure is represented by a function, the Boolean operations here can be handled in a simple implicit way, which greatly simplifies the algorithm and improves the efficiency.
- the parametric design of engraving work is divided into the following steps:
- the original curved surface is divided into polygons by the method of Voronoi diagram, and the engraved parts are limited in the polygonal area obtained by the division, thereby determining the number of engraved parts and the position of the central point.
- the maximum axial length of the super ellipsoid should be determined according to the polygon.
- the axial length range of the super ellipsoid should be:
- D i is the shortest distance from the centroid q i of polygon P i to the boundary of the polygon
- d 0 is the minimum printing accuracy. Limitations in the invention and is an optimization variable for adjustable control.
- Boolean operations can be transformed into the following simple functional operations:
- ⁇ s min(SDF, ⁇ 1 , ⁇ 2 ,..., ⁇ NC )(10)
- ⁇ s is the functional description of the thin shell structure after digging out the engraved parts, satisfying:
- the invention adopts the flexibility optimization model, combines with the mechanical index to drive the parametric design of the engraving of the thin shell structure, and optimizes the distribution and shape of the engraving holes under the specified volume, so that it has the maximum structural rigidity.
- the flexibility minimization optimization model is the most common problem form, which reflects the relationship between the basic mechanical properties and the shape of the structure. It takes the minimum strain energy as the goal and takes the volume as the constraint to optimize the parameter variables to be determined.
- the specific problem form is as follows:
- ⁇ is the design domain, defined as the smallest bounding box occupied by the thin shell structure
- f is the body force
- s is defined in the Riemannian boundary
- ⁇ The surface force on s
- u is the displacement field
- v is the test function defined on the design domain ⁇
- U ad ⁇ v
- v 0 ⁇ on ⁇ u
- H 1 is The first-order Sobolev space
- ⁇ is the second-order linear strain tensor
- E is the fourth-order elastic tensor, which is determined by the elastic modulus and Poisson's ratio of the material.
- H(x) is defined as:
- the problem is discretized based on the finite element subdivision method, and then the numerical optimization method is used to automatically solve the problem to obtain the optimal parameter variable value.
- the method of multi-grid is adopted in the present invention to speed up the calculation of mechanical response analysis.
- the design domain ⁇ is divided into two types of uniform hexagonal volume grids with different precisions—coarse elements and fine elements.
- Coarse elements are mainly used to interpolate to generate displacement field functions and construct stiffness matrices, while fine elements are mainly used to represent models and perform integral calculations.
- the element stiffness matrix defined above is:
- ⁇ i is the area occupied by the i-th coarse element
- B is the strain matrix
- D i is the constitutive matrix
- n b represents the number of fine elements in the coarse element
- E ij is the elastic modulus value at the node
- D 0 is the constitutive matrix of the full material element under constant Young's modulus
- x ij is the position coordinate of the integration point inside the fine element
- v b is the volume of the fine element.
- I is the strain energy of the engraving model
- U is the offset vector
- F is the nodal force vector
- N b is the total number of fine elements in the solution domain
- V is the volume fraction of the model in the design domain
- ⁇ s is the value of the shape description function ⁇ s at the lth node of the jth fine unit.
- ⁇ is a very small positive number, and ⁇ is determined by the precision of fine elements in finite element division.
- the numerical optimization method can be used to iteratively solve the constructed optimization problem to obtain the optimized parameters.
- the effective set method is used to solve the optimization problem, and the sensitivity information of the objective function and the constraint function on the optimization variable is required, namely
- L i is the parameter of the i-th engraved part
- N s is the number of coarse units
- N b is the number of fine units
- U k is the displacement vector of the k-th thick unit
- K 0 is the local stiffness matrix of the kth coarse element fully filled with material
- v b is the volume of the fine element.
- the present invention belongs to the intersection field of computer-aided design and mechanical manufacturing. Facing the needs of 3D printing and industrialization, a parametric modeling method is proposed on the thin shell structure, and the surface of the structure is engraved and hollowed out, which can be applied to architecture, medicine and fields of engineering design.
- this kind of engraving and hollowing out design can realize the lightweight of the structure, and reduce the consumption of structural materials under the premise of ensuring the mechanical properties; for the biological or medical field, this kind of engraving and hollowing out design can be beneficial to the heat dissipation of functional structures, Cell migration, etc., has very important application value.
- the present invention proposes an implicit modeling method, transforms the engraving design of the structural surface into a parametric design method, and establishes a complete system from design to optimization to production. It expresses, analyzes, optimizes and stores the structure in a form, which is very efficient and convenient. Because of the implicit method, the functional analysis and optimization frameworks can be cleverly coupled, greatly reducing the complexity of calculations, improving efficiency, shortening the entire design and optimization cycle, and meeting the requirements of production and optimization. Daily needs. The present invention also has very strong expansibility.
- Fig. 1 is a flow chart of parametric engraving design and optimization based on thin shell structure.
- Figure 2 is a diagram of the parametric engraving design and optimization results based on the thin shell structure.
- Fig. 2 (a) the input original thin shell structure; (b) the sculpted part designed by the hyperellipsoid; (c) the Voronoi subdivision of the original mesh surface; (d) the optimization result of the sculpted design.
- the first step is to obtain the thin shell structure and make it implicit.
- the specific method is to calculate the vertex normal vector of the surface first, and offset S 0 along the positive and negative directions of the normal vector at equal intervals
- a thin shell structure S T with thickness h 0 is obtained.
- Then compute the directed distance field SDF of ST in the whole design domain ⁇ as its implicit representation.
- Boolean operations can also be reduced to implicit operations:
- ⁇ s min(SDF, ⁇ 1 , ⁇ 2 , . . . , ⁇ NC )
- SDF is the functional description of the thin shell structure, is the shape description function of NC carved parts, ⁇ s is the function representation of the final structure, satisfying:
- the carved parts are used to carve and dig holes on the thin shell.
- the multi-grid method is used to discretize the above problem, and the design domain ⁇ is evenly divided into two sets of grid units with different thicknesses, that is, first divided into coarse units, and then each coarse unit is subdivided into fine units.
- Coarse elements are used to interpolate the displacement field function, and fine elements are used to describe the model and perform integral calculations.
- the discrete form of the optimization problem can be obtained:
- I is the strain energy of the engraving model
- U is the offset vector
- F is the nodal force vector
- N b is the total number of fine elements in the solution domain
- V is the volume fraction of the model in the design domain
- ⁇ s is the value of the shape description function ⁇ s at the lth node of the jth fine unit.
- the optimal solution of the optimization problem can be obtained.
- the engraved hollow thin shell with the smallest energy and the largest structural flexibility under the volume specified by the user is obtained.
- the structure reduces the mass of the object and reduces the consumption of materials under the premise of satisfying certain force conditions.
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Abstract
一种基于薄壳结构的参数化雕刻设计方法，在输入任意一种三角网格表示的流形曲面后，通过偏移一定的厚度得到薄壳结构。计算曲面的Voronoi剖分得到Voronoi多边形分布图，确定雕刻部件的个数、中心点位置和容许的最大尺寸。根据曲面的法向确定雕刻部件的摆放方向。基于薄壳结构的参数化雕刻设计方法结合力学中的结构柔度优化模型，以全局结构刚度最大化为目标，以体积为约束，构建雕刻优化设计建模问题，给出对应的离散化形式。再利用有效集解优化算法对优化问题进行求解，得到在指定体积下刚度最大的雕刻设计结果。基于薄壳结构的参数化雕刻设计方法提出了完整的薄壳结构的参数化雕刻设计与优化框架，具有高效、通用、鲁棒性强等优点，大大缩短了设计与优化周期，丰富了结构设计的多样性。
Description
本发明属于计算机辅助设计和机械制造领域，主要内容为基于薄壳结构提出了一种参数化建模方法，进行结构表面的雕刻镂空设计和制造，可应用于建筑、医学和工程设计等领域。
随着增材制造技术的不断发展和普及，越来越多形状各异、功能繁多的复杂结构被设计出来满足人们的个性化需求和设计美学的发展，在这其中，参数化设计方法受到越来越多的关注和应用，其就是将结构转化为参数或函数形式表示，通过调整参数去控制整个建模过程，实现设计的自动化。用户可以根据不同的需求去设计参数调整的方法，比如结合力学方面进行拓扑优化，设计出轻量化的模型；结合热学的知识，根据温度变化对模型进行优化设计等等。近些年，有很多基于薄壳结构的雕刻设计工作涌现出来，一方面提高了设计的多样性，有美学的价值；另一方面，有工程力学、生物医学、热力学等多方面的实际意义。然而目前大部分相关的工作都只是基于薄壳结构提出了雕刻挖洞的设计方案，并没有一套完整的优化方法能够根据用户的不同需求进行优化设计，得到满足不同需求的个性化优化设计结果。
本发明利用隐式建模的方法，在薄壳结构上进行参数化雕刻设计，结合力学领域的结构柔度最小化优化模型，提出了一整套完整的设计和优化建模框架。所优化出来的结构能够在用户规定的体积之下，具有最大的结构刚度。所研发的发明算法利用函数表示进行分析、优化和存储，有很高的效率，而且能够轻松扩展到不同方面的应用，具有普适性，且算法也具有很强的鲁棒性。
发明内容
本发明目的是提出了一整套完整的设计和优化建模框架。
本发明采用的技术方案是：
一种基于薄壳结构的参数化雕刻设计方法，步骤如下：
(一)隐式化建模方法
本发明是基于隐式建模的方法进行结构的表示和优化的，都是采用函数的形式表示薄壳结构和雕刻部件，使得它们之间的融合(布尔操作)变得简单易行，具有很高的效率和广泛的通用性，并且能够兼容力学分析和优化框架。
1.薄壳结构的隐式化
薄壳结构就是具有一定厚度的层状结构，其厚度远小于整体模型的其它尺寸。本发明中，输入的是一个任意形状的流形曲面S
0，其一般是由三角网格表示的。然后计算曲面的顶点法向量，将曲面沿着法向量正负方向分别等距偏移
这样就可以得到厚度为h
0的薄壳结构S
T。为了用隐式的方法对薄壳结构进行表示，接下来计算整个设计域Ω中薄壳结构的有向距离场SDF作为其函数描述。即满足：
其中，x＝(x,y,z)∈Ω为设计域中的任何一点。
2.雕刻部件的隐式化
雕刻部件作为独立于输入薄壳结构的模型，通过与薄壳结构简单的布尔操作实现雕刻挖洞。本发明中，雕刻部件应该具有下面两个特征：1)能够用函数形式进行隐式表示；2)具有可控参数，参数能够对雕刻部件的形状、尺寸大小、摆放位置和方向等进行调节。有各种各样的模型满足以上两个特征，本发明选用超椭球体及其组合作为雕刻部件。
超椭球体的函数表示为：
其中，p为偶数，可以控制超椭球体的形状；(x,y,z)∈Ω为设计域中的任一点，(x
0，y
0，z
0)为超椭球体的中心位置坐标，将其限制在薄壳结构的中心面上，并通 过它调整超椭球体在薄壳结构上的位置。L
1，L
2和L
3可以控制超椭球体三个轴的长度，用来决定雕刻部件的尺寸大小，当三个轴长相等时，雕刻部件退化为一个超球体。旋转矩阵R
0＝{R
ij}
3×3可以调节超椭球体局部坐标系的旋转，用来控制超椭球体的摆放方向。
在确定了一个超椭球体的位置、大小和摆放方向之后，可以通过在超椭球体局部坐标系的xy-平面上对超椭球体进行等角度旋转实现多个超椭球体的组合。具体地，如果设置一个位置处有n个超椭球体进行组合，则超椭球体要旋转的角度分别是
对于第k个椭球体，计算旋转矩阵
复合上之前的原始旋转矩阵R
kR
0就可以得到新的超椭球体φ
0，然后通过布尔操作就可以得到多个超椭球体的融合，因为这里都采用了函数表示的隐式化方法，所以布尔操作可以转化为简单的函数操作：
到此为止，雕刻部件超椭球体的可控参数有{x
0，y
0，z
0，L
1，L
2，L
3，R
0，k}，通过调节这些参数就可以控制超椭球体在薄壳结构上的位置，尺寸大小，摆放方向等。选用不同的参数，就可以得到不同的雕刻挖洞效果，用户可以根据不同的需求应用去参数优化设计，得到满足个性化要求的结构。
3.参数化雕刻设计
有了薄壳结构和雕刻部件的函数表示之后，只需要通过简单的布尔操作就可以完成在薄壳结构上的雕刻挖洞设计。而因为结构的表示都采用了函数表示，所以这里的布尔操作可以用简单的隐式化方式来处理，大大简化了算法，提高了效率。
雕刻工作的参数化设计分为以下几步：
首先确定雕刻部件在薄壳结构上的分布，即确定雕刻部件的个数及其中心点的位置。本发明中通过Voronoi图的方法对原始曲面进行多边形分割，将雕刻部件限制在分割得到的多边形区域中，从而确定雕刻部件的个数及其中心点的位置。在指定了Voronoi分割的精度之后，对原始曲面S
0进行自适应多边形剖分，得到多边形集合
确定了有NC个雕刻部件，并以多边形的质心
为雕刻部件的中心位置。
然后确定雕刻部件的局部坐标系，即确定原始旋转矩阵
在得到雕刻部件的中心点位置之后，计算出该点处关于原始曲面S
0的外法向量
以外法向量为每个雕刻部件的新z轴计算旋转矩阵
接着确定雕刻部件的类型，指定由n个超椭球体进行融合得到雕刻部件之后，根据公式(4)计算xy-平面旋转矩阵
则每个超椭球体的函数表达为
其中，为了将超椭球体限制在所属的多边形区域中，要根据多边形来确定超椭球体的轴长最大值，同时为了满足实际打印需求，则超椭球体的轴长范围应为:
其中，D
i为多边形P
i的质心q
i到多边形边界的最短距离，d
0为最小打印精度。本发明中限制
和
为可调节控制的优化变量。
最后将雕刻挖洞的操作转化为薄壳结构和雕刻部件之间的布尔操作，即从薄壳结构上去除属于雕刻部件的部分，即可得到最后的雕刻结果。因为薄壳结构和雕刻部件都是用函数隐式表示的，所以布尔操作可以转化为以下简单的函 数操作：
φ
s＝min(SDF，φ
1，φ
2，...，φ
NC)⑽
这样，φ
s就是薄壳结构挖去雕刻部件之后结构的函数描述，满足：
然后采用传统的Marching Cube算法从形状描述函数φ
s提取结构的三角网格表示。
以上就完成了整个雕刻的参数化设计过程，可以发现整个处理方法都是采用函数隐式表示的方式，大大简化了算法复杂度，提高了效率。设计的结构形状丰富多样，可控性强，而且能够轻松拓展到多种应用中去。
(二)优化问题的建模及求解
本发明采用柔度优化模型，结合力学指标驱动薄壳结构雕刻的参数化设计，在指定的体积下，优化雕刻孔洞的分布和形状，使其具有最大的结构刚度。
1.优化问题建模
力学应用中，柔度最小化优化模型是最常见的问题形式，它反映了基本力学性能和结构形状之间的关系。其是以应变能最小为目标，以体积为约束去优化待确定的参数变量。结合上面介绍的雕刻参数化设计方法，具体问题形式如下：
使得
其中，Ω为设计域，定义为薄壳结构所占的最小包围盒，x＝(x，y，z)∈Ω为设计域中任意一点，f为体积力，s为定义在黎曼边界Γ
s上的面力，u是位移场，v是定义在设计域Ω上的测试函数，U
ad＝{v|v∈H
1(Ω
M)，在Γ
u上v＝0}，H
1为一阶索伯列夫空间，ε为二阶线性应变张量，E为四阶弹性张量，由材料的弹性模量和泊松比决定。
为定义在狄利克雷边界Γ
u上的位移约束，
为体积约束值，Heaviside函数H(x)定义为：
2.问题离散化
有了上述问题的连续形式，接下来基于有限元剖分的方法对问题进行离散化，然后用数值优化方法进行自动的求解，得到最优化的参数变量值。为了提高优化求解的效率，本发明中采用了多重网格的方法来加快力学响应分析的计算。具体地，将设计域Ω划分为两种不同精度的均匀六边形体网格——粗单元和细单元。粗单元主要用来插值生成位移场函数，构建刚度矩阵，细单元主要用来表示模型和进行积分的计算。对于第i个粗单元，定义在上面的单元刚度矩阵为：
其中，Ω
i是第i个粗单元所占的区域，B为应变矩阵，D
i为本构矩阵，n
b表示粗单元内细单元的个数，E
ij为节点处的弹性模量值，D
0为常值杨氏模量下满材料单元的本构矩阵，x
ij为细单元内部积分点的位置坐标，v
b为细单元的体积。 通过这种方式求得多有的局部单元刚度矩阵之后，就可以整合得到总体刚度矩阵K，然后就可以得到优化问题(12-13)的离散形式：
使得
其中，I为雕刻模型的应变能，U为偏移量向量，F为节点力向量，N
b为求解域中总的细单元的个数，V为模型占设计域的体积分数，
为指定的体积约束，
为第j个细单元的第l个节点处形状描述函数φ
s的值。
其中，a为一个很小的正数，η由有限元剖分时细单元的精度决定。
3.求解优化问题
有了上面优化问题的离散形式，就可以采用数值优化方法来对构建的最优化问题进行迭代地求解，得到最优化的参数，这里需要优化的参数一共有2＊NC个，即
本发明中采用有效集的方法去求解优化问题，需要目标函数和约束函数关于优化变量的敏感度信息，即
其中，L
i为第i个雕刻部件的参数，N
s为粗单元的个数，N
b为细单元的个数，U
k为第k个粗单元的位移向量，
为第k个粗单元中第j个细单元的第l个节点处φ
s的值，K
0为第k个粗单元满材料填充下的局部刚度矩阵，v
b为细单元的体积。然后将目标函数值、约束函数值以及一阶梯度信息传递给解优化算法中进 行迭代地求解，就可以得到优化问题(16-17)的局部最优解。将得到的最优化参数值传入参数化雕刻设计方法中，就可以获得最优化的雕刻模型，其在用户指定的体积下，具有最大的结构刚度。
本发明属于计算机辅助设计和机械制造的交叉领域，面向3D打印和工业化需求，在薄壳结构上提出了一种参数化建模方法，对结构表面进行雕刻镂空设计，可应用于建筑、医学和工程设计等领域。对于工业领域，这种雕刻镂空设计可以实现结构的轻量化，在保证力学性能的前提下，减少结构材料的消耗；而对于生物或者医学领域，这种雕刻镂空设计可以有利于功能结构的散热、细胞迁移等，具有十分重要的应用价值。本发明提出了一种隐式化建模方法，将结构表面的雕刻设计转化为参数化设计的方式，建立了一套完整的从设计到优化再到制作生产的完整系统，该系统完全用函数形式进行结构的表示、分析、优化和存储，表现得非常的高效，便捷。因为采用了隐式的方法，功能分析和优化框架之间能够巧妙地进行耦合，很大程度上降低了计算的复杂度，提高了效率，缩短了整个设计与优化的周期，并能够满足生产和生活需求。本发明还具有很强的可扩展性，一方面表现在雕刻部件的多样性，只要满足发明所指出的特征，都可以用来在薄壳结构上进行雕刻设计；另一方面，该框架可以很容易扩展应用到热力学、生物学、声学和光学等领域。
图1是基于薄壳结构的参数化雕刻设计与优化流程图。
图2是基于薄壳结构的参数化雕刻设计与优化结果图。
图2中：(a)输入的原始薄壳结构；(b)由超椭球体设计的雕刻部件；(c)对原始网格曲面进行Voronoi剖分；(d)雕刻设计优化结果。
以下结合附图和技术方案，进一步说明本发明的具体实施方式。
本发明具体实施步骤如下：
1.薄壳结构的构造及其隐式化
在输入三角网格表示的流形曲面S
0并指定厚度之后，第一步要做的是得到薄壳结构并对其隐式化。具体做法是先计算曲面的顶点法向量，将S
0沿着法向量正反方向分别等距偏移
得到厚度为h
0的薄壳结构S
T。然后计算整个设计域Ω中S
T的有向距离场SDF作为其隐式表示。
2.对S
0进行Voronoi剖分
指定剖分精度，根据精度对S
0进行Voronoi剖分，求出Voronoi多边形的质心
作为雕刻部件的中心坐标。计算每个多边形质心到多边形边界的最短距离，并结合打印精度确定雕刻部件的轴长范围
3.雕刻部件的构建及其隐式化
计算中心点处的外法向量，并以此求出旋转矩阵R
0＝{R
ij}
3×3，从而确定雕刻部件的摆放方向。假设雕刻部件采用4个超椭球体融合的方式得到，则4个超椭球体的旋转角度分别是
和π，根据旋转角度确定了xy-平面上的旋转矩阵分量之后，就可以得到每个超椭球体的形状描述函数φ
k。于是，雕刻部件的隐式函数表示就是：
其中，
就是第i个中心点处第j个分超椭球体的形状描述函数，每个超椭球体的轴长L
1和L
2都是可优化的参数，L
3对应z-轴方向的轴长，确定为轴长允许的最大值
4.雕刻设计过程
利用布尔操作将薄壳结构与雕刻部件相交的部分挖去，完成雕刻设计工作。因为薄壳结构和雕刻部件都有隐式的函数表示，布尔操作也可以简化为隐式操作：
φ
s＝min(SDF，φ
1，φ
2，...，φ
NC)
其中，SDF为薄壳结构的函数描述，
是NC个雕刻部件的形状描述函数，φ
s即为最终结构的函数表示，满足：
其中，x＝(x,y,z)∈Ω为设计域中的任何一点。
5.优化模型构建
以结构刚度最大化为目标，以体积为约束，用雕刻部件在薄壳上进行雕刻挖洞设计，需要优化的参数一共有2＊NC个，即
通过优化使得在给定约束材料体积的前提下，结构的全局刚度最强。首先采用多重网格的方法对上述问题进行离散化，将设计域Ω均匀分为两套粗细不同的网格单元，即先分为粗单元，然后将每个粗单元细分为细单元，用粗单元去插值位移场函数，用细单元去描述模型且进行积分计算。计算所有的局部单元刚度矩阵，然后整合成总体的刚度矩阵K之后，就可以得到优化问题的离散形式：
使得
其中，I为雕刻模型的应变能，U为偏移量向量，F为节点力向量，N
b为求解域中总的细单元的个数，V为模型占设计域的体积分数，
为指定的体积约束，
为第j个细单元的第l个节点处形状描述函数φ
s的值。
6.优化问题求解
求出目标函数和约束函数关于优化变量的敏感度信息：
将敏感度信息代入数值解优化算法——有效集法中去，即可得到优化问题的最优解，这样就得到了满足用户指定体积之下，能量最小即结构柔度最大的雕刻 镂空薄壳结构，在满足一定受力条件的前提下减小了物体的质量，减少了材料的消耗。
Claims (1)
- 一种基于薄壳结构的参数化雕刻设计方法，其特征在于，步骤如下：(一)薄壳结构和雕刻部件的隐式表示输入由三角网格表示的任意形状的流形曲面S 0；然后计算流形曲面的顶点法向量，将流形曲面沿着法向量正负方向分别等距偏移得到厚度为h 0的薄壳结构S T；接下来计算整个设计域Ω中薄壳结构的有向距离场SDF作为其函数描述；选用n个超球体的融合来构建形状丰富的雕刻部件；先在指定精度下计算流形曲面S 0的Voronoi剖分，用Voronoi的多边形来确定雕刻部件的中心位置
和轴长范围
通过计算雕刻部件中心点的外法向量确定雕刻部件局部坐标系的旋转矩阵R 0＝{R ij} 3×3，从而确定雕刻部件的摆放方向；如果该中心点处确定有n个超椭球体构建雕刻部件，则根据等分原则确定旋转角
然后计算旋转矩阵R kR 0确定这n个超椭球体的旋转方向，从而得到雕刻部件的函数表示：
其中，就是第i个中心点处第j个分超椭球体的形状描述函数，每个超椭球体的轴长L 1，L 2和L 3都是可优化的参数；
(二)基于隐式表示的参数化雕刻设计雕刻设计就是在薄壳结构上挖去雕刻部件所占的部分，采用布尔操作来处理，因为薄壳结构和雕刻部件都有隐式的函数表示，布尔操作简化为隐式操作：φ s＝min(SDF,φ 1,φ 2,...,φ NC)其中，SDF为薄壳结构的函数描述，是NC个雕刻部件的形状描述函数，φ s即为最终结构的函数表示，满足：
其中，x＝(x,y,z)∈Ω为设计域中的任何一点；(三)优化模型构建及其求解以结构刚度最大化为目标，以体积为约束，用雕刻部件在薄壳结构上进行雕刻挖洞设计，需要优化的参数一共有2*NC个，即通过优化使得在给定约束材料体积的前提下，结构的全局刚度最强；采用有限元分析的方法对优化问题进行求解，首先对上述问题进行离散化；采用多重网格的方法，在保证计算精度的同时提高计算效率，将设计域Ω均匀分为两套粗细不同的网格单元，即先分为粗单元，然后将每个粗单元细分为细单元，用粗单元去插值位移场函数，用细单元去描述模型且进行积分计算；计算所有的局部单元刚度矩阵，然后整合成总体的刚度矩阵K后，得到优化问题的离散形式：
使得KU＝F
其中，I为雕刻模型的应变能，K为总刚度矩阵，U为偏移量向量，F为节点力向量，N b为求解域中总的细单元的个数，V为模型占设计域的体积分数，H(x)为Heaviside函数，η为Heaviside函数的正则化参数，为指定的体积约束，
为第j个细单元的第l个节点处形状描述函数φ s的值；
然后求出目标函数和约束函数关于优化变量的敏感度信息：
其中，L i为第i个雕刻部件的参数，N s为粗单元的个数，N b为细单元的个数，U k为第k个粗单元的位移向量，为第k个粗单元中第j个细单元的第l个节点处φ s的值，K 0为第k个粗单元满材料填充下的局部刚度矩阵,v b为细单元的体积；将敏感度信息代入数值解优化算法——有效集法中去，即得到优化问题的最优解，这样就得到了满足用户指定体积之下，能量最小即结构柔度最大的雕刻镂空薄壳结构，在满足一定受力条件的前提下减小了物体的质量，减少了材料的消耗。
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