CN111428397B - Topological optimization design method considering additive manufacturing structure self-supporting constraint - Google Patents

Topological optimization design method considering additive manufacturing structure self-supporting constraint Download PDF

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CN111428397B
CN111428397B CN202010131080.XA CN202010131080A CN111428397B CN 111428397 B CN111428397 B CN 111428397B CN 202010131080 A CN202010131080 A CN 202010131080A CN 111428397 B CN111428397 B CN 111428397B
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邹君
冯振宇
解江
陈琨
张悦超
柴崇博
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Civil Aviation University of China
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Abstract

本发明公开了一种考虑增材制造结构自支撑约束的拓扑优化设计方法，属于结构优化设计技术领域。基于SIMP方法和结构自支撑数学模型，通过惩罚函数表征结构的自支撑特性并作为显示约束条件，采用自支撑约束灵敏度过滤促进支撑结构演化，得到了考虑增材制造结构自支撑约束的拓扑优化设计方法。该方法通过支撑结构渐进演化使最终拓扑优化结果实现结构自支撑，避免通过二次设计增加支撑结构，从而节省材料成本，缩短设计周期。此外，该方法完全适用于连续体结构拓扑优化，灵活性高，计算代价小，可作为扩展功能植入现有的拓扑优化框架中。

Figure 202010131080

The invention discloses a topology optimization design method considering the self-supporting constraints of additive manufacturing structures, and belongs to the technical field of structure optimization design. Based on the SIMP method and the structural self-supporting mathematical model, a penalty function is used to characterize the self-supporting properties of the structure and serve as a display constraint. The self-supporting constraint sensitivity filter is used to promote the evolution of the support structure, and a topology optimization design considering the self-supporting constraints of the additive manufacturing structure is obtained. method. The method makes the final topology optimization result realize the structure self-support through the gradual evolution of the support structure, and avoids adding the support structure through the secondary design, thereby saving the material cost and shortening the design cycle. In addition, the method is fully suitable for topology optimization of continuum structures, with high flexibility and low computational cost, and can be implanted into existing topology optimization frameworks as an extended function.

Figure 202010131080

Description

一种考虑增材制造结构自支撑约束的拓扑优化设计方法A topology optimization design method considering self-supporting constraints of additively manufactured structures

技术领域technical field

本发明属于结构优化设计相关领域，涉及一种考虑增材制造结构自支撑约束的拓扑优化设计方法。The invention belongs to the related field of structural optimization design, and relates to a topology optimization design method considering the self-supporting constraints of additive manufacturing structures.

背景技术Background technique

拓扑优化是一种先进的结构设计方法，其能通过利用数值优化技术在满足一定约束条件下寻找到最优的材料布局。与传统的尺寸优化和形状优化相比，极大地拓展了设计空间，从而能够设计出创新性、高性能的结构布局形式。自80年代后期提出拓扑优化概念以来，拓扑优化理论与方法已得到广泛关注和飞速发展。典型的拓扑优化方法包括变密度法、渐进法、均匀化方法等。在消除棋盘格式、网格依赖性等数值不稳定现象方面也发展了多种技术，最常用的是密度过滤或灵敏度过滤技术。但由于拓扑优化结果往往几何构型复杂，采用传统的制造工艺制造非常困难，因此需要设计人员基于制造技术或经验对优化结果进行二次设计以满足可制造性，从而失去了产品在轻量性、高性能上的优势。Topology optimization is an advanced structural design method, which can find the optimal material layout under certain constraints by using numerical optimization techniques. Compared with the traditional size optimization and shape optimization, the design space is greatly expanded, so that innovative and high-performance structural layout forms can be designed. Since the concept of topology optimization was proposed in the late 1980s, topology optimization theory and methods have received extensive attention and developed rapidly. Typical topology optimization methods include variable density method, asymptotic method, homogenization method and so on. Various techniques have also been developed to eliminate numerical instabilities such as checkerboard format, grid dependence, etc. The most commonly used technique is density filtering or sensitivity filtering. However, because the topology optimization results are often complex in geometric configuration, it is very difficult to manufacture by traditional manufacturing processes. Therefore, designers need to re-design the optimization results based on manufacturing technology or experience to meet the manufacturability, thus losing the product's lightness. , high performance advantage.

增材制造的出现很好地解决了拓扑优化制造难题。与传统制造方法不同，增材制造方法是通过材料逐层累加的方法来制造实体零件。相对于传统的材料切削加工技术，增材制造能实现“自由制造”，解决许多过去难以制造的复杂结构零件的成形，可大大减少加工工序，缩短加工周期，降低研发制造成本。但增材制造并非可以完全自由地进行零部件设计，存在一些独特的制造约束。拓扑优化技术应用于增材制造的最大瓶颈是拓扑优化算法中没有考虑增材制造约束。The emergence of additive manufacturing has solved the problem of topology optimization manufacturing well. Unlike traditional manufacturing methods, additive manufacturing methods create solid parts by adding materials layer by layer. Compared with traditional material cutting processing technology, additive manufacturing can realize "free manufacturing" and solve the forming of many complex structural parts that were difficult to manufacture in the past, which can greatly reduce the processing procedures, shorten the processing cycle, and reduce the R&D and manufacturing costs. But additive manufacturing does not offer complete freedom in component design, and there are some unique manufacturing constraints. The biggest bottleneck in the application of topology optimization technology to additive manufacturing is that the additive manufacturing constraints are not considered in the topology optimization algorithm.

在增材制造过程中，当悬挑结构下表面倾斜角度大于一定值时，在逐层制造过程中容易发生结构坍塌，此时需要在其下方增加支撑结构，同时将上层热量快速传导到基板，以降低温度梯度，减小热变形。支撑结构的使用不仅增加了设计难度，还导致加工时间及成本增加，而且在后期去除时工艺难度较大，破坏了结构的最优性。研究表明，当下表面倾斜角度小于一定值时，结构可以实现自支撑。因此，更好的方法是通过改进设计使结构能完全实现自支撑，从而无需增加支撑结构，由此来降低材料和时间成本。In the additive manufacturing process, when the inclination angle of the lower surface of the cantilever structure is greater than a certain value, the structure is prone to collapse during the layer-by-layer manufacturing process. At this time, a support structure needs to be added below it, and the heat of the upper layer is quickly conducted to the substrate. In order to reduce the temperature gradient and reduce thermal deformation. The use of the support structure not only increases the design difficulty, but also increases the processing time and cost, and the process is difficult to remove in the later stage, which destroys the optimality of the structure. Studies have shown that the structure can be self-supporting when the inclination angle of the lower surface is less than a certain value. Therefore, a better approach is to improve the design to make the structure fully self-supporting so that no additional support structures are required, thereby reducing material and time costs.

针对上述问题，迫切需要开发一种考虑考虑增材制造结构自支撑约束的拓扑优化设计方法，使拓扑优化结果实现结构自支撑，从而零件无需增加支撑结构就能直接加工，节省材料成本，缩短设计周期。In view of the above problems, it is urgent to develop a topology optimization design method that considers the self-supporting constraints of the additive manufacturing structure, so that the topology optimization results can realize the structure self-supporting, so that the parts can be directly processed without adding support structures, saving material costs and shortening the design. cycle.

发明内容SUMMARY OF THE INVENTION

为了解决上述问题，本发明的目的在于提供一种考虑增材制造结构自支撑约束的拓扑优化设计方法。In order to solve the above problems, the purpose of the present invention is to provide a topology optimization design method that considers the self-supporting constraints of additive manufacturing structures.

为了达到上述目的，本发明提供的考虑增材制造结构自支撑约束的拓扑优化设计方法包括按顺序进行的下列步骤：In order to achieve the above object, the topology optimization design method considering the self-supporting constraints of the additive manufacturing structure provided by the present invention includes the following steps in sequence:

1)建立优化模型，定义几何模型、载荷和边界条件，基于SIMP方法框架定义设计变量、目标函数和约束条件，并对单元密度、材料属性参数、结构体分比约束、结构自支撑惩罚函数、优化算法参数进行设置；1) Establish an optimization model, define geometric models, loads and boundary conditions, define design variables, objective functions and constraints based on the SIMP method framework, and determine the element density, material property parameters, structural body fraction constraints, structural self-supporting penalty functions, Optimization algorithm parameters are set;

2)对上述优化模型中的单元密度设计变量进行线性密度过滤而得到单元中间密度；2) Perform linear density filtering on the cell density design variables in the above-mentioned optimization model to obtain the cell intermediate density;

3)采用Heaviside函数对上述单元中间密度进行过滤而得到单元物理密度；3) adopt the Heaviside function to filter the above-mentioned unit intermediate density to obtain the unit physical density;

4)在上述单元物理密度的基础上进行有限元分析、体积约束和结构自支撑约束响应分析，得到结构柔度、结构材料总体积和结构自支撑约束惩罚函数值；4) On the basis of the physical density of the above elements, carry out finite element analysis, volume constraint and structural self-supporting constraint response analysis, and obtain the structural flexibility, the total volume of structural materials and the structural self-supporting constraint penalty function values;

5)对上述结构柔度、结构材料总体积和自支撑约束惩罚函数的函数值进行灵敏度分析，并采用灵敏度过滤算子对自支撑约束惩罚函数的函数值的灵敏度进行过滤；5) Sensitivity analysis is performed on the above-mentioned structural flexibility, the total volume of structural materials and the function value of the self-supporting constraint penalty function, and the sensitivity filter operator is used to filter the sensitivity of the function value of the self-supporting constraint penalty function;

6)采用优化算法对优化模型进行优化求解，更新单元密度设计变量和自支撑约束值，并判断是否收敛，若不收敛返回步骤2)进行算法迭代，若收敛则优化结束，最终输出拓扑优化结果。6) Use the optimization algorithm to optimize and solve the optimization model, update the element density design variables and the self-supporting constraint value, and judge whether to converge, if not, return to step 2) to perform algorithm iteration, if it converges, the optimization is over, and the topology optimization result is finally output. .

在步骤1)中，所述的建立优化模型，定义几何模型、载荷和边界条件，基于SIMP方法框架定义设计变量、目标函数和约束条件，并对单元密度、材料属性参数、结构体分比约束、结构自支撑惩罚函数、优化算法参数进行设置的具体步骤如下：In step 1), the optimization model is established, the geometric model, load and boundary conditions are defined, design variables, objective functions and constraints are defined based on the SIMP method framework, and the element density, material property parameters, structure ratio constraints are The specific steps for setting the structural self-supporting penalty function and optimization algorithm parameters are as follows:

基于SIMP方法框架，以结构柔度最小作为优化目标，将离散化的单元相对密度作为设计变量，以结构体分比约束和结构自支撑约束惩罚函数值作为约束条件，建立优化模型，优化模型的数学表达式为：Based on the SIMP method framework, taking the minimum structural flexibility as the optimization objective, the discretized element relative density as the design variable, and the structural volume fraction constraint and the structural self-supporting constraint penalty function value as the constraint conditions, the optimization model is established. The mathematical expression is:

Figure BDA0002395791720000031
Figure BDA0002395791720000031

其中，c为结构柔度；where c is the structural flexibility;

ρ为单元密度设计变量；ρ is the element density design variable;

Figure BDA0002395791720000032
为过滤后的单元物理密度；
Figure BDA0002395791720000032
is the filtered unit physical density;

U和K分别为总体节点位移向量和总体刚度矩阵；U and K are the overall nodal displacement vector and overall stiffness matrix, respectively;

ue和k0分别为单元位移向量和单元刚度矩阵； ue and k 0 are the element displacement vector and element stiffness matrix, respectively;

F为结构外载荷向量；F is the external load vector of the structure;

N为单元数量；N is the number of units;

V和V0分别为结构材料总体积和设计区域总体积；V and V 0 are the total volume of structural materials and the total volume of the design area, respectively;

f为结构材料体分比约束值；f is the structural material volume fraction constraint value;

Figure BDA0002395791720000033
为结构自支撑约束惩罚函数；
Figure BDA0002395791720000033
is the structural self-supporting constraint penalty function;

ε为自支撑约束值；ε is the self-supporting constraint value;

Figure BDA0002395791720000041
为密度惩罚函数，
Figure BDA0002395791720000042
对中间密度进行惩罚，其中p为惩罚因子，Emin为最小弹性模量，E0为实体单元弹性模量。
Figure BDA0002395791720000041
is the density penalty function,
Figure BDA0002395791720000042
Penalize the intermediate density, where p is the penalty factor, E min is the minimum elastic modulus, and E 0 is the elastic modulus of the solid element.

在步骤2)中，所述的对优化模型中的单元密度设计变量进行线性密度过滤而得到单元中间密度的方法是：In step 2), the described method for performing linear density filtering on the element density design variables in the optimization model to obtain the element intermediate density is:

过滤后的单元中间密度ρ*是对过滤半径rmin的圆形范围内所有单元的密度进行加权平均得到的，具体公式如下：The filtered cell median density ρ * is obtained by weighting the average density of all cells within the circular range of the filter radius r min , and the specific formula is as follows:

Figure BDA0002395791720000043
Figure BDA0002395791720000043

其中，Ne是以单元e为中心，过滤半径为rmin的圆形范围内的单元集合，Hei为单元i和单元e之间的权重系数，Δ(e,i)为单元i和单元e之间的距离。Among them, Ne is the unit set within a circle with the unit e as the center and the filter radius is r min , He ei is the weight coefficient between unit i and unit e, Δ(e, i) is unit i and unit the distance between e.

在步骤3)中，所述的采用Heaviside函数对单元中间密度进行过滤而得到单元物理密度的方法是：In step 3), the method that the described adopting Heaviside function to filter the cell intermediate density and obtain the cell physical density is:

采用Heaviside函数对单元中间密度ρ*进行过滤而得到单元物理密度

Figure BDA0002395791720000044
具体公式如下：The cell physical density is obtained by filtering the cell intermediate density ρ * using the Heaviside function
Figure BDA0002395791720000044
The specific formula is as follows:

Figure BDA0002395791720000045
Figure BDA0002395791720000045

其中，β为控制参数。Among them, β is the control parameter.

在步骤4)中，所述的在单元物理密度的基础上进行有限元分析、体积约束和结构自支撑约束响应分析，得到结构柔度、结构材料总体积和结构自支撑约束惩罚函数值的方法是：In step 4), the finite element analysis, volume constraint and structural self-supporting constraint response analysis are performed on the basis of the physical density of the element, and the method for obtaining the structural flexibility, the total volume of the structural material and the structural self-supporting constraint penalty function value Yes:

基于单元物理密度

Figure BDA0002395791720000046
进行有限元分析得到结构柔度c，对单元物理密度
Figure BDA0002395791720000047
求和得到结构材料总体积V；根据自支撑模型识别无支撑单元，计算结构自支撑约束惩罚函数
Figure BDA00023957917200000413
的函数值
Figure BDA0002395791720000049
并将其作为自支撑约束条件，所述自支撑约束条件为结构自支撑约束惩罚函数
Figure BDA00023957917200000410
的函数值
Figure BDA00023957917200000411
小于等于自支撑约束值ε：
Figure BDA00023957917200000412
based on element physical density
Figure BDA0002395791720000046
Carry out finite element analysis to obtain the structural flexibility c, and the physical density of the element
Figure BDA0002395791720000047
The total volume V of the structural material is obtained by summing up; the unsupported elements are identified according to the self-supporting model, and the penalty function of the structural self-supporting constraint is calculated.
Figure BDA00023957917200000413
the function value of
Figure BDA0002395791720000049
and take it as the self-supporting constraint, which is the structural self-supporting constraint penalty function
Figure BDA00023957917200000410
the function value of
Figure BDA00023957917200000411
Less than or equal to the self-supporting constraint value ε:
Figure BDA00023957917200000412

所述的自支撑模型定义为：各实体单元密度不能大于下方支撑区域单元的最大密度。不考虑边界情况，每个单元由下方3个单元提供支撑，具体公式如下：The self-supporting model is defined as: the density of each solid unit cannot be greater than the maximum density of the unit in the support area below. Regardless of boundary conditions, each unit is supported by the 3 units below. The specific formula is as follows:

Figure BDA0002395791720000051
Figure BDA0002395791720000051

其中，

Figure BDA0002395791720000052
和
Figure BDA0002395791720000053
分别为单元(i,j)物理密度及其支撑区域单元中所有单元的最大密度，i和j分别为单元行号和列号。由于最大值函数不满足可导性要求，不能采用梯度优化算法。因此采用基于P范数的光滑近似函数代替最大值函数以满足可导性要求，具体公式如下：in,
Figure BDA0002395791720000052
and
Figure BDA0002395791720000053
are the physical density of cell (i, j) and the maximum density of all cells in the cell in the support region, respectively, i and j are cell row and column numbers, respectively. Since the maximum value function does not meet the requirements of differentiability, the gradient optimization algorithm cannot be used. Therefore, a smooth approximation function based on the P norm is used to replace the maximum value function to meet the differentiability requirements. The specific formula is as follows:

Figure BDA0002395791720000054
Figure BDA0002395791720000054

其中，参数pn用于控制与最大值函数之间的误差；Among them, the parameter p n is used to control the error between the maximum function;

结构自支撑约束惩罚函数

Figure BDA0002395791720000055
的函数值
Figure BDA0002395791720000056
采用如下公式进行计算：Structural Self-Support Constraint Penalty Function
Figure BDA0002395791720000055
the function value of
Figure BDA0002395791720000056
Calculate using the following formula:

Figure BDA0002395791720000057
Figure BDA0002395791720000057

其中，Mu为违反自支撑约束的单元集合。where Mu is the set of elements that violate the self-supporting constraint.

对于边界情况，自支撑约束响应分析时虚拟在单元物理密度矩阵左右两侧增加1层0密度单元，在下方增加一行值为1的单元代表基板，此时可同样采用公式(5)开展分析。For the boundary case, in the self-supporting constraint response analysis, a layer of 0-density cells is added to the left and right sides of the physical density matrix of the cell, and a row of cells with a value of 1 is added below to represent the substrate. At this time, formula (5) can also be used for analysis.

在步骤5)中，所述的对结构柔度、结构材料总体积和自支撑约束惩罚函数的函数值进行灵敏度分析，并采用灵敏度过滤算子对自支撑约束惩罚函数的函数值的灵敏度进行过滤的方法是：In step 5), the sensitivity analysis is performed on the structural flexibility, the total volume of structural materials and the function value of the self-supporting constraint penalty function, and the sensitivity filter operator is used to filter the sensitivity of the function value of the self-supporting constraint penalty function The method is:

所述的灵敏度分析采用链式法则进行分析，具体如下：The sensitivity analysis described is carried out using the chain rule, as follows:

Figure BDA0002395791720000058
Figure BDA0002395791720000058

其中，φ为所分析的响应变量，后两项分别为Heaviside密度过滤和线性密度过滤方程的导数，分别是对公式(2)和(3)进行求导而得到的。Among them, φ is the analyzed response variable, and the last two terms are the derivatives of the Heaviside density filtering and linear density filtering equations, which are obtained by derivation of formulas (2) and (3), respectively.

结构柔度c对单元物理密度

Figure BDA0002395791720000061
的灵敏度采用经典伴随法公式计算得到，结构材料总体积V对单元物理密度
Figure BDA0002395791720000062
的灵敏度为1。Structural flexibility c versus element physical density
Figure BDA0002395791720000061
The sensitivity of is calculated by the classical adjoint method formula, the total volume V of the structural material is related to the physical density of the unit
Figure BDA0002395791720000062
The sensitivity is 1.

对于结构自支撑约束惩罚函数

Figure BDA0002395791720000063
的函数值
Figure BDA0002395791720000064
的灵敏度分析，在不考虑边界情况下每个单元可为上层3个单元提供支撑；对违反自支撑约束的单元，采用增加下方支撑单元密度的方式来减小结构自支撑约束惩罚函数
Figure BDA0002395791720000065
的函数值；自支撑约束惩罚函数
Figure BDA0002395791720000066
的函数值
Figure BDA0002395791720000067
对各单元的灵敏度按下式进行计算：Penalty function for structural self-supporting constraints
Figure BDA0002395791720000063
the function value of
Figure BDA0002395791720000064
According to the sensitivity analysis of , each unit can provide support for the upper 3 units without considering the boundary; for the unit that violates the self-supporting constraint, the penalty function of the structural self-supporting constraint is reduced by increasing the density of the supporting unit below.
Figure BDA0002395791720000065
The function value of ; the self-supporting constraint penalty function
Figure BDA0002395791720000066
the function value of
Figure BDA0002395791720000067
The sensitivity of each unit is calculated as follows:

Figure BDA0002395791720000068
Figure BDA0002395791720000068

公式右边3项分别为单元物理密度

Figure BDA0002395791720000069
变化时对左上方、正上方和右上方的3个被支撑单元的影响，具体公式如下：The three items on the right side of the formula are the physical density of the unit.
Figure BDA0002395791720000069
The influence of the change on the three supported units at the upper left, the upper right and the upper right, the specific formula is as follows:

Figure BDA00023957917200000610
Figure BDA00023957917200000610

对于边界情况，灵敏度分析时虚拟在单元物理密度矩阵左右两侧增加2层0密度单元，上方增加1层0密度单元，此时可同样采用公式(8)和(9)开展分析。For the boundary case, in the sensitivity analysis, two layers of 0-density cells are added to the left and right sides of the cell physical density matrix, and one layer of zero-density cells is added above.

为促进支撑结构演化，利用灵敏度过滤算子对自支撑约束惩罚函数

Figure BDA00023957917200000611
的函数值
Figure BDA00023957917200000612
的灵敏度进行过滤，过滤后的灵敏度是对该单元上方过滤半径rmin的半圆形范围内所有单元的灵敏度进行加权平均得到的，具体公式如下：In order to promote the evolution of the support structure, the sensitivity filter operator is used for the self-support constraint penalty function
Figure BDA00023957917200000611
the function value of
Figure BDA00023957917200000612
The sensitivity after filtering is obtained by the weighted average of the sensitivity of all units within the semicircular range of the filtering radius r min above the unit. The specific formula is as follows:

Figure BDA00023957917200000613
Figure BDA00023957917200000613

其中，Ne是以单元e为中心，过滤半径为rmin的上方半圆形范围内的单元集合，Hei为单元i和单元e之间的权重系数，Δ(e,i)为单元i和单元e之间的距离，单元i的行号不小于单元e的行号。Among them, Ne is the unit set in the upper semicircle with the filter radius r min as the center of unit e , He ei is the weight coefficient between unit i and unit e, Δ(e, i) is unit i The distance between unit e and the row number of unit i is not less than the row number of unit e.

在步骤6)中，所述的采用优化算法对优化模型进行优化求解，更新单元密度设计变量和自支撑约束值，并判断是否收敛，若不收敛返回步骤2)进行算法迭代，若收敛则优化结束，最终输出拓扑优化结果的方法是：In step 6), the optimization algorithm is used to optimize and solve the optimization model, update the element density design variable and the self-supporting constraint value, and judge whether to converge, if not, return to step 2) to perform algorithm iteration, and if converged, optimize At the end, the method to finally output the topology optimization result is:

所述的优化算法采用MMA优化算法，为使优化过程更加稳定，使自支撑约束值ε随优化过程逐渐减小，具体公式如下：The described optimization algorithm adopts the MMA optimization algorithm. In order to make the optimization process more stable, the self-supporting constraint value ε gradually decreases with the optimization process. The specific formula is as follows:

Figure BDA0002395791720000071
Figure BDA0002395791720000071

其中，loop为循环迭代次数，m和ε0均为常数，NMu为识别的无支撑单元数量。where loop is the number of loop iterations, m and ε 0 are both constants, and N Mu is the number of unsupported units identified.

本发明提供的考虑增材制造结构自支撑约束的拓扑优化设计方法具有如下有益效果：The topology optimization design method considering the self-supporting constraints of the additive manufacturing structure provided by the present invention has the following beneficial effects:

基于SIMP方法框架，结合结构自支撑数学模型，通过惩罚函数表征结构的自支撑特性并作为显示约束条件，采用自支撑约束灵敏度过滤促进支撑结构演化。能够分析结构的自支撑约束特性并进行控制，通过支撑结构渐进演化使最终拓扑优化结果实现结构自支撑，从而可直接打印制造，同时使结构力学性能最优。避免了通过二次设计增加支撑结构与后期去除，有助于节省材料成本，缩短设计周期。与现有的考虑增材制造结构自支撑约束的拓扑优化设计方法相比，本方法通过结构边界支撑结构渐进演化的方式实现结构自支撑，不考虑单元无支撑时的连锁影响，因此该方法计算代价更小。此外，本方法灵活性高，完全适用于连续体结构拓扑优化，可作为扩展功能植入现有的拓扑优化框架中。Based on the SIMP method framework, combined with the structural self-supporting mathematical model, the self-supporting property of the structure is characterized by a penalty function and used as a display constraint, and the self-supporting constraint sensitivity filter is used to promote the evolution of the support structure. The self-supporting constraint characteristics of the structure can be analyzed and controlled, and the final topology optimization result can realize the structure self-supporting through the gradual evolution of the support structure, so that the structure can be directly printed and manufactured, and the mechanical properties of the structure can be optimized at the same time. It avoids adding support structures and removing them later by secondary design, which helps to save material costs and shorten the design cycle. Compared with the existing topology optimization design method that considers the self-supporting constraints of additive manufacturing structures, this method realizes the structural self-supporting through the progressive evolution of the structural boundary support structure, and does not consider the interlocking effect when the element is unsupported. Therefore, this method calculates less expensive. In addition, the method is highly flexible, fully suitable for topology optimization of continuum structures, and can be implanted into existing topology optimization frameworks as an extended function.

附图说明Description of drawings

图1是本发明提供的考虑增材制造结构自支撑约束的拓扑优化设计方法流程图；1 is a flowchart of a topology optimization design method considering the self-supporting constraints of additive manufacturing structures provided by the present invention;

图2是本发明中支撑单元示意图；2 is a schematic diagram of a support unit in the present invention;

图3是本发明中自支撑约束灵敏度过滤示意图，左侧两图分别为不采用灵敏度过滤可能导致的弱支撑和无支撑示意图，右图为自支撑约束半圆形灵敏度过滤示意图；3 is a schematic diagram of self-supporting constraint sensitivity filtering in the present invention, the left two figures are respectively a schematic diagram of weak support and no support that may result from not using sensitivity filtering, and the right figure is a schematic diagram of self-supporting constraint semicircular sensitivity filtering;

图4是本发明中增材制造支撑结构设计与结构自支撑设计示意图，其中图4(a)为零件悬挑部分倾斜角度过大无法打印制造；图4(b)为通过在下方增加支撑结构满足可制造要求；图4(c)为通过本发明方法使结构实现自支撑而可直接打印制造；Fig. 4 is a schematic diagram of the support structure design and structural self-supporting design of additive manufacturing in the present invention, wherein Fig. 4(a) shows that the inclination angle of the overhanging part of the part is too large to be printed and manufactured; Fig. 4(b) shows that the support structure is added below Meet the manufacturability requirements; Figure 4(c) shows that the structure can be directly printed and manufactured by making the structure self-supporting by the method of the present invention;

图5是本发明实施例的结构设计模型；Fig. 5 is the structural design model of the embodiment of the present invention;

图6是本发明实施例不考虑自支撑约束拓扑优化结果示意图；6 is a schematic diagram of a topology optimization result that does not consider self-supporting constraints according to an embodiment of the present invention;

图7是本发明实施例考虑自支撑约束拓扑优化结果示意图。FIG. 7 is a schematic diagram of a topology optimization result considering self-supporting constraints according to an embodiment of the present invention.

具体实施方式Detailed ways

如图1所示，本发明提供的考虑增材制造结构自支撑约束的拓扑优化设计方法包括按顺序进行的下列步骤：As shown in FIG. 1, the topology optimization design method considering the self-supporting constraints of the additive manufacturing structure provided by the present invention includes the following steps in sequence:

1)建立优化模型，定义几何模型、载荷和边界条件，基于SIMP方法框架定义设计变量、目标函数和约束条件，并对单元密度、材料属性参数、结构体分比约束、结构自支撑惩罚函数、优化算法参数进行设置；1) Establish an optimization model, define geometric models, loads and boundary conditions, define design variables, objective functions and constraints based on the SIMP method framework, and determine the element density, material property parameters, structural body fraction constraints, structural self-supporting penalty functions, Optimization algorithm parameters are set;

具体步骤如下：Specific steps are as follows:

基于SIMP方法框架，以结构柔度最小作为优化目标，将离散化的单元相对密度作为设计变量，以结构体分比约束和结构自支撑约束惩罚函数值作为约束条件，建立优化模型，优化模型的数学表达式为：Based on the SIMP method framework, taking the minimum structural flexibility as the optimization objective, the discretized element relative density as the design variable, and the structural volume fraction constraint and the structural self-supporting constraint penalty function value as the constraint conditions, the optimization model is established. The mathematical expression is:

Figure BDA0002395791720000091
Figure BDA0002395791720000091

其中，c为结构柔度；where c is the structural flexibility;

ρ为单元密度设计变量；ρ is the element density design variable;

Figure BDA0002395791720000092
为过滤后的单元物理密度；
Figure BDA0002395791720000092
is the filtered unit physical density;

U和K分别为总体节点位移向量和总体刚度矩阵；U and K are the overall nodal displacement vector and overall stiffness matrix, respectively;

ue和k0分别为单元位移向量和单元刚度矩阵； ue and k 0 are the element displacement vector and element stiffness matrix, respectively;

F为结构外载荷向量；F is the external load vector of the structure;

N为单元数量；N is the number of units;

V和V0分别为结构材料总体积和设计区域总体积；V and V 0 are the total volume of structural materials and the total volume of the design area, respectively;

f为结构材料体分比约束值；f is the structural material volume fraction constraint value;

Figure BDA0002395791720000093
为结构自支撑约束惩罚函数；
Figure BDA0002395791720000093
is the structural self-supporting constraint penalty function;

ε为自支撑约束值；ε is the self-supporting constraint value;

Figure BDA0002395791720000096
为密度惩罚函数，
Figure BDA0002395791720000095
对中间密度进行惩罚，其中p为惩罚因子，Emin为最小弹性模量，E0为实体单元弹性模量。
Figure BDA0002395791720000096
is the density penalty function,
Figure BDA0002395791720000095
Penalize the intermediate density, where p is the penalty factor, E min is the minimum elastic modulus, and E 0 is the elastic modulus of the solid element.

2)对上述优化模型中的单元密度设计变量进行线性密度过滤而得到单元中间密度；2) Perform linear density filtering on the cell density design variables in the above-mentioned optimization model to obtain the cell intermediate density;

采用线性密度过滤的目的是避免出现棋盘格现象以及优化结果的网格依赖性，过滤后的单元中间密度ρ*是对过滤半径rmin的圆形范围内所有单元的密度进行加权平均得到的，具体公式如下：The purpose of using linear density filtering is to avoid the checkerboard phenomenon and the grid dependence of the optimization results. The intermediate density ρ * of the filtered cells is obtained by the weighted average of the densities of all cells within the circular range of the filter radius r min , The specific formula is as follows:

Figure BDA0002395791720000101
Figure BDA0002395791720000101

其中，Ne是以单元e为中心，过滤半径为rmin的圆形范围内的单元集合，Hei为单元i和单元e之间的权重系数，Δ(e,i)为单元i和单元e之间的距离。Among them, Ne is the unit set within a circle with the unit e as the center and the filter radius is r min , He ei is the weight coefficient between unit i and unit e, Δ(e, i) is unit i and unit the distance between e.

3)采用Heaviside函数对上述单元中间密度进行过滤而得到单元物理密度；3) adopt the Heaviside function to filter the above-mentioned unit intermediate density to obtain the unit physical density;

采用线性密度过滤后，在结构边界处容易出现多层中间密度灰色单元，因此采用Heaviside函数对上述单元中间密度ρ*进行过滤而得到单元物理密度

Figure BDA0002395791720000102
以消除这些中间密度单元，具体公式如下：After linear density filtering is used, multi-layer intermediate density gray cells are prone to appear at the structural boundary. Therefore, the Heaviside function is used to filter the intermediate density ρ * of the above cells to obtain the physical density of the cell.
Figure BDA0002395791720000102
To eliminate these intermediate density cells, the specific formula is as follows:

Figure BDA0002395791720000103
Figure BDA0002395791720000103

其中，β为控制参数，其值越大Heaviside函数就越趋近于阶跃函数。Among them, β is a control parameter, and the larger the value is, the closer the Heaviside function is to the step function.

4)在上述单元物理密度的基础上进行有限元分析、体积约束和结构自支撑约束响应分析，得到结构柔度、结构材料总体积和结构自支撑约束惩罚函数值；4) On the basis of the physical density of the above elements, carry out finite element analysis, volume constraint and structural self-supporting constraint response analysis, and obtain the structural flexibility, the total volume of structural materials and the structural self-supporting constraint penalty function values;

基于单元物理密度

Figure BDA0002395791720000104
进行有限元分析得到结构柔度c，对单元物理密度
Figure BDA0002395791720000105
求和得到结构材料总体积V。根据自支撑模型识别无支撑单元，计算结构自支撑约束惩罚函数
Figure BDA0002395791720000106
的函数值
Figure BDA0002395791720000107
并将其作为自支撑约束条件，所述自支撑约束条件为结构自支撑约束惩罚函数
Figure BDA0002395791720000108
的函数值
Figure BDA0002395791720000109
小于等于自支撑约束值ε：
Figure BDA00023957917200001010
based on element physical density
Figure BDA0002395791720000104
Carry out finite element analysis to obtain the structural flexibility c, and the physical density of the element
Figure BDA0002395791720000105
The summation yields the total volume V of structural material. Identify unsupported elements according to the self-supporting model, and calculate the structural self-supporting constraint penalty function
Figure BDA0002395791720000106
the function value of
Figure BDA0002395791720000107
and take it as the self-supporting constraint, which is the structural self-supporting constraint penalty function
Figure BDA0002395791720000108
the function value of
Figure BDA0002395791720000109
Less than or equal to the self-supporting constraint value ε:
Figure BDA00023957917200001010

所述的自支撑模型定义为：各实体单元密度不能大于下方支撑区域单元的最大密度。不考虑边界情况，如图2所示，每个单元由下方3个单元提供支撑，具体公式如下：The self-supporting model is defined as: the density of each solid unit cannot be greater than the maximum density of the unit in the support area below. Regardless of the boundary conditions, as shown in Figure 2, each unit is supported by the 3 units below. The specific formula is as follows:

Figure BDA00023957917200001011
Figure BDA00023957917200001011

其中，

Figure BDA00023957917200001012
和
Figure BDA00023957917200001013
分别为单元(i,j)物理密度及其支撑区域单元中所有单元的最大密度，i和j分别为单元行号和列号。由于最大值函数不满足可导性要求，不能采用梯度优化算法。因此采用基于P范数的光滑近似函数代替最大值函数以满足可导性要求，具体公式如下：in,
Figure BDA00023957917200001012
and
Figure BDA00023957917200001013
are the physical density of cell (i, j) and the maximum density of all cells in the cell in the support region, respectively, i and j are cell row and column numbers, respectively. Since the maximum value function does not meet the requirements of differentiability, the gradient optimization algorithm cannot be used. Therefore, a smooth approximation function based on the P norm is used to replace the maximum value function to meet the differentiability requirements. The specific formula is as follows:

Figure BDA0002395791720000111
Figure BDA0002395791720000111

其中，参数pn用于控制与最大值函数之间的误差，其值越大，与最大值函数之间的误差越小，但非线性程度增加。Among them, the parameter p n is used to control the error between the function and the maximum value. The larger the value, the smaller the error between the function and the maximum value, but the degree of nonlinearity increases.

结构自支撑约束惩罚函数

Figure BDA0002395791720000112
的函数值
Figure BDA0002395791720000113
采用如下公式进行计算：Structural Self-Support Constraint Penalty Function
Figure BDA0002395791720000112
the function value of
Figure BDA0002395791720000113
Calculate using the following formula:

Figure BDA0002395791720000114
Figure BDA0002395791720000114

其中，Mu为违反自支撑约束的单元集合。结构自支撑约束惩罚函数的函数值

Figure BDA0002395791720000115
是关于单元物理密度和该密度与其最大支撑单元密度差有关的函数，其值越大，表明结构违反自支撑约束程度越严重。由于结构边界处容易出现较多的密度接近0的单元，因此单元物理密度
Figure BDA0002395791720000116
的指数采用0.5，以减小低密度单元的干扰。where Mu is the set of elements that violate the self-supporting constraint. Function value of the structural self-supporting constraint penalty function
Figure BDA0002395791720000115
is a function of the physical density of the element and the difference between this density and its maximum supporting element density, and the larger the value, the more serious the violation of the self-supporting constraint by the structure. Since more cells with a density close to 0 are likely to appear at the structural boundary, the physical density of the cell is
Figure BDA0002395791720000116
The exponent of 0.5 is adopted to reduce the interference of low-density cells.

对于边界情况，自支撑约束响应分析时虚拟在单元物理密度矩阵左右两侧增加1层0密度单元，在下方增加一行值为1的单元代表基板，此时可同样采用公式(5)开展分析。自支撑约束响应分析不考虑单元无支撑时的连锁影响，因此仅识别出靠近结构几何边界附近的无支撑单元。通过支撑结构渐进演化的方式使最终优化结果实现结构自支撑。For the boundary case, in the self-supporting constraint response analysis, a layer of 0-density cells is added to the left and right sides of the physical density matrix of the cell, and a row of cells with a value of 1 is added below to represent the substrate. At this time, formula (5) can also be used for analysis. A self-supporting restraint response analysis does not consider cascading effects when the element is unsupported, so only unsupported elements close to the geometric boundary of the structure are identified. Through the gradual evolution of the support structure, the final optimization result realizes the structure self-supporting.

5)对上述结构柔度、结构材料总体积和自支撑约束惩罚函数的函数值进行灵敏度分析，并采用灵敏度过滤算子对自支撑约束惩罚函数的函数值的灵敏度进行过滤；5) Sensitivity analysis is performed on the above-mentioned structural flexibility, the total volume of structural materials and the function value of the self-supporting constraint penalty function, and the sensitivity filter operator is used to filter the sensitivity of the function value of the self-supporting constraint penalty function;

所述的灵敏度分析采用链式法则进行分析，具体如下：The sensitivity analysis described is carried out using the chain rule, as follows:

Figure BDA0002395791720000117
Figure BDA0002395791720000117

其中，φ为所分析的响应变量，后两项分别为Heaviside密度过滤和线性密度过滤方程的导数，分别是对公式(2)和(3)进行求导而得到的。Among them, φ is the analyzed response variable, and the last two terms are the derivatives of the Heaviside density filtering and linear density filtering equations, which are obtained by derivation of formulas (2) and (3), respectively.

结构柔度c对单元物理密度

Figure BDA0002395791720000121
的灵敏度采用经典伴随法公式计算得到，结构材料总体积V对单元物理密度
Figure BDA0002395791720000122
的灵敏度为1。Structural flexibility c versus element physical density
Figure BDA0002395791720000121
The sensitivity of is calculated by the classical adjoint method formula, the total volume V of the structural material is related to the physical density of the unit
Figure BDA0002395791720000122
The sensitivity is 1.

对于结构自支撑约束惩罚函数

Figure BDA0002395791720000123
的函数值
Figure BDA0002395791720000124
的灵敏度分析，在不考虑边界情况下每个单元可为上层3个单元提供支撑。对违反自支撑约束的单元，采用增加下方支撑单元密度的方式来减小结构自支撑约束惩罚函数
Figure BDA0002395791720000125
的函数值。自支撑约束惩罚函数
Figure BDA0002395791720000126
的函数值
Figure BDA00023957917200001213
对各单元的灵敏度按下式进行计算：Penalty function for structural self-supporting constraints
Figure BDA0002395791720000123
the function value of
Figure BDA0002395791720000124
Sensitivity analysis of , each unit can provide support for the upper 3 units without considering the boundary. For elements that violate the self-supporting constraint, the penalty function of the structural self-supporting constraint is reduced by increasing the density of the supporting elements below.
Figure BDA0002395791720000125
the function value. Self-supporting constraint penalty function
Figure BDA0002395791720000126
the function value of
Figure BDA00023957917200001213
The sensitivity of each unit is calculated as follows:

Figure BDA0002395791720000128
Figure BDA0002395791720000128

公式右边3项分别为单元物理密度

Figure BDA0002395791720000129
变化时对左上方、正上方和右上方的3个被支撑单元的影响。具体公式如下：The three items on the right side of the formula are the physical density of the unit.
Figure BDA0002395791720000129
The effect on the 3 supported units in the upper left, upper right and upper right when changing. The specific formula is as follows:

Figure BDA00023957917200001210
Figure BDA00023957917200001210

对于边界情况，灵敏度分析时虚拟在单元物理密度矩阵左右两侧增加2层0密度单元，上方增加1层0密度单元，此时可同样采用公式(8)和(9)开展分析。For the boundary case, in the sensitivity analysis, two layers of 0-density cells are added to the left and right sides of the cell physical density matrix, and one layer of zero-density cells is added above.

在支撑结构渐进演化过程中会出现密度渐变区域，其灵敏度值较小，使得在优化中无法有效支撑或者弱支撑，如图3所示。为促进支撑结构演化，利用灵敏度过滤算子对自支撑约束惩罚函数

Figure BDA00023957917200001211
的函数值
Figure BDA00023957917200001212
的灵敏度进行过滤，过滤后的灵敏度是对该单元上方半圆形过滤半径rmin范围内所有单元的灵敏度进行加权平均得到的，具体公式如下：During the gradual evolution of the support structure, there will be a density gradient area, and its sensitivity value is small, which makes it impossible to support effectively or weakly in the optimization, as shown in Figure 3. In order to promote the evolution of the support structure, the sensitivity filter operator is used for the self-support constraint penalty function
Figure BDA00023957917200001211
the function value of
Figure BDA00023957917200001212
The sensitivity after filtering is obtained by the weighted average of the sensitivity of all units within the range of the semicircular filtering radius r min above the unit. The specific formula is as follows:

Figure BDA0002395791720000131
Figure BDA0002395791720000131

其中，Ne是以单元e为中心，过滤半径为rmin的上方半圆形范围内的单元集合，如图3所示。Hei为单元i和单元e之间的权重系数，Δ(e,i)为单元i和单元e之间的距离，单元i的行号不小于单元e的行号。Among them, Ne is the unit set in the upper semicircle with the unit e as the center and the filter radius r min , as shown in Figure 3. He ei is the weight coefficient between unit i and unit e, Δ(e, i) is the distance between unit i and unit e, and the row number of unit i is not less than the row number of unit e.

6)采用优化算法对优化模型进行优化求解，更新单元密度设计变量和自支撑约束值，并判断是否收敛，若不收敛返回步骤2)进行算法迭代，若收敛则优化结束，最终输出拓扑优化结果。6) Use the optimization algorithm to optimize and solve the optimization model, update the element density design variables and the self-supporting constraint value, and judge whether to converge, if not, return to step 2) to perform algorithm iteration, if it converges, the optimization is over, and the topology optimization result is finally output. .

本发明中的优化算法采用MMA优化算法，为使优化过程更加稳定，使自支撑约束值ε随优化过程逐渐减小，具体公式如下：The optimization algorithm in the present invention adopts the MMA optimization algorithm. In order to make the optimization process more stable, the self-supporting constraint value ε is gradually reduced with the optimization process. The specific formula is as follows:

Figure BDA0002395791720000132
Figure BDA0002395791720000132

其中，loop为循环迭代次数，m和ε0均为常数，NMu为识别的无支撑单元数量。可见当迭代次数小于m2时结构自支撑约束阈值ε影响较小，当迭代次数大于m2时自支撑约束值ε迅速降低，从而对优化过程进行控制使结构最终实现自支撑。where loop is the number of loop iterations, m and ε 0 are both constants, and N Mu is the number of unsupported units identified. It can be seen that when the number of iterations is less than m 2 , the self-supporting constraint threshold ε of the structure has little effect, and when the number of iterations is greater than m 2 , the self-supporting constraint value ε decreases rapidly, so that the optimization process is controlled so that the structure is finally self-supporting.

图4是本发明中增材制造支撑结构设计与结构自支撑设计示意图；如图4(a)所示，当悬挑结构下表面倾斜角度大于一定值时无法打印制造，需要在其下方增加支撑结构，如图4(b)所示。采用本发明的设计方法可使下表面倾斜角度满足临界值要求，结构从而实现自支撑，如图4(c)所示，此时可以直接打印制造。Figure 4 is a schematic diagram of the additive manufacturing support structure design and structural self-support design in the present invention; as shown in Figure 4(a), when the inclination angle of the lower surface of the cantilever structure is greater than a certain value, it cannot be printed and manufactured, and a support needs to be added below it The structure is shown in Figure 4(b). By adopting the design method of the present invention, the inclination angle of the lower surface can meet the critical value requirement, and the structure can be self-supporting, as shown in Fig. 4(c), and can be directly printed at this time.

以下以MBB梁结构的拓扑优化设计进行进一步说明，设计域为150×50的矩形，左侧边水平方向和右下角竖直方向位移约束，向下集中力作用在左上角，如图5所示。打印方向从下到上，基板位于设计域底边。参数设置如下：材料弹性模量E0＝1，Emin＝1e-9，泊松比v＝0.3，载荷大小为F＝1，SIMP插值参数p为3，参数pn＝80，初始单元相对密度0.5，材料体分比约束值f为0.5，过滤半径rmin为单元尺寸的3倍，Heaviside函数参数β初始值为1且每隔100个循环增加1倍，常数ε0＝0.005，常数m＝10。The topology optimization design of the MBB beam structure is further described below. The design domain is a rectangle of 150×50, the left side is constrained in the horizontal direction and the lower right corner is vertically displaced, and the downward concentrated force acts on the upper left corner, as shown in Figure 5. . The printing direction is from bottom to top, and the substrate is located at the bottom edge of the design domain. The parameters are set as follows: material elastic modulus E 0 =1, E min =1e-9, Poisson's ratio v = 0.3, load magnitude is F = 1, SIMP interpolation parameter p is 3, parameter p n =80, the initial element relative to The density is 0.5, the material volume fraction constraint value f is 0.5, the filter radius r min is 3 times the element size, the initial value of the Heaviside function parameter β is 1 and increases by 1 times every 100 cycles, the constant ε 0 =0.005, the constant m =10.

图6—图7分别为不考虑自支撑约束和考虑自支撑约束时的拓扑优化设计结果。Figures 6 to 7 show the design results of topology optimization without considering the self-supporting constraints and when considering the self-supporting constraints, respectively.

本领域的技术人员容易理解，以上所述具体实施案例仅是针对本发明做出的举例说明，并不用以限制本发明，凡在本发明精神和原则之内所做的任何修改、替换以及改进，均应包含在本发明的保护范围之内。Those skilled in the art can easily understand that the above-mentioned specific implementation cases are only examples for the present invention, and are not intended to limit the present invention. Any modifications, replacements and improvements made within the spirit and principles of the present invention , should be included within the protection scope of the present invention.

Claims (2)

1. A topological optimization design method taking into account self-supporting constraints of an additive manufacturing structure, the topological optimization design method taking into account self-supporting constraints of the additive manufacturing structure comprising the following steps performed in sequence:
1) establishing an optimization model, defining a geometric model, a load and boundary conditions, defining design variables, a target function and constraint conditions based on an SIMP method frame, and setting unit density, material attribute parameters, structural body ratio constraint, a structural self-supporting penalty function and optimization algorithm parameters;
2) performing linear density filtering on the unit density design variables in the optimization model to obtain unit intermediate density;
3) filtering the cell intermediate density by adopting a Heaviside function to obtain a cell physical density;
4) carrying out finite element analysis, volume constraint and structure self-supporting constraint response analysis on the basis of the unit physical density to obtain a structure flexibility, a structure material total volume and a structure self-supporting constraint penalty function value;
5) carrying out sensitivity analysis on the structural flexibility, the structural material total volume and the function value of the self-supporting constraint penalty function, and filtering the sensitivity of the function value of the self-supporting constraint penalty function by adopting a sensitivity filter operator;
6) carrying out optimization solution on the optimization model by adopting an optimization algorithm, updating the unit density design variable and the self-supporting constraint value, judging whether convergence occurs or not, returning to the step 2) for algorithm iteration if convergence does not occur, finishing optimization if convergence occurs, and finally outputting a topology optimization result;
the method is characterized in that: in step 4), the method for performing finite element analysis, volume constraint and structure self-supporting constraint response analysis on the basis of the unit physical density to obtain the structure flexibility, the total volume of the structure material and the structure self-supporting constraint penalty function value comprises the following steps:
based on physical density of cells
Figure FDA0003526732750000011
Performing finite element analysis to obtain structural flexibility c, and comparing the physical density of the unit
Figure FDA0003526732750000012
Summing to obtain the total volume V of the structural material; identifying the unsupported unit according to the self-supporting model, and calculating the self-supporting constraint penalty function of the structure
Figure FDA0003526732750000013
Function value of
Figure FDA0003526732750000014
And taking the self-supporting constraint condition as a self-supporting constraint penalty function of the structure
Figure FDA0003526732750000015
Function value of
Figure FDA0003526732750000016
Less than or equal to the self-supporting constraint value ε:
Figure FDA0003526732750000017
the self-supporting model is defined as: the density of each solid unit cannot be greater than the maximum density of the units in the lower support area; regardless of the boundary condition, each cell is supported by the lower 3 cells, and the specific formula is as follows:
Figure FDA0003526732750000021
wherein,
Figure FDA0003526732750000022
and
Figure FDA0003526732750000023
respectively, the physical density of the unit (i, j) and the maximum density of all units in the unit of the support area, wherein i and j are respectively a unit row number and a unit column number; since the maximum function does not meet the conductibility requirement, a gradient optimization algorithm cannot be adopted; a smooth approximate function based on the P norm is adopted to replace a maximum function to meet the conductibility requirement, and the specific formula is as follows:
Figure FDA0003526732750000024
wherein the parameter pnFor controlling the error between the function of the maximum value;
structural self-supporting constrained penalty function
Figure FDA0003526732750000025
Function value of
Figure FDA0003526732750000026
The following formula is used for calculation:
Figure FDA0003526732750000027
wherein Mu is a set of units violating the self-supporting constraint;
for the boundary condition, virtually adding 1 layer of 0 density unit on the left and right sides of the unit physical density matrix during the self-supporting constraint response analysis, adding a row of unit with the value of 1 below to represent a substrate, and then carrying out the analysis by adopting the formula (5) in the same way.
2. The method of topologically optimally designing considering additive manufacturing structure self-support constraints of claim 1, wherein: in step 5), the method for performing sensitivity analysis on the structural flexibility, the structural material total volume and the function value of the self-supporting constraint penalty function and filtering the sensitivity of the function value of the self-supporting constraint penalty function by using a sensitivity filter operator comprises the following steps:
the sensitivity analysis adopts a chain rule for analysis, and specifically comprises the following steps:
Figure FDA0003526732750000028
wherein phi is the analyzed response variable, the latter two terms are derivatives of the Heaviside density filtering and linear density filtering equations respectively, and are obtained by differentiating the formulas (2) and (3);
physical density of structure compliance c vs. unit
Figure FDA0003526732750000031
The sensitivity of the structure material is calculated by adopting a classic adjoint method formula, and the total volume V of the structure material is opposite to the physical density of the unit
Figure FDA0003526732750000032
The sensitivity of (3) is 1;
constraining penalty function for structural self-support
Figure FDA0003526732750000033
Function value of
Figure FDA0003526732750000034
Each cell can provide support for the upper 3 cells without considering the boundary; for units violating self-supporting constraint, adding lower supporting sheetMethod for reducing structure self-supporting constraint penalty function by using element density
Figure FDA0003526732750000035
The function value of (a); self-supporting constrained penalty function
Figure FDA0003526732750000036
Function value of
Figure FDA0003526732750000037
The sensitivity for each unit was calculated as follows:
Figure FDA0003526732750000038
the right 3 terms of the formula are the cell physical densities
Figure FDA0003526732750000039
The influence on the 3 supported units at the upper left, right above and upper right when changing is as follows:
Figure FDA00035267327500000310
for the boundary condition, 2 layers of 0 density units are added on the left side and the right side of the physical density matrix of the unit virtually during sensitivity analysis, 1 layer of 0 density unit is added above the physical density matrix of the unit virtually, and then the analysis can be carried out by adopting the formulas (8) and (9) in the same way;
to promote support structure evolution, self-supporting constraint penalty function is restrained by sensitivity filter operator
Figure FDA00035267327500000311
Function value of
Figure FDA00035267327500000312
Is filtered, the filtered sensitivity being to the cellUpper filter radius rminThe sensitivity of all the units in the semicircular range is weighted and averaged, and the specific formula is as follows:
Figure FDA00035267327500000313
Hei＝max(0,rmin-Δ(e,i))
wherein N iseCentered on unit e, filter radius rminIn the upper semicircular range of (1), HeiAnd delta (e, i) is the distance between the unit i and the unit e, and the row number of the unit i is not less than that of the unit e.
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CN111859671B (en) * 2020-07-21 2021-06-22 南京理工大学 A Shape Preserving Topology Optimization Method Considering Dangling Feature Constraints
CN112149243B (en) * 2020-09-08 2024-04-19 华中科技大学 Flexible driving mechanism design method based on progressive evolution topology updating algorithm
CN112643047B (en) * 2020-12-08 2022-11-18 首钢集团有限公司 A hollow structure unsupported CMT arc forming method
CN112721172B (en) * 2020-12-14 2022-02-08 武汉纺织大学 Additive manufacturing-oriented self-supporting structure design method and related self-supporting structure optimization algorithm
CN112966410B (en) * 2021-02-03 2022-12-09 西安交通大学 A Topology Optimization Method for Additive Manufacturing Self-supporting Structure Adapted to Variable Critical Angle
CN112765865B (en) * 2021-02-04 2022-05-20 上海交通大学 Support structure design method for controlling metal powder bed additive manufacturing thermal deformation
CN113051796B (en) * 2021-03-19 2022-10-21 湖南科技大学 Structural topology optimization design method applied to additive manufacturing
CN113704921B (en) * 2021-08-31 2024-02-20 上海理工大学 Lightweight design method for shell of electric wheel rotor

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN106650026A (en) * 2016-11-24 2017-05-10 浙江大学 Self-supporting structure design method oriented to three-dimensional printing
CN107025340A (en) * 2017-03-30 2017-08-08 华中科技大学 A kind of self-supporting network structure method of topological optimization design suitable for increasing material manufacturing
CN109002598A (en) * 2018-06-29 2018-12-14 华中科技大学 Consider the self-supporting micro-structure Topology Optimization Method at overhanging angle and minimum dimension constraint
CN110502822A (en) * 2019-08-15 2019-11-26 燕山大学 A topology-optimized design method for self-supporting structures for additive manufacturing

Family Cites Families (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US10525628B2 (en) * 2016-04-28 2020-01-07 Wisconsin Alumni Research Foundation Systems and methods for controlling support structures in manufacturing
BE1024514B1 (en) * 2016-09-30 2018-03-21 Materialise N.V. Systems and methods for improving buildability in additive production.
US11084223B2 (en) * 2018-08-20 2021-08-10 Lawrence Livermore National Security, Llc Optimal toolpath generation system and method for additively manufactured composite materials

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN106650026A (en) * 2016-11-24 2017-05-10 浙江大学 Self-supporting structure design method oriented to three-dimensional printing
CN107025340A (en) * 2017-03-30 2017-08-08 华中科技大学 A kind of self-supporting network structure method of topological optimization design suitable for increasing material manufacturing
CN109002598A (en) * 2018-06-29 2018-12-14 华中科技大学 Consider the self-supporting micro-structure Topology Optimization Method at overhanging angle and minimum dimension constraint
CN110502822A (en) * 2019-08-15 2019-11-26 燕山大学 A topology-optimized design method for self-supporting structures for additive manufacturing

Non-Patent Citations (3)

* Cited by examiner, † Cited by third party
Title
"Efficient topology optimization in MATLAB using 88 lines of code";Erik Andreassen 等;《Structural and Multidisciplinary Optimization》;20101027;第1-16页 *
"基于特定微结构的增材制造非均质多层级结构构型设计方法";廖王程;《中国优秀硕士学位论文全文数据库 工程科技I辑》;20200215(第02期);第1-4章 *
"考虑连通性与结构特征约束的増材制造结构拓扑优化方法";李取浩;《中国博士学位论文全文数据库 工程科技II辑》;20180915(第09期);第1-5章 *

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