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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CN111428397A (en
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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

The invention discloses a topological optimization design method considering self-supporting constraint of an additive manufacturing structure, and belongs to the technical field of structural optimization design. Based on the SIMP method and the structure self-supporting mathematical model, the self-supporting characteristic of the structure is represented by a penalty function and is used as a display constraint condition, the self-supporting constraint sensitivity is adopted to filter and promote the evolution of the support structure, and the topology optimization design method considering the self-supporting constraint of the additive manufacturing structure is obtained. According to the method, the final topological optimization result is structurally self-supported through gradual evolution of the supporting structure, and the supporting structure is prevented from being increased through secondary design, so that the material cost is saved, and the design period is shortened. In addition, the method is completely suitable for the topology optimization of the continuum structure, has high flexibility and low calculation cost, and can be implanted into the existing topology optimization framework as an extended function.

Description

Topological optimization design method considering additive manufacturing structure self-supporting constraint
Technical Field
The invention belongs to the field related to structural optimization design, and relates to a topological optimization design method considering additive manufacturing structure self-supporting constraint.
Background
Topology optimization is an advanced structural design method that can find the optimal material layout by using numerical optimization technology under certain constraint conditions. Compared with the traditional size optimization and shape optimization, the design space is greatly expanded, so that an innovative and high-performance structural layout form can be designed. Since the topology optimization concept is proposed in the late 80 s, the topology optimization theory and method have been widely concerned and rapidly developed. Typical topology optimization methods include variable density methods, progressive methods, homogenization methods, and the like. Various techniques have also been developed to eliminate numerical instabilities such as checkerboard patterns, grid dependencies, etc., the most common being density filtering or sensitivity filtering techniques. However, since the topological optimization result is often complex in geometric configuration and is very difficult to manufacture by using the conventional manufacturing process, a designer needs to design the optimization result secondarily based on the manufacturing technology or experience to meet the manufacturability, thereby losing the advantages of the product in terms of light weight and high performance.
The advent of additive manufacturing has well addressed the topology optimization manufacturing challenges. Unlike conventional manufacturing methods, additive manufacturing methods manufacture solid parts by a method in which materials are added layer by layer. Compared with the traditional material cutting processing technology, the additive manufacturing can realize 'free manufacturing', solves the problem of forming of a plurality of complex structural parts which are difficult to manufacture in the past, and can greatly reduce the processing procedures, shorten the processing period and reduce the research and development manufacturing cost. However, additive manufacturing is not completely free to design parts and there are some unique manufacturing constraints. The biggest bottleneck of the application of the topology optimization technology to additive manufacturing is that additive manufacturing constraints are not considered in the topology optimization algorithm.
In the vibration material disk manufacturing process, when structure lower surface inclination of encorbelmenting is greater than a definite value, take place the structure easily and collapse in the successive layer manufacturing process, need increase bearing structure below this moment, conduct the base plate with upper heat fast simultaneously to reduce temperature gradient, reduce heat altered shape. The use of the support structure not only increases the design difficulty, but also leads to increased processing time and cost, and the process difficulty is higher when the support structure is removed at a later stage, thereby destroying the optimality of the structure. Studies have shown that the structure can be self-supporting when the lower surface tilt angle is less than a certain value. Therefore, it is better to make the structure completely self-supporting by improving the design, so that it is not necessary to add a support structure, thereby reducing the material and time costs.
In view of the above problems, there is an urgent need to develop a topology optimization design method considering additive manufacturing structure self-supporting constraint, so that the topology optimization result realizes structure self-supporting, and thus parts can be directly processed without adding a supporting structure, thereby saving material cost and shortening design cycle.
Disclosure of Invention
In order to solve the above problem, it is an object of the present invention to provide a topology optimization design method that takes into account the self-supporting constraints of an additive manufacturing structure.
In order to achieve the above object, the present invention provides a topology optimization design method considering self-supporting constraints of an 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) and (3) 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.
In step 1), the establishing of the optimization model, the defining of the geometric model, the load and the boundary condition, the defining of the design variables, the objective function and the constraint conditions based on the framework of the SIMP method, and the setting of the unit density, the material property parameters, the structural component ratio constraint, the structural self-supporting penalty function and the optimization algorithm parameters comprise the following specific steps:
based on the SIMP method framework, the minimum structural flexibility is used as an optimization target, the discrete unit relative density is used as a design variable, the structural component ratio constraint and the structural self-supporting constraint penalty function value are used as constraint conditions, an optimization model is established, and the mathematical expression of the optimization model is as follows:
Figure BDA0002395791720000031
wherein c is the structural flexibility;
rho is a unit density design variable;
Figure BDA0002395791720000032
is the filtered unit physical density;
u and K are respectively a total node displacement vector and a total rigidity matrix;
ueand k0Respectively a unit displacement vector and a unit stiffness matrix;
f is a structural external load vector;
n is the number of units;
v and V0Respectively representing the total volume of the structural material and the total volume of the design area;
f is a structural material volume ratio constraint value;
Figure BDA0002395791720000033
constraining a penalty function for the structural self-support;
epsilon is a self-supporting constraint value;
Figure BDA0002395791720000041
in order to be a density penalty function,
Figure BDA0002395791720000042
to intermediate densityPenalty, where p is a penalty factor, EminAt a minimum modulus of elasticity, E0Is the modulus of elasticity of the solid element.
In step 2), the method for obtaining the unit intermediate density by performing linear density filtering on the unit density design variables in the optimization model comprises:
filtered cell median density ρ*Is to the filtration radius rminIs obtained by performing weighted average on the densities of all the units in the circular range, and the specific formula is as follows:
Figure BDA0002395791720000043
wherein N iseCentered on unit e, filter radius rminOf the circular range of cells, HeiIs the weight coefficient between cell i and cell e, and Δ (e, i) is the distance between cell i and cell e.
In step 3), the method for obtaining the physical density of the cell by filtering the cell intermediate density by using the Heaviside function is as follows:
using the Heaviside function to the cell intermediate density ρ*Filtering to obtain unit physical density
Figure BDA0002395791720000044
The specific formula is as follows:
Figure BDA0002395791720000045
wherein β is a control parameter.
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 BDA0002395791720000046
Performing finite element analysis to obtain structural flexibility c, and comparing the physical density of the unit
Figure BDA0002395791720000047
Summing to obtain the total volume V of the structural material; identifying the non-support unit according to the self-support model, and calculating the structure self-support constraint penalty function
Figure BDA00023957917200000413
Function value of
Figure BDA0002395791720000049
And taking the self-supporting constraint condition as a self-supporting constraint penalty function of the structure
Figure BDA00023957917200000410
Function value of
Figure BDA00023957917200000411
Less than or equal to the self-supporting constraint value ε:
Figure BDA00023957917200000412
the self-supporting model is defined as follows: the density of each solid unit cannot be greater than the maximum density of the units of the underlying support area. Regardless of the boundary condition, each cell is supported by the lower 3 cells, and the specific formula is as follows:
Figure BDA0002395791720000051
wherein the content of the first and second substances,
Figure BDA0002395791720000052
and
Figure BDA0002395791720000053
the physical density of cell (i, j) and the maximum density of all cells in its support area cell, i and j being the cell row number and column number, respectively. ByThe maximum function does not meet the guiding requirement, and a gradient optimization algorithm cannot be adopted. Therefore, a smooth approximation function based on the P norm is adopted to replace the maximum function to meet the conductibility requirement, and the specific formula is as follows:
Figure BDA0002395791720000054
wherein the parameter pnFor controlling the error between the function of the maximum value;
structural self-supporting constrained penalty function
Figure BDA0002395791720000055
Function value of
Figure BDA0002395791720000056
The following formula is used for calculation:
Figure BDA0002395791720000057
where Mu is the set of units that violate the self-supporting constraint.
For the boundary condition, during the self-supporting constrained response analysis, 1 layer of 0 density unit is virtually added on the left side and the right side of the unit physical density matrix, and a row of unit with the value of 1 is added below the unit to represent the substrate, and then the analysis can be carried out by adopting the formula (5) in the same way.
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 BDA0002395791720000058
where φ is the analyzed response variable, the latter two terms are derivatives of the Heaviside Density Filter and Linear Density Filter equations, respectively, derived from equations (2) and (3), respectively.
Physical density of structure compliance c vs. unit
Figure BDA0002395791720000061
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 BDA0002395791720000062
The sensitivity of (3) is 1.
Constraining penalty function for structural self-support
Figure BDA0002395791720000063
Function value of
Figure BDA0002395791720000064
Each cell can provide support for the upper 3 cells without considering the boundary; for units violating self-supporting constraint, reducing structure self-supporting constraint penalty function by increasing density of lower supporting units
Figure BDA0002395791720000065
The function value of (a); self-supporting constrained penalty function
Figure BDA0002395791720000066
Function value of
Figure BDA0002395791720000067
The sensitivity for each unit was calculated as follows:
Figure BDA0002395791720000068
the right 3 terms of the formula are the cell physical densities
Figure BDA0002395791720000069
The influence on the 3 supported units at the upper left, right above and upper right when changing is as follows:
Figure BDA00023957917200000610
for the boundary condition, in the sensitivity analysis, 2 layers of 0 density units are virtually added on the left and right sides of the physical density matrix of the unit, and 1 layer of 0 density unit is added on the upper side of the physical density matrix of the unit, 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 BDA00023957917200000611
Function value of
Figure BDA00023957917200000612
Is filtered with the sensitivity of the unit above the filter radius rminThe sensitivity of all the units in the semicircular range is weighted and averaged, and the specific formula is as follows:
Figure BDA00023957917200000613
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.
In step 6), the method for optimizing and solving the optimization model by using the optimization algorithm, updating the unit density design variable and the self-supporting constraint value, judging whether convergence occurs, returning to step 2) for algorithm iteration if convergence does not occur, ending optimization if convergence occurs, and finally outputting a topology optimization result comprises the following steps:
the optimization algorithm adopts an MMA optimization algorithm, and in order to make the optimization process more stable, the self-supporting constraint value epsilon is gradually reduced along with the optimization process, the specific formula is as follows:
Figure BDA0002395791720000071
wherein loop is the loop iteration number, m and epsilon0Are all constant, NMuThe identified number of unsupported elements.
The topological optimization design method considering the self-supporting constraint of the additive manufacturing structure has the following beneficial effects:
based on an SIMP method framework, a structure self-supporting mathematical model is combined, the self-supporting characteristic of the structure is represented through a penalty function and is used as a display constraint condition, and the self-supporting constraint sensitivity is adopted to filter and promote the evolution of the support structure. The self-supporting constraint characteristic of the structure can be analyzed and controlled, and the final topological optimization result is structurally self-supported through gradual evolution of the supporting structure, so that direct printing and manufacturing can be realized, and the mechanical property of the structure is optimal. The secondary design is avoided, the supporting structure is increased, the later-stage removal is avoided, the material cost is saved, and the design period is shortened. Compared with the existing topological optimization design method considering the self-supporting constraint of the additive manufacturing structure, the method realizes the self-supporting of the structure in a mode of gradual evolution of the structure boundary supporting structure, and does not consider the linkage influence when the unit is not supported, so the method has lower calculation cost. In addition, the method is high in flexibility, is completely suitable for the topology optimization of the continuum structure, and can be implanted into the existing topology optimization framework as an extended function.
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FIG. 1 is a flow chart of a topology optimization design method provided by the present invention that takes into account the self-supporting constraints of an additive manufacturing structure;
FIG. 2 is a schematic view of a support unit according to the present invention;
FIG. 3 is a schematic diagram of self-supporting constrained sensitivity filtering in the present invention, wherein the left two diagrams are respectively a schematic diagram of weak support and no support which may result from not using sensitivity filtering, and the right diagram is a schematic diagram of self-supporting constrained semicircular sensitivity filtering;
FIG. 4 is a schematic diagram of an additive manufacturing support structure design and a structural self-supporting design according to the present invention, wherein FIG. 4(a) shows that the overhanging portion of the part is too inclined to be printed; FIG. 4(b) is a diagram of meeting manufacturability requirements by adding support structures below; FIG. 4(c) is a diagram of a direct-print fabrication by the method of the present invention to render the structure self-supporting;
FIG. 5 is a structural design model of an embodiment of the present invention;
FIG. 6 is a diagram illustrating a result of topology optimization without considering self-supporting constraints according to an embodiment of the present invention;
FIG. 7 is a diagram illustrating a result of a topology optimization considering self-supporting constraints according to an embodiment of the present invention.
Detailed Description
As shown in fig. 1, the topology optimization design method considering the self-supporting constraint of the additive manufacturing structure provided by the invention comprises the following steps 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;
the method comprises the following specific steps:
based on the SIMP method framework, the minimum structural flexibility is used as an optimization target, the discrete unit relative density is used as a design variable, the structural component ratio constraint and the structural self-supporting constraint penalty function value are used as constraint conditions, an optimization model is established, and the mathematical expression of the optimization model is as follows:
Figure BDA0002395791720000091
wherein c is the structural flexibility;
rho is a unit density design variable;
Figure BDA0002395791720000092
is a filtered unitPhysical density;
u and K are respectively a total node displacement vector and a total rigidity matrix;
ueand k0Respectively a unit displacement vector and a unit rigidity matrix;
f is a structural external load vector;
n is the number of units;
v and V0Respectively representing the total volume of the structural material and the total volume of the design area;
f is a structural material volume ratio constraint value;
Figure BDA0002395791720000093
constraining a penalty function for the structural self-support;
epsilon is a self-supporting constraint value;
Figure BDA0002395791720000096
in order to be a density penalty function,
Figure BDA0002395791720000095
punishing the intermediate density, wherein p is a punishment factor, EminAt a minimum modulus of elasticity, E0Is the modulus of elasticity of the solid element.
2) Performing linear density filtering on the unit density design variables in the optimization model to obtain unit intermediate density;
the purpose of adopting linear density filtering is to avoid checkerboard phenomenon and grid dependency of optimized result, and the filtered unit intermediate density rho*Is to the filtration radius rminIs obtained by performing weighted average on the densities of all the units in the circular range, and the specific formula is as follows:
Figure BDA0002395791720000101
wherein N iseCentered on unit e, filter radius rminWithin a circular range ofSet of elements, HeiIs the weight coefficient between cell i and cell e, and Δ (e, i) is the distance between cell i and cell e.
3) Filtering the cell intermediate density by adopting a Heaviside function to obtain a cell physical density;
after linear density filtering, multiple layers of intermediate density gray cells are easy to appear at the structure boundary, so the Heaviside function is adopted to perform the intermediate density rho of the cells*Filtering to obtain unit physical density
Figure BDA0002395791720000102
To eliminate these intermediate density units, the specific formula is as follows:
Figure BDA0002395791720000103
wherein, β is a control parameter, and the larger the value of β is, the more the Heaviside function approaches to the step function.
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;
based on physical density of cells
Figure BDA0002395791720000104
Performing finite element analysis to obtain structural flexibility c, and comparing the physical density of the unit
Figure BDA0002395791720000105
And 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 BDA0002395791720000106
Function value of
Figure BDA0002395791720000107
And using it as a self-supporting constraint, said self-supportingSupport constraint condition is structure self-support constraint penalty function
Figure BDA0002395791720000108
Function value of
Figure BDA0002395791720000109
Less than or equal to the self-supporting constraint value ε:
Figure BDA00023957917200001010
the self-supporting model is defined as: the density of each solid unit cannot be greater than the maximum density of the units of the underlying support area. Regardless of the boundary conditions, as shown in fig. 2, each cell is supported by the lower 3 cells, and the specific formula is as follows:
Figure BDA00023957917200001011
wherein the content of the first and second substances,
Figure BDA00023957917200001012
and
Figure BDA00023957917200001013
the physical density of cell (i, j) and the maximum density of all cells in its support area cell, i and j being the cell row number and column number, respectively. Since the maximum function does not meet the guiding requirements, a gradient optimization algorithm cannot be employed. Therefore, a smooth approximation function based on the P norm is adopted to replace the maximum function to meet the conductibility requirement, and the specific formula is as follows:
Figure BDA0002395791720000111
wherein the parameter pnThe larger the value of the error between the control and the maximum function is, the smaller the error between the control and the maximum function is, but the degree of nonlinearity is increased.
Structural self-supporting constrained penalty function
Figure BDA0002395791720000112
Function value of
Figure BDA0002395791720000113
The following formula is used for calculation:
Figure BDA0002395791720000114
where Mu is the set of units that violate the self-supporting constraint. Function value of structure self-supporting constraint penalty function
Figure BDA0002395791720000115
Is a function of the physical density of the cells and the difference between that density and its maximum supported cell density, the greater the value, the more severe the structural violation of the self-supporting constraint. Since more cells with density close to 0 are likely to appear at the structure boundaries, the cell physical density
Figure BDA0002395791720000116
The index of (2) is 0.5 to reduce interference of low density cells.
For the boundary condition, during the self-supporting constrained response analysis, 1 layer of 0 density unit is virtually added on the left side and the right side of the unit physical density matrix, and a row of unit with the value of 1 is added below the unit to represent the substrate, and then the analysis can be carried out by adopting the formula (5) in the same way. The self-supporting constrained response analysis does not account for the chain effects of the elements when unsupported, and therefore only identifies unsupported elements near the structural geometry boundaries. And the final optimization result is enabled to realize the self-support of the structure through a mode of gradual evolution of the support structure.
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;
the sensitivity analysis adopts a chain rule for analysis, and specifically comprises the following steps:
Figure BDA0002395791720000117
where φ is the analyzed response variable, the latter two terms are derivatives of the Heaviside Density Filter and Linear Density Filter equations, respectively, derived from equations (2) and (3), respectively.
Physical density of structure compliance c vs. unit
Figure BDA0002395791720000121
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 BDA0002395791720000122
The sensitivity of (3) is 1.
Constraining penalty function for structural self-support
Figure BDA0002395791720000123
Function value of
Figure BDA0002395791720000124
Each cell can provide support for the upper 3 cells without regard to the boundary. For units violating self-supporting constraint, reducing structure self-supporting constraint penalty function by increasing density of lower supporting units
Figure BDA0002395791720000125
The function value of (1). Self-supporting constrained penalty function
Figure BDA0002395791720000126
Function value of
Figure BDA00023957917200001213
The sensitivity for each unit was calculated as follows:
Figure BDA0002395791720000128
the right 3 terms of the formula are the cell physical densities
Figure BDA0002395791720000129
Influence on the 3 supported units at the upper left, right above and upper right when changed. The specific formula is as follows:
Figure BDA00023957917200001210
for the boundary condition, in the sensitivity analysis, 2 layers of 0 density units are virtually added on the left and right sides of the physical density matrix of the unit, and 1 layer of 0 density unit is added on the upper side of the physical density matrix of the unit, and then the analysis can be carried out by adopting the formulas (8) and (9) in the same way.
In the gradual evolution process of the support structure, a density gradient area occurs, and the sensitivity value is small, so that the support cannot be effectively supported or weakly supported in the optimization process, as shown in fig. 3. To promote support structure evolution, self-supporting constraint penalty function is restrained by sensitivity filter operator
Figure BDA00023957917200001211
Function value of
Figure BDA00023957917200001212
Is filtered with a sensitivity of a semi-circular filtering radius r above the unitminThe sensitivity of all units in the range is weighted and averaged, and the specific formula is as follows:
Figure BDA0002395791720000131
wherein N iseCentered on element e, filter radius rminThe upper semi-circular area of (a) as shown in fig. 3. 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.
6) And (3) 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 optimization algorithm in the invention adopts an MMA optimization algorithm, and in order to make the optimization process more stable, the self-supporting constraint value epsilon is gradually reduced along with the optimization process, the specific formula is as follows:
Figure BDA0002395791720000132
wherein loop is the loop iteration number, m and epsilon0Are all constant, NMuThe identified number of unsupported elements. It can be seen that when the number of iterations is less than m2The influence of a structural self-supporting constraint threshold epsilon is small, and when the iteration number is more than m2And the self-supporting constraint value epsilon is rapidly reduced, so that the optimization process is controlled, and the structure is finally self-supported.
FIG. 4 is a schematic diagram of an additive manufacturing support structure design and a structural self-supporting design according to the present invention; as shown in fig. 4(a), when the lower surface of the cantilever structure is inclined at an angle greater than a certain value, printing and manufacturing cannot be performed, and a support structure needs to be added below the cantilever structure, as shown in fig. 4 (b). By adopting the design method of the invention, the lower surface inclination angle can meet the requirement of a critical value, and the structure can realize self-supporting, as shown in figure 4(c), and the printing and manufacturing can be directly carried out at the moment.
Further explained below by the topological optimization design of the MBB beam structure, the design domain is a 150 × 50 rectangle, the displacement of the left side in the horizontal direction and the right bottom corner in the vertical direction is restricted, and downward concentrated force acts on the left top corner, as shown in FIG. 5. The printing direction is from bottom to top, and the base plate is located at the bottom edge of the design field. The parameters are set as follows: modulus of elasticity E of the material0＝1，Emin1e-9, poisson's ratio v 0.3, load size F1, SIMP interpolation parameter p 3, parameter pn80 initial unit relative density 0.5, material volume fraction constraint f 0.5, filter radius rminHeav 3 times the cell sizeThe initial value of the parameter beta of the iside function is 1 and increases by 1 time every 100 cycles, with a constant epsilon00.005 and 10.
FIGS. 6-7 show the results of the topology optimization design without and with self-supporting constraints taken into account, respectively.
It will be appreciated by those skilled in the art that the above embodiments are merely illustrative and not restrictive, and that any modifications, substitutions and improvements made within the spirit and scope of the present invention are intended to be included within the 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 the content of the first and second substances,
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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