CN110502822A - A topology-optimized design method for self-supporting structures for additive manufacturing - Google Patents

Abstract

The invention discloses a topology optimization design method for a self-supporting structure for additive manufacturing, which belongs to the related technical field of structure optimization design. The "four-unit method" is used to calculate and constrain the overhang angle α, and the overhang angle constraint is combined with the SIMP method. In combination, the method for obtaining the overhang structure that is a self-supporting structure in additive manufacturing can effectively suppress the appearance of overhang structures that need to be supported, avoid the addition of support structures, effectively reduce the consumption and cost of consumables, and improve the surface quality of the workpiece; The calculation method is completely applicable to the continuum structure, with high flexibility and wide application range.

Description

A topology-optimized design method for self-supporting structures for additive manufacturing

technical field

The invention relates to a topology optimization design method for a self-supporting structure for additive manufacturing, and belongs to the technical field of structure optimization design.

Background technique

Topology optimization is an effective method to realize the lightweight design of workpiece structure. However, because the optimized workpiece is often complex in structure, most topology optimization is only used for the conceptual design of the structure, and it is difficult to carry out the later processing stage due to the limitation of traditional processing technology. In recent years, with the rise and development of additive manufacturing technology, domestic and foreign scholars and industrial designers generally believe that additive manufacturing technology has significant advantages such as strong ability to process complex components, short processing cycles, and no need for tooling and molds. Limited by the workpiece structure, combining it with topology optimization is the best way to achieve lightweight parts.

The forming principle of additive manufacturing technology is "lamination manufacturing", that is, parts are formed by adding materials layer by layer. However, there are still manufacturing constraints in the additive manufacturing process, of which the overhanging structure constraints are the most serious. In additive manufacturing, the model is sliced by the software and then printed layer by layer. In order to avoid printing collapse, it is required that there is enough material under each part of each layer after the model is sliced. If the supporting material under the layer is insufficient, it will form. For the overhang structure, auxiliary supports must be added artificially to ensure successful printing. These supports are formed during processing and removed during post-processing, resulting in serious waste of raw materials and time costs, and even damage to the surface of the workpiece during removal. Scholars at home and abroad have found that whether the overhanging structure needs auxiliary support can be judged by the overhanging angle, which is the angle between the overhanging surface of the workpiece and the horizontal plane, and found that there is an overhanging threshold angle. When the overhanging angle is greater than or equal to the overhanging threshold angle, the overhanging structure can be self-supporting, and high printing quality can be achieved without adding auxiliary supports; when the overhanging angle is smaller than the overhanging threshold angle, auxiliary supports need to be added.

In view of the above problems, there is an urgent need to develop a topology-optimized design method for self-supporting structures for additive manufacturing. In the process of topology optimization, the calculated overhang angle is constrained so that it is always greater than or equal to the overhang threshold angle, that is, to ensure that the overhang structure is always a self-supporting structure, so that the workpiece can be directly processed without support, which greatly saves costs and saves time.

SUMMARY OF THE INVENTION

The technical problem to be solved by the present invention is to provide a topology optimization design method for a self-supporting structure for additive manufacturing, which can calculate and constrain the overhang angle, obtain a overhang structure that is always a self-supporting structure, avoid the addition of support structures, and effectively The consumption and cost of consumables are reduced, and the surface quality of the workpiece is improved. At the same time, the invention is completely applicable to the continuum structure, has high flexibility and wide application range, and can be implanted into the topology optimization framework as an extension module, which is simple and easy to implement.

In order to solve the above technical problems, the technical solution adopted in the present invention is: a topology optimization design method for a self-supporting structure for additive manufacturing, calculating and constraining the overhang angle α through the "four-unit method", and constraining the overhang angle. Combined with the SIMP method, a method for overhanging structures that are self-supporting structures in additive manufacturing is obtained.

The further improvement of the technical solution of the present invention is: comprise the following steps:

A Establish the geometric model of the part, define the load and boundary conditions, define the design variables, objective function and constraint function based on the SIMP density-stiffness interpolation model, and initialize the element density, material property parameters, material volume fraction, and optimization algorithm parameters;

B takes the relative density of the unit body as the design variable, and obtains the intermediate density of the unit body through a linear density filter

C through the non-linear density filter and the intermediate density of the unit cell Get the physical density of a cell D obtains the overhang angle α through the "four-unit method", takes the overhang angle as the overhang angle constraint function, applies the overhang angle constraint condition, and obtains the overhang angle constraint equation. The overhang angle constraint condition is that the overhang angle is greater than or equal to the overhang threshold angle, namely : α≥α ₀ ;

E Calculate the sensitivity of the objective function and the constraint function to the design variables, and obtain the sensitivity control equation;

F according to the physical density of the unit cell Solve the overhang angle constraint equation and the mathematical expression of the SIMP density-stiffness interpolation model, obtain the structural response, and calculate the objective function value, constraint function value and sensitivity value;

G uses the obtained sensitivity value to judge whether the algorithm converges. If not, it returns to step (2) to perform algorithm iteration. If it converges, the algorithm iteration ends and the final topology optimization result is output.

A further improvement of the technical solution of the present invention is: in the step A, based on the SIMP density-stiffness interpolation model, the relative density of each discrete unit body is used as a design variable, and the overall stiffness of the macro structure is used as an objective function, and the optimization objective makes the overall macro structure The stiffness is maximized, that is, the macroscopic structural flexibility is minimized, and the structural volume fraction is used as a volume constraint function. The mathematical expression of the model is:

find: ρ=(ρ ₁ ,ρ ₂ ,...,ρ _nele ) ^T (1)

min: C=F ^T U (2)

s.t.: K(ρ)U=F (3)

0≤ρ _i ≤1 i=1,2,…,nele (5)

Among them, ρ=(ρ ₁ ,ρ ₂ ,...,ρ _nele ) ^T is the discretized relative density of each unit cell, and ρ is a continuous variable of [0,1];

C is the macroscopic structural flexibility;

F is the load vector;

U is the node displacement vector;

K is the overall stiffness of the macrostructure;

ρ _i is the relative density of the i-th unit cell;

v _i is the volume fraction of the i-th unit body;

V is the structural volume fraction;

to set the structure volume fraction;

Formula (2) is the objective equation, formula (3) is the control equation, and formula (4) is the volume constraint equation.

A further improvement of the technical solution of the present invention is: the step B adopts a linear density filter, which is specifically as follows:

N _e ={m|‖x _m -x _e ‖≤R} (7)

w(x _m )=R-‖x _m -x _e ‖ (8)

in, is the intermediate density of the unit body e, that is, the density of the relative density of the unit body e after linear filtering;

R is the linear density filter radius;

x _m is the centroid coordinate of the cell m.

A further improvement of the technical solution of the present invention is: in the step C, a nonlinear density filter is used to Filter to Physical Density details as follows:

Among them, β is the filtering degree of the filter,

η is the threshold parameter, which is obtained by the bisection method to keep the use of materials before and after the nonlinear filter unchanged. The formula is as follows:

The further improvement of the technical solution of the present invention is: the "four-unit method" in the step D is:

When using the SIMP method for topology optimization, there is a density uniform transition region with a cell density from 0 to 1 at the boundary of the SIMP density-stiffness interpolation model, and the density uniform transition region has a density gradient and the density gradient is perpendicular to the SIMP density-stiffness interpolation. The boundary of the model, it can be seen that there is a density contour perpendicular to the density gradient in the uniform density transition area and the density contour is parallel to the model boundary.

The further improvement of the technical solution of the present invention is: the concrete steps of the "four-unit method" are as follows:

Select a unit in the transition area of uniform density and its adjacent three units to form a "field"-shaped quadruple. Starting from the selected unit, record the density at the center of the four units in turn. As ρ ₁ , ρ ₂ , ρ ₃ , ρ ₄ , the density distribution in the density transition area is a linear function, and the density contour is calculated according to the density of the selected four units, then the density contour and the horizontal direction The included angle is the overhang angle α.

The further improvement of the technical solution of the present invention is: the specific calculation process of the "four-unit method" is as follows:

a. The boundary structure of the SIMP density-stiffness interpolation model is divided into upper boundary, vertical boundary and lower boundary, of which the lower boundary is the overhanging structure and only the overhanging structure needs to consider overhanging constraints during processing. Therefore, determine the boundary where the four-unit body is located. structure:

That is, to satisfy formula (12), the selected quadruple is located at the lower boundary:

b. Calculate the overhang angle α:

The further improvement of the technical solution of the present invention is: the constraint equation of the suspension angle is:

The further improvement of the technical solution of the present invention is: the sensitivity control equation in the step E is:

Owing to having adopted the above-mentioned technical scheme, the technical progress that the present invention obtains is:

By combining the overhang angle constraint with the traditional SIMP topology optimization method, the present invention can prevent the overhang structure that needs to be supported from appearing, does not need to artificially add additional supports and remove supports during the additive manufacturing process, shortens the manufacturing cycle, saves consumables, and reduces costs , improve the surface quality of the workpiece; at the same time, the invention is completely suitable for the continuum structure, has high flexibility, wide application range, and is easily implanted into the topology optimization framework as an expansion module, which is simple and easy to implement.

This application is based on the SIMP density-stiffness interpolation model. First, the linear density filter is used to obtain the intermediate density of the unit body, and then the nonlinear density filter and the intermediate density of the unit body are used to obtain the physical density of the unit body, which can avoid the optimization process. Checkerboard phenomenon and grid dependence of optimization results.

Description of drawings

Fig. 1 is the flow chart of the present invention;

Figure 2 is a schematic diagram of a suspension structure and a support structure of the present invention;

Fig. 3 is the topological optimization structure schematic diagram that the present invention does not impose the suspension constraint;

Fig. 4 is a partial enlarged view of A2 in Fig. 3 of the present invention;

Fig. 5 is the explanatory diagram of "four-unit method" at place B in Fig. 4 of the present invention;

FIG. 6 is a schematic diagram of the topology optimization structure after the pendant constraint is imposed in the present invention.

Detailed ways

Below in conjunction with embodiment, the present invention is described in further detail:

A topology optimization design method for self-supporting structures used in additive manufacturing, calculating and constraining the overhang angle α through the "four-unit method", and combining the overhang angle constraint with the SIMP method to obtain a method for the overhang structure of the self-supporting structure . As shown in Fig. ₂ , the overhanging structure of the printed solid is divided into two parts by the overhanging threshold angle α0. The thick solid line part is the overhanging structure with the overhanging angle α greater than or equal to the overhanging threshold angle α0, and there is _no need to add auxiliary support during printing to achieve self-support; the dashed part is the overhanging structure with the overhanging angle α smaller than the overhanging threshold angle _α0 , which needs to be printed when printing. Add auxiliary supports to ensure successful printing.

As shown in Figure 1, it includes the following steps:

A Establish the geometric model of the part, define the load and boundary conditions, define the design variables, objective function and constraint function based on the SIMP density-stiffness interpolation model, and initialize the element density, material property parameters, material volume fraction, and optimization algorithm parameters;

In the step A, based on the SIMP density-stiffness interpolation model, the discretized relative density of each unit body is used as a design variable, and the overall stiffness of the macro structure is used as the objective function, and the optimization goal is to maximize the overall stiffness of the macro structure, that is, the flexibility of the macro structure. To minimize, taking the structural volume fraction as a constraint function, the mathematical expression of the model is:

find: ρ=(ρ ₁ ,ρ ₂ ,...,ρ _nele ) ^T (1)

min: C=F ^T U (2)

s.t.: K(ρ)U=F (3)

0≤ρ _i ≤1 i=1,2,…,nele (5)

Among them, ρ=(ρ ₁ ,ρ ₂ ,...,ρ _nele ) ^T is the discretized relative density of each unit cell, and ρ is a continuous variable of [0,1];

C is the macroscopic structural flexibility;

F is the load vector;

U is the node displacement vector;

K is the overall stiffness of the macrostructure;

ρ _i is the relative density of the i-th unit cell;

v _i is the volume fraction of the i-th unit body;

V is the structural volume fraction;

to set the structure volume fraction;

Formula (2) is the objective equation, formula (3) is the control equation, and formula (4) is the constraint equation.

B takes the relative density of the unit body as the design variable, and obtains the intermediate density of the unit body through a linear density filter

The step B adopts a linear density filter, which is as follows:

N _e ={m|‖x _m -x _e ‖≤R} (7)

w(x _m )=R-‖x _m -x _e ‖ (8)

in, is the intermediate density of the unit body e, that is, the density of the relative density of the unit body e after linear filtering;

R is the linear density filter radius;

x _m is the centroid coordinate of the cell m.

C through the non-linear density filter and the intermediate density of the unit cell Get the physical density of a cell

Traditional SIMP generally only uses linear density filters. In order to avoid the checkerboard phenomenon in the optimization process and the grid dependence of the optimization results, the present invention further adopts a nonlinear density filter to further filtered to physical density details as follows:

Among them, β is the filtering degree of the filter,

η is the threshold parameter, which is obtained by the bisection method to keep the use of materials before and after the nonlinear filter unchanged. The formula is as follows:

D obtains the overhang angle α through the "four-element method", takes the overhang angle as the overhang angle constraint function, applies the overhang angle constraint condition, and obtains the overhang angle constraint equation, the overhang angle constraint condition is that the overhang angle is greater than or equal to the overhang threshold angle, namely : α≥α ₀ ;

As shown in Figures 3 to 5, when the SIMP method is used for topology optimization, the boundary of the SIMP density-stiffness interpolation model has a uniform density transition region with a cell density from 0 to 1. The uniform density transition region has a density gradient and The density gradient is perpendicular to the boundary of the SIMP density-stiffness interpolation model. It can be seen that there is a density contour perpendicular to the density gradient in the uniform density transition region and the density contour is parallel to the model boundary.

The "four-unit method" uses this principle to select a unit in the transition area of uniform density and its adjacent three units to form a "field"-shaped four-unit. From the selected unit At the beginning, the density at the center point of the four units is recorded as ρ ₁ , ρ ₂ , ρ ₃ , ρ ₄ in turn, and the density distribution in the density transition area is a linear function, which is calculated according to the density of the selected four units. density contour, the angle between the density contour and the horizontal direction is the overhang angle α.

The specific calculation process is as follows:

a. The boundary structure of the SIMP density-stiffness interpolation model is divided into upper boundary, vertical boundary and lower boundary, of which the lower boundary is the overhanging structure and only the overhanging structure needs to consider overhanging constraints during processing. Therefore, determine the boundary where the four-unit body is located. structure:

That is, to satisfy formula (12), the selected quadruple is located at the lower boundary:

b. Calculate the overhang angle α:

Applying the overhang angle constraint, the overhang angle constraint equation is obtained as:

Table 1. Density of the center of a quaternary

According to Fig. 3, it can be estimated that the overhang angles at points A1, A2, and A3 are about 30°. According to the center density data of the four-unit body given in Table 1, the points A1, A2, and A3 are calculated in combination with formula (13). The overhang angles are 31.4°, 31.8°, and 32.6°, respectively, which proves that the proposed overhang angle calculation method is accurate and effective.

E calculates the sensitivity of the objective function and the constraint function to the design variables, and obtains the sensitivity control equation (15):

F according to the physical density of the unit cell Solve the overhang angle constraint equation and the mathematical expression of the SIMP density-stiffness interpolation model, obtain the structural response, and calculate the objective function value, constraint function value and sensitivity value;

G uses the obtained sensitivity value to judge whether the algorithm converges. If not, it returns to step (2) to perform algorithm iteration. If it converges, the algorithm iteration ends and the final topology optimization result is output.

Figure 5 shows the topology optimization results after adding overhang constraints.

The specific implementation cases described in this specification are only examples for the present invention. Those skilled in the art can make various modifications and additions to the specific implementation cases described, but will not deviate from the spirit of the present invention or go beyond the scope defined by the appended claims.

Claims (10)

1. A method of topologically optimal design of a self-supporting structure for additive manufacturing, characterized by: the method for calculating and constraining the overhang angle alpha through a four-unit body method, and combining the overhang angle constraint with a SIMP method to obtain the overhang structure which is a self-supporting structure in the additive manufacturing.

2. The method of topologically optimized design of a self-supporting structure for additive manufacturing of claim 1, wherein: the method comprises the following steps:

a, establishing a part geometric model, defining load and boundary conditions, defining design variables, a target function and a constraint function based on a SIMP density-rigidity interpolation model, and initializing unit body density, material attribute parameters, material volume fraction and optimization algorithm parameters;

b, taking the relative density of the unit body as a design variable, and acquiring the intermediate density of the unit body through a linear density filter

C pass through nonlinear Density Filter and intermediate Density of Unit cellsObtaining physical Density of Unit bodies

D, obtaining an overhang angle alpha through a four-unit body method, taking the overhang angle as an overhang angle constraint function, and applying an overhang angle constraint condition to obtain an overhang angle constraint equation, wherein the overhang angle constraint condition is that the overhang angle is greater than or equal to an overhang threshold angle: alpha is more than or equal to alpha₀；

E, calculating the sensitivity of the target function and the constraint function to the design variable to obtain a sensitivity control equation;

f according to physical Density of Unit cellSolving a suspension angle constraint equation and a SIMP density-rigidity interpolation model mathematical expression to obtain a structural response, and calculating an objective function value, a constraint function value and a sensitivity value;

and G, judging whether the algorithm is converged or not by using the obtained sensitivity value, if not, returning to the step (2) for algorithm iteration, if so, ending the algorithm iteration, and outputting a final topology optimization result.

3. The method of claim 2, wherein the method comprises: in the step A, based on a SIMP density-rigidity interpolation model, the relative density of each discrete unit body is used as a design variable, the overall rigidity of the macro structure is used as a target function, the target is optimized to maximize the overall rigidity of the macro structure, namely the flexibility of the macro structure is minimized, the volume fraction of the structure is used as a volume constraint function, and the mathematical expression of the model is as follows:

find：ρ＝(ρ₁,ρ₂,……,ρ_nele)^T (1)

min∶C＝F^TU (2)

s.t.:K(ρ)U＝F (3)

0≤ρ_i≤1 i＝1,2,…,nele (5)

where ρ ═ p (ρ)₁,ρ₂,……,ρ_nele)^TIs the discretized relative density of each unit body, and rho is [0,1 ]]A continuous variable of (a);

c is the flexibility of the macrostructure;

f is a load vector;

u is a node displacement vector;

k is the integral rigidity of the macrostructure;

ρ_irelative density of the ith unit cell;

v_iis the volume fraction of the ith unit cell;

v is the structural volume fraction;

to set the structural volume fraction;

equation (2) is the objective equation, equation (3) is the governing equation, and equation (4) is the volume constraint equation.

4. The method of claim 2, wherein the method comprises: the step B adopts a linear density filter, and specifically comprises the following steps:

N_e＝{m|‖x_m-x_e‖≤R} (7)

w(x_m)＝R-‖x_m-x_e‖ (8)

wherein,is the median density of unit e, i.e. the relative density of unit e after linear filtering;

r is the linear density filter radius;

x_mis the centroid coordinate of the unit cell m.

5. The method of claim 2, wherein the method comprises: in the step C, a nonlinear density filter is adoptedFiltering to physical densityThe method comprises the following specific steps:

wherein, beta is the filtering degree of the filter,

eta is a threshold parameter and is obtained by a bisection method, and is used for keeping the use of materials before and after the nonlinear filter unchanged, and the formula is as follows:

6. the method of claim 2, wherein the method comprises: the four-unit method in the step D comprises the following steps:

when the SIMP method is used for topology optimization, a density uniform transition region with the unit volume density of 0-1 exists at the boundary of the SIMP density-rigidity interpolation model, a density gradient exists in the density uniform transition region, and the density gradient is perpendicular to the boundary of the SIMP density-rigidity interpolation model, so that a density contour line perpendicular to the density gradient exists in the density uniform transition region, and the density contour line is parallel to the model boundary.

7. The method of claim 6, wherein the method comprises: the specific steps of the four-unit body method are as follows:

selecting a certain unit body in a transition area with uniform density and three adjacent unit bodies to form a four-unit body in a shape of Chinese character tian, and sequentially recording the density of the central points of the four unit bodies as rho from the selected unit body₁、ρ₂、ρ₃、ρ₄And the density distribution in the density transition region is a linear function, a density contour line is calculated according to the densities of the four selected unit bodies, and the included angle between the density contour line and the horizontal direction is the overhang angle alpha.

8. The method of claim 7, wherein the method comprises: the specific calculation process of the four-unit body method is as follows:

a. the boundary structure of the SIMP density-rigidity interpolation model is divided into an upper boundary, a vertical boundary and a lower boundary, wherein the lower boundary is a suspension structure, and suspension constraint conditions need to be considered only when the suspension structure is processed, so that the boundary structure where the four-unit body is located is judged:

that is, the formula (12) is satisfied, that is, the selected four-unit body is located at the lower boundary:

b. calculating the overhang angle α:

9. the method of claim 8, wherein the method comprises: the suspension angle constraint equation is:

10. a method of topologically optimised design of a self-supporting structure for additive manufacturing according to claim 3, characterised in that: the sensitivity control equation in the step E is: