CN107391824B - Topological optimization design method of self-supporting structure in additive manufacturing - Google Patents

Topological optimization design method of self-supporting structure in additive manufacturing Download PDF

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CN107391824B
CN107391824B CN201710558812.1A CN201710558812A CN107391824B CN 107391824 B CN107391824 B CN 107391824B CN 201710558812 A CN201710558812 A CN 201710558812A CN 107391824 B CN107391824 B CN 107391824B
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张卫红
周璐
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Northwestern Polytechnical University
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Abstract

The invention discloses a topological optimization design method of a self-supporting structure in additive manufacturing, which is used for solving the technical problem of poor practicability of the traditional topological optimization design method of the self-supporting structure. The technical scheme is that a certain number of closed polygonal holes are arranged in an entity design area, and the topological layout evolution of the structure is driven through the actions of movement, deformation, fusion, shrinkage, expansion and the like of the holes. In addition, the relative position between each polygon vertex is controlled, so that the inclination angle of the suspended part in the overall structure is larger than the critical angle value, and the self-support of the structure is realized. Compared with the design method of the background art, the self-supporting sacrificial fabric has less flexibility for realizing the structure. In addition, the design variables are independent of the units, the number is small, the calculation amount is small, and convergence is easy. In addition, the structural boundary designed by the invention is smooth, no intermediate density unit exists in the structure, the post-processing processes such as model reconstruction and the like are facilitated for engineering designers, and the practicability is good.

Description

Topological optimization design method of self-supporting structure in additive manufacturing
Technical Field
The invention relates to a topological optimization design method of a self-supporting structure, in particular to a topological optimization design method of a self-supporting structure in additive manufacturing.
Background
Because the additive manufacturing adopts a mode of stacking cross sections layer by layer to construct an object, for some suspended parts in a model, common printing equipment of fused deposition type, photocuring molding, digital light processing molding, selective laser melting and the like needs to add extra supports under the parts to prevent the structure from deforming and collapsing, and the supports are removed after printing is finished to obtain the final structure. The presence of these supports not only results in waste of raw materials and increased manufacturing costs, but also the post-treatment process of removing the supports reduces the surface finish of the structure and increases the length of the manufacturing process. Therefore, the self-supporting property of the structure is considered in the structural topology optimization design, namely, the requirement that the structure can be printed completely without adding extra support is met while the optimal structure is found, and the method has very important engineering application value.
The document "Gaynor AT, Guest JK. topology Optimization designing Optimization algorithms: engineering scientific information in additive manufacturing through design, structural and Multidisciplicating Optimization,2016,54(5): 1157-. The document states that this structure is self-supporting during printing when the design structure has all dangling boundaries inclined at an angle with respect to the printing reference plane that is greater than a critical angle value specified by the printing apparatus. The method proposed in the literature is based on the density method, requiring the presence of a non-empty cell not only for a non-zero variation of the cell density, but also for a sufficient distribution of material in the support zone of the lower part of the cell. In each optimization iteration step, the method provided by the literature needs to traverse each unit layer by layer from bottom to top, and judge whether the unit exists according to the numerical value of the unit density variable and the average density value of the supporting area units, so as to obtain the integral topology of the structure.
While the methods proposed in the literature result in self-supporting structures, the structures sacrifice a lot of compliance in order to be self-supporting. In addition, the design variables are large in number, large in calculation amount, slow in convergence and low in efficiency. The existence of the intermediate unit makes the result of the design ambiguous and the boundary jagged, which greatly increases the workload of reconstruction and is difficult to apply to engineering practice.
Disclosure of Invention
In order to overcome the defect that the existing topological optimization design method of the self-supporting structure is poor in practicability, the invention provides a topological optimization design method of the self-supporting structure in additive manufacturing. The method arranges a certain number of closed polygonal holes in an entity design area, and drives the topological layout evolution of the structure through the actions of movement, deformation, fusion, reduction, expansion and the like of the holes. In addition, the relative position between each polygon vertex is controlled, so that the inclination angle of the suspended part in the overall structure is larger than the critical angle value, and the self-support of the structure is realized. Compared with the design method of the background art, the self-supporting sacrificial fabric has less flexibility for realizing the structure. In addition, the design variables are independent of the units, the number is small, the calculation amount is small, and convergence is easy. In addition, the structural boundary designed by the invention is smooth, no intermediate density unit exists in the structure, the post-processing processes such as model reconstruction and the like are facilitated for engineering designers, and the practicability is good.
The technical scheme adopted by the invention for solving the technical problems is as follows: a topological optimization design method of a self-supporting structure in additive manufacturing is characterized by comprising the following steps:
step one, for a polygon with n sides, the included angle of the connecting lines of any two adjacent vertexes and the central point is required to be equal. And sequentially solving a critical ratio of the distances from the two adjacent vertexes to the central point, wherein the critical ratio enables the included angle between the edge where the two vertexes are located and the reference plane to be a critical angle value. If the critical ratio exists, the corresponding proportional relation is used as a design variable for controlling the relative position between the adjacent vertexes of the polygon. Otherwise, the design variable is defined as the distance from the vertex to the central point. In addition, the design variables also include center point coordinates to control the movement of the polygon.
And step two, initially distributing m n-edge holes on the solid region omega, and giving an initial value to the design variable.
And step three, obtaining a level set function of each edge of each polygon through n vertex coordinates of the polygon. Utilizing the KS function to realize the Boolean intersection operation of the level set function of all edges of each polygon to obtain the level set function of each polygon, wherein the calculation formula is as follows:
Figure BDA0001346513660000021
wherein p is a parameter of the KS function, and wherein p is<0, representing a Boolean crossover operation. Phi is ai，jAs a function of the level set of the ith edge in the jth polygon. PhijAs a function of the level set of the jth polygon.
Level set function of monolithic structure
Figure BDA0001346513660000022
Also implemented by the KS function:
Figure BDA0001346513660000023
wherein p >0 in the formula represents Boolean operation, and the minus sign represents a hole at the polygon.
And step four, dispersing omega by adopting a fixed grid, and defining load and boundary conditions at the same time.
Step five, defining the topology optimization problem as follows:
Min J＝FTU
Figure BDA0001346513660000031
wherein J, V and
Figure BDA0001346513660000032
representing structural compliance, total volume and maximum volume constraints. K. F and U represent the overall stiffness matrix, the overall load vector and the displacement vector of the structure, respectively. Here, H is a Heaviside function used to screen the integration points involved in the calculation, and x is the coordinates of the integration points. d ═ d (d)1，d2，d3，...，dm×(n+2)) Is a design variable vector, dtDenotes the tth design variable, t 1, 2.., m × (n +2), the upper and lower limits of which are respectively tdAnd
Figure BDA0001346513660000033
the design variables include three types of coordinates of the center point of the polygon, distances from the center point to the vertex and proportions between the distances.
And sixthly, carrying out primary finite element analysis on the established model, and respectively carrying out sensitivity analysis on the target function and the constraint function. And then, an optimization algorithm GCMMA is selected from a structure optimization platform Boss-Quattro for optimization design to obtain an optimization result.
The invention has the beneficial effects that: the method arranges a certain number of closed polygonal holes in an entity design area, and drives the topological layout evolution of the structure through the actions of movement, deformation, fusion, reduction, expansion and the like of the holes. In addition, the relative position between each polygon vertex is controlled, so that the inclination angle of the suspended part in the overall structure is larger than the critical angle value, and the self-support of the structure is realized. Compared with the design method of the background art, the self-supporting sacrificial fabric has less flexibility for realizing the structure. In addition, the design variables are independent of the units, the number is small, the calculation amount is small, and convergence is easy. In addition, the structural boundary designed by the invention is smooth, no intermediate density unit exists in the structure, the post-processing processes such as model reconstruction and the like are facilitated for engineering designers, and the practicability is good.
The method obtains a design result after 100 steps of iteration in the embodiment, and has high convergence rate. The area of the initial structure is 29.91, the flexibility is 5800.88, the flexibility of the optimized structure is 3679.95, the area is 20.00, and the flexibility is reduced by 36.56%. Without considering self-support, the structural compliance obtained with the same initial structural optimization was 3634.28, the area was also 20.00, so achieving structural self-support only sacrificed 1.25% compliance. In contrast, in the same example of the document in the background art, the compliance of the designed structure considering the self-supporting performance of the structure is increased by 6% compared with the compliance of the designed structure not considering the self-supporting performance, which is much larger than that of the embodiment. In addition, the result structure boundary of this embodiment is clear, and does not have middle density, and engineering practicality is strong.
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
Drawings
FIG. 1 is a schematic diagram of the geometric dimensions and boundary conditions of a model in an embodiment of the method of the present invention.
FIG. 2 is a diagram illustrating an initial layout of polygonal holes in an embodiment of the method of the present invention.
FIG. 3 is a diagram of the design results of an embodiment of the method of the present invention.
Detailed Description
Reference is made to fig. 1-3. The topological optimization design method of the self-supporting structure in additive manufacturing takes a cantilever beam structure under fixed load as an example to illustrate the method. The cantilever beam structure has the length of 8 and the height of 5, the Young modulus of 1 and the Poisson ratio of 0.3. The distribution of material in the cantilever beam is designed to minimize compliance, with a total material usage volume fraction of up to 50%. The method comprises the following specific steps:
1. design variables are defined. In this embodiment, 24-sided holes are adopted, and the included angles of the connection lines from any adjacent vertex to the central point are required to be equal, so that the included angle can be calculated to be 15 °. And sequentially solving a critical ratio of the distances from the two adjacent vertexes to the central point, wherein the critical ratio enables the included angle between the edge where the two vertexes are located and the reference plane to be a critical angle value. Wherein, if there are solutions to the 16 critical ratios, the corresponding proportional relationship is used as a design variable for controlling the relative position between the adjacent vertexes of the polygon. If the other 8 critical ratios do not exist, the corresponding boundary is not a suspended part, the inclination angle of the boundary does not need to be limited, and the design variable is defined as the distance from the vertex to the central point. In addition, the design variables also include center point coordinates to control the movement of the polygon.
2. 17 24-sided holes were initially distributed over an 8 x 5 solid area Ω, and the initial shape of each hole was given as a square with a side length of 0.92, given the initial values of the design variables.
3. And constructing a level set function. Obtaining a level set function of each edge according to the vertex coordinates of each polygon, and performing Boolean intersection operation on all edges of each polygon by using a KS function to obtain the level set function of each polygon, wherein the calculation formula is as follows:
Figure BDA0001346513660000041
wherein p is a parameter of the KS function, and p is-7, which represents a Boolean operation representation. Phi is ai，jAs a function of the level set of the ith edge in the jth polygon. PhijAs a function of the level set of the jth polygon.
Level set function of monolithic structure
Figure BDA0001346513660000042
Comprises the following steps:
Figure BDA0001346513660000043
wherein p is 7, representing boolean and operation. The minus sign indicates holes where the polygon is.
4. Omega is discretized using a rectangular grid of 0.1 x 0.1. The boundary conditions are defined as: and fixing the left end node of the cantilever beam, and applying a load along the negative direction of the y axis at the midpoint of the right end of the cantilever beam, wherein the load is 10.
5. Defining a topology optimization problem:
Min J＝FTU
Figure BDA0001346513660000051
wherein J, V and
Figure BDA0001346513660000052
representing structural compliance, total volume and maximum volume constraints. K. F and U represent the overall stiffness matrix, the overall load vector and the displacement vector of the structure, respectively. H is the Heaviside function used to screen the integration points involved in the calculation, and x is the integration point coordinates. d ═ d (d)1，d2，d3，...，d17×(24+2)) Is a design variable vector, dtDenotes the tth design variable, t 1, 2.., 17 × (24+2), whose upper and lower limits are respectively tdAnd
Figure BDA0001346513660000053
the design variables of this embodiment include three types of coordinates of the center point of the polygon, the distance from the center point to the vertex, and the ratio between the distances. Wherein the abscissa and the ordinate of the center point are both defined as [0,10 ]]The center point to vertex distance ranges from 0,5]The upper limit of the proportional design variable is the corresponding critical ratio value, and the minimum value is greater than 0.
6. Finite element analysis and optimization solution. Finite element analysis is carried out on the model, and the sensitivity of the objective function and the sensitivity of the constraint function are solved through Matlab programming, and then an optimization algorithm GCMMA is selected from a structure optimization platform Boss-Quattro to carry out optimization design on the model, so that an optimization result is obtained.
The method obtains a design result after 100 steps of iteration in the embodiment, and has high convergence rate. The area of the initial structure is 29.91, the flexibility is 5800.88, the flexibility of the optimized structure is 3679.95, the area is 20.00, and the flexibility is reduced by 36.56%. Without considering self-support, the structural compliance obtained with the same initial structural optimization was 3634.28, the area was also 20.00, so achieving structural self-support only sacrificed 1.25% compliance. In contrast, in the same example of the document in the background art, the compliance of the designed structure considering the self-supporting performance of the structure is increased by 6% compared with the compliance of the designed structure not considering the self-supporting performance, which is much larger than that of the embodiment. In addition, the design variables of the embodiment are few, the calculated amount is small, the structure boundary of the result is clear, the intermediate density does not exist, and the engineering practicability is strong.

Claims (1)

1. A topological optimization design method of a self-supporting structure in additive manufacturing is characterized by comprising the following steps:
step one, for a polygon with n sides, the included angle of the connecting lines of any two adjacent vertexes and a central point is required to be equal; sequentially solving a critical ratio of the distances from two adjacent vertexes to the central point, wherein the critical ratio enables the included angle between the edge where the two vertexes are located and the reference plane to be a critical angle value; if the critical ratio exists, the corresponding proportional relation is used as a design variable for controlling the relative position between the adjacent vertexes of the polygon; otherwise, the design variable is defined as the distance from the vertex to the central point; in addition, the design variables also include center point coordinates to control the movement of the polygon;
step two, initially distributing m n-edge holes in an entity region omega, and giving an initial value to a design variable;
step three, obtaining a level set function of each edge of each polygon through n vertex coordinates of the polygon; utilizing the KS function to realize the Boolean intersection operation of the level set function of all edges of each polygon to obtain the level set function of each polygon, wherein the calculation formula is as follows:
Figure FDA0002370821620000011
wherein p is a parameter of the KS function, and wherein p is<0, representing a Boolean crossover operation; phi is ai,jIs the level set function of the ith side in the jth polygon; phijA level set function for the jth polygon;
level set function of monolithic structure
Figure FDA0002370821620000012
Also implemented by the KS function:
Figure FDA0002370821620000013
wherein, p >0 in the formula represents Boolean operation, and the negative sign represents a hole at the polygon;
dispersing omega by adopting a fixed grid, and defining load and boundary conditions at the same time;
step five, defining the topology optimization problem as follows:
Min J＝FTU
Figure FDA0002370821620000014
wherein J, V and
Figure FDA0002370821620000015
representing structural compliance, total volume and maximum volume constraints; K. f and U respectively represent an overall stiffness matrix, an overall load vector and a displacement vector of the structure; h is a Heaviside function used for screening integration points involved in calculation, and x is the coordinates of the integration points; d ═ d (d)1,d2,d3,...,dm×(n+2)) Is a design variable vector, dtDenotes the tth design variable, t 1, 2.., m × (n +2), the lower and upper limits of which are respectivelyd tAnd
Figure FDA0002370821620000021
the design variables comprise three types of polygon center point coordinates, distances from the center point to the vertex and proportions among the distances;
step six, carrying out primary finite element analysis on the established model, and respectively carrying out sensitivity analysis on the target function and the constraint function; and then, an optimization algorithm GCMMA is selected from a structure optimization platform Boss-Quattro for optimization design to obtain an optimization result.
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CN111814383B (en) * 2020-07-25 2022-05-31 西北工业大学 B-spline density method-based self-supporting structure topology optimization design method
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