CN111444640B - Structural topology optimization method considering inclination constraint of additive manufacturing - Google Patents
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Abstract
The invention discloses a structural topology optimization method considering inclination constraint of additive manufacturing, and aims to solve the problem of self-support in the process of structural additive manufacturing. The method comprises the steps of firstly establishing a unit density updating iteration rule meeting inclination angle constraint of additive manufacturing, then adding the rule into a topological optimization model after smooth approximation treatment, constructing a topological optimization mathematical model considering the inclination angle constraint of the additive manufacturing by taking the minimum flexibility of a structure as an optimization target, and solving the optimization model by adopting an OC algorithm to obtain an optimal configuration. The examples provided demonstrate that a continuous body structure topology optimization method that takes into account additive manufacturing tilt angle constraints is effective that is capable of controlling the tilt angle of the topology configuration, enabling additive manufacturing directly without adding additional support structures.
Description
Technical Field
The invention belongs to the technical field related to structural topology optimization design, and particularly relates to a structural topology optimization method considering additive manufacturing inclination constraint.
Background
The additive manufacturing is a subverted manufacturing technology, and compared with the traditional subtractive manufacturing mode, the additive manufacturing technology realizes the preparation of the structure through the mode of accumulating materials layer by layer, and can realize the rapid forming of the complex structure due to the unique manufacturing mechanism, thereby having great advantages. With the development of the age, additive manufacturing technology is mature, and is widely applied to the fields of military, medical treatment, aerospace, electronics and the like. However, in the additive manufacturing process, there are still many manufacturing constraints, for example, in the process of manufacturing a large overhanging structure, a supporting structure needs to be added, otherwise, collapse, failure, and other phenomena of the structure occur, and the supporting structure needs to satisfy the characteristics of printability, easy removal, less material consumption, and the like, so that the printing time and cost are saved. Therefore, in the additive manufacturing process, how to reduce the supporting structure and even remove the supporting structure to achieve self-supporting of the configuration, so that the parts are manufactured smoothly, and the method has great research significance.
The topology optimization technology is to search the optimal layout of materials in a determined design domain, so that certain performance of a structure is optimal, and the topology optimization technology is the research direction with the highest potential in the field of structure optimization design. The topology optimization method can obtain an innovative configuration with special performance, however, the topological structure is often complex, the traditional manufacturing process is difficult to process and even cannot manufacture, the problem in the aspect is exactly solved by the appearance of the additive manufacturing technology, and the two materials can be combined to better play respective advantages. Because of the special manufacturing principle of additive manufacturing, the structure to be printed can be printed only when being fully supported, and the inclination angle of the structure must be larger than the critical value of the printable inclination angle of the equipment on a macroscopic scale, otherwise, collapse phenomenon can occur, and the required structure cannot be manufactured. Therefore, the structural topology optimization method considering the inclination constraint of additive manufacturing is researched, and the technical requirement is very high.
Disclosure of Invention
Aiming at the improvement requirement of the prior art, the invention provides a structural topology optimization method considering the inclination constraint of additive manufacturing, which follows the basic principle of additive manufacturing, establishes a unit density updating iteration rule meeting the inclination constraint of additive manufacturing, enables the structure to macroscopically meet the inclination constraint of additive manufacturing, avoids adding additional supporting structures, and enables the optimized topological structure to be directly printed.
In order to achieve the above object, the present invention provides a structural topology optimization method considering inclination constraint of additive manufacturing, comprising the steps of:
step one: defining initial conditions, namely structural design domain, load conditions, constraint conditions and related properties of materials, uniformly dispersing the materials into each unit to obtain initial unit density distribution;
step two: firstly, obtaining the density distribution of an initial structure by using a density filtering method, and then updating an iteration rule according to the unit density of the inclination constraint of additive manufacturing to obtain printable density distribution meeting the inclination constraint;
step three: carrying out finite element analysis on the structure to obtain the whole displacement field of the structure;
step four: constructing a topological optimization mathematical model considering inclination constraint of additive manufacturing, and calculating to obtain an objective function value and sensitivity of design variables;
step five: updating the design variables by using an OC algorithm (Modified Optimization Criteria methods) to obtain the latest density distribution;
step six: judging whether the objective function is converged or not according to the convergence condition, outputting the topological configuration if the convergence condition is met, ending the iterative computation, otherwise, turning to the second step to continue the iterative computation.
Further, the unit density updating iteration rule considering the inclination constraint of the additive manufacturing is as follows:
wherein i, j are the positions of the unit in the vertical direction and the horizontal direction respectively, and ζ (i, j) is the printable density value of the unit at (i, j) after each iteration; x (i, j) is the initial density value of the cell at (i, j) before each iteration; Γ (i, j) is the maximum value of printable density of each cell within the cell support region at (i, j); ψ (i, j) is the minimum value of the maximum printable densities of the layers in the unit support area at (i, j), and n is the number of layers of the support area.
Further, the inclination angle constraint obtained by selecting one layer in the support area is 45 degrees, and the iterative model is as follows:
ξ(i,j)＝min(x(i,j),Γ(i,j))(2)
Γ(i,j)＝max(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1))(3)
further, the inclination angle constraint obtained by selecting two layers in the support area is 63 degrees, and the iterative model is as follows:
ξ(i,j)＝min(x(i,j),Ψ(i,j))(4)
Ψ(i,j)＝min(Γ 1 (i,j),Γ 2 (i,j))(5)
Γ 1 (i,j)＝max(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1))(6)
Γ 2 (i,j)＝max(ξ(i-2,j-1),ξ(i-2,j),ξ(i-2,j+1))(7)
wherein Γ is n (i, j) is the maximum printable density of each cell in the i-n layers within the cell support region at (i, j).
Further, selecting two layers of support areas and three units of each layer, and obtaining an iteration model constrained by the inclination angle of 63 degrees as follows:
where x is the initial density of the unit, ψ is the smaller value of the maximum printable density of each layer in the unit support area Γ 1 i 、Γ 2 i Is the maximum printable density of each cell in each layer, ζ 1 、ξ 2 、ξ 3 、
Is the printable structure density, n s Is the number of units per layer of the unit support region, n s =3; epsilon and P are parameters that adjust the smooth approximation model, ζ 0 Taking 0.5.
Further, the model that considers the additive manufacturing tilt constraints is:
ξ(i,j)＝smin(x(i,j),Ψ(i,j))(12)
Ψ(i,j)＝smin(Γ 1 (i,j),Γ 2 (i-1,j))(13)
Γ 1 (i,j)＝smax(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1))(14)
Γ 2 (i,j)＝smax(ξ(i-2,j-1),ξ(i-2,j),ξ(i-2,j+1))(15)
wherein smin and smax take the approximate form of formulas (8) and (10), respectively, it being noted that the printable density of the i-th layer is closely related to the printable densities of the 1 st to i-1 st layers,
a short form of a smin symbol, and a single subscript indicates an index symbol of the number of layers.
Further, the topological optimization mathematical model taking into account the additive manufacturing dip constraints is:
wherein c is an objective function, i.e., the overall compliance value of the structure; k is the overall stiffness matrix; u is integral displacement; u (u) e Is a unit displacement matrix; k (k) e Is a matrix of cell stiffness; f is an external load vector; v (x) is the volume after structure optimization; v (V) 0 Is the initial volume of the structure; f is a given volume fraction; zeta type toy i For the tilt constraint equation of the structure i.e. tilt constraint of additive manufacturing,
a short form of a smin symbol, and a single subscript indicates an index symbol of the number of layers.
Further, the sensitivity of the objective function with respect to the design variables can be derived from the following formula.
Wherein lambda is i For Lagrangian multiplier vectors, for the first layer and the second layer, define
And is also provided with
Further, the design variables are updated using an OC algorithm:
where m is a positive movement limit,
is a scale factor, eta is a numerical damping coefficient, and x min Is a lower value limit; x is x new And (5) iteratively optimizing the updated solution for the optimized rule operator.
In summary, compared with the prior art, the technical scheme has the following advantages:
(1) The structural topology optimization method considering the inclination angle constraint of the additive manufacturing follows the basic principle of layer-by-layer stacking of the additive manufacturing, and establishes a unit density iteration rule meeting the inclination angle constraint of the additive manufacturing from a microscopic angle so that the structure macroscopically meets the inclination angle constraint of the additive manufacturing.
(2) The structural topology optimization method considering the inclination angle constraint of additive manufacturing can be suitable for a continuous structure, the optimized topological configuration can be directly printed, an additional supporting structure is not needed to be added, the self-supporting of the structure can be realized, and therefore the manufacturing cost is reduced.
Drawings
FIG. 1 is a flow chart of a structural topology optimization method of the present invention that takes into account additive manufacturing tilt constraints.
FIG. 2 is a schematic diagram of the present invention regarding the cell density update iteration rule.
Fig. 3 is a schematic diagram of the design domain of a Michell beam structure designed according to the present invention.
FIG. 4 is a Michell Liang Tapu optimized configuration without tilt constraints resulting from topology optimization of an example of the invention.
FIG. 5 is a Michell Liang Tapu optimized configuration with dip constraints resulting from an example topology optimization of the present invention.
FIG. 6 is a graph of the magnitude of the angle formed by the Michell Liang Tapu configuration of the example topology optimization of the present invention with respect to the boundary of the horizontal plane and the horizontal plane.
Fig. 7 is a Michell beam print model with tilt angle constraints obtained by 3D printing for an example of the present invention.
FIG. 8 is a Michell beam printing model without inclination constraint obtained by 3D printing according to an example of the present invention.
Detailed Description
The invention will be further described with reference to the drawings and examples in order to better explain the technical scheme, objects and advantages of the invention. Furthermore, the specific examples described herein are intended to be illustrative of the invention only and are not intended to be limiting.
As shown in fig. 1, the invention provides a structural topology optimization method considering inclination constraint of additive manufacturing, which mainly comprises the following steps:
step one: defining initial conditions, namely structural design domain, load conditions, constraint conditions and related properties of materials, uniformly dispersing the materials into each unit to obtain initial density distribution;
step two: firstly, obtaining the density distribution of an initial structure by using a density filtering method, and then updating an iteration rule according to the unit density of the inclination constraint of additive manufacturing to obtain printable density distribution meeting the inclination constraint; as shown in fig. 2, the cell density update iteration rule taking into account the additive manufacturing dip constraint, the mathematical model is:
wherein i, j are the positions of the unit in the vertical direction and the horizontal direction respectively, and ζ (i, j) is the printable density value of the unit at (i, j) after each iteration; x (i, j) is the initial density value of the cell at (i, j) before each iteration; Γ (i, j) is the maximum value of printable density of each cell within the cell support region at (i, j); ψ (i, j) is the minimum value of the maximum printable densities of the layers in the unit support area at (i, j), and n is the number of layers of the support area.
At this time, the inclination angle constraint obtained by selecting one layer in the support area is 45 degrees, and the iterative model is as follows:
ξ(i,j)＝min(x(i,j),Γ(i,j))(2)
Γ(i,j)＝max(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1))(3)
the inclination constraint obtained by selecting two layers in the support area is 63 degrees, and the iterative model is as follows:
ξ(i,j)＝min(x(i,j),Ψ(i,j))(4)
Ψ(i,j)＝min(Γ 1 (i,j),Γ 2 (i,j))(5)
Γ 1 (i,j)＝max(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1))(6)
Γ 2 (i,j)＝max(ξ(i-2,j-1),ξ(i-2,j),ξ(i-2,j+1))(7)
wherein Γ is n (i, j) is the maximum printable density of each cell in the i-n layers within the cell support region at (i, j).
Selecting two layers of support areas and three units of each layer, and obtaining an iteration model constrained by 63 degrees of inclination angles as follows:
where x is the initial density of the unit, ψ is the smaller value of the maximum printable density of each layer in the unit support area Γ 1 i 、Γ 2 i Is the maximum printable density of each cell in each layer, ζ 1 、ξ 2 、ξ 3 、
Is the printable structure density, n s Is the number of units per layer of the unit support region, n s =3; epsilon and P are parameters that adjust the smooth approximation model, ζ 0 Taking 0.5.
The model that considers the additive manufacturing tilt constraints is:
ξ(i,j)＝smin(x(i,j),Ψ(i,j))(12)
Ψ(i,j)＝smin(Γ 1 (i,j),Γ 2 (i-1,j))(13)
Γ 1 (i,j)＝smax(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1))(14)
Γ 2 (i,j)＝smax(ξ(i-2,j-1),ξ(i-2,j),ξ(i-2,j+1))(15)
wherein smin and smax take the approximate form of formulas (8) and (10), respectively, it being noted that the printable density of the i-th layer is closely related to the printable densities of the 1 st to i-1 st layers,
a short form of a smin symbol, and a single subscript indicates an index symbol of the number of layers.
Step three: carrying out finite element analysis on the structure to obtain the whole displacement field of the structure; in this embodiment, ku=f, where K is a global stiffness matrix and F is an external load vector, and the obtained global displacement matrix U includes the displacement U of each node e 。
Step four: constructing a topological optimization mathematical model considering inclination constraint of additive manufacturing, and calculating to obtain an objective function value and sensitivity of design variables;
first, the topology optimization mathematical model taking into account additive manufacturing dip constraints is:
wherein c is an objective function, i.e., the overall compliance value of the structure; k is the overall stiffness matrix; u is integral displacement; u (u) e Is a unit displacement matrix; k (k) e Is a matrix of cell stiffness; f is an external load vector; v (x) is the volume after structure optimization; v (V) 0 Is the initial volume of the structure; f is a given volume fraction; zeta type toy i For the tilt constraint equation of the structure i.e. tilt constraint of additive manufacturing,
a short form of a smin symbol, and a single subscript indicates an index symbol of the number of layers.
Second, the sensitivity of the objective function with respect to the design variables can be derived from the following formula.
Wherein lambda is i For the lagrangian multiplier vector,for the first layer and the second layer, define
And->
Then->
Step five: updating the design variables by using an OC algorithm (Modified Optimization Criteria methods) to obtain the latest density distribution;
Step six: judging whether the objective function is converged or not according to the convergence condition, outputting the topological configuration if the convergence condition is met, ending the iterative computation, otherwise, turning to the second step to continue the iterative computation.
Referring to fig. 3 to 8, the present invention is further described below with respect to the design of a michel beam structure. As shown in fig. 3, which is a schematic diagram of a design domain of a Michell beam structure, the Michell beam has a length of l=180 and a height of h=90, and a vertical downward concentrated force is applied to the center of the lower boundary of the bottom of the beam structure, wherein the lower left corner of the beam structure is fixed, and the lower right corner is simply supported. The elastic modulus of the material was 1, poisson ratio was 0.3, concentrated load f=1 (all amounts are relative values, dimensionless), and given volume fraction was 0.5. Based on finite element theory, a four-node unit is adopted to disperse a design domain into a grid model of 180X90, the minimum flexibility of the structure is used as an optimization target, a horizontal plane is defined as a substrate, and a vertical upward direction is the printing direction of the structure. Unit density update iteration rule smoothingThe relevant parameters of the approximation model are set to epsilon=10 -4 ，P＝40，ξ 0 ＝0.5。
The Michell Liang Tapu optimization design without inclination constraint and with inclination constraint is respectively carried out, the obtained topological configurations are shown in fig. 4 and 5, and it can be seen from the figures that the topological configurations without inclination constraint and with inclination constraint are completely different, the topological configurations with inclination constraint are mainly represented by the topological configurations with inclination constraint, the included angles of the components and the horizontal plane are relatively large, the topological configurations without inclination constraint have small angles of the components, even the situation of approximately zero exists, and the completely suspended structure cannot be directly used for additive manufacturing. In order to more intuitively represent the minimum inclination of the topology structure capable of being controlled by the added inclination constraint, typical inclinations of the topology structure without the inclination constraint and with the inclination constraint are measured respectively, the angle numbers are shown in fig. 4 (b) and fig. 5 (b), the sizes of all the angle values are shown in fig. 6, and all the angle values are rounded.
To further verify the feasibility of the topology optimization method proposed herein to accommodate the pitch constraints of the additive manufactured components, it was demonstrated that the topologically derived structure could be directly subjected to additive manufacturing. 3D printing is respectively carried out on topological configurations of the Michell beam with or without inclination constraint and the cantilever beam with the hole. The length of the Michell beam and the cantilever beam with the holes is set to be 120mm, the width of the cantilever beam is set to be 60mm, the thickness of the cantilever beam is set to be 10mm, and the model is respectively prepared by adopting a fused deposition modeling technology without adding an external supporting structure. Fig. 7 is a Michell beam printing model with inclination constraints obtained by 3D printing, and fig. 8 is a Michell beam printing model without inclination constraints obtained by 3D printing.
Through example analysis and 3D printing on the configuration, the structural topology optimization method considering the inclination constraint of additive manufacturing can prove to be feasible, and the inclination of the topology configuration can be controlled. And establishing a unit density iteration rule meeting inclination angle constraint of the additive manufacturing from a microscopic angle, so that the structure macroscopically meets the inclination angle constraint of the additive manufacturing. The method can be suitable for a continuous structure, the optimized topological configuration can be directly printed, an additional supporting structure is not required to be added, and the self-supporting of the structure can be realized, so that the manufacturing cost is reduced.
The above is a specific step of the present invention, and does not limit the protection scope of the present invention; any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included within the scope of the present invention.
Claims (8)
1. A structural topology optimization method taking into account additive manufacturing dip angle constraints, comprising the steps of:
step one: defining initial conditions, namely structural design domain, load conditions, constraint conditions and related properties of materials, uniformly dispersing the materials into each unit to obtain initial unit density distribution;
step two: updating iteration rules according to unit density of inclination constraint of additive manufacturing to obtain printable density distribution meeting inclination constraint;
step three: carrying out finite element analysis on the structure to obtain the whole displacement field of the structure;
step four: constructing a topological optimization mathematical model considering inclination constraint of additive manufacturing, and calculating to obtain an objective function value and sensitivity of design variables;
step five: updating the design variables by using an OC algorithm (Modified Optimization Criteria methods) to obtain the latest density distribution;
step six: judging whether the objective function is converged or not according to the convergence condition, outputting the topological configuration if the convergence condition is met, ending the iterative computation, otherwise, turning to the second step to continue the iterative computation;
the unit density updating iteration rule considering the inclination constraint of additive manufacturing is as follows:
wherein i, j are the positions of the unit in the vertical direction and the horizontal direction respectively, and ζ (i, j) is the printable density value of the unit at (i, j) after each iteration; x (i, j) is the initial density value of the cell at (i, j) before each iteration; Γ (i, j) is the maximum value of printable density of each cell within the cell support region at (i, j); ψ (i, j) is the minimum value of the maximum printable densities of the layers in the unit support area at (i, j), and n is the number of layers of the support area.
2. A structural topology optimization method taking into account additive manufacturing dip constraints as recited in claim 1, wherein: the inclination angle constraint obtained by selecting one layer in the support area is 45 degrees, and the iterative model is as follows:
ξ(i,j)＝min(x(i,j),Γ(i,j))(2)
Γ(i,j)＝max(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1))(3)。
3. a structural topology optimization method taking into account additive manufacturing dip constraints according to claim 2, wherein: the inclination constraint obtained by selecting two layers in the support area is 63 degrees, and the iterative model is as follows:
ξ(i,j)＝min(x(i,j),Ψ(i,j))(4)
Ψ(i,j)＝min(Γ 1 (i,j),Γ 2 (i,j)) (5)
Γ 1 (i,j)＝max(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1)) (6)
Γ 2 (i,j)＝max(ξ(i-2,j-1),ξ(i-2,j),ξ(i-2,j+1)) (7)
wherein Γ is n (i, j) is the maximum printable density of each cell in the i-n layers within the cell support region at (i, j).
4. A structural topology optimization method taking into account additive manufacturing dip constraints according to claim 3, wherein: selecting two layers of support areas and three units of each layer, and obtaining an iteration model constrained by 63 degrees of inclination angles as follows:
where x is the initial density of the unit, ψ is the smaller value of the maximum printable density of each layer in the unit support area Γ 1 i 、Γ 2 i Is the maximum printable density of each cell in each layer, ζ 1 、ξ 2 、ξ 3 、
Is the printable structure density, n s Is the number of units per layer of the unit support region, n s =3; epsilon and P are parameters that adjust the smooth approximation model, ζ 0 Taking 0.5.
5. The structural topology optimization method taking into account additive manufacturing dip constraints of claim 4, wherein: the model that considers the additive manufacturing tilt constraints is:
ξ(i,j)＝smin(x(i,j),Ψ(i,j)) (12)
Ψ(i,j)＝smin(Γ 1 (i,j),Γ 2 (i-1,j)) (13)
Γ 1 (i,j)＝smax(ξ(i-1,j-1),ξ(i-1,j),ξ(i-1,j+1)) (14)
Γ 2 (i,j)＝smax(ξ(i-2,j-1),ξ(i-2,j),ξ(i-2,j+1)) (15)
wherein smin and smax take the approximate form of formulas (8) and (10), respectively, it should be noted that the i-th layer is printableThe density is closely related to the printable densities of layers 1 through i-1,
a short form of a smin symbol, and a single subscript indicates an index symbol of the number of layers.
6. The structural topology optimization method taking into account additive manufacturing dip constraints of claim 5, wherein: the topological optimization mathematical model considering the inclination constraint of additive manufacturing is as follows:
wherein c is an objective function, i.e., the overall compliance value of the structure; k is the overall stiffness matrix; u is integral displacement; u (u) e Is a unit displacement matrix; k (k) e Is a matrix of cell stiffness; f is an external load vector; v (x) is the volume after structure optimization; v (V) 0 Is the initial volume of the structure; f is a given volume fraction; zeta type toy i For the tilt constraint equation of the structure i.e. tilt constraint of additive manufacturing,
a short form of a smin symbol, and a single subscript indicates an index symbol of the number of layers.
7. The structural topology optimization method taking into account additive manufacturing dip constraints of claim 6, wherein: the sensitivity of the objective function to the design variables can be deduced from the following formula:
wherein lambda is i For Lagrangian multiplier vectors, for the first layer and the second layer, define
And->
Then
I is an identity matrix.
8. The structural topology optimization method taking into account additive manufacturing dip constraints of claim 7, wherein: updating design variables by using an OC algorithm:
where m is a positive movement limit,
is a scale factor, eta is a numerical damping coefficient, and x min Is a lower value limit; x is x e new And (5) iteratively optimizing the updated solution for the optimized rule operator. />
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