CN115659619A - Geometric topological optimization and additive manufacturing based integrated method - Google Patents
Geometric topological optimization and additive manufacturing based integrated method Download PDFInfo
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Abstract
The invention relates to a method based on geometric topological optimization and additive manufacturing integration; firstly, a geometric model is constructed by NURBS, and simultaneously topology optimization is carried out by adopting an isogeometric analysis method, so that a smooth optimization result is obtained. Drawing, rotating, mirroring, array and other functions are compiled based on the principle of graphics, isosurface functions in MATLAB software are used for connecting spatial three-dimensional volume data equivalent points of an optimization result, and surface and vertex data in an obtained curved surface are returned in a structural body. These data are then concatenated to form a triangular patch geometry, i.e., STL format, and the resulting STL file is directly available for 3D printing. The whole process can be directly operated in MATLAB without being converted into other three-dimensional drawing software for post-processing, so that the working efficiency from optimization design to 3D printing is improved.
Description
Technical Field
The invention relates to the technical field of additive manufacturing, in particular to an isogeometric topological optimization and additive manufacturing integrated method.
Background
Currently, topology optimization is of great interest due to its highly innovative configuration. In popular terms, topological optimization is to find out where the interior of the structure needs a filling material to obtain the best performance under certain constraint conditions. However, the topology-optimized structure is very complex and difficult to manufacture, which greatly reduces the feasibility of engineering application. But with the rise of additive manufacturing, it is a feasible matter to manufacture complex topologies.
The existing method for integrating topology optimization and additive manufacturing is to establish a model in three-dimensional software, introduce the model into finite element analysis software to perform topology optimization on the model, and output the result of the topology optimization as a model which can be directly used for 3D printing. However, most of the existing finite element analysis software is based on classical finite elements, the result after topology optimization is very rough, conversion among a plurality of software is needed, and the working efficiency is very low.
Compared with the topological optimization using classical finite elements, the topological optimization result based on isogeometry is very smooth and has very high manufacturability, however, the topological optimization based on isogeometry is performed in MATLAB in the whole process, and how to output the result after isogeometry topological optimization in MATLAB as a model which can be used for 3D printing is a problem to be researched. Therefore, it is necessary to develop a method that can convert the iso-geometric topological optimization results into a model that can be directly used for 3D printing.
Disclosure of Invention
The invention provides an isogeometric topological optimization and additive manufacturing based integrated method, which is characterized by comprising the following steps of:
step 1, obtaining an optimal topological structure by adopting an NURBS (non-uniform rational B-spline) construction model;
step 2, after a two-dimensional optimization result of the equal geometric topological optimization is obtained, post-processing is carried out on the two-dimensional optimization result, and an STL file which can be directly used for 3D printing is output;
step 3, the result of the geometric topological optimization is 3-dimensional, stretching, rotating, mirroring and array are not needed, and the STL file is directly output through post-processing;
step 4, the output STL file is used for 3D printing, and the mechanical property of the printed model is checked;
the method also comprises the following steps in the step 1:
step 11, constructing a geometric model by using NURBS;
step 12, seeking an optimal layout of materials in a design domain through topological optimization;
step 13, adopting a Shelpfder function to smooth;
step 14, constructing a density distribution function and solving a structural response by using an isogeometric method;
step 15, calculating the sensitivity;
and step 16, updating the density of the control points through an OC optimization criterion method.
Further, in step 11, the geometric model two-dimensional NURBS surface may be approximated as:
wherein, the first and the second end of the pipe are connected with each other,
denotes NURBS base function, P i,j Representing the control points, η and ξ represent the two directions of the parameter space, respectively.
Further, in step 12, in the topology optimization process, the density of the control points is used as the design variable to be evolved, and the mathematical formula of the isogeometric topology optimization method for minimizing the compliance is expressed as:
find:ρ i,j (i＝1，2，…，n；j＝1，2，…，m)
wherein ρ i,j Representing the control density, J representing the objective function, u representing the displacement field,
a control density function representing the topology of the structure,
representing an elasticity matrix, epsilon representing strain, and omega representing a design domain; n, m represents the number of control points in two parameter directions; g is volume constraint, u is displacement field under IGA solution, and G represents Dirichlet boundary gamma D The above-specified displacement vector, δ u, is the motion-allowable displacement space H 1 A virtual displacement field in (Ω); v. of 0 Represents the volume fraction of the solid material, a (u, δ u) represents the bilinear energy function, and l (δ u) represents the linear load function.
Further, in step 13, the node densities are smoothed using a shepherd's function, each node density being equal to the average of all node densities within the current node density local support domain:
wherein the content of the first and second substances,
denotes the smoothed control density, Ψ denotes the shepherd function,
mu represents the number of nodes in the current node density local support domain in two parameter directions.
Further, in step 14, the NURBS basis function is linearly combined with the smoothed node density to construct a density distribution function:
solving the structural response using the IGA method, omega, in physical space e In (e), the IGA cell stiffness matrix may be represented as:
wherein, B T Denotes the transpose of the geometric matrix, B denotes the geometric matrix, Ω e Representing the design domain of the cell and D representing the elastic matrix.
In isogeometric analysis, the element stiffness matrix involves two coordinate transformations, and the calculation of the element stiffness matrix can be expressed as:
wherein, J 1 ，J 2 Respectively representing two mapped jacobian matrices. And adopting Gaussian integration to the formula, and replacing function integration with numerical integration to obtain a detailed numerical value of the unit stiffness matrix:
wherein ξ i And η j Representing the coordinates of the Gaussian points in the parameter space, ω i ，ω j Representing the weight.
Further, in step 15, the sensitivity calculation is performed by taking the first derivative of the objective function and the volume constraint with respect to the density distribution function:
wherein γ represents a penalty parameter，D 0 A matrix of elastic tensors representing the density of the solid.
Using the chain derivation method, the derivative of the density distribution function with respect to the initial control density is obtained:
and (3) deducing a first derivative of the target function and the constraint function relative to the initial control density, and completing final sensitivity analysis:
further, in step 16, the heuristic update scheme of the OC optimization criterion method is:
wherein the content of the first and second substances,
a representation of the update factor is made,
denotes the density value, p, of the k-th iteration min Denotes the minimum density value, p max Denotes the maximum density value, t denotes the motion limit, λ (k) A lagrange multiplier representing the kth iteration; mu is a very small positive number to avoid a denominator of zero.
Furthermore, in step 2, stretching, rotating, mirroring and array functions are written by using the principle of graphics; connecting the spatial three-dimensional volume data equivalent points of the optimization result by using an isosurface function in MATLAB, and returning the obtained data in the curved surface in the structural body; these data are concatenated to form the STL format of the triangular patch geometry.
Further, in step 2, the stretched length is:
L＝ht×min(nelx，nely)
where ht is a coefficient based on stretch modeling, and nelx and nely represent the number of cells in the x and y directions, respectively.
Further, in step 2, the topology rotates around the rotation axis with the rotation axis as the center, S represents the distance of the topology from the rotation axis:
S＝lr×nelx
where lr is a coefficient based on rotational modeling.
The invention achieves the following beneficial effects:
the existing method for integrating topology optimization and additive manufacturing is to firstly establish a model by using three-dimensional software, then introduce the model into finite element analysis software to carry out topology optimization on the model, and then output the result of the topology optimization into the model which can be directly used for 3D printing. However, most of the existing finite element analysis software is based on classical finite elements, the result after topology optimization is very rough, conversion among a plurality of software is needed, and the working efficiency is very low.
In the invention, an isogeometric method is used for replacing a finite element to carry out topology optimization, but the whole process of the isogeometric topology optimization is carried out in MATLAB, so that a post-processing program is written, and the result after the isogeometric topology optimization is output as a model which can be used for 3D printing. The whole process does not need other three-dimensional software for post-processing, and the working efficiency is greatly improved. The results after optimization are very smooth and extremely high in manufacturability, as shown in the figure.
Drawings
FIG. 1 is an analytical flow chart of an integrated approach based on iso-geometric topological optimization and additive manufacturing in accordance with the present invention;
FIG. 2 is a schematic diagram of a result of geometric topological optimization of a cantilever beam based on an integrated geometric topological optimization and additive manufacturing method according to the present invention;
FIG. 3 is a schematic diagram of a stretched cantilever beam in an integrated approach based on iso-geometric topological optimization and additive manufacturing in accordance with the present invention;
FIG. 4 is a schematic diagram of a result of geometric topological optimization of a cantilever beam based on an integrated geometric topological optimization and additive manufacturing method according to the present invention;
FIG. 5 is a schematic view of a cantilever beam rotated by 20 degrees in an integrated method based on iso-geometric topological optimization and additive manufacturing according to the present invention;
FIG. 6 is a schematic diagram of the result of the geometric topology optimization of a Michell structure in an integrated method based on geometric topology optimization and additive manufacturing according to the present invention;
FIG. 7 is a schematic diagram of the results after mirroring of the Michell structure in an integrated method based on iso-geometric topology optimization and additive manufacturing according to the present invention;
FIG. 8 is a schematic structural diagram of an initial optimization in an integrated iso-geometric topology optimization and additive manufacturing based method according to the present invention;
FIG. 9 is a schematic diagram of a post-array structure in an integrated iso-geometric topology optimization and additive manufacturing based method according to the present invention;
fig. 10 is a schematic diagram of an STL file based on post-array output in an integrated iso-geometric topology optimization and additive manufacturing method in accordance with the present invention.
Detailed Description
The technical solution of the present invention will be described in more detail with reference to the accompanying drawings, and the present invention includes, but is not limited to, the following embodiments.
As shown in fig. 1, the present embodiment provides an integrated method based on iso-geometric topological optimization and additive manufacturing, and meanwhile, avoids the problem that the optimized result is rough when using the conventional finite element for topological optimization. According to the method, the traditional finite element is replaced by using geometric analysis methods such as NURBS (non-uniform rational B-spline), iterative solution is carried out by taking the density of control points as a design variable, then an optimized result is obtained, and the optimized result is subjected to post-processing to be output as an STL (standard template library) file which can be directly used for 3D (three-dimensional) printing, so that the integration of CAD (computer aided design), CAE (computer aided engineering) and additive manufacturing is realized.
The specific implementation steps of the method based on the integration of the geometric topological optimization and the additive manufacturing are as follows:
step 1, geometric topological optimization is performed:
specifically, a NURBS construction model is adopted, the density rho of control points is used as a design variable, the minimum compliance J is used as a target function, a Shelpferd function is adopted for smoothing, then sensitivity calculation is carried out, and the density of the control points is updated through an OC optimization criterion method to obtain an optimal topological structure.
The geometric topological optimization comprises the following steps:
step 11, constructing a geometric model by using NURBS; the two-dimensional NURBS surface may be approximated as:
wherein, the first and the second end of the pipe are connected with each other,
denotes NURBS basis function, P i,j Representing the control points, η and ξ represent the two directions of the parameter space, respectively.
Step 12, a topology optimization problem; the basic objective of topology optimization is to find the optimal layout of materials in the design domain, in the optimization process, the density of control points is used as the design variable to be evolved, and the mathematical formula of the isogeometric topology optimization method for compliance minimization is expressed as:
wherein ρ i,j Representing the control density, J representing the objective function, u representing the displacement field,
a control density function representing the topology of the structure,
representing an elasticity matrix, epsilon representing strain, and omega representing a design domain; n, m represents the number of control points in two parameter directions; g is volume constraint, u is displacement field under IGA solution, and G represents Dirichlet boundary gamma D The above-specified displacement vector, δ u, is the motion-allowable displacement space H 1 A virtual displacement field in (Ω); v. of 0 Represents the volume fraction of the solid material, a (u, δ u) represents the bilinear energy function, and l (δ u) represents the linear load function.
Step 13, adopting a shepherd function to carry out smoothing;
the node density was smoothed using Shepard approximation function. The basic principle is as follows: each node density is equal to the average of all node densities within the current node density local support domain.
Wherein the content of the first and second substances,
denotes the smoothed control density, Ψ denotes the shepherd function,
mu represents the number of nodes in the current node density local support domain in two parameter directions.
Step 14, constructing a density distribution function and solving a structural response by using an isogeometric method; the NURBS basis function is linearly combined with the smoothed node density to construct a density distribution function.
Then, the structural response is solved by using an IGA method, and the structure response is in a physical space omega e In (3), the IGA cell stiffness matrix may be expressed as:
wherein, B T Denotes the transpose of the geometric matrix, B denotes the geometric matrix, Ω e Representing the design domain of the cell and D representing the elastic matrix.
In isogeometric analysis, the cell stiffness matrix involves two coordinate transformations, respectively Y: (
To omega e ): parameter space to physical space; z (
To
): parent space to parameter space. Thus, the calculation of the cell stiffness matrix can be expressed as:
wherein, J 1 ，J 2 Respectively representing two mapped jacobian matrices. And adopting Gaussian integration to the formula, and replacing function integration with numerical integration to obtain a detailed numerical value of the unit stiffness matrix:
wherein ξ i And η j Representing the coordinates of the Gaussian points in the parameter space, ω i ，ω j Representing the weight.
Step 15, calculating the sensitivity; from equation (2), the first derivative of the objective function and the volume constraint with respect to the density distribution function can be obtained:
wherein γ represents a penalty parameter, S 0 A matrix of elastic tensors representing the density of the solid.
The density distribution function includes the smoothing mechanism of Shepard function and the linear combination of NURBS basis function and smooth control density. Using the chain derivation method, the derivative of the density distribution function with respect to the initial control density can be obtained as follows:
then, the first derivative of the objective function and the constraint function relative to the initial control density can be deduced, and the final sensitivity analysis is completed. The final form is as follows:
step 16, updating the density of the control points by an OC optimization criterion method; the OC method is to construct a Lagrange function of an optimization problem by introducing a K-T condition of a topological optimization problem, construct an optimization criterion according to physical properties and structural characteristics of materials, and indirectly optimize the structural characteristics by iterative updating of design variables and Lagrange multipliers. The heuristic update scheme is as follows:
wherein the content of the first and second substances,
the value of the update factor is represented by,
denotes the density value, p, of the k-th iteration min Denotes the minimum density value, p max Means of maximumLarge density value, t represents a motion limit, λ (k) Representing the lagrangian multiplier for the kth iteration. Mu is a very small positive number to avoid a denominator of zero.
And 2, after a two-dimensional optimization result of the isogeometric topological optimization is obtained, post-processing is carried out on the two-dimensional optimization result, and the STL file which can be directly used for 3D printing is output. The general flow of the post-treatment is as follows: firstly, writing functions of stretching, rotating, mirroring, array and the like by using a graphics principle, then connecting spatial three-dimensional volume data equivalent points of an optimization result by using an isosurface function in MATLAB, returning data (a surface and a vertex) in an obtained curved surface in a structural body, and then connecting the data to form a triangular patch geometry, namely an STL format.
Stretching;
as shown in fig. 2, taking the result of the geometric topology optimization of the two-dimensional cantilever as an example, a stretching command is performed on the two-dimensional cantilever, and the output STL file is shown in fig. 3. Wherein the length of stretching:
L＝ht×min(nelx，nely) (13)
where ht is a coefficient based on stretch modeling, nelx and nely represent the number of cells in the x and y directions, respectively.
Rotating:
taking the rotation axis as the center, the topology rotates around it, as shown in fig. 4, S represents the distance from the topology to the rotation axis, and the calculation formula is as follows:
S＝lr×nelx (14)
where lr is a coefficient based on rotational modeling. Or taking the two-dimensional cantilever beam optimization result with equal geometry as an example, rotating the two-dimensional cantilever beam optimization result by 20 degrees around the rotating shaft, and outputting the STL file as shown in the attached figure 5.
Mirroring:
in engineering problems, there are many symmetric structures, which we generally reduce to half for analysis. Therefore, the mirror command is very practical for this symmetrical structure. As shown in FIG. 6, left, bottom, right, and top represent mirror images along the left, bottom, right, and top, respectively. Taking the result of the geometric topology optimization of the two-dimensional Michell structure as an example, the use of only the mirror image command is not enough to generate the three-dimensional STL file, and the three-dimensional STL file needs to be stretched first and then mirrored. To facilitate the effect, the Michell structure is mirrored along the left part, and the output STL file is shown in fig. 7.
Array:
when analyzing dot matrix materials or periodically arranged materials, array commands are required. As shown in fig. 8-9, the structure is firstly arrayed into 3 rows and 2 columns, and then the structure is subjected to post-stretching treatment, and the output STL file is shown in fig. 10.
And 3, if the optimization result is 3-dimensional, directly outputting the STL file through post-processing without stretching, rotating, mirroring and arraying.
And 4, using the output STL file for 3D printing, and checking the mechanical property of the printed model.
The method for integrating geometric topological optimization and additive manufacturing is described in detail, and many examples are applied in the method to explain the principle and the embodiment of the invention, and the description of the examples is only used to help understand the method and the core idea of the invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.
Claims (10)
1. An isogeometric topological optimization and additive manufacturing based integration method is characterized by comprising the following steps of:
step 1, obtaining an optimal topological structure by adopting an NURBS (non-uniform rational B-spline) construction model;
step 2, after a two-dimensional optimization result of the equal geometric topological optimization is obtained, post-processing is carried out on the two-dimensional optimization result, and an STL file which can be directly used for 3D printing is output;
step 3, the result of the geometric topological optimization is 3-dimensional, stretching, rotating, mirroring and array are not needed, and the STL file is directly output through post-processing;
step 4, using the output STL file for 3D printing, and checking the mechanical property of the printed model;
the method also comprises the following steps in the step 1:
step 11, constructing a geometric model by using NURBS;
step 12, seeking an optimal layout of materials in a design domain through topological optimization;
step 13, adopting a shepherd function to carry out smoothing;
step 14, constructing a density distribution function and solving a structural response by using an isogeometric method;
step 15, calculating the sensitivity;
and step 16, updating the density of the control points through an OC optimization criterion method.
2. The integrated method for geometric-based topological optimization and additive manufacturing according to claim 1, wherein in step 11, the geometric model two-dimensional NURBS surface can be approximated as:
wherein the content of the first and second substances,
denotes NURBS base function, P i,j Representing the control points, η and ξ represent the two directions of the parameter space, respectively.
3. The integrated method based on isogeometric topological optimization and additive manufacturing according to claim 2, wherein in step 12, the density of the control points is used as the design variable to be evolved in the topological optimization process, and the mathematical formula of the isogeometric topological optimization method for minimizing the flexibility is expressed as:
find:ρ i,j (i＝1，2，…，n；j＝1，2，…，m)
Min：
S.t:
where ρ is i,j Representing the control density, J representing the objective function, u representing the displacement field,
a control density function representing the topology of the structure,
representing an elasticity matrix, epsilon representing strain, and omega representing a design domain; n, m represents the number of control points in two parameter directions; g is volume constraint, u is displacement field under IGA solution, and G represents Dirichlet boundary gamma D The above-specified displacement vector, δ u, is the motion-allowable displacement space H 1 A virtual displacement field in (Ω); v. of 0 Represents the volume fraction of the solid material, a (u, δ u) represents the bilinear energy function, and l (δ u) represents the linear load function.
4. The integrated method based on equi-geometric topological optimization and additive manufacturing of claim 3, wherein in step 13, the node densities are smoothed by a Shelpfer function, each node density being equal to an average value of all node densities within the current node density local support domain:
wherein the content of the first and second substances,
denotes the smoothed control density, Ψ denotes the shepherd function,
mu represents the number of nodes in the current node density local support domain in two parameter directions.
5. The integrated method for geometric topology optimization and additive manufacturing according to claim 4, wherein in step 14, the NURBS basis function is linearly combined with the smoothed node density to construct a density distribution function:
solving the structural response using the IGA method, omega, in physical space e In (3), the IGA cell stiffness matrix may be expressed as:
wherein, B T Denotes the transpose of the geometric matrix, B denotes the geometric matrix, Ω e Representing the design domain of the cell and D representing the elastic matrix.
In isogeometric analysis, the element stiffness matrix involves two coordinate transformations, and the calculation of the element stiffness matrix can be expressed as:
wherein, J 1 ，J 2 Respectively representing two mapped jacobian matrices. And adopting Gaussian integration to the formula, and replacing function integration with numerical integration to obtain a detailed numerical value of the unit stiffness matrix:
wherein ξ i And η j Representing the coordinates of the Gaussian points in the parameter space, ω i ，ω j Representing the weight.
6. The integrated method for geometric-based topological optimization and additive manufacturing according to claim 5, wherein in step 15, the sensitivity calculation is performed by obtaining the first derivative of the objective function and the volume constraint with respect to the density distribution function:
wherein γ represents a penalty parameter, D 0 A matrix of elastic tensors representing the density of the solid;
using the chain derivation method, the derivative of the density distribution function with respect to the initial control density is obtained:
and (3) deducing a first derivative of the target function and the constraint function relative to the initial control density, and completing final sensitivity analysis:
7. the integrated method based on equi-geometric topological optimization and additive manufacturing of claim 6, wherein in step 16, the heuristic update scheme of the OC optimization criterion method is as follows:
wherein，
The value of the update factor is represented by,
denotes the density value, p, of the k-th iteration min Denotes the minimum density value, p max Denotes the maximum density value, t denotes the motion limit, λ (k) A lagrange multiplier representing the kth iteration; mu is a very small positive number to avoid a denominator of zero.
8. The integrated method based on the equi-geometric topological optimization and the additive manufacturing of claim 1, wherein in the step 2, stretching, rotating, mirroring and array functions are written by using a graphic principle; connecting the spatial three-dimensional volume data equivalent points of the optimization result by using an isosurface function in MATLAB, and returning the obtained data in the curved surface in the structural body; these data are concatenated to form the STL format of the triangular patch geometry.
9. The integrated method based on the equi-geometric topological optimization and the additive manufacturing according to claim 8, wherein in the step 2, the stretching length is as follows:
L＝ht×min(nelx，nely)
where ht is a coefficient based on stretch modeling, and nelx and nely represent the number of cells in the x and y directions, respectively.
10. The integrated method based on equi-geometric topological optimization and additive manufacturing of claim 8, wherein in step 2, the topological structure rotates around the rotation axis with the rotation axis as the center, and S represents the distance from the topological structure to the rotation axis:
S＝lr×nelx
where lr is a coefficient based on rotational modeling.
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