CN112685945A - Magnetic-structure multi-physical-field topological optimization design method for additive manufacturing - Google Patents
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Abstract
The application discloses a magnetic-structure multi-physical-field topology optimization design method for additive manufacturing, which comprises the following steps: step 10, analyzing and obtaining a structure displacement vector and a magnetic field vector based on a structure field and magnetic field material interpolation model and a finite element analysis method by combining the unit printing density of an object to be printed, establishing a target function according to the structure displacement vector and the magnetic field vector, and establishing a magnetic-structure multi-physical field topology optimization model by combining a volume constraint condition; and 20, according to the magnetic-structure multi-physical topology optimization model, combining the objective function and the constraint to unit design density sensitivity, iteratively updating the unit design density space through an MMA algorithm, and when the relative error of the objective function in the magnetic-structure multi-physical field topology optimization model is judged to be smaller than a preset threshold value, printing the object to be printed according to the updated unit design density space. Through the technical scheme in this application, when taking into account magnetic field and structure field performance, realized the self-supporting of structure and printed, avoided the use of supporting material.
Description
Technical Field
The application relates to the technical field of engineering structures and analysis, in particular to a magnetic-structure multi-physical-field topological optimization design method for additive manufacturing.
Background
In recent years, with the development of automation, intellectualization, and weight reduction of equipment, the demand for integrated design of parts represented by new energy automobile motors, electromagnetic actuators, and the like has been increasing, and these parts are required to satisfy not only the performance requirements of magnetic fields but also the mechanical performance requirements such as rigidity and strength. By developing the topological optimization design of the magnetic-structure multi-physical field, the performance of the magnetic field and the structure field is effectively improved under the condition of light weight.
The development of the additive manufacturing technology provides manufacturing guarantee for the topological design configuration of the magnetic-structure multi-physical field. Although additive manufacturing greatly improves the design and manufacturing freedom over subtractive or isomaterial manufacturing.
However, there are still some manufacturability constraints, such as the need to add support material to print when the topological configuration exceeds the maximum suspension constraint, resulting in unnecessary material and post-processing costs.
The documents "Garbaldi M, Gerada C, Ashcroft I A. free-Form Design of electric Machine Rotor for Production Using Additive manufacturing. journal of Mechanical design.2019,141(7): 1-13" describe a topology optimization method oriented to magnetic-Mechanical field coupling and use this method for designing the Rotor of an electric Machine. The objective function of the topological optimization design in the article is the minimum mechanical field flexibility and the minimum rotor magnetic field energy, the constraint condition is the volume fraction, and the finally obtained rotor structure greatly improves the torque performance and simultaneously reduces the rotor quality. However, when multi-physical fields are considered, additive manufacturability constraints are not considered, and the problem that a supporting material is needed in the additive manufacturing process exists, so that the material cost and the post-treatment cost are increased.
The document "Langelaar m.an additive manufacturing filter for polarization Optimization of print-ready designs, structural and Multidisciplinary Optimization,2017,55(3): 871-83" introduces an additive manufacturing filter embedded in the SIMP method, which can convert the blueprint density into the print density through a mapping function, and the finally obtained configuration does not violate the maximum suspension constraint of 45 °, so that self-supporting printing can be realized. The work mainly considers the topological optimization design of the performance of the mechanical field for additive manufacturing, but the sensitivity optimization operation is complex and the requirement on the performance of computer hardware is high.
Disclosure of Invention
The purpose of this application lies in: the additive manufacturing performance of the complex equipment magnetic-structure multi-physical-field topological optimization configuration is improved, and additive manufacturing constraints such as no support are fully considered in the construction of a topological optimization model. The additive manufacturing constraint is embedded into the magnetic-structure topological optimization design model to realize the self-supporting printing of the structure, improve the performance of the structure, lighten the quality of the structure and reduce the manufacturing cost of the structure. The performance of a magnetic field and a mechanical field can be effectively improved while the light weight is considered.
The technical scheme of the application is as follows: the method for the topological optimization design of the magnetic-structure multi-physics field facing to the additive manufacturing is provided, and comprises the following steps: step 10, taking the unit printing density of the object to be printed as interpolation, calculating an integral displacement vector and an integral magnetic vector potential vector, generating constraint conditions according to the integral displacement vector and the integral magnetic vector potential vector, and establishing a magnetic-structure multi-physical-field topological optimization model, wherein the unit printing density is determined by a unit design density space; and 20, iteratively updating the unit design density space according to the sensitivity and the magnetic-structure multi-physical-field topological optimization model, and printing the object to be printed according to the updated unit design density space when the relative error of the objective function in the magnetic-structure multi-physical-field topological optimization model is judged to be smaller than a preset threshold value.
In any one of the above technical solutions, further, step 10 specifically includes: step 11, carrying out finite element mesh division on a design domain of an object to be printed, and filtering a unit design density space of the finite element mesh to generate a unit printing density space, wherein the unit printing density space is a vector matrix and comprises a plurality of unit printing densities; step 12, interpolating the elastic modulus and the magnetic permeability of each finite element grid according to the unit printing density in the unit printing density space to obtain the unit elastic modulus and the unit magnetic permeability; step 13, respectively correcting the structural field initial unit stiffness matrix and the static magnetic field initial stiffness matrix according to the unit elastic modulus and the unit magnetic permeability, and calculating an integral displacement vector and an integral magnetic vector potential vector according to the corrected structural field unit stiffness matrix and the static magnetic field stiffness matrix; and 14, calculating a multi-physical-field problem optimization objective function according to the integral displacement vector and the integral magnetic vector potential vector, and establishing a magnetic-structure multi-physical-field topological optimization model by combining with a volume constraint condition.
In any of the above technical solutions, further, the unit design density space and the unit print density space are vector matrices.
In any of the above technical solutions, further, the unit print density space
The calculation formula of (2) is as follows:
in the formula, ρsRepresenting the cell support domain, P is a first parameter, ε is a second parameter,
supporting the domain p for the cellsThe unit print density of the kth unit in (1), Q is a third parameter.
In any of the above embodiments, further, the modulus of elasticity of the unit
And cell permeability
The calculation formula of (2) is as follows:
wherein,
for the unit print density obtained after filtration, E0Is the modulus of elasticity, constant E, of the materialminIs a constant, vrIs the relative permeability of the material, P _ s is the structural field penalty parameter, and P _ m is the magnetic field penalty parameter.
In any one of the above technical solutions, further, the magnetic-structure multi-physics field topology optimization model includes an objective function and a constraint condition, the sensitivity is a sensitivity of the objective function to a cell design density space, and a calculation formula of the sensitivity is as follows:
in the formula, ρsd,j-1And ρsd,jCell design density vectors representing the jth-1 and jth rows of the support domain, respectively, c being the objective function, ρjA density vector is designed for the jth cell in the cell design density space,
for the jth unit print density vector, n, in the unit print density spacesP is the first parameter, ε is the second parameter, and Q is the third parameter, which is the number of elements in the support domain.
The beneficial effect of this application is:
according to the technical scheme, the performance of a magnetic field and a mechanical field can be greatly improved while the light weight is realized through the magnetic-structure multi-physical-field topological optimization technology, and the multi-physical-field topological optimization and additive manufacturing are successfully combined through introducing additive manufacturing constraint, so that the structure obtained through the magnetic-structure multi-physical-field topological optimization can be used for printing an object to be printed without the help of a supporting material, and the material cost and the post-processing cost are reduced.
The optimization design method provided by the application has the advantages of higher stability, higher convergence rate and fewer intermediate density units in topology optimization, and can be well popularized to the application of topology optimization design of other multi-physical fields for additive manufacturing.
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The advantages of the above and/or additional aspects of the present application will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
fig. 1 is a schematic flow diagram of a magnetic-structure multi-physics field topology optimization design method for additive manufacturing according to an embodiment of the present application;
FIG. 2 is a schematic view of an object to be printed under structured field-magnetic field effects according to one embodiment of the present application;
FIG. 3 is a schematic diagram of a density space according to an embodiment of the present application;
FIG. 4 is a schematic diagram of a result of topology optimization under single physical field action and multiple physical field action according to an embodiment of the present application;
FIG. 5 is a schematic illustration of printing results under different printing directions according to one embodiment of the present application.
Detailed Description
In order that the above objects, features and advantages of the present application can be more clearly understood, the present application will be described in further detail with reference to the accompanying drawings and detailed description. It should be noted that the embodiments and features of the embodiments of the present application may be combined with each other without conflict.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present application, however, the present application may be practiced in other ways than those described herein, and therefore the scope of the present application is not limited by the specific embodiments disclosed below.
As shown in fig. 1 and fig. 2, this embodiment provides a magnetic-structure multi-physical field topology optimization design method for additive manufacturing, which includes dividing a finite element mesh into design domains, setting an initial unit design density space, obtaining a unit print density space by combining with mapping of an additive manufacturing filter, interpolating an elastic modulus and a magnetic permeability on the basis of the unit print density space, obtaining a response of a magnetic field machine and a mechanical field by using a finite element method, constructing a multi-physical field topology optimization model on the basis, constructing an objective function by using a weighting method, and finally solving the magnetic-structure multi-physical field topology optimization model by using an optimization criterion method to obtain final material distribution, so as to implement magnetic-structure multi-physical field topology optimization design.
The method comprises the following steps:
In this embodiment, step 10 specifically includes:
step 11, carrying out finite element mesh division on the design domain of the object to be printed, filtering the unit design density space rho of the finite element mesh, and generating a unit printing density space
Wherein the unit design density space rho and the unit print density space
Is a vector matrix comprising a plurality of elements, i.e. rho ═ rhoe],
ρeThe density of any cell in the cell design density space, p, referred to as the cell design density,
printing a density space for a cell
The density of any one cell in (a), referred to as the cell print density;
specifically, as shown in fig. 3, the armature portion of the object to be printed (actuator) is set as a design field which is subjected to both the magnetic field and the structural field, and the design field is used to determine the boundary conditions of the magnetic-structural field effect. And carrying out finite element mesh division on the design domain from top to bottom, dividing the design domain into 30 multiplied by 50 meshes, wherein the mesh type is a quadrilateral unit, and taking each finite element mesh as a unit.
It should be noted that the structural field in the present embodiment may be a mechanical field.
In this embodiment, the cell design density ρ corresponding to each finite element mesh in the cell design density space ρ is giveneThe initial value of (2) is set to 0.6. Setting the printing direction from bottom to top, filtering the unit design density space rho to obtain a unit printing density space
The filtering method in this embodiment is not limited, and filtering may be performed by using an additive manufacturing filter.
In this embodiment, the unit design density space ρ is filtered layer by layer from the lowest layer to the highest layer, so that the unit print density space ρ is filtered layer by layer
The density of the middle and bottom layer units is the density of the bottom layer units in the unit design density space rho, and the unit printing density space
The specific calculation formula is as follows:
in the formula, ρsRepresenting the element support field, i.e. the area where the next layer of elements is needed to support the element (finite element mesh), the first parameter P is the control smoothness parameter and the second parameter ε is the control approximation parameter, which can be taken as 60 and 10, respectively-3，nsTo support the number of cells in the domain, here taking the value 3,
supporting the domain p for the cellsThe unit print density of the kth unit in (1), the third parameter Q is calculated as follows:
through the calculation of the third parameter Q, the accuracy of the calculation of the unit support domain can be improved, and the subsequently acquired unit elastic modulus is ensured
And cell permeability
The reliability of the method is beneficial to improving the precision of the magnetic-structure multi-physical-field topological optimization model, the accuracy of the spatial calculation of the printing density of the unit of the object to be printed is ensured, the object to be printed can be printed without the help of a supporting material, and the material cost and the post-processing cost are reduced.
Step 12, printing a density space according to the units
Unit printing density of
Interpolating the elastic modulus E and the magnetic permeability v of each finite element grid (unit) to obtain the elastic modulus of the unit
And cell permeability
In this embodiment, a topological optimization algorithm is established based on the SIMP framework, and material interpolation is performed on the elastic modulus and the magnetic permeability of the material to obtain the elastic modulus and the magnetic permeability for the unit printing density space
Function of (2), in particular the interpolated cell modulus of elasticity
And cell permeability
The calculation formula of (a) is as follows:
wherein,
for the unit print density obtained after filtration, E0Is made of woodModulus of elasticity, constant E of the materialminIs a small elastic modulus value for avoiding singularity of the whole rigidity matrix of the structural field, and the value is 10-3；vrIs the relative permeability of the material, P _ s is the structural field penalty parameter, and P _ m is the magnetic field penalty parameter.
Step 13, according to the elastic modulus of the unit
And permeability of the cell
And respectively correcting the structural field initial unit stiffness matrix and the static magnetic field initial stiffness matrix, and calculating an integral displacement vector and an integral magnetic vector potential vector according to the corrected structural field unit stiffness matrix and static magnetic field stiffness matrix.
From the interpolated unit modulus of elasticity
And cell permeability
Correcting the structural field initial unit stiffness matrix and the static magnetic field initial unit stiffness matrix, wherein the corresponding calculation formula is as follows:
wherein,
is a matrix of structural field-initiated cell stiffness,
for initial stiffness of static magnetic fieldThe matrix, both of which can be derived from the energy principle in the finite element method.
It should be noted that after the finite element mesh division is performed, the nodes of each finite element mesh, that is, the vertices of the finite element mesh, are numbered, and the specific numbering mode is not limited in this embodiment.
Respectively carrying out the rigidity matrix k of the modified structural field units according to the node numbers of the finite element gridse,sStatic magnetic field stiffness matrix ke,mAssembling to obtain corresponding integral rigidity matrix, and solving structural field and static magnetic field control equation to obtain structural field response-integral displacement vector
And magnetostatic field response-the overall magnetic vector potential vector
The corresponding calculation formula is:
in the formula,
to be the assembled structural field bulk stiffness matrix,
is the global displacement vector, F is the global force load vector,
to be the assembled static magnetic field global stiffness matrix,
is the integral magnetic vector potential vector, and P is the integral excitation vector;
and 14, calculating a multi-physical-field problem optimization objective function according to the integral displacement vector and the integral magnetic vector potential vector, and establishing a magnetic-structure multi-physical-field topological optimization model by combining volume constraint conditions, wherein the magnetic-structure multi-physical-field topological optimization model comprises the objective function and the constraint conditions.
In this embodiment, the objective function min c is a normalized function with minimum mechanical compliance and minimum magnetic compliance, and the constraint conditions at least include a physical field control equation, a component volume constraint and a unit design density constraint, where the physical field control equation includes a structural field and a static magnetic field control equation, and is determined by the overall displacement vector and the overall magnetic vector potential vector, and the calculation formula of the magnetic-structural multi-physical field topology optimization model is as follows:
0<ρmin≤ρe≤1
in the formula, cmehFor the value of the objective function of the structure field, cmagAs a value of the magnetic field objective function, ueIs a unit displacement vector, aeThe unit magnetic vector potential vectors can be respectively displaced from the whole vector through node numbering of the finite element grids according to the finite element principle
And the integral magnetic vector potential vector
Middle extraction of rhoeIs the cell design density, ke,sIs a structural field unit stiffness matrix, ke,mA matrix of static magnetic field stiffness is generated,
Cref_mechand Cref_magIs a normalized coefficient, N is the number of units in the design domain, which in this embodiment is 1500; v. ofeAnd V0Respectively unit volume and allowable volume; rhominDesigning a lower density limit for the unit, usually a small value, to prevent singularity of the stiffness matrix; w is a1And w2The weighting coefficients can be adjusted according to actual engineering problems.
In this example, a method of weighting factor adjustment is shown, namely the value of the objective function c according to the structure fieldmechObjective function value c of magnetic fieldmagCalculating a weight coefficient, wherein the corresponding calculation formula is as follows:
the present embodiment also shows a sensitivity calculation method.
Specifically, the sensitivity in the topology optimization process is crucial, and it is the basis for spatial updating of the cell design density, and because of the existence of the additive manufacturing filter, the sensitivity of the objective function to the initial density needs to be calculated by the chain rule:
where c is the objective function, where,
is the cell density after filtration, peIs the design cell density. In the formula,
the following needs to be determined by the concomitant method:
in the formula
The project calculation is more complicated because each line of elements in the unit printing density space is a function of all units in the unit design density space in the lower layer support domain, the top layer of the laying layer is the 1 st layer, the bottom layer is the nth layer, and the derivative of the ith line in the unit printing density space to the jth line in the unit design density space is calculated:
in the formula, deltaijIs a kronecker symbol, i ═ j, δ ij1 or i ≠ j, δ ij0. The ith row in the cell print density space is a function of the cell design densities of all rows in the lower level in the cell design density space, so that the above equation holds only for i ≦ j, when i is>At the time of j, the number of the first,
in this embodiment, the formula for calculating the sensitivity of the objective function to the cell design density space is as follows:
unknown in the formula
And
the calculation formula of (2) is as follows:
in the formula, ρsd,j-1And ρsd,jThe cell design density vectors for the jth-1 and jth rows of the support field are represented, respectively.
And 20, iteratively updating the unit design density space through an MMA algorithm according to the sensitivity and the magnetic-structure multi-physical-field topological optimization model, and printing the object to be printed according to the updated unit design density space when the relative error of the objective function in the magnetic-structure multi-physical-field topological optimization model is judged to be smaller than a preset threshold value.
Specifically, the unit design density space rho is iteratively updated through an optimization criterion method according to the sensitivity and a magnetic-structure multi-physical field topology optimization model, wherein the sensitivity is the sensitivity of a target function to the unit design density space, and can be calculated and determined through the calculation formula.
If the relative error delta of the objective function is smaller than a preset threshold value and can be set to be 0.1%, the unit design density space rho is considered to be converged, iteration is stopped, and the object to be printed is printed according to the updated unit design density space rho; if not, the objective functionLet rho be if the relative error delta of number is greater than or equal to 0.1%k＝ρk+1And returning to the step 11, and solving the next round of unit design density space rho until a convergence condition is reached to obtain a final unit design density space. The calculation formula of the relative error delta of the objective function is as follows:
in the formula, n is the number of iterations.
FIG. 4 shows the topological configurations (w) obtained by taking into account only the structural field (mechanical field) in each case1＝1,w20), topological configuration obtained taking into account the effect of the magnetic field only (w)1＝0,w21) and both fields act simultaneously (w)1≠0,w2Not equal to 1) obtained topological configuration. From the topological configuration obtained, the end result can satisfy the maximum suspension angle constraint of 45 °, and therefore the additive manufacturing constraint. Fig. 5 shows the magnetic-structure multi-physical field topological optimization configuration obtained under different printing directions.
The final output configuration of the invention is the optimized configuration presented by the unit in the unit printing density space, and due to the embedded additive manufacturing filter, all parts violating the maximum suspension angle of-45 degrees are filtered out, thereby ensuring that the structure can realize self-supporting printing.
The technical scheme of the application is described in detail in the above with reference to the accompanying drawings, and the application provides a magnetic-structure multi-physical-field topology optimization design method for additive manufacturing, which includes: step 10, taking the unit printing density of the object to be printed as interpolation, calculating an integral displacement vector and an integral magnetic vector potential vector, generating constraint conditions according to the integral displacement vector and the integral magnetic vector potential vector, and establishing a magnetic-structure multi-physical field topology optimization model; and 20, iteratively updating the unit design density space according to the sensitivity and the magnetic-structure multi-physical-field topological optimization model, and printing the object to be printed according to the updated unit design density space when the relative error of the objective function in the magnetic-structure multi-physical-field topological optimization model is judged to be smaller than a preset threshold value. Through the technical scheme in this application, realized the self-supporting printing of structure, avoided the use of supporting material.
The steps in the present application may be sequentially adjusted, combined, and subtracted according to actual requirements.
The units in the device can be merged, divided and deleted according to actual requirements.
Although the present application has been disclosed in detail with reference to the accompanying drawings, it is to be understood that such description is merely illustrative and not restrictive of the application of the present application. The scope of the present application is defined by the appended claims and may include various modifications, adaptations, and equivalents of the invention without departing from the scope and spirit of the application.
Claims (6)
1. An additive manufacturing-oriented magnetic-structure multi-physical field topological optimization design method is characterized by comprising the following steps:
step 10, taking the unit printing density of the object to be printed as interpolation, calculating an integral displacement vector and an integral magnetic vector potential vector, generating constraint conditions according to the integral displacement vector and the integral magnetic vector potential vector, and establishing a magnetic-structure multi-physical field topology optimization model, wherein the unit printing density is determined by a unit design density space;
and 20, iteratively updating the unit design density space according to the sensitivity and the magnetic-structure multi-physical-field topological optimization model, and printing the object to be printed according to the updated unit design density space when the relative error of the objective function in the magnetic-structure multi-physical-field topological optimization model is judged to be smaller than a preset threshold value.
2. The additive manufacturing-oriented magnetic-structure multi-physical-field topological optimization design method according to claim 1, wherein the step 10 specifically comprises:
step 11, performing finite element mesh division on the design domain of the object to be printed, and filtering a unit design density space of the finite element mesh to generate a unit print density space, wherein the unit print density space is a vector matrix and comprises a plurality of unit print densities;
step 12, interpolating the elastic modulus and the magnetic permeability of each finite element grid according to the unit printing density in the unit printing density space to obtain a unit elastic modulus and a unit magnetic permeability;
step 13, respectively correcting the structural field initial unit stiffness matrix and the static magnetic field initial stiffness matrix according to the unit elastic modulus and the unit magnetic permeability, and calculating an integral displacement vector and an integral magnetic vector potential vector according to the corrected structural field unit stiffness matrix and the static magnetic field stiffness matrix;
and 14, calculating a multi-physical-field problem optimization objective function according to the integral displacement vector and the integral magnetic vector potential vector, and establishing the magnetic-structure multi-physical-field topological optimization model by combining with a volume constraint condition.
3. The additive manufacturing oriented magnetic-structure multiphysics field topological optimization design method of claim 2, wherein the cell design density space and the cell print density space are vector matrices.
4. Additive manufacturing oriented magnetic-structure multi-physical field topology optimization design method according to claim 2 or 3, wherein the unit print density space
The calculation formula of (2) is as follows:
in the formula, ρsRepresenting the cell support domain, P is a first parameter, ε is a second parameter,
supporting the domain p for the cellsThe unit print density of the kth unit in (1), Q is a third parameter.
5. The additive manufacturing-oriented magnetic-structure multi-physical-field topological optimization design method of claim 2, wherein the unit elastic modulus
And permeability of the cell
The calculation formula of (2) is as follows:
wherein,
for the unit print density obtained after filtration, E0Is the modulus of elasticity, constant E, of the materialminIs a constant, vrIs the relative permeability of the material, P _ s is the structural field penalty parameter, and P _ m is the magnetic field penalty parameter.
6. The additive manufacturing-oriented magnetic-structure multi-physical-field topological optimization design method according to claim 1, wherein the objective function and the constraint condition are included in the magnetic-structure multi-physical-field topological optimization model, the sensitivity is a sensitivity of the objective function to the unit design density space, and a calculation formula of the sensitivity is as follows:
in the formula, ρsd,j-1And ρsd,jCell design density vectors representing the jth-1 and jth rows of the support domain, respectively, c being the objective function, ρjDesigning a density vector for a jth cell in the cell design density space,
for the jth unit print density vector, n, in said unit print density spacesP is the first parameter, ε is the second parameter, and Q is the third parameter, which is the number of elements in the support domain.
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| CN114356252A (en) * | 2021-12-27 | 2022-04-15 | 金蝶软件（中国）有限公司 | Dynamic field printing method, system and related device |
| CN114741798A (en) * | 2022-03-10 | 2022-07-12 | 湖南大学 | Motor rotor structure topology optimization method considering electromagnetic and mechanical properties |
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