CN113158354A - Turbine disk topology optimization method based on sensitivity and dynamic gray scale inhibition - Google Patents
Turbine disk topology optimization method based on sensitivity and dynamic gray scale inhibition Download PDFInfo
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Abstract
The invention provides a topological optimization method based on sensitivity and dynamic gray level suppression, which is used for solving the problems of difficulty and inaccuracy in solving the topological optimization of a turbine disc under inertial load and fixed load and gray level units in the topological structure optimization process. The method comprises the following steps of (1) establishing a topological optimization design domain of a radial section of a turbine disc; (2) designing a topological optimization mathematical model; (3) for elastic modulus E (x)i) Punishment is carried out; (4) obtaining a displacement matrix U of the section of the turbine disc according to finite element calculation, and further calculating a target function-structural strain energy C; (5) calculating the sensitivity; (6) designing an updating iterative formula according to an optimization criterion method; (7) a method for suppressing load sensitivity; (8) move and moveA state gray scale suppression method; (9) and judging convergence. The three-spoke-plate turbine disk designed according to the invention has the following beneficial effects: the centrifugal load of the turbine blades on the wheel rim and the centrifugal load of the turbine disc are effectively borne, and the arc-shaped structure of the rear spoke plate and the support structure is beneficial to bearing the pneumatic tangential load transferred by the turbine blades on the wheel rim; the arrangement of the holes enables the turbine disc to be light; the requirements of strength and rigidity are met.
Description
Technical Field
The invention belongs to the technical field of aeroengines and high-pressure gas turbines, and particularly relates to a topological optimization method based on sensitivity and dynamic gray scale inhibition. The asymmetrical three-spoke structure of the turbine disk connecting the hub and the wheel rim improves the capability of the turbine disk for bearing centrifugal load, radial load and tangential load. The overall stress distribution of the turbine disk is improved. Meanwhile, due to the existence of the holes, the mass of the turbine disk is also reduced.
Background
The traditional turbine disk adopts a single-spoke structure, and the current single-spoke turbine disk reaches the structural design limit, so that the further development of an engine is severely limited. With the development of 3D printing technology, the manufacturing bottleneck of the turbine disk structure with novel design is overcome, and the innovative design of the turbine disk with higher bearing capacity and lighter weight becomes one of the core technologies of aircraft engine and gas turbine design.
A new type of double-spoke turbine disk distinguished from the previous single-spoke disk is taught in Twin-web rotor disk (U.S. Pat. No. 5,961,287, 1999-10-5). The turbine disk is composed of a front and a rear wheel webs in the axial direction, and the two wheel webs enclose a central disk cavity. Compared with a single-spoke turbine disk, the double-spoke turbine disk can reduce weight 1/4 compared with a single-spoke turbine disk on the premise of meeting the requirements of strength and rigidity of the disk. Due to the outstanding advantages exhibited by double-disc Turbine disks, double-disc Turbine disks are designated as a future development trend of high-pressure Turbine disk structures in the united states' developed Integrated High Performance Turbine Engine Technology (IHPTET) [ R ]. Gas Turbine disks, Archived, 2006) for the Integrated High Performance Turbine Engine Technology (IHPTET).
The typical structure of a double-web turbine disk proposed by the patent Twin-web rotor disk (U.S. Pat. No. 5,961,287, 1999-10-5) generally has a symmetrical structural form in a radial section, while different types of aircraft engines and high-pressure gas turbines have different radial and tangential forces of the turbine disk under centrifugal load and aerodynamic load, the load state is very complicated and has no symmetry, and a symmetrical structure bears complicated asymmetric load, and the stress uniformity is necessarily affected. The double-web structure is not an optimal structure for the turbine disk. The load condition of the turbine disk in operation is further researched, and the design of the turbine disk with uniform stress and light weight under the condition of meeting the requirements on strength and rigidity has important significance for improving the performance of aeroengines and gas turbines.
With the expansion of the application field of structural topology optimization, the research on the structural topology optimization method under various load conditions draws attention in the engineering field. Inertial loads, which are common design-related loads, exist in many structures. In large civil constructions, for example, the self-weight of the structure is generally one of the important loads that are not negligible; in steam turbines and aerospace vehicles, structures bear inertial loads caused by acceleration or angular velocity, for example, the aeroengine turbine needs to bear great centrifugal force when working, and a bearing structure of the aircraft needs to bear the inertial loads caused by the acceleration in a launching stage.
However, when the method is based on variable density topological optimization, firstly, a traditional SIMP interpolation model can generate a material auxiliary effect under an inertial load, secondly, the whole flexibility of the structure under the inertial load has no monotonicity to the structure parameters (unit density), and the sensitivity of the whole flexibility is not constant to a negative value, so that the solution algorithm of the optimization criterion method can not solve the topological optimization problem under the inertial load condition. Thirdly, the gray suppression in the solving process is too strong, so that the solution is converged to an unoptimized solution, and too weak, the solution is not converged or the gray units are too many. The invention provides a load sensitivity suppression method and a dynamic gray suppression method by using an EAMP interpolation model. Based on the invention, a three-spoke-plate turbine disk optimization structure is obtained, and a structural topology optimization design method which solves the problems of non-monotonous sensitivity and many gray level units and unclear boundaries of a topological structure in inertial load topology optimization on the basis of an optimization criterion method and is suitable for inertial loads is provided.
Disclosure of Invention
In view of the above technical problems, the present invention provides a topological optimization method based on sensitivity and dynamic gray scale suppression, and aims to provide a turbine disk structure adapted to the working environment of aircraft engines and gas turbines, which improves the stress distribution uniformity of the turbine disk under centrifugal load and aerodynamic load, improves the bearing capacity of the turbine disk, and reduces the structural weight of the turbine disk body.
In order to solve the technical problems, the technical scheme of the invention is as follows:
the utility model provides a three radials turbine disk, turbine disk includes rim 1, wheel hub 2 and connects rim 1 and wheel hub 2's radials structure, the radials structure includes preceding radials 3, well radials and back radials, rim 1, lower extreme connection wheel hub 2 are connected to the upper end of preceding radials 3, well radials and back radials.
Further, the middle spoke plate and the rear spoke plate are arranged in a crossed mode.
Further, the middle spoke plate and the rear spoke plate are arranged in a crossed mode to form a middle spoke plate upper portion 7, a middle spoke plate lower portion 4, a rear spoke plate upper portion 6 and a rear spoke plate lower portion 5, the middle spoke plate upper portion 7 and the rear spoke plate upper portion 6 are connected with the wheel rim 1, and the middle spoke plate lower portion 4 and the rear spoke plate lower portion 5 are connected with the wheel hub 2.
Further, a first hole 8 is defined among the front wheel disk 3, the rear wheel disk upper portion 6 and the middle wheel disk lower portion 4, a third hole 10 is defined among the middle wheel disk upper portion 7, the rear wheel disk upper portion 6 and the wheel rim 1, and a second hole 9 is defined among the middle wheel disk lower portion 4, the rear wheel disk lower portion 5 and the wheel hub 2.
Further, the rear spoke plate is of an arc-shaped structure.
Further, the three-spoke plate turbine disk is manufactured by 3D printing.
A three-spoke-plate turbine disk comprises a rim 1, a hub 2 and a spoke plate structure for connecting the rim 1 and the hub 2, wherein the spoke plate structure comprises a front spoke plate 3, a middle spoke plate and a rear spoke plate, the upper ends of the front spoke plate 3, the middle spoke plate and the rear spoke plate are connected with the rim 1, the lower ends of the front spoke plate, the middle spoke plate and the rear spoke plate are connected with the hub 2, and the middle spoke plate and the rear spoke plate are arranged in a crossed manner; the middle spoke plate and the rear spoke plate are arranged in a crossed mode to form a middle spoke plate upper portion 7, a middle spoke plate lower portion 4, a rear spoke plate upper portion 6 and a rear spoke plate lower portion 5, the middle spoke plate upper portion 7 and the rear spoke plate upper portion 6 are connected with the rim 1, and the middle spoke plate lower portion 4 and the rear spoke plate lower portion 5 are connected with the hub 2; a first hole 8 is defined among the front wheel disk 3, the rear wheel disk upper part 6 and the middle wheel disk lower part 4, a third hole 10 is defined among the middle wheel disk upper part 7, the rear wheel disk upper part 6 and the wheel rim 1, and a second hole 9 is defined among the middle wheel disk lower part 4, the rear wheel disk lower part 5 and the wheel hub 2; the rear spoke plate is of an arc-shaped structure.
An engine has the three-spoke turbine disk.
A design method of a three-spoke-plate turbine disk based on topology optimization for manufacturing the three-spoke-plate turbine disk is characterized in that: comprises the following steps of (a) carrying out,
(1) establishing a topological optimization design domain of a radial section of the turbine disc, wherein the length L and the width D of the design domain are designed; setting a non-topological optimization design domain on the rim of the turbine disc according to the size of the turbine blade; the radius of a rotating shaft of the turbine disc is r, and the rotating speed is omega; one side of the rotating shaft of the turbine disc is fixedly restricted in displacement; global centrifugal force F1Radial component F of centrifugal force and aerodynamic load of impeller2Component F of aerodynamic load in tangential direction3(ii) a Density of the material is rho0Modulus of elasticity of E0(ii) a A volume constraint fraction of f;
(2) designing a topological optimization mathematical model: a topological optimization mathematical model based on a variable density method and taking structural flexibility as a target is as shown in a formula (1)
Wherein x is the relative density of cells of a finite element in a design domain; n is the total number of units; c is structural flexibility; K. u and F are respectively a structural integral rigidity matrix, a displacement vector and a load vector; v. ofiIs the unit volume; v0F is the initial design domain volume and volume fraction respectively; x is the number ofminIs the lower limit of the design variable;
(3) applying an EAMP material interpolation model, wherein the mathematical expression is as formula (2), and based on the model, according to the relative density of the unit and the elastic modulus E (x) of the uniti) Comprises the following steps:
(4) obtaining a displacement matrix U of the section of the turbine disc according to finite element calculation, and further calculating a target function, namely structural strain energy C;
(5) sensitivity calculation, sensitivity of the objective function to cell density.
Wherein
(6) And according to an optimization criterion method, the solution iteration of the mathematical model (1) formula is as follows:
wherein k is iteration times, eta is a damping coefficient, and m is a moving step length;
for iterative solution of the format, λ1Is a lagrange multiplier; expression (8) is easily derived based on the Lagrangian function and the K-T condition:
(7) a method for suppressing load sensitivity is provided. Adding a load sensitivity inhibition factor in the expression (8); the target function is kept monotonous through the inhibition effect of the factor; the expression after increasing the load sensitivity inhibitor is shown in formula (9):
wherein the content of the first and second substances,
the value of s is a load sensitivity inhibitor as shown in the following formula (10)
The value range of l can be from extremely small to infinite, different values of l represent different degrees of inhibition on load sensitivity, and the smaller the l, the smaller the inhibition degree, namely the more the load sensitivity is kept; the greater the l, the greater the degree of inhibition, i.e. the less the load sensitivity retention;
(8) a dynamic gray scale suppression method is provided. Based on the gray scale suppression method, the new iteration format (11) is as follows:
wherein q is a dynamic gray scale suppression factor that changes with changes in the number of optimization steps, as shown in the following formula (12);
q＝1+δk (12)
wherein, δ is the inhibition step length, which can be taken as different values according to different specific optimization problems, generally 0.02 or less, and is related to the iteration step number of the optimization problem;
(9) and convergence judgment: judging whether the updated design variable-unit density reaches convergence according to the convergence condition formula (13); if the convergence condition is not met, continuing to perform the next step of loop iteration until the convergence condition is reached;
wherein the content of the first and second substances,
and
the i-cell densities at the k +1 and k times, respectively, and ε is the convergence accuracy and may be generally 0.001 or less.
Compared with the prior art, the invention has the following beneficial effects: the centrifugal load of the turbine blades on the wheel rim and the centrifugal load of the turbine disc are effectively borne, and the arc-shaped structure of the rear spoke plate and the support structure is beneficial to bearing the pneumatic tangential load transferred by the turbine blades on the wheel rim; the arrangement of the holes enables the turbine disc to be light; the requirements of strength and rigidity are met.
Drawings
FIG. 1 is a schematic view of a three-hole three-spoke turbine disk according to the present invention;
FIG. 2 is a topological optimized geometric model of the present invention;
FIG. 3 illustrates an EAMP material interpolation model of the present invention;
FIG. 4 is a flow chart for optimizing the present invention;
FIG. 5 is a schematic view of a four-hole three-spoke turbine disk according to the present invention;
FIG. 6 is a schematic view of a five-hole three-spoke turbine disk according to the present invention;
FIG. 7 is a conventional single-disk turbine disk configuration, a typical double-disk turbine disk, and a triple-disk turbine disk configuration of the present invention;
FIG. 8 is a finite element analysis of a conventional single-disk turbine disk structure;
FIG. 9 is a finite element analysis of a typical double radial plate turbine disk configuration;
FIG. 10 is a turbine disk structure of three-web structure 1, which is a finite element analysis;
FIG. 11 three spoke plate configuration 2 turbine disk configuration finite element analysis
The wheel rim 1, the hub 2, the front wheel disk 3, the middle wheel disk lower part 4, the rear wheel disk lower part 5, the rear wheel disk upper part 6, the middle wheel disk upper part 7, the first hole 8, the second hole 9, the third hole 10, the first fork 11, the second fork 12 and the fourth hole 13 are shown in the figure.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments.
In the description of the present invention, it is to be understood that the terms "center", "length", "width", "thickness", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", "axial", "circumferential", and the like, indicate orientations or positional relationships based on those shown in the drawings, and are used merely for convenience of description and for simplicity of description, and do not indicate or imply that the device or element being referred to must have a particular orientation, be constructed and operated in a particular orientation, and thus, are not to be considered as limiting the present invention.
In the present invention, unless otherwise expressly specified or limited, the terms "disposed," "mounted," "connected," and "fixed" are to be construed broadly and may, for example, be fixedly connected or detachably connected; may be a mechanical connection; may be directly connected or indirectly connected through an intermediate. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
Furthermore, the terms "first", "second", etc. are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature.
Example one
As shown in FIG. 1, a three radials turbine disk, turbine disk includes rim 1, hub 2 and connects rim 1 and hub 2's radials structure, the radials structure includes preceding radials 3, well radials and back radials, rim 1, lower extreme connection hub 2 are connected to the upper end of preceding radials 3, well radials and back radials.
Example two
On the basis of the first embodiment, it is further provided that the middle spoke plate and the rear spoke plate are arranged in a crossed manner.
EXAMPLE III
On the basis of the first embodiment, it is further provided that the middle spoke plate and the rear spoke plate are arranged in a crossed mode to form a middle spoke plate upper portion 7, a middle spoke plate lower portion 4, a rear spoke plate upper portion 6 and a rear spoke plate lower portion 5, the middle spoke plate upper portion 7 and the rear spoke plate upper portion 6 are connected with the wheel rim 1, and the middle spoke plate lower portion 4 and the rear spoke plate lower portion 5 are connected with the wheel hub 2.
Example four
On the basis of the third embodiment, in the fourth embodiment, a first hole 8 is defined among the front wheel disk 3, the rear wheel disk upper part 6 and the middle wheel disk lower part 4, a third hole 10 is defined among the middle wheel disk upper part 7, the rear wheel disk upper part 6 and the wheel rim 1, and a second hole 9 is defined among the middle wheel disk lower part 4, the rear wheel disk lower part 5 and the wheel hub 2.
EXAMPLE five
On the basis of the fourth embodiment, the fifth embodiment is that the rear spoke plate is of an arc-shaped structure.
The three spokes in the above embodiment can be made by 3D printing, and based on the above improvement on the turbine disk, the turbine disk in the present invention can be applied to an aircraft engine.
Because the turbine disk needs to bear the centrifugal load generated by high-speed rotation and simultaneously needs to bear larger radial force and tangential force brought by the turbine blades, compared with the common single-spoke plate and double-spoke plate structure, the three-spoke plate turbine disk not only adds a middle spoke plate, but also adds an oblique supporting structure. Through radials and bearing structure in increasing, strengthened the holistic bearing capacity of turbine dish for the holistic comprehensive displacement of turbine dish and stress all effectively reduce. The stress distribution of the turbine disk is improved, and the service life of the turbine disk is prolonged.
Aiming at the three-spoke-plate turbine disk, the design method of the invention is as follows:
1. establishing a topological optimization design domain of a radial section of the turbine disc as shown in FIG. 2, wherein the length L and the width D of the design domain are taken according to the actual engineering value; the turbine disk rim is set as a non-topological optimization design domain according to the size of the turbine blades, namely, the size (black part in fig. 2) of the turbine disk rim where the turbine blades are installed is the non-topological optimization design domain; the radius of a rotating shaft of the turbine disc is r, and the rotating speed is omega; one side of the rotating shaft of the turbine disc is fixedly restricted in displacement; global centrifugal force F1Radial component F of centrifugal force and aerodynamic load of impeller2Component F of aerodynamic load in tangential direction3(ii) a Density of the material is rho0Modulus of elasticity of E0(ii) a A volume constraint fraction of f;
2. designing a topological optimization mathematical model: a topological optimization mathematical model based on a variable density method and taking structural flexibility as a target is as shown in a formula (1)
Wherein x is the relative density of cells of a finite element in a design domain; n is the total number of units; c is structural flexibility; K. u and F are respectively a structural integral rigidity matrix, a displacement vector and a load vector; v. ofiIs the unit volume; v0F is the initial design domain volume and volume fraction respectively; x is the number ofminIs the lower limit of the design variable;
3. an EAMP material interpolation model is applied, the mathematical expression is as formula (2), and based on the model, the relative density of units and the elastic modulus E (x) of the units are calculatedi) Comprises the following steps:
4. obtaining a displacement matrix U of the section of the turbine disc according to finite element calculation, and further calculating an objective function, namely structural strain energy C;
5. sensitivity calculation, sensitivity of the objective function to cell density.
Wherein
6. According to the optimization criterion method, the solution iteration of the mathematical model (1) formula:
wherein k is iteration times, eta is a damping coefficient, and m is a moving step length;
solving the format for iteration; lambda [ alpha ]1For the Lagrangian multiplier, expression (8) is easily derived based on the Lagrangian function and the K-T condition:
7. a load sensitivity suppression method is provided. Adding a load sensitivity inhibition factor in the expression (8); the target function is kept monotonous through the inhibition effect of the factor; the expression after increasing the load sensitivity inhibitor is shown in formula (9):
wherein the content of the first and second substances,
the value of s is a load sensitivity inhibitor as shown in the following formula (10)
The value range of l can be from extremely small to infinite, different values of l represent different degrees of inhibition on load sensitivity, and the smaller the l, the smaller the inhibition degree, namely the more the load sensitivity is kept; the greater the l, the greater the degree of inhibition, i.e. the less the load sensitivity retention;
8. a dynamic gray scale suppression method is provided. Based on the gray scale suppression method, the new iteration format (11) is as follows:
wherein q is a dynamic gray scale suppression factor that changes with changes in the number of optimization steps, as shown in the following formula (12);
q＝1+δk (12)
wherein, δ is the inhibition step length, which can be taken as different values according to different specific optimization problems, generally 0.02 or less, and is related to the iteration step number of the optimization problem;
9. and (3) convergence judgment: judging whether the updated design variable-unit density reaches convergence according to the convergence condition formula (13); if the convergence condition is not met, continuing to perform the next step of loop iteration until the convergence condition is reached;
wherein the content of the first and second substances,
and
the i-cell densities at the k +1 and k times, respectively, and ε is the convergence accuracy and may be generally 0.001 or less.
Compared with the traditional single-spoke plate and double-spoke plate turbine disk, the invention is suitable for more complicated load conditions, the additional middle spoke plate and the supporting structure improve the circumferential, radial and axial bearing capacity of the connecting part of the spoke plate, the wheel rim and the wheel hub, improve the stress distribution uniformity of the turbine disk, and simultaneously reduce the mass of the turbine disk due to the existence of the holes.
As shown in fig. 7, a conventional single-spoke and double-spoke turbine disk structure and two three-spoke structures based on the present invention are shown, wherein two three-spoke structures are designed under the condition of table by using the topology optimization method of the present invention: the three-spoke plate structure 1 has the same mass (1 percent of reduction) as a single-spoke plate structure, and has the same mass (2 percent of reduction) as a typical double-spoke plate structure; the triple-web structure 2 is reduced in weight by 11% compared to a single-web structure and by 12% compared to a typical double-web structure.
Through finite element calculation, the comprehensive displacement, the maximum VM stress of a radial plate and the overall average stress of various turbine disks are compared as follows:
table 1 comparison of a conventional single-web turbine disk with a three-web structure 1
Table 2 comparison of a conventional single-web turbine disk with a three-web structure 2
Table 3 typical double spoke turbine disk compared to the triple spoke configuration 1
Table 4 comparison of a typical double-spoke turbine disk with a triple-spoke configuration 2
From the above table, it can be seen that when the mass of a typical Twin-web is comparable to that of a conventional single-web, the combined displacement, maximum VM stress, and overall average stress of the Twin-web are reduced, indicating that the performance of a Twin-web turbine disk is improved over that of a conventional single-web turbine disk based on the teachings of patent Twin-web rotor disk (U.S. Pat. No. 5,961,287, 1999-10-5). As can be seen from tables 1 and 3, when the overall mass of the three-web plate 1 is substantially equivalent to that of the conventional single-web plate and the typical double-web plate, the comprehensive displacement, the maximum VM stress and the overall average stress are greatly reduced. As can be seen from tables 2 and 4, when the overall mass of the three-web plate 2 is reduced by 11% and 12% respectively as compared with the conventional single-web plate and the typical double-web plate, the combined displacement, maximum VM stress, and overall average stress of the three-web plate 2 are also reduced to a large extent. It can thus be seen that the three-web turbine disk of the present invention has better performance characteristics than conventional single-web and typical double-web turbine disks, with the excellent results being seen from the finite element analysis results in fig. 8, 9, 10, and 11.
Referring to fig. 5 and 6, the number of holes may be three, or four, five or more, and only structural changes are made without departing from the essence of the tri-web.
Compared with the prior art, the invention has the following beneficial effects: the centrifugal load of the turbine blades on the wheel rim and the centrifugal load of the turbine disc are effectively borne, and the arc-shaped structure of the rear spoke plate and the support structure is beneficial to bearing the pneumatic tangential load transferred by the turbine blades on the wheel rim; the arrangement of the holes enables the turbine disc to be light; the requirements of strength and rigidity are met.
Finally, it should be noted that: although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that modifications may be made to the embodiments or portions thereof without departing from the spirit and scope of the invention.
Claims (1)
1. A topological optimization method based on sensitivity and dynamic gray scale inhibition is characterized in that: comprises the following steps.
(1) Establishing a topological optimization design domain of a radial section of the turbine disc, wherein the length L and the width D of the design domain are designed; setting a non-topological optimization design domain on the rim of the turbine disc according to the size of the turbine blade; the radius of a rotating shaft of the turbine disc is r, and the rotating speed is omega; one side of the rotating shaft of the turbine disc is fixedly restricted in displacement; global centrifugal force F1Radial component F of centrifugal force and aerodynamic load of impeller2Component F of aerodynamic load in tangential direction3(ii) a Density of the material is rho0Modulus of elasticity of E0(ii) a A volume constraint fraction of f;
(2) designing a topological optimization mathematical model: a topological optimization mathematical model based on a variable density method and taking structural flexibility as a target is as shown in a formula (1)
Wherein x is the relative density of cells of a finite element in a design domain; n is the total number of units; c is structural flexibility; K. u and F are respectively a structural integral rigidity matrix, a displacement vector and a load vector; v. ofiIs the unit volume; v0F is the initial design domain volume and volume fraction respectively; x is the number ofminIs the lower limit of the design variable;
(3) applying an EAMP material interpolation model, wherein the mathematical expression is as formula (2), and based on the model, according to the relative density of the unit and the elastic modulus E (x) of the uniti) Comprises the following steps:
(4) obtaining a displacement matrix U of the section of the turbine disc according to finite element calculation, and further calculating a target function, namely structural strain energy C;
(5) sensitivity calculation, sensitivity of the objective function to cell density.
Wherein
(6) And according to an optimization criterion method, the solution iteration of the mathematical model (1) formula is as follows:
wherein k is iteration times, eta is a damping coefficient, and m is a moving step length;
solving the format for iteration; expression (8) is easily derived based on the Lagrangian function and the K-T condition:
(7) a method for suppressing load sensitivity is provided. Adding a load sensitivity inhibition factor in the expression (8); the target function is kept monotonous through the inhibition effect of the factor; the expression after increasing the load sensitivity inhibitor is shown in formula (9):
wherein the content of the first and second substances,
the value of s is a load sensitivity inhibitor as shown in the following formula (10)
The value range of l can be from extremely small to infinite, different values of l represent different degrees of inhibition on load sensitivity, and the smaller the l, the smaller the inhibition degree, namely the more the load sensitivity is kept; the greater the l, the greater the degree of inhibition, i.e. the less the load sensitivity retention;
(8) a dynamic gray scale suppression method is provided. Based on the gray scale suppression method, the new iteration format (11) is as follows:
wherein q is a dynamic gray scale suppression factor that changes with changes in the number of optimization steps, as shown in the following formula (12);
q＝1+δk (12)
wherein, δ is the inhibition step length, which can be taken as different values according to different specific optimization problems, generally 0.02 or less, and is related to the iteration step number of the optimization problem;
(9) and convergence judgment: judging whether the updated design variable-unit density reaches convergence according to the convergence condition formula (13); if the convergence condition is not met, continuing to perform the next step of loop iteration until the convergence condition is reached;
wherein the content of the first and second substances,
and
the i-cell densities at the k +1 and k times, respectively, and ε is the convergence accuracy and may be generally 0.001 or less.
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