CN117892414B - Structural robustness topology optimization design method for anisotropic material under simple harmonic excitation - Google Patents
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Abstract
The invention provides a topological optimization design method for structural robustness of an anisotropic material under simple harmonic excitation, which comprises the steps of defining a design domain and design conditions of the anisotropic material structure under the simple harmonic excitation; solving the structural response of the anisotropic material under the action of the uncertainty simple harmonic excitation; establishing a robust dynamic topology optimization mathematical model by taking a weighted sum of a mean value and a standard deviation of dynamic compliance as an objective function and taking the volume fraction as a constraint; solving an optimization objective function and a constraint function, solving sensitivity information, and correcting to obtain corrected sensitivity information; updating the design variables, judging whether the solving optimization algorithm meets the convergence condition, and if so, obtaining the optimal topological configuration of the anisotropic material structure. The anisotropic material structure obtained by the invention can bear component force generated by uncertain simple harmonic excitation, and has better robustness.
Description
Technical Field
The invention relates to the technical field of uncertainty topology optimization design, in particular to a structural robustness topology optimization design method for an anisotropic material under simple harmonic excitation.
Background
Along with the improvement of the requirements of engineering structures on material performance, the anisotropic material is widely applied to the fields of bridges, rail transit, aerospace and the like. Compared with the traditional single-phase material, the anisotropic material has excellent mechanical property and excellent structural property, including high strength, high rigidity/weight ratio, fatigue resistance, corrosion resistance, thermal stability and the like. At present, the topology optimization design research of the anisotropic material structure ignores the requirement of the anisotropic material structure on dynamic performance, and the service life of the structure is easy to reduce or even damage the structure.
Compared with the factors such as material properties, boundary constraint conditions and the like of the structure, the external load state has greater influence on the structural topology optimization design. Most engineering structures, such as aerospace vehicles, ships, bridges, etc., are subjected to various time-varying loads during operation, and a complex time-varying periodic excitation is typically approximately decomposed into a superposition of several simple harmonic excitations. However, simple harmonic excitation suffers from amplitude and direction uncertainties. Therefore, the research on the topology optimization design of the anisotropic material junction under the uncertain simple harmonic excitation has very important theoretical and practical significance.
In the prior art, the existing structural dynamic topology optimization research on anisotropic materials under simple harmonic excitation is mainly carried out under deterministic conditions, and the influence of uncertain simple harmonic excitation on an optimization result is ignored. And the uncertainty analysis process is complex, the calculation of the dynamic response analysis of the structure is more complex, and even the nested double-loop solution is involved, so that the calculation efficiency is low, and the problem of large-scale engineering design is difficult to solve.
Disclosure of Invention
Based on this, the present invention aims to provide a topological optimization design method for structural robustness of anisotropic materials under simple harmonic excitation, so as to at least solve the above-mentioned shortcomings in the prior art.
The invention provides a topological optimization design method for structural robustness of an anisotropic material under simple harmonic excitation, which comprises the following steps:
defining a design domain and design conditions of an anisotropic material structure under simple harmonic excitation;
Performing orthogonal decomposition on the uncertain simple harmonic excitation, and performing second-order Taylor expansion on the uncertain simple harmonic excitation to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation;
Punishing a unit stiffness and a mass matrix of the anisotropic material structure, and obtaining an anisotropic material structure response under the uncertain simple harmonic excitation based on a load uncertainty factor, a plurality of finite units, the unit stiffness and the mass matrix;
Establishing a robust dynamic topology optimization mathematical model of the anisotropic material structure by taking a weighted sum of a dynamic compliance mean value of the anisotropic material structure and a standard deviation of the dynamic compliance of the anisotropic material structure as an objective function and taking the volume fraction of the anisotropic material structure as a constraint;
Solving an optimization objective function and a constraint function of the anisotropic material structure based on the robustness dynamic topology optimization mathematical model, solving sensitivity information of the optimization objective function and the constraint function, and correcting the sensitivity information to obtain corrected sensitivity information;
And updating design variables by adopting a moving asymptotic algorithm, judging whether a solving and optimizing result meets a convergence condition, and if so, obtaining the optimal topological configuration of the anisotropic material structure.
Compared with the prior art, the invention has the beneficial effects that: and the second-order Taylor expansion is carried out on the uncertain simple harmonic excitation, a large number of samples are not needed to be adopted for carrying out uncertainty analysis on the anisotropic material structure, and the complexity of topology optimization solution is reduced. The topological structure of the anisotropic material structure obtained by topological optimization can bear component force generated by uncertain simple harmonic excitation, and has better robustness.
Further, the step of orthogonally decomposing the uncertain simple harmonic excitation and performing second-order taylor expansion on the uncertain simple harmonic excitation to quantify the uncertainty of the amplitude and direction of the simple harmonic excitation includes:
Carrying out orthogonal decomposition on the uncertain simple harmonic excitation in two directions, and representing the load amplitude and the load direction of the uncertain simple harmonic excitation as a deterministic mean value and a standard deviation composition representing random disturbance;
And performing second-order Taylor expansion on the uncertain simple harmonic excitation based on a second-order perturbation method so as to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation.
Further, the step of punishing the cell stiffness and the mass matrix of the anisotropic material structure, and obtaining the anisotropic material structure response under the uncertain simple harmonic excitation based on the load uncertainty factor, the finite cells, the cell stiffness and the mass matrix comprises the following steps:
punishment of unit stiffness and a mass matrix of the anisotropic material structure based on an anisotropic material interpolation model;
and obtaining the uncertainty response of the anisotropic material structure under the simple harmonic excitation according to the load uncertainty factor, the finite elements, the element rigidity and the mass matrix and the structure simple harmonic vibration finite element analysis.
Further, the expression of the robust dynamic topology optimization mathematical model of the anisotropic material structure is as follows:
；
In the method, in the process of the invention, Representing a cell density array,/>Representing an angular array of elementary fibers,/>The function of the optimization objective is represented as,Representing dynamic compliance expectations,/>Representing dynamic compliance variance, i.e., dynamic compliance desire/>And dynamic compliance variance/>Weighted sum of/(v)As a weighting coefficient,/>Is a dynamic stiffness matrix,/>For displacement vector/>Taylor series coefficient of,/>For simple harmonic excitation/>Taylor expansion term of,/>For optimized material volume,/>For the initial material volume,/>To the allowable material volume ratio,/>Representing cell density,/>For the design variable minimum,/>For unit fiber angle,/>Is the number of finite element units.
Further, the step of solving the sensitivity information of the optimization objective function and the constraint function and correcting the sensitivity information includes:
introducing Lagrangian multipliers to solve sensitivity information of the optimization objective function and the constraint function;
the sensitivity information is corrected by a mapping filtering method based on a Heaviside function.
Further, after the step of determining whether the solution optimization result meets the convergence condition, the method further includes:
If the uncertainty of the simple harmonic excitation is not satisfied, carrying out orthogonal decomposition on the uncertain simple harmonic excitation, and carrying out second-order Taylor expansion on the uncertain simple harmonic excitation so as to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation; punishing a unit stiffness and a mass matrix of the anisotropic material structure, and obtaining an anisotropic material structure response under the uncertain simple harmonic excitation based on a load uncertainty factor, a plurality of finite units, unit stiffness and the mass matrix; establishing a robust dynamic topology optimization mathematical model of the anisotropic material structure by taking a weighted sum of a dynamic compliance mean value of the anisotropic material structure and a standard deviation of the dynamic compliance of the anisotropic material structure as an objective function and taking the volume fraction of the anisotropic material structure as a constraint and the uncertain response; solving an optimization objective function and a constraint function of the anisotropic material structure based on the robustness dynamic topology optimization mathematical model, solving sensitivity information of the optimization objective function and the constraint function, and correcting the sensitivity information to obtain corrected sensitivity information; and updating design variables by adopting a moving asymptotic algorithm until the convergence condition is met, and obtaining the optimal topological configuration of the anisotropic material structure.
Drawings
FIG. 1 is a flow chart of an anisotropic material structure robustness topology optimization design method under simple harmonic excitation in an embodiment of the invention;
FIG. 2 is a topology optimization design domain of an L-beam structure in an embodiment of the invention;
FIG. 3 is a dynamic zone directed topology optimization result of an L-beam structure in an embodiment of the present invention;
fig. 4 is a dynamic robustness topology optimization result of an L-beam structure in an embodiment of the present invention.
The invention will be further described in the following detailed description in conjunction with the above-described figures.
Detailed Description
Example 1
Referring to fig. 1, a topology optimization design method for structural robustness of an anisotropic material under simple harmonic excitation in an embodiment of the invention is shown, and the method includes steps S1 to S7:
S1, defining a design domain and design conditions of an anisotropic material structure under simple harmonic excitation;
It should be noted that in this embodiment, the design conditions include simple harmonic excitation of the anisotropic material structure, boundary conditions, initial values of design variables, filter radius, material properties, and volume constraints.
S2, carrying out orthogonal decomposition on the uncertain simple harmonic excitation, and carrying out second-order Taylor expansion on the uncertain simple harmonic excitation so as to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation;
Specifically, the step S2 includes steps S21 to S22:
S21, carrying out orthogonal decomposition on the uncertain simple harmonic excitation in two directions, and representing the load amplitude and the load direction of the uncertain simple harmonic excitation as a deterministic mean value and a standard deviation composition representing random disturbance;
It will be appreciated that, in this embodiment, Representing simple harmonic excitation, decomposing the simple harmonic excitation into/>, by orthogonal decompositionDirection/>And/>Direction/>Is,/>And/>Can be expressed as/>Individual node/>Direction/>And size/>Wherein/>And/>The expression of (2) is:
；
；
It should be explained that the simple harmonic excitation is represented by the load amplitude mean value of certainty And standard deviation/>, which represents the magnitude of the random disturbanceComposition, load direction/>Can be represented by the load direction mean/>, which represents certaintyAnd standard deviation/>, representing random disturbance directionThe specific expression is:
；
。
S22, performing second-order Taylor expansion on the uncertain simple harmonic excitation based on a second-order perturbation method so as to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation;
it should be explained that the expansion expression for performing the second-order taylor expansion on the uncertain simple harmonic excitation based on the second-order perturbation method is as follows:
；
Where F represents the simple harmonic excitation, ，/>，/>，/>，，/>，/>Representing simple harmonic excitation magnitude,/>Representing the direction of simple harmonic excitation.
S3, punishing unit rigidity and a mass matrix of the anisotropic material structure, and obtaining anisotropic material structure response under the uncertain simple harmonic excitation based on load uncertainty factors, a plurality of finite units, the unit rigidity and the mass matrix;
Specifically, the step S3 includes steps S31 to S32:
S31, punishing unit rigidity and quality matrix of the anisotropic material structure based on an anisotropic material interpolation model;
s32, obtaining an uncertainty response of the anisotropic material structure under simple harmonic excitation based on a load uncertainty factor, a plurality of finite elements, element rigidity and a mass matrix and according to structure simple harmonic vibration finite element analysis;
It should be explained that, when the improved solid anisotropic material punishment model is adopted to punish the elastic modulus of the main direction of the anisotropic material, the expression of the two-dimensional orthotropic material elastic matrix in the main direction coordinate system of the fiber is as follows:
；
In the method, in the process of the invention, Representing an elastic matrix,/>、/>、/>Respectively represent the longitudinal modulus, the transverse modulus and the shear modulus of the punishment anisotropic material of the improved punishment model of the solid anisotropic material,/>、/>Respectively representing longitudinal poisson ratio and transverse poisson ratio, wherein the expressions of longitudinal modulus, transverse modulus and shear modulus are as follows:
；
In the method, in the process of the invention, Representing cell density,/>Representing the direction angle of the unit fiber,/>Representing the cell stiffness matrix penalty factor,/>、/>、/>Representing the longitudinal modulus, transverse modulus, shear modulus, respectively, of a filled solid material unit,/>、/>Respectively, the longitudinal poisson's ratio and the transverse poisson's ratio.
Then the elastic matrix in the global coordinate systemIs that
Wherein,Representing an elastic matrix in a global coordinate system,/>Representing a transfer matrix,/>Elastic modulus matrix,/>A transpose matrix representing the transfer matrix; the transfer matrix expression is:
；
the cell stiffness matrix and the cell mass matrix can be obtained through the elastic matrix, and are respectively expressed as:
；
；
In the method, in the process of the invention, Representing a cell stiffness matrix,/>Representing a cell quality matrix,/>Representing the thickness of the structure,/>Representing material density,/>Is a matrix of form functions,/>Is a strain displacement matrix,/>Is an elastic matrix under global coordinates,/>Representing the area of the cell,/>Representing the differential sign.
The integral rigidity matrix and the integral mass matrix can be obtained through the unit rigidity matrix and the unit mass matrix, and the expressions of the integral rigidity matrix and the integral mass matrix are respectively as follows:
；
；
In the method, in the process of the invention, Representing the overall stiffness matrix,/>Representing an overall quality matrix,/>For designing the number of finite elements divided in a domain,/>Represents the/>Density of individual units,/>Represents the/>Angle design variable of individual units characterizing fiber angle direction,/>Representing the cell quality matrix penalty coefficients. Penalty coefficients for preventing local modal phenomena in low cell density regionsAnd/>The values are all 3.
It is worth to say that, under the effect of simple harmonic excitation, the continuum is forced to vibrate, and the vibration equation expression is:
；
In the method, in the process of the invention, 、/>、/>Respectively, an overall mass matrix, a stiffness matrix, a damping matrix,/>Representing displacement,/>Express speed,/>Representing acceleration,/>Is the simple harmonic excitation to which the system is subjected,/>And/>The expressions are respectively:
；
；
In the method, in the process of the invention, Is the amplitude of simple harmonic excitation,/>Frequency of simple harmonic excitation,/>As the magnitude of the displacement is,Then, it is possible to obtain:
；
Order the For dynamic stiffness/>The structural balance equation can be reduced to:
；
In the method, in the process of the invention, Represents the damping coefficient,/>Representing the frequency of the simple harmonic excitation.
S4, taking a weighted sum of a dynamic compliance mean value of the anisotropic material structure and a standard deviation of the dynamic compliance of the anisotropic material structure as an objective function, and taking the volume fraction of the anisotropic material structure as a constraint to establish a robust dynamic topology optimization mathematical model of the anisotropic material structure;
It should be explained that, in this embodiment, the expression of the robust dynamic topology optimization mathematical model of the anisotropic material structure is:
；
In the method, in the process of the invention, Representing a cell density array,/>Representing an angular array of elementary fibers,/>The function of the optimization objective is represented as,Representing dynamic compliance expectations,/>Representing dynamic compliance variance, i.e., dynamic compliance desire/>And dynamic compliance variance/>Weighted sum of/(v)As a weighting coefficient,/>Is a dynamic stiffness matrix,/>For displacement vector/>Taylor series coefficient of,/>For simple harmonic excitation/>Taylor expansion term of,/>For optimized material volume,/>For the initial material volume,/>To the allowable material volume ratio,/>Representing cell density,/>For the design variable minimum,/>For unit fiber angle,/>Is the number of finite element units.
S5, solving an optimization objective function and a constraint function of the anisotropic material structure based on the robustness dynamic topological optimization mathematical model, solving sensitivity information of the optimization objective function and the constraint function, and correcting the sensitivity information to obtain corrected sensitivity information;
Specifically, the step S5 includes steps S51 to S52:
S51, introducing Lagrangian multipliers to solve sensitivity information of the optimization objective function and the constraint function;
S52, correcting the sensitivity information based on a mapping filtering method of a Heaviside function;
It should be explained that, since the material properties and cell node coordinates of the anisotropic material structure are not affected by the load uncertainty factors, the overall dynamic stiffness matrix of the structure To be deterministic, the structural dynamic equilibrium equation can be expressed as:
；
Structural displacement array The Taylor series expansion of the second order is performed as follows:
；
The available displacement vector The taylor series coefficients of (a) are:
；
Flexibility of structure The method can obtain:
；
In the case of flexibility By/>、/>、/>、/>、/>、/>、/>、/>、/>、/>、/>、/>、/>And/>Composition of/>, wherein、/>、/>、/>、/>、/>、/>、/>、/>、/>、/>、/>Determined as a known quantity, so the flexibility/>Can be expressed as uncertain parameter/>And/>The superscript T denotes the sign of the matrix transpose operation. Assume that the amplitude and direction of the simple harmonic excitation follow a gaussian normal distribution, and that the amplitude magnitude/>And direction/>Both uncertain parameters are independently distributed, wherein，/>Then calculate/>And/>The mean and variance of the higher order function can be found by:
；
；
In the method, in the process of the invention, And/>Representing higher order coefficients. If the magnitude of the simple harmonic excitation is acted/>And direction/>Instead of following a gaussian normal distribution, the random distribution can be transformed into a gaussian normal distribution by Box-Cox power transformation.
Expected value of compliance of anisotropic material structureThe expression of (2) is:
；
In the method, in the process of the invention, Representing the softness of the structure,/>Representing coincidence of expected value operations,/>Representing a simple harmonic excitation of the wave-guide,Representing a displacement array, superscript/>Symbol representing matrix transposition operation,/>Representing simple harmonic excitation magnitude,/>Representing the simple harmonic excitation direction,/>Is displacement/>Taylor expansion term of (1), wherein/>，For simple harmonic excitation/>Taylor expansion term of (1), wherein/>，/>，，/>，/>，/>，/>。
The desired expression for compliance can be expressed as:
；
Wherein:
；
In the method, in the process of the invention, Simplified term representing expected value expression,/>Indicating the coincidence of the desired value operation,Representing simple harmonic excitation magnitude,/>Representing the simple harmonic excitation direction,/>For simple harmonic excitation/>Taylor expansion term of (1), wherein/>，/>，/>，/>，/>，/>，。
Similarly, the variance of softness is obtainedThe variance expression of the compliance is:
；
Wherein:
；
；
；
；
；
；
In the method, in the process of the invention, Simplified term representing variance expression,/>Representing the coincidence of variance operations,/>Representing simple harmonic excitation magnitude,/>Indicates the simple harmonic excitation direction, superscript/>The symbols representing the matrix transpose operation are,For simple harmonic excitation/>Taylor expansion term of (1), wherein/>，/>，，/>，/>，/>，/>。
In this embodiment, the lagrangian multiplier is introduced, and the expression for obtaining the sensitivity of the optimization target to the unit density is:
；
In the method, in the process of the invention, Representing an optimized objective function,/>Representing cell density,/>Indicating the desire for dynamic compliance,Representing dynamic compliance variance,/>As a weighting coefficient,/>Representing the overall dynamic stiffness matrix of the device,Simplified term representing expected value expression,/>A simplified term representing a variance expression,Representing Lagrangian multiplier,/>For simple harmonic excitation/>Taylor expansion term of (1), superscript/>Symbol representing matrix transposition operation,/>Is displacement/>Is a taylor expansion term of (c).
To eliminate the derivative term of the displacement vector on the unit density, lagrangian multiplier can be usedThe assignment is:
；
The sensitivity of the objective function to the design variables can be reduced to:
；
substituting the structural balance equation to obtain the sensitivity of the objective function to the design variable as follows:
；
In the method, in the process of the invention, 、/>、/>Respectively, an overall mass matrix, a stiffness matrix, a damping matrix,/>Represents the damping coefficient of the damping device,Representing the frequency of simple harmonic excitation,/>As a weighting coefficient,/>Representing dynamic compliance variance,/>As the weighting coefficient(s),Representing the overall dynamic stiffness matrix,/>Simplified term representing expected value expression,/>Simplified term representing variance expression,/>Representing Lagrangian multiplier,/>Is excited by simple harmonicTaylor expansion term of (1), superscript/>Symbol representing matrix transposition operation,/>Is displacement/>Is a taylor expansion term of (c).
To avoid deriving the fiber angle by the displacement vector, the fiber angle is calculated byThe coefficient of the term is 0, and Lagrange multiplier is assigned as:
；
the sensitivity of the objective function to fiber angle can be reduced to:
；
Substituting the structural balance equation to obtain the sensitivity of the objective function to the fiber angle as follows:
；
In the method, in the process of the invention, 、/>、/>Respectively, an overall mass matrix, a stiffness matrix, a damping matrix,/>Represents the damping coefficient of the damping device,Representing the frequency of simple harmonic excitation,/>As a weighting coefficient,/>Representing dynamic compliance variance,/>As the weighting coefficient(s),Representing the overall dynamic stiffness matrix,/>Simplified term representing expected value expression,/>Simplified term representing variance expression,/>Represents Lagrangian multiplier, superscript/>Symbol representing matrix transposition operation,/>Is displacement/>A taylor expansion term of (2);
It is noted that when correction is performed on the sensitivity information by using a mapping filtering method based on the Heaviside function, the unit density values are concentrated toward both ends of 0 and 1.
S6, updating design variables by adopting a moving asymptotic algorithm, judging whether the solving and optimizing result meets a convergence condition, and if so, obtaining the optimal topological configuration of the anisotropic material structure;
It is worth to say that if the solution optimization result does not meet the convergence condition, orthogonal decomposition is carried out on the uncertain simple harmonic excitation, and second-order taylor expansion is carried out on the uncertain simple harmonic excitation so as to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation; punishing a unit stiffness and a mass matrix of the anisotropic material structure, and obtaining an anisotropic material structure response under the uncertain simple harmonic excitation based on a load uncertainty factor, a plurality of finite units, unit stiffness and the mass matrix; establishing a robust dynamic topology optimization mathematical model of the anisotropic material structure by taking a weighted sum of a dynamic compliance mean value of the anisotropic material structure and a standard deviation of the dynamic compliance of the anisotropic material structure as an objective function and taking the volume fraction of the anisotropic material structure as a constraint and the uncertain response; solving an optimization objective function and a constraint function of the anisotropic material structure based on the robustness dynamic topology optimization mathematical model, solving sensitivity information of the optimization objective function and the constraint function, and correcting the sensitivity information to obtain corrected sensitivity information; and updating design variables by adopting a moving asymptotic algorithm until the convergence condition is met, and obtaining the optimal topological configuration of the anisotropic material structure.
It should be explained that, in the present embodiment, the expression of the convergence condition is:
；
In the method, in the process of the invention, For/>Cell density vector for the next iteration,/>For/>Cell density vector for the next iteration.
In this embodiment, in order to further verify the effectiveness of the topology optimization design method for structural robustness of anisotropic materials under simple harmonic excitation, the present invention is explained by taking an L-shaped beam structure as an example.
The design domain and boundary conditions of the L-beam structure are shown in FIG. 2, and F in FIG. 2 represents the applied load. The L-shaped beam structure has a design domain size of 60mm by 60mm, is divided into 80 by 80 finite element grids, and has a hollow area with a size of 40mm by 40mm at the right upper end of the structure. The magnitude of the simple harmonic force is Simple harmonic excitation frequency/>=200Rad/s, excitation direction standard deviation/>=0.3. The anisotropic materials are all selected as glass fiber reinforced epoxy resin, and the parameters are/>=39GPa，/>=8.4GPa，/>=4.2GPa，/>=0.26, Material density was 2.54×10 3kg/m3. The initial value of the unit density variable is set to 0.4, the initial value of the unit fiber angle variable is set to 0, and the minimum filter radius/>Set to 1.5 and 0.4 parts by volume.
Fig. 3 and fig. 4 are respectively an L-shaped beam structure dynamic deterministic topology result and an L-shaped beam structure dynamic robust topology optimization result, and compared with the dynamic deterministic topology optimization result, the configuration of the anisotropic material L-shaped beam structure obtained by dynamic robust topology optimization is greatly different, and a V-shaped thin rod supporting structure appears in the left side area, because the structure needs more supporting structures to bear component force generated by uncertain harmonic excitation. In addition, the expected value and standard deviation of the structural dynamic compliance obtained by structural robustness topological optimization design of the anisotropic material under simple harmonic excitation are smaller than the dynamic deterministic topological optimization result, and the structural dynamic compliance is better in robustness.
In summary, the structural robustness topological optimization design method of the anisotropic material under the simple harmonic excitation in the embodiment of the invention adopts a second order perturbation method to quantify the uncertainty load, utilizes an anisotropic material interpolation model to punish the stiffness matrix and the quality matrix, obtains the structural uncertainty response under the simple harmonic excitation according to the simple harmonic vibration finite element analysis, uses the weighted sum of the structural dynamic compliance mean value and the standard deviation as an objective function, uses the structural volume fraction as a constraint, establishes an anisotropic material structural robustness dynamic topological optimization mathematical model under the simple harmonic excitation, introduces Lagrange multipliers to carry out the solution sensitivity analysis, and adopts a moving asymptotic optimization algorithm to solve the topological optimization problem. According to the invention, the structural robustness topological optimization design of the anisotropic material under simple harmonic excitation is carried out by adopting the perturbation method, and the structural anisotropic material obtained by topological optimization can bear component force generated by the simple harmonic excitation and has better robustness.
In the description of the present specification, a description referring to terms "one embodiment," "some embodiments," "examples," "specific examples," or "some examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiments or examples. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
The foregoing examples illustrate only a few embodiments of the invention and are described in detail herein without thereby limiting the scope of the invention. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the invention, which are all within the scope of the invention. Accordingly, the scope of protection of the present invention is to be determined by the appended claims.
Claims (3)
1. The topological optimization design method for structural robustness of anisotropic materials under simple harmonic excitation is characterized by comprising the following steps:
defining a design domain and design conditions of an anisotropic material structure under simple harmonic excitation;
Performing orthogonal decomposition on the uncertain simple harmonic excitation, and performing second-order Taylor expansion on the uncertain simple harmonic excitation to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation;
Punishing a unit stiffness and a mass matrix of the anisotropic material structure, and obtaining an anisotropic material structure response under the uncertain simple harmonic excitation based on a load uncertainty factor, a plurality of finite units, the unit stiffness and the mass matrix;
The step of punishing the unit rigidity and the mass matrix of the anisotropic material structure and obtaining the anisotropic material structure response under the uncertain simple harmonic excitation based on the load uncertainty factors, a plurality of finite units, the unit rigidity and the mass matrix comprises the following steps of:
punishment of unit stiffness and a mass matrix of the anisotropic material structure based on an anisotropic material interpolation model;
Obtaining an uncertainty response of the anisotropic material structure under simple harmonic excitation based on a load uncertainty factor, a plurality of finite elements, element rigidity and a mass matrix and according to structure simple harmonic vibration finite element analysis;
Establishing a robust dynamic topology optimization mathematical model of the anisotropic material structure by taking a weighted sum of a dynamic compliance mean value of the anisotropic material structure and a standard deviation of the dynamic compliance of the anisotropic material structure as an objective function and taking the volume fraction of the anisotropic material structure as a constraint;
The expression of the robust dynamic topology optimization mathematical model of the anisotropic material structure is as follows:
；
In the method, in the process of the invention, Representing a cell density array,/>Representing an angular array of elementary fibers,/>Representing an optimized objective function,/>Representing dynamic compliance expectations,/>Representing dynamic compliance variance, i.e., dynamic compliance desire/>And dynamic compliance variance/>Weighted sum of/(v)As a weighting coefficient,/>Is a dynamic stiffness matrix,/>Is a displacement vectorTaylor series coefficient of,/>For simple harmonic excitation/>Taylor expansion term of,/>For optimized material volume,/>For the initial material volume,/>To the allowable material volume ratio,/>Representing cell density,/>For the design variable minimum,/>For unit fiber angle,/>Is the number of finite element units;
Solving an optimization objective function and a constraint function of the anisotropic material structure based on the robustness dynamic topology optimization mathematical model, solving sensitivity information of the optimization objective function and the constraint function, and correcting the sensitivity information to obtain corrected sensitivity information;
The step of solving the sensitivity information of the optimization objective function and the constraint function and correcting the sensitivity information comprises the following steps:
introducing Lagrangian multipliers to solve sensitivity information of the optimization objective function and the constraint function;
correcting the sensitivity information by a mapping filtering method based on a Heaviside function;
And updating design variables by adopting a moving asymptotic algorithm, judging whether a solving and optimizing result meets a convergence condition, and if so, obtaining the optimal topological configuration of the anisotropic material structure.
2. The topological optimization design method for structural robustness of anisotropic material under simple harmonic excitation according to claim 1, wherein the steps of orthogonally decomposing the uncertain simple harmonic excitation and performing second-order taylor expansion on the uncertain simple harmonic excitation to quantify the uncertainty of the amplitude and direction of the simple harmonic excitation comprise:
Carrying out orthogonal decomposition on the uncertain simple harmonic excitation in two directions, and representing the load amplitude and the load direction of the uncertain simple harmonic excitation as a deterministic mean value and a standard deviation composition representing random disturbance;
And performing second-order Taylor expansion on the uncertain simple harmonic excitation based on a second-order perturbation method so as to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation.
3. The method for topological optimization design of structural robustness of anisotropic material under simple harmonic excitation according to claim 1, wherein after the step of judging whether the solution optimization result meets the convergence condition, the method further comprises:
If the uncertainty of the simple harmonic excitation is not satisfied, carrying out orthogonal decomposition on the uncertain simple harmonic excitation, and carrying out second-order Taylor expansion on the uncertain simple harmonic excitation so as to quantify the uncertainty of the amplitude and the direction of the simple harmonic excitation; punishing a unit stiffness and a mass matrix of the anisotropic material structure, and obtaining an anisotropic material structure response under the uncertain simple harmonic excitation based on a load uncertainty factor, a plurality of finite units, unit stiffness and the mass matrix; establishing a robust dynamic topology optimization mathematical model of the anisotropic material structure by taking a weighted sum of a dynamic compliance mean value of the anisotropic material structure and a standard deviation of the dynamic compliance of the anisotropic material structure as an objective function and taking the volume fraction of the anisotropic material structure as a constraint; solving an optimization objective function and a constraint function of the anisotropic material structure based on the robustness dynamic topology optimization mathematical model, solving sensitivity information of the optimization objective function and the constraint function, and correcting the sensitivity information to obtain corrected sensitivity information; and updating design variables by adopting a moving asymptotic algorithm until the convergence condition is met, and obtaining the optimal topological configuration of the anisotropic material structure.
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