CN117892414A - 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:
；
Where denotes a cell density array,/> denotes a cell fiber angle array,/> denotes an optimization objective function, denotes a dynamic compliance desire,/> denotes a dynamic compliance variance, i.e., a weighted sum of dynamic compliance desire/> and dynamic compliance variance/> ,/> is a weighting coefficient,/> is a dynamic stiffness matrix,/> is a taylor series coefficient of displacement vector/> ,/> is a taylor expansion term of simple harmonic excitation/> ,/> is an optimized material volume,/> is an initial material volume,/> is an allowable material volume ratio,/> denotes a cell density,/> is a minimum design variable,/> is a cell fiber angle, and/> is a number of finite element cells.
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 is understood that, in the present embodiment, represents simple harmonic excitation, and the simple harmonic excitation is decomposed by orthogonal decomposition into two forces of/> direction/> and/> direction/> ,/> and/> may be expressed as a function of direction/> and size/> on the/> node/> , where the expressions of/> and/> are:
；
；
It should be explained that the simple harmonic excitation consists of a load amplitude mean representing certainty and a standard deviation/> representing the magnitude of random disturbance, and the load direction/> may consist of a load direction mean/> representing certainty and a standard deviation/> representing the direction of random disturbance, with the specific expressions:
；
。
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, ,/>,/>,/>,,/>,/> represents the magnitude of the simple harmonic excitation, and/() represents the direction of the 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:
；
wherein denotes an elastic matrix,/> 、/>、/> denotes longitudinal modulus, transverse modulus and shear modulus of the improved solid anisotropic material punishment model punishment anisotropic material, and/> 、/> denotes longitudinal poisson ratio and transverse poisson ratio, respectively, wherein the longitudinal modulus, transverse modulus and shear modulus are expressed as follows:
；
Where denotes the cell density,/> denotes the cell fiber direction angle,/> denotes the cell stiffness matrix penalty factor,/> 、/>、/> denotes the longitudinal modulus, transverse modulus, shear modulus, respectively, of a filled solid material cell,/> 、/> denotes the longitudinal poisson ratio, respectively, transverse poisson ratio.
Then the elastic matrix in the global coordinate system is
Wherein denotes an elastic matrix in a global coordinate system,/> denotes a transfer matrix,/> elastic modulus matrix,/> denotes a transpose matrix of 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:
；
；
Where denotes a cell stiffness matrix,/> denotes a cell mass matrix,/> denotes a thickness of the structure,/> denotes a material density,/> is a shape function matrix,/> is a strain displacement matrix,/> is an elastic matrix in global coordinates,/> denotes a region of the cell, and/> denotes a differential symbol.
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:
；
；
Where denotes the overall stiffness matrix,/> denotes the overall mass matrix,/> denotes the number of finite elements divided within the design domain,/> denotes the density of the/> elements,/> denotes the angular design variable characterizing the angular direction of the fiber of the/> elements, and/> denotes the element mass matrix penalty factor. In order to prevent local modal phenomenon in the low cell density region, penalty coefficients and/> are both 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:
；
Wherein 、/>、/> is the overall mass matrix, stiffness matrix, damping matrix,/> represents displacement,/> represents velocity,/> represents acceleration,/> is the simple harmonic excitation to which the system is subjected, and the expressions/> and/> are respectively:
；
；
Wherein is the amplitude of simple harmonic excitation,/> is the frequency of simple harmonic excitation,/> is the amplitude of displacement, , and then the following can be obtained:
；
Let be dynamic stiffness/> , the structural balance equation can be reduced to:
；
Where denotes the damping coefficient and/() denotes 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:
；
Where denotes a cell density array,/> denotes a cell fiber angle array,/> denotes an optimization objective function, denotes a dynamic compliance desire,/> denotes a dynamic compliance variance, i.e., a weighted sum of dynamic compliance desire/> and dynamic compliance variance/> ,/> is a weighting coefficient,/> is a dynamic stiffness matrix,/> is a taylor series coefficient of displacement vector/> ,/> is a taylor expansion term of simple harmonic excitation/> ,/> is an optimized material volume,/> is an initial material volume,/> is an allowable material volume ratio,/> denotes a cell density,/> is a minimum design variable,/> is a cell fiber angle, and/> is a number of finite element cells.
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 the cell node coordinates of the anisotropic material structure are not affected by the load uncertainty factor, the overall dynamic stiffness matrix of the structure is determined, and the structural dynamic balance equation can be expressed as:
；
The structural displacement array performs a second order taylor series expansion as:
；
The taylor series coefficients of the available displacement vectors are:
；
the flexibility of the structure can be obtained:
；
where compliance consists of />、/>、/>、/>、/>、/>、/>、/>、/>、/>、/>、/>、/> and/> , where />、/>、/>、/>、/>、/>、/>、/>、/>、/>、/>、/> is determined as a known quantity, so compliance/> can be expressed as a function of the uncertain parameters/> and/> , with the superscript T representing the sign of the matrix transpose operation. Assuming that the amplitude and direction of the simple harmonic excitation follow a gaussian normal distribution, and that the amplitude magnitude/> and the direction/> two uncertain parameters are distributed independently, wherein ,/>, the mean and variance of the higher order functions/> and/> can be calculated by the following formula:
；
；
Where and/> represent higher order coefficients. If the applied simple harmonic excitation magnitude/> and direction/> do not follow a gaussian normal distribution, the random distribution can be transformed into a gaussian normal distribution by Box-Cox power transformation.
The expected value of the compliance of the anisotropic material structure is expressed as:
；
Wherein denotes the compliance of the structure,/> denotes the coincidence of the desired value operation,/> denotes the simple harmonic excitation, denotes the displacement array, superscript/> denotes the sign of the matrix transposition operation,/> denotes the simple harmonic excitation magnitude,/> denotes the simple harmonic excitation direction,/> is the taylor expansion term of displacement/> , wherein/> , is the taylor expansion term of simple harmonic excitation/> , wherein />,/>,,/>,/>,/>,/>.
The desired expression for compliance can be expressed as:
；
Wherein:
；
wherein denotes a simplified term of the expected value expression,/> denotes coincidence of the expected value operation, denotes a simple harmonic excitation magnitude,/> denotes a simple harmonic excitation direction,/> is a taylor expansion term of simple harmonic excitation/> , wherein />,/>,/>,/>,/>,/>,.
Similarly, the variance of the compliance is obtained, and the variance expression of the compliance is:
；
Wherein:
；
；
；
；
；
；
Wherein denotes a simplified term of the variance expression,/> denotes a coincidence of the variance operation,/> denotes a simple harmonic excitation size,/> denotes a simple harmonic excitation direction, a superscript/> denotes a sign of the matrix transposition operation, is a taylor expansion term of the simple harmonic excitation/> , 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:
；
Where denotes an optimization objective function,/> denotes a unit density,/> denotes a dynamic compliance desire, denotes a dynamic compliance variance,/> is a weighting coefficient,/> denotes an overall dynamic stiffness matrix, denotes a simplified term of a desired value expression,/> denotes a simplified term of a variance expression, denotes a lagrangian multiplier,/> is a taylor expansion term of simple harmonic excitation/> , superscript/> denotes a sign of a matrix transposition operation, and/> is a taylor expansion term of displacement/> .
To eliminate the displacement vector derivative of cell density, the lagrangian multiplier may be assigned as:
；
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:
；
Wherein 、/>、/> is an overall mass matrix, stiffness matrix, damping matrix,/> represents damping coefficient, represents frequency of simple harmonic excitation,/> is a weighting coefficient,/> represents dynamic compliance variance,/> is a weighting coefficient, represents overall dynamic stiffness matrix,/> represents simplified term of desired value expression,/> represents simplified term of variance expression,/> represents lagrangian multiplier,/> is taylor expansion term of simple harmonic excitation , superscript/> represents sign of matrix transposition operation, and/> is taylor expansion term of displacement/> , respectively.
In order to avoid deriving the fiber angle by the displacement vector so that the coefficient of terms is 0, the 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:
；
Wherein 、/>、/> is an overall mass matrix, a stiffness matrix, and a damping matrix, wherein/> represents a damping coefficient, represents a frequency of simple harmonic excitation,/> is a weighting coefficient,/> represents a dynamic compliance variance,/> is a weighting coefficient, represents an overall dynamic stiffness matrix,/> represents a simplified term of a desired value expression,/> represents a simplified term of a variance expression,/> represents a lagrangian multiplier, superscript/> represents a sign of a matrix transposition operation, and/> is a taylor expansion term of displacement/> , respectively;
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:
；
Where is the cell density vector for the/> iteration and/> is the cell density vector for the/> 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 , the simple harmonic excitation frequency/> =200 rad/s, and the standard deviation of the excitation direction/> =0.3. The anisotropic materials were all selected as glass fiber reinforced epoxy resins with parameters/> =39GPa,/>=8.4GPa,/>=4.2GPa,/> =0.26 and material densities of 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, the minimum filter radius/> is set to 1.5, and the volume fraction is set to 0.4.
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 (6)
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;
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.
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 topological optimization design method for robustness of an anisotropic material structure under simple harmonic excitation according to claim 1, wherein the step of penalizing a cell stiffness and a mass matrix of the anisotropic material structure and solving an anisotropic material structure response under the uncertain simple harmonic excitation based on a load uncertainty factor, a plurality of finite cells, the cell stiffness and the mass matrix comprises:
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.
4. The method for topologically optimizing the structural robustness of an anisotropic material under simple harmonic excitation according to claim 1, wherein the expression of the dynamic topological optimization mathematical model of the structural robustness of the anisotropic material is:
；
Where denotes a cell density array,/> denotes a cell fiber angle array,/> denotes an optimization objective function,/> denotes a dynamic compliance desire,/> denotes a dynamic compliance variance, i.e., a weighted sum of dynamic compliance desire/> and dynamic compliance variance/> ,/> is a weighting coefficient,/> is a dynamic stiffness matrix,/> is a taylor series coefficient of displacement vector ,/> is a taylor expansion term of simple harmonic excitation/> ,/> is an optimized material volume,/> is an initial material volume,/> is an allowable material volume ratio,/> denotes a cell density,/> is a minimum design variable,/> is a cell fiber angle,/> is a number of finite element cells.
5. The topological optimization design method for structural robustness of anisotropic material under simple harmonic excitation according to claim 1, wherein the steps of solving the sensitivity information of the optimization objective function and the constraint function and correcting the sensitivity information comprise:
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.
6. 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 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.
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