CN114329664B - Iso-geometric topology optimization method for composite material and application thereof - Google Patents
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Abstract
The invention belongs to the field of material structure optimization, and particularly relates to an isogeometric topology optimization method for a composite material and application thereof, wherein the isogeometric topology optimization method comprises the following steps: (1) Implicit description of parameterized level set topology based on non-uniform rational B-splines (NURBS); (2) Multiphase level set (NM-LS) multiphase material distribution description; (3) an adaptive gaussian integration method capable of fusing topology information; (4) a multiphase material stiffness maximization topology optimization model; (5) multiphase material stiffness maximization sensitivity analysis; (6) And updating multiple design variables of the multiphase material and solving the problem. The method provided by the invention constructs a heterogeneous level set (NM-LS) model based on non-uniform rational B-splines (NURBS) to describe the distribution of materials, realizes the optimal design of the composite material, effectively eliminates a plurality of numerical problems generated by linear interpolation in the design and analysis, and improves the numerical precision and iterative stability.
Description
Technical Field
The invention belongs to the field of material structure optimization, and particularly relates to an isogeometric topology optimization method for a composite material and application thereof.
Background
Composite continuum is finding increasing engineering applications due to its advanced, superior and multifunctional properties. However, how to reasonably distribute multiple materials in one field will present more challenges to the design. Traditional methods rely largely on human insight or intuition that, through trial and error, the ability to find suitable multi-material designs may be lost. In recent years, with the widespread development of topology optimization, it is considered as a numerical tool for finding an optimal material layout in a design domain.
In recent years, research on composite materials is gradually in progress, an LSM method suitable for multiple materials is provided, an implicit boundary representation model is provided in the LSM, and a high-dimensional Level Set Function (LSF) is utilized to describe a structure boundary, so that the definition and smoothness of the structure boundary are ensured. In a numerical implementation, alternative material methods are employed to calculate the physical properties of the cells that pass through the structure boundaries. The phantom material process obscures the structural boundaries and interfaces, resulting in loss of definition of the structural boundaries and interfaces. Meanwhile, the unknown structural response in the elastic equilibrium equation is calculated by adopting a finite element method, and the effectiveness and efficiency of the LSM are seriously affected by the defects. Due to its good properties, IGA was introduced into topology optimization to develop an Isogeometric Topology Optimization (ITO) method. Combining LSM and IGA, the NURBS basis functions can be directly employed to construct a level set function. However, research on how to develop an efficient and effective IGA-based multi-material topology optimized LSM remains lacking. Although many ITO methods using IGA and level sets have been developed in recent years, there are three fundamental numerical problems that limit the further development and application of ITO: 1) How to develop an optimization model with high numerical precision and accurate structural geometric expression in optimization, and replace finite elements by a substitute material method; 2) IGA-based LSM three-dimensional structural designs are lacking; 3) How to develop an lsa-based composite structure LSM with powerful design capabilities.
Disclosure of Invention
Aiming at the defects and improvement demands of the prior art, the invention provides an isogeometric topological optimization method for a composite material and application thereof, and aims to eliminate a plurality of numerical problems generated by linear interpolation in design and analysis and improve the numerical precision and iterative stability of the structural design of the composite material.
To achieve the above object, according to one aspect of the present invention, there is provided a composite-oriented isogeometric topology optimization method, including:
s1, carrying out parameterized level set topology implicit description on a composite material topology based on NURBS to obtain a NURBS parameterized level set function;
S2, establishing a multiphase level set model describing multiphase material distribution according to the NURBS parameterized level set function, and calculating to obtain the elastic performance of each point in a design domain;
S3, dividing the geometric unit cut by the structural boundary into four equal subunits by adopting a self-adaptive Gaussian integration method, and if one subunit of the four subunits is still cut by the structural boundary, further subdividing the subunit into four subunits, wherein the process is continuously carried out until a predefined highest level is reached; solving all the equigeometric unit stiffness matrixes K e based on the elastic performance of each point in the design domain;
S4, constructing a multiphase material stiffness maximization topology optimization model by adopting the elastic performance of each point in the design domain and the stiffness matrixes K e of all the geometric units and taking the multiphase material stiffness maximization as a target;
S5, performing iterative updating on the level set discrete values in the topological optimization model by calculating an objective function and sensitivity to obtain an optimized composite material topological optimization configuration.
Further, the implementation manner of the step S1 includes:
S1.1, defining M level set functions phi θ for representing the distribution of M+1 phases, wherein the M+1 phases comprise M materials and a cavity phase, and obtaining an initial composite material topology:
Wherein D is a reference field containing all allowable shapes; omega θ is a design field with positive values of the level set function; Γ θ is the design domain of the zero level set; m represents the total number of types of materials in the composite material;
S1.2, introducing a pseudo time variable t into a level set function phi θ, differentiating time variables on two sides of phi θ (x, t) =0 to obtain dynamic evolution of a structural boundary, wherein the dynamic evolution is used for representing the propulsion of the topological structure boundary and is represented as a first-order Hamilton-Jacobi partial differential equation:
In the method, in the process of the invention, A normal velocity field that is a level set function Φ θ; by normal velocity field/>Derived structural boundary dynamic motion/>Equal to solving a feasible solution of the Hamilton-Jacobi partial differential equation;
S1.3, parameterizing each level set function phi θ in S1.1 correspondingly to obtain a discrete form thereof, namely, a NURBS parameterized level set function:
In the method, in the process of the invention, Is the discrete level set value of control point (i, j) th,/>Is a bivariate NURBS basis function, ζ and η are corresponding parameter coordinates in a parameter coordinate system, p and q are orders in two parameter directions of ζ and η, and m and n are total numbers of control points in two directions;
And based on S1.2, obtaining a normal velocity field of the level set function phi θ The method comprises the following steps:
Further, the implementation manner of the step S2 includes:
establishing a multiphase level set model of the composite material according to the NURBS parameterized level set function, wherein the multiphase level set model is expressed as:
Wherein E (x, phi) is the elastic properties of each point in the design domain comprising different materials; the level set function Φ 1 determines the areas of solid material and cavity phases in the design domain; the level set function Φ 2 determines the distribution of the first material, i.e. Φ 1 >0 and Φ 2 < 0 when θ=1; the level set function Φ 3 determines the distribution of the second and third materials in the region shown by the level set function Φ 2 >0, i.e., Φ 1＞0,Φ2＞0&(Φ3＜0forθ＝2orΦ3＞0forθ＝3);Eθ is the constitutive property of the θ -th material, H (Φ θ) is the Heaviside function of the level set function Φ θ, θ=1, 2.
Further, in the step S3, the two-dimensional numerical equation of the stiffness matrix K e of all the isogeometric units is as follows:
Wherein J 1 gives the mapping of the dual-cell parent space to the parameter space; j 2 gives the mapping of parameter space to physical space; omega k and omega l are the corresponding product weights in the directions of the two parameters of xi and eta; e (phi k,l) is the elastic property of the level set function phi k,l; nu and Nv are the number of Gaussian product points in the directions of two parameters, namely xi and eta respectively; b e is a unit strain-displacement matrix; ζ, η are coordinates of the grid cell corresponding under the parameter coordinate system.
Further, in the step S4, the multiphase material stiffness maximizing topology optimization model is expressed as:
Wherein J is the average structure flexibility, E is the elastic property of the material calculated by the multiphase level set model; Discrete level set values for the theta material of control point (i, j); Φ is a set of level set functions, including Φ θ, θ=1, 2, … M; representation/> A volume constraint function; h (Φ θ) is the Heaviside function of the level set function Φ θ, θ=1, 2,. -%, M; /(I)Is/>A predefined volume fraction; /(I)And/>Upper and lower bounds of discrete level set values, respectively; u is the global displacement field in the design domain; v represents the virtual displacement field, and both u and v belong to a kinematically allowable displacement space H 1;
The unknown displacement field u is calculated by solving a linear elastic equilibrium equation comprising a bilinear energy function a and a linear load function l, and the specific form is as follows:
where f represents physical strength and h represents boundary traction.
The volume constraints of all materials are defined in an implicit wayFor example, when m=3, G 1 is the total volume fraction of the three materials, G 2 is the total volume fraction of the second and third materials, and G 3 is the volume fraction of the third material.
Further, the step S5 includes:
s5.1, initializing design variables;
S5.2, substituting design variables into M parameterized level set functions phi θ (x) and a multiphase level set model of the composite material, and calculating a displacement field by Ku=F so as to calculate an objective function and sensitivity;
S5.3, updating the level set discrete value by adopting an MMA method And (3) repeatedly executing the design variables until the iteration termination condition is reached, and obtaining an optimized topological optimization configuration of the composite material according to the objective function and the sensitivity calculated in the step S5.2 in the last iteration.
Further, the sensitivity is obtained by analyzing partial derivatives of the objective function and the constraint condition with respect to time, wherein the partial derivatives of the objective function and the constraint condition with respect to time are expressed as:
where δ is the derivative of the Heaviside function with respect to the level set function, i.e
The invention also provides a topological structure of the composite material, which is obtained by adopting the isogeometric topological optimization method facing the composite material.
The invention also provides a computer readable storage medium comprising a stored computer program, wherein the computer program when run by a processor controls a device in which the storage medium is located to perform a composite-oriented isogeometric topology optimization method as described above.
In general, through the above technical solutions conceived by the present invention, the following beneficial effects can be obtained:
(1) The invention provides an isogeometric topology and shape optimization method of composite material structural design, which aims at the problem that the existing LSM three-dimensional structural design method based on IGA is not available, and develops an IGA method adopting self-adaptive Gaussian integration to solve the unknown response (displacement field) of a composite material structure, wherein a Q4IGA unit cut by a parent domain structure boundary is firstly divided into four equal subcells, and if one of the four subcells is still cut by the boundary, the four subcells are further subdivided. This process will continue until a predefined maximum level is reached. The self-adaptive process gathers Gaussian intersection points around the structure boundary, so that the numerical precision is improved sufficiently, and a plurality of numerical problems caused by linear interpolation in design and analysis can be effectively eliminated.
(2) The present invention provides an isogeometric topology and shape optimization method for composite structural design that describes the distribution of multiple materials in a composite by constructing a heterogeneous rational B-spline (NURBS) based multiphase level set (NM-LS) model that consists of implicit expression of level sets, NURBS parameterized level sets and their development (embodied in the construction of the NM-LS model), accurately represents structural geometry with NURBS, and still uses the same NURBS basis functions in the analysis to construct solution space. The same NURBS basis function parameterized representation structure topology level set function is adopted in each iteration, the high uniformity of the basis function effectively keeps clear and smooth description of the structure boundary, the loss of brittleness characteristics in linear interpolation is avoided, the problems of instability, boundary blurring and the like caused in the optimization process of the traditional classical method are solved, the clear boundary is obtained, and the effectiveness and the efficiency of LSM are improved;
(3) The isogeometric topology and shape optimization method of the composite material structural design provided by the invention integrates three models of structural geometry (firstly, geometric structure construction is carried out on the whole structure before the NM-LS model is constructed), numerical analysis and topological description through the same NURBS basis function, and improves numerical precision and iterative stability in the composite material optimization process.
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FIG. 1 is a flow chart of an isogeometric topology optimization method for a composite material provided by an embodiment of the invention;
FIG. 2 is a schematic illustration of a multiphase level set (NM-LS) model of two NURBS-based materials provided by an embodiment of the present invention;
FIG. 3 is a schematic diagram of comparing a general Gaussian integration method and an adaptive Gaussian integration method according to an embodiment of the present invention, wherein (a) is a schematic diagram of the general Gaussian integration method and (b) is a schematic diagram of the adaptive Gaussian integration method;
Fig. 4 is a schematic diagram of an optimized structure design of a three-dimensional Michelle type structure according to an embodiment of the present invention, where (a) is a schematic diagram of a level set function 1 with positive three dimensions, (b) is a schematic diagram of a level set function 2 with positive three dimensions, and (c) is a schematic diagram of distribution of two materials in a design field.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
The invention provides an isogeometric topology optimization method for a composite material, which is shown in a figure 1 and comprises the following steps:
(1) Constructing a geometric design model by using NURBS basis functions, and accurately describing the structural geometry of the composite material; refining and establishing an IGA grid according to the input h/p/k; corresponding loads and boundary conditions of the three-dimensional Michelle type structure of the multiphase material are established.
(2) Defining initial discrete values of the level set, and representing by implicit boundary descriptions of the level set: in the composite topology description, M level set functions are defined to represent the distribution of M+1 phases (including M materials and cavity phases),
Where D is a reference field containing all allowable shapes, Ω θ is a design field with a level set function positive value Φ θ, Γ θ corresponds to a zero level set, and M represents the total number of classes of material.
When representing the progression of the structure boundary by a level set function, it is necessary to introduce a pseudo-time variable t in the level set function. By differentiating the time variable on both sides Φ θ (x, t) =0, the dynamic evolution of the structural boundary can be expressed by the first order Hamilton-Jacobi partial differential equation (H-J PDEs) as:
In the middle of The normal velocity field is a level set function Φ θ. Thus, dynamic motion/>, of structural boundaries derived from normal velocity fieldsEqual to the feasible solution to the H-J partial differential equation.
Parameterizing the level set function accordingly NURBS, yielding a discrete version thereof:
Wherein the method comprises the steps of Is the discrete level set value of control point (i, j)/>Is a bivariate NURBS basis function, wherein (ζ, η) is a corresponding parameter coordinate under a parameter coordinate system, (i, j) is a corresponding control point number, and a normal speed corresponding to the level set function is:
(3) Constructing an NM-LS model of the composite material according to the NURBS parameterized level set function phi θ (x) in the step (2), and obtaining the material elasticity performance of each E point by the following formula for the design domain of one material, two materials and three materials:
Wherein the level set function Φ 1 determines the areas of solid materials and cavity phases in the design field; the level set function Φ 2 determines the distribution of the first material, Φ 1＞0&Φ2 < 0, for θ=1; the level set function Φ 3 determines the distribution of the second and third materials in the region indicated by the level set function Φ 2 > 0, namely:
Φ1＞0,Φ2＞0&(Φ3＜0forθ＝2orΦ3＞0forθ＝3)；
E θ is the constitutive property of the theta material, H (Φ θ) corresponds to the Heaviside function of the theta level set function, theta = 1, 2.
(4) The unknown response of the structure is solved by adopting the self-adaptive Gaussian product method on the model established in the steps above, and the method is mainly used for solving the IGA unit stiffness matrix K e. The IGA unit is first divided into four equal sub-units by structural boundary cutting, and if one of the four units is still boundary-cut, the unit is further subdivided into four sub-units. This process continues until a predefined highest level is reached (3 in this embodiment), as shown in fig. 3 (b), which is a process of the general gaussian product method in fig. 3 (a). In order to effectively solve the overall stiffness matrix K, first, an IGA cell stiffness matrix K e needs to be calculated, and a specific two-dimensional mathematical equation is as follows:
Wherein J 1 and J 2 give the mapping of the dual-cell parent space to the parameter space and the NURBS-based parameter space to the physical space, respectively; wherein omega k and omega l are corresponding product weights, nu and Nv are the number of Gaussian product points in two parameter directions respectively, and B e is a unit strain-displacement matrix.
(3) After solving the displacement field in the steps, calculating the structural flexibility minimization objective function and constraint condition of the composite material of the structure, wherein the ITSO formula of the corresponding composite material is as follows:
wherein J is the average flexibility of the structure, E is the elastic performance of the material calculated by the NM-LS model, Is a discrete level set value for control point (i, j). Φ is a set of level set functions including Φ θ, θ=1, 2, … m./>Representation/>The volume constraint function, H (Φ θ), corresponds to the Heaviside function of the θ -level set function, θ=1, 2. Wherein/>Is/>A predefined volume fraction. /(I)And/>Upper and lower bounds of discrete level set values. u is the global displacement field in the design field, v is the virtual displacement field, and all belong to a kinematically allowed displacement space H 1;
Wherein the unknown displacement field u is calculated by solving a linear elastic equilibrium equation comprising a bilinear energy function a and a linear load function l. The specific form is as follows:
where f is the physical strength and h represents the boundary traction. Finally, it should be noted that the volumetric constraints of all materials are defined in an implicit manner For example, when m=3, G 1 is the total volume fraction of the three materials, G 2 is the total volume fraction of the second and third materials, and G 3 is the volume fraction of the third material.
(4) The partial derivative of the objective function with respect to time is calculated by accompanying sensitivity analysis,
Where δ is the derivative of the Heaviside function with respect to the level set function, i.e
(5) Updating level set discrete values by MMA methodThe method comprises the steps of substituting updated design variables into M parameterized level set functions phi θ (x) and an NM-LS model representing composite materials in a design domain, updating a displacement field by the updated composite material model, updating and calculating an objective function and sensitivity calculation to obtain new control variables, judging whether convergence conditions are met (the node density difference between two continuous iterations is smaller than a certain value), and continuing the iteration process if the convergence conditions are not met, so that the topology optimization configuration of the optimized composite materials is obtained.
The results of the above steps of the present invention are described in detail below in conjunction with one embodiment shown in fig. 4:
(1) The structural design domain of the composite material to be optimized is a 32 multiplied by 22 area;
(2) The related data of the three-dimensional Michell structure in NURBS parameterization and IGA are shown in table 1, and the related data comprises a polynomial index of a NURBS base function, the number of the NURBS base function, the number of isogeometric analysis units, node vectors in three parameter directions and the number of control points in the three parameter directions, and a geometric design model is constructed by utilizing the NURBS base function according to the parameters; the maximum consumption of the material 1 and the material 2 is 0.08 and 0.07 respectively, the penalty coefficient is 3, and the maximum value of the volume fraction is set to be 15%;
TABLE 1 data relating to three-dimensional Michell-type Structure in NURBS parameterization and IGA
(3) Defining initial discrete values of the level set and constructing 2 level set functions to represent the distribution of 3 phases (including 2 materials and cavity phases) as follows:
Implicit boundary description representation using level sets: in the composite topology description, 2 level set functions (m=2) are defined;
where D is a reference field containing all allowable shapes, Ω θ is a design field with a level set function positive value Φ θ, Γ θ corresponds to a zero level set. By differentiating the time variable on both sides Φ θ (x, t) =0, the dynamic evolution of the structural boundary can be expressed by the first order Hamilton-Jacobi partial differential equation (H-J PDEs) as:
In the middle of The normal velocity field is a level set function Φ θ. Thus, the dynamic motion of the structural boundary derived from the normal velocity field is equal to the feasible solution to the H-J partial differential equation.
Parameterizing the level set function accordingly NURBS, yielding a discrete version thereof:
Wherein the method comprises the steps of Is the discrete level set value of control point (i, j) th,/>The normal velocity corresponding to the level set function is as follows:
(4) And (3) establishing an NM-LS model of the composite material according to the NURBS parameterized level set function, so as to obtain an NM-LS scheme of two materials as shown in figure 2. For this example design domain, the material elastic properties for each E point can be obtained by:
When M=2, E (x, Φ) =H (Φ 1)[(1-H(Φ2))E1+H(Φ2)E2 ]
Wherein the level set function Φ 1 determines the areas of solid material and cavity phase in the design area, the level set function Φ 2 determines the distribution of the first material, i.e., Φ 1＞0&Φ2＜0,forθ＝1.Eθ is the constitutive property of the θ th material, H (Φ θ) corresponds to the Heaviside function of the θ th level set function, θ=1, 2.
(5) The unknown response of the structure is solved by adopting the adaptive Gaussian product method as shown in the (b) of fig. 3, and is mainly used for solving the IGA unit stiffness matrix. In order to effectively solve the overall stiffness matrix K, first, an IGA cell stiffness matrix K e needs to be calculated, and a specific two-dimensional mathematical equation is as follows:
Wherein J 1 and J 2 give the mapping of the dual-cell parent space to the parameter space and the NURBS-based parameter space to the physical space, respectively; wherein omega k and omega l are corresponding product weights, nu and Nv are the number of Gaussian product points in two parameter directions respectively, and B e is a unit strain-displacement matrix.
(6) Calculating the structural flexibility minimization objective function and constraint condition of the structural composite material, wherein the ITSO formula of the corresponding composite material is as follows:
wherein J is the average flexibility of the structure, E is the elastic performance of the material calculated by the NM-LS model, Is a discrete level set value for control point (i, j). Φ is a set of level set functions including Φ θ, θ=1, 2, … m./>Representation/>The volume constraint function, H (Φ θ), corresponds to the Heaviside function of the θ -level set function, θ=1, 2. Wherein/>Is/>A predefined volume fraction. /(I)And/>Upper and lower bounds of discrete level set values. u is the global displacement field in the design field, v is the virtual displacement field, and all belong to a kinematically allowed displacement space H 1;
Wherein the unknown displacement field u is calculated by solving a linear elastic equilibrium equation comprising a bilinear energy function a and a linear load function l. The specific form is as follows:
where f is the physical strength and h represents the boundary traction. Finally, it should be noted that the volumetric constraints of all materials are defined in an implicit manner For example, when m=3, G 1 is the total volume fraction of the three materials, G 2 is the total volume fraction of the second and third materials, and G 3 is the volume fraction of the third material.
(7) The partial derivative of the objective function with respect to time is calculated by accompanying sensitivity analysis,
Where δ is the derivative of the Heaviside function with respect to the level set function, i.e
(8) Updating level set discrete values by MMA methodThe method comprises the steps of substituting updated design variables into M parameterized level set functions phi θ (x) and an NM-LS model representing composite materials in a design domain, updating a displacement field by the updated composite material model, updating and calculating an objective function and sensitivity calculation to obtain new control variables, judging whether convergence conditions are met, and continuing the iterative process if the convergence conditions are not met, so that the topology optimization configuration of the optimized composite materials is obtained. The convergence criterion is that the difference in node density between two consecutive iterations is less than 1% L-infinity, or a maximum of 150 iteration steps is reached.
The results of the optimization of the three-dimensional Michell-type structure of the two materials and voids are shown in fig. 4, including a three-dimensional view of the positive level set function 1 in fig. 4 (a), and a three-dimensional view of the positive level set function 2 in fig. 4 (b). The layout of the various materials is represented according to the definition of the NM-LS model, with the final optimized two material topologies shown in FIG. 4 (c). The final optimization diagram shows that the distribution of the material 1, the material 2 and the cavity phase in the optimized topological structure is reasonable. Wherein the material 1 is mainly arranged in areas with boundary and load conditions, and the material 2 mainly creates independent stress members within the design domain, due to the comparability of young's modulus of the material 1 and the material 2. In an optimised multiphase topology, the more rigid material 1 acts to form a load bearing member and the material 2 acts to transfer forces. In addition, the NURBS parameterization mechanism in LSM can accurately represent the multi-material topology structure in each iteration, proving the effectiveness and efficiency of the method of unifying the structural geometry model by using NURBS parameterization level set topology optimization and IGA.
The results show that all three iteration curves have stable tracks, and the effectiveness of the method in the structural design of the composite material is proved. During the first 80 iterations, the volume fraction gradually decreases during a very stable process. After the volume fraction reaches a specified value, the optimizer tends to adjust the structural boundary, keeping the main part of the structural topology unchanged, and optimizing the relevant performance. Some details of the corresponding topology optimization process can be clearly observed from the intermediate design of the provided three-dimensional Michell-type structure. The stable topological optimization process and the rapid convergence property show the effectiveness of the method for optimizing the composite material structure containing various materials and cavities.
In summary, the invention (an isogeometric topology and shape optimization method for composite materials) aims to eliminate a plurality of numerical problems generated by linear interpolation in design and analysis, solve the problems of instability, boundary blurring and the like caused by the conventional classical method in the optimization process, and improve the effectiveness and efficiency of LSM. And the numerical precision and iteration stability of the structural design of the composite material are improved. On one hand, aiming at the problem that the existing LSM three-dimensional structure design method based on IGA is not available, the invention develops a geometric topology optimization (ITSO) method such as a level set method for solving the unknown response of the composite material structure by adopting an IGA method of self-adaptive Gaussian integration, and the like, so that a plurality of numerical problems caused by linear interpolation in design and analysis can be effectively eliminated; on the other hand, the invention describes the distribution of various materials in the composite material by constructing a heterogeneous level set (NM-LS) model based on non-uniform rational B-splines (NURBS), wherein the NM-LS model consists of implicit expression of the level set, NURBS parameterized level set and development thereof, solves the problems of instability, boundary blurring and the like caused by the traditional method in the optimization process, obtains clear boundaries and improves the effectiveness and efficiency of LSM.
In addition, the invention integrates the same NURBS basis functions in three models of structural geometry, numerical analysis and topological description, and improves the numerical precision and iterative stability in the optimization process of the composite material. The same NURBS basis function is adopted when the geometric model is constructed, the numerical analysis and the topology description are carried out, and errors are reduced when the NURBS basis function can be used for constructing the geometric model and carrying out related numerical analysis and topology description based on the basis function, so that the numerical precision and the iteration stability are improved.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (9)
1. The isogeometric topology optimization method for the composite material is characterized by comprising the following steps of:
s1, carrying out parameterized level set topology implicit description on a composite material topology based on NURBS to obtain a NURBS parameterized level set function;
S2, establishing a multiphase level set model describing multiphase material distribution according to the NURBS parameterized level set function, and calculating to obtain the elastic performance of each point in a design domain;
S3, dividing the geometric unit cut by the structural boundary into four equal subunits by adopting a self-adaptive Gaussian integration method, and if one subunit of the four subunits is still cut by the structural boundary, further subdividing the subunit into four subunits, wherein the process is continuously carried out until a predefined highest level is reached; solving all the equigeometric unit stiffness matrixes K e based on the elastic performance of each point in the design domain;
S4, constructing a multiphase material stiffness maximization topology optimization model by adopting the elastic performance of each point in the design domain and the stiffness matrixes K e of all the geometric units and taking the multiphase material stiffness maximization as a target;
S5, performing iterative updating on the level set discrete values in the topological optimization model by calculating an objective function and sensitivity to obtain an optimized composite material topological optimization configuration.
2. The method for optimizing isogeometric topology for a composite material according to claim 1, wherein the implementation manner of the step S1 includes:
S1.1, defining M level set functions phi θ for representing the distribution of M+1 phases, wherein the M+1 phases comprise M materials and a cavity phase, and obtaining an initial composite material topology:
Wherein D is a reference field containing all allowable shapes; omega θ is a design field with positive values of the level set function; Γ θ is the design domain of the zero level set; m represents the total number of types of materials in the composite material;
S1.2, introducing a pseudo time variable t into a level set function phi θ, differentiating time variables on two sides of phi θ (x, t) =0 to obtain dynamic evolution of a structural boundary, wherein the dynamic evolution is used for representing the propulsion of the topological structure boundary and is represented as a first-order Hamilton-Jacobi partial differential equation:
In the method, in the process of the invention, A normal velocity field that is a level set function Φ θ; by normal velocity field/>Derived dynamic motion of structural boundariesEqual to solving a feasible solution of the Hamilton-Jacobi partial differential equation;
S1.3, parameterizing each level set function phi θ in S1.1 correspondingly to obtain a discrete form thereof, namely, a NURBS parameterized level set function:
In the method, in the process of the invention, Is the discrete level set value of control point (i, j) th,/>Is a bivariate NURBS basis function, ζ and η are corresponding parameter coordinates in a parameter coordinate system, p and q are orders in two parameter directions of ζ and η, and m and n are total numbers of control points in two directions;
And based on S1.2, obtaining a normal velocity field of the level set function phi θ The method comprises the following steps:
3. The method for optimizing the isogeometric topology of a composite material according to claim 2, wherein the implementation manner of the step S2 includes:
establishing a multiphase level set model of the composite material according to the NURBS parameterized level set function, wherein the multiphase level set model is expressed as:
Wherein E (x, phi) is the elastic properties of each point in the design domain comprising different materials; the level set function Φ 1 determines the areas of solid material and cavity phases in the design domain; the level set function Φ 2 determines the distribution of the first material, i.e. Φ 1 > 0 and Φ 2 < 0 when θ=1; the level set function Φ 3 determines the distribution of the second and third materials in the region shown by the level set function Φ 2 > 0; e θ is the constitutive property of the θ -th material, H (Φ θ) is the Heaviside function of the level set function Φ θ, θ=1, 2.
4. A method of composite-oriented isogeometric topology optimization as recited in claim 3, wherein the two-dimensional numerical equation of all isogeometric unit stiffness matrices K e is as follows:
Wherein J 1 gives the mapping of the dual-cell parent space to the parameter space; j 2 gives the mapping of parameter space to physical space; omega k and omega l are the corresponding product weights in the directions of the two parameters of xi and eta; e (phi k,l) is the elastic property of the level set function phi k,l; nu and Nv are the number of Gaussian product points in the directions of two parameters, namely xi and eta respectively; b e is a unit strain-displacement matrix; ζ, η are coordinates of the grid cell corresponding under the parameter coordinate system.
5. The method of composite-oriented isogeometric topology optimization of claim 4, wherein in step S4, the multiphase material stiffness maximizing topology optimization model is expressed as:
Wherein J is the average structure flexibility, E is the elastic property of the material calculated by the multiphase level set model; Discrete level set values for the theta material of control point (i, j); Φ is a set of level set functions, including Φ θ, θ=1, 2, … M; /(I) Representation ofA volume constraint function; h (Φ θ) is the Heaviside function of the level set function Φ θ, θ=1, 2,. -%, M; /(I)Is/>A predefined volume fraction; /(I)And/>Upper and lower bounds of discrete level set values, respectively; u is the global displacement field in the design domain; v represents the virtual displacement field, and both u and v belong to a kinematically allowable displacement space H 1;
The unknown displacement field u is calculated by solving a linear elastic equilibrium equation comprising a bilinear energy function a and a linear load function l, and the specific form is as follows:
where f represents physical strength and h represents boundary traction.
6. The method for geometric topological optimization of a composite material according to claim 5, wherein the step S5 comprises:
s5.1, initializing design variables;
s5.2, substituting design variables into M parameterized level set functions phi θ (x) and a multiphase level set model of the composite material, and calculating a displacement field by Ku=F so as to calculate an objective function and sensitivity;
S5.3, updating the level set discrete value by adopting an MMA method And (3) repeatedly executing the design variables until the iteration termination condition is reached, and obtaining an optimized topological optimization configuration of the composite material according to the objective function and the sensitivity calculated in the step S5.2 in the last iteration.
7. The method for geometric topological optimization of a composite material according to claim 6, wherein the sensitivity is obtained by analyzing partial derivatives of an objective function and a constraint condition with respect to time, respectively, wherein the partial derivatives of the objective function and the constraint condition with respect to time are expressed as:
where δ is the derivative of the Heaviside function with respect to the level set function, i.e Phi θ is the discrete level set value of the theta-th material; j is the average structure compliance, used to characterize the objective function expression.
8. A topology of a composite material, characterized in that it is obtained by an isogeometric topology optimization method for composite materials according to any of claims 1 to 7.
9. A computer readable storage medium, characterized in that the computer readable storage medium comprises a stored computer program, wherein the computer program, when run by a processor, controls a device in which the storage medium is located to perform a composite oriented isogeometric topology optimization method according to any one of claims 1 to 7.
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