CN111259589B - Continuous frequency constraint topology optimization design method considering breakage-safety - Google Patents
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Abstract
The invention discloses a continuous frequency constraint topology optimization design method considering breakage-safety, which comprises the steps of firstly establishing a finite element model based on a continuous base structure; determining the number of damage working conditions, and inputting the central coordinate value of an initial damage area and the parameters of the damage area; inputting frequency constraint optimization parameters to form a frequency constraint optimization model under the condition of considering damage-safety; sequentially carrying out modal analysis on the damage-safety structure containing the damage working condition, and extracting a modal analysis result of the damage condition of each structure; establishing an approximate continuous mathematical optimization list of a topological optimization model considering frequency constraint under a damage-safety condition; solving the optimization model by adopting a mathematical programming algorithm; and carrying out inversion treatment on the optimal topological configuration to obtain the optimal topological configuration and the fundamental frequency of the optimal configuration. The invention can obviously improve the redundancy of the structure and provides a reference for the topological optimization problem of considering damage-safety in the dynamics field.
Description
Technical Field
The invention belongs to the technical field of engineering structural design, and particularly relates to a continuous body frequency constraint topology optimization design method under the condition of considering damage and safety.
Background
Topology optimization is to find the optimal force transmission path of a structure by taking material distribution as an optimization object under given external load and boundary conditions, and the optimization method is gradually developed along with the high-speed development of finite element theory and computer software technology. Compared with size optimization and shape optimization, the structural topological optimization can determine more design parameters, can remarkably save materials and obtain greater economic benefits, can provide comprehensive conceptual design for engineering designers in the initial stage of structural design, and has great attraction to the engineering designers when the optimal scheme is applied to the layout of the structure.
The traditional topological structure obtained through topological optimization is usually an optimal force transmission path under the action of external force, the performance of the material is fully exerted, but the optimal configuration is prone to be a statically determined structure, the defect is that the structure lacks redundancy, and in this case, the local damage of the structure is easy to cause the overall damage. In fields with higher safety requirements, such as aerospace, ships, high-speed trains, bridges, ultra-high buildings and the like, the high sensitivity of the topological structure to local damage can make the economic effect of the topological structure redundant and the safety guarantee is insufficient, and the topological structure is often regarded as economical but lacks the safety guarantee in the engineering field with higher safety, so that the optimization of damage-safety is considered to be paid attention.
The engineering structure generally needs to consider the vibration problem, the response of the structure under the dynamic load is related to the natural frequency and the external load frequency of the structure, and in extreme cases, when the external load frequency is very close to the natural frequency of a certain order of the structure, the structure can generate resonance phenomenon. Therefore, when the structural power design is carried out, the natural frequency of the structure needs to be limited, and the design of aircrafts such as rockets and missiles has strict requirements on the natural frequency of the structure, so that the topological optimization design of the structural frequency has important significance.
Aiming at the problem of traditional frequency constraint topological optimization design of a continuum structure, the invention considers the design of a damage-safety concept, and provides a frequency constraint topological optimization design method of the continuum under the condition of considering damage-safety. The result of conventional frequency topology optimization is similar to a skeleton structure, and although the topology maximizes the utilization rate of materials under the condition of considering frequency, the conventional topology structure can be completely destroyed in the face of local damage (caused by collision, explosion and fatigue failure), and has disastrous results. After the design concept of damage-safety is introduced, the continuous body frequency constraint topology optimization design method under the damage-safety condition is combined with the structural modal finite element analysis technology, and the condition that the inside of the structure is damaged is considered in the frequency constraint topology optimization, so that the problem that the traditional frequency optimization structural component is too efficient can be effectively solved. The continuous frequency constraint topology optimization design method under the condition of considering damage-safety not only can realize the design purpose of traditional frequency topology optimization, but also can obviously improve the redundancy of the optimal topology configuration, and after any position of the optimal structure is damaged, the frequency of the structure is near the constraint frequency, and the structure can still work normally. The invention is beneficial to reducing the sensitivity of the optimal structure frequency to local damage and improving the redundancy of the optimal structure, and has important theoretical significance and wide engineering application value in the aspect of optimizing the structural power in various fields such as aviation, aerospace, automobiles, bridges, civil engineering and the like.
Disclosure of Invention
Aiming at the problem of frequency constraint topology optimization design of the traditional continuum structure, the invention considers the design concept of breakage-safety, adopts the strategy of hierarchical laying to approximate random and infinite breakage working conditions, and combines the design concept of breakage-safety with the frequency constraint topology optimization for the first time. The continuous body frequency constraint topology optimization design method taking frequency as a constraint object under the condition of damage-safety can obviously improve the redundancy of an optimal structure, obviously reduce the sensitivity of the optimal structure frequency to local damage, and has extremely strong theoretical significance and application value in the field of structure power topology optimization. In order to achieve the above purpose, the invention adopts the following technical scheme:
the topological optimization design method for the continuous body frequency constraint under the condition of considering breakage-safety comprises the following steps:
firstly, determining a continuum base structure and establishing a finite element model;
secondly, determining the number of damage working conditions, and inputting initial damage region center coordinate values and damage region parameters in a finite element model;
thirdly, inputting frequency constraint optimization parameters to form a frequency constraint optimization model under the condition of considering damage-safety;
fourthly, carrying out modal analysis on the damage-safety structure containing the damage working conditions in sequence, and extracting modal analysis results of all the damage working conditions;
fifthly, establishing an approximate continuous mathematical optimization list of a topological optimization model considering frequency constraint under a damage-safety condition;
sixthly, solving the optimization model by adopting a mathematical programming algorithm;
and seventh, carrying out inversion treatment on the optimal topological configuration to obtain the optimal topological configuration and the fundamental frequency of the optimal configuration.
Compared with the prior art, the invention has the advantages that:
compared with the traditional structural frequency topology optimization design, the structural damage-safety design concept is fully considered, the damage working condition is added in the optimization design, the redundancy of the optimal structure can be obviously improved, the sensitivity of the optimal structure to local damage is obviously reduced, and the method has extremely strong theoretical significance and application value in the structural power topology optimization field.
Drawings
FIG. 1 is a flow chart of a continuous body frequency constrained topology optimization design taking into account breakage-safety conditions.
Fig. 2 is a basic structure and optimal topology taking into account breakage-safety conditions.
Fig. 3 is a diagram of a topology optimization process taking into account breakage-safety conditions.
Fig. 4 is an iteration graph, (a) is a total volume iteration curve of the structure, and (b) is a fundamental frequency iteration curve of each structural failure condition.
Detailed Description
As shown in fig. 1, the invention provides a continuous body frequency constraint topology optimization design method under the condition of considering breakage-safety, and the specific solution is as follows:
the first step, determining a continuum base structure and establishing a finite element model:
firstly, determining the maximum design boundary of a continuum structure to be optimized based on design requirements, namely determining a base structure, and determining a damaged design domain and a non-damaged design domain according to certain performance requirements; secondly, based on an MSC.Patran software platform, a basic structure geometric model is established, and finite element grids are divided; finally defining materials, endowing unit attributes, and applying boundary conditions; finally, setting a modal analysis output option.
Secondly, determining the number of damage working conditions, and inputting an initial damage area center coordinate value and a damage area parameter:
in a main program written based on MSC.Patran software built-in language PCL (Patran Command Language), firstly setting the number of damaged working conditions, wherein the number of damaged working conditions can be determined according to actual problems; then inputting a central two-dimensional coordinate value of the initial damaged area; then inputting damaged area parameters: the square of the diagonal of the damaged area and the square of the side of the damaged area are half, so that the initial damaged area of the square can be determined.
Thirdly, inputting frequency constraint optimization parameters to form a frequency constraint optimization model considering damage-safety conditions:
inputting constraint parameters including constraint fundamental frequency size, convergence accuracy and filter radius according to the design requirement of the natural frequency performance of the structure; the size, shape and frequency constraints of the damaged area are determined, and a topological optimization model taking fundamental frequency as constraint and the minimum structural weight as an optimization target can be established, wherein the damage-safety is considered:
wherein: t represents a topological variable vector, t i Characterizing the existence or non-existence of the unit for the topological variable of the ith unit; t E N The representation topology variable t is a vector belonging to an n-dimensional European space; w represents the weight of the structure and,an initial weight for the ith cell; f (f) w (t i )、f k (t i )、f m (t i ) The weight filter function, the rigidity matrix filter function and the quality matrix filter function are respectively proposed in an ICM (Independent, continuous, mapping) method; lambda (lambda) l Characteristic value corresponding to the fundamental frequency representing the first damaged condition,/->Representing a characteristic value constraint value corresponding to the fundamental frequency of the first damaged working condition; l represents the number of damaged working conditions; n represents the total number of units;
fourthly, carrying out modal analysis on the damage-safety structure containing the damage working conditions in sequence, and extracting modal analysis results of all the damage working conditions:
the base structure containing a local damage area of the structure is generally called as a structure damage condition, and the structure can be sequentially damaged from an initial damage area according to the number of damage working conditions in a PCL main program, and each structure damage condition is subjected to modal analysis. Sequentially extracting the modal analysis result of each structural damage condition unit, wherein the modal analysis result is U respectively il And V il ，Represents the fundamental mode strain energy, + of cell i under the first failure condition>Represents the fundamental mode kinetic energy of cell i in the first failure condition. In this way, the derivative of the corresponding fundamental frequency characteristic value to each design variable, namely sensitivity analysis result, can be calculated by extracting the modal strain energy and the modal kinetic energy of the unit under a certain damage condition calculated by modal analysis, and the mechanical performance parameter of the structural unit is provided for establishing an explicit optimization equation.
Fifthly, establishing an approximate continuous mathematical optimization list of a topological optimization model considering frequency constraint under damage-safety conditions:
taking the design variables asγ k Is the power of the filter function of the unit stiffness, here gamma k =5; and based on methods such as sensitivity analysis and Taylor expansion, the optimization model is subjected to explicit processing, and a corresponding quadratic programming explicit equation is obtained as follows:
wherein α=γ w /γ k ，γ w =2 is the power exponent of the weight filter function, γ k =5 is the power exponent of the unit stiffness filter function; d, d il And e l Is a constant coefficient of the explicit constraint inequality; v is the number of iterations;is the difference between the modal strain energy and the modal kinetic energy of the ith unit under the first damaged working condition and the ith iteration; when->When d=1, when +.>When d= -1; />Is the upper design variable limit.
In order to make the topological variable t take 0 or 1 as much as possible, adding a discrete condition and forming a single target with the original target according to linear weighting, and omitting a constant term after twice approximation of the target to obtain the target:
the approximate continuous mathematical optimization column is updated as follows:
sixth, adopting a mathematical programming algorithm to solve the optimization model:
considering that the number of design variables in the optimization model (4) is far greater than the number of constraints, the design variables are converted into dual models based on Coulomb-Tack conditions, and then the dual models are solved by adopting a sequence quadratic programming algorithm. If the obtained solution meets the convergence accuracy condition, outputting an optimal solution, and performing a seventh step; if the convergence accuracy condition is not met, modifying the topological variable, returning to the fourth step, and carrying out finite element analysis and optimization model processing on the damage condition of the next round until the convergence accuracy condition is met.
Seventh, inversion processing is carried out on the optimal topological configuration, and the optimal topological configuration and the fundamental frequency of the optimal configuration are obtained:
and selecting a proper threshold to invert the topological variable according to the topological variable cloud chart of the topological optimization result, so as to realize the mapping of the topological optimization variable from continuous to discrete 0/1 and obtain the optimal topological configuration and the corresponding fundamental frequency.
The invention discloses a continuous body frequency constraint topology optimization design method under the condition of considering damage-safety, which comprises the following steps: (1) Determining a continuum base structure and establishing a finite element model; (2) Determining the number of damage working conditions, and inputting the central coordinate value of an initial damage area and the parameters of the damage area; (3) Inputting frequency constraint optimization parameters to form a frequency constraint optimization model under the condition of considering damage-safety; (4) Sequentially carrying out modal analysis on the damage-safety structure containing the damage working conditions, and extracting modal analysis results of all the damage working conditions; (5) Establishing an approximate continuous mathematical optimization list of a topological optimization model considering frequency constraint under a damage-safety condition; (6) Solving the optimization model by adopting a mathematical programming algorithm; (7) And carrying out inversion treatment on the optimal topological configuration to obtain the optimal topological configuration and the fundamental frequency of the optimal configuration. The invention can effectively solve the topological optimization design problem of the continuous body frequency constraint under the damage-safety condition, and compared with the traditional frequency topological optimization, the invention can obviously improve the redundancy of the structure and provide a reference for the topological optimization problem of the damage-safety in the dynamics field.
Example:
the following describes in detail the implementation of the invention with reference to an example of frequency topology optimization taking into account the breakage-safety conditions.
Firstly, establishing a basic structure with the size shown in (a) of fig. 2 based on MSC.Patran, wherein the basic structure is a rectangular plate with the size of 200mm multiplied by 100mm multiplied by 9mm, one end of the basic structure is fixed, a concentrated mass block is arranged at the middle position of the other end of the basic structure, and the mass block is 0.001g; the elastic modulus of the material is 68890MPa, the Poisson ratio is 0.3, and the density is 1.0X10 -9 Mg/mm 3 The method comprises the steps of carrying out a first treatment on the surface of the Divided intoThe left boundary of 20X 40 rectangular units adopts fixed constraint, and simultaneously constrains the rotation of all the units in the x and y directions and the translation of all the units in the z direction; the first order frequency of the base structure is 1958Hz, and the frequency constraint is that the first order frequency is greater than 900Hz; the finite element model is shown in fig. 2 (b). Considering 8 damage conditions, the damage areas are uniformly distributed in the base structure in square shapes with the side length of 50mm×50mm, as shown in fig. 2 (c). Finally, selecting output unit strain energy in the modal analysis setting options;
and secondly, setting damage working condition parameters in a main program written based on PCL language. Firstly, setting the number of damaged working conditions to be equal to 8; then inputting the central two-dimensional coordinate values (25, 25) of the initial damaged area; then inputting damaged area parameters: the square of half of the diagonal line of the damaged area is 225, and the square of half of the side length of the damaged area is 225, so that the initial damaged area of the square can be determined;
and thirdly, inputting frequency constraint optimization parameters in the optimization main program. Frequency optimization parameters: the frequency constraint direction is more than or equal to 900Hz; the convergence accuracy was 0.001 and the filter radius was 7.5. The size, shape and frequency constraints of the damaged area are determined;
fourthly, carrying out modal analysis on the damage condition of the initial structure by adopting MSC. Nastran, sequentially carrying out modal analysis on the damage condition of the structure by FOR circulation, automatically extracting unit modal kinetic energy and modal strain energy by a program, and providing mechanical performance parameters of a structural unit FOR establishing an explicit optimization equation;
fifthly, establishing an approximate continuous mathematical optimization column type of a topological optimization model considering frequency constraint under the damage-safety condition based on methods such as sensitivity analysis, taylor expansion and the like;
and sixthly, solving the optimization model by adopting a mathematical programming algorithm. If the given convergence condition is met, outputting an optimization result, if the convergence condition is not met, modifying the topological variable, updating the finite element model, returning to the fourth step, and carrying out finite element analysis and optimization model processing of the damage condition of the next round until the convergence condition is met;
and seventh, inverting the topological variable to obtain the optimal topological configuration.
Frequency topology optimization with consideration of breakage-safety conditions example optimal topology is shown in fig. 2 (e), the optimal structure after inversion of the threshold value selection of 0.95 is shown in fig. 2 (f), and specific values of the optimal topology are shown in table 1. Compared with the traditional frequency optimization optimal topological structure in the step (d) of fig. 2, the redundancy of the structure can be obviously improved, and the sensitivity of the optimal structure to local damage can be obviously reduced. As can be seen from the topology optimization iteration history of fig. 3, the path that sees the frequency response is preserved; meanwhile, as can be seen from the iteration curve of the target volume in FIG. 3, the total volume of the structure under the condition of considering damage is finally and stably converged to 78180mm after 31 iterations 3 . As can be seen from the iteration curve of fundamental frequency of each structural failure condition in fig. 3, the fundamental frequency of each structural failure condition tends to converge after iteration. Therefore, the method provided by the invention can realize the continuous body frequency constraint topology optimization design considering the damage-safety condition.
Table 1 frequency topology optimization example optimal topology results considering breakage-safety conditions
Claims (6)
1. The continuous body frequency constraint topology optimization design method considering breakage-safety is characterized by comprising the following steps of:
determining a damaged design domain and a non-damaged design domain of a continuum structure, establishing a corresponding finite element model, and setting boundary conditions and material attribute parameters;
secondly, inputting the number of damage working conditions, inputting the central coordinate value of an initial damage area into the finite element model, and simultaneously inputting the parameters of the damage area, so that the position and the shape of the initial damage area are determined;
inputting frequency constraint optimization parameters including the size, convergence accuracy and filtering radius of constraint fundamental frequency to form a frequency constraint optimization model under the condition of considering breakage-safety;
fourth, the base structure containing a localized damage region of the structure is referred to as a structure damage condition; starting from an initial damage area, sequentially carrying out damage on the structure according to the number of damage working conditions, carrying out modal analysis on each structure damage condition, extracting a modal analysis result of each structure damage condition unit, and providing mechanical performance parameters of the structure units for establishing an explicit optimization equation;
fifthly, forming an approximate continuous mathematical optimization column type taking frequency as constraint and the minimum structural weight as a target in consideration of breakage-safety;
sixthly, solving the continuous mathematical optimization column by adopting a mathematical programming algorithm;
and seventh, inverting the topological variable to obtain an optimal topological configuration and a corresponding fundamental frequency.
2. The optimization design method of continuous frequency constraint topology considering breakage-safety according to claim 1, wherein the first step is specifically:
firstly, determining the maximum design boundary of a continuum structure to be optimized based on design requirements, namely determining a base structure, and determining a damaged design domain and a non-damaged design domain according to performance requirements; secondly, based on an MSC.Patran software platform, a basic structure geometric model is established, and finite element grids are divided; finally defining materials, endowing unit attributes, and applying boundary conditions; finally, setting a modal analysis output option.
3. The optimization design method of the continuum frequency constraint topology considering breakage-safety according to claim 2, wherein the second step is specifically:
in a main program written based on the built-in language PCL of MSC.Patran software, firstly setting the number of damaged working conditions, and determining the number of damaged working conditions according to actual problems; then inputting a central two-dimensional coordinate value of the initial damaged area; then inputting damaged area parameters: the square of half the diagonal line of the damaged area and the square of half the side length of the damaged area determine the initial damaged area of the square.
4. The optimization design method of continuous frequency constraint topology considering breakage-safety according to claim 3, wherein the third step is specifically:
inputting constraint parameters including constraint fundamental frequency size, convergence accuracy and filter radius according to the design requirement of the natural frequency performance of the structure; the size, shape and frequency constraints of the damaged area are determined, and a topological optimization model taking fundamental frequency as constraint and the minimum structural weight as an optimization target is established in consideration of damage-safety:
wherein: t represents a topological variable vector, t i Characterizing the existence or non-existence of the unit for the topological variable of the ith unit; t E N The representation topology variable t is a vector belonging to an n-dimensional European space; w represents the weight of the structure and,an initial weight for the ith cell; f (f) w (t i )、f k (t i )、f m (t i ) The weight filter function, the rigidity matrix filter function and the quality matrix filter function are respectively provided in the ICM method; lambda (lambda) l Characteristic value corresponding to the fundamental frequency representing the first damaged condition,/->Representing a characteristic value constraint value corresponding to the fundamental frequency of the first damaged working condition; l represents the number of damaged working conditions; n represents the total number of units.
5. The optimization design method of continuous frequency constraint topology considering breakage-safety according to claim 4, wherein the fourth step is specifically:
achieving secondary initial breakage in PCL main programStarting a region, sequentially damaging the structures according to the number of damage working conditions, and carrying out modal analysis on the damage condition of each structure; sequentially extracting the modal analysis result of each structural damage condition unit, wherein the modal analysis result is U respectively il And V il ，Represents the fundamental mode strain energy, + of cell i under the first failure condition>The fundamental frequency modal kinetic energy of the unit i under the first damaged condition is represented; and (3) extracting the unit modal strain energy and modal kinetic energy under a certain damage condition calculated by modal analysis, calculating the derivative of the corresponding fundamental frequency characteristic value on each design variable, namely a sensitivity analysis result, and providing mechanical performance parameters of the structural unit for establishing an explicit optimization equation.
6. The optimization design method of continuous frequency constraint topology considering breakage-safety according to claim 5, wherein the fifth step is specifically:
taking the design variables asγ k Is the power of the filter function of the unit stiffness, here gamma k =5; and based on a sensitivity system and a Taylor expansion method, carrying out explicit processing on the optimization model to obtain a corresponding quadratic programming explicit equation, wherein the method comprises the following steps:
wherein α=γ w /γ k ，γ w =2 is the power exponent of the weight filter function, γ k =5 is the power exponent of the unit stiffness filter function; d, d il And e l Is a constant coefficient of the explicit constraint inequality; v is the number of iterations;is the difference between the modal strain energy and the modal kinetic energy of the ith unit under the first damaged working condition and the ith iteration; when->When d=1, when +.>When d= -1; />Is the upper limit of the design variable;
in order to make the topological variable t take 0 or 1 as much as possible, adding a discrete condition and forming a single target with the original target according to linear weighting, and omitting a constant term after twice approximation of the target to obtain the target:
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