Discrete continuous optimization variable interpolation calculation methodTechnical Field
The application belongs to the field of anisotropic material direction and topology combined optimization, and particularly relates to a discrete continuous optimization variable interpolation calculation method.
Background
When the anisotropic material direction and topology combined optimization is carried out, the problem of continuous material direction angle optimization, namely material direction conversion matrix calculation, is involved, and the problem easily falls into a local optimal solution in the material direction angle interval optimization process. The research finds that the problems can be solved through a discrete interval technology, but the related design variables of related research are more, and the influence on the topology optimization efficiency is larger. The application provides a new discrete continuous interpolation mode for the material direction and topology collaborative optimization design problem by means of the concept of weight function in the finite element method.
Disclosure of Invention
The application aims to provide a discrete continuous optimization variable interpolation calculation method, which aims to solve the problems that the prior discrete related design variables are more and the influence on topology optimization efficiency is larger.
The technical scheme of the application is that the discrete continuous optimization variable interpolation calculation method comprises the following steps:
Determining the period of the material direction conversion matrix, and angle the material direction according to the period of the material direction conversion matrixThe optimization of (a) is converted into four intervals, and a material direction conversion matrix is established;
introducing a shape function as a weight function, and calculating the weight functions of the rigidity of the four interval materials;
defining the initial rigidity of the anisotropic material, and then determining the material rigidity of four sections by combining a material direction conversion matrix;
calculating the rigidity of the macrostructure according to the material rigidity and the weight function of the four sections;
Acquiring the relation between the optimized real material direction and the discrete variable and the continuous variable based on the continuous variable;
And setting an objective function based on strain energy, and iteratively updating discrete variables, continuous variables and material density in the objective function with minimum strain energy of the objective function as targets to obtain the optimal material direction and topology, thereby completing interpolation calculation.
Preferably, the period of the material direction conversion matrix isAngle of material directionThe optimizing conversion of (a) is performed into four sections, each section has the size ofThe material direction conversion matrix is obtained as follows:
。
Preferably, the weight function satisfiesWeight function of four interval material rigidities、、AndThe method comprises the following steps of:
;
;
;
;
in the formula,AndAre discrete variables.
Preferably, the anisotropic material has an initial stiffnessThe definition is as follows:
;
in the formula,、、、、Respectively the elastic constants of the anisotropic materials in different directions;
Material stiffness in four zones、、AndThe method comprises the following steps of:
;
Where T is the material direction conversion matrix and TT is the transpose of the material direction conversion matrix.
Preferably, the macrostructure stiffnessThe method comprises the following steps:
;
in the formula,As a continuous variable, the number of the variables,Material information of each section is superimposed in the form of a continuous weight function,The method comprises the following steps:, is of the range length of (2)。
Preferably, the optimized real material direction and the optimized real material direction、AndThe relation between the two is:
;
in the formula,As a continuous variable, the number of the variables,To optimize the actual material direction.
Preferably, a material interpolation calculation method is adopted to establish a topological optimization mathematical model and an optimization column, and a gradient descent method is utilized to iteratively update discrete variables in an objective functionContinuous variableAnd material density。
The discrete continuous optimization variable interpolation calculation method provides a new discrete continuous interpolation mode by means of the concept of the weight function in the finite element method when the anisotropic material direction and topology are optimized in a combined mode, so that design variables can be reduced, optimization efficiency can be improved, and the material direction can be prevented from falling into a local optimal solution for the collaborative optimization design of the composite material structure material direction and topology.
Drawings
In order to more clearly illustrate the technical solution provided by the present application, the following description will briefly refer to the accompanying drawings. It will be apparent that the figures described below are merely some embodiments of the application.
FIG. 1 is a flow chart of anisotropic material direction and topology collaborative optimization;
FIG. 2 is a general flow chart of the present application;
FIG. 3 is a schematic diagram of the discrete optimization of the material direction angle in the basic coordinate system according to the present application;
FIG. 4 is a schematic view of the application effect of the L-beam verification example of the present application;
Fig. 5 is a schematic diagram of the application effect of the simple beam verification example of the present application.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
A discrete continuous optimization variable interpolation calculation method is shown in figure 1, which is an anisotropic material direction and topology collaborative optimization flow chart, the method is mainly applied to interpolation calculation, and discrete continuous operation of the material direction is carried out in the link, wherein the discrete continuous operation comprises conversion of the material direction and combination of the material direction and an optimal design variable.
The anisotropic material has all or part of physical, chemical and other properties in different directions, and has different performance values.
As shown in fig. 2, the method comprises the following steps:
step S100, determining the period of the material direction conversion matrix, and angling the material direction according to the period of the material direction conversion matrixThe optimization of (a) is converted into four intervals, and a material direction conversion matrix is established;
The material direction conversion matrix is defined as follows, and it can be seen that the period of the material direction conversion matrix isIf the material direction angle optimization is performed based on a gradient algorithm in the topology optimization, a local optimal solution can be easily obtained. As shown in figure 3 ①-④, the material is oriented at an angleThe optimizing conversion of (a) is performed in four sections, and each section is only of the sizeThis greatly reduces the risk that the dashed line direction in fig. 3 is the direction of the orthotropic material.
The material direction conversion matrix T is:
。
step S200, introducing a shape function as a weight function, and calculating the weight functions of the rigidity of the four interval materials.
Preferably, the weight function satisfiesWeight function of four interval material rigidities、、AndThe method comprises the following steps of:
;
;
;
;
in the formula,AndAre discrete variables.
Step S300, initial rigidity definition of the anisotropic material is carried out, and then material rigidity of four sections is determined by combining a material direction conversion matrix.
Initial stiffness of anisotropic materialThe definition is as follows:
;
in the formula,、、、、The elastic constants in different directions in the anisotropic material are respectively calculated by engineering constant Young modulus and Poisson's ratio.
Material stiffness in four zones、、AndThe method comprises the following steps of:
;
Where T is the material direction conversion matrix and TT is the transpose of the material direction conversion matrix.
And step S400, calculating the rigidity of the macrostructure according to the material rigidity and the weight function of the four sections.
Macrostructural stiffnessThe method comprises the following steps:
;
in the formula,As a continuous variable, the number of the variables,Material information of each section is superimposed in the form of a continuous weight function,The method comprises the following steps:, is of the range length of (2)。
Step S500, obtaining the optimized real material direction and the optimized real material direction based on the continuous variable、AndThe relation between the two is:
;
in the formula,To optimize the actual material direction.
Step S600, obtaining the macroscopic structural rigidity of four sections, and combining the real material direction and the real material direction、AndThe relation between the materials is established by adopting a material interpolation calculation method to establish a topological optimization mathematical model and an optimization list, and then an objective function based on strain energy is setAs an objective functionWith minimum strain energy as target, iteratively updating discrete variables in an objective function using a gradient descent methodContinuous variableAnd material densityAnd obtaining the optimal material direction and the topology shape, and finishing interpolation calculation. As shown in fig. 4 and 5, wherein fig. 4 is an L-beam verification example application effect, and fig. 5 is a simple beam verification example application effect.
In summary, when the anisotropic material direction and topology combined optimization is performed, a new discrete continuous interpolation mode is provided by means of the concept of the weight function in the finite element method, so that on one hand, design variables can be reduced, optimization efficiency is improved, on the other hand, the material direction can be prevented from falling into a local optimal solution, and the method is used for collaborative optimization design of the composite material structure material direction and topology.
Finally, it should be noted that in the drawings of the disclosed embodiments of the present invention, only the structures related to the embodiments of the present invention are referred to, and other structures may refer to the general design, so that the same embodiment and different embodiments of the present invention may be combined with each other without conflict;
finally, the foregoing description of the preferred embodiment of the invention is provided for the purpose of illustration only, and is not intended to limit the invention to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention.