CN114091124A - Preparation method of negative thermal expansion metamaterial sandwich board based on topological optimization design - Google Patents

Preparation method of negative thermal expansion metamaterial sandwich board based on topological optimization design Download PDF

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CN114091124A
CN114091124A CN202111263692.5A CN202111263692A CN114091124A CN 114091124 A CN114091124 A CN 114091124A CN 202111263692 A CN202111263692 A CN 202111263692A CN 114091124 A CN114091124 A CN 114091124A
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叶红玲
张行
魏南
隋允康
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Abstract

A preparation method of a negative thermal expansion metamaterial sandwich plate based on topological optimization design comprises the following steps: (1) establishing an overall structural model of the negative thermal expansion metamaterial sandwich plate; (2) establishing a metamaterial base structure finite element model for filling the sandwich board; (3) inputting basic material parameters, constraint values and a metamaterial base structure finite element model; (4) calculating the values of the equivalent thermal expansion coefficient and the equivalent elastic modulus of the metamaterial and the corresponding partial derivatives; (5) establishing an explicit optimization model, and solving through linear programming; (6) judging whether the equivalent thermal expansion coefficient convergence precision is met or not, and returning to the fourth step if the equivalent thermal expansion coefficient convergence precision is not met; if so, inverting the optimal continuous topological variable until the requirement of equivalent elastic modulus is met to obtain an optimal topological structure; (7) modeling the filling metamaterial, and filling the periodic array of negative thermal expansion metamaterial between the clamping plates to form the negative thermal expansion sandwich plate. The invention has more design freedom and better performance.

Description

Preparation method of negative thermal expansion metamaterial sandwich board based on topological optimization design
Technical Field
The invention relates to a sandwich board filling structure material layout design, which is suitable for a structural concept design considering thermal expansion compensation.
Background
Topology optimization is the most challenging research field in structure optimization design, and is an innovative design method. The continuum structure optimization aims to search the optimal material distribution in a design domain under the condition of meeting the constraint, and realize the design of target performance. The topological optimization is designed according to the existence of each unit in the structure after finite element division, has more design freedom degrees compared with size optimization and shape optimization, and can break through the dependence on experience in design, thereby gaining the favor of designers.
Thermal expansion is a common phenomenon in engineering application, and large deformation is easily generated at high temperature, so that the working precision of the structure is affected. Local deformations tend to cause stress concentrations after accumulation, especially in extreme environments where thermal loads tend to be cyclic loads and therefore dangerous. In some projects, the requirement on precision is very high, so how to eliminate the influence caused by thermal expansion is very important; at present, the aerospace mostly adopts composite materials to compensate thermal deformation, but the cost is higher and the adjustable range is smaller. The microstructure of the bi-material is less researched, and particularly, after the two materials are combined, the characteristics far exceeding those of the basic material can be generated, the characteristics which are not possessed by the conventional material, such as negative thermal expansion performance, and the whole structure shrinks when the temperature rises. Filling the metamaterial between the double-layer plates can obtain the sandwich plate with negative thermal expansion for compensating thermal expansion deformation.
At present, most metamaterials filled in the negative thermal expansion sandwich plate are designed through experience or intuition, the degree of freedom of design can be greatly improved by adopting a topological optimization method to design the metamaterials, and meanwhile, the metamaterial is not limited by thinking and positioning, can be used for designing metamaterial structures in different forms, and can meet certain constraint. Therefore, metamaterial topology optimization considering thermal deformation of the dual material is necessary.
The invention provides a negative thermal expansion metamaterial sandwich board based on topological optimization design. The negative thermal expansion metamaterial of the sandwich layer is designed based on a homogenization theory and a topological optimization method. The topological optimization model takes the equivalent thermal expansion coefficient as a target and the equivalent elastic modulus as a constraint, explicit processing and solving are carried out on the target and the constraint based on Taylor expansion, and the threshold value is selected by utilizing the dichotomy in the inversion process, so that the blindness of threshold value selection is eliminated. Simulation proves that the sandwich board filled with the metamaterial can realize negative thermal expansion characteristics.
Disclosure of Invention
Aiming at the topological optimization design problem of the thermal expansion coefficient of the filled metamaterial in the sandwich plate, compared with the design mode depending on experience, the design method has more design freedom degrees and meets more engineering requirements. The development of metamaterials is an important direction of material science, and more extraordinary performance designs are provided for engineering. The invention provides a design method of a negative thermal expansion sandwich plate with more universality; designing a sandwich metamaterial structure by means of a topological optimization method, and taking an equivalent thermal expansion coefficient and an equivalent elastic modulus in a homogenization theory as targets and constraints; gradually solving by means of a linear programming method to obtain a final structure of the metamaterial; periodic boundary conditions are added in the design process, so that the sandwich board has better performance. Therefore, the topological optimization design method effectively improves the design efficiency and saves the design cost.
The optimization process is shown in fig. 1. The concrete solution is as follows:
firstly, establishing a negative thermal expansion metamaterial sandwich plate overall structure model
The sandwich plate structure is divided into three layers, wherein the upper layer and the lower layer are made of common materials, the middle layer is filled with arrayed negative thermal expansion metamaterials, the upper plate and the lower plate are used for protecting the sandwich layer and are used for being connected with other structures, and the sandwich layer provides negative thermal expansion deformation.
Secondly, establishing a metamaterial base structure finite element model for filling the sandwich board;
based on the MATLAB software platform, an initial geometric model of the structure is built up by means of matrices, each value of which represents a topological variable of one element, represented in MATLAB by topological variable vectors r and s containing N elements, the values of which determine the material properties of the finite element, N usually being taken to be 2500 for representing 50 × 50 elements. Two topological variables r ═ r1,r2,r3,…rN]And s ═ s1,s2,s3,…sN]。riAnd siThe topology design variables representing the ith cell determine the presence and type of the cell. Wherein r isi(i ═ 1,2,3 … N) and siThe values of (i ═ 1,2,3 … N) are all between 0 and 1. r isi0 or r i1 indicates that the ith unit is material-free or material-containing; s i0 or s i1 represents that the ith unit is a material I or a material II; the row and column size of the matrix represents the unit number of the structure in the direction, and the finite element model is identified by reading the matrix size;
inputting basic material parameters, constraint values and a metamaterial base structure finite element model;
inputting the modulus of elasticity E of the base material at the corresponding position of the input end in the function(1)、E(2)Poisson ratio v(1)、v(2)And coefficient of thermal expansion alpha(1)、α(2)The superscript (1) indicates that the material attribute corresponds to the material number one, and the superscript (2) indicates that the material attribute corresponds to the material number two. And identifying the material property of each unit through r and s input in the second step, wherein the formula is as follows:
Figure BDA0003322471020000021
Figure BDA0003322471020000022
Figure BDA0003322471020000023
wherein EiIs the modulus of elasticity, v, of the i-th elementiIs the elastic modulus of the i-th cell, αiIs the coefficient of thermal expansion of the ith cell. r isiAnd siDetermining the lower limit of the equivalent elastic modulus of the sandwich metamaterial according to actual requirements by representing the topological variable of the ith unitEThe lower limit is typically 1% to 10% of the modulus of elasticity of the weaker material in the base material. And inputting the finite element model of the metamaterial-based structure constructed in the second step in a matrix form.
Fourthly, calling an MATLAB program to carry out homogenization analysis, and calculating the equivalent thermal expansion coefficient, the equivalent elastic modulus and the corresponding partial derivative;
and assembling a basic unit stiffness matrix according to the material attributes and the metamaterial-based structure read in the third step by using a Matlab program, and numbering the units and the nodes according to the geometric information. Under the assumption of periodicity, the unit stiffness matrix is combined with the numbering matrix using kron's command to assemble the overall stiffness matrix in the periodic array. And the sparse command is used for storing the total rigid matrix into a sparse matrix, so that the storage space is saved and the solving speed is accelerated. And applying total load, applying forces in all directions to each unit, solving displacement through a static equilibrium equation, applying local load to a single unit, and solving a local displacement field. Value of global strain field
Figure BDA0003322471020000024
Figure BDA0003322471020000031
And the value of the local strain field
Figure BDA0003322471020000032
The superscript 0 indicates that the strain is calculated for the local cell. Applying unit heat load to obtain the value of the thermal strain field
Figure BDA0003322471020000033
The superscript a indicates that the strain is due to thermal loading. Since the two elastic moduli respectively represent the macroscopicityAnd microscopic, so the index of the elasticity tensor is denoted by ijkl and pqrs, respectively. And ij and kl represent two independent dimensions of the macro, pq and rs represent two independent dimensions of the micro, and the equivalent thermal expansion coefficient and the equivalent elastic modulus are calculated by multiplying two by two
Figure BDA0003322471020000034
Figure BDA0003322471020000035
Figure BDA0003322471020000036
Wherein
Figure BDA0003322471020000037
Is the equivalent modulus of elasticity of the structure;
Figure BDA0003322471020000038
is the equivalent thermal stress tensor;
Figure BDA0003322471020000039
is the equivalent coefficient of thermal expansion. And Y is the volume of the negative thermal expansion metamaterial, and the modulus Y of the unit volume is adopted for calculation when the calculation is carried out, considering that the volume is always positive during calculation. EpqrsIs the program basis EiThe calculated elasticity tensor matrix.
Based on the adjoint method, calculating the objective function pair variable riSensitivity of (2)
Figure BDA00033224710200000310
Figure BDA00033224710200000311
Figure BDA00033224710200000312
Calculating objective function pair variable siSensitivity of (2)
Figure BDA00033224710200000313
Figure BDA00033224710200000314
Figure BDA00033224710200000315
Fifthly, establishing an explicit optimization model and solving through linear programming;
firstly, establishing an optimization model for designing the negative thermal expansion metamaterial, and taking the minimum thermal expansion coefficient as a target
Figure BDA00033224710200000316
Where r and s are topological variable vectors representing material properties. ENIndicating that r and s are within the space of N positive integer design variables. In order to ensure the orthogonality of the structure, the objective function alpha (r, s) is set as the thermal expansion coefficient in the horizontal direction
Figure BDA00033224710200000317
And coefficient of thermal expansion in the vertical direction
Figure BDA00033224710200000318
A sum of the formula
Figure BDA0003322471020000041
Secondly, the model is subjected to explicit expression based on Taylor expansion, and the target can be expressed as the Taylor expansion according to the equivalent thermal expansion coefficient, the equivalent elastic modulus value and the corresponding partial derivative calculated in the fourth step
Figure BDA0003322471020000042
Where the superscript of the band indicates a fixed value for the corresponding function before the symbol is optimized. Non-viable cells*Indicating that the previous partial derivative is the value before optimization.
For linear programming solution, it is written in the form of matrix multiplication
Figure BDA0003322471020000043
Where T is the transposed symbol. Taylor expansion of the constraint is also performed
Figure BDA0003322471020000044
Then the constraint can be written as
Figure BDA0003322471020000045
Therefore, a standard linear programming model is obtained as follows
Figure BDA0003322471020000046
Wherein [ F]Is the first order partial derivative of the objective function,
Figure BDA0003322471020000047
[A]then the partial derivative of the constraints in the linear programming model,
Figure BDA0003322471020000048
b is a constraint value
Figure BDA0003322471020000049
The superscript in the row indicates a fixed value for the corresponding function before the symbol is optimized. r and s represent values of the topological variables before optimization.
Figure BDA00033224710200000410
The equivalent elastic modulus of the negative thermal expansion metamaterial before optimization is shown.
Sixthly, judging whether the thermal expansion coefficient convergence precision is met or not, and returning to the fourth step if the thermal expansion coefficient convergence precision is not met; if so, inverting the optimal continuous topological variable until the requirement of equivalent elastic modulus is met to obtain an optimal topological structure;
and judging whether the convergence meets the convergence precision of the thermal expansion coefficient, if not, updating the finite element model for the next iteration, if so, utilizing a dichotomy theory, reducing a search area by a successive dichotomy threshold space method, and searching an optimal inversion threshold value to invert the intermediate topological variable so as to obtain an optimal topological structure.
Seventhly, modeling the filling metamaterial, and filling the periodic array of negative thermal expansion metamaterial between the clamping plates to form a negative thermal expansion sandwich plate;
by extracting the characteristics of the optimized model, a smooth negative thermal expansion metamaterial model is established, the metamaterial is filled into the plate to obtain the negative thermal expansion sandwich plate, and the negative thermal expansion performance of the sandwich plate is verified in finite element software.
Compared with the prior art, the invention has the advantages that:
(1) the adjustable thermal expansion metamaterial structure is optimally designed by adopting a topological optimization method, the parameters of the structure optimization are not optimized, and the adjustable thermal expansion metamaterial has more design freedom and better performance;
(2) the metamaterial design based on the homogenization method can consider the influence of the arrangement mode of the periodic arrays of the metamaterial in the sandwich plate on the negative thermal expansion performance of the sandwich plate, and has important influence on the design of the sandwich plate.
Drawings
FIG. 1 is a flow chart of a method for topological optimization design of a tunable thermal expansion metamaterial.
FIG. 2 is a base structure of a metamaterial.
FIG. 3 is an optimized block diagram of a metamaterial.
FIG. 4 is an iterative history plot of the coefficient of thermal expansion of the metamaterial with the equivalent elastic modulus.
FIG. 5 is a negative expansion sandwich plate structure filled with metamaterials.
FIG. 6 is a finite element simulation result of a negative thermal expansion sandwich plate
Detailed Description
The following describes the detailed implementation steps of the present invention with reference to a topological optimization design algorithm of the tunable thermal expansion metamaterial.
The method comprises the following steps of firstly, establishing a negative thermal expansion metamaterial sandwich plate overall structure model, wherein the upper layer and the lower layer are made of structural steel, and the negative thermal expansion metamaterial is made of two basic materials of aluminum and the structural steel.
Secondly, establishing a finite element model of the metamaterial base structure of which the base structure is as shown in FIG. 2, dividing the base structure into 50 × 50 units, wherein the units are four-node rectangular units which become 1 × 1;
thirdly, defining the elastic modulus of the material as E(1)＝7,E(2)21, Poisson's ratio is 3, coefficient of thermal expansion α(1)＝5，α (2)1, the elastic modulus constraint is 0.3, and the above information is input into the MATLAB function;
and fourthly, calling an MATLAB program to calculate the equivalent elastic modulus, the equivalent thermal expansion coefficient and the corresponding sensitivity, arranging the information into a standard linear programming format, inputting the standard linear programming format into a linear programming program to solve, and obtaining an optimization result.
Fifthly, if the optimization result meets the convergence condition, outputting the optimal continuous result; and if the optimization result does not meet the convergence condition, modifying the unit topological variables, returning to the fourth step, and continuing to perform homogenization analysis and solution until the convergence condition is met.
And sixthly, obtaining the optimal topological structure based on the inversion of the intermediate topological variables by the dichotomy.
The optimal results for the thermal expansion coefficients are shown in table 1. From fig. 3, which is an optimal topology, and fig. 4, which is an iteration history, it can be seen that the iteration process satisfies the constraint condition and is stable and convergent. Therefore, the metamaterial design method based on topological optimization has effectiveness and feasibility.
TABLE 1 optimal topology results
Figure BDA0003322471020000061
And seventhly, modeling the filling metamaterial, and filling the periodic array of negative thermal expansion metamaterial between the clamping plates to form the negative thermal expansion sandwich plate shown in the figure 5, wherein the upper layer and the lower layer are made of common materials, and the middle layer is made of a metamaterial structure. The finite element results in FIG. 6 show that the boundary mean deformation is-4.5884 × 10-7m, showing that the structure has negative thermal expansion properties.

Claims (8)

1. A preparation method of a negative thermal expansion metamaterial sandwich plate based on topological optimization design is characterized by comprising the following steps:
firstly, establishing an overall structural model of a negative thermal expansion metamaterial sandwich plate;
secondly, establishing a metamaterial base structure finite element model for filling the sandwich board;
inputting basic material parameters, constraint values and a metamaterial base structure finite element model;
fourthly, calling an MATLAB program to carry out homogenization analysis, and calculating the equivalent thermal expansion coefficient and the equivalent elastic modulus of the metamaterial and the corresponding partial derivative;
fifthly, establishing an explicit optimization model and solving through linear programming;
sixthly, judging whether the equivalent thermal expansion coefficient convergence precision is met or not, and returning to the fourth step if the equivalent thermal expansion coefficient convergence precision is not met; if so, inverting the optimal continuous topological variable until the requirement of equivalent elastic modulus is met to obtain an optimal topological structure;
and seventhly, modeling the filling metamaterial, and filling the periodic array of negative thermal expansion metamaterial between the clamping plates to form the negative thermal expansion sandwich plate.
2. The production method according to claim 1,
establishing an overall structural model of the negative thermal expansion metamaterial sandwich plate; the sandwich plate consists of an upper solid steel plate layer, a lower solid steel plate layer and a middle negative thermal expansion metamaterial, wherein the negative thermal expansion metamaterial consists of two basic materials, namely aluminum and structural steel.
3. The method for preparing a metamaterial structure finite element model for sandwich panel filling according to the claim 1, wherein the finite element model of the metamaterial structure for sandwich panel filling in the second step is realized by the following steps:
establishing a geometric model of an initial base structure, dividing the geometric model into a finite element model with N units, and representing the finite element model in MATLAB by topological variable vectors r and s containing N elements, wherein the values of the topological variables determine the material properties of the finite element, and N is generally 2500 used for representing 50 × 50 units; two topological variables r ═ r1,r2,r3,…rN]And s ═ s1,s2,s3,…sN]；riAnd siA topology design variable representing the ith cell, determining the presence and type of the cell; wherein r isi(i ═ 1,2,3 … N) and si(i ═ 1,2,3 … N) each has a value between 0 and 1; r isi0 or ri1 indicates that the ith unit is material-free or material-containing; si0 or si1 indicates that the ith unit is material one or material two, respectively.
4. The method for preparing a metamaterial as claimed in claim 1, wherein the third step is to input the base material parameters, the constraint values and the finite element model of the metamaterial base structure, and the method is implemented by:
inputting the modulus of elasticity E of the base material in a function(1)、E(2)Poisson ratio v(1)、v(2)And coefficient of thermal expansion alpha(1)、α(2)The actual value of (1) indicates that the material attribute corresponds to the first material, and (1) is labeled2) Indicating that the material attribute corresponds to material II; determining the lower limit of the equivalent elastic modulus of the sandwich metamaterial according to actual requirementsEThe lower limit is usually 1 to 10 percent of the elastic modulus of a weaker material in a base material; inputting the finite element model of the metamaterial base structure constructed in the second step in a matrix form; the material parameters were assigned to the structure by the following formula:
Ei＝ri 3[E(1)(1-si 3)+E(2)si 3](i＝1,2,3,…,N) (1)
νi＝ri 3[ν(1)(1-si 3)+ν(2)si 3](i＝1,2,3,…,N) (2)
αi＝ri 3[α(1)(1-si 3)+α(2)si 3](i＝1,2,3,…,N) (3)
wherein Ei、viAnd alphaiThe elastic modulus, poisson's ratio and thermal expansion coefficient of the ith cell, respectively.
5. The method according to claim 1, wherein the step four calls a MATLAB program to perform a homogenization analysis, and calculates the equivalent thermal expansion coefficient and the equivalent elastic modulus and the corresponding partial derivatives thereof, and the specific implementation process is as follows:
reading the initial structure by the MATLAB program, automatically applying periodic boundary conditions and loads to the whole structure by the program, calculating and extracting the value of the output global strain field by the program
Figure FDA0003322471010000021
And the value of the local strain field
Figure FDA0003322471010000022
Superscript 0 indicates that the strain is calculated for the local cell; applying unit heat load to obtain the value of the thermal strain field
Figure FDA0003322471010000023
The superscript a indicates that the strain is due to thermal loading; since the two elastic moduli represent the macro and the micro, respectively, the subscripts of the elastic tensor are denoted by ijkl and pqrs, respectively; and ij and kl represent two independent dimensions of the macro, pq and rs represent two independent dimensions of the micro, and the equivalent thermal expansion coefficient and the equivalent elastic modulus are calculated by multiplying two by two
Figure FDA0003322471010000024
Figure FDA0003322471010000025
Figure FDA0003322471010000026
Wherein
Figure FDA0003322471010000027
The equivalent elastic modulus of the structure is shown, and the superscript H represents that the equivalent elastic modulus is obtained by calculation through a homogenization method;
Figure FDA0003322471010000028
is the equivalent thermal stress tensor;
Figure FDA0003322471010000029
is the equivalent coefficient of thermal expansion; y is the volume of the negative thermal expansion metamaterial, and the model | Y | of the unit volume is adopted for calculation when the calculation is participated in considering that the total volume is a positive value during the calculation; epqrsIs the program basis EiV and viA calculated elasticity tensor matrix; based on the adjoint method, calculating the first-order partial derivative of the objective function to the variable to obtain the variable riSensitivity of (2)
Figure FDA00033224710100000210
And to variable siSensitivity of (2)
Figure FDA00033224710100000211
6. The preparation method according to claim 1, wherein the establishing of the explicit optimization model in the fifth step is solved by linear programming, and the specific implementation process is as follows:
establishing a negative thermal expansion metamaterial optimization model taking a thermal expansion coefficient as a target,
Figure FDA00033224710100000212
where r and s are topological variable vectors representing material properties; eNRepresenting an N-dimensional design space corresponding to the N units in the second step; the standard function α (r, s) is set to the coefficient of thermal expansion α in the horizontal direction1 HAnd coefficient of thermal expansion in the vertical direction
Figure FDA00033224710100000213
Summing; the taylor expansion based on the values of the equivalent thermal expansion coefficient and the equivalent elastic modulus calculated in the fourth step and the corresponding partial derivatives can represent the target as
Figure FDA0003322471010000031
Wherein the superscript of band represents a fixed value of the corresponding function before the symbol is optimized; non-viable cells*Indicating that the previous partial derivative is the value before optimization.
Therefore, a standard linear programming model is obtained as follows
Figure FDA0003322471010000032
Wherein [ F]Is the first order partial derivative of the objective function,
Figure FDA0003322471010000033
[A]then the partial derivative of the constraints in the linear programming model,
Figure FDA0003322471010000034
b is a constraint value
Figure FDA0003322471010000035
T is a transposed symbol; r and s represent values of topological variables before optimization;
Figure FDA0003322471010000036
the equivalent elastic modulus of the negative thermal expansion metamaterial before optimization is shown.
7. The method according to claim 1, wherein the sixth step is performed to determine whether or not the equivalent thermal expansion coefficient convergence accuracy is satisfied, and if not, the fourth step is performed; if so, inverting the optimal continuous topological variable until the requirement of equivalent elastic modulus is met to obtain an optimal topological structure:
and judging whether the convergence meets the convergence precision of the equivalent thermal expansion coefficient, if not, updating the finite element model for the next iteration, and if so, dispersing the intermediate topological variable by adopting a dichotomy optimal inversion threshold selection method to further obtain the optimal topological metamaterial structure meeting the requirement.
8. The method of claim 1, wherein the seventh step models the filler metamaterial, fills a periodic array of negative thermal expansion metamaterial between the sandwich plates to form a negative thermal expansion sandwich plate, and verifies the negative thermal expansion sandwich plate by finite element calculation.
CN202111263692.5A 2021-10-26 2021-10-26 Preparation method of negative thermal expansion metamaterial sandwich board based on topological optimization design Pending CN114091124A (en)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114582445A (en) * 2022-03-10 2022-06-03 湖南大学 Metamaterial optimization method, system and equipment

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114582445A (en) * 2022-03-10 2022-06-03 湖南大学 Metamaterial optimization method, system and equipment

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