A topology optimization method for thermally driven micro-nano compliant mechanisms considering scale effects

Abstract

The present invention provides a topology optimization method for a thermally actuated micro-nano compliant mechanism that considers scale effects. The method comprises defining design conditions for the thermally actuated micro-nano compliant mechanism and setting material property indicators; establishing a finite element analysis model of the thermally actuated micro-nano compliant mechanism; obtaining a structural displacement response; establishing a topology optimization model for the thermally actuated micro-nano compliant mechanism; calculating an optimization objective function and constraint sensitivity information; smoothing the sensitivity information; solving the optimization problem of the thermally actuated micro-nano compliant mechanism using a moving progressive optimization algorithm, and determining whether convergence conditions of the moving progressive algorithm are met. If so, outputting an optimal topology configuration of the thermally actuated micro-nano compliant mechanism that considers scale effects. The thermally actuated micro-nano compliant mechanism obtained by topology optimization in the present invention exhibits a significant scale effect. As a scale parameter related to the characteristic length increases, the topology configuration of the thermally actuated micro-nano compliant mechanism changes, indicating that the scale effect becomes more significant.

Description

Technical Field

[0001] The present invention relates to the technical field of compliant mechanism optimization design, and in particular to a topology optimization method for a thermally driven micro-nano compliant mechanism considering scale effects. Background Art

[0002] Thermally actuated compliant mechanisms utilize thermal expansion and deformation to generate motion and force at the output end. Compared to other actuation methods, thermally actuated compliant mechanisms can output larger forces and displacements and offer advantages such as ease of control and integration. With the continuous development and application of micro- and nanotechnology, the scale of thermal actuators is becoming increasingly smaller. However, the topological optimization design of large-scale thermally actuated compliant mechanisms based on traditional dielectric theory fails to accurately reflect the mechanical properties at the microscale. Therefore, it is necessary to consider scale effects when designing thermally actuated compliant mechanisms.

[0003] To account for scale effects, most studies consider incorporating higher-order elastic theories, with couple stress theory being the most widely used. Although higher-order elastic theories have been applied to many microstructure designs, their application to topology optimization remains largely unexplored. Furthermore, existing research primarily focuses on using compliance as the objective function. The application of couple stress theory to the topology optimization design of thermally actuated micro- and nano-compliant mechanisms is urgently needed.

[0004] In summary, in the existing technology, the results obtained by using traditional medium theory for topological optimization design of large-scale thermally driven compliant mechanisms cannot truly reflect the mechanical properties at the microscale. Summary of the Invention

[0005] Based on this, the purpose of the present invention is to provide a topology optimization method for thermally driven micro-nano compliant mechanisms taking into account scale effects, so as to address the deficiencies in the above-mentioned prior art.

[0006] The present invention provides a topology optimization method for a thermally driven micro-nano compliant mechanism taking into account scale effects, the method comprising:

[0007] Defining the design conditions of the thermally driven micro-nano compliant mechanism and setting various material property indicators of the thermally driven micro-nano compliant mechanism;

[0008] A finite element analysis model of the thermally driven micro-nano compliant mechanism is established based on modified couple stress and the introduction of non-classical equivalent stress;

[0009] The relationship between the design conditions and the material property indicators is represented by a penalty model, an equivalent nodal thermal load expression is derived based on the relationship, and a thermal-solid coupled finite element equilibrium equation is further solved to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism;

[0010] Taking the maximization of the output displacement of the thermally driven micro-nano compliant mechanism as the objective function and the volume of the thermally driven micro-nano compliant mechanism as the constraint, a mathematical model for topological optimization of the thermally driven micro-nano compliant mechanism is established;

[0011] Calculating the output end displacement of the optimization objective function of the thermally driven micro-nano compliant mechanism and the sensitivity information of the constraint volume according to the mathematical model of the topology optimization of the thermally driven micro-nano compliant mechanism;

[0012] Using sensitivity filtering technology to correct the optimization objective function and the sensitivity information, and using Heaviside mapping function to smooth the sensitivity information;

[0013] A moving asymptotic optimization algorithm is used to solve the optimization problem of the thermally driven micro-nano compliant mechanism, and it is determined whether the convergence condition of the moving asymptotic optimization algorithm is met. If so, the optimal topological configuration of the thermally driven micro-nano compliant mechanism is output.

[0014] Compared with the prior art, the beneficial effects of the present invention are: based on the modified couple stress theory, a size-dependent non-classical equivalent stress is introduced to establish a finite element analysis model of the thermally driven micro-nano compliant mechanism, and the structural displacement response of the thermally driven micro-nano compliant mechanism is obtained by solving the thermo-solid coupling finite element equilibrium equation. The optimization objective function and the sensitivity information of the constraints are calculated by the topology optimization model of the thermally driven micro-nano compliant mechanism considering the scale effect, and then the convergence condition is judged whether it is met by the moving asymptotic algorithm, so that the optimal topological configuration of the thermally driven micro-nano compliant mechanism can be obtained, and the scale effect of the thermally driven micro-nano compliant mechanism can be effectively displayed.

[0015] Furthermore, the steps of defining the design conditions of the thermally driven micro-nano compliant mechanism and setting various material property indicators of the thermally driven micro-nano compliant mechanism include:

[0016] defining the design domain, boundary conditions, and applied loads of the thermally driven micro-nano compliant mechanism;

[0017] The material elastic modulus, Poisson's ratio, thermal expansion coefficient, number of finite elements, initial value of element density, sensitivity filtering radius and scale parameters of the thermally driven micro-nano compliant mechanism are set.

[0018] Furthermore, the step of establishing a finite element analysis model of the thermally driven micro-nano compliant mechanism based on the modified couple stress and the introduction of non-classical equivalent stress includes:

[0019] Based on the modified couple stress theory and the classical medium theory, the scale effect of the thermally driven micro-nano compliant mechanism at the microscale is described;

[0020] Non-classical equivalent stress is introduced, and the scale effect of the thermally driven micro-nano compliant mechanism is characterized by a scale parameter to obtain a finite element analysis model of the thermally driven micro-nano compliant mechanism.

[0021] Furthermore, the step of representing the relationship between the design conditions and the material property indicators based on a penalty model, deriving an equivalent node thermal load expression based on the relationship, and further solving the thermo-solid coupled finite element equilibrium equation to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism includes:

[0022] An improved solid isotropic material penalty model is used to represent the relationship between the material elastic modulus and the initial value of the element density in the material property index, and a node load expression is derived based on the relationship;

[0023] A finite element equilibrium equation is solved according to the node load expression, and a finite element analysis is performed on the structure of the thermally driven micro-nano compliant mechanism based on the finite element equilibrium equation to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism.

[0024] Furthermore, the mathematical model of the thermally driven micro-nano compliant mechanism topology optimization is expressed as:

[0025] ;

[0026] Where, represents the output displacement of the mechanism, represents a constant, where , Indicates the number of units, Indicates the The design variable unit density of the unit, 、 、 They represent displacement vector one, displacement vector two, and displacement vector three respectively. represents the global stiffness matrix of the design domain, represents the equivalent nodal thermal load vector, Indicates the Unit elastic modulus Elastic modulus of solid material The relationship function between represents the temperature difference between a certain moment and the initial moment, represents the linear expansion coefficient of the material, represents the elastic modulus, Indicates the Poisson's ratio of the material, with a superscript represents the transpose symbol, is the transformation matrix that transforms the element stiffness matrix into the global stiffness matrix, is the design domain, is the strain-displacement matrix of classical mechanics, is the strain-displacement matrix of the couple stress theory, is the elastic matrix of the classical theoretical unit, is the elastic matrix of the couple stress theory element, is the constraint function, is the unit volume, is the initial volume of the mechanism, is the volume constraint of the mechanism, Represents the cell design domain.

[0027] Furthermore, the step of calculating the output end displacement of the optimization objective function of the thermally driven micro-nano compliant mechanism and the sensitivity information of the constraint volume according to the mathematical model of the topology optimization of the thermally driven micro-nano compliant mechanism includes:

[0028] Calculating the output end displacement of the thermally driven micro-nano compliant mechanism according to the mutual strain energy of the mechanism, and calculating the structural volume of the thermally driven micro-nano compliant mechanism by unit density to obtain the optimization target and volume constraint;

[0029] The objective function output displacement maximization of the optimization target and the sensitivity information of the volume constraint are calculated based on the mathematical model of the topology optimization of the thermally driven micro-nano compliant mechanism.

[0030] Furthermore, the steps of correcting the optimization objective function and the sensitivity information using sensitivity filtering technology and smoothing the sensitivity information using a Heaviside mapping function include:

[0031] Using sensitivity filtering technology to correct the optimization objective function and the sensitivity information;

[0032] The sensitivity information is smoothed using a Heaviside mapping function so that the cell density is concentrated toward both ends of a preset interval, thereby reducing the occurrence of intermediate grayscale cells in the topological configuration.

[0033] Furthermore, after the step of determining whether the convergence condition of the moving asymptotic optimization algorithm is satisfied, the method further includes:

[0034] If the convergence condition of the moving asymptotic optimization algorithm is not satisfied, the finite element analysis model of the thermally driven micro-nano compliant mechanism is repeatedly executed based on the modified couple stress and the introduction of non-classical equivalent stress; the relationship between the design conditions and the material property indicators is represented based on the penalty model, the equivalent node thermal load expression is derived according to the relationship, and the thermal-solid coupling finite element equilibrium equation is further solved to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism; the output displacement maximization of the thermally driven micro-nano compliant mechanism is used as the objective function, and the volume of the thermally driven micro-nano compliant mechanism is used as the constraint to establish the thermally driven micro-nano compliant mechanism. A mathematical model for topology optimization of a thermally driven micro-nano compliant mechanism is provided; the output end displacement of the optimization objective function of the thermally driven micro-nano compliant mechanism and the sensitivity information of the constraint volume are calculated according to the mathematical model for topology optimization of the thermally driven micro-nano compliant mechanism; the optimization objective function and the sensitivity information are corrected by using sensitivity filtering technology, and the sensitivity information is smoothed by using a Heaviside mapping function; the optimization problem of the thermally driven micro-nano compliant mechanism is solved by using a moving asymptotic optimization algorithm, and whether the convergence condition of the moving asymptotic optimization algorithm is met is judged until the optimal topological configuration of the thermally driven micro-nano compliant mechanism is output. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 This is a flow chart of a topology optimization method for a thermally driven micro-nano compliant mechanism considering scale effects in an embodiment of the present invention;

[0036] Figure 2 Schematic diagram of the design domain, loads, and boundary conditions of a thermally driven micro-nano compliant mechanism according to an embodiment of the present invention;

[0037] Figure 3 The topological optimization results of the thermally driven micro-nano compliant mechanism in the embodiment of the present invention are based on classical theory without considering the scale effect;

[0038] Figure 4 The modified couple stress theory is used for the thermally driven micro-nano compliant mechanism in the embodiment of the present invention, and the topology optimization result considering the scale effect is obtained.

[0039] The following specific embodiments will further illustrate the present invention in conjunction with the above-mentioned drawings. DETAILED DESCRIPTION

[0040] See also Figure 1 , which shows a topology optimization method for a thermally driven micro-nano compliant mechanism considering scale effects in an embodiment of the present invention, the method includes steps S1 to S7:

[0041] S1, defining the design conditions of the thermally driven micro-nano compliant mechanism and setting various material property indicators of the thermally driven micro-nano compliant mechanism;

[0042] Specifically, the step S1 includes steps S11 to S12:

[0043] S11, defining the design domain, boundary conditions, and applied load of the thermally driven micro-nano compliant mechanism;

[0044] S12, setting the material elastic modulus, Poisson's ratio, thermal expansion coefficient, number of finite elements, initial value of element density, sensitivity filtering radius and scale parameters of the thermally driven micro-nano compliant mechanism.

[0045] S2, establishing a finite element analysis model of the thermally driven micro-nano compliant mechanism based on modified couple stress and introduction of non-classical equivalent stress;

[0046] Specifically, step S2 includes steps S21 to S22:

[0047] S21, based on the modified couple stress theory and the classical medium theory to describe the scale effect of the thermally driven micro-nano compliant mechanism at the microscale;

[0048] It can be understood that the modified couple stress theory is a high-order elastic theory derived from the principle of minimum potential energy. According to this theory, in a two-dimensional problem with isotropic plane stress, the displacement field includes the element displacement vector and the unit microscopic rotation displacement vector , is the unit microscopic rotation displacement vector, 、 They represent the unit displacement vector in the x-direction and the unit displacement vector in the y-direction respectively. The components of the Cauchy strain tensor are , the components of the symmetric curvature tensor are Correspondingly, the components of the Cauchy stress tensor are , the components of the modified couple stress tensor are In summary, the stress and strain components of a two-dimensional element are composed of five components. The stress-strain matrix considering couple stress is:

[0049] ;

[0050] ;

[0051] Where, is the stress matrix, They are respectively the stress component in the x direction, the stress component in the y direction, the shear stress component, the couple stress component in the x direction, and the couple stress component in the y direction. is the strain matrix, They are the x-direction strain component, y-direction strain component, shear strain component, x-direction curvature component, and y-direction curvature component, respectively. The superscript T indicates the transposition symbol.

[0052] The stress-strain relationship is:

[0053] ;

[0054] Where, is the stress matrix, is the elasticity matrix, is the strain matrix.

[0055] In this embodiment, only two-dimensional design problems are considered, and the microscopic rotation angle is constrained to be equal to the macroscopic rotation angle, that is, = , so the rotation can be calculated from the displacement as follows:

[0056] ;

[0057] Where, is the unit microscopic rotation displacement vector, Represents the unit displacement vector For independent variables Partial derivatives, unit displacement vectors For independent variables The partial derivative of .

[0058] This gives the two components of the curvature:

[0059] ;

[0060] Where, is the curvature component in the x direction, is the curvature component in the y direction, is the unit displacement vector For independent variables The second-order partial derivative of is the unit displacement vector For independent variables and independent variables The second-order partial derivative of is the unit displacement vector For independent variables and independent variables The second-order partial derivative of is the unit displacement vector For independent variables The second-order partial derivative of .

[0061] S22, introducing a non-classical equivalent stress and characterizing the strength of the scale effect of the thermally driven micro-nano compliant mechanism with a scale parameter to obtain a finite element analysis model of the thermally driven micro-nano compliant mechanism;

[0062] It can be understood that the strain-displacement relationship for isotropic plane stress problems based on the modified couple stress theory is:

[0063] ;

[0064] ;

[0065] Where, is the strain matrix, They are the x-direction strain component, y-direction strain component, shear strain component, x-direction curvature component, and y-direction curvature component, respectively. is the curvature matrix, are the two components of the unit displacement vector, is the unit displacement vector, , is the unit microscopic rotation displacement vector, is the symbol of partial derivative.

[0066] The corresponding constitutive equation is:

[0067] ;

[0068] ;

[0069] Where, is the stress matrix, They are respectively the stress component in the x direction, the stress component in the y direction, the shear stress component, the couple stress component in the x direction, and the couple stress component in the y direction. is the strain matrix, They are the x-direction strain component, y-direction strain component, shear strain component, x-direction curvature component, and y-direction curvature component, respectively. is the curvature matrix, is the couple stress matrix, the superscript T represents the transposition symbol, G represents the shear modulus, and is given by Given, is the elastic modulus of the material, is the Poisson's ratio of the material.

[0070] It should be considered that the constitutive relation introduces an additional material parameter , called the material characteristic length parameter. This parameter is an important material parameter that reflects the size effect and does not exist in classical elastic theory. In actual calculations, The value of is determined experimentally. The total constitutive matrix From the classical elastic constitutive matrix and the couple stress constitutive matrix Composition, the relationship is:

[0071] ;

[0072] Where, is the total constitutive matrix, is the classical elastic constitutive matrix, is the couple stress constitutive matrix.

[0073] S3, representing the relationship between the design condition and the material property index based on a penalty model, deriving an equivalent nodal thermal load expression based on the relationship, and further solving a thermal-solid coupled finite element equilibrium equation to obtain a structural displacement response of the thermally driven micro-nano compliant mechanism;

[0074] Specifically, step S3 includes steps S31 to S32:

[0075] S31, using an improved solid isotropic material penalty model to represent the relationship between the material elastic modulus and the initial value of the element density in the material property index, and deriving a node load expression based on the relationship;

[0076] It can be understood that the SIMP material interpolation model is used to describe the relationship between material properties and design variables x, and the expression is:

[0077] ;

[0078] Where, Indicates the Unit elastic modulus Elastic modulus of solid material The relationship function between , the specific expression is: , represents the unit density design variable, represents the initial unit density design variable, represents the minimum unit density design variable, represents the penalty coefficient. In this embodiment, The value of is 3;

[0079] Thermal strain occurs when the temperature changes , whose expression is:

[0080] ;

[0081] Where, represents thermal strain, represents the linear expansion coefficient of the material, represents the initial temperature, Indicates the steady-state temperature of the compliant mechanism, with the superscript represents the transpose symbol;

[0082] When considering uniform temperature changes, an imaginary temperature load applied to each node can be equivalent to a node thermal load, which is equivalent to a node force. Its expression is:

[0083] ;

[0084] Where, represents the temperature load related to the element density, represents the strain matrix, represents the elasticity matrix, represents the design domain, represents thermal strain, Represents the thickness of the thermally driven compliant mechanism.

[0085] The equivalent nodal thermal load of the element is expressed as:

[0086] ;

[0087] Where, represents the unit equivalent node thermal load, represents the strain matrix, represents the elasticity matrix, represents the linear expansion coefficient of the material, represents the initial temperature, represents the steady-state temperature of the compliant mechanism, represents the elastic modulus, represents the temperature difference between a certain moment and the initial moment, Represents the Poisson's ratio of the material, and the superscript T represents the transposition symbol.

[0088] S32, solving a finite element equilibrium equation according to the node load expression, and performing a finite element analysis on the structure of the thermally actuated micro-nano compliant mechanism based on the finite element equilibrium equation to obtain a structural displacement response of the thermally actuated micro-nano compliant mechanism;

[0089] It can be understood that in this embodiment, in the structural field, the finite element equilibrium equation is:

[0090] ;

[0091]

[0092] Where, represents the load array, represents the overall stiffness matrix of the structure, represents the overall nodal displacement array, is the transformation matrix that transforms the element stiffness matrix into the global stiffness matrix, is the design domain, represents the unit design domain, is the strain-displacement matrix of classical mechanics, is the strain-displacement matrix of the couple stress theory, is the elastic matrix of the classical theoretical unit, is the elastic matrix of the couple stress theory element, with the superscript Represents the transpose symbol.

[0093] S4, establishing a mathematical model for topology optimization of the thermally driven micro-nano compliant mechanism with maximizing the output displacement of the thermally driven micro-nano compliant mechanism as an objective function and with the volume of the thermally driven micro-nano compliant mechanism as a constraint;

[0094] In this embodiment, the mathematical model for topology optimization of the thermally driven micro-nano compliant mechanism is expressed as:

[0095] ;

[0096] Where, represents the output displacement of the mechanism, represents a constant, where , Indicates the number of units, Indicates the The design variable unit density of the unit, 、 、 They represent displacement vector one, displacement vector two, and displacement vector three respectively. represents the global stiffness matrix of the design domain, represents the equivalent nodal thermal load vector, Indicates the Unit elastic modulus Elastic modulus of solid material The relationship function between represents the temperature difference between a certain moment and the initial moment, represents the linear expansion coefficient of the material, represents the elastic modulus, Indicates the Poisson's ratio of the material, with a superscript represents the transpose symbol, is the transformation matrix that transforms the element stiffness matrix into the global stiffness matrix, is the design domain, is the strain-displacement matrix of classical mechanics, is the strain-displacement matrix of the couple stress theory, is the elastic matrix of the classical theoretical unit, is the elastic matrix of the couple stress theory element, is the constraint function, is the unit volume, is the initial volume of the mechanism, is the volume constraint of the mechanism, Represents the cell design domain.

[0097] S5, calculating the output end displacement of the optimization objective function of the thermally driven micro-nano compliant mechanism and the sensitivity information of the constraint volume according to the mathematical model of the topology optimization of the thermally driven micro-nano compliant mechanism;

[0098] Specifically, step S5 includes steps S51 to S52:

[0099] S51, calculating the output end displacement of the thermally driven micro-nano compliant mechanism based on the mutual strain energy of the mechanism, and calculating the structural volume of the thermally driven micro-nano compliant mechanism by unit density to obtain an optimization target and volume constraint;

[0100] S52, calculating the objective function output displacement maximization of the optimization target and the sensitivity information of the volume constraint according to the mathematical model of the topology optimization of the thermally driven micro-nano compliant mechanism;

[0101] It can be understood that in this embodiment, in the structural thermal field of the compliant mechanism, the displacement field of the structure It can be calculated by the following formula:

[0102] ;

[0103] Where, represents the heat load array, represents the total stiffness matrix of the modified couple stress theory, represents the overall node displacement array, and taking the derivative of both sides of the formula we can get:

[0104] ;

[0105] represents the temperature load related to the unit density. By substituting and taking the derivative, we can get:

[0106] ;

[0107] The adjoint matrix equation is:

[0108] ;

[0109] Where, represents a unit vector, represents the total stiffness matrix of the modified couple stress theory, represents the adjoint vector of the displacement, represents the elastic modulus of the solid material, represents the elastic modulus of the empty phase material;

[0110] Therefore, it can be deduced that:

[0111] ;

[0112] Therefore, the upper limit of displacement can be obtained respectively and displacement lower limit Sensitivity to design variables and ;

[0113] Finally, the sensitivity of the output displacement of the thermally driven micro-nano compliant mechanism to the design variables can be obtained:

[0114] ;

[0115] Where, 、 Represent the upper limit of displacement Sensitivity to design variables, lower limit of displacement Sensitivity to design variables. and displacement lower limit The sensitivity of the objective function can be obtained.

[0116] The sensitivity of the constraint, i.e., the volume fraction, to the element density is obtained as:

[0117] ;

[0118] Where, Indicates the unit volume fraction, represents the cell density, represents the symbol of partial derivative, Represents a volume constraint.

[0119] S6, using sensitivity filtering technology to modify the optimization objective function and the sensitivity information, and using a Heaviside mapping function to smooth the sensitivity information;

[0120] Specifically, step S6 includes steps S61 to S62:

[0121] S61, using sensitivity filtering technology to correct the optimization objective function and the sensitivity information;

[0122] S62, using a Heaviside mapping function to smooth the sensitivity information so that the cell density is concentrated at both ends of a preset range to reduce the appearance of intermediate grayscale cells in the topological configuration;

[0123] It can be understood that the formula of the Heaviside mapping function is:

[0124] ;

[0125] Where, represents the processed design variables, represents the threshold parameter, Represents the parameter that controls the smoothness of the change, Indicates the relative density of the material, Represents the symbol of the hyperbolic tangent function.

[0126] It should be explained that, in this embodiment, the preset interval range is 0-1. By concentrating the cell density toward the ends of 0-1, the occurrence of intermediate grayscale cells in the topological configuration can be reduced.

[0127] S7, using a moving asymptotic optimization algorithm to solve the optimization problem of the thermally actuated micro-nano compliant mechanism, and determining whether a convergence condition of the moving asymptotic optimization algorithm is satisfied, and if so, outputting an optimal topological configuration of the thermally actuated micro-nano compliant mechanism;

[0128] Furthermore, if the convergence condition of the mobile asymptotic optimization algorithm is not satisfied, the finite element analysis model of the thermally driven micro-nano compliant mechanism is repeatedly executed based on the modified couple stress and the introduction of non-classical equivalent stress; the relationship between the design conditions and the material property indicators is represented based on the penalty model, the equivalent node thermal load expression is derived according to the relationship, and the thermal-solid coupling finite element equilibrium equation is further solved to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism; the output displacement maximization of the thermally driven micro-nano compliant mechanism is used as the objective function, and the volume of the thermally driven micro-nano compliant mechanism is used as the constraint to establish a thermally driven micro-nano compliant mechanism. A mathematical model for topology optimization of a micro-nano compliant mechanism; calculating the output end displacement of the optimization objective function of the thermally driven micro-nano compliant mechanism and the sensitivity information of the constraint volume based on the mathematical model for topology optimization of the thermally driven micro-nano compliant mechanism; using sensitivity filtering technology to correct the optimization objective function and the sensitivity information, and using a Heaviside mapping function to smooth the sensitivity information; using a moving asymptotic optimization algorithm to solve the optimization problem of the thermally driven micro-nano compliant mechanism, and judging whether the convergence condition of the moving asymptotic optimization algorithm is met, until the optimal topological configuration of the thermally driven micro-nano compliant mechanism is output.

[0129] In order to further verify the effectiveness of the topology optimization method of thermally driven micro-nano compliant mechanisms considering scale effects, a thermally driven micro-nano compliant mechanism is taken as an example to explain.

[0130] The design domain, boundary conditions, input and output of thermally driven micro-nano compliant mechanisms are as follows: Figure 2 As shown, in the entire compliant mechanism, the upper and lower left sides are fixed, and the output displacement is at the midpoint of the right side. Due to the symmetry of the mechanism, only the lower half is used for design analysis. Similarly, it is discretized into 20,000 planar quadrilateral elements for calculation. When the temperature rises, the left side of the mechanism deforms and squeezes, causing the output position structure to move to the right, thus achieving a thermal actuation effect.

[0131] In this example, the compliant mechanism size is , the elastic modulus of the thermally driven material At 100 GPa, Poisson's ratio , thermal expansion coefficient , the volume is set to 0.15, the elastic modulus of the empty phase material for , minimum filter radius Set to 4 to output the position and spring stiffness. for In the Heaviside mapping function of the thermal actuator, the iterative control parameter The number of steps to double the value is 50 steps, The initial value is 1, the maximum value is 16, and the parameter The value is 0.5. In order to prevent the material from converging too quickly to both sides at the beginning of the iteration, set the penalty coefficient The initial value is 0.5, the maximum value is 3, and the penalty coefficient is 30 steps per iteration. The value of is increased by 0.5. The iteration stop condition is set as follows: the total number of steps exceeds 500, or the change in cell density is less than 0.001.

[0132] Without considering the scale effect, the thermally driven micro-nano compliant mechanism is obtained by topology optimization using classical medium theory, such as Figure 3 As shown, the output displacement of the thermally driven micro-nano compliant mechanism is 1.5684, and the volume constraint is 0.150.

[0133] Consider the scale effect and set the scale parameter Under the condition of , the thermally driven micro-nano compliant mechanism is obtained by topology optimization using modified couple stress theory, such as Figure 4 As shown, the output displacement of the thermally driven micro-nano compliant mechanism is 1.6037, the volume constraint is 0.150, and the mechanism has an obvious scale effect.

[0134] In summary, the topology optimization method of the thermally driven micro-nano compliant mechanism considering the scale effect in the above-mentioned embodiment of the present invention is based on the modified couple stress theory, introduces a size-related non-classical equivalent stress, establishes a finite element analysis model of the thermally driven micro-nano compliant mechanism, and obtains the structural displacement response of the thermally driven micro-nano compliant mechanism by solving the thermo-solid coupling finite element equilibrium equation. The optimization objective function and the sensitivity information of the constraints are calculated by the topology optimization model of the thermally driven micro-nano compliant mechanism considering the scale effect, and then the convergence condition is judged whether it is met by the moving asymptotic algorithm, so that the optimal topological configuration of the thermally driven micro-nano compliant mechanism can be obtained, and the scale effect of the thermally driven micro-nano compliant mechanism can be effectively displayed.

[0135] Throughout this specification, reference to terms such as "one embodiment," "some embodiments," "examples," "specific examples," or "some examples" means that a specific feature, structure, material, or characteristic described in conjunction with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

[0136] The above-described embodiments merely illustrate several implementations of the present invention, and while their descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that a person skilled in the art would be able to make numerous variations and improvements without departing from the spirit of the present invention, all of which fall within the scope of protection of the present invention. Therefore, the scope of protection of the present invention shall be determined by the appended claims.

Claims

1. A topology optimization method for thermally driven micro-nano compliant mechanisms considering scale effects, characterized in that: The method comprises: Defining design conditions for a thermally driven micro-nano compliant mechanism and setting various material property indicators of the thermally driven micro-nano compliant mechanism; A finite element analysis model of the thermally driven micro-nano compliant mechanism is established based on modified couple stress and the introduction of non-classical equivalent stress; The relationship between the design conditions and the material property indicators is represented by a penalty model, an equivalent nodal thermal load expression is derived based on the relationship, and a thermal-solid coupled finite element equilibrium equation is further solved to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism; Taking the maximization of the output displacement of the thermally driven micro-nano compliant mechanism as the objective function and the volume of the thermally driven micro-nano compliant mechanism as the constraint, a mathematical model for the topology optimization of the thermally driven micro-nano compliant mechanism is established. The expression of the mathematical model for the topology optimization of the thermally driven micro-nano compliant mechanism is: ； Where, represents the output displacement of the mechanism, represents a constant, where , Indicates the number of units, Indicates the The design variable unit density of the unit, 、 、 They represent displacement vector one, displacement vector two, and displacement vector three respectively. represents the global stiffness matrix of the design domain, represents the equivalent nodal thermal load vector, Indicates the Unit elastic modulus Elastic modulus of solid material The relationship function between represents the temperature difference between a certain moment and the initial moment, represents the linear expansion coefficient of the material, represents the elastic modulus, Indicates the Poisson's ratio of the material, with a superscript represents the transpose symbol, is the transformation matrix that transforms the element stiffness matrix into the global stiffness matrix, is the design domain, is the strain-displacement matrix of classical mechanics, is the strain-displacement matrix of the couple stress theory, is the elastic matrix of the classical theoretical unit, is the elastic matrix of the couple stress theory element, is the constraint function, is the unit volume, is the initial volume of the mechanism, is the volume constraint of the mechanism, represents the unit design domain; Calculating the output end displacement of the optimization objective function of the thermally driven micro-nano compliant mechanism and the sensitivity information of the constraint volume according to the mathematical model of the topology optimization of the thermally driven micro-nano compliant mechanism; Using sensitivity filtering technology to correct the optimization objective function and the sensitivity information, and using Heaviside mapping function to smooth the sensitivity information; A moving asymptotic optimization algorithm is used to solve the optimization problem of the thermally driven micro-nano compliant mechanism, and it is determined whether the convergence condition of the moving asymptotic optimization algorithm is met. If so, the optimal topological configuration of the thermally driven micro-nano compliant mechanism is output.

2. The method for topology optimization of thermally driven micro-nano compliant mechanisms considering scale effects according to claim 1, characterized in that: The steps of defining the design conditions of the thermally driven micro-nano compliant mechanism and setting various material property indicators of the thermally driven micro-nano compliant mechanism include: defining the design domain, boundary conditions, and applied loads of the thermally driven micro-nano compliant mechanism; The material elastic modulus, Poisson's ratio, thermal expansion coefficient, number of finite elements, initial value of element density, sensitivity filtering radius and scale parameters of the thermally driven micro-nano compliant mechanism are set.

3. The method for topology optimization of thermally driven micro-nano compliant mechanisms considering scale effects according to claim 1, characterized in that: The steps of establishing a finite element analysis model of the thermally driven micro-nano compliant mechanism based on modified couple stress and introduction of non-classical equivalent stress include: Based on the modified couple stress theory and the classical medium theory, the scale effect of the thermally driven micro-nano compliant mechanism at the microscale is described; Non-classical equivalent stress is introduced, and the scale effect of the thermally driven micro-nano compliant mechanism is characterized by a scale parameter to obtain a finite element analysis model of the thermally driven micro-nano compliant mechanism.

4. The method for topology optimization of thermally driven micro-nano compliant mechanisms considering scale effects according to claim 1, characterized in that: The steps of expressing the relationship between the design conditions and the material property indicators based on a penalty model, deriving an equivalent node thermal load expression based on the relationship, and further solving the thermal-solid coupling finite element equilibrium equation to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism include: An improved solid isotropic material penalty model is used to represent the relationship between the material elastic modulus and the initial value of the element density in the material property index, and a node load expression is derived based on the relationship; A finite element equilibrium equation is solved according to the node load expression, and a finite element analysis is performed on the structure of the thermally driven micro-nano compliant mechanism based on the finite element equilibrium equation to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism.

5. The method for topology optimization of thermally driven micro-nano compliant mechanisms considering scale effects according to claim 1, characterized in that: The step of calculating the output end displacement of the optimization objective function of the thermally driven micro-nano compliant mechanism and the sensitivity information of the constraint volume according to the mathematical model of the topology optimization of the thermally driven micro-nano compliant mechanism comprises: Calculating the output end displacement of the thermally driven micro-nano compliant mechanism according to the mutual strain energy of the mechanism, and calculating the structural volume of the thermally driven micro-nano compliant mechanism by unit density to obtain the optimization target and volume constraint; The objective function output displacement maximization of the optimization target and the sensitivity information of the volume constraint are calculated based on the mathematical model of the topology optimization of the thermally driven micro-nano compliant mechanism.

6. The method for topology optimization of thermally driven micro-nano compliant mechanisms considering scale effects according to claim 1, characterized in that: The steps of correcting the optimization objective function and the sensitivity information using the sensitivity filtering technology and smoothing the sensitivity information using the Heaviside mapping function include: Using sensitivity filtering technology to correct the optimization objective function and the sensitivity information; The sensitivity information is smoothed using a Heaviside mapping function so that the cell density is concentrated toward both ends of a preset interval, thereby reducing the occurrence of intermediate grayscale cells in the topological configuration.

7. The method for topology optimization of thermally driven micro-nano compliant mechanisms considering scale effects according to claim 1, characterized in that: After the step of determining whether the convergence condition of the moving asymptotic optimization algorithm is satisfied, the method further includes: If the convergence condition of the moving asymptotic optimization algorithm is not satisfied, the finite element analysis model of the thermally driven micro-nano compliant mechanism is repeatedly executed based on the modified couple stress and the introduction of non-classical equivalent stress; the relationship between the design conditions and the material property indicators is represented based on the penalty model, the equivalent node thermal load expression is derived according to the relationship, and the thermal-solid coupling finite element equilibrium equation is further solved to obtain the structural displacement response of the thermally driven micro-nano compliant mechanism; the output displacement maximization of the thermally driven micro-nano compliant mechanism is used as the objective function, and the volume of the thermally driven micro-nano compliant mechanism is used as the constraint to establish the thermally driven micro-nano compliant mechanism. A mathematical model for topology optimization of a thermally driven micro-nano compliant mechanism is provided; the output end displacement of the optimization objective function of the thermally driven micro-nano compliant mechanism and the sensitivity information of the constraint volume are calculated according to the mathematical model for topology optimization of the thermally driven micro-nano compliant mechanism; the optimization objective function and the sensitivity information are corrected by using sensitivity filtering technology, and the sensitivity information is smoothed by using a Heaviside mapping function; the optimization problem of the thermally driven micro-nano compliant mechanism is solved by using a moving asymptotic optimization algorithm, and whether the convergence condition of the moving asymptotic optimization algorithm is met is judged until the optimal topological configuration of the thermally driven micro-nano compliant mechanism is output.

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