A reliability optimization method for flexoelectric cantilever beam structure based on double-layer nesting method

Abstract

The present invention discloses a reliability optimization method for a flexoelectric cantilever beam structure based on a double-layer nesting method, which belongs to the technical field of flexoelectric effect. The method comprises the following steps: establishing a deterministic optimization design model for a polymorphic flexoelectric cantilever beam; establishing a reliability optimization design model for a polymorphic flexoelectric cantilever beam with parameter uncertainty based on a double-layer nesting method; performing reliability optimization on a polymorphic flexoelectric cantilever beam with parameter uncertainty based on a double-layer nesting method; and solving statistical moments such as mean and variance of reliability optimization constraints based on a Monte Carlo simulation sampling method. The invention provides a reliability optimization method for a flexoelectric cantilever beam structure based on a double-layer nesting method, which can ensure that the structure has optimal structural performance while meeting specified reliability requirements, that is, the optimal flexoelectric cantilever beam structure size can make the flexoelectric effect more significant, making it more potential in the development of micro-nano devices, and having a broader application prospect in the next generation of micro-nano electromechanical system equipment.

Description

Technical Field

[0001] The present invention relates to the technical field of flexoelectric effect, and in particular to a reliability optimization method for a flexoelectric cantilever beam structure based on a double-layer nesting method. Background Art

[0002] Multi-physics coupling effects are widely present in nature. The mechanical behavior of materials is not only affected by external loads. Due to their unique internal structure, their deformation is often coupled with various physical quantities such as heat, electricity, and magnetism. The emergence of multi-field problems has brought about a large amount of interdisciplinary integration and promoted the rapid development of materials science. As a classic multi-physics coupling effect, the electromechanical coupling effect includes piezoelectric effect, electrostrictive effect, electrorheological effect, etc. It reflects the conversion relationship between the mechanical energy and electrical energy of materials and is widely present in ceramic materials, polymer films, and even biological tissues. The study of electromechanical coupling effects has promoted the design and development of various intelligent devices such as new sensors, actuators, and energy harvesters. The flexoelectric effect is a more universal electromechanical coupling effect than the piezoelectric effect. As the main structural unit for flexoelectric signal output, the flexoelectric cantilever has broad application prospects in the next generation of micro-nano electromechanical systems.

[0003] The flexoelectric effect has two notable characteristics: first, it is not restricted by crystal symmetry and exists in all dielectric materials; second, it exhibits significant size effects. However, its application in practical engineering is far less mature than that of the piezoelectric effect. Improving the flexoelectric signal output and designing high-performance flexoelectric materials are urgent challenges. Unclear parameters during the design process of flexoelectric materials or the operation of flexoelectric structures can lead to excessively low flexoelectric output voltage or potential. While uncertainty typically occurs as small numerical fluctuations around the nominal value, the coupling of multiple factors can lead to significant variations in output performance. To quantify the impact of uncertainty on the output performance of flexoelectric cantilever beams, reliability theory has gained traction. In deterministic design optimization, structural loads, dimensional parameters, and design requirements are all given in a deterministic form. While this simplifies the structural optimization process, it fails to account for the impact of uncertainty, often resulting in optimization results that are more radical than the actual results. With the continued advancement of structural reliability analysis methods, reliability-based optimization is poised to become a mainstream trend in optimization design. Therefore, the uncertainty optimization design of the flexoelectric cantilever beam structure parameters is of great significance to the subsequent output performance and development of the flexoelectric cantilever beam, and is conducive to the application and development of the flexoelectric cantilever beam structure.

[0004] In order to make up for the shortcomings of deterministic optimization design, reliability optimization design considering uncertainty has emerged one after another. Once the reliability optimization model is established, reliability optimization design becomes a mathematical problem. Since the model contains probabilistic constraints, it is impossible to directly use existing optimization algorithms for calculation. At present, all reliability optimization methods adopt a conversion strategy, that is, in the iterative process, the reliability constraints are first converted into deterministic constraints in a certain way, thereby converting the probabilistic constraint optimization problem into a conventional deterministic optimization problem, and then using conventional optimization algorithms to solve the problem. Reliability optimization methods mainly include three types of methods: (1) double-loop method (double-layer method), (2) single-loop method, and (3) decoupling method. Among them, the double-loop method uses two nested optimization loops: design optimization loop (outer layer) and reliability analysis loop (inner layer). Each time the design optimization loop needs to perform reliability constraint evaluation, the latter, namely the reliability analysis loop, needs to be called. Summary of the Invention

[0005] The purpose of the present invention is to provide a reliability optimization method for a flexural electric cantilever beam structure based on a double-layer nesting method, which solves the problem that the structure cannot be guaranteed to achieve optimal structural performance under the premise of meeting specified reliability requirements.

[0006] To achieve the above object, the present invention provides a reliability optimization method for a flexoelectric cantilever beam structure based on a double-layer nesting method, S1, establishing a deterministic optimization design model for a polymorphic flexoelectric cantilever beam;

[0007] S2. Establish a reliability optimization design model for polymorphic flexoelectric cantilever beams with parameter uncertainty based on a double-layer nesting method;

[0008] S3. Reliability optimization of polymorphic flexoelectric cantilever beam with parameter uncertainty based on double-layer nesting method;

[0009] S4. Based on the Monte Carlo simulation sampling method, the statistical moments such as mean and variance of the reliability optimization constraints are solved.

[0010] Preferably, in step S1, a polymorphic flexoelectric cantilever beam model is constructed, design variables and design parameters are determined, initial parameter design values of design variables, optimization objectives and constraint functions are selected, a deterministic optimization design model of a polymorphic flexoelectric cantilever beam is established, and design parameters (μ 31 、ɑ 33 , E), determine the design variables (B, L, h), and the deterministic optimization models of the three flexoelectric cantilever beam structures are as follows:

[0011] The deterministic optimization model of the output voltage in the open circuit state is:

[0012] Given: z = [z1, z2, z3] T

[0013] Solution: x = [x1, x2, x3] T

[0014] Optimization:

[0015] Constraints are satisfied: Q(x,z)≥74.6×10 -12 , d(x,z)≥174×10 -12

[0016] 0.016≤x1≤0.024, 0.0005≤x2≤0.0015, 0.001≤x3≤0.003,

[0017] Where, z is the design parameter, which is defined as a fixed value in the optimization section because of its small value. It is the flexoelectric coefficient, dielectric constant and Young's modulus of the flexoelectric cantilever beam; x is the design variable, which is the length, thickness and width of the flexoelectric cantilever beam. ——Optimization objective function, the output voltage of the beam in the open circuit state; Q(x,z)——charge constraint function, 74.6×10 -12 is the lower boundary of the constraint; d(x,z)——effective piezoelectric coefficient constraint function, 174×10 -12 is the lower bound of the constraint;

[0018] The deterministic optimization model of output charge under short-circuit state is:

[0019] Given: z = [z1, z2, z3] T

[0020] Solution: x = [x1, x2, x3] T

[0021] Optimization:

[0022] Satisfy the constraints: d(x,z)≥174×10 -12 0.016≤x1≤0.024, 0.0005≤x2≤0.0015, 0.001≤x3≤0.003.

[0023] Where, Q(x,z) is the optimization objective function, and the output charge of the beam in the short-circuit state;

[0024] ——voltage constraint function, 0.235 is the lower bound of the constraint;

[0025] d(x,z)——effective piezoelectric coefficient constraint function, 174×10 -12 is the lower bound of the constraint.

[0026] The deterministic optimization model of the effective piezoelectricity under short-circuit state is:

[0027] Given: z = [z1, z2, z3] T

[0028] Solution: x = [x1, x2, x3] T

[0029] Optimization: d(x,z)

[0030] Constraints are satisfied: Q(x,z)≥74.6×10 -12 ,

[0031] 0.016≤x1≤0.024, 0.0005≤x2≤0.0015, 0.001≤x3≤0.003.

[0032] Where, d(x,z) is the optimization objective function, and the effective piezoelectric coefficient of the flexoelectric cantilever beam under the short-circuit state;

[0033] ——voltage constraint function, 0.235 is the lower bound of the constraint;

[0034] Q(x,z)——charge confinement function, 74.6×10 -12 is the lower bound of the constraint.

[0035] 4. Preferably, in step S2, a reliability optimization design model of a polymorphic flexoelectric cantilever beam with parameter uncertainty is established. Under the condition that the structural reliability is not less than 99.9%, the reliability optimization design of the flexoelectric cantilever beam structure is performed. The response values of the optimization objective function and the constraint performance function obtained by the deterministic optimization in step S1, and the corresponding design variable values are used as the initial values of the mean values of the reliability optimization design variables, so that the deterministic optimization design model established in step S1 is converted into a reliability optimization design model. The reliability optimization models of the three flexoelectric cantilever beams are as follows:

[0036] The output voltage reliability optimization model considering parameter uncertainty in the open circuit state is:

[0037] Given: z = [z1, z2, z3] T

[0038] Solution:

[0039] Optimization:

[0040] Satisfy the constraints:

[0041]

[0042]

[0043]

[0044] Where, Indicates the mean value of the output voltage;

[0045] The output charge reliability optimization model considering parameter uncertainty under short-circuit state is:

[0046] Given: z = [z1, z2, z3] T

[0047] Solution:

[0048] Optimization:

[0049] Satisfy the constraints:

[0050]

[0051]

[0052]

[0053] Where, represents the mean value of the output charge;

[0054] The effective piezoelectric reliability optimization model considering parameter uncertainty under short-circuit state is:

[0055] Given: z = [z1, z2, z3] T

[0056] Solution:

[0057] Optimization:

[0058] Satisfy the constraints:

[0059]

[0060]

[0061]

[0062] Where, represents the mean value of the effective piezoelectric coefficient.

[0063] Preferably, in steps S3 and S4, based on the reliability optimization model in step S2, the reliability optimization objective function, the response value of the constraint performance function and the corresponding design variable mean μ are solved. x , μx The output response values were calculated by substituting them into the voltage output model under open circuit state, the charge output model under short circuit state and the effective piezoelectric coefficient model respectively. The statistical moments such as the mean and standard deviation of the constraint performance function were calculated by using the Monte Carlo simulation sampling method.

[0064] Preferably, in steps S3 and S4, the reliability of the flexoelectric cantilever structure is optimized by a reliability cycle of the Monte Carlo simulation sampling method, and the expressions for solving the mean and variance by the Monte Carlo simulation method are shown in formulas (4) and (5):

[0065]

[0066]

[0067] The integral formula for the probability of not meeting the constraints using the Monte Carlo simulation method is shown in the following formula (6):

[0068]

[0069] Where f(X) is the joint probability density function of the variables, F is the domain that does not meet the constraints, and P f is the probability of not satisfying the constraint.

[0070] Therefore, the present invention adopts a flexoelectric cantilever beam structure reliability optimization method based on a double-layer nesting method using the above structure, which has the following beneficial effects:

[0071] The present invention can ensure that the structure has optimal structural performance while meeting the specified reliability requirements, that is, the optimal flexoelectric cantilever beam structure size can make the flexoelectric effect more significant, making it more promising in the development of micro-nano devices and having broader application prospects in the next generation of micro-nano electromechanical system equipment.

[0072] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 This is a schematic diagram of the deformation and output signal of a flexoelectric cantilever beam according to a reliability optimization design method of a flexoelectric cantilever beam with parameter uncertainty based on a double-layer nesting method of the present invention;

[0074] Figure 2 This is a line graph of the optimization results of the open-circuit voltage model of a reliability optimization design method for a flexural electric cantilever beam structure with parameter uncertainty based on a double-layer nesting method of the present invention;

[0075] Figure 3The present invention is a method for optimizing the reliability of a flexural electric cantilever beam structure with parameter uncertainty based on a double-layer nesting method, and a frequency histogram of the output voltage model optimization results comparison;

[0076] Figure 4 This is a line graph of the optimization results of the short-circuit charge model of a reliability optimization design method for a flexural electric cantilever beam structure with parameter uncertainty based on a double-layer nesting method of the present invention;

[0077] Figure 5 The present invention is a parameter uncertainty flexoelectric cantilever beam structure reliability optimization design method based on a double-layer nesting method, and outputs a charge model optimization result comparison frequency histogram;

[0078] Figure 6 This is a line graph of the optimization results of the effective piezoelectric coefficient model of a reliability optimization design method for a flexural electric cantilever beam structure with parameter uncertainty based on a double-layer nesting method of the present invention;

[0079] Figure 7 The present invention discloses a reliability optimization design method for a flexural electric cantilever beam structure with parameter uncertainty based on a double-layer nesting method, and compares the frequency histogram of the optimization results of the effective piezoelectric coefficient model. DETAILED DESCRIPTION

[0080] The technical solution of the present invention is further described below with reference to the accompanying drawings and embodiments.

[0081] Unless otherwise defined, the technical or scientific terms used in the present invention shall have the usual meanings understood by persons of ordinary skill in the field to which the present invention belongs. The words "first", "second" and similar terms used in the present invention do not indicate any order, quantity or importance, but are only used to distinguish different components. Words such as "include" or "comprise" mean that the elements or objects preceding the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. Words such as "connect" or "connected" are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "Up", "down", "left", "right" and the like are only used to indicate relative positional relationships. When the absolute position of the object being described changes, the relative positional relationship may also change accordingly.

[0082] Example

[0083] As shown in the figure, the present invention provides a reliability optimization method for flexoelectric cantilever beam structures based on a double-layer nesting method. The output performance of flexoelectric cantilever beam structures is affected by the actual working environment. At the same time, the beam structure has unavoidable factors such as design, manufacturing, assembly and installation, as well as the dispersion of the material itself, which leads to parameter uncertainty in the beam structure. Under the combined effects of random working environment and parameter uncertainty, the output of flexoelectric cantilever beams in different states will vary greatly. Combining accumulated engineering experience and data, the structural parameters of the flexoelectric cantilever beam are used as design variables x = [x1, x2, x3] T , the performance parameters of the material are the design parameters z=[z1,z2,z3] T The distribution of design variables and design parameters of the given flexoelectric cantilever beam structure is shown in Table 1:

[0084] Table 1 Distribution of design variables and design parameters of flexoelectric cantilever beam structure

[0085]

[0086] Step S1: establishing a deterministic optimization design model for a polymorphic flexoelectric cantilever beam;

[0087] First, a polymorphic flexoelectric cantilever model was established to determine the design parameters (μ 31 、ɑ 33 , E), determine the design variables (B, L, h), refer to the actual flexoelectric cantilever structure to give the initial parameter design values as shown in Table 2, establish a deterministic optimization design model of the polymorphic flexoelectric cantilever, and perform deterministic optimization to solve the optimal variable values of the flexoelectric cantilever structure. The flexoelectric cantilever model is as follows Figure 1 As shown, the length of the piezoelectric beam is L, the width is B, and the height is h.

[0088] (1) The deterministic optimization model of the output voltage in the open circuit state is:

[0089] Given: z = [z1, z2, z3] T

[0090] Solution: x = [x1, x2, x3] T

[0091] Optimization:

[0092] Constraints are satisfied: Q(x,z)≥74.6×10 -12 , d(x,z)≥174×10 -12 0.016≤x1≤0.024, 0.0005≤x2≤0.0015, 0.001≤x3≤0.003.

[0093] Where, z is the design parameter, which is defined as a fixed value in the optimization section because of its small value. It is the flexoelectric coefficient, dielectric constant and Young's modulus of the flexoelectric cantilever beam; x is the design variable, which is the length, thickness and width of the flexoelectric cantilever beam. ——Optimization objective function, the output voltage of the beam in the open circuit state; Q(x,z)——charge constraint function, 74.6×10 -12 is the lower boundary of the constraint; d(x,z)——effective piezoelectric coefficient constraint function, 174×10 -12 is the lower bound of the constraint.

[0094] (2) The deterministic optimization model of output charge under short-circuit state is:

[0095] Given: z = [z1, z2, z3] T

[0096] Solution: x = [x1, x2, x3] T

[0097] Optimization:

[0098] Satisfy the constraints: d(x,z)≥174×10 -12 0.016≤x1≤0.024, 0.0005≤x2≤0.0015, 0.001≤x3≤0.003.

[0099] Where, Q(x,z) is the optimization objective function, and the output charge of the beam in the short-circuit state;

[0100] ——voltage constraint function, 0.235 is the lower bound of the constraint;

[0101] d(x,z)——effective piezoelectric coefficient constraint function, 174×10 -12 is the lower bound of the constraint.

[0102] (3) The deterministic optimization model of the effective piezoelectric coefficient under short-circuit state is:

[0103] Given: z = [z1, z2, z3] T

[0104] Solution: x = [x1, x2, x3] T

[0105] Optimization: d(x,z) (3)

[0106] Constraints are satisfied: Q(x,z)≥74.6×10 -12 , 0.016≤x1≤0.024, 0.0005≤x2≤0.0015, 0.001≤x3≤0.003.

[0107] Where, d(x,z) is the optimization objective function and the effective piezoelectric coefficient of the beam under the short-circuit state;

[0108] ——voltage constraint function, 0.235 is the lower bound of the constraint;

[0109] Q(x,z)——charge confinement function, 74.6×10 -12 is the lower bound of the constraint.

[0110] Table 2 Initial parameter design values

[0111]

[0112] Step S2: establishing a reliability optimization design model for a polymorphic flexoelectric cantilever beam with parameter uncertainty based on a double-layer nesting method;

[0113] Considering that the mean of the design variables can better reflect the real results, the mean of the original design variables is used and As a new design variable, a reliability optimization design model of a polymorphic flexoelectric cantilever beam considering parameter uncertainty is established. The reliability optimization models of the output voltage in the open circuit state, the output charge in the short circuit state, and the effective piezoelectric coefficient in the short circuit state are obtained as follows:

[0114] (1) The reliability optimization model of the output voltage in the open circuit state is:

[0115] Given: z = [z1, z2, z3] T

[0116] Solution:

[0117] Optimization:

[0118] Satisfy the constraints:

[0119]

[0120]

[0121]

[0122] Where, Indicates the mean value of the output voltage.

[0123] (2) The reliability optimization model of output charge under short-circuit state is:

[0124] Given: z = [z1, z2, z3] T

[0125] Solution:

[0126] Optimization:

[0127] Satisfy the constraints:

[0128]

[0129]

[0130]

[0131] Where, Represents the mean value of the output charge.

[0132] (3) The reliability optimization model of the effective piezoelectric coefficient under short-circuit state is:

[0133] Given: z = [z1, z2, z3] T

[0134] Solution:

[0135] Optimization:

[0136] Satisfy the constraints:

[0137]

[0138]

[0139]

[0140] Where, represents the mean value of the effective piezoelectric coefficient.

[0141] Step S3: performing reliability optimization on the polymorphic flexoelectric cantilever beam with parameter uncertainty based on a double-layer nesting method;

[0142] A reliability optimization design method for a flexoelectric cantilever beam structure with parameter uncertainty based on a double-layer nesting method is used to optimize the reliability of the flexoelectric cantilever beam structure under the condition that the structural reliability is not less than 99.9%. That is, by optimizing the external dimensional parameters of the flexoelectric cantilever beam structure, the structural reliability is guaranteed, and the output performance of the beam structure in different states (open circuit and short circuit) is optimized. Simple random sampling is used to complete the initial sampling of design variables and design parameters. With the design variables as independent control parameters, the sequential quadratic programming algorithm (SQP) is selected to solve the response values of the deterministic optimization objective function and the constraint performance function, as well as the corresponding design variable values. The Monte Carlo simulation method is used to calculate the mean and standard deviation of the constraint performance function, and its failure probability is calculated. The design variables obtained by deterministic optimization are used as the initial points of the mean value of the reliability optimization design variables of the polymorphic flexoelectric cantilever beam structure, and the response values of the reliability optimization objective function and the constraint performance function and the corresponding design variable mean value μ are obtained. x The generalized mathematical models of both deterministic optimization design and reliability optimization design are as follows:

[0143]

[0144]

[0145] Where Pr{·} is the probability operator, X is the random input variable, g i (X,d) is the i-th performance function, is the i-th reliability objective constraint, h j (·) is the jth deterministic constraint function, m is the number of probabilistic constraints, M is the number of deterministic constraints, and n d is the number of design variables d. In reliability optimization design, it is considered that g i The region where (X,d)≤0 is the failure domain.

[0146] The reliability optimization of the flexoelectric cantilever beam structure is carried out using a double-layer nesting method. The outer layer performs deterministic optimization to solve the optimal variable value of the structure, and the inner layer performs a reliability cycle based on the Monte Carlo sampling method. The Monte Carlo simulation method solves the mean μ and variance σ 2 The expressions of are shown in formula (9) and formula (10):

[0147]

[0148]

[0149] Where f(x) is the variable x={x1,x2,…,x n}, N is the number of samples, and n is the number of variables.

[0150] If the constraints given by the reliability optimization design model are not met, the Monte Carlo simulation sampling method is used to solve the following probability integral formula:

[0151]

[0152] Where f(X) is the joint probability density function of the variables, F is the domain that does not meet the constraints, and P f is the probability of not satisfying the constraint.

[0153] Step S4: Calculate statistical moments such as mean and variance of reliability optimization constraints based on Monte Carlo simulation sampling method.

[0154] The design variable mean μ obtained in step S3 is x Substitute them into the voltage output model under open circuit, the charge output model under short circuit and the effective piezoelectric coefficient model respectively, calculate the output response value, and use the Monte Carlo simulation sampling method to calculate the above three reliability optimization constraint performance functions (such as Q(x,z), d(x,z)) and other statistical moments such as the mean and standard deviation. Then, the output performance is optimized and analyzed, and the results are analyzed as follows:

[0155] (1) Reliability optimization analysis of output voltage in open circuit state

[0156] Combining the above theories, the length, thickness, and width of the flexure electric cantilever beam structure are calculated under the condition of electrical open circuit after deterministic optimization and reliability optimization, as well as the output voltage results, as shown in Tables 3 and 4 below:

[0157] Table 3 Output voltage optimization model calculation results

[0158]

[0159] Table 4 Output voltage determination optimization and reliability optimization results

[0160]

[0161] As shown in Tables 3 and 4, the initial voltage output of the structure is 0.2373V. After deterministic optimization, the output voltage reaches 0.2850V. After reliability optimization, the output voltage is stable at around 0.55V. At this time, the optimal length, thickness and width are 0.0182m, 0.0014m and 0.0026m respectively. Under this condition, the output voltage reaches 0.5618V. Figure 2As shown in the figure, it can be seen intuitively that the output voltage after deterministic optimization is slightly higher than the initial output voltage. After increasing the number of samples in the reliability optimization constraint, the output voltage gradually stabilizes around 0.56V.

[0162] The statistical moment indicators of the output voltage under different sizes are calculated by Monte Carlo simulation sampling method as shown in Table 5:

[0163] Table 5 Statistical moments of output voltage of flexoelectric cantilever beam structure

[0164]

[0165] It can be seen more clearly from Table 5 that the output voltage value after reliability optimization is about 2.36 times the initial voltage and 1.87 times that of deterministic optimization. Although the variance is larger, the magnitude is very small and does not affect the stability of the output of the flexoelectric cantilever beam structure. The output voltage model optimization results are compared with the frequency histogram as shown in the figure below. Figure 3 shown.

[0166] (2) Reliability optimization analysis of output charge under short-circuit state

[0167] In the short-circuit state, the output charge of the initial condition is 75pC, which can be further compared with the optimized results. The length, thickness, and width of the flexoelectric cantilever structure after deterministic optimization and reliability optimization, as well as the output charge results, are shown in Tables 6 and 7 below:

[0168] Table 6 Output charge optimization model calculation results

[0169]

[0170] Table 7 Deterministic optimization and reliability optimization results

[0171]

[0172] It can be seen from Tables 6 and 7 that the deterministic optimization results of the charge are the same as the initial results, so it is necessary to optimize the reliability of the charge model. As the sample size increases, the results of the reliability optimization stabilize at around 126pC. Through reliability optimization, the output charge of the flexure electric cantilever structure in this article is improved. By comparing the data in the table, it is found that the optimal length, thickness and width are 0.0195m, 0.0015m and 0.0022m respectively. Under this condition, the output charge reaches 126.92pC. Comparison of the optimization results of the short-circuit charge model of the reliability optimization design method of the flexure electric cantilever structure with parameter uncertainty based on the double-layer nesting method Figure 4As shown in the figure, it can be seen that the initial charge is the same as the value after deterministic optimization. After increasing the number of samples in the reliability optimization constraint, the output charge gradually stabilizes around 126pC / N.

[0173] The statistical moment indicators of the output charge at different sizes are calculated by Monte Carlo simulation sampling method as shown in Table 8:

[0174] Table 8 Statistical moments of output charge of flexoelectric cantilever structure

[0175]

[0176] It can be seen more clearly from Table 8 that the output charge value after reliability optimization is about 1.962 times the initial charge and the charge after deterministic optimization. Although the variance is larger, the magnitude is very small and does not affect the stability of the flexoelectric cantilever beam output. The output charge model optimization results are compared with the frequency histogram as shown in the figure. Figure 5 shown.

[0177] (3) Reliability optimization analysis of effective piezoelectric coefficient under short-circuit state

[0178] Under the short-circuit state, the effective piezoelectric coefficient of the initial condition is 183.21pC / N, which can be further compared with the optimized results. The length, thickness, and width of the flexoelectric cantilever structure after deterministic optimization and reliability optimization, as well as the output results of the effective piezoelectric coefficient, are compared in Tables 9 and 10 below:

[0179] Table 9 Calculation results of effective piezoelectric coefficient optimization model

[0180]

[0181] Table 10 Results of effective piezoelectric coefficient determination optimization and reliability optimization

[0182]

[0183] It can be seen from Tables 9 and 10 that the deterministic optimization results are the same as the initial results, so it is necessary to perform reliability optimization on the effective piezoelectric coefficient model. As the sample size increases, the results of the reliability optimization stabilize at around 200pC / N. Generally, when the effective piezoelectric coefficient is greater than 200pC / N, the piezoelectric performance is better. Through reliability optimization, the effective piezoelectric coefficient of the flexural electric cantilever beam structure is improved. By comparing the data in the table, it is found that the optimal length, thickness and width are 0.01978m, 0.00096m and 0.00228m respectively. Under this condition, the effective piezoelectric coefficient reaches 202.63pC / N. Comparison of the optimization results of the effective piezoelectric coefficient model of the reliability optimization design method of the flexural electric cantilever beam structure with parameter uncertainty based on the double-layer nesting method. Figure 6As shown in the figure, it can be seen that the initial effective piezoelectric coefficient is the same as the value after deterministic optimization. After increasing the number of samples in the reliability optimization constraint, the effective piezoelectric coefficient gradually stabilizes around 200pC / N.

[0184] The statistical moment index of the effective piezoelectric coefficient under different sizes is calculated by Monte Carlo simulation sampling method as shown in Table 11:

[0185] Table 11 Statistical moments of effective piezoelectric coefficients of flexoelectric cantilever beam structures

[0186]

[0187] It can be seen more clearly from Table 11 that the output charge value after reliability optimization is about 1.1 times the initial effective piezoelectric coefficient and the effective piezoelectric coefficient after deterministic optimization. Although the variance is larger, the magnitude is very small and does not affect the stability of the output of the flexoelectric cantilever structure. The comparison of the frequency histogram of the effective piezoelectric coefficient model optimization results is shown in the figure below. Figure 7 shown.

[0188] Based on the above analysis, in the open circuit state, the output voltage is initially 0.2373V, which reaches 0.2850V after deterministic optimization and 0.5618V after reliability optimization. At this time, the length, thickness and width of the flexoelectric cantilever structure are 0.0182m, 0.0014m and 0.0026m respectively; in the short circuit state, the output charge is initially 75.00pC, which is still 75.00pC after deterministic optimization and reaches 12 after reliability optimization. 6.92pC, at which point the length, thickness, and width of the flexoelectric cantilever structure are 0.0195m, 0.0015m, and 0.0022m, respectively. The effective piezoelectric coefficient is initially 183.21pC / N, which remains 183.21pC / N after deterministic optimization and reaches 202.63pC / N after reliability optimization. At this point, the length, thickness, and width of the flexoelectric cantilever structure are 0.01978m, 0.00096m, and 0.00228m, respectively.

[0189] Therefore, the present invention adopts the above-mentioned flexoelectric cantilever beam structure reliability optimization method based on the double-layer nesting method, which can ensure that the structure has optimal structural performance while meeting the specified reliability requirements. That is, the optimal flexoelectric cantilever beam structure size can make the flexoelectric effect more significant, making it more potential in the development of micro-nano devices, and has a broader application prospect in the next generation of micro-nano electromechanical system equipment.

[0190] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A reliability optimization method for flexoelectric cantilever beam structure based on double-layer nesting method, characterized in that: The following steps are involved: S1. Establish a deterministic optimization design model for a polymorphic flexoelectric cantilever beam; S2. Establish a reliability optimization design model for a polymorphic flexoelectric cantilever beam with parameter uncertainty based on a double-layer nesting method. By establishing deterministic optimization models for the output voltage in the open-circuit state, the output charge in the short-circuit state, and the effective piezoelectricity in the short-circuit state, the reliability optimization design of the flexoelectric cantilever beam structure is performed under the condition that the structural reliability is not less than 99.9%. S3. Reliability optimization of a polymorphic flexoelectric cantilever beam with parameter uncertainty is performed based on a double-layer nesting method. The reliability optimization objective function, the response value of the constraint performance function, and the corresponding design variable mean are solved through the reliability optimization model. These functions are substituted into the voltage output model in the open circuit state, the charge output model in the short circuit state, and the effective piezoelectric coefficient model, respectively, to calculate the output response value. S4. The statistical moments for solving the reliability optimization constraints based on the Monte Carlo simulation sampling method are mean, variance, standard deviation, kurtosis, and skewness.

2. The method for optimizing the reliability of a flexoelectric cantilever beam structure based on a double-layer nesting method according to claim 1, characterized in that: In step S1, a polymorphic flexoelectric cantilever model is constructed, design variables and design parameters are determined, initial parameter design values of design variables, optimization objectives and constraint functions are selected, a deterministic optimization design model of the polymorphic flexoelectric cantilever is established, and design parameters are determined ( 、 、 ), determine the design variables ( B 、 L 、 h ), the deterministic optimization models of the three flexoelectric cantilever beam structures are as follows, The deterministic optimization model of the output voltage in the open circuit state is: Given: Solution: Optimization: Satisfy the constraints: , , ， ， ， Where, - Design parameters, which are defined as fixed values in the optimization chapter due to their small values, are the flexoelectric coefficient, dielectric constant and Young's modulus of the flexoelectric cantilever beam; ——Design variables, which are the length, thickness and width of the flexoelectric cantilever beam; ——Optimize the objective function, the output voltage of the beam in the open circuit state; ——charge confinement function, is the lower bound of the constraint; ——effective piezoelectric coefficient constraint function, is the lower bound of the constraint; The deterministic optimization model of output charge under short-circuit state is: Given: Solution: Optimization: Satisfy the constraints: , , ， ， ； Where, ——Optimize the objective function, the output charge of the beam in the short-circuit state; ——voltage constraint function, is the lower bound of the constraint; ——effective piezoelectric coefficient constraint function, is the lower bound of the constraint; The deterministic optimization model of the effective piezoelectricity under short-circuit state is: Given: Solution: Optimization: Satisfy the constraints: , , , ， ； Where, ——Optimization objective function, effective piezoelectric coefficient of flexoelectric cantilever beam under short-circuit state; ——voltage constraint function, is the lower bound of the constraint; ——charge confinement function, is the lower bound of the constraint.

3. The method for optimizing the reliability of a flexoelectric cantilever beam structure based on a double-layer nesting method according to claim 2, characterized in that: In step S2, a reliability optimization design model of a polymorphic flexoelectric cantilever beam with parameter uncertainty is established. Under the condition that the structural reliability is not less than 99.9%, the reliability optimization design of the flexoelectric cantilever beam structure is performed. The response values of the optimization objective function and the constraint performance function obtained by the deterministic optimization in step S1, and the corresponding design variable values are used as the initial values of the mean values of the reliability optimization design variables, so that the deterministic optimization design model established in step S1 is converted into a reliability optimization design model. The reliability optimization models of the three flexoelectric cantilever beams are as follows: The output voltage reliability optimization model considering parameter uncertainty in the open circuit state is: Given: Solution: Optimization: (1) Satisfy the constraints: ， ， ， ， Where, Indicates the mean value of the output voltage; The output charge reliability optimization model considering parameter uncertainty under short-circuit state is: Given: Solution: Optimization: (2) Satisfy the constraints: ， ， ， ， ， Where, represents the mean value of the output charge; The effective piezoelectric reliability optimization model considering parameter uncertainty under short-circuit state is: Given: Solution: Optimization: (3) Satisfy the constraints: ， ， ， ； Where, represents the mean value of the effective piezoelectric coefficient.

4. The method for optimizing the reliability of a flexoelectric cantilever beam structure based on a double-layer nesting method according to claim 3, characterized in that: In steps S3 and S4, based on the reliability optimization model in step S2, the reliability optimization objective function, the response value of the constraint performance function and the corresponding design variable mean are solved. ,Will The output response values were substituted into the voltage output model under the open circuit state, the charge output model under the short circuit state and the effective piezoelectric coefficient model respectively, and the statistical moments of the constraint performance function, including mean, variance, standard deviation, kurtosis and skewness, were calculated using the Monte Carlo simulation sampling method.

5. The method for optimizing the reliability of a flexoelectric cantilever beam structure based on a double-layer nesting method according to claim 4, characterized in that: In steps S3 and S4, the reliability of the flexoelectric cantilever structure is optimized through the reliability cycle of the Monte Carlo simulation sampling method. The expressions for solving the mean and variance by the Monte Carlo simulation method are shown in Equations (4) and (5): （4） （5） The integral formula for the probability of not meeting the constraints using the Monte Carlo simulation method is shown in the following formula (6): （6） in, is the joint probability density function of the variables, F is the domain that does not satisfy the constraints, is the probability of not satisfying the constraint.

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