A method for determining self-adaptive Rayleigh damping in fluid-structure coupling simulation of an impulse water turbine

Abstract

The present application belongs to the technical field of hydraulic machinery engineering, and particularly relates to a method for determining self-adaptive Rayleigh damping in fluid-structure coupling simulation of an impulse water turbine, which comprises the following steps: setting a target frequency interval and a target damping ratio before simulation; establishing a Rayleigh damping model based on a mass proportionality coefficient and a stiffness proportionality coefficient; constructing a composite optimization objective function containing an overall frequency band error term, a key frequency point accuracy term and a high frequency penalty term; under the condition of a fixed weight factor λ , taking partial derivatives of the objective function J ( α , β ) with respect to α and β respectively and setting them to zero to obtain an analytical solution of the parameters; then, self-adaptively adjusting λ according to the comparison result of the key frequency point error and a preset threshold; and finally, inputting the optimized α and β into the fluid-structure coupling simulation model of the impulse water turbine. The method takes into account both the overall frequency band average error and the key frequency point local accuracy, realizes self-adaptive optimization of the damping parameters through analytical solution, and significantly improves the stability, convergence and result accuracy of the simulation calculation.

Description

Technical Field

[0001] This invention belongs to the field of hydraulic machinery engineering technology, and in particular relates to an adaptive Rayleigh damping determination method for fluid-structure interaction simulation of an impulse turbine. Background Technology

[0002] In numerical calculations and fluid-structure interaction analyses of large hydraulic machinery such as impulse turbines, the Rayleigh damping model is often used to simulate the damping characteristics of the structure. This model, through the combined effect of mass and stiffness proportionality factors, can approximately describe the damping behavior of the structure within a certain frequency range. However, the selection of damping parameters lacks a unified standard, and their rationality directly affects the convergence of the simulation and the accuracy of the results.

[0003] Existing methods for determining damping parameters mainly fall into two categories: The first is the "two-frequency matching method," which involves matching the damping ratio to the target value at two specified frequency points to obtain the parameters. This method is simple to implement, but it only guarantees the fitting accuracy at two frequency points, with larger errors at other frequencies in the interval. The second is the "least squares integral method," which solves for the parameters by reducing the squared deviation of the damping ratio from the target value over the entire frequency interval. This method is superior to the two-point method in terms of overall average error, but its fixed weight distribution often results in large deviations at the ends of the interval or at key modal frequencies. Furthermore, it lacks a mechanism to suppress the risk of high-frequency amplification, leading to distortion of simulation results at critical locations.

[0004] In fluid-structure interaction simulations of impulse turbines, the nozzle-flush interaction excites several specific frequency components. The damping ratio accuracy of these frequency components is crucial for residual force prediction and operational stability. While existing methods can reduce overall error, they cannot simultaneously address the local accuracy constraints at critical frequencies, often resulting in a situation where "the overall error is small, but the deviation at critical frequencies is significant," directly impacting the assessment of the unit's operating status and stability. Therefore, there is an urgent need for a Rayleigh damping optimization method that can ensure a small average error across the entire frequency band while maintaining accuracy at critical frequencies and suppressing high-frequency distortion. Summary of the Invention

[0005] The purpose of this invention is to propose an adaptive Rayleigh damping determination method for fluid-structure interaction simulation of impulse turbines. This method constructs a composite objective function consisting of an overall frequency band error term, a key frequency accuracy term, and a penalty term, and introduces an adaptive weight adjustment mechanism to achieve a balance between minimizing the average error across the entire frequency band and constraining the accuracy at key frequencies. This overcomes the problems of key frequency distortion, limitations in full-band fitting, and insufficient numerical stability in existing methods, providing a more stable and reliable method for determining damping parameters in fluid-structure interaction simulation of impulse turbines.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] The present invention provides a method for determining adaptive Rayleigh damping in fluid-structure interaction simulation of an impulse turbine, comprising:

[0008] Step 1: Before simulation, set the target frequency range [ω1, ω2] and the target damping ratio η. The target damping ratio is a basic structural property and is usually a constant.

[0009] Step 2: Establish a Rayleigh damping model based on the mass proportionality coefficient α and the stiffness proportionality coefficient β;

[0010] Step 3: Construct a composite optimization objective function J(α, β) consisting of the following three parts to characterize the overall deviation between Rayleigh damping and the target damping ratio: (1) the overall frequency band error term of Rayleigh damping and the target damping ratio within the target frequency range [ω1, ω2]; (2) the key frequency accuracy term of Rayleigh damping and the target damping ratio η at key frequencies within at least three target frequency ranges [ω1, ω2]; (3) the stiffness scaling factor penalty term used to suppress high-frequency amplification.

[0011] Step 4: Adjust the weighting factor λ in the composite optimization objective function so that the result of the key frequency point accuracy term is less than or equal to the error threshold ε. e This is used to determine the value of the weighting factor λ;

[0012] Step 5: Based on the determination of λ in Step 4, take the partial derivatives of the composite optimization objective function J(α, β) with respect to α and β respectively and set them to zero to obtain the optimized mass proportionality coefficient α and stiffness proportionality coefficient β.

[0013] Step 6: Input the optimized mass scaling factor α and stiffness scaling factor β obtained in Step 5 into the fluid-structure interaction simulation of the impulse turbine to improve the calculation stability and result accuracy.

[0014] As a further preferred option, in step 2, the mathematical expression of the Rayleigh damping ratio model is:

[0015]

[0016] Where ω is the frequency, α is the mass proportionality coefficient, and β is the stiffness proportionality coefficient, and α and β are non-negative to avoid numerical instability.

[0017] As a further preferred option, in step 3, the mathematical expression of the composite optimization objective is:

[0018]

[0019] Where W(ω) is the frequency band weighting function, which can be a constant of 1 to indicate equal weighting; ω1 and ω2 are the endpoint frequencies of the target frequency range; dω is the frequency differential element; m represents the number of key frequency points; ω kλ is the frequency corresponding to the key frequency point, η is the target damping ratio, and γ is the penalty term coefficient used to suppress high-frequency amplification effects.

[0020] As a further preferred embodiment, the solution of the key frequency point accuracy term includes at least the values ​​at the following three frequency points: the two endpoints ω1 and ω2 of the frequency interval and the center frequency of the interval (ω1+ω2) / 2.

[0021] As a further preferred option, the weighting factor λ in step 4 is adaptively determined through iteration or binary search, so that...

[0022]

[0023] Where ε e The preset error threshold is preferably within a range of 10. -5 ~10 -3 .

[0024] As a further preferred option, in step 5, the partial derivative of the composite optimization objective function is taken and set to zero to obtain the optimized mass proportionality coefficient α and stiffness proportionality coefficient β.

[0025] .

[0026] This invention proposes an adaptive Rayleigh damping determination method for fluid-structure interaction simulation of impact turbines. It combines full-band average error control with local accuracy constraints at key frequencies to form a composite optimization criterion that takes into account both overall fitting and local control. This method differs from existing methods that rely solely on two-frequency matching or single least-squares fitting, and has higher relevance and applicability.

[0027] Compared with traditional methods that only guarantee average accuracy, this invention can control the damping ratio error at key frequency points such as endpoints and nozzle-water bucket excitation within a preset threshold while ensuring the overall damping ratio fitting accuracy across the frequency range. This improves the reliability of simulation results at key physical points and reduces the risk of numerical divergence or result distortion.

[0028] This invention introduces an adaptive adjustment mechanism of the weighting factor λ into the composite error objective function, enabling the method to automatically balance the overall accuracy and local constraints according to the actual error situation, avoiding the arbitrariness of manually selecting parameters, and ensuring the applicability and robustness of the method under different working conditions.

[0029] This invention further introduces a penalty term related to the β coefficient into the objective function, which can be optimized by combining the unique nozzle-bucket excitation power spectrum characteristics of the impulse turbine, and effectively suppress the abnormal amplification of damping parameters in the high-frequency band, ensuring the stability and applicability of the method in complex numerical environments.

[0030] This method can complete the analytical solution of damping parameters based solely on conventional numerical calculation results. The calculation process does not rely on complex iterative optimizers, making it efficient and concise. Furthermore, the obtained damping parameters can be applied to different operating conditions of similar impulse turbines, demonstrating significant engineering application value. Attached Figure Description

[0031] Figure 1 This is a schematic diagram of the geometric structure of the impulse turbine provided by the present invention;

[0032] Figure 2 This is a comparison chart of the damping ratio at different frequencies between the traditional Rayleigh damping parameter solution method provided by this invention and the method of this invention;

[0033] Figure 3 This is a comparison chart of the absolute error of the damping ratio at different frequencies between the traditional Rayleigh damping parameter solution method provided by this invention and the method of this invention.

[0034] Figure 4 This is a comparison chart of vibration displacement results obtained by using the present invention and traditional methods in fluid-structure interaction simulation of an impact turbine, provided by the present invention.

[0035] Figure 5 This is a flowchart of the method provided by the present invention. Detailed Implementation

[0036] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with the accompanying drawings and embodiments. The described embodiments are only a part of the embodiments of this invention, and not all of them. Other embodiments obtained by those skilled in the art without creative effort should also fall within the protection scope of this invention.

[0037] This invention provides a method for determining adaptive Rayleigh damping in fluid-structure interaction simulation of an impulse turbine, comprising the following steps:

[0038] Step 1: Input frequency range and target damping ratio. Set the target frequency range [ω1, ω2] required for the fluid-structure interaction simulation of the impulse turbine, and determine the target damping ratio η. The damping ratio is a fundamental structural property and is usually a constant.

[0039] Step 2: Establish a Rayleigh damping model based on the mass proportionality coefficient α and the stiffness proportionality coefficient β:

[0040]

[0041] Where ω is the frequency, α is the mass proportionality coefficient, and β is the stiffness proportionality coefficient, and α and β are non-negative to avoid numerical instability.

[0042] Step 3: Construct a composite optimization objective function J(α, β) consisting of the following three parts to characterize the overall deviation between Rayleigh damping and the target damping ratio: (1) the overall frequency band error term of Rayleigh damping and the target damping ratio within the target frequency range [ω1, ω2]; (2) the key frequency accuracy term of Rayleigh damping and the target damping ratio η at key frequencies within at least three target frequency ranges [ω1, ω2]; (3) the stiffness scaling factor penalty term used to suppress high-frequency amplification.

[0043]

[0044] Where W(ω) is the frequency band weighting function, which can be a constant of 1 to indicate equal weighting; ω1 and ω2 are the endpoint frequencies of the target frequency range; dω is the frequency differential element; m represents the number of key frequency points; ω k λ is the frequency corresponding to the key frequency point, η is the target damping ratio, and γ is the penalty term coefficient used to suppress high-frequency amplification effects.

[0045] Step 4: Adjust the weighting factor λ so that the result of the key frequency point accuracy term is less than or equal to the error threshold ε. e The value of the weighting factor λ is determined accordingly. The solution of the key frequency point accuracy term includes at least the values ​​at the following three frequency points: the two endpoints ω1 and ω2 of the frequency interval and the center frequency (ω1+ω2) / 2 of the interval.

[0046]

[0047] Where ε e A preset error threshold is set, which is determined by the user according to engineering requirements and can be flexibly adjusted under different working conditions. The method of this invention does not limit the numerical range of the threshold; in specific embodiments, a value of 10 is preferred. -5 ~10 -3 The magnitude is to ensure that the error at key frequency points is controlled within an acceptable range.

[0048] Step 5: Take the partial derivative of the composite optimization objective function and set it to zero to obtain the optimized mass proportionality coefficient α and stiffness proportionality coefficient β.

[0049]

[0050] Step 6: Output and apply the optimized mass scaling factor α and stiffness scaling factor β to the fluid-structure interaction simulation of the impact turbine. This ensures the accuracy of the damping ratio fitting and the control effect at key frequencies during the overall calculation process, thereby improving the simulation convergence and the reliability of the results.

[0051] The overall process of this invention is as follows: Figure 5 The diagram illustrates the overall steps of this method and the logical relationships between each step.

[0052] In this embodiment of the invention, the method includes the following steps:

[0053] Step 1: In this embodiment, for an impulse turbine with a rated speed of 375 r / min (e.g. Figure 1 The diagram shown is a geometrical schematic of an impulse turbine (used to illustrate the applicable objects and typical application scenarios of the method of this invention). The first five dry modal frequencies are calculated, where the first and fifth frequencies are 243.21 Hz and 348.68 Hz, respectively, corresponding to an angular frequency range of 1528.1 rad / s to 2190.8 rad / s. These frequencies are set as ω1 and ω2. The target damping ratio η is taken as 0.03, which is a fundamental material property. Step 2: Establish the Rayleigh damping model. Based on the Rayleigh damping principle, a damping ratio model is established:

[0054]

[0055] Where α is the mass proportionality coefficient and β is the stiffness proportionality coefficient. This model can describe the frequency-dependent damping ratio characteristics at different angular frequencies.

[0056] Step 3: Construct a composite error objective function. To ensure both the overall fitting effect and the accuracy of key frequencies, the following composite error objective function is constructed:

[0057]

[0058] Where W(ω) is the frequency band weighting function, which can be a constant of 1 to indicate equal weighting; ω1 and ω2 are the endpoint frequencies of the target frequency range; dω is the frequency differential element; m represents the number of key frequency points; ω k λ is the frequency corresponding to the key frequency point, η is the target damping ratio, and γ is the penalty term coefficient used to suppress high-frequency amplification effects.

[0059] In this embodiment, W(ω) is a constant of 1 to represent equal weights, but the weights can also be set according to actual engineering needs. γ is preferably 10. -6 , is a small positive number used to prevent high-frequency numerical divergence caused by an excessively large stiffness proportionality coefficient.

[0060] Step 4: Adjust the weighting factor λ so that the result of the key frequency point accuracy term is less than or equal to the error threshold ε. e This is used to determine the value of the weighting factor λ.

[0061]

[0062] The key frequencies are the two ends of the interval and the midpoint of the band of interest, at 295.95 Hz. ε e 10 preferred -5 ~10 -3 The magnitude of ε in this embodimente Take 10 -4 This is used to constrain the fitting error of the damping ratio at key frequency points. It should be understood that the method of this invention does not limit the specific value of the threshold; it can be flexibly adjusted according to actual needs under different operating conditions.

[0063] Step 5: Take the partial derivative of the composite optimization objective function and set it to zero to obtain the optimized mass proportionality coefficient α and stiffness proportionality coefficient β.

[0064]

[0065] Based on the principle of minimizing error, the final solution results are: α = 54.83, β = 1.60 × 10⁻⁶. -5 ,like Figure 2 and Figure 3 As shown.

[0066] Step 6: Apply the obtained damping parameters α and β to the fluid-structure interaction transient simulation of the impulse turbine. The simulation results are as follows: Figure 4 As shown, compared with the traditional Rayleigh damping determination method, the vibration displacement amplitude obtained by this method is slightly reduced, and the numerical convergence process is more stable.

[0067] Therefore, the method of the present invention not only maintains a small average error over the entire frequency band, but also effectively reduces the risk of large endpoint deviations and vibration calculation distortion in existing methods, thereby improving the stability and reliability of the fluid-structure interaction simulation results of the impact turbine.

[0068] The calculation process of this invention is based on analytical matrix solving, which is highly efficient. Furthermore, this method does not rely on additional experimental measurements; it can optimize damping parameters using only conventional numerical simulation data, thus possessing strong engineering applicability.

[0069] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for determining adaptive Rayleigh damping in fluid-structure interaction simulation of an impulse turbine, characterized in that, include: Step 1: Before simulation, set the target frequency range [ω1, ω2] and the target damping ratio η, where the target damping ratio is a basic structural property; Step 2: Establish a Rayleigh damping model based on the mass proportionality coefficient α and the stiffness proportionality coefficient β; Step 3: Construct a composite optimization objective function J(α, β) consisting of the following three parts to characterize the overall deviation between Rayleigh damping and the target damping ratio: (1) the overall frequency band error term of Rayleigh damping and the target damping ratio within the target frequency range [ω1, ω2]; (2) the key frequency accuracy term of Rayleigh damping and the target damping ratio η at key frequencies within at least three target frequency ranges [ω1, ω2]; (3) the stiffness scaling factor penalty term used to suppress high-frequency amplification. Step 4: Adjust the weighting factor λ in the composite optimization objective function so that the result of the key frequency point accuracy term is less than or equal to the error threshold ε. e This is used to determine the value of the weighting factor λ; Step 5: Based on the determination of λ in Step 4, take the partial derivatives of the composite optimization objective function J(α, β) with respect to α and β respectively and set them to zero to obtain the optimized mass proportionality coefficient α and stiffness proportionality coefficient β. Step 6: Input the optimized mass scaling factor α and stiffness scaling factor β obtained in Step 5 into the fluid-structure interaction simulation of the impulse turbine to improve the calculation stability and result accuracy.

2. The method for determining adaptive Rayleigh damping in fluid-structure interaction simulation of an impulse turbine according to claim 1, characterized in that, In step 2, the mathematical expression of the Rayleigh damping model is: ； Where ω is the frequency, α is the mass proportionality coefficient, and β is the stiffness proportionality coefficient, and α and β are non-negative to avoid numerical instability.

3. The method for determining adaptive Rayleigh damping in fluid-structure interaction simulation of an impulse turbine according to claim 2, characterized in that, In step 3, the mathematical expression for the composite optimization objective is: ； Where W(ω) is the frequency band weighting function, which can be a constant of 1 to indicate equal weighting; ω1 and ω2 are the endpoint frequencies of the target frequency range; dω is the frequency differential element; m represents the number of key frequency points; ω k λ is the frequency corresponding to the key frequency point, η is the target damping ratio, and γ is the penalty term coefficient used to suppress high-frequency amplification effects.

4. The method for determining adaptive Rayleigh damping in fluid-structure interaction simulation of an impulse turbine according to claim 3, characterized in that, The solution of the critical frequency point accuracy term includes at least the values ​​at the following three frequency points: the two endpoints ω1 and ω2 of the frequency interval and the center frequency of the interval (ω1+ω2) / 2.

5. The method for determining adaptive Rayleigh damping in fluid-structure interaction simulation of an impulse turbine according to claim 3, characterized in that, In step 4, the weighting factor λ is adaptively determined through iteration or binary search, such that: ； Where ε e The preset error threshold has a range of 10. -5 ~10 -3 .

6. The method for determining adaptive Rayleigh damping in fluid-structure interaction simulation of an impulse turbine according to claim 3, characterized in that, In step 5, the partial derivative of the composite optimization objective function is calculated and set to zero to obtain the optimized mass proportionality coefficient α and stiffness proportionality coefficient β. 。

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