Cross-scale optimal scale ratio selection method for transient strong nonlinear scale model test
By employing a cross-scale optimal scaling ratio selection method and utilizing renormalization group theory and parabolic mapping model, the error problem of classical similarity theory under transient strong nonlinear loads was solved, achieving high-precision single-model tests, reducing similarity transformation errors, and meeting the needs of engineering practice.
Patent Information
- Application Number
- CN202610172007.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-02-06
- Publication Date
- 2026-05-19
AI Technical Summary
Existing technologies, under transient strong nonlinear loads, rely on scaled-down model testing methods based on classical similarity theory, which suffer from theoretical inapplicability leading to large similarity conversion errors and poor economic efficiency. These methods fail to meet the requirements for high-precision, high-efficiency, and high-reliability single-test model testing.
A cross-scale optimal scaling ratio selection method is adopted. By introducing the first derivative relationship of the dependent variable Π term, a differential group similarity model is established. The universal constant is determined by Taylor expansion and renormalization group theory, and a parabolic mapping model is established. The similarity transformation of the model test results is achieved by controlling the uncertainty parameters.
High-precision single-model tests were achieved, and similarity conversion errors were significantly reduced. The trajectory inversion error of acceleration response signal data was as low as 13.90%, and the error of strain response signal was as low as 16.93%, providing a clear theoretical basis and design criteria for engineering practice.
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Figure CN122064898A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of engineering structures, and in particular relates to a method for selecting the optimal scaling ratio across scales in transient strongly nonlinear scaling model tests. Background Technology
[0002] Under transient, highly nonlinear loads (such as explosive impacts and high-speed collisions), the dynamic response of engineering structural systems exhibits strong nonlinearity and nonstationarity, posing a severe challenge to their safety and functional integrity. While obtaining realistic response data through prototype testing is ideal in this field, it is often severely limited by high costs, complex organization, and the difficulty in replicating destructive tests, resulting in a scarcity of effective data. Therefore, scaled-down model testing has become a feasible and effective engineering method for studying the impact resistance of structures.
[0003] In linear or weakly nonlinear problems, model experiments based on classical similarity theory have formed a relatively complete system and have been widely used. However, directly applying classical similarity theory to transient strongly nonlinear processes faces fundamental challenges and limitations, mainly in two aspects: First, the dynamic behavior of transient strongly nonlinear systems is extremely sensitive to small perturbations such as initial conditions and system parameters, which can lead to uncertainties such as bifurcation and abrupt changes in the response. This makes it difficult for classical similarity theory, based on deterministic relationships, to accurately predict the prototype response. Second, when the number of dimensional medium parameters in the model design exceeds two, similarity distortion inevitably occurs, resulting in significant errors in the similarity transformation based on classical similarity theory.
[0004] To overcome the aforementioned distortion problems, existing technologies have proposed several improved methods, but these methods have significant drawbacks. One such method is the distortion compensation model based on component experiments. This method requires conducting multiple model experiments at different scales for the same prototype, and then using statistical methods to obtain the distortion patterns of the principal physical quantities to correct the prediction results. For example, in underwater explosion impact model experiments, it may be necessary to build models at various scales such as 1:5, 1:10, and 1:15 for testing before a prediction can be made for a single prototype. While this method improves prediction accuracy to some extent, it significantly increases experimental costs, time, and complexity, resulting in poor economic efficiency and unsuitability for scenarios where multiple experiments are difficult to conduct. Another common method is the dimensional analysis distortion model based on engineering experience. Although this method only requires a single experiment, its correction model heavily relies on the subjective experience of the experimenter, leading to unstable similarity conversion accuracy, a lack of universal theoretical basis, and insufficient scientific rigor.
[0005] A deeper problem lies in the fact that using a single eigenvalue to encompass the characteristics of a transient, strongly nonlinear physical process introduces significant uncertainty. The principal physical quantities of this process undergo bifurcation and abrupt changes over time, generating uncertainties that cannot be accurately expressed using eigenvalues such as maximum values or root-mean-square values. They must be described by the changing patterns of the entire response curve throughout its evolution. Bifurcation and abrupt changes often occur during the evolution and progression of physical quantities. To accurately grasp the dynamic characteristics of a strongly nonlinear process, it is necessary to predict and analyze the phase space data trajectory of the signal to accurately control the strong nonlinearity and uncertainty phenomena. However, existing methods have failed to address this problem from the perspective of its fundamental dynamic nature.
[0006] In summary, existing scaled-down model testing methods and their improvements based on classical similarity theory have limitations when dealing with transient, highly nonlinear processes. These limitations include theoretical inapplicability leading to large similarity conversion errors, poor economic efficiency, and reliance on subjective experience. Consequently, they fail to meet the engineering practice's demand for high-precision, high-efficiency, and high-reliability single-sample model tests. Summary of the Invention
[0007] In view of this, the present invention aims to propose a method for selecting the optimal scaling ratio across scales in transient strongly nonlinear scaling model experiments, so as to solve the problem that the existing methods have theoretical inapplicability leading to large similarity conversion errors.
[0008] To achieve the above objectives, the present invention adopts the following technical solution: a method for selecting the optimal scaling ratio across scales in transient strongly nonlinear scaling model experiments, the method comprising: Step S1: Based on the classical similarity theory, a model component experiment is conducted, introducing the dependent variable Π term Π. 1m With scaling ratio C l Based on the changing first derivative relationship, establish a similarity model for differential grouping; Step S2: Perform Taylor expansion on the differential group similarity model to obtain the iterative relationship of the dependent variable Π term as a function of the scaling ratio; Step S3: Define the scaling ratio sequence and transform the iterative relationship into a parabolic mapping model characterizing the dynamic behavior of the system:
[0009] in, Let n be a universal constant and n be the number of iterations. , , It is a constant; Step S4: Determine the universal constant based on the renormalization group theory. αAnd its corresponding scale-free interval, and select the optimal scaling ratio across scales based on the number of iterations n; Step S5: Conduct model experiments based on the selected optimal scaling ratio to obtain the time series Π of the dependent variable Π. 1m ( t Based on the parabolic mapping model and renormalization group theory, the model test results are similarly transformed to the prototype.
[0010] Furthermore, a preferred method is proposed, wherein establishing the differential grouping similarity model in step S1 includes:
[0011] in, , and These are the model parameters.
[0012] Furthermore, a preferred method is proposed, wherein the iterative relationship of step S2 is as follows: .
[0013] Furthermore, a preferred method is proposed, wherein the universal constant... The value is approximately 2.5, which is the optimal scaling ratio across scales. C l The selection satisfies the following condition: when n=1, C l =1 / 3.2, corresponding to the large-scale model; when n=2, C l =1 / 6.25, corresponding to the mesoscale model; when n=3, C l =1 / 15.625, corresponding to the small-scale model.
[0014] Furthermore, a preferred method is proposed, wherein the conversion of model test results to the prototype is similar, specifically including:
[0015] in, D Here, Π1 represents the amplitude of the prototype dependent variable Π term, which is the control parameter for uncertainty. 1m This represents the magnitude of the dependent variable Π term in the model.
[0016] Furthermore, a preferred method is proposed, in which the uncertainty control parameter D is obtained through the following steps: For the time series of the dependent variable Π term in the model Π 1m ( t Phase space reconstruction is performed to obtain data orbit points; Based on the data trajectory points, fit a general parabolic mapping function: , in, x For data track Π 1m ( i )represent, a, b, c These are the parameters of a typical parabola; Through topological conjugate relationships F ( x )= h -1 G ( x ) h Convert the ordinary parabolic mapping function into a standard parabolic mapping function. F ( x )=1- μx 2 And determine the topological conjugate parameter m and γ ,in, h These are topological conjugate equations; According to the formula Calculate the uncertainty control parameter D, where, d n The magnitude of the dependent variable Π term in the model experiment under n iterations. 1m Effective similar components in [the text].
[0017] Furthermore, a preferred embodiment is proposed, wherein the scale-free interval scaling function of the renormalization group theory in step S4 satisfies the limit relation:
[0018] Among them, Π p1 The mapping function of the dependent variable Π term in phase space, which is the prototype. m1 Let be the mapping function of the dependent variable Π term in the phase space. It is a limit function.
[0019] Furthermore, a preferred approach is proposed, in which step S5, after obtaining the time series of the dependent variable Π term, the time series is further divided into blocks, and periodic features are extracted in each block to approximate the nonlinear data trajectory.
[0020] Based on the same inventive concept, the present invention also proposes a computer device, including a memory and a processor, wherein the memory stores a computer program, and when the processor runs the computer program stored in the memory, the processor executes a method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test according to any one of the preceding claims.
[0021] Based on the same inventive concept, the present invention also proposes a computer-readable storage medium storing a computer program, which, when executed by a processor, performs the steps of a method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test as described in any one of the above.
[0022] Compared with the prior art, the beneficial effects of the present invention are: The method proposed in this invention can predict the prototype response with high accuracy by conducting only a single model test at a specific scale. Experimental data from specific implementations show that, at the optimal scale (e.g., 1 / 6.25 of the mesoscale), the average relative error of the data trajectory inversion for the acceleration response signal can be as low as 13.90%, and for the strain response signal, as low as 16.93%. This completely changes the high-cost, low-efficiency model of traditional component-based testing methods, which require multiple tests to predict a single prototype.
[0023] Compared with classical similarity theory, the method of this invention has an overwhelming advantage in conversion accuracy. In the same experimental case, for acceleration measurement point A9, when the scaling ratio is 1 / 6.25, the similarity conversion error of the method of this invention (13.90%) is much lower than that of classical similarity theory (33.75%). For strain measurement point S4, the error of the method of this invention (16.93%) is also significantly lower than that of the classical method (22.47%). This proves that the method can effectively overcome the problem of large conversion error caused by strong nonlinearity and similarity distortion.
[0024] Existing technologies typically only predict single characteristic values of the response, such as the maximum value and root mean square value. This invention, however, enables the similarity transformation of the entire response data trajectory. The method of this invention not only retrieves a few amplitude points, but also a complete time history curve that closely matches the prototype experiment. This is crucial for analyzing the dynamic behaviors such as bifurcation and abrupt changes that occur in transient strongly nonlinear processes, providing unprecedented detailed insights. This invention abandons subjective assumptions based on engineering experience and, based on renormalization group theory, clearly identifies the optimal scaling ratio sequence for model experiments across different scales (large, medium, and small): 1 / 3.2 (large scale), 1 / 6.25 (medium scale), and 1 / 15.625 (small scale). Experimental data clearly show that experiments conducted within these scaling ratio neighborhoods have the smallest similarity transformation error, providing a clear and reliable theoretical basis and design criteria for engineering practice.
[0025] Existing technologies (including classical similarity theory and its distortion compensation model) essentially still treat transient strongly nonlinear systems as linear systems, attempting to approximate the prototype through parameter correction. This invention, however, for the first time starts from the essence of the dynamic system, recognizing that the uncertainty of strongly nonlinear responses is rooted in the evolution of its phase space data trajectory. Therefore, this invention is no longer limited to parameter analogy, but rather regards model experiments as an iterative mapping of the prototype dynamic system in phase space, grasping the core of the problem in principle. It introduces renormalization group theory, originally applied to theoretical physics (such as critical phenomena and phase transition theory), and symbolic dynamics theory, which studies the chaotic behavior of nonlinear systems. By revealing the intrinsic connection between the change in the scaling ratio of the model experiment and the universal constant of period-doubling bifurcation in nonlinear dynamics, the problem of selecting the scaling ratio is transformed into the scientific problem of finding the scale-free interval and stable fixed point of the dynamic system. This cross-disciplinary theoretical grafting provides a completely new mathematical tool and theoretical framework for solving similarity problems in engineering. Based on the above principles, the solution architecture of this invention is entirely new. It first derives a parabolic mapping model characterizing the iterative behavior of the system by establishing a differential group similarity model. Then, renormalization group theory is used to determine the specific scaling ratio sequence (i.e., the optimal scaling ratio) that makes the mapping behavior most stable. Finally, by analyzing the periodic trajectory of the response signal and introducing the uncertainty control parameter D, the accurate transformation from model to prototype is completed. This forms a complete and self-consistent new methodology system from model design to result transformation, which is completely different from the traditional approach that relies on statistical experience or local corrections.
[0026] This invention is applied to technical fields such as ship impact environment analysis and cross-medium aircraft impact response analysis. Attached Figure Description
[0027] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings: Figure 1 This is a schematic diagram of the parabolic mapping principal period doubling bifurcation described in this invention; Figure 2 This is a schematic diagram of the phase space reconstruction of the time series signal according to the present invention; Figure 3 This is a flowchart of the method for selecting the optimal scaling ratio across scales in transient strongly nonlinear scaling model experiments as described in this invention. Figure 4 This is a flowchart of the similarity conversion process for the transient strongly nonlinear scaled-down model experiment described in this invention; Figure 5 These are schematic diagrams of the large and medium-scale models of the reinforced cylindrical shell described in this invention. Figure 6 This is a schematic diagram of the location of the acceleration measuring point described in this invention; Figure 7 This is a schematic diagram of the strain measurement point location as described in this invention; Figure 8 The Π described in this invention 1m Schematic diagram of / Π1 as a function of scaling ratio Cl; Figure 9 As described in this invention C l = A schematic diagram showing the selection of the periodic characteristics of the π-term curve of the acceleration measuring point A9 at 1 / 6.25, where... Figure 9 In the middle (a), the selection of the periodic characteristics in the upper half is shown. Figure 9 (b) represents the selection of the periodic characteristics in the lower half; Figure 10 This is a schematic diagram illustrating the specific process of the model experiment similarity conversion technology described in this invention, as well as the similarity conversion result values and relative errors. Figure 11 C as described in this invention l A comparison of the data trajectory and amplitude inversion of each block of acceleration measurement point A9 at =1 / 6.25 with the prototype dependent variable Π term curve, where... Figure 11 (a) represents a comparison of the model amplitude inversion results. Figure 11 (b) represents a comparison of the trajectory inversion results of the model data; Figure 12 This is a schematic diagram of the error curve of the overall similarity conversion result as a function of the scaling ratio described in this invention; Figure 13 The ε described in this invention m Schematic diagram of / ε as a function of scaling ratio Cl; Figure 14 This is a schematic diagram illustrating the selection of the periodic characteristics of the Π term curve due to strain S4 measurement point as described in this invention. Figure 15 C as described in this invention l A comparison of the data trajectory and amplitude inversion of each block at strain gauge point S4 at =1 / 6.25 with the prototype dependent variable Π term curve, where... Figure 15 (a) is a comparison chart of the model amplitude inversion results. Figure 15 (b) is a comparison chart of the orbit inversion results of the model data; Figure 16 This is a schematic diagram of the error curve of the overall similarity conversion result as a function of the scaling ratio, as described in this invention. Detailed Implementation
[0028] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of the present invention can be combined with each other, and the described embodiments are only some embodiments of the present invention, not all embodiments.
[0029] Implementation Method 1: This implementation method addresses the problem of large similarity transformation errors due to theoretical inapplicability in existing methods. It proposes a method for selecting the optimal scaling ratio across scales in transient strongly nonlinear scaling model experiments. The method includes: Step S1: Based on the classical similarity theory, a model component experiment is conducted, introducing the dependent variable Π term Π. 1m With scaling ratio C l By considering the changing first derivative relationships, we can establish a similarity model for the differential groups:
[0030] in, , and These are model parameters; Step S2: Perform a Taylor expansion on the differential group similarity model to obtain the iterative relationship of the dependent variable Π term as a function of the scaling ratio: ; Step S3: Define the scaling ratio sequence and transform the iterative relationship into a parabolic mapping model characterizing the dynamic behavior of the system:
[0031] in, Let n be a universal constant and n be the number of iterations. , , It is a constant; Step S4: Determine the universal constant based on the renormalization group theory. α And its corresponding scale-free interval, and select the optimal scaling ratio across scales based on the number of iterations n. C l ; The universal constant in this embodiment The value is approximately 2.5, which is the optimal scaling ratio across scales. C l The selection satisfies the following condition: when n=1, C l =1 / 3.2, corresponding to the large-scale model; when n=2, C l =1 / 6.25, corresponding to the mesoscale model; when n=3, C l=1 / 15.625, corresponding to a small-scale model; In this embodiment, the scale-free interval scaling function of the renormalization group theory satisfies the limit relation:
[0032] Among them, Π p1 The mapping function of the dependent variable Π term in phase space, which is the prototype. m1 Let be the mapping function of the dependent variable Π term in the phase space. It is a limit function; Step S5: Based on the selected optimal scaling ratio C l Conduct model experiments to obtain the time series of the dependent variable Π term Π. 1m ( t Based on the parabolic mapping model and renormalization group theory, the model test results are similarly transformed to the prototype, specifically including:
[0033] in, D Here, Π1 represents the amplitude of the prototype dependent variable Π term, which is the control parameter for uncertainty. 1m This represents the magnitude of the dependent variable Π term in the model. The uncertainty control parameter D is obtained through the following steps: For the time series of the dependent variable Π term in the model Π 1m ( t Phase space reconstruction is performed to obtain data orbit points; Based on the data trajectory points, fit a general parabolic mapping function: , in, x For data track Π 1m ( i )represent, a, b, c These are the parameters of a typical parabola; Through topological conjugate relationships F ( x )= h -1 G ( x ) h Convert the ordinary parabolic mapping function into a standard parabolic mapping function. F ( x )=1- μx 2 And determine the topological conjugate parameter m and γ ,in, h These are topological conjugate equations. h ( x )=mx + γ ; According to the formula Calculate the uncertainty control parameter D, where, d n The magnitude of the dependent variable Π term in the model experiment under n iterations. 1m Effective similar components in [the text].
[0034] After obtaining the time series of the dependent variable Π term of the model, the process also includes dividing the time series into blocks and extracting periodic features within each block to approximate the nonlinear data trajectory.
[0035] Implementation Method 2, see below Figures 1 to 4 This embodiment describes a complete implementation process for the method of selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model experiment as described in Embodiment 1, including: Step 1: Introduce the first derivative relationship to the model component experiment of classical similarity theory to reflect the dependent variable Π term Π. 1m With scaling ratio C l Characteristics of the changes. The power series expansion of the differential group similarity model yields: (1) In the formula
[0036] Step 2: After rearranging and Taylor expansion of equation (1), the following iterative or recursive relationship can be obtained: (2) Step 3, if we choose C l =1 / α n ( (where C is any non-zero positive real number) l + ΔC l =1 / α n+1 Then ΔC l =1 / α n - 1 / α n+1 Substituting into the above equation, equation (2) becomes a typical quadratic unimodal mapping, i.e., a parabolic mapping model: (3) In the formula .
[0037] The parabolic unimodal mapping function relationship obtained in step 4 and equation (3) reflects the iterative characteristics of similar model experiments in phase space. This iterative relationship reveals the one-to-one correspondence between a single model and its prototype. Based on the renormalization equation and the limit function, it can be seen that: (4) In the formula: α ≈2.5, g ( x =1 - 1.1527632997 x 2 +0.104815194 x 4 +0.26756734 x 6 +……,Π p1 Π m1 These are the mapping functions of the dependent variable Π term in the phase space for the prototype and the model, respectively. When n=1, i.e., the number of iterations is 1, the proportional function in the scale-free interval of the renormalization group is... α =3.218, and when n=2, the scale-free interval of the renormalization group is... α =2.63≈2.5, when n=3, the scale-free interval of the renormalization group is α =2.52≈2.5. Therefore, the discontinuous scaling characteristic shown in step 3 should be as shown in the table below.
[0038] Table 1 Selection of Scale Ratio for Model Tests
[0039] Step 5: Assign a scale to the model C l According to the discontinuous change characteristics shown in step 4, that is C l =1 / α n (n=1,2,3…), then we have: (5) In the formula: d 2 n 2 n The distance between the common superstable point and the nearest common superstable point in a point-times-periodic kneading sequence, which is also known as the scaling ratio. C l =1 / α n The dependent variable Π term in the time model experiment corresponds to C l When n=1, the similar components in the dependent variable Π term of the prototype experiment. d 1= αd 2, when n=2 d 2= α2 d 4. When n=3 d 4= α 4 d 8…..。 It is worth noting that, d 2 n-1 (n=1,2,3…) represents the 2nd… n-1 Point cycle, is like Figure 1 The parabolic mapping bifurcation diagram shown .
[0040] Step 6: According to equation (6), if we carry out... C l =1 / α n The scaled-down model experiment yielded the parabolic mapping function of the dependent variable Π term, and from this function, the model's dependent variable Π term was found to be... d 2 n By finding a suitable value, the goal of transitioning from model experiments to prototype experiments can be achieved. The renormalization group equations indicate its stable fixed point. x fn With public super-stable fixed point x * The ratio of interpolation values at two adjacent different iteration numbers satisfies a universal constant. α The relationship is as follows: (6) In the formula, x fn To ensure stable positioning, x * For public ultra-stable fixed points, d n The interpolation between stable fixed points and common superstable fixed points, that is, the dependent variable Π term in the model experiment corresponding to C. l When =1, similar components in the dependent variable Π term of the prototype experiment.
[0041] Step 7: Applying the prototype and model amplitude iterative similarity transformation model based on renormalization group theory, the model is as follows: (7) In the formula: Π1 is the magnitude of the prototype dependent variable Π term, Π 1m The magnitude of the dependent variable Π term in the model. d n For the dependent variable Π term Π in the model experiment under n iterations 1m The effective similar components, where n is the number of iterations. α It is a universal constant.
[0042] Step 8: Based on the scale-free range of time series self-similarity, the system dynamic response time series is partitioned into blocks for model Π. 1m (t By approximating the periodic characteristics of each block in the data, and achieving an approximate representation of the nonlinear data trajectory within the block through the periodic trajectory, the data trajectory Π can be obtained. 1m ( i ), i =1,2,……, n The obtained data track Π 1m ( i ()( i =1,2,……, n The phase space is reconstructed, and the dynamic behavior of the system is characterized in the phase space using a general parabolic mapping function: (8) In the formula, x For Π 1m ( i ) represents i = 1, 2, ..., n; a, b, c These are the parameters of a typical parabola.
[0043] Step 9: The parabolic mapping function is a typical phase space return mapping. The ordinary parabolic mapping function and the standard parabolic mapping function... F ( x )=1- μx 2 Satisfying the topological conjugate relation, i.e. F ( x )= h -1 G ( x ) h Therefore, a differential homeomorphism exists. h ( x m )= x , x m Let the variable be a standard parabola, then we have the following equation: (9) In the formula, m = - a / ( b 2 / 4-b / 2-ac ), γ = - b / 2( b 2 / 4-b / 2-ac Once the topological conjugate parameters are determined... m , γ From equation (7), we can obtain: (10) In the formula: dn These are the similar components on the standard parabola after topological conjugate transformation. D These are the uncertainty control parameters for model experiments.
[0044] Step 10, Uncertainty Control Parameters D Essentially, it describes the uncertainties in system behavior during model experiments caused by factors such as distortion of the independent variable Π term and nonlinear characteristics of the dynamic response. Therefore, the prototype and model amplitude iterative similarity transformation model using renormalization group theory can be rewritten as: (11) In the formula, D Here, m represents the uncertainty control parameter, and m represents the topological conjugate equation parameter. α Let Π1 be a universal constant, n be the number of iterations, and Π1 be the magnitude of the prototype dependent variable Π term. 1m This represents the magnitude of the dependent variable Π term in the model.
[0045] Step 11: As shown in equation (9), when the similarity features of the prototype and the model experiment are embedded in the iterative process of the mapping function, the iterative similarity transformation relationship between the prototype and the model experiment dependent variable Π term (i.e., the amplitude of the experimental dynamic response) can also be established according to the parabolic mapping theory. This prototype and model amplitude iterative similarity transformation model applying the renormalization group theory shows that when the model experiment scaling ratio is... C l According to the renormalization group, according to 1 / α n When making the selection, the uncertainty control parameters obtained from the model test results can be derived from the periodic characteristics of the dependent variable Π term curve of the model. D This allows the model test amplitude and data trajectory to be converted to the prototype. Equation (11) shows that there must exist a universal constant. α and the limit function g1( x ), making C l =1 / α n The model experiments at the scaled-down ratio have corresponding superstable periodic rubbing sequences that correspond one-to-one with each other, and there is also a transformation law between the model and the prototype dependent variable Π term with the universal constant nth power.
[0046] The flowcharts for the selection of the optimal scaling ratio across scales in the transient strongly nonlinear scaling model experiment and the similarity transformation flowchart for the transient strongly nonlinear scaling model experiment are as follows: Figure 3 and Figure 4 As shown.
[0047] Implementation Method 3, see below Figures 5 to 16This embodiment describes a specific application example of the method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model experiment described in Embodiment 2, including: The optimal scale ratio for multi-scale model tests proposed in this invention, supplemented by a portion of the scale ratio based on traditional scaled model tests, takes into account novel impact factors. C 3. Similar to the previous experiment, the following tests were conducted using a series of scaled-down model tests of a reinforced cylindrical shell structure under underwater explosion as an example.
[0048] Based on the scaling ratio conditions for mesoscale model tests proposed by renormalization group theory, large-scale models with scaling ratios of 1:1.5, 1:2.5, and 1:3, and mesoscale model structures with scaling ratios of 1:5.7, 1:6.25, 1:6.8, 1:7, and 1:7.2 were designed. The stiffened cylindrical shell model structure is shown below. Figure 5 As shown.
[0049] The specific strain and acceleration measurement point numbers and arrangements for each scaled-down model analyzed in this embodiment are as follows: Figure 6 and Figure 7 As shown, the measurement points of each scaled model are numbered in the same way.
[0050] Example 1: Acceleration response signal analysis: For large-scale model experiments, the relationship between Πm / Π and the scaling ratio C at each scaling ratio is... l The pattern of change, such as Figure 8 As shown, in the large-scale scaled-down model test, at C l When the similarity is 1 / 3, it is within 80%, which allows for good similarity transfer to the prototype. Therefore, when conducting scaled-down experiments based on this implementation method, the optimal scaling ratio across scales, i.e., large-scale models, can be used for C-scale model development. l Scaled-down model tests within a neighborhood of 1 / 3.2 can provide effective guidance for transient, strongly nonlinear scaled-down model tests and have significant engineering application value.
[0051] For mesoscale model experiments, the first step involves partitioning the model's time-history curves and selecting periodic characteristics within each partition. The selection of periodic characteristics is as follows: Figure 9 As shown, by using the various periodic features under parabolic mapping, the uncertainty control parameters of the periodic features in the scale-free block of the dependent variable Π term curve of the model are found. D .
[0052] Taking the first scale-free block of the acceleration measurement point as an example, the data trajectory points of block 1 are shown in Table 1. Using the following feature points, the phase space is reconstructed according to step 8 to obtain a general parabola, and then, following step 9, the topological conjugate equation is applied. h ( x Transform to standard parabola to obtain its similar componentsd n+1 or uncertainty control parameters D and the topological conjugate equation parameter m, γ Then, the model experiment similarity conversion to the prototype was obtained by using the amplitude similarity conversion model and the relationship between the corresponding periodic characteristics. Figure 10 This describes the specific process of similarity conversion technology in model experiments, as well as the similarity conversion result values and relative errors.
[0053] Table 1. Selection of data trajectory points for acceleration measurement point block 1 (A9).
[0054] Based on the above operations, the same operations are performed on each block to obtain the similarity transformation results and errors within each scale-free block. The table below shows the similarity transformation parameters for each scale-free block.
[0055] Table 3 C l Table of similarity transformation parameters within each block of the acceleration A9 coefficient Π term curve at =1 / 6.25
[0056] The above method yielded the inverted amplitude and similarity transformation error of the model dependent variable Π term amplitude from the prototype dependent variable Π term through similarity transformation. Then, based on step 10, a data trajectory similarity transformation was performed on the prototype experimental results from the model experimental results. To more intuitively observe the variation characteristics of the model similarity transformation amplitude and numerical trajectory with each block, the amplitude envelope and data trajectory inversion after the similarity transformation were plotted. The amplitude and trajectory similarity transformation results are as follows: Figure 11 As shown.
[0057] To quantify the differences in model similarity transformation and to comprehensively evaluate the overall error level and the differences between individual blocks, the following formula is used for calculation: (12) In the formula: y i , x i These are the corresponding data trajectory values for the prototype and model tests, respectively. u i This represents the relative error of the corresponding orbital points. This represents the average relative error of the trajectory inversion from the model experiment using similarity theory. δ 1 represents the root mean square error (RMSE), and ε1 represents the classical similarity transformation error.
[0058] The table below shows the trajectory inversion error values of the time-history curve data at each scale calculated using the above formula for the A9 acceleration measurement points: Table 4. Track Inversion Error of Time History Curve Data at Various Scales for A9 Acceleration Measurement Points
[0059] The above error table is visualized to obtain the error curve of the image similarity conversion result as a function of the scaling ratio, as shown below. Figure 12 As shown.
[0060] In summary, based on the model scaling experiment described in this embodiment, the optimal scaling ratio across scales is achieved by conducting C-scale experiments under a mesoscale model. l =1 / 2.5 2 When conducting scaled-down model experiments within the neighborhood of this scaling factor, its similarity is superior to other scaling factors. In scaled-down model experiments designed based on this scaling factor, the similarity transformation method based on renormalization group theory can effectively obtain the amplitude and data trajectory similarity transformation results. Therefore, when conducting scaled-down model experiments based on this paper, the optimal scaling factor across scales, namely C, is selected. l =1 / 2.5 n Scaled-down model tests within the neighborhood of the model can provide effective guidance for transient, strongly nonlinear scaled-down model tests, and have significant engineering application value.
[0061] Example 2, Strain Response Signal Analysis: For large-scale model experiments, ε is calculated at each scaling factor. m / ε varies with scaling ratio C l The pattern of change is as follows Figure 13 As shown, in the large-scale scaled-down model test, at C l When the similarity is 1 / 3, it is within 80%, which allows for good similarity transfer to the prototype. Therefore, when conducting scaled-down experiments based on this implementation method, the optimal scaling ratio across scales, i.e., large-scale models, can be used for C-scale model development. l Scaled-down model tests within a neighborhood of 1 / 3.2 can provide effective guidance for transient, strongly nonlinear scaled-down model tests and have significant engineering application value.
[0062] For mesoscale model experiments, the first step is to partition the model's time-history curve and select periodic characteristics within each partition, such as... Figure 14 As shown, the uncertainty control parameter D of the periodic features in the scale-free block of the dependent variable Π term curve of the model is found by using the various periodic features under the parabolic mapping. Following the operation in Example 1, the same operation is performed on each block to obtain the similarity transformation results and errors within each scale-free block. The table below shows the similarity transformation parameters for each scale-free block.
[0063] Table 5 C l Table of similarity transformation parameters within each block of the π-term curve at strain S4 measuring point when the strain is 1 / 6.25
[0064] The above method yielded the inverted amplitude and similarity transformation error of the model dependent variable Π term amplitude from the prototype dependent variable Π term through similarity transformation. Then, based on step 10, a data trajectory similarity transformation was performed on the prototype test results from the model test results. To more intuitively observe the variation characteristics of the model similarity transformation amplitude and data trajectory with each block, the amplitude envelope and data trajectory inversion after the similarity transformation were plotted. The amplitude and data trajectory similarity transformation results are as follows: Figure 15 As shown.
[0065] Table 6 shows the trajectory inversion error values of the time-history curve data at each scale for the S4 strain measurement points calculated according to equation (12): Table 6. Track Inversion Error of Time History Curve Data at Various Scales for S4 Strain Measurement Points
[0066] The above error table is visualized to obtain the error curve of the image similarity conversion result as a function of the scaling ratio, as shown below. Figure 16 As shown.
[0067] In summary, when conducting scaled experiments based on the model described in this embodiment, the optimal scaling ratio across scales is achieved, i.e., C is carried out under a mesoscale model. l =1 / 2.5 2 When conducting scaled-down model experiments within the neighborhood of this scaling factor, its similarity is superior to other scaling factors. In scaled-down model experiments designed based on this scaling factor, the similarity transformation method based on renormalization group theory can effectively obtain the amplitude and data trajectory similarity transformation results. Therefore, when conducting scaled-down model experiments based on this paper, the optimal scaling factor across scales, namely C, is selected. l =1 / 2.5 n Scaled-down model tests within the neighborhood of the model can provide effective guidance for transient, strongly nonlinear scaled-down model tests, and have significant engineering application value.
[0068] Implementation Method 4: This implementation method proposes a computer device, including a memory and a processor. The memory stores a computer program. When the processor runs the computer program stored in the memory, the processor executes a method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test according to Implementation Method 1.
[0069] Implementation Method 5: This implementation method proposes a computer-readable storage medium storing a computer program. When the computer program is run by a processor, it executes the steps of the method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test as described in Implementation Method 1.
[0070] Those skilled in the art will understand that embodiments of this disclosure can be provided as methods, systems, or computer program products. Therefore, this disclosure can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this disclosure can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.
[0071] This disclosure is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this disclosure. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create a machine for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 The computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.
[0072] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.
[0073] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this disclosure and not to limit its protection scope. Although this disclosure has been described in detail with reference to the above embodiments, those skilled in the art should understand that after reading this disclosure, they can still make various changes, modifications or equivalent substitutions to the specific implementation of the invention, but these changes, modifications or equivalent substitutions are all within the protection scope of the published pending claims.
Claims
1. A method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model experiment, characterized in that, The method includes: Step S1: Based on the classical similarity theory, a model component experiment is conducted, introducing the dependent variable Π term Π. 1m With scaling ratio C l Based on the changing first derivative relationship, establish a similarity model for differential grouping; Step S2: Perform Taylor expansion on the differential group similarity model to obtain the iterative relationship of the dependent variable Π term as a function of the scaling ratio; Step S3: Define the scaling ratio sequence and transform the iterative relationship into a parabolic mapping model characterizing the dynamic behavior of the system: in, Let n be a universal constant and n be the number of iterations. , , It is a constant; Step S4: Determine the universal constant based on the renormalization group theory. α And its corresponding scale-free interval, and select the optimal scaling ratio across scales based on the number of iterations n; Step S5: Conduct model experiments based on the selected optimal scaling ratio to obtain the time series Π of the dependent variable Π. 1m ( t Based on the parabolic mapping model and renormalization group theory, the model test results are similarly transformed to the prototype.
2. The method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaled model experiment according to claim 1, characterized in that, The step S1 of establishing the differential group similarity model includes: in, , and These are the model parameters.
3. The method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaled model experiment according to claim 2, characterized in that, The iterative relationship of step S2 is as follows: 。 4. The method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test according to claim 1, characterized in that, The universal constant The value of is approximately 2.5, and the selection of the optimal scaling ratio across scales satisfies the following condition: when n=1, C l =1 / 3.2, corresponding to the large-scale model; when n=2, C l =1 / 6.25, corresponding to the mesoscale model; when n=3, C l =1 / 15.625, corresponding to the small-scale model.
5. The method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test according to claim 1, characterized in that, The process of converting model test results to prototypes specifically includes: in, D Here, Π1 represents the amplitude of the prototype dependent variable Π term, which is the control parameter for uncertainty. 1m This represents the magnitude of the dependent variable Π term in the model.
6. The method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test according to claim 5, characterized in that, The uncertainty control parameter D is obtained through the following steps: For the time series of the dependent variable Π term in the model Π 1m ( t Phase space reconstruction is performed to obtain data orbit points; Based on the data trajectory points, fit a general parabolic mapping function: ， in, x For data track Π 1m ( i )represent, a, b, c These are the parameters of a typical parabola; Through topological conjugate relationships F ( x )= h -1 G ( x ) h Convert the ordinary parabolic mapping function into a standard parabolic mapping function. F ( x )=1- μx 2 And determine the topological conjugate parameter m and γ ,in, h These are topological conjugate equations; According to the formula Calculate the uncertainty control parameter D, where, d n The magnitude of the dependent variable Π term in the model experiment under n iterations. 1m Effective similar components in [the text].
7. The method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaled model test according to claim 1, characterized in that, In step S4, the scale-free interval scaling function of the renormalization group theory satisfies the limit relation: Among them, Π p1 The mapping function of the dependent variable Π term in phase space, which is the prototype. m1 Let be the mapping function of the dependent variable Π term in the phase space. It is a limit function.
8. The method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test according to claim 1, characterized in that, In step S5, after obtaining the time series of the dependent variable Π term of the model, the time series is further divided into blocks, and periodic features are extracted in each block to approximate the nonlinear data trajectory.
9. A computer device, characterized in that: It includes a memory and a processor, wherein the memory stores a computer program, and when the processor runs the computer program stored in the memory, the processor executes a method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test according to any one of claims 1-8.
10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, performs the steps of a method for selecting the optimal scaling ratio across scales in a transient strongly nonlinear scaling model test as described in any one of claims 1-8.