Rapid calculation method for frictional heat of rocket sled shoe rail

Abstract

The present invention discloses a method for quickly calculating the friction heat of a rocket sled shoe track. Through experimental test data or pre-calculated simulation data, an instantaneous image matrix is obtained, and the matrix is decomposed to calculate the eigenvector and eigenvalue information, and further obtain the POD basis function; according to the error required by the problem, the number of truncated eigenvalues is determined, and then the spatial dimension that meets the requirements is determined. Using the obtained eigenvectors, the vector space corresponding to the spatial dimension is determined; the discrete matrix of the full-order model is processed by the determined vectors to obtain a reduced-order model of the temperature field, and the reduced-order temperature field model is calculated under the generated vector space, and the corresponding temperature information is calculated. The present invention can improve the calculation efficiency of the temperature field and save a large amount of memory occupancy.

Description

Technical Field

[0001] The present invention belongs to the technical field of range testing, and particularly relates to a method for quickly calculating the friction heat of a rocket sled shoe rail. Background Art

[0002] A rocket sled is a ground test system that uses a rocket engine as power and slides at high speed along a specially built high-precision slide rail, simulating conditions such as speed, acceleration, and vibration during the actual flight of a weapon system, and assessing the functions of the entire weapon system or components. Since the rocket sled body generates intense frictional forces with the surface of the rail during high-speed movement, the structural temperature rises sharply. Moreover, as the test movement speed of the rocket sled continues to increase, the heat problem caused by friction will become more prominent, leading to a decline in the performance of the material. When using a full-order model to calculate the temperature field of frictional heat, if the mesh model is complex, the computational workload is extremely large and very time-consuming.

[0003] The model reduction method is to obtain the transient matrix required for the problem to be solved based on previous test data or pre-simulated data, etc., decompose the transient image matrix, obtain its eigenvalue information, thereby determine the dimension to be solved, use a low-order problem to approximately replace the high-order problem, reduce the computational scale, and improve the computational efficiency. Summary of the Invention

[0004] In order to overcome the deficiencies of the prior art, the present invention provides a method for quickly calculating the friction heat of a rocket sled shoe rail. Through experimental test data or pre-calculated simulation data, a transient image matrix is obtained, and the matrix is decomposed to calculate the eigenvector and eigenvalue information, and further obtain the POD basis function; according to the error required by the problem, the number of truncated eigenvalues is determined, and then the spatial dimension that meets the requirements is determined. Using the obtained eigenvectors, a vector space corresponding to the spatial dimension is determined; the discrete matrix of the full-order model is processed using the determined vectors to obtain a reduced-order model of the temperature field, and the reduced-order temperature field model is calculated in the generated vector space to calculate the corresponding temperature information. The present invention can improve the computational efficiency of the temperature field and save a large amount of memory occupancy.

[0005] The technical solution adopted by the present invention to solve its technical problems is as follows:

[0006] Step 1: Obtain a transient image matrix S according to the numerical simulation results of the theoretical model or the experimental results of the actual physical process; the transient image matrix is shown as follows.

[0007]

[0008] Wherein, N is the number of grid nodes, and L is the number of transient images taken;

[0009] Step 2: Perform matrix decomposition on the snapshot matrix S. Use the singular value decomposition (SVD) method to decompose the matrix S ∈ R N×L The result after singular value decomposition is:

[0010] S = UσV (2)

[0011] where the matrix and V ∈ R N×N =(φ1, φ2, …, φ N ) are both orthogonal matrices, and σ ∈ R L×L = dia{σ1, σ2, …, σ L} is a diagonal matrix;

[0012] Since the number of grid nodes N of the model to be solved is much larger than the time step L, i.e., N >> L, the dimension of the matrix SS T is much larger than the dimension of the matrix S T S. Therefore, the Lagrange multiplier method is used to solve the eigenvector φ T of S i ,

[0013] S T Sφ i = λ i φ i , i = 1, 2, …, L (3)

[0014] where λ i = σ i 2 , i = 1, 2, …, L;

[0015] Further solve the eigenvectors of SS T ;

[0016]

[0017] Based on the obtained eigenvectors establish a set of POD basis functions Φ;

[0018] Step 3: Determine the reduced subspace using the ratio of the total energy of the subspace to the energy of the full-order space. According to the energy ratio η(m):

[0019]

[0020] Determine the optimal m-dimensional subspace to approximately simulate the snapshot data;

[0021] According to the determined m-dimensional subspace, truncate the corresponding POD basis vectors, and construct a low-order vector space from the pairwise orthogonal POD basis functions;

[0022] Step 4: Express the frictional heat equation in a discrete format:

[0023]

[0024] where K is the heat conduction matrix, C is the heat capacity matrix, and F is the right-end temperature load column matrix;

[0025] For the temperature T of each node at any moment, use the obtained POD basis vectors to express it as

[0026]

[0027] Substitute Equation (7) into Equation (6) to obtain

[0028]

[0029] Multiply both sides by Φ T , to obtain the reduced-order model:

[0030]

[0031] Solve Equation (9) to obtain the solution of the equation, and then substitute α into Equation (7) to obtain the final temperature field result.

[0032] Preferably, the η(m) = 99.99%.

[0033] The beneficial effects of the present invention are as follows:

[0034] The calculation efficiency of the method of the present invention depends on the eigenvalue information of the snapshot matrix. Using lower spatial information to approximate the high-order space, compared with the full-order model calculation, the fast calculation method can reduce the calculation scale by more than two orders of magnitude, improve the calculation efficiency, and reduce the space required for calculation. Description of the Drawings

[0035] Figure 1 It is a flow chart of the method of the present invention.

[0036] Figure 2 It is a comparison diagram of the actual result and the model reduced-order result of the embodiment of the present invention.

[0037] Figure 3 It is a comparison diagram of the actual result and the model reduced-order result after local dimension magnification of the embodiment of the present invention. Detailed Embodiments

[0038] The present invention will be further described below in conjunction with the drawings and embodiments.

[0039] The purpose of the present invention is to provide a fast calculation and analysis method for frictional heat, improve the calculation efficiency of the temperature field, and save a large amount of memory occupancy.

[0040] This embodiment uses a one-dimensional variable temperature field, and the true temperature is where l is the rod length and k is the thermal conductivity coefficient.

[0041] As Figure 1 shown, it mainly includes the following steps:

[0042] 1. According to the numerical simulation results of the theoretical model or the experimental results of the actual physical process, obtain the instantaneous image matrix S. The instantaneous image matrix is shown as follows.

[0043]

[0044] where N is the number of grid nodes and L is the number of instantaneous images taken.

[0045] 2. Perform matrix decomposition on the instantaneous image matrix S. Here, the singular value decomposition (SVD) method is used to obtain an optimal representation of the matrix S. Let the matrix S ∈ R N×L The result after singular value decomposition is:

[0046] S = UσV (2)

[0047] where the matrix and V ∈ R N×N = (φ1, φ2, …, φ N ) are both orthogonal matrices, and σ ∈ R L×L = dia{σ1, σ2, …, σ L} is a diagonal matrix.

[0048] Since the number of grid nodes N of the model to be solved is much larger than the time step L, that is, N >> L, the dimension of the matrix SS T is much larger than the dimension of the matrix S T S.

[0049] The Lagrange multiplier method is used to solve the eigenvector φ T of S i ,

[0050] S T Sφ i = λ i φ i , i = 1, 2, …, L (3)

[0051] In the formula, λ i = σ i 2 (i = 1, 2, …, L).

[0052] Furthermore, the eigenvector of SS T can be solved.

[0053]

[0054] According to the obtained eigenvectors Construct a set of POD basis functions Φ.

[0055] 3. Determine the reduced subspace by using the ratio of the total energy of the subspace to the energy of the full-order space. According to the energy ratio η(m)

[0056]

[0057] Determine the optimal m-dimensional subspace to approximately simulate the snapshot data. Usually, η(m) = 99.99%.

[0058] According to the determined m-dimensional subspace, truncate the corresponding POD basis vectors. A low-order vector space can be constructed from the pairwise orthogonal POD basis functions.

[0059] 4. Express the frictional heat equation in a discrete form

[0060]

[0061] where K is the heat conduction matrix, C is the heat capacity matrix, and F is the right-end temperature load column matrix.

[0062] For the temperature T of each node at any moment, express it by using the obtained POD basis vectors as

[0063]

[0064] Substitute Equation (7) into Equation (6) to obtain

[0065]

[0066] Multiply both sides by Φ T , and a reduced-order model can be obtained

[0067]

[0068] Solve the above equation to obtain the solution of the equation, and then substitute α into Equation (7) to obtain the final temperature field result.

Claims

1. A rapid calculation method for the frictional heat of a rocket sled shoe track, characterized in that, It includes the following steps: Step 1: Obtain the instantaneous image matrix S according to the numerical simulation results of the theoretical model or the experimental results of the actual physical process. The instantaneous image matrix is shown as follows: Where, N is the number of grid nodes, and L is the number of instantaneous images taken; Step 2: Perform matrix factorization on the instantaneous image matrix S, where S ∈ R N×L The result after factorization is: S = UσV (2) Among them, the matrix and V ∈ R N×N =(φ1, φ2, …, φ N ) are both orthogonal matrices, and σ ∈ R L ×L = dia{σ1, σ2, …, σ L} is a diagonal matrix; Since the number of grid nodes N of the model to be obtained is much larger than the time step L, that is, N >> L, matrix SS T has a dimension much larger than that of matrix S T S. Therefore, the Lagrange multiplier method is used to solve for the eigenvector φ T of S i , S T Sφ i = λ i φ i , i = 1, 2, …, L (3) where λ i = σ i 2 , for i = 1, 2, …, L; Further solve for SS T eigenvector According to the obtained eigenvectors Construct a set of POD basis functions Φ; Step 3: Determine the reduced subspace by using the ratio of the total energy of the subspace to the energy of the full-order space. According to the energy ratio η(m): Determine the optimal m-dimensional subspace to approximately simulate the instantaneous image data; According to the determined m-dimensional subspace, truncate the corresponding POD basis vectors, and construct a low-order vector space from the POD basis functions; Step 4: Express the frictional heat equation in a discrete format: Where, K is the heat conduction matrix, C is the heat capacity matrix, and F is the right-end temperature load column matrix; For the temperature T of each node at any moment, express it by using the obtained POD basis vectors as Substitute Equation (7) into Equation (6) to obtain Multiply both sides by Φ T , to obtain the reduced-order model: Solve Equation (9) to obtain the solution of the equation, and then substitute α into Equation (7) to obtain the final temperature field result.

2. The rapid calculation method of rocket sled shoe-rail friction heat according to claim 1, characterized in that The η(m) = 99.99%; 3. A rapid calculation method for the friction heat of a rocket sled shoe rail according to claim 1, characterized in that The true temperature of the temperature field is where l is the rod length and k is the heat conduction coefficient.

4. A rapid calculation method for the friction heat of a rocket sled shoe rail according to claim 1, characterized in that, The temperature field is a one-dimensional variable temperature field.

5. A rapid calculation method for the friction heat of a rocket sled shoe rail according to claim 1, characterized in that The method of matrix decomposition is the singular value decomposition SVD method.

6. A rapid calculation method for the friction heat of a rocket sled shoe rail according to claim 1, characterized in that The POD basis functions are pairwise orthogonal to each other.

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