A method for accelerating convergence of CFD solution based on adaptive nonlinear dynamic dimension reduction
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-05-26
- Publication Date
- 2026-08-11
AI Technical Summary
对于强非线性流动特征，如激波的突变、边界层的分离等复杂现象，POD无法有效捕获其非线性特征结构，导致降维表示精度不足，进而影响收敛求解的结果；其次，POD降维得到的子空间是固定的，在整个迭代过程中保持不变
[0061] This invention presents an accelerated convergence method for CFD solutions based on adaptive nonlinear dynamic dimensionality reduction. It replaces the traditional linear POD method with a nonlinear encoder-decoder network, enabling the learning of the nonlinear manifold structure of flow field data. Through nonlinear transformation of multi-layer neural networks, the network can capture strongly nonlinear flow characteristics such as shock waves and boundary layer separation. In multiple test cases, compared to the POD method, the reconstruction error of nonlinear dimensionality reduction is reduced by 30%-50%, providing a more accurate low-dimensional representation for subsequent optimization.
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of computational fluid dynamics, specifically relating to a method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction. Background Technology
[0002] Computational fluid dynamics (CFD) is widely used in numerical simulation of flow fields in engineering fields such as aerospace and automotive. Iterative methods are the most commonly used solution approach for steady-state flow fields. However, for complex flow problems, such as transonic flow, high Reynolds number turbulence, eddy shedding, and shock wave boundary layer disturbances—all highly nonlinear flow phenomena—traditional iterative methods often converge slowly or even fail to converge, resulting in long computation times and high costs, severely limiting the efficiency of CFD technology in engineering design.
[0003] In recent years, researchers have proposed several methods to accelerate convergence. The first category is dimensionality reduction methods based on Proper Orthogonal Decomposition (POD), which is currently the most effective and widely used approach. The POD method extracts the main modes of the flow field through singular value decomposition, projects the high-dimensional flow field onto a low-dimensional subspace, and optimizes the residuals in the low-dimensional space, thereby accelerating convergence and significantly reducing the difficulty of solving the optimization problem. The second category is initial field generation methods based on neural networks. These methods directly predict near-convergent flow fields as initial fields by training deep learning models, reducing the number of iterations. However, neural network methods require building large flow field datasets for model training. For new flow configurations or parameter combinations, it is often necessary to re-collect data and retrain the model, resulting in high data acquisition costs and long training cycles. In practical engineering applications, these methods face problems such as difficulty in dataset construction and insufficient generalization ability, limiting their engineering applicability.
[0004] In contrast, POD-based online dimensionality reduction optimization methods do not require pre-training and can utilize flow field snapshots generated during the current iteration for online optimization, offering better engineering applicability. However, existing POD-based convergence acceleration methods have some technical drawbacks. First, POD is essentially a linear dimensionality reduction method, performing mode decomposition based on the linear correlation of the data. For highly nonlinear flow characteristics, such as shock wave abrupt changes and boundary layer separation, POD cannot effectively capture their nonlinear characteristic structures, resulting in insufficient accuracy of the dimensionality reduction representation, which in turn affects the convergence solution results. Second, the subspace obtained by POD dimensionality reduction is fixed and remains unchanged throughout the iteration process. However, during CFD iterative solutions, the flow field characteristics evolve continuously with each iteration, especially in the early and late stages of convergence, where the main characteristics of the flow field may change significantly. A fixed dimensionality reduction subspace cannot adapt to such dynamic changes, limiting the applicability of the acceleration method. Third, the dimensionality reduction dimension of POD is usually a pre-set fixed value, requiring expert judgment; too low a dimension will result in the loss of important information, while too high a dimension increases computational costs. In practical applications, the optimal dimensionality reduction dimension changes continuously with the convergence state, making it difficult to maintain optimal performance at different stages with a fixed dimension. Secondly, existing methods lack the embedding of physical constraints when optimizing in low-dimensional subspaces. While the optimized solution mathematically reduces the residual, it may violate fundamental physical laws such as mass conservation, momentum conservation, and energy conservation, leading to poor physical consistency and low reliability of the solution results. Finally, key decisions such as triggering dimensionality reduction optimization criteria and adjusting optimization parameters still rely on human experience and repeated trials, lacking intelligent adaptive mechanisms. Different triggering strategies and parameter settings are required for different flow problems, indicating insufficient versatility and robustness of existing methods. Summary of the Invention
[0005] The problem to be solved by this invention is to realize a CFD accelerated convergence method with adaptive decision-making capability, and to propose a CFD solution accelerated convergence method based on adaptive nonlinear dynamic dimensionality reduction.
[0006] To achieve the above objectives, the present invention provides the following technical solution:
[0007] A method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction includes the following steps:
[0008] S1. Construct a nonlinear encoder and decoder network;
[0009] S2. Set the initial CFD iteration parameters;
[0010] S3. Perform standard CFD iteration, call the CFD solver to obtain the current flow field, and calculate the residual vector and residual norm;
[0011] S4. Determine whether the convergence condition is met based on the residual norm. If the condition is met, output the result; otherwise, proceed to the next step.
[0012] S5. Establish an adaptive triggering mechanism;
[0013] S6. Based on the adaptive triggering mechanism established in step S5, determine whether dimensionality reduction optimization is triggered. If it is, perform dimensionality reduction optimization and then update the initial value of the iteration. If it is not, directly update the initial value of the iteration.
[0014] S7. Based on the updated initial values for the iteration, return to step S3 to perform standard CFD iterations until the convergence condition is met.
[0015] Furthermore, the specific implementation method of step S1 includes the following steps:
[0016] S1.1. The encoder is configured with a multi-layer fully connected neural network structure to map the high-dimensional flow field to a low-dimensional latent space. The mapping function is:
[0017]
[0018] Where u is the input flow field vector with dimension ? , representing the flow field variables of all nodes on the CFD mesh; Let be a low-dimensional latent representation vector with dimension . ; The encoder mapping function is composed of multiple fully connected neural networks; it satisfies... ;
[0019] S1.2. Configure the decoder to reconstruct the low-dimensional latent representation vector into a high-dimensional flow field. The reconstruction function is:
[0020]
[0021] in, To reconstruct the flow field vector; This is the decoder mapping function, which is composed of multiple layers of neural networks.
[0022] Furthermore, in step S2, the initial flow field is set. Initial values are obtained using uniform inflow, empirical formulas, or results from previous calculations; an iteration counter is set. This is used to track the current iteration step and initialize the residual history array for subsequent determination of the convergence trend.
[0023] Furthermore, step S3 calls the CFD solver to perform one or more standard iterations to obtain the first... Step flow field ; Calculate the first The residual vector of the flow field at each step norm of the residual .
[0024] Furthermore, the convergence condition for step S4 is: Then it is considered that the iteration converges, where, This is the preset convergence tolerance.
[0025] Furthermore, the adaptive triggering mechanism in step S5 is as follows:
[0026] Rule 1 - Triggered at fixed intervals, every fixed number of steps Trigger an optimization. Pick step;
[0027] Rule 2 - Residual Decline Rate Triggered, defined as:
[0028]
[0029] in, R is the residual decrease rate, representing the relative change in residual between two adjacent iterations; )|| is the first The residual norm of the step; ||R( )|| is the first The residual norm of the step;
[0030] when Time-triggered optimization, among which As a preset threshold, take ;
[0031] Rule 3 - Residual Stagnation Trigger: When consecutive steps Optimization is triggered when the residual does not decrease. Take 5-10 steps;
[0032] Rule 4 - Residual oscillation trigger: Determine whether the change in residual exceeds the threshold.
[0033] If any of the above rules are met, dimensionality reduction optimization will be triggered.
[0034] Furthermore, the specific implementation method of dimensionality reduction optimization in step S6 includes the following steps:
[0035] S6.1. Collect recent data The flow field snapshots of each step constitute a snapshot sequence S=[ , , ..., ], Total number of snapshots;
[0036] S6.2. Input the flow field snapshot into the encoder to obtain a low-dimensional latent representation vector;
[0037] S6.3. First, the snapshot sequence Perform singular value decomposition To obtain singular values ;
[0038] The computational dimension is The energy percentage of singular values:
[0039]
[0040] in, The percentage of energy represents the proportion of the energy before the energy level is determined. The proportion of energy contained in each mode to the total energy; For the first There are 1 singular values, and the square of the singular value represents the energy of the corresponding mode.
[0041] The dimensions are adaptively adjusted based on the energy percentage and the residual descent rate:
[0042] like < Then add dimensions to obtain the adjusted dimensions. ;
[0043] like > and Then, by reducing the dimensionality, we obtain... = max(r -Δr, Otherwise, maintain the current dimension and obtain... = ;in, To adjust the step size, and These are the upper and lower bounds of the dimension, respectively.
[0044] S6.4. Low-dimensional space optimization: Constructing an optimization problem in the latent space and finding the optimization variables. This minimizes the residual of the reconstructed flow field; the optimization problem is formulated as follows:
[0045]
[0046] in, To optimize variables, the dimension is... ; Let be the objective function, and let represent the squared residual of the reconstructed flow field after correction. The radius of the trust domain is used to prevent over-correction from causing divergence. For physical constraint operators;
[0047] Trust domain radius The expression for adaptive adjustment based on the current residuals and potential dimensions is:
[0048]
[0049] in, Let the initial trust domain radius be , For adjustment coefficients;
[0050] The optimal correction amount is obtained by solving the optimization problem using the L-BFGS-B algorithm or the trust region method. Calculate the optimized latent representation The optimized flow field is obtained by reconstructing it through the decoder. ;
[0051] S6.5. Physical constraint projection, applying the optimized flow field Project onto the physically constrained manifold to ensure that the fundamental physical laws are satisfied;
[0052] Specific constraints include: mass conservation constraint, which, for incompressible flow, satisfies zero divergence. Momentum conservation constraints satisfy the Navier-Stokes equations; energy conservation constraints satisfy the energy equations; boundary condition constraints ensure that the flow field at the boundary satisfies the given boundary conditions.
[0053] Using the Lagrange multiplier method or projection operator Project onto the manifold that satisfies the constraints;
[0054] S6.6. Online learning updates, based on collected snapshot sequences. Update the encoder-decoder network parameters online and set the total loss function as follows:
[0055]
[0056] in, To reconstruct the loss, For residual loss, For physical constraint loss, For regularization loss, , , These are the weighting coefficients for residual loss, physical constraint loss, and regularization loss, respectively.
[0057] The network parameters are updated using gradient descent, and the learning rate is... The system adaptively adjusts based on residual changes, employs a sliding window strategy to maintain a fixed-size snapshot buffer, and samples from the buffer for training each time. The sampling strategy can be uniform sampling or sampling based on the importance of the residuals.
[0058] Furthermore, in step S6, dimensionality reduction optimization is performed, and then the initial values of the iterations are updated to the optimized flow field. As the new initial value for the iteration, set ;
[0059] If dimensionality reduction optimization is not triggered, the standard CFD iteration result is used as... Update the initial value during iteration.
[0060] The beneficial effects of this invention are:
[0061] This invention presents an accelerated convergence method for CFD solutions based on adaptive nonlinear dynamic dimensionality reduction. It replaces the traditional linear POD method with a nonlinear encoder-decoder network, enabling the learning of the nonlinear manifold structure of flow field data. Through nonlinear transformation of multi-layer neural networks, the network can capture strongly nonlinear flow characteristics such as shock waves and boundary layer separation. In multiple test cases, compared to the POD method, the reconstruction error of nonlinear dimensionality reduction is reduced by 30%-50%, providing a more accurate low-dimensional representation for subsequent optimization.
[0062] This invention presents an accelerated convergence method for CFD solving based on adaptive nonlinear dynamic dimensionality reduction, which achieves dynamic evolution of the reduced subspace through an online learning mechanism. As the CFD iteration progresses, the encoder-decoder network continuously learns new flow field snapshots and adaptively adjusts network parameters, enabling the reduced subspace to follow the continuous changes in flow field characteristics. Compared to fixed subspace methods, the dynamically evolving subspace increases the overall convergence speed by 2 times, especially in the early stages of convergence when flow field characteristics change drastically, where the acceleration effect is even more significant.
[0063] This invention presents an accelerated convergence method for CFD solutions based on adaptive nonlinear dynamic dimensionality reduction. It employs multiple adaptive triggering rules and intelligent criteria for dimensionality reduction optimization, avoiding unnecessary computational overhead. The optimization frequency is reduced during the stable convergence phase, and timely intervention occurs during the difficult convergence phase. Compared to a fixed-frequency triggering strategy, the adaptive triggering mechanism reduces computational overhead by approximately 20%-30%, while maintaining or even improving the acceleration effect.
[0064] This invention presents an accelerated convergence method for CFD solutions based on adaptive nonlinear dynamic dimensionality reduction. This method dynamically adjusts the potential spatial dimension according to the complexity of the flow field using an adaptive dimensionality adjustment rule based on energy proportion. A higher dimension is used in the early stages of convergence when the flow field is complex, while the dimension is reduced in the later stages when the flow field tends to stabilize, thus ensuring both representation accuracy and computational efficiency.
[0065] This invention presents an accelerated convergence method for CFD solutions based on adaptive nonlinear dynamic dimensionality reduction. This method embeds physical constraints such as mass, momentum, and energy conservation during the low-dimensional optimization process. Through constrained optimization and physical projection, it ensures that the optimized solution satisfies fundamental physical laws. Compared to unconstrained optimization methods, the physical consistency error is reduced to below 10%, improving the reliability and credibility of the computational results. Attached Figure Description
[0066] Figure 1 This is a flowchart of a CFD solution acceleration convergence method based on adaptive nonlinear dynamic dimensionality reduction as described in this invention. Detailed Implementation
[0067] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only for explaining the invention and are not intended to limit the invention; that is, the described specific embodiments are merely a part of the embodiments of the invention, and not all of them. The components of the specific embodiments of the invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations, and the invention may also have other embodiments.
[0068] Therefore, the following detailed description of specific embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected specific embodiments of the invention. All other specific embodiments obtained by those skilled in the art based on these specific embodiments without inventive effort are within the scope of protection of this invention.
[0069] To further understand the invention's content, features, and effects, the following specific embodiments are provided, along with accompanying drawings. Figure 1 Detailed explanation is as follows:
[0070] Example 1:
[0071] A method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction includes the following steps:
[0072] S1. Construct a nonlinear encoder and decoder network;
[0073] Furthermore, the specific implementation method of step S1 includes the following steps:
[0074] S1.1. The encoder is configured with a multi-layer fully connected neural network structure to map the high-dimensional flow field to a low-dimensional latent space. The mapping function is:
[0075]
[0076] Where u is the input flow field vector with dimension ? , representing the flow field variables of all nodes on the CFD mesh; Let be a low-dimensional latent representation vector with dimension . ; The encoder mapping function is composed of multiple fully connected neural networks; it satisfies... ;
[0077] S1.2. Configure the decoder to reconstruct the low-dimensional latent representation vector into a high-dimensional flow field. The reconstruction function is:
[0078]
[0079] in, To reconstruct the flow field vector; This is the decoder mapping function, which is composed of multiple layers of neural networks.
[0080] Furthermore, taking a four-layer network as an example, the transformation relationship between each layer of the encoder is expressed as follows:
[0081]
[0082]
[0083]
[0084]
[0085] in, , , This is the output vector of the intermediate hidden layer; , , , Here are the weight matrices for each layer; , , , These are the bias vectors for each layer; For non-linear activation functions, the ReLU function is typically used. Alternatively, the Tanh function. Through multiple layers of nonlinear transformation, the network can learn complex nonlinear mapping relationships.
[0086] The decoder employs a network structure symmetrical to the encoder, progressively restoring the low-dimensional representation to a high-dimensional flow field layer by layer. The initial latent dimension is set to... The algorithm will be adaptively adjusted based on the convergence results.
[0087] S2. Set the initial CFD iteration parameters;
[0088] Furthermore, in step S2, the initial flow field is set. Initial values are obtained using uniform inflow, empirical formulas, or results from previous calculations; an iteration counter is set. This is used to track the current iteration step and initialize the residual history array for subsequent determination of the convergence trend.
[0089] S3. Perform standard CFD iteration, call the CFD solver to obtain the current flow field, and calculate the residual vector and residual norm;
[0090] Furthermore, step S3 calls the CFD solver to perform one or more standard iterations to obtain the first... Step flow field ; Calculate the first The residual vector of the flow field at each step norm of the residual .
[0091] Furthermore, the L2 norm or the infinite norm is usually used.
[0092] S4. Determine whether the convergence condition is met based on the residual norm. If the condition is met, output the result; otherwise, proceed to the next step.
[0093] Furthermore, the convergence condition for step S4 is: Then it is considered that the iteration converges, where, This is the preset convergence tolerance. It is determined based on accuracy requirements. Usually taken arrive If convergence is not achieved, proceed with the subsequent steps.
[0094] S5. Establish an adaptive triggering mechanism;
[0095] Furthermore, the adaptive triggering mechanism in step S5 is as follows:
[0096] Rule 1 - Triggered at fixed intervals, every fixed number of steps Trigger an optimization. Pick step;
[0097] Rule 2 - Residual Decline Rate Triggered, defined as:
[0098]
[0099] in, R is the residual decrease rate, representing the relative change in residual between two adjacent iterations; )|| is the first The residual norm of the step; ||R( )|| is the first The residual norm of the step;
[0100] when Time-triggered optimization, among which As a preset threshold, take ;
[0101] Rule 3 - Residual Stagnation Trigger: When consecutive steps Optimization is triggered when the residual does not decrease. Take 5-10 steps;
[0102] Rule 4 - Residual oscillation trigger: Determine whether the change in residual exceeds the threshold.
[0103] If any of the above rules are met, dimensionality reduction optimization will be triggered.
[0104] S6. Based on the adaptive triggering mechanism established in step S5, determine whether dimensionality reduction optimization is triggered. If it is, perform dimensionality reduction optimization and then update the initial value of the iteration. If it is not, directly update the initial value of the iteration.
[0105] Furthermore, the specific implementation method of dimensionality reduction optimization in step S6 includes the following steps:
[0106] S6.1. Collect recent data The flow field snapshots of each step constitute a snapshot sequence S=[ , , ..., ], Total number of snapshots; the snapshot sequence contains historical information about the flow field evolution, helping to capture the main characteristics of the flow. Window size. Typically, 20-50 steps are taken, which contains enough information without excessively increasing the computational load.
[0107] S6.2. Input the flow field snapshot into the encoder to obtain a low-dimensional latent representation vector;
[0108] Furthermore, the current flow field Input to the encoder to obtain a low-dimensional latent representation. Potential representation By compressing the main information of the flow field and optimizing it in a low-dimensional space, the computational complexity can be significantly reduced.
[0109] S6.3. First, the snapshot sequence Perform singular value decomposition To obtain singular values ;
[0110] The computational dimension is The energy percentage of singular values:
[0111]
[0112] in, The percentage of energy represents the proportion of the energy before the energy level is determined. The proportion of energy contained in each mode to the total energy; For the first There are 1 singular values, and the square of the singular value represents the energy of the corresponding mode.
[0113] The dimensions are adaptively adjusted based on the energy percentage and the residual descent rate:
[0114] like < Then add dimensions to obtain the adjusted dimensions. ;
[0115] like > and Then, by reducing the dimensionality, we obtain... = max(r -Δr, Otherwise, maintain the current dimension and obtain... = ;in, To adjust the step size, and These are the upper and lower bounds of the dimension, respectively.
[0116] Furthermore, the dimension is adaptively adjusted based on the energy percentage and the residual decrease rate: if < (e.g., 0.95) indicates that the current dimension is insufficient to represent the main features of the flow field, so the dimension is increased: ;like > (e.g., 0.99) and This indicates that the current dimension is redundant and can be reduced. = max(r - Δr, Otherwise, maintain the current dimension. = r. Where To adjust the step size (usually 5-10). and These are the upper and lower limits of the dimension.
[0117] S6.4. Low-dimensional space optimization: Constructing an optimization problem in the latent space and finding the optimization variables. This minimizes the residual of the reconstructed flow field; the optimization problem is formulated as follows:
[0118]
[0119] in, To optimize variables, the dimension is... ; Let be the objective function, and let represent the squared residual of the reconstructed flow field after correction. The radius of the trust domain is used to prevent over-correction from causing divergence. For physical constraint operators;
[0120] Trust domain radius The expression for adaptive adjustment based on the current residuals and potential dimensions is:
[0121]
[0122] in, Let the initial trust domain radius be , For adjustment coefficients;
[0123] The optimal correction amount is obtained by solving the optimization problem using the L-BFGS-B algorithm or the trust region method. Calculate the optimized latent representation The optimized flow field is obtained by reconstructing it through the decoder. ;
[0124] S6.5. Physical constraint projection, applying the optimized flow field Project onto the physically constrained manifold to ensure that the fundamental physical laws are satisfied;
[0125] Specific constraints include: mass conservation constraint, which, for incompressible flow, satisfies zero divergence. Momentum conservation constraints satisfy the Navier-Stokes equations; energy conservation constraints satisfy the energy equations; boundary condition constraints ensure that the flow field at the boundary satisfies the given boundary conditions.
[0126] Using the Lagrange multiplier method or projection operator Project onto the manifold that satisfies the constraints;
[0127] S6.6. Online learning updates, based on collected snapshot sequences. Update the encoder-decoder network parameters online and set the total loss function as follows:
[0128]
[0129] in, To reconstruct the loss, For residual loss, For physical constraint loss, For regularization loss, , , These are the weighting coefficients for residual loss, physical constraint loss, and regularization loss, respectively. , , .
[0130] Furthermore, reconstruction loss Measure the reconstruction accuracy of the network; residual loss Encourage online learning of flow fields with low residuals; physical constraint loss Penalizes reconstructions that violate physical laws; regularization loss This helps prevent network overfitting.
[0131] The network parameters are updated using gradient descent, and the learning rate is... The system adaptively adjusts based on residual changes, employs a sliding window strategy to maintain a fixed-size snapshot buffer, and samples from the buffer for training each time. The sampling strategy can be uniform sampling or sampling based on the importance of the residuals.
[0132] Furthermore, in step S6, dimensionality reduction optimization is performed, and then the initial values of the iterations are updated to the optimized flow field. As the new initial value for the iteration, set ;
[0133] If dimensionality reduction optimization is not triggered, the standard CFD iteration result is used as... Update the initial value during iteration.
[0134] S7. Based on the updated initial values for the iteration, return to step S3 to perform standard CFD iterations until the convergence condition is met.
[0135] Furthermore, update the iteration counter. Record the residual history, return to step S3 to continue iterating until the convergence condition is met.
[0136] Furthermore, in step S1, the encoder-decoder network adopts an autoencoder (AE) or variational autoencoder (VAE) architecture. For the AE architecture, the encoder network structure is as follows: The decoder network structure is For the VAE architecture, the encoder output layer additionally outputs a mean vector and a variance vector, and the latent representation is obtained through reparameterization techniques. (Network dimension) For the flow field dimension, As a potential dimension, the intermediate dimensions decrease / increase layer by layer, achieving a smooth dimension transformation.
[0137] Furthermore, a pre-training step is included before step S1: the encoder-decoder network is pre-trained offline using initial flow field snapshots or historical case data. Pre-training enables the network to possess preliminary flow field representation capabilities. Pre-training data can come from previous cases, empirical databases, or fast computation results using coarse meshes. The pre-training loss function simultaneously considers reconstruction accuracy and residual minimization, balancing the weights of both. After pre-training, online fine-tuning is performed during actual CFD iterations to adapt to the current flow field characteristics.
[0138] Furthermore, in step S6.4, the physical constraints include: for incompressible flow, the mass conservation constraint is expressed as the divergence of the velocity field being zero. The velocity field is projected onto a subspace with zero divergence using a projection operator; for compressible flow, the mass conservation constraint is expressed as a continuity equation. Momentum conservation constraints are achieved through residual penalty terms in the Navier-Stokes equations; energy conservation constraints are achieved through residual penalty terms in the energy equations; boundary condition constraints are achieved by forcing the fulfillment of given values or gradient conditions at boundary nodes.
[0139] Furthermore, in step S6.6, the online learning employs an incremental learning strategy, using only a small number of newly added snapshots for each update to avoid retraining the entire network. Specifically, a fixed-size snapshot buffer is maintained, and the buffer is updated using a first-in-first-out (FIFO) strategy. During each update, a small batch (batch size of...) is randomly sampled from the buffer. Gradient updates are performed. The learning rate uses a decay strategy, gradually decreasing as iterations progress. The initial learning rate is typically [value missing]. The attenuation factor is .
[0140] Example 2:
[0141] This embodiment is based on Embodiment 1 and is illustrated with the following example:
[0142] A method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction includes the following steps:
[0143] Constructing an encoder-decoder network: An autoencoder architecture is adopted. The encoder is a four-layer fully connected network, with the number of neurons in each layer as follows: ,in The flow field dimension (corresponding to 100,000 grid nodes). The initial latent dimensions are given. The decoder is a mirror image of the encoder. The activation function used is ReLU.
[0144] Pre-training was performed using 100 flow field snapshots from historical examples, with a training period of 50 epochs and a learning rate of 0.001. The Adam optimizer was used. After pre-training, the network reconstruction error was reduced to below 5%.
[0145] Initialize CFD solution: Taking the transonic flow of the NACA0012 airfoil as an example, the incoming Mach number... Reynolds number Angle of attack The RANS equations were solved using the finite volume method, and the Spalart-Allmaras model was used for the turbulence. The mesh size was 100,000 nodes. A uniform inflow was used as the initial flow field. Convergence tolerance was specified. .
[0146] Initiating the iteration: The first 50 steps use standard iterations to warm up and allow the flow field to develop initially. From step 51 onwards, an adaptive acceleration strategy is activated.
[0147] Set trigger parameters: fixed period Step; Residual descent rate threshold Number of stalled steps step.
[0148] when During the step, the residual decrease rate This triggers dimensionality reduction optimization. Collect the flow field snapshots from the last 40 steps.
[0149] Perform SVD decomposition on the snapshot matrix to calculate the energy percentage. Current dimension. At that time, the proportion of energy If the judgment dimensions are insufficient, increase the dimensions to... .
[0150] Construct an optimization problem in a 25-dimensional potential space, with an initial trust region radius. The L-BFGS-B algorithm was used to solve the problem, and the optimal correction was obtained after 20 iterations. .
[0151] The decoded and reconstructed flow field is optimized and projected onto a flow field subspace with zero divergence, ensuring mass conservation. The optimized residuals are derived from... Down to .
[0152] The network is updated online using the latest 40 snapshots, from which 32 snapshots are randomly sampled for training for 3 epochs with a learning rate of 0.0005.
[0153] The optimized flow field is then used for further iterations. In subsequent iterations, the triggering frequency and potential dimensions are dynamically adjusted based on the convergence results.
[0154] Final result: The traditional method requires 800 iterations and takes 10 hours of computation; this method requires 200 iterations and takes 2 hours of computation. The speedup is 5 times that of the traditional method.
[0155] It should be noted that relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.
[0156] Although this application has been described above with reference to specific embodiments, various modifications can be made and components can be replaced with equivalents without departing from the scope of this application. In particular, as long as there is no structural conflict, the features in the specific embodiments disclosed in this application can be combined with each other in any way. The lack of an exhaustive description of these combinations in this specification is merely for the sake of brevity and resource conservation. Therefore, this application is not limited to the specific embodiments disclosed herein, but includes all technical solutions falling within the scope of the claims.
Claims
1. A method for accelerating convergence of CFD solution based on adaptive nonlinear dynamic dimension reduction, characterized in that, Includes the following steps: S1. Construct a nonlinear encoder and decoder network; S2. Set the initial CFD iteration parameters; S3. Perform standard CFD iteration, call the CFD solver to obtain the current flow field, and calculate the residual vector and residual norm; S4. Determine whether the convergence condition is met based on the residual norm. If the condition is met, output the result; otherwise, proceed to the next step. S5. Establish an adaptive triggering mechanism; S6. Based on the adaptive triggering mechanism established in step S5, determine whether dimensionality reduction optimization is triggered. If it is, perform dimensionality reduction optimization and then update the initial value of the iteration. If it is not, directly update the initial value of the iteration. S7. Based on the updated initial values for the iteration, return to step S3 to perform standard CFD iterations until the convergence condition is met.
2. The method of claim 1, wherein, The specific implementation method of step S1 includes the following steps: S1.
1. The encoder is configured with a multi-layer fully connected neural network structure to map the high-dimensional flow field to a low-dimensional latent space. The mapping function is: Where u is the input flow field vector with dimension ? , representing the flow field variables of all nodes on the CFD mesh; Let be a low-dimensional latent representation vector with dimension . ; The encoder mapping function is composed of multiple fully connected neural networks; it satisfies... ; S1.
2. Configure the decoder to reconstruct the low-dimensional latent representation vector into a high-dimensional flow field. The reconstruction function is: in, To reconstruct the flow field vector; This is the decoder mapping function, which is composed of multiple layers of neural networks.
3. The method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction according to claim 2, characterized in that, In step S2, the initial flow field is set. Initial values are obtained using uniform inflow, empirical formulas, or results from previous calculations; an iteration counter is set. This is used to track the current iteration step and initialize the residual history array for subsequent determination of the convergence trend.
4. The method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction according to claim 3, characterized in that, Step S3 calls the CFD solver to perform one or more standard iterations to obtain the first step. Step flow field ; Calculate the first The residual vector of the flow field at each step norm of the residual .
5. The method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction according to claim 4, characterized in that, The convergence condition for step S4 is: Then it is considered that the iteration converges, where, This is the preset convergence tolerance.
6. The method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction according to claim 5, characterized in that, The specific details of the adaptive triggering mechanism in step S5 are as follows: Rule 1 - Triggered at fixed intervals, every fixed number of steps Trigger an optimization. Pick step; Rule 2 - Residual Decline Rate Triggered, defined as: in, R is the residual decrease rate, representing the relative change in residual between two adjacent iterations; )|| is the first The residual norm of the step; ||R( )|| is the first The residual norm of the step; when Time-triggered optimization, among which As a preset threshold, take ; Rule 3 - Residual Stagnation Trigger: When consecutive steps Optimization is triggered when the residual does not decrease. Take 5-10 steps; Rule 4 - Residual oscillation trigger: Determine whether the change in residual exceeds the threshold. If any of the above rules are met, dimensionality reduction optimization will be triggered.
7. The method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction according to claim 6, characterized in that, The specific implementation method of dimensionality reduction optimization in step S6 includes the following steps: S6.
1. Collect recent data The flow field snapshots of each step constitute a snapshot sequence S=[ , , ..., ], Total number of snapshots; S6.
2. Input the flow field snapshot into the encoder to obtain a low-dimensional latent representation vector; S6.
3. First, the snapshot sequence Perform singular value decomposition To obtain singular values ; The computational dimension is The energy percentage of singular values: in, The percentage of energy represents the first... The proportion of energy contained in each mode to the total energy; For the first There are 1 singular value, and the square of the singular value represents the energy of the corresponding mode; The dimensions are adaptively adjusted based on the energy percentage and the residual descent rate: like < Then add dimensions to obtain the adjusted dimensions. ; like > and Then, by reducing the dimensionality, we obtain... = max(r - Δr, Otherwise, maintain the current dimension and obtain... = ;in, To adjust the step size, and These are the upper and lower bounds of the dimension, respectively. S6.
4. Low-dimensional space optimization: Constructing an optimization problem in the latent space and finding the optimization variables. This minimizes the residual of the reconstructed flow field; the optimization problem is formulated as follows: in, To optimize variables, the dimension is... ; Let be the objective function, and let represent the squared residual of the reconstructed flow field after correction. The radius of the trust domain is used to prevent over-correction from causing divergence. For physical constraint operators; Trust domain radius The expression for adaptive adjustment based on the current residuals and potential dimensions is: in, Let the initial trust domain radius be , For adjustment coefficients; The optimal correction amount is obtained by solving the optimization problem using the L-BFGS-B algorithm or the trust region method. Calculate the optimized latent representation The optimized flow field is obtained by reconstructing it through the decoder. ; S6.
5. Physical constraint projection, applying the optimized flow field Project onto the physically constrained manifold to ensure that the fundamental physical laws are satisfied; Specific constraints include: mass conservation constraint, which, for incompressible flow, satisfies zero divergence. Momentum conservation constraints satisfy the Navier-Stokes equations; energy conservation constraints satisfy the energy equations; boundary condition constraints ensure that the flow field at the boundary satisfies the given boundary conditions. Using the Lagrange multiplier method or projection operator Project onto the manifold that satisfies the constraints; S6.
6. Online learning updates, based on collected snapshot sequences. Update the encoder-decoder network parameters online and set the total loss function as follows: in, To reconstruct the loss, For residual loss, For physical constraint loss, For regularization loss, , , These are the weighting coefficients for residual loss, physical constraint loss, and regularization loss, respectively. The network parameters are updated using gradient descent, and the learning rate is... The system adaptively adjusts based on residual changes, employs a sliding window strategy to maintain a fixed-size snapshot buffer, and samples from the buffer for training each time. The sampling strategy can be uniform sampling or sampling based on the importance of the residuals.
8. The method for accelerating convergence in CFD solving based on adaptive nonlinear dynamic dimensionality reduction according to claim 7, characterized in that, In step S6, dimensionality reduction optimization is performed, and then the initial values of the iterations are updated to the optimized flow field. As the new initial value for the iteration, set ; If dimensionality reduction optimization is not triggered, the standard CFD iteration result is used as... Update the initial value during iteration.