Partial differential equation rapid solving method and solver based on grade radial basis function

Abstract

The invention provides a partial differential equation rapid solving method and solver based on a grade radial basis function, and relates to the technical field of computers and numerical calculation, and the method comprises the steps: obtaining a to-be-solved partial differential equation and a calculation region; the calculation area is jointly defined by a central point set, a configuration point set and an evaluation point set; applying a nested relationship constraint to the central point set to construct a nested sequence, so that a low-level point set covers a global domain, and a high-level point set realizes coarse-scale global representation and fine-scale local focusing through local encryption; assembling a radial basis function with a self-adaptive support radius for each scale level on the basis of applying a nested relationship constraint to the central point set to obtain a variable support set kernel function; establishing a grade radial basis function test space by adopting a grade construction method; and based on the level radial basis function heuristic space, solving the to-be-solved partial differential equation through a level radial basis function assembly point method. According to the scheme, the partial differential equation can be rapidly solved in a meshless mode.

Description

Technical Field

[0001] This invention relates to the fields of computer and numerical computing technology, and in particular to a fast solution method and solver for partial differential equations based on hierarchical radial basis functions. Background Technology

[0002] In science and engineering, partial differential equations (PDEs) are a core mathematical tool for characterizing complex physical phenomena and key processes, widely used in modeling and simulation in many disciplines such as fluid mechanics, solid mechanics, and electromagnetism. However, due to the nonlinearity, geometric complexity, or irregularity of boundary conditions inherent in the problems themselves, most PDEs are difficult or even impossible to solve analytically. Therefore, developing efficient and high-precision numerical solution methods has always been a core issue and cutting-edge direction in scientific computing and engineering simulation. Against this backdrop, developing numerical solvers for PDEs that combine fast computation capabilities with high accuracy for complex geometric regions has become a key challenge driving technological progress in related fields.

[0003] Currently, numerical methods for solving partial differential equations can be broadly categorized into two types: mesh-based methods and meshless methods. Mesh-based methods primarily include the finite difference method, the finite element method, and the finite volume method. Mesh-based methods discretize the continuous computational domain into interconnected mesh elements and construct approximate solutions on these elements. These methods have a well-established theoretical framework with a rigorous theoretical foundation and clear convergence and stability analyses, thus finding widespread application in the numerical simulation of numerous scientific and engineering problems. However, the limitations of these methods stem from their strong dependence on the mesh. When dealing with complex geometries (such as porous media and internal flow channels in fluid machinery), generating a high-quality mesh is itself a specialized and time-consuming preprocessing task. The mesh quality directly determines the computational accuracy and stability, and manual intervention is costly. When facing large deformation problems (such as metal forming and soft tissue deformation), the mesh may suffer severe distortion, leading to a sharp reduction in the time step or even computational interruption. This often necessitates complex mesh re-distribution or mapping techniques, introducing additional errors and computational burden. When performing dynamic adaptive encryption (such as shock wave and phase change interface capture), the mesh density needs to be adjusted in real time during the simulation. This not only involves complex algorithms, but also requires maintaining the demapping accuracy and conservation between the old and new meshes, making it difficult to implement and resulting in significant computational overhead.

[0004] To overcome the bottlenecks in the aforementioned complex application scenarios, meshless methods, which do not rely on fixed mesh topology, have shown greater adaptability and flexibility in dealing with complex geometry and large deformation problems, and have gradually developed into an important numerical simulation framework that has attracted much attention. However, there is still a lack of specific solutions for partial differential equations based on meshless methods. Summary of the Invention

[0005] In view of this, and to address the above shortcomings, a fast solution method and solver for partial differential equations based on hierarchical radial basis functions is proposed to achieve fast solution of partial differential equations in a meshless manner.

[0006] In a first aspect, the present invention provides a fast solution method for partial differential equations based on hierarchical radial basis functions, including:

[0007] Obtain the partial differential equation to be solved and the computational domain; wherein, the computational domain is jointly defined by the set of center points, the set of placement points, and the set of evaluation points;

[0008] Nesting constraints are imposed on the set of center points to construct a nested sequence, so that the low-level point set covers the global domain and the high-level point set achieves coarse-scale global representation and fine-scale local focus through local encryption;

[0009] Based on the nested relation constraints imposed on the center point set, a radial basis function with an adaptive support radius is assembled for each scale level to obtain a variable support kernel function;

[0010] The hierarchical construction method is used to establish the test space of hierarchical radial basis functions;

[0011] Based on the proposed space of the hierarchical radial basis functions, the partial differential equation to be solved is obtained by the assembly point method of the hierarchical radial basis functions.

[0012] Preferably, the step of applying nesting relation constraints to the set of center points and constructing a nested sequence includes:

[0013] Apply nesting constraints to the set of center points X to construct a nested sequence. ,satisfy ; where, the set of center points for each layer is defined as , This is the set of new center points at level j compared to level j-1.

[0014] Preferably, the radial basis function for each scale level assembly with an adaptive support radius includes:

[0015] For the newly added set of center points in the j-th layer any node in Assemble a kernel function for it. Its support radius is determined by the shape parameters of this scale. Dynamic control is represented as follows: ;

[0016] in, Let d be a compactly supported radial basis function, d be the spatial dimension, and x be the region. Configuration points on Center point.

[0017] Preferably, the shape parameters The fill distance between the node and the current layer satisfies the following relationship:

[0018] in It is a constant. For the first The fill distance of each horizontal node.

[0019] Preferably, the step of establishing the hierarchical radial basis function trial space using the hierarchical construction method includes:

[0020] Define a subspace that grows from the newly added kernel function. , means as follows:

[0021] Where span is the set of all linear combinations of these vectors;

[0022] The space is explored by layer-by-layer subspace construction and the establishment of a complete hierarchical radial basis function. This results in a large support domain for low-level basis functions, which are responsible for global coarse approximation, while a small support domain for high-level basis functions, which focus on local details; as shown below: .

[0023] Preferably, when the partial differential equation to be solved is a two-dimensional time-independent partial differential equation, the step of solving the partial differential equation using the rank radial basis function assembly point method includes:

[0024] The two-dimensional time-independent partial differential equation is expressed as follows: (1)

[0025] Where L and B are the interior and boundary differential operators, respectively. Let be the function to be solved. and Represent the source term and boundary function, respectively;

[0026] Based on the trial space of hierarchical radial basis functions , consisting of a set of hierarchical radial basis functions For the function to be solved To approximate, it can be represented as follows: (2)

[0027] in, Indicates scale level. This represents the number of center points on the i-th level. The coefficients to be determined are: Add a set of center points for the i-th layer Nodes in;

[0028] Let Y1 and Y2 be the internal and boundary point sets, respectively. Substituting equation (2) into equation (1), the equation is discretized as follows: (3)

[0029] in, Indicates a configuration point;

[0030] Formula (3) can be written in matrix form as follows: (4)

[0031] Where LA and bdy represent the internal configuration matrix and the boundary configuration matrix, respectively;

[0032] MATLAB is called to solve formula (4), output the coefficient c to be solved, and the function to be solved is obtained based on the coefficient c.

[0033] Preferably, when the partial differential equation to be solved is a two-dimensional time-dependent partial differential equation, the step of solving the partial differential equation using the rank radial basis function assembly point method includes:

[0034] The two-dimensional time-dependent partial differential equation is expressed as follows: (5)

[0035] Where L and B are the interior and boundary differential operators, respectively, L t For time differential operators, Let be the function to be solved. and Represent the source term and boundary function, respectively;

[0036] The partial differential equation is discretized using the finite difference method, as follows: (6)

[0037] Substituting formula (2) into formula (6) makes the equation discretized as follows: (7)

[0038] Furthermore, formula (7) can be written in matrix form as follows: (8)

[0039] Use MATLAB to solve equation (8) and output the coefficients to be determined. And based on the coefficient to be determined The numerical solution of the function to be solved at time n+1 is obtained.

[0040] Secondly, the present invention provides a fast solver for partial differential equations based on hierarchical radial basis functions, comprising: an equation acquisition module, a nested relation application module, a radial basis function assembly module, a trial space establishment module, and an equation solving module;

[0041] The equation acquisition module is configured to acquire the partial differential equation to be solved and the computational domain; wherein, the computational domain is jointly defined by the center point set, the configuration point set, and the evaluation point set;

[0042] The nesting relationship application module is configured to apply nesting relationship constraints to the set of center points and construct a nested sequence so that the low-level point set covers the global domain and the high-level point set achieves coarse-scale global representation and fine-scale local focus through local encryption.

[0043] The radial basis function assembly module is configured to assemble radial basis functions with adaptive support radii for each scale level based on the nested relationship constraints applied to the center point set, thereby obtaining a variable support kernel function;

[0044] The trial space establishment module is configured to establish a hierarchical radial basis function trial space using a hierarchical construction method.

[0045] The equation solving module is configured to solve the partial differential equation to be solved using the assembly point method of the hierarchical radial basis function, based on the hierarchical radial basis function trial space.

[0046] Thirdly, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed in a computer, causes the computer to perform any of the methods described in the first aspect.

[0047] Fourthly, the present invention provides a computing device including a memory and a processor, wherein the memory stores executable code, and when the processor executes the executable code, it implements any of the methods described in the first aspect.

[0048] As can be seen from the above technical solution, the fast solution method and solver for partial differential equations based on hierarchical radial basis functions provided in this invention includes obtaining the partial differential equation to be solved and the computational domain jointly defined by the center point set, the configuration point set, and the evaluation point set. Then, nesting constraints are applied to the center point set to construct a nested sequence, so that the low-level point set covers the global domain, and the high-level point set achieves coarse-scale global representation and fine-scale local focusing through local refinement. Further, based on the nesting constraints applied to the center point set, radial basis functions with adaptive support radii are assembled for each scale level to obtain variable-support kernel functions. Then, a hierarchical construction method is used to establish a hierarchical radial basis function trial space. Finally, based on the hierarchical radial basis function trial space, the partial differential equation to be solved is solved using the hierarchical radial basis function assembly point method. Therefore, this solution, by introducing a multi-scale framework and variable-support radial basis functions, utilizes the local compact support characteristics of the kernel function to transform the discrete system into a sparse structure. This meshless solution method significantly reduces matrix storage and single-operation overhead. At the same time, it enables the solver to theoretically possess excellent scalability for handling large-scale complex geometric problems, and provides an effective way to achieve fast and high-precision engineering calculations in practice. Attached Figure Description

[0049] Figure 1 The flowchart illustrates a method for rapidly solving partial differential equations based on hierarchical radial basis functions, as provided in this embodiment of the invention.

[0050] Figure 2 This invention provides a fast solver for partial differential equations based on hierarchical radial basis functions.

[0051] Figure 3 This is a comparison of the hierarchical radial basis function (right) of the present invention with the hierarchical basis function of the finite element method (left).

[0052] Figure 4 This is a diagram showing the distribution of complex geometric domains and gridless points in this invention.

[0053] Figure 5 For multi-subdomain partitioning and hierarchical radial basis function point layout.

[0054] Figure 6 This is a velocity distribution diagram at different times based on the Chorin vortex decay test of the present invention using large eddy simulation.

[0055] Figure 7 This is a comparison of the CPU time of the parallel algorithm and the serial algorithm of this invention.

[0056] Figure 8 This is a schematic diagram of the computational domain and collocation points for a backward stepped flow experiment.

[0057] Figure 9 This invention uses the Smagorinsky model to simulate the flow field state at different times in a backward stepped flow.

[0058] Figure 10 This invention uses the Smagorinsky model to simulate the velocity distribution at different times in a backward stepped flow. Detailed Implementation

[0059] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0060] like Figure 1 As shown, this embodiment of the invention provides a fast solution method for partial differential equations based on hierarchical radial basis functions. This method may include the following steps:

[0061] Step 101: Obtain the partial differential equation to be solved and the computational domain; wherein, the computational domain is jointly defined by the set of center points, the set of placement points, and the set of evaluation points;

[0062] Step 102: Apply nesting relationship constraints to the set of center points and construct a nested sequence so that the low-level point set covers the global domain and the high-level point set achieves coarse-scale global representation and fine-scale local focus through local encryption;

[0063] Step 103: Based on the nested relation constraints applied to the center point set, assemble radial basis functions with adaptive support radii for each scale level to obtain variable support kernel functions;

[0064] Step 104: Establish the hierarchical radial basis function trial space using the hierarchical construction method;

[0065] Step 105: Based on the proposed space of the hierarchical radial basis functions, solve the partial differential equation to be solved using the assembly point method of the hierarchical radial basis functions.

[0066] In this embodiment, by introducing a nested point set multi-scale framework and a variable-support radial basis function, the discrete system is transformed into a sparse structure by utilizing the local compact support property of the kernel function. This meshless solution method significantly reduces matrix storage and single-operation overhead. Simultaneously, it theoretically endows the solver with excellent scalability for handling large-scale complex geometric problems and provides an effective approach for achieving fast and high-precision engineering calculations in practice.

[0067] The following provides a more detailed explanation of each step in the fast solution method for partial differential equations based on hierarchical radial basis functions provided by this invention.

[0068] For step 101, obtain the partial differential equation to be solved and the computational domain; wherein, the computational domain is jointly defined by the center point set, the configuration point set and the evaluation point set;

[0069] In this embodiment, the partial differential equation to be solved can be either a two-dimensional non-time-dependent partial differential equation or a two-dimensional time-dependent partial differential equation. That is, the solution method provided by this scheme can be applied to both time-dependent and time-independent partial differential equations.

[0070] For step 102, nesting relationship constraints are applied to the set of center points to construct a nested sequence so that the low-level point set covers the global domain and the high-level point set achieves coarse-scale global representation and fine-scale local focus through local encryption.

[0071] In this step, we consider constructing a structure that satisfies... The nested point sets provide a spatial basis for multi-scale approximation. Lower-level point sets cover the global domain, while higher-level point sets achieve "coarse-scale global representation + fine-scale local focus" through local refinement, avoiding the grid dependency of traditional methods. Specifically, in the fast solution architecture of this invention, the discretization of the computational domain is defined by three types of point sets with different functions: the center point set X, used to generate radial basis functions; the configuration point set Y, used to discretize partial differential equations and boundary conditions; and the evaluation point set E, used for the output and visualization of the final solution. All three types of point sets can use scattered nodes, unrestricted by any grid structure, thus possessing inherent flexibility in handling complex geometries. In particular, to achieve multi-scale solutions, strict nesting constraints are imposed on the center point set X. For example, a nested sequence is constructed. ,satisfy The set of center points for each layer is defined as follows: and This refers to the set of center points added at level j compared to level j-1. (Level j center points) A globally sparse distribution is used to quickly capture low-frequency trends in solutions; a new set of center points is subsequently added. Based on the error estimate or physical field gradient of the previous layer's solution, the density can be adaptively increased in key regions (such as the boundary layer and near singularities). This nested center point generation strategy lays the foundation for constructing a hierarchical radial basis function space and is the key to driving the entire multi-scale solver to achieve efficient "coarse-to-fine" solutions.

[0072] For step 103, based on the nested relation constraints applied to the center point set, a radial basis function with an adaptive support radius is assembled for each scale level to obtain a variable support kernel function;

[0073] In this step, we consider defining a multi-scale nested center point set. Based on this, a radial basis function with an adaptive support radius, i.e., a variable support kernel function, is assembled for each scale level j. Specifically, for the newly added set of center points in the j-th layer... any node in Assemble a kernel function for it. Its support radius is determined by the shape parameters of this scale. Dynamic control: ;

[0074] in, Let d be a compactly supported radial basis function, d be the spatial dimension, and x be the region. Configuration points on Center point. Shape parameters The fill distance from the current layer's nodes satisfies the following relationship: ;

[0075] in, It is a constant. For the first The filling distance of nodes at each level. The core of this multi-scale scheme lies in the fact that at a coarser scale, the kernel function has a larger support domain to capture the global trend of the solution; while at a finer scale, the support domain of the kernel function adaptively shrinks to capture local features. This idea of ​​adaptive scaling of the support radius with node density effectively reduces the impact of [the problem], and is one of the core mechanisms that enables this solver to achieve both high accuracy and high computational efficiency.

[0076] For step 104, the hierarchical construction method is used to establish the hierarchical radial basis function test space;

[0077] In this embodiment, a hierarchical radial basis function (RBF) trial space is constructed. The trial space is spanned by RBFs of different levels. Lower-level RBFs have large support domains and are responsible for global coarse approximation, while higher-level RBFs have small support domains and focus on local details. This framework effectively improves computational accuracy and significantly reduces CPU time. Specifically, it is based on a defined multi-scale node set. With variable support kernel function This step details the hierarchical construction method used in the invention to establish the final trial function space. The core idea of ​​this method is to construct a multi-level sequence of function spaces by introducing kernel functions associated with newly added nodes layer by layer, thus laying the foundation for multi-scale solution algorithms. First, the subspace spanned by the newly added kernel functions at this scale is defined. : ;

[0078] Here, span is the set of all linear combinations of these vectors.

[0079] Secondly, based on this, a complete hierarchical radial basis function trial space is constructed through the layer-by-layer summation of subspaces. : ;

[0080] This idea only requires introducing new basis functions related to the newly added nodes at each layer, so that the structural sparsity of the discrete system can be inherited and controlled layer by layer while maintaining the approximation ability, thus theoretically ensuring the optimal balance between computational accuracy and efficiency.

[0081] Step 105: Based on the proposed space of the hierarchical radial basis functions, solve the partial differential equation to be solved using the assembly point method of the hierarchical radial basis functions.

[0082] In this step, we will take two-dimensional time-independent partial differential equations that are independent of time and two-dimensional time-dependent partial differential equations that are dependent on time as examples to explain the solution of this step in detail.

[0083] (1) The partial differential equation to be solved is a two-dimensional, time-independent partial differential equation.

[0084] When the partial differential equation to be solved is a two-dimensional non-time-dependent partial differential equation, it is expressed as follows: (1)

[0085] Where L and B are the interior and boundary differential operators, respectively. Let be the function to be solved. and Represent the source term and boundary function, respectively;

[0086] Based on the trial space of hierarchical radial basis functions , consisting of a set of hierarchical radial basis functions For the function to be solved To approximate, it can be represented as follows: (2)

[0087] in, Indicates scale level. This represents the number of center points on the i-th level. The coefficients to be determined are: Add a set of center points for the i-th layer Nodes in;

[0088] Let Y1 and Y2 be the internal and boundary point sets, respectively. Substituting equation (2) into equation (1), the equation is discretized as follows: (3)

[0089] in, Indicates a configuration point;

[0090] Formula (3) can be written in matrix form as follows: (4)

[0091] Where LA and bdy represent the internal configuration matrix and the boundary configuration matrix, respectively, they can be represented as follows: ; ; ; ;

[0092] Furthermore, MATLAB is called to solve formula (4), output the coefficients to be solved c, and the function to be solved is obtained based on the coefficients to be solved c.

[0093] (2) The partial differential equation to be solved is a two-dimensional time-dependent partial differential equation.

[0094] When the partial differential equation to be solved is a two-dimensional time-dependent partial differential equation, the two-dimensional time-dependent partial differential equation is expressed as follows: (5)

[0095] Where L and B are the interior and boundary differential operators, respectively, L t For time differential operators, Let be the function to be solved. and Represent the source term and boundary function, respectively;

[0096] The partial differential equation is discretized using the finite difference method. Taking the fully implicit scheme as an example, the equation can be discretized as follows: (6)

[0097] Substituting formula (2) into formula (6) makes the equation discretized as follows: (7)

[0098] Furthermore, formula (7) can be written in matrix form as follows: (8)

[0099] Where A is the block matrix, represented as follows: ;

[0100] Furthermore, the linear equation system of formula (8) is solved by calling the " / " command in MATLAB, and the coefficients to be determined are output. and the coefficient to be determined Substitute the function to be solved The numerical solution at time n+1 is obtained.

[0101] Furthermore, after the solution is completed, the solution can be visualized and its accuracy verified. Specifically, this is done when solving a linear system to obtain the coefficient vector. Then, by substituting into the approximation expression (2), any evaluation point on the computational domain can be reconstructed. Numerical solution at [location] To comprehensively evaluate the effectiveness and superiority of the fast solver proposed in this invention, systematic post-processing is required, mainly including visualization, quantitative error analysis, and CPU time comparison analysis.

[0102] Firstly, in terms of visualization, after obtaining the numerical solution... Subsequently, a multi-mode visualization strategy was adopted to address the physical meaning of solution fields for different types of partial differential equations: for scalar fields (such as temperature and pressure), their global distribution and gradient changes were displayed using two-dimensional / three-dimensional contour maps and topographic maps; for vector fields (such as velocity and stress), their direction and intensity were presented using vector arrow diagrams or streamline diagrams to characterize the evolution of eddies; for eigenvalue or frequency domain problems, modal shape contour maps were used to reveal the vibration patterns. This process not only visually verifies the continuity and boundary fit of the solution, but also provides crucial evidence for qualitative comparison with reference solutions and identification of non-physical oscillations.

[0103] Secondly, in obtaining numerical solutions With analytical solution Following this, rigorous quantitative error analysis is necessary to accurately evaluate the solver's numerical performance. The fundamental significance of this process lies in transforming the method's accuracy from a qualitative judgment into a quantifiable and comparable objective indicator, thereby providing data support for algorithm verification, parameter optimization, and the reliability of results in engineering applications. This is achieved by defining the maximum absolute error as an error metric. The root mean square (RMS) error is used to calculate the worst-case deviation over the region and the overall approximation accuracy. Quantitative error analysis provides a solid basis for verifying the convergence of the solver, evaluating its engineering applicability, and making objective comparisons with other methods. Therefore, it is an indispensable key step in judging whether the solver has high reliability.

[0104]

[0105]

[0106] Finally, a comparative analysis of CPU time highlights the core advantages of this solver from a computational efficiency perspective. Compared to the high computational complexity caused by dense matrices in traditional global collocation methods, this method introduces a multi-scale framework and variable-support radial basis functions, utilizing the local compact support properties of kernel functions to transform discrete systems into sparse structures, significantly reducing matrix storage and single-operation overhead. This theoretically endows the solver with excellent scalability for handling large-scale complex geometric problems and provides an effective approach for achieving fast and high-precision engineering calculations in practice.

[0107] like Figure 2 As shown, the present invention also provides a fast solver for partial differential equations based on hierarchical radial basis functions, including: an equation acquisition module 201, a nested relation application module 202, a radial basis function assembly module 203, a trial space establishment module 204, and an equation solving module 205;

[0108] The equation acquisition module 201 is configured to acquire the partial differential equation to be solved and the computational domain; wherein, the computational domain is jointly defined by the center point set, the configuration point set, and the evaluation point set;

[0109] The nesting relationship application module 202 is configured to apply nesting relationship constraints to the set of center points and construct a nested sequence so that the low-level point set covers the global domain and the high-level point set achieves coarse-scale global representation and fine-scale local focus through local encryption.

[0110] The radial basis function assembly module 203 is configured to assemble radial basis functions with adaptive support radii for each scale level based on the nested relationship constraints applied to the center point set, thereby obtaining a variable support kernel function;

[0111] The trial space establishment module 204 is configured to establish a hierarchical radial basis function trial space using a hierarchical construction method.

[0112] The equation solving module 205 is configured to solve the partial differential equation to be solved using the assembly point method of the hierarchical radial basis function based on the hierarchical radial basis function trial space.

[0113] The effects of the present invention will be further explained below with reference to the accompanying drawings.

[0114] Figure 3This image compares hierarchical radial basis functions (right) with finite element hierarchical basis functions (left). Both types of basis functions follow multi-scale evolution, a characteristic that allows them to achieve multi-scale modeling through hierarchical progression, combining coarse-scale global representation with fine-scale local refinement. This effectively balances computational efficiency with solution accuracy, making them suitable for multi-scale problems in fields such as fluid mechanics and solid mechanics. Compared to finite element hierarchical radial basis functions, hierarchical radial basis functions possess smoothness characteristics that allow for arbitrary-order differentiability. Compared to piecewise linear basis functions in finite elements, they can more accurately capture the higher-order derivative characteristics of physical fields, such as vorticity fields in fluid mechanics and stress gradient fields in solid mechanics. In terms of computational efficiency, hierarchical radial basis functions achieve true multi-scale adaptation through radially symmetric basis function distribution, completing local refinement without global mesh reconstruction, significantly reducing computational costs.

[0115] Figure 4 Several typical complex geometric regions and their corresponding meshless point distributions are presented. The geometric domains in the upper row all exhibit significant irregularities, such as abrupt changes in boundary curvature, discontinuous or abrupt boundaries, thin-walled structures with spatial curves, and non-fixed topological morphology. These characteristics pose significant challenges to traditional meshing methods. The meshless point distributions presented in the lower row clearly demonstrate the high flexibility of this method in complex regions. Points can achieve a fully adaptive spatial layout based on geometric features and solution requirements, such as automatic densification in regions with large boundary curvature, natural alignment along thin-walled structures, and maintaining reasonable sparsity at internal holes or gaps. This characteristic not only completely eliminates the topological constraints of structured meshes but also dynamically adjusts node density based on local physical properties, thereby achieving optimal distribution of computational resources without manual intervention. The "arbitrariness" and "adaptability" of point distribution are the core advantages of meshless methods in handling complex geometric problems, enabling them to effectively adapt to geometric and topological complexities that are difficult for traditional meshing methods to handle while maintaining high accuracy.

[0116] Figure 5 This paper demonstrates the multi-subdomain partitioning and hierarchical radial basis function (RBF) collocation layout based on a domain decomposition method. Hexagonal stars represent physical boundary points, circles represent virtual boundary points, and solid points represent internal collocation points. For example... Figure 5 As shown, the domain decomposition method divides a large region into multiple smaller regions for solution and uses artificially constructed virtual boundaries to facilitate information exchange between adjacent subdomains, effectively reducing the computational load.

[0117] Figure 6 The velocity magnitude distribution at different times is shown in the Chorin vortex decay test based on large eddy simulation. Figure 6The numerical results accurately capture the dynamic evolution of characteristic high-frequency oscillation modes in the velocity field: in the initial stage, the field exhibits a significant high-frequency wave structure, whose spatial scale and amplitude characteristics are highly consistent with the analytical solution; as time evolves, the velocity magnitude gradually decreases. This numerical behavior is consistent with the theoretical predictions, fully demonstrating the computational effectiveness, reliability, and stability of the domain decomposition-based hierarchical radial basis function method.

[0118] Figure 7 It shows the time step (Above) and (Bottom) Comparison of CPU time between parallel computing (blue bar) and serial computing (red bar), and the corresponding speedup ratio for different numbers of center points (green line, right vertical axis). The graph shows that the CPU time for both methods increases exponentially with the number of center points, but the serial algorithm grows significantly faster than the parallel algorithm. This is because the increased number of iterations required for convergence at larger time steps increases CPU time. While the overhead of communication and synchronization may mask the benefits of parallel computing at smaller point sets, its computational efficiency becomes increasingly apparent as the problem size increases.

[0119] Figure 8 The computational domain and multi-subdomain point distribution diagram of the backward stepped flow experiment are shown, where hexagons represent physical boundary points, circles represent virtual boundary points, and solid dots represent internal points. This invention does not rely on a mesh; it only requires node information within the boundary and computational domain to perform numerical solutions. This provides a discrete foundation for high-precision simulation of complex flow phenomena such as the evolution of separated eddies and pressure fluctuations in backward stepped flow.

[0120] Figure 9 The streamline diagrams of the backward step flow at different time points are presented, with a Reynolds number Re = 100. A total of 5866 center points were set up across the entire flow domain for numerical simulation. To ensure numerical stability during the time progression, a fixed time step was used for time discretization. In the backward step flow, a single main vortex forms downstream of the step at time T = 1, characterized by a closed return streamline. As time evolves, this vortex gradually expands in both the flow direction and vertical direction. By time T = 4-5, the vortex growth tends to stabilize, entering a quasi-steady-state stage—its size and streamline morphology no longer change, indicating that the flow has reached a steady state under the given Reynolds number Re = 100.

[0121] Figure 10This paper presents velocity maps at different times for a backward stepped flow at a Reynolds number Re = 100, simulated using a domain decomposition hierarchical radial basis function method. The flow forms a local recirculation region downstream of the step at time T = 1. Over time, this recirculation region expands along the flow direction and gradually develops into a stable quasi-steady-state structure. This invention accurately captures the evolution of eddies through the domain decomposition strategy and the multi-scale representation capability of hierarchical basis functions, verifying its comprehensive advantages in geometric adaptability, dynamic process capture accuracy, and numerical stability in simulating complex separated flows.

[0122] Table 1

[0123] Table 1 shows the solution of two-dimensional flower domains using the hierarchical radial basis function method (see Table 1). Figure 4 The RMS error and convergence order of the Helmholtz equation with up-variable wavenumber are presented. Even for complex computational domains, the RMS error at different wavenumbers decreases significantly with increasing number of centroids. However, under the same node density, higher wavenumbers correspond to larger RMS errors. Nevertheless, the hierarchical radial basis function method maintains high accuracy when dealing with high-frequency fluctuations, especially in cases of dense node distribution.

[0124] Table 2

[0125] Table 2 shows the solution of the three-dimensional S-shaped pipe domain using the hierarchical radial basis function method (see Table 2). Figure 4 The RMS error and convergence order of the Helmholtz equation with up-variable wavenumber are presented. This invention can be easily extended to high-dimensional problems; even for complex three-dimensional computational domains, the RMS error at different wavenumbers shows a significant decreasing trend as the number of centroids increases. The hierarchical radial basis function method still exhibits good numerical stability and accuracy preservation in three-dimensional high-frequency wave simulations, especially in regions with dense node distribution.

[0126] This specification also provides a computer-readable storage medium having a computer program stored thereon, which, when executed in a computer, causes the computer to perform the methods in any of the embodiments of the specification.

[0127] This specification also provides a computing device, including a memory and a processor, wherein the memory stores executable code, and when the processor executes the executable code, it implements the method in any of the embodiments of the specification.

[0128] The device embodiments provided by the present invention are based on the same inventive concept as the method embodiments in this specification. For details, please refer to the description in the method embodiments of this specification, which will not be repeated here.

[0129] The modules or units in the device of this invention can be merged, divided, and deleted according to actual needs. The above-disclosed embodiments are merely preferred embodiments of the present invention and should not be construed as limiting the scope of the invention. Those skilled in the art will understand that implementing all or part of the processes of the above embodiments and making equivalent changes according to the claims of this invention still fall within the scope of the invention.

Claims

1. A fast solution method for partial differential equations based on hierarchical radial basis functions, characterized in that, include: Obtain the partial differential equation to be solved and the computational domain; wherein, the computational domain is jointly defined by the set of center points, the set of placement points, and the set of evaluation points; Nesting constraints are imposed on the set of center points to construct a nested sequence, so that the low-level point set covers the global domain and the high-level point set achieves coarse-scale global representation and fine-scale local focus through local encryption; Based on the nested relation constraints imposed on the center point set, a radial basis function with an adaptive support radius is assembled for each scale level to obtain a variable support kernel function; The hierarchical construction method is used to establish the test space of hierarchical radial basis functions; Based on the proposed space of the hierarchical radial basis functions, the partial differential equation to be solved is obtained by the assembly point method of the hierarchical radial basis functions.

2. The method for fast solution of partial differential equations based on hierarchical radial basis functions according to claim 1, characterized in that, The step of applying nested relation constraints to the set of center points and constructing a nested sequence includes: Apply nesting constraints to the set of center points X to construct a nested sequence. ,satisfy ; where, the set of center points for each layer is defined as , This is the set of new center points at level j compared to level j-1.

3. The method for fast solution of partial differential equations based on hierarchical radial basis functions according to claim 2, characterized in that, The radial basis function for each scale level assembly with an adaptive support radius includes: For the newly added set of center points in the j-th layer any node in Assemble a kernel function for it. Its support radius is determined by the shape parameters of this scale. Dynamic control is represented as follows: ； in, Let d be a compactly supported radial basis function, d be the spatial dimension, and x be the region. Configuration points on Center point.

4. The method for fast solution of partial differential equations based on hierarchical radial basis functions according to claim 3, characterized in that, The shape parameters The fill distance between the node and the current layer satisfies the following relationship: ； in It is a constant. For the first The fill distance of each horizontal node.

5. The method for fast solution of partial differential equations based on hierarchical radial basis functions according to claim 4, characterized in that, The method of establishing a hierarchical radial basis function trial space using a hierarchical construction method includes: Define a subspace that grows from the newly added kernel function. , means as follows: ； Where span is the set of all linear combinations of these vectors; The space is explored by layer-by-layer subspace construction and the establishment of a complete hierarchical radial basis function. This results in a large support domain for low-level basis functions, which are responsible for global coarse approximation, while a small support domain for high-level basis functions, which focus on local details; as shown below: 。 6. The method for fast solution of partial differential equations based on hierarchical radial basis functions according to claim 5, characterized in that, When the partial differential equation to be solved is a two-dimensional, time-independent partial differential equation, the step of solving the partial differential equation using the rank radial basis function assembly point method includes: The two-dimensional time-independent partial differential equation is expressed as follows: （1） Where L and B are the interior and boundary differential operators, respectively. Let be the function to be solved. and Represent the source term and boundary function, respectively; Based on the trial space of hierarchical radial basis functions , consisting of a set of hierarchical radial basis functions For the function to be solved To approximate, it can be represented as follows: （2） in, Indicates scale level. This represents the number of center points at the i-th level. The coefficients to be determined are: Add a set of center points for the i-th layer Nodes in; Let Y1 and Y2 be the internal and boundary point sets, respectively. Substituting equation (2) into equation (1), the equation is discretized as follows: （3） in, Indicates a configuration point; Formula (3) can be written in matrix form as follows: （4） Where LA and bdy represent the internal configuration matrix and the boundary configuration matrix, respectively; MATLAB is called to solve formula (4), output the coefficient c to be solved, and the function to be solved is obtained based on the coefficient c.

7. The method for fast solution of partial differential equations based on hierarchical radial basis functions according to claim 6, characterized in that, When the partial differential equation to be solved is a two-dimensional time-dependent partial differential equation, the step of solving the partial differential equation using the rank radial basis function assembly point method includes: The two-dimensional time-dependent partial differential equation is expressed as follows: （5） Where L and B are the interior and boundary differential operators, respectively, L t For time differential operators, Let be the function to be solved. and Represent the source term and boundary function, respectively; The partial differential equation is discretized using the finite difference method, as follows: （6） Substituting formula (2) into formula (6) makes the equation discretized as follows: （7） Furthermore, formula (7) can be written in matrix form as follows: （8） Use MATLAB to solve equation (8) and output the coefficients to be determined. And based on the coefficient to be determined The numerical solution of the function to be solved at time n+1 is obtained.

8. A fast solver for partial differential equations based on hierarchical radial basis functions, characterized in that, include: The module includes: equation acquisition module, nested relation application module, radial basis function assembly module, trial space establishment module, and equation solving module. The equation acquisition module is configured to acquire the partial differential equation to be solved and the computational domain; wherein, the computational domain is jointly defined by the center point set, the configuration point set, and the evaluation point set; The nesting relationship application module is configured to apply nesting relationship constraints to the set of center points and construct a nested sequence so that the low-level point set covers the global domain and the high-level point set achieves coarse-scale global representation and fine-scale local focus through local encryption. The radial basis function assembly module is configured to assemble radial basis functions with adaptive support radii for each scale level based on the nested relationship constraints applied to the center point set, thereby obtaining a variable support kernel function; The trial space establishment module is configured to establish a hierarchical radial basis function trial space using a hierarchical construction method. The equation solving module is configured to solve the partial differential equation to be solved using the assembly point method of the hierarchical radial basis function, based on the hierarchical radial basis function trial space.

9. A computer-readable storage medium having a computer program stored thereon, which, when executed in a computer, causes the computer to perform the method described in any one of claims 1-7.

10. A computing device, comprising a memory and a processor, wherein executable code is stored in the memory, and when the processor executes the executable code, it implements the method of any one of claims 1-7.

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