CN-121996884-A - Quick partial differential equation solving method and quick partial differential equation solving device based on grade radial basis function

CN121996884ACN 121996884 ACN121996884 ACN 121996884ACN-121996884-A

Abstract

The invention provides a quick partial differential equation solving method and a quick partial differential equation solver based on a hierarchical radial basis function, which relate to the technical field of computers and numerical computation, and comprise the steps of obtaining a partial differential equation and a computing area to be solved; the method comprises the steps of defining a calculation area by a central point set, a configuration point set and an evaluation point set, applying a nested relation constraint to the central point set to construct a nested sequence, enabling a low-level point set to cover a global area, enabling a high-level point set to realize coarse-scale global representation and fine-scale local focusing through local encryption, horizontally assembling radial basis functions with self-adaptive supporting radiuses for each scale on the basis of the nested relation constraint applied to the central point set to obtain a variable support kernel function, establishing a level radial basis function heuristic space by adopting a level construction method, and solving partial differential equations to be solved through the level radial basis function assembly point method on the basis of the level radial basis function heuristic space. The scheme can realize the quick solution of the partial differential equation in a gridless mode.

Inventors

LIU ZHIYONG
XU QIUYAN
YANG JIYE
LIU HAOWEI

Assignees

宁夏大学

Dates

Publication Date: 20260508
Application Date: 20260127

Claims (10)

1. A partial differential equation quick solving method based on a hierarchical radial basis function is characterized by comprising the following steps: Acquiring a partial differential equation to be solved and a calculation area, wherein the calculation area is defined by a center point set, a configuration point set and an evaluation point set together; a nested relation constraint is applied to the center point set, a nested sequence is constructed, 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; On the basis of applying the nested relation constraint to the center point set, horizontally assembling a radial basis function with a self-adaptive supporting radius for each scale to obtain a variable support kernel function; Establishing a hierarchical radial basis function heuristic space by adopting a hierarchical construction method; and solving the partial differential equation to be solved by a hierarchical radial basis function assembly point method based on the hierarchical radial basis function heuristic space.
2. The method of claim 1, wherein said applying a nested relationship constraint to said set of center points to construct a nested sequence comprises: Applying a nested relation constraint to the center point set X to construct a nested sequence Satisfies the following conditions Wherein each layer center point set is defined as , A set of center points for the j-th level as compared to the j-1-th level.
3. The quick solution method of partial differential equation based on hierarchical radial basis functions according to claim 2, wherein said fitting a radial basis function with an adaptive support radius for each scale level comprises: Newly added set of center points for layer j Any node in (a) To which a kernel function is fitted The supporting radius of the support is determined by the shape parameter of the scale Dynamic control, expressed as follows: ; Wherein, the For tightly-packed radial basis functions, d is the spatial dimension, x is the region At the point of the configuration on the upper surface, Is the center point.
4. A method for rapid solution of partial differential equations based on hierarchical radial basis functions as claimed in claim 3, wherein said shape parameters The contact filling distance with the current layer satisfies the following relation: ; Wherein the method comprises the steps of Is a constant value, and the constant value is a constant value, Is the first Filling distance of the nodes under the level.
5. The method for quickly solving partial differential equations based on hierarchical radial basis functions according to claim 4, wherein the step of establishing a hierarchical radial basis function heuristic space by using a hierarchical construction method comprises the steps of: Defining subspaces grown by newly added kernel functions The expression is as follows: ; Where span is the set of all linear combinations of these vectors; Layer-by-layer and construction of complete hierarchical radial basis function heuristic space through subspaces The support domain of the low-level basic function is made large and is responsible for global coarse approximation, the support domain of the high-level basic function is small and local detail is focused, and the method is expressed as follows: 。
6. The method for quickly solving partial differential equations based on the hierarchical radial basis functions according to claim 5, wherein when the partial differential equations to be solved are two-dimensional time-independent partial differential equations, the method for solving the partial differential equations to be solved by the hierarchical radial basis function fitting point method comprises: the two-dimensional time-independent partial differential equation is expressed as follows: (1) wherein L and B are the internal and boundary differential operators, respectively, For the function to be solved, And Respectively representing a source item and a boundary function; Heuristic space based on hierarchical radial basis function From a set of hierarchical radial basis functions Function to be solved The approximation is performed as follows: (2) Wherein, the Representing the level of the scale, Indicating the number of center points on the ith level, For the coefficients to be found, the coefficients, Newly added center point set for the ith layer Is a node in (a); let Y 1 and Y 2 be the internal configuration point set and the boundary configuration point set, respectively, substituting equation (2) into equation (1) to make the equation be discretized: (3) Wherein, the Representing a configuration point; the formula (3) is written as a matrix form as follows: (4) wherein LA, bdy represent an internal configuration matrix and a boundary configuration matrix, respectively; And (3) calling MATLAB to solve the formula (4), outputting a coefficient c to be solved, and obtaining a function to be solved according to the coefficient c to be solved.
7. The method for quickly solving partial differential equations based on the hierarchical radial basis functions according to claim 6, wherein when the partial differential equations to be solved are two-dimensional time-dependent partial differential equations, the solving the partial differential equations to be solved by the hierarchical radial basis function fitting point method includes: The two-dimensional time-dependent partial differential equation is expressed as follows: (5) Wherein L and B are inner and boundary differential operators, respectively, L t is a time differential operator, For the function to be solved, And Respectively representing a source item and a boundary function; The partial differential equation is discretized by a finite difference method, and is expressed as follows: (6) Substituting equation (2) into equation (6) to make the equation discrete as: (7) Further, the formula (7) is written as the following matrix form: (8) Calling MATLAB to solve the formula (8) and outputting coefficients to be solved And according to the coefficient to be solved And obtaining a numerical solution of the n+1 time of the function to be solved.
8. The partial differential equation quick solver based on the hierarchical radial basis function is characterized by comprising an equation acquisition module, a nested relation application module, a radial basis function assembly module, a heuristic space establishment module and an equation solving module; the equation acquisition module is configured to acquire a partial differential equation to be solved and a calculation area, wherein the calculation area is defined by a center point set, a configuration point set and an evaluation point set together; The nested relation applying module is configured to apply nested relation constraint to the center point set, and 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; the radial basis function assembling module is configured to horizontally assemble radial basis functions with self-adaptive supporting radiuses for each scale on the basis of applying nested relation constraint to the center point set, so as to obtain a variable support kernel function; The heuristic space establishing module is configured to establish a hierarchical radial basis function heuristic space by adopting a hierarchical construction method; the equation solving module is configured to solve the partial differential equation to be solved through a hierarchical radial basis function assembly point method based on the hierarchical radial basis function heuristic space.
9. A computer readable storage medium having stored thereon a computer program which, when executed in a computer, causes the computer to perform the method of any of claims 1-7.
10. A computing device comprising a memory having executable code stored therein and a processor, which when executing the executable code, implements the method of any of claims 1-7.

Description

Quick partial differential equation solving method and quick partial differential equation solving device based on grade radial basis function Technical Field The invention relates to the technical field of computers and numerical computation, in particular to a quick partial differential equation solving method and a quick partial differential equation solving device based on a hierarchical radial basis function. Background In the scientific and engineering fields, partial differential equations are core mathematical tools for describing complex physical phenomena and key processes, and are widely applied to modeling and simulation of various disciplines such as fluid mechanics, solid mechanics, electromagnetics and the like. However, most partial differential equations are difficult or even impossible to solve due to the nonlinearity of the problem itself, geometric complexity, or boundary condition irregularities. Therefore, developing a high-efficiency and high-precision numerical solution method is always a core subject and a leading edge direction in the fields of scientific calculation and engineering simulation. Under the background, developing a partial differential equation numerical solver with both rapid computing capability and high-precision guarantee for complex geometric areas has become a key challenge for promoting technical progress in the related fields. At present, the numerical solution method of partial differential equation can be mainly divided into two major categories of grid-based method and grid-free method. Among them, the grid-based method mainly has a finite difference method, a finite element method, a finite volume method, and the like. Grid-based methods discretize a continuous computational region into interconnected grid cells and construct an approximate solution on these cells. The method forms a quite mature theoretical system, has strict theoretical basis and clear analysis of convergence and stability, and is widely applied to numerical simulation of a plurality of scientific and engineering problems. However, the limitations of such methods are arising from their strong dependence on the grid. When complex geometric shapes (such as porous media and internal flow channels of fluid machinery) are processed, the generation of high-quality grids is a professional and time-consuming pretreatment work, the grid quality directly determines the calculation precision and stability, and the manual intervention cost is high. When faced with large deformation problems (such as metal forming and soft tissue deformation), the grid may be severely distorted, resulting in a sharp reduction of the time step and even a calculation interruption, and often complicated grid repartition or mapping techniques are required, thereby introducing additional errors and calculation amount. When dynamic self-adaptive encryption (such as shock wave and phase change interface capture) is carried out, the grid density needs to be adjusted in real time in the simulation process, so that the algorithm is complex, the demapping precision and conservation between the new grid and the old grid need to be maintained, the implementation difficulty is high, and the calculation cost is obvious. In order to break through the bottleneck in the complex application scene, the mesh-free method shows stronger adaptability and flexibility in coping with complex geometric and large deformation problems because the mesh-free method does not need to rely on a fixed mesh topological structure, and gradually develops into an important numerical simulation framework with great attention, but a specific partial differential equation solving scheme based on the mesh-free method is still lacking. Disclosure of Invention In view of the above, a quick solution method and a quick solution device for partial differential equations based on a hierarchical radial basis function are provided to realize quick solution for partial differential equations in a gridless manner. In a first aspect, the present invention provides a quick solution method for partial differential equations based on a hierarchical radial basis function, including: Acquiring a partial differential equation to be solved and a calculation area, wherein the calculation area is defined by a center point set, a configuration point set and an evaluation point set together; a nested relation constraint is applied to the center point set, a nested sequence is constructed, 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; On the basis of applying the nested relation constraint to the center point set, horizontally assembling a radial basis function with a self-adaptive supporting radius for each scale to obtain a variable support kernel function; Establishing a hierarchical radial basis function heuristic space by adopting a hierarchical