Numerical method and device for accurately applying essential boundary conditions for meshless method

Abstract

The invention discloses a numerical method and device for accurately applying essential boundary conditions for a meshless method, and relates to the technical field of computational mechanics, and the method comprises the steps: obtaining a displacement boundary curve and a force boundary curve of a mechanical problem solving domain; based on the displacement boundary curve and the force boundary curve, a B-spline basis function is adopted to construct a boundary area; constructing a boundary basis function based on the boundary region; constructing an allowable displacement solution of a to-be-solved mechanical problem based on the boundary basis function; the allowable displacement solution satisfies essential boundary conditions on the displacement boundary curve. According to the method, the essential boundary conditions can be accurately and stably applied in mechanical problem solving based on the meshless method.

Description

Technical Field

[0001] The present application relates to the field of computational mechanics technology, and in particular to a numerical method and device for accurately applying essential boundary conditions to a meshless method. Background Art

[0002] Imposing essential boundary conditions is a problem that almost all numerical simulation methods must face. The traditional finite element method can directly impose essential boundary conditions on the boundary nodes, but the finite element method based on structured grids cannot directly impose essential boundary conditions through the unit nodes because the unit boundaries and the solution domain boundaries generally do not coincide. The nodes of most numerical methods such as meshless methods and wavelet numerical methods do not have interpolation characteristics, and essential boundary conditions cannot be directly imposed through the nodes.

[0003] Lagrange multiplier method, penalty function method, modified variational principle method and other weak forms based on correction can only approximately impose essential boundary conditions, and the numerical accuracy cannot be guaranteed. The solution of the finite element coupling method can only maintain C at the boundary. 0 Continuous, not suitable for numerical methods with high continuity requirements.

[0004] Another approach involves imposing essential boundary conditions using implicit equations. This approach uses R-functions to construct implicit equations for the domain boundaries and, based on these equations, constructs a solution structure that satisfies various essential boundary conditions. This approach requires an extremely narrow boundary region near the boundary to achieve high-precision numerical results, which can compromise overall numerical stability. Summary of the Invention

[0005] The purpose of this application is to provide a numerical method and device for accurately applying essential boundary conditions for meshless methods, which can accurately and stably apply the numerical values ​​of essential boundary conditions in solving mechanical problems based on meshless methods.

[0006] To achieve the above objectives, this application provides the following solutions.

[0007] In a first aspect, the present application provides a numerical method for accurately applying essential boundary conditions for a meshless method. The numerical method for accurately applying essential boundary conditions for a meshless method includes the following steps.

[0008] Obtain the displacement boundary curve and force boundary curve of the mechanical problem solution domain.

[0009] A boundary region is constructed based on the displacement boundary curve and the force boundary curve using a B-spline basis function.

[0010] Based on the boundary region, a boundary basis function is constructed.

[0011] Based on the boundary basis functions, an allowable displacement solution of the mechanical problem to be solved is constructed; the allowable displacement solution satisfies the essential boundary conditions on the displacement boundary curve.

[0012] In a second aspect, the present application provides a computer device comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-mentioned numerical method for accurately applying essential boundary conditions for the gridless method.

[0013] In a third aspect, the present application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the aforementioned numerical method for accurately applying essential boundary conditions for the gridless method.

[0014] In a fourth aspect, the present application provides a computer program product, comprising a computer program, which, when executed by a processor, implements the aforementioned numerical method for accurately applying essential boundary conditions for the meshless method.

[0015] According to the specific embodiments provided in this application, this application discloses the following technical effects.

[0016] The present application provides a numerical method and device for accurately applying essential boundary conditions for a meshless method. First, the displacement boundary curve and force boundary curve of the mechanical problem solution domain are obtained. Accurately obtaining these boundary curves can ensure that subsequent calculations are consistent with the actual situation, making the calculation results more reliable. It provides an accurate basis for the subsequent determination of boundary areas and a series of operations. Secondly, based on the displacement boundary curve and the force boundary curve, the boundary area is constructed using B-spline basis functions. The B-spline basis function has good local controllability and smoothness, can fit complex boundary areas well, and improve the accuracy and flexibility of boundary representation. Then, based on the boundary area, a boundary basis function is constructed, which can accurately describe the distribution of displacement and force at the boundary, and provide powerful functional support for the subsequent construction of displacement solutions that meet the boundary conditions. In addition, the construction of the boundary basis function helps to improve the precision and accuracy of problem solving, and its good mathematical properties also facilitate subsequent complex mathematical operations and analysis. Finally, based on the boundary basis functions, an allowable displacement solution of the mechanical problem to be solved is constructed; the allowable displacement solution satisfies the essential boundary conditions on the displacement boundary curve, so that the calculated displacement solution is consistent with the actual situation at the boundary, ensuring the rationality and physical significance of the entire calculation problem. This provides a displacement field that conforms to the actual boundary constraints for subsequent further mechanical analysis and solution, helping to improve the reliability and accuracy of the entire calculation process. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0018] Figure 1 This is a diagram of the application environment of a numerical method for accurately applying essential boundary conditions to a meshless method in one embodiment of the present application.

[0019] Figure 2 A flowchart of a numerical method for accurately applying essential boundary conditions to a meshless method is provided in accordance with an embodiment of the present application.

[0020] Figure 3 A schematic diagram of a given boundary area within a two-dimensional area provided in an embodiment of the present application.

[0021] Figure 4 Schematic diagram of the calculation model of a plate with a central circular hole provided in one embodiment of the present application.

[0022] Figure 5 A schematic diagram of a boundary area selected in a calculation model provided in an embodiment of the present application.

[0023] Figure 6 A schematic diagram illustrating a comparison of stress values ​​along the left boundary provided in an embodiment of the present application.

[0024] Figure 7 A schematic diagram of strain energy error comparison provided in one embodiment of the present application.

[0025] Figure 8 A schematic diagram of the structure of a computer device provided in one embodiment of the present application. DETAILED DESCRIPTION

[0026] The following will be combined with the drawings in the embodiments of this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are only part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this application.

[0027] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0028] The numerical method for accurately applying essential boundary conditions to the meshless method provided in the embodiment of the present application can be applied to Figure 1In the application environment shown. Among them, the terminal 102 communicates with the server 104 through the network. The data storage system can store the data that the server 104 needs to process. The data storage system can be set up separately, integrated on the server 104, or placed on the cloud or other servers. The terminal 102 can send the obtained displacement boundary curve and force boundary curve of the mechanical problem solution domain to the server 104. After the server 104 receives the displacement boundary curve and the force boundary curve, for the displacement boundary curve and the force boundary curve, the server 104 uses the B-spline basis function to construct the boundary area based on the displacement boundary curve and the force boundary curve; based on the boundary area, construct the boundary basis function; based on the boundary basis function, construct the allowable displacement solution of the mechanical problem to be solved; the allowable displacement solution satisfies the essential boundary conditions on the displacement boundary curve. The server 104 can feedback the obtained allowable displacement solution to the terminal 102. In addition, in some embodiments, the numerical method for accurately applying the essential boundary conditions for the gridless method can also be implemented separately by the server 104 or the terminal 102. For example, the terminal 102 can directly apply the numerical values ​​of the essential boundary conditions to the displacement boundary curve and the force boundary curve, or the server 104 can obtain the displacement boundary curve and the force boundary curve of the calculation area from the data storage system, and accurately apply the numerical values ​​of the essential boundary conditions to the displacement boundary curve and the force boundary curve.

[0029] The terminal 102 may be, but is not limited to, various desktop computers, laptop computers, smart phones, and tablet computers. The server 104 may be implemented as an independent server or a server cluster consisting of multiple servers, or a cloud server.

[0030] In an exemplary embodiment, Figure 2 As shown, a numerical method for accurately applying essential boundary conditions to the meshless method is provided. The method is executed by a computer device, specifically a computer device such as a terminal or a server, or a terminal and a server. In the embodiment of the present application, the method is applied to Figure 1 The server 104 in the example is used for explanation, and the steps include the following steps S1 to S4.

[0031] S1: Obtain the displacement boundary curve and force boundary curve of the mechanical problem solution domain.

[0032] S2: constructing a boundary region using a B-spline basis function based on the displacement boundary curve and the force boundary curve.

[0033] S3: Constructing a boundary basis function based on the boundary area.

[0034] S4: Based on the boundary basis function, construct an allowable displacement solution of the mechanical problem to be solved; the allowable displacement solution satisfies the essential boundary conditions on the displacement boundary curve.

[0035] In this application, the construction of relevant trial functions and test functions in a selected boundary region is first considered, and the selection of the boundary region is not subject to any special restrictions. B-spline basis functions are used to approximate the corresponding boundary curve and the selected boundary region. In the boundary region, the B-spline basis functions used to describe the boundary region are also used to construct the boundary basis functions. At the same time, relevant weight functions are introduced to transform the global basis functions and boundary basis functions in the boundary region. The transformed relevant basis functions maintain high-order continuity and can at least reconstruct linear polynomials. The solution constructed based on the transformed basis functions can automatically meet the essential boundary conditions, and the constructed solution is complete. Even if the solution near the boundary has complex features, the numerical accuracy near the boundary and the stability of the overall solution can be ensured. Compared with existing methods, the solution constructed using the method of this application can accurately meet various complex essential boundary conditions. At the same time, the constructed boundary basis functions are complete, and no additional restrictions are required on the overall approximation of the solution region. There are no special restrictions on the selection of the boundary region, which can ensure the accuracy and stability of the overall numerical solution.

[0036] Space R d (d represents the spatial dimension) can be constructed using linear combinations of B-spline basis functions. The coefficients of the basis functions are called control points. Given n B-spline basis functions N i,p , i=1,2,3,…,n ， and the corresponding control point B i ∈R d , i = 1, 2, 3, ..., n, the piecewise polynomial B-spline curve can be constructed as shown below.

[0037]

[0038] Where C(ξ) is the piecewise polynomial B-spline curve; N i,p (ξ) is the B-spline basis function; B i are control points; n is the number of B-spline basis functions.

[0039] In two-dimensional space, the physical coordinates of the curve are as follows.

[0040]

[0041] Among them, (x i ,y i ) is the control point B i 's coordinates.

[0042] In general numerical methods, the approximate solution defined in the region Ω can be expressed as follows.

[0043]

[0044] Among them, φ i (x) is the basis function; ci is the variable to be solved; u(x) is the approximate solution of the entire solution area.

[0045] For two-dimensional problems, the boundary region can be described by two curves, such as Figure 3 As shown. The B-spline basis function is used to approximate these two curves. For the curve Γ u and Γ u′ , the approximate expressions are as follows.

[0046]

[0047]

[0048] Among them, (x u (ξ),y u (ξ)) is the curve Γ u The physical coordinates of the control points corresponding to the B-spline basis functions; For control points Coordinates in two-dimensional space; (x u′ (ξ),y u′ (ξ)) is the curve Γ u′ The physical coordinates of the control points corresponding to the B-spline basis functions; For control points Coordinates in two-dimensional space; N i,p (ξ) is the B-spline basis function; (ξ,η) is the coordinate in the two-dimensional parameter space; n is the number of B-spline basis functions.

[0049] Boundary region Ω b It can be expressed by the following formula.

[0050]

[0051] Among them, (x(ξ,η), y(ξ,η)) is the representation of the boundary region in the two-dimensional parameter space.

[0052] The boundary basis functions can be reconstructed in the boundary region as shown below.

[0053]

[0054] L k (η)=η k-1 (9).

[0055] in, is the boundary basis function; N i,p (ξ) is the B-spline basis function; L k(η) is the polynomial basis function; k is a constant greater than 1; (ξ,η) is the coordinate in the two-dimensional parameter space.

[0056] The following is an example of solving an elastic mechanics problem to illustrate the implementation process of this application. The governing equations of the elastic static problem are as follows.

[0057] ▽·σ+b=0 in the region Ω (10).

[0058] σ=C:ε (11).

[0059] ε=▽ S u (12).

[0060] Where σ is the Cauchy stress; b is the body force; C is the elastic tensor; ε is the strain; u is the displacement; ▽ S is the symmetric part of the gradient operator.

[0061] The essential and natural boundary conditions are shown below.

[0062]

[0063] Among them, Γ u and Γ t are the given displacement and force boundaries respectively; m is the unit external normal vector of the region Ω; is a given displacement; For a given force.

[0064] The expressions of the trial function space u and the test function space δu defined on the region Ω are as follows.

[0065]

[0066] The global approximate basis function is expressed as d is the dimension of the space.

[0067] The allowed displacement solution can be constructed as follows.

[0068]

[0069] Among them, u i In the boundary region Ω b The allowable displacement solution in ; is the boundary region Ω b Local approximation within; For the region Ω-Ω b The local approximation within is the component of the global approximate basis function, and the global approximate basis function can be selected from the basis functions of general numerical methods (including the finite element method).

[0070] For the u The essential boundary conditions on It should be broken down into the following two parts.

[0071]

[0072] in, is the boundary region Ω b Local approximation within; is the variable that satisfies the homogeneous boundary condition on the displacement boundary curve; are variables related to the essential boundary conditions.

[0073] The expression in the two-dimensional parameter space is as follows.

[0074]

[0075] in, for Representation in two-dimensional parameter space; for Representation in two-dimensional parameter space; for Representation in two-dimensional parameter space; w(η) is the basic weight function; are the control points that satisfy the homogeneous boundary conditions on the displacement boundary curve; is the B-spline basis function of the boundary region in the two-dimensional parameter space; (ξ,η) is the coordinate in the two-dimensional parameter space; n is the number of B-spline basis functions; is the control point under the essential boundary conditions, is a constant, and is used to construct the following approximation; is a two-dimensional boundary basis function.

[0076]

[0077] in, is the essential boundary condition; is an approximation of the given displacement boundary condition.

[0078] Obviously, the u constructed above i Satisfied in Γ u The essential boundary conditions on The modified global basis function also satisfies high-order continuity.

[0079] The present application also provides an application scenario, which applies the above-mentioned numerical method for accurately applying essential boundary conditions for the meshless method. Specifically: the numerical method for accurately applying essential boundary conditions for the meshless method provided in this embodiment can be applied in the stress calculation and analysis scenario of the plate. The stress calculation and analysis scenario of the plate includes: a boundary curve acquisition link, a boundary area determination link, a boundary basis function construction link and an allowable displacement solution construction link; first, the displacement boundary curve and the force boundary curve of the calculation area are obtained; secondly, the boundary area is determined based on the displacement boundary curve and the force boundary curve; the boundary area is a curve approximated by the B-spline basis function; then, based on the boundary area, a boundary basis function is constructed; finally, based on the boundary basis function, an allowable displacement solution is constructed; the allowable displacement solution satisfies the essential boundary conditions of the displacement boundary curve.

[0080] The following is a practical example to verify the effectiveness of this application. Consider the infinite plate with a central circular hole. The plate is subjected to a uniform unit tension along the axis at infinity. The radius of the circular hole is set to . The calculation model for this problem takes a finite plate as Figure 4 The exact solution to this problem is shown below.

[0081]

[0082] The exact solution is used to calculate the forces on the model's force boundaries (right and top), and symmetric boundary conditions are used on the left and bottom boundaries.

[0083] In this embodiment, uniform node distribution is adopted, and the node spacing is w=w x =w y The boundary area selected is as follows Figure 5 As shown, the width is represented by t. Figure 6 Displays the stress along the left boundary (σ x ) values. The relevant approximate parameters are w = 0.05 and t / w = 0.2. It can be seen that the numerical simulation results are very close to the exact solution even in the stress concentration area.

[0084] Figure 7 A plot of the strain energy error is shown. The x-axis represents the element size. Different boundary width to mesh size ratios were used in the calculations, and the convergence rate is nearly identical for each ratio. The error is relatively small when the ratio is larger. This result demonstrates that the new method improves the accuracy of the boundary region while maintaining overall numerical accuracy and stability.

[0085] In an exemplary embodiment, a computer device is provided. The computer device may be a server or a terminal. The internal structure diagram thereof may be as follows: Figure 8As shown. The computer device includes a processor, a memory, an input / output interface (Input / Output, abbreviated as I / O) and a communication interface. The processor, the memory and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The database of the computer device is used to store the displacement boundary curve and the force boundary curve of the domain for solving mechanical problems. The input / output interface of the computer device is used to exchange information between the processor and an external device. The communication interface of the computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor, a numerical method for accurately applying essential boundary conditions for the gridless method is implemented.

[0086] Those skilled in the art will understand that Figure 8 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than shown in the figure, or combine certain components, or have a different component arrangement.

[0087] In an exemplary embodiment, a computer device is further provided, including a memory and a processor. The memory stores a computer program, and the processor implements the above method embodiments when executing the computer program.

[0088] In an exemplary embodiment, a computer-readable storage medium is provided, storing a computer program, which implements the above-mentioned method embodiments when executed by a processor.

[0089] In an exemplary embodiment, a computer program product is provided, including a computer program. When the computer program is executed by a processor, the above method embodiments are implemented.

[0090] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use and processing of relevant data must comply with relevant regulations.

[0091] Those skilled in the art will understand that all or part of the processes in the above-mentioned embodiment methods can be implemented by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to memory, database or other media used in the embodiments provided in this application may include at least one of non-volatile and volatile memory. Non-volatile memory may include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory may include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM may be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM).

[0092] The databases involved in the various embodiments provided herein may include at least one of a relational database and a non-relational database. Non-relational databases may include, but are not limited to, distributed databases based on blockchains. The processors involved in the various embodiments provided herein may include, but are not limited to, general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic units, data processing logic units based on quantum computing, and the like.

[0093] The technical features of the above embodiments can be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0094] This document uses specific examples to illustrate the principles and implementation methods of this application. The description of the above examples is only intended to help understand the method and core concept of this application. At the same time, for those skilled in the art, based on the concept of this application, there may be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. A numerical method for accurately imposing essential boundary conditions for meshless methods, characterized in that: The numerical method for accurately applying essential boundary conditions for the meshless method includes: Obtain the displacement boundary curve and force boundary curve of the mechanical problem solution domain; Constructing a boundary region using a B-spline basis function based on the displacement boundary curve and the force boundary curve; Based on the boundary area, constructing a boundary basis function; Based on the boundary basis functions, an allowable displacement solution of the mechanical problem to be solved is constructed; the allowable displacement solution satisfies the essential boundary conditions on the displacement boundary curve.

2. The numerical method for accurately applying essential boundary conditions for meshless methods according to claim 1, characterized in that: The expression of the boundary area is: Among them, (x u (ξ),y u (ξ)) is the curve Γ u The physical coordinates of the control points corresponding to the B-spline basis functions; For control points Coordinates in two-dimensional space; (x u′ (ξ),y u′ (ξ)) is the curve Γ u′ The physical coordinates of the control points corresponding to the B-spline basis functions; For control points Coordinates in two-dimensional space; N i,p (ξ) is the B-spline basis function; (ξ,η) is the coordinate in the two-dimensional parameter space; n is the number of B-spline basis functions.

3. The numerical method for accurately applying essential boundary conditions for meshless methods according to claim 2, characterized in that: The expression of the boundary area in the two-dimensional parameter space is: Among them, (x(ξ,η), y(ξ,η)) is the representation of the boundary region in the two-dimensional parameter space.

4. The numerical method for accurately applying essential boundary conditions for meshless methods according to claim 1, characterized in that: The expression of the boundary basis function is: L k (n)=n k-1 ; in, is the boundary basis function; N i,p (ξ) is the B-spline basis function; L k (η) is the polynomial basis function; k is a constant greater than 1; (ξ,η) is the coordinate in the two-dimensional parameter space.

5. The numerical method for accurately applying essential boundary conditions for meshless methods according to claim 1, characterized in that: The expression of the allowable displacement solution is: Among them, u i In the boundary region Ω b The allowable displacement solution in ; is the boundary region Ω b Local approximation within; For the region Ω-Ω b Local approximation within .

6. The numerical method for accurately applying essential boundary conditions for meshless methods according to claim 5, characterized in that: The boundary region Ω b The calculation formula of the local approximation within is: in, is the boundary region Ω b Local approximation within; is the variable that satisfies the homogeneous boundary condition on the displacement boundary curve; are variables related to the essential boundary conditions.

7. The numerical method for accurately applying essential boundary conditions for meshless methods according to claim 6, characterized in that: The boundary region Ω b The local approximation in the two-dimensional parameter space is expressed as: w(n)=1-6n 2 +8th 3 -3rd 4 0≤ζ≤1.0; in, for Representation in two-dimensional parameter space; for Representation in two-dimensional parameter space; for Representation in two-dimensional parameter space; w(η) is the basic weight function; are the control points that satisfy the homogeneous boundary conditions on the displacement boundary curve; is the B-spline basis function of the boundary region in the two-dimensional parameter space; (ξ,η) is the coordinate in the two-dimensional parameter space; n is the number of B-spline basis functions; is the control point under essential boundary conditions; is a two-dimensional boundary basis function.

8. A computer device comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the numerical method for accurately applying essential boundary conditions for a meshless method according to any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the numerical method for accurately applying essential boundary conditions to the meshless method according to any one of claims 1 to 6 is implemented.

10. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the numerical method for accurately applying essential boundary conditions to the meshless method according to any one of claims 1 to 6 is implemented.

Images