A smooth finite element numerical calculation method based on digital twin

Abstract

The present invention belongs to the field of computer technology, and discloses a smooth finite element numerical calculation method based on digital twins, the method comprising: obtaining solid material data and polygonal unit data of solid materials; dividing each of the polygonal units in the target range into a plurality of smooth units; calculating the stiffness matrix of each of the smooth units, and obtaining the stiffness matrix of the polygonal units; obtaining the stiffness matrix and load matrix of the solid material based on the stiffness matrix of the polygonal units; performing matrix operations according to the stiffness matrix and the load matrix of the solid material, and obtaining the displacement approximate solution matrix U of the static elastic problem of the solid material. It can greatly improve the solution accuracy and energy convergence speed of the finite element without increasing the calculation cost, and can eliminate the restrictions on the shape of the bilinear isoparametric element, more flexibly discretize the problem domain, and solve the problem of the traditional finite element method's restrictions on the shape of the mapping unit.

Description

Technical Field

[0001] The present invention relates to the field of computer technology, and in particular to a smooth finite element numerical calculation method based on digital twins. Background Art

[0002] After more than half a century of development, the finite element method has become a powerful and widely used numerical simulation technology in engineering and science. Mapping elements (such as isoparametric elements) play a very important role in finite element analysis. At present, the basic requirement for using mapping elements is that the elements must be convex and no severe distortion is allowed to ensure that a one-to-one coordinate correspondence is established between the physical coordinates associated with the element and the natural coordinates. Specifically, for a four-node element using a mapped bilinear shape function, a necessary condition is that any internal angle should not be greater than 180° in theory. In numerical implementation, the positivity of the determinant of the Jacobian matrix should be checked frequently to avoid severe distortion of the element. In addition to the complexity of implementation, the quadrature rule required for high-order approximate function problems will greatly increase the amount of calculation.

[0003] Therefore, how to overcome the limitation of the traditional finite element method on the shape of the mapping unit and discretize the problem domain more flexibly has become a problem that needs to be solved urgently by those skilled in the art. Summary of the invention

[0004] The embodiment of the present invention provides a smooth finite element numerical calculation method based on digital twins to solve the problem of limitations of the traditional finite element method on the shape of the mapping unit. In order to have a basic understanding of some aspects of the disclosed embodiments, a simple summary is given below. This summary is not a general review, nor is it intended to identify key / important components or describe the scope of protection of these embodiments. Its only purpose is to present some concepts in a simple form as a preface to the detailed description that follows.

[0005] According to a first aspect of an embodiment of the present invention, a smooth finite element numerical calculation method based on digital twin is provided.

[0006] In one embodiment, a smooth finite element numerical algorithm for a digital twin includes:

[0007] A smooth finite element numerical calculation method based on digital twin, characterized in that the method comprises:

[0008] Acquire solid material data and polygonal unit data of solid materials;

[0009] Dividing each of the polygonal units within the target range into a plurality of smooth units;

[0010] Calculating the stiffness matrix of each smooth unit and obtaining the stiffness matrix of the polygonal unit;

[0011] Obtaining a solid material stiffness matrix and a load matrix based on the stiffness matrix of the polygonal unit;

[0012] Matrix operations are performed based on the stiffness matrix and load matrix of the solid material to obtain the displacement approximate solution matrix U for the static elastic problem of the solid material.

[0013] Furthermore, obtaining solid material data of the solid material specifically includes:

[0014] Get Young's modulus and Poisson's ratio for a solid material.

[0015] Furthermore, obtaining polygonal unit data of solid materials specifically includes:

[0016] Get the element number, element node number, and node coordinates of a polygon element.

[0017] Furthermore, the polygonal unit is a quadrilateral unit.

[0018] Furthermore, each of the polygonal units within the target range is divided into a plurality of smooth units, specifically:

[0019] Each quadrilateral element within the target range is divided into four smooth elements.

[0020] Further, the stiffness matrix of each smooth unit is calculated, and the stiffness matrix of the polygonal unit is obtained, which specifically includes:

[0021] Each of the smooth units is looped to determine the unit area and normal of each smooth unit, and a smoothing operation is performed based on the unit area and the normal to obtain a smooth unit stiffness matrix.

[0022] Furthermore, a smoothing operation is performed based on the unit area and the normal, specifically including:

[0023] Determine the definition domain of the static elastic problem of the solid material, and obtain the outer boundary normal and boundary displacement of the definition domain based on preset boundary conditions;

[0024] Obtaining a displacement gradient of the definition domain based on the outer normal of the boundary and the boundary displacement;

[0025] A smoothing operation is performed on the displacement gradient of each smoothing unit in the domain.

[0026] Furthermore, the determining of the definition domain of the static elastic problem of the solid material and obtaining the boundary normal and boundary displacement of the definition domain based on preset boundary conditions specifically include:

[0027] Assuming the definition domain Ω of the two-dimensional static elasticity problem, the boundary of the definition domain is calculated based on the following formula:

[0028] σ ij +b j =0 in Ω

[0029] Γ=Γ u +Γ t

[0030]

[0031] Among them, σ ij is the stress tensor, b j is the physical force, Γ is the boundary of the definition domain Ω, Γ u is the forced displacement boundary, Γ t is the traction boundary, is an empty set;

[0032] At this time, the boundary conditions are:

[0033] σ ij n j =t i onΓ t

[0034]

[0035] Among them, t i is the traction force, n j is the outer normal of the boundary, is the forced displacement, u i is the boundary Γ u displacement.

[0036] Furthermore, the obtaining of the displacement gradient of the definition domain based on the outer normal of the boundary and the boundary displacement specifically includes:

[0037] The following formula is used for variational weak processing:

[0038]

[0039] in, is the symmetric part of the displacement gradient, D ijkl is the material matrix, δ is the δ function, and k and l are the node numbers.

[0040] A smoothing operation is performed on the displacement gradient of each smoothing cell within each element using the following formula:

[0041]

[0042] Among them, Φ is a smooth function, u h(x C ) is the displacement after smoothing operation, x C is the smooth unit coordinate;

[0043] According to the distribution integral, the right side of the above formula is transformed into:

[0044]

[0045] Where n(x) is the normal vector outside the boundary;

[0046] The combined formulas yield:

[0047]

[0048] Among them, Γ C is a smooth cell boundary, is the displacement gradient after smoothing operation.

[0049] Furthermore, the variational weak formula satisfies the following conditions:

[0050]

[0051] Among them, B I is the standard gradient matrix, N I is the shape function matrix.

[0052] Furthermore, the specific form of the smooth function Φ is as follows:

[0053]

[0054] Among them, Ω C is a smooth unit, A C For smooth unit Ω C The area, u h (x C ) is the displacement after smoothing operation, x C are smooth unit coordinates.

[0055] Furthermore, the smoothing strain is calculated using the following formula:

[0056]

[0057] in, is the smooth strain matrix, is the smooth strain, d I is the elastic modulus.

[0058] Furthermore, the smooth strain matrix is ​​calculated using the following formula:

[0059]

[0060]

[0061] Among them, n k (x) is the normal vector outside the boundary.

[0062] Furthermore, the stiffness matrix based on the polygonal unit is used to obtain a solid material stiffness matrix and a load matrix, which specifically includes:

[0063] The smooth unit stiffness matrices are combined to obtain the stiffness matrix of the polygonal unit, and the boundary conditions of the static elastic problem of the solid material are loaded to obtain the final solid material stiffness matrix K and load matrix F.

[0064] According to a second aspect of an embodiment of the present invention, a computer device is provided.

[0065] In some embodiments, the computer device includes a memory and a processor, the memory stores a computer program, and the processor implements the steps of the above algorithm when executing the computer program.

[0066] According to a third aspect of an embodiment of the present invention, a computer-readable storage medium is provided.

[0067] In some embodiments, the computer storage medium includes one or more program instructions, and the one or more program instructions are used to execute the method described above.

[0068] The technical solution provided by the embodiment of the present invention may have the following beneficial effects:

[0069] The digital twin-based smooth finite element numerical calculation method provided by the present invention obtains the solid material data and polygonal unit data of the solid material, and divides each of the polygonal units in the target range into multiple smooth units; then calculates the stiffness matrix of each of the smooth units, and obtains the stiffness matrix of the polygonal unit, and obtains the stiffness matrix and load matrix of the solid material based on the stiffness matrix of the polygonal unit; and performs matrix operations based on the stiffness matrix and load matrix of the solid material to obtain the displacement approximate solution matrix U of the static elastic problem of the solid material. Thus, without increasing the calculation cost, the solution accuracy and energy convergence speed of the finite element are greatly improved, and the restrictions on the shape of the bilinear isoparametric element can be eliminated, the problem domain can be more flexibly discretized, and the problem of the traditional finite element method's restrictions on the shape of the mapping unit is solved.

[0070] It is to be understood that the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention.

[0072] Figure 1 is a flow chart of a smooth finite element numerical calculation method based on digital twins according to an exemplary embodiment;

[0073] Figure 2 is a schematic structural diagram of a cantilever beam according to an exemplary embodiment;

[0074] Figure 3a is a schematic diagram showing a comparison of the shift norms of the algorithm of the present invention and the traditional finite element method according to an exemplary embodiment;

[0075] Figure 3b is a schematic diagram showing energy norm comparison between the method of the present invention and the traditional finite element method according to an exemplary embodiment;

[0076] Figure 3c is a schematic diagram showing a comparison of the calculation time between the algorithm of the present invention and the traditional finite element method according to an exemplary embodiment;

[0077] Figure 4a is a schematic diagram of regular polygon mesh division according to an exemplary embodiment;

[0078] Figure 4b is a schematic diagram of irregular polygon mesh division according to an exemplary embodiment;

[0079] Figure 4c is a schematic diagram showing a comparison between the calculation results and analytical solutions of the algorithm of the present invention under different grids according to an exemplary embodiment;

[0080] Figure 5 The figure is a schematic diagram showing the structure of a computer device according to an exemplary embodiment. DETAILED DESCRIPTION

[0081] The following description and the accompanying drawings fully disclose specific embodiments herein, enabling those skilled in the art to practice them. Parts and features of some embodiments may be included in or replace parts and features of other embodiments. The scope of the embodiments herein includes the entire scope of the claims and all available equivalents of the claims. In this document, the terms "first", "second", etc. are only used to distinguish one element from another, without requiring or implying any actual relationship or order between these elements. In fact, the first element can also be called the second element, and vice versa. Moreover, the terms "comprising", "including" or any other variant thereof are intended to cover non-exclusive inclusion, such that a structure, device or equipment comprising a series of elements not only includes those elements but also other elements not explicitly listed, or elements inherent to such structure, device or equipment. Without further limitation, an element defined by the statement "comprising one..." does not exclude the existence of additional identical elements in the structure, device or equipment comprising the said element. The various embodiments herein are described in a progressive manner, with each embodiment highlighting the differences from other embodiments. The same or similar parts among the various embodiments can be referred to each other.

[0082] In this document, the orientation or positional relationships indicated by terms such as "longitudinal", "transverse", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. are based on the orientation or positional relationships shown in the drawings. They are only for the convenience of describing this document and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus should not be construed as a limitation on the present invention. In the description herein, unless otherwise specified and defined, the terms "installed", "connected", "joined" shall be understood in a broad sense. For example, it can be a mechanical connection or an electrical connection, or it can be the communication inside two elements. It can be directly connected or indirectly connected through an intermediate medium. For those of ordinary skill in the art, the specific meanings of the above terms can be understood according to specific circumstances.

[0083] In this document, unless otherwise stated, the term "plurality" means two or more.

[0084] In this document, the character " / " indicates that the objects before and after are in an "or" relationship. For example, A / B means: A or B.

[0085] In this document, the term "and / or" is a description of the associated relationship of an object, indicating that three relationships can exist. For example, A and / or B means: A or B, or, A and B these three relationships.

[0086] In a specific embodiment, the smooth finite element numerical calculation method based on digital twin provided by the present invention projects the strain field onto one or a group of constant fields based on the smooth unit in the mapping unit. Figure 1 As shown, the method comprises the following steps:

[0087] S1: Acquire solid material data and polygonal unit data of solid materials.

[0088] For ease of operation, polygonal unit data is specifically quadrilateral unit data. Specifically, solid material data and quadrilateral unit data belong to static elastic problem data of solid materials, which are input numerical data. Among them, solid material data includes Young's modulus and Poisson's ratio, and quadrilateral unit data includes unit number, unit node number and node coordinates.

[0089] S2: Divide each polygonal unit in the target range into a plurality of smooth units. Based on the above, step S2 specifically divides each quadrilateral unit in the target range into four smooth units.

[0090] S3: Calculate the stiffness matrix of each of the smooth units and obtain the stiffness matrix of the polygonal units.

[0091] Specifically, step S3 is to loop over the smooth units to determine the unit area and normal of each smooth unit, and perform a smoothing operation based on the unit area and normal to obtain a smooth unit stiffness matrix.

[0092] In step S3, the smoothing operation includes the following steps:

[0093] S31: Determine the domain of static elastic problems of solid materials;

[0094] S32: obtaining the outer boundary normal and boundary displacement of the definition domain based on the preset boundary conditions;

[0095] S33: obtaining a displacement gradient of the definition domain based on the outer normal of the boundary and the boundary displacement;

[0096] S34: performing a smoothing operation on the displacement gradient of each smooth unit in the definition domain.

[0097] Specifically, in a specific scenario, to illustrate the principle of this method, assume that the definition domain of the two-dimensional static elastic problem is Ω, and consider the elastic static problem on the definition domain Ω, whose control equation is as follows:

[0098] σ ij +b j =0inΩ (1)

[0099] Γ=Γ u+Γ t (2)

[0100]

[0101] Among them, σ ij is the stress tensor, b j is the physical force, Γ is the boundary of the definition domain Ω, Γ u is the forced displacement boundary, Γ t is the traction boundary, Is an empty set.

[0102] At this time, the boundary conditions are as follows:

[0103] σ ij n j =t i onΓ t (4)

[0104]

[0105] Among them, t i is the traction force, n j is the outer normal of the boundary, is the forced displacement, u i is the boundary Γ u displacement.

[0106] Furthermore, the variational weak form of the above formula (1) is as follows:

[0107]

[0108] in, is the symmetric part of the displacement gradient, D ijkl is the material matrix, δ is the δ function, and k and l are the node numbers.

[0109] In order to ensure the convergence of the numerical solution, the linear accuracy of the weak form solution should be ensured, so the following conditions must be met:

[0110]

[0111] Among them, B I is the standard gradient matrix, N I is the shape function matrix.

[0112] The smoothing operation is performed on the displacement gradient of each smoothing unit within each element, as shown in the following equation:

[0113]

[0114] Among them, Φ is a smooth function, the specific form is as follows:

[0115]

[0116] Among them, Ω C is a smooth unit, A C For smooth unit Ω C The area, u h (x C ) is the displacement after smoothing operation, x C are smooth unit coordinates.

[0117] According to the distribution integral, the right side of formula (9) can be transformed into:

[0118]

[0119] Where n(x) is the normal vector outside the boundary.

[0120] Combining formulas (9) and (10), we can get:

[0121]

[0122] Among them, Γ C is a smooth cell boundary, is the displacement gradient after smoothing operation.

[0123] The smoothing function converts the element surface integral into the element boundary line integral. Similarly, the smoothed strain can be obtained by the following formula:

[0124]

[0125] in, is the smooth strain matrix, is the smooth strain, d I is the elastic modulus.

[0126] For two-dimensional problems, is defined as follows:

[0127]

[0128]

[0129] Among them, n k (x) is the normal vector outside the boundary.

[0130] If a line integral is performed along each boundary using a Gaussian point, then It can be calculated by the following formula:

[0131]

[0132] Among them, x i For boundary segments Gaussian integral point on and Boundary line segments The length and external normal of For boundary segments Gaussian integration point on .

[0133] S4: Obtaining a solid material stiffness matrix and a load matrix based on the stiffness matrix of the polygonal unit.

[0134] The step S4 specifically combines the smooth unit stiffness matrices to obtain the stiffness matrix of the quadrilateral unit, loads the boundary conditions of the static elastic problem of the solid material, and obtains the final stiffness matrix K and load matrix F of the solid material.

[0135] S5: Perform matrix operations based on the stiffness matrix K of the solid material and the load matrix F to obtain the displacement approximate solution matrix U of the static elastic problem of the solid material.

[0136] Specifically, the stiffness matrix of the smoothed unit can be composed of the stiffness matrices of all smooth units of the unit, that is, matrix operations are performed based on the stiffness matrix K of the solid material and the load matrix F to obtain the displacement approximate solution matrix U of the static elastic problem of the solid material.

[0137] Among them, the solid material stiffness matrix is ​​calculated according to formula (16):

[0138]

[0139] Among them, K e is the stiffness matrix, is the smooth strain matrix, and D is the material matrix of the solid material.

[0140] Taking a cantilever beam as an example, a specific embodiment of the smooth finite element numerical calculation method based on digital twin provided by the present invention is given below.

[0141] like Figure 2 As shown in the figure, a cantilever beam with a length of L and a height of D is subjected to a parabolic traction force at the free end. The relevant parameters are: E = 3.0 × 10 7 kPa, v = 0.3, D = 12m, L = 48m, P = 1000N. Assuming that the cantilever beam has unit thickness, the exact solution of its displacement is as follows:

[0142]

[0143]

[0144]

[0145] The corresponding exact stress solution is as follows:

[0146]

[0147] σ 22 (x,y)=0 (21)

[0148]

[0149] In order to verify the convergence speed of the algorithm of the present invention, two paradigms are used, namely the displacement paradigm e d and energy paradigm e :

[0150]

[0151]

[0152] like Figure 3a and Figure 3b As shown, the smooth finite element numerical algorithm (SFEM) of the present invention is compared with the convergence speed of the traditional finite element method (FEM). Both methods achieve equivalent convergence speeds in displacement and energy, and the displacement of the method of the present invention is more accurate than that of the traditional finite element method. Compared with using S1, the energy convergence of SFEM using S2 is about twice as fast. In scheme 1 (S1), SC / GP=4 is used to calculate the stiffness matrix (displacement) and stress (energy), while in scheme 2 (S2), SC / GP=4 is only used to calculate the stiffness matrix (displacement), and SC / GP=1 is used for post-processing of stress and energy. In scheme 3 (S3), SC / GP=1 is always used).

[0153] The calculation time of the smooth finite element numerical algorithm (SFEM) of the present invention is similar to that of the traditional finite element method. Figure 3c As shown, when the number of discrete elements is not very large, both methods require almost the same time. However, as the number of elements increases, the calculation time of the smooth finite element numerical algorithm of the present invention is less than that of the traditional finite element method.

[0154] In order to demonstrate the function of the smooth finite element numerical algorithm of the present invention with complex shape elements, the beam is divided into regular elements and irregular polygonal elements, such as Figure 4a and Figure 4b The settlement result is then plotted together with the exact solution in Figure 4c The calculation results are consistent with the exact solution results. However, irregular polygonal elements cannot be used in traditional finite element method.

[0155] The smooth finite element numerical algorithm of the present invention greatly improves the solution accuracy and energy convergence speed of the finite element without increasing the calculation cost, and can eliminate the restrictions on the shape of bilinear isoparametric elements and discretize the problem domain more flexibly.

[0156] In one embodiment, a computer device is provided. The computer device may be a server, and its internal structure diagram may be as follows: Figure 5 As shown. The computer device includes a processor, a memory and a network interface connected through a system bus. Among them, 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 static information and dynamic information data. The network 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, the steps in the above algorithm embodiment are implemented.

[0157] Those skilled in the art will understand that Figure 5 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present invention, and does not constitute a limitation on the computer device to which the solution of the present invention is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine certain components, or have a different arrangement of components.

[0158] In one embodiment, a computer device is further provided, including a memory and a processor, wherein a computer program is stored in the memory, and the processor implements the steps in the above algorithm embodiment when executing the computer program.

[0159] In one embodiment, a computer-readable storage medium is provided, on which a computer program is stored. When the computer program is executed by a processor, the steps in the above algorithm embodiment are implemented.

[0160] Those skilled in the art will appreciate that all or part of the processes in the above-mentioned embodiments of the algorithm 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 algorithms. Among them, any reference to memory, storage, database or other media used in the embodiments provided by the present invention can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory or optical memory, etc. Volatile memory can include random access memory (RAM) or external cache memory. As an illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM).

[0161] The present invention is not limited to the structures which have been described above and shown in the drawings, and various modifications and changes may be made without departing from the scope thereof. The scope of the present invention is limited only by the appended claims.

Claims

1. A smooth finite element numerical calculation method based on digital twins, It is characterized in that The method comprises: Acquire solid material data and polygonal unit data of solid materials; Dividing each of the polygonal units within the target range into a plurality of smooth units; Calculating the stiffness matrix of each of the smooth units and obtaining the stiffness matrix of the polygonal units; Obtaining a solid material stiffness matrix and a load matrix based on the stiffness matrix of the polygonal unit; Matrix operations are performed based on the stiffness matrix and load matrix of the solid material to obtain the displacement approximate solution matrix U for the static elastic problem of the solid material; the unit stiffness matrix after smoothing is composed of the stiffness matrices of all smooth units of the unit, that is, matrix operations are performed based on the stiffness matrix K and load matrix F of the solid material to obtain the displacement approximate solution matrix U for the static elastic problem of the solid material; Among them, the solid material stiffness matrix is ​​calculated according to formula (16): Among them, K e is the stiffness matrix, is the smooth strain matrix, D is the material matrix of the solid material, A C For smooth unit Ω C area.

2. The smooth finite element numerical calculation method according to claim 1, It is characterized in that Acquiring solid material data of solid materials specifically includes: Get Young's modulus and Poisson's ratio for a solid material.

3. The smooth finite element numerical calculation method according to claim 1, It is characterized in that Obtaining polygonal unit data of solid materials specifically includes: Get the element number, element node number, and node coordinates of a polygon element.

4. The smooth finite element numerical calculation method according to claim 3, It is characterized in that The polygonal unit is a quadrilateral unit.

5. The smooth finite element numerical calculation method according to claim 4, It is characterized in that Each of the polygonal units within the target range is divided into a plurality of smooth units, specifically: Each quadrilateral element within the target range is divided into four smooth elements.

6. The smooth finite element numerical calculation method according to any one of claims 1 to 5, It is characterized in that Calculating the stiffness matrix of each smooth unit and obtaining the stiffness matrix of the polygonal unit specifically includes: Each of the smooth units is looped to determine the unit area and normal of each smooth unit, and a smoothing operation is performed based on the unit area and the normal to obtain a smooth unit stiffness matrix.

7. The smooth finite element numerical calculation method according to claim 6, It is characterized in that Perform smoothing operations based on cell area and normals, including: Determine the definition domain of the static elastic problem of the solid material, and obtain the outer boundary normal and boundary displacement of the definition domain based on preset boundary conditions; Obtaining a displacement gradient of the definition domain based on the outer normal of the boundary and the boundary displacement; A smoothing operation is performed on the displacement gradient of each smoothing unit in the domain.

8. The smooth finite element numerical calculation method according to claim 7, It is characterized in that The determining of the definition domain of the static elastic problem of the solid material and obtaining the outer boundary normal and boundary displacement of the definition domain based on preset boundary conditions specifically include: Assuming the definition domain Ω of the two-dimensional static elasticity problem, the boundary of the definition domain is calculated based on the following formula: σ ij +b j =0 in Ω C=C u +C t Among them, σ ij is the stress tensor, b j is the physical force, Γ is the boundary of the definition domain Ω, Γ u is the forced displacement boundary, Γ t is the traction boundary, is an empty set; At this time, the boundary conditions are: s ij n j =t i onG t Among them, t i is the traction force, n j is the outer normal of the boundary, is the forced displacement, u i is the boundary Γ u displacement.

9. The smooth finite element numerical calculation method according to claim 8, It is characterized in that The step of obtaining the displacement gradient of the definition domain based on the outer normal of the boundary and the boundary displacement specifically includes: The following formula is used for variational weak processing: in, is the symmetric part of the displacement gradient, D ijkl is the material matrix, δ is the δ function, k and l are the node numbers; A smoothing operation is performed on the displacement gradient of each smoothing cell within each element using the following formula: Among them, Φ is a smooth function, u h (x C ) is the displacement after smoothing operation, x C is the smooth unit coordinate; According to the distribution integral, the right side of the above formula is transformed into: Where n(x) is the normal vector outside the boundary; The combined formulas yield: Among them, Γ C is a smooth cell boundary, is the displacement gradient after the smoothing operation.

10. The smooth finite element numerical calculation method according to claim 9, It is characterized in that The variational weak formula satisfies the following conditions: Among them, B I is the standard gradient matrix, N I is the shape function matrix.

11. The smooth finite element numerical calculation method according to claim 9, It is characterized in that The specific form of the smooth function Φ is as follows: Among them, Ω C is a smooth unit, A C For smooth unit Ω C The area, u h (x C ) is the displacement after smoothing operation, x C are smooth unit coordinates.

12. The smooth finite element numerical calculation method according to claim 9, It is characterized in that The smooth strain is calculated using the following formula: in, is the smooth strain matrix, is the smooth strain, d I is the elastic modulus.

13. The smooth finite element numerical calculation method according to claim 12, It is characterized in that The smooth strain matrix is ​​calculated using the following formula: Among them, n k (x) is the normal vector outside the boundary.

14. The smooth finite element numerical calculation method according to claim 6, It is characterized in that The solid material stiffness matrix and the load matrix are obtained based on the stiffness matrix of the polygonal unit, specifically including: The smooth unit stiffness matrices are combined to obtain the stiffness matrix of the polygonal unit, and the boundary conditions of the static elastic problem of the solid material are loaded to obtain the final solid material stiffness matrix K and load matrix F.

15. A computer device comprising a memory and a processor, wherein the memory stores a computer program. It is characterized in that When the processor executes the computer program, the steps of the method according to any one of claims 1 to 14 are implemented.

16. A computer-readable storage medium, It is characterized in that The computer storage medium includes one or more program instructions, and the one or more program instructions are used to execute the method according to any one of claims 1-14.

Images