C1 Continuous B-Spline Surface / Volume Numerical Shape Function Construction Method for Ultra-Large-Scale Heterogeneous Structure Simulation

Abstract

The present invention discloses a method for constructing a C1 continuous B-spline surface / body numerical shape function for ultra-large-scale heterogeneous structure simulation. By carefully designing an intermediate interpolation surface / body that covers the boundaries of coarse elements, this method realizes important properties such as C1 continuity and element segmentation of the numerical shape function. The numerical shape function is constructed based on a coarse grid, which greatly reduces the overall simulation degrees of freedom, lowers the computational cost, can reflect the internal fine-grained material information, maintains the high precision of the simulation results, has strong parallelism, and has great potential in the simulation of ultra-large-scale heterogeneous structures. The C1 continuity of this numerical shape function is crucial for the continuity of solutions of physical quantities involving first-order partial derivatives (such as stress / strain), which is an area not yet studied. This method is applicable to multiple simulation fields such as statics, dynamics, heat flow, and magnetic field analysis, and its advantages in stability, simulation accuracy, and computational efficiency have been verified in statics / dynamics simulations.

Description

Technical Field

[0001] The present invention relates to the technical field of advanced simulation of ultra-large-scale heterogeneous structures, and particularly to a method for constructing a C1 continuous B-spline surface / volume numerical shape function for ultra-large-scale heterogeneous structure simulation. Background Art

[0002] Heterogeneous structures, such as alloys, reinforced heterogeneous materials, multi-material composites, silicon-based materials, and biological structures, have wide applications in fields such as the automotive industry, aerospace, mechanical engineering, mobile phones, and biomedicine. In addition, heterogeneous structures can describe complex CAD models and porous models, avoiding the cumbersome or unstable defect handling or mesh generation processes, and are of great value in embedded simulation methods. Therefore, efficiently and accurately simulating the physical behavior of ultra-large-scale heterogeneous structures is a highly challenging problem in various simulation fields (such as applied mechanics, thermotics, and electromagnetics). However, the microscopic-scale material heterogeneity and tiny geometric features of the structures make it extremely challenging to find excellent numerical methods.

[0003] Traditional finite element method (FEM) and even the recent finite cell method (FCM) require a sufficiently fine mesh to resolve the fine-grained heterogeneity of ultra-large-scale structures to capture the details of heterogeneous structures for the calculation of the overall solution or the integration of element matrices, resulting in an excessive computational load. Therefore, there is an urgent need to develop a method that can both ensure computational accuracy and improve efficiency.

[0004] Numerical coarsening, or the multi-scale finite element method (MsFEM), is considered an effective method for solving the challenges of heterogeneous structure simulation. Different from the widely studied numerical homogenization method, numerical coarsening attempts to construct coarsened shape (or basis) functions on a coarse mesh to represent the solution space. It constructs shape functions for each coarse element and seeks the fine mesh solution in the finite-dimensional space spanned by these shape functions. The continuity of the shape function, especially at the element boundaries, is crucial for determining the quality of the simulation solution. For example, it is very critical in generating a continuous stress / strain field without post-processing, achieving higher accuracy and convergence speed, or avoiding cell-crossing artifacts in the material point method (MPM) in large non-linear deformations. In addition, it is also applicable to several numerical analysis problems involving second-order derivatives, such as Kirchhoff-Love shell theory and strain gradient elasticity methods. However, although various numerical shape functions have been proposed and the importance of C1 continuity has been frequently mentioned, so far, no numerical shape function with C1 continuity has been proposed. Summary of the Invention

[0005] To address the problems existing in the above heterogeneous structures and numerical coarsening, the present invention proposes a method for constructing C1 continuous B-spline surface / volume numerical shape functions for ultra-large-scale heterogeneous structure simulation. The present invention constructs a new numerical shape function with C1 continuity using two-dimensional / three-dimensional free-form B-spline surfaces / volumes for linear and nonlinear physical simulations of heterogeneous structures. As a verification, it is applied in the Material Point Method (MPM), abbreviated as coarsened MPM (cMPM), demonstrating performance advantages in avoiding element crossing noise and continuous stress fields in large non-linear deformations.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A method for constructing C1 continuous B-spline surface / volume numerical shape functions for ultra-large-scale heterogeneous structure simulation, comprising the following steps:

[0008] 1) Construct a coarse node-boundary displacement transformation. Introduce coarse nodes on the coarse background grid, adjust the number of coarse nodes by balancing the requirements between accuracy and efficiency, construct an intermediate interpolation surface / volume covering the boundaries of the coarse elements on the coarse elements, and sample at the boundary detail points to obtain a matrix-form coarse node-boundary displacement transformation;

[0009] 2) Construct a boundary-internal displacement transformation. Based on the displacement values of the boundary detail points, solve the equilibrium equations within the coarse elements to establish a displacement mapping relationship from the boundary detail points to the internal detail points, and realize the capturing ability of the shape function for the heterogeneous materials of the fine grid;

[0010] 3) Construct B-spline surface / volume shape functions with C1 continuity. Use the displacement mapping relationship between the above-obtained coarse nodes and detail points as control points, construct B-spline surface / volume shape functions for each coarse element, and the shape functions satisfy C1 continuity and element divisibility between the coarse elements;

[0011] In the above technical solution, in step 1), the simulation region is discretized using a regular Cartesian coarse background grid, and coarse nodes are introduced on each coarse element;

[0012] There are two methods for introducing coarse nodes, corresponding to two types of coarse elements: B-spline coarse elements and boundary B-spline coarse elements;

[0013] In the B-spline coarse element D α Among them, select all corner points located on D α and its adjacent elements as coarse nodes The number of coarse nodes is r = 16;

[0014] In the boundary B-spline coarse element D α Among them, select the endpoints located on the edges of D α and its adjacent edges as coarse nodes The number of coarse nodes is r = 12.

[0015] Construct the coarse node-boundary displacement transformation as follows:

[0016] For each coarse element D α Construct an intermediate interpolation surface / volume

[0017]

[0018] where Q I is a displacement component in the coarse node displacement vector Q I , and ψ I (x) is the basis function. Combine all the x and y components to obtain the following matrix form,

[0019]

[0020] For each boundary detail point x k , k = 1, 2, …, n b Calculate the matrix ψ α (x) and rearrange it row by row to obtain the coarse node-boundary displacement transformation Ψ α ,

[0021]

[0022] which is used to implement the mapping from the coarse node displacement Q α to the boundary detail point displacement q b . The different positions (i.e., the number and location) of the coarse nodes determine the different basis functions ψ α (x) and the corresponding IIS and matrix mapping Ψ α . This method proposes two strategies for constructing the intermediate interpolation surface / volume IIS, respectively for two types of coarse elements: B-spline coarse elements and boundary B-spline coarse elements.

[0023] Construct the intermediate interpolation surface / volume for B-spline coarse elements. Define a B-spline basis function B I (x) at each coarse node, then is expressed as,

[0024]

[0025] Since the coarse nodes are selected as the corner points of adjacent coarse elements, the obtained above will span the boundary of the coarse element D α , covering the boundary detail points and ensuring the C1 continuity of the constructed numerical shape functions between coarse elements.

[0026] Construct an intermediate interpolation surface / body for the boundary B-spline coarse element, construct B-spline curves on each edge, and stretch them along the direction of adjacent edges to obtain a local coverage of the boundary detail points

[0027]

[0028] where x k is the boundary detail point, C 0 is the B-spline curve on the edge e k nearest to x 0 , X0 and X1 are the two endpoints of the edge e 0 , C 1 and C 2 are the B-spline curves on the adjacent edges e 1 and e 2 , x i is the projection point of x k onto the edge e i (i = 0, 1, 2), and the weight a is calculated according to the relative distance ratio of x k ,

[0029]

[0030] where H is the size of the coarse element and s is the size of the fine element. Through the above stretching operation, it will also cross the boundary of the coarse element D α to cover the boundary detail points, ensuring the C1 continuity of the constructed numerical shape functions between coarse elements.

[0031] Compared with the aforementioned B-spline coarse element, the use of the boundary B-spline coarse element reduces the number of coarse nodes and can further improve the calculation efficiency with relatively less accuracy loss.

[0032] The step 2) of constructing the boundary-internal displacement transformation is specifically as follows:

[0033] The boundary-internal mapping matrix M α maps the boundary detail point displacement q α to the internal detail point displacement q b by solving the static equilibrium equation of the local boundary constraints within the coarse element D i . It takes the coarse node-boundary displacement transformation as the displacement constraint, and the static equilibrium equation is:

[0034]

[0035] where K b , K i , K bi , K ib are submatrices of the tangential stiffness matrix, fb and f i is a sub - vector of the load. By solving q in the second row of the above equation i , while assuming f i = 0, we get

[0036]

[0037] where q i is the displacement vector of the detail points, I is the identity matrix, and M α is the calculated boundary - interior mapping matrix.

[0038] The step 3) constructs the B - spline surface / volume shape functions as follows:

[0039] In the coarse element D α , the numerical shape function is represented as the B - spline surface / volume Φ α (x):

[0040] Φ α (x)= B α (x)p α

[0041] where B α (x) is the B - spline basis function matrix, and p α is the matrix that controls the shape of the B - spline shape function Φ α (x). Through the linear combination of the numerical shape function Φ α (x) and the coarse node displacement Q α , the displacement u(x) at any point x ∈ D α can be obtained:

[0042] u(x)= Φ α (x)Q α = B α (x)p α Q α , x ∈ D α .

[0043] To reflect the material distribution of the coarse element, p α will be expressed as the coarse node - detail point displacement mapping, that is, the product of the coarse node - boundary mapping Ψ α and the boundary - interior mapping M α ,

[0044]

[0045] where r is the number of coarse nodes, is the number of detail points, and n b is the number of boundary nodes.

[0046] The obtained shape functions satisfy C1 continuity, material sensitivity, and element partitionability. Applying them to the Galerkin simulation (MPM is adopted in the present invention) can improve the simulation accuracy and obtain a continuous stress / strain field.

[0047] The beneficial effects of the present invention are as follows:

[0048] 1) The present invention constructs numerical shape functions with C1 continuity for the first time. By using free-form B-spline surfaces / solids, it can capture the fine-grained heterogeneous material distribution within coarse elements, thereby achieving high-precision simulation solutions and continuous physical fields (such as the stress / strain field in the field of mechanical simulation).

[0049] 2) In coarse elements, in addition to the degrees of freedom at the vertices, additional degrees of freedom on the boundaries are introduced, and the number setting can balance different requirements for simulation accuracy or efficiency.

[0050] 3) The numerical shape functions are in matrix form, considering the anisotropy of the materials inside the coarse elements. The shape functions satisfy the important property of unit partitionability, avoiding some non-physical simulation results.

[0051] 4) For each coarse element, the numerical shape functions are constructed based on two consecutive mappings from the coarse nodes to the boundary detail points and then to the internal detail points. This process only requires solving a small-scale linear system, with high computational efficiency.

[0052] 5) The numerical shape functions can eliminate the element-crossing noise caused by the discontinuous derivatives of the shape functions in traditional MPM and greatly improve the computational efficiency. Description of the Drawings

[0053] Figure 1 The placement methods of coarse nodes for two types of coarse elements: B-spline coarse elements and boundary B-spline coarse elements, and their corresponding intermediate interpolation surfaces.

[0054] Figure 2 The flow schematic diagram of the method of the present invention: (a) introducing coarse nodes on the coarse element → (b) constructing the intermediate interpolation surface IIS to obtain the coarse node-boundary mapping → (c) solving the local equilibrium equation to construct the boundary-internal mapping → (d) constructing the B-spline surface / solid shape function → (e) obtaining a continuous stress field through simulation.

[0055] Figure 3 The boundary conditions of the bending beam experiment and the meshes of different methods: a. Boundary conditions; b. Fine mesh used in the BSMPM simulation; c. Coarse mesh used in the BSMPM; d. Coarse mesh used in the cMPM.

[0056] Figure 4 The deformation results of the bending beam experiment. Detailed Implementation Manner

[0057] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0058] The present invention provides a method for constructing a C1 continuous shape function on a non - linear heterogeneous structure. It uses free - form B - spline surfaces / volumes, can capture the fine - scale heterogeneous material distribution within coarse grid cells, realize continuous physical quantity distribution (such as stress / strain fields), maintain simulation accuracy, and at the same time greatly improve the simulation efficiency. These numerical shape functions satisfy the important properties of element division and are efficiently calculated through sequential coarse - grid boundary mapping and boundary - interior mapping. Only a small - scale linear system needs to be solved for each coarse grid cell, with a small amount of calculation. The application and testing of the numerical shape functions in the simulation of non - linear heterogeneous structures based on MPM show high precision and high efficiency. The C1 continuity of the numerical shape functions is crucial for the continuity of stress / strain solutions, etc., which is an area not yet studied.

[0059] The method of the present invention is as follows:

[0060] Discretize the simulation region using a regular coarse background grid to obtain coarse cells D α , for the coarse cell D α Two methods for introducing coarse nodes are proposed, corresponding to two types of coarse cells: B - spline coarse cells and boundary B - spline coarse cells, as Figure 1 shown.

[0061] In the B - spline coarse cell D α , select the corner points located on D α and its adjacent cells as coarse nodes The number of coarse nodes is r = 16. In the boundary B - spline coarse cell D α , select the end points located on the sides of D α and its adjacent sides as coarse nodes The number of coarse nodes is r = 12. The constructed numerical shape function is expressed as a quadratic B - spline surface / volume Φ α (x):

[0062] Φ α (x)=B α (x)p α

[0063] where B α (x) is the quadratic B - spline basis function matrix, and p α is the matrix that controls the shape of the B - spline shape function Φ α (x). Since the p - order B - spline shape function has (p - 1) - order continuity within its domain of definition, the quadratic B - spline surface / volume shape function constructed above is continuous within the coarse cell D αIt is C1 continuous inside.

[0064] Define Q α as the coarse node displacements (or degrees of freedom DOFs) to be calculated. Through the numerical shape function Φ α (x) and the linear combination of the coarse node displacements Q α , the displacement u(x) at any point x ∈ D α can be obtained.

[0065] u(x) = Φ α (x)Q α = B α (x)p α Q α , x ∈ D α .

[0066] On the other hand, consider the discrete coarse element D α = X × Y with node vectors X and Y. Its detail points are In the fine grid, the detail point displacements q at the detail points α can be calculated globally for the structure Ω or locally for the element D α . For any point x ∈ D α , the high-fidelity displacement can be expressed as the linear combination of q α and the B-spline basis function B α (x) as follows:

[0067]

[0068] Considering the above equation, the coarse element shape function Φ α (x), or the control matrix p α , needs to satisfy the following equation:

[0069]

[0070] Therefore, the control matrix p α of the numerical shape function needs to be constructed as the mapping from the coarse node displacements Q α to the detail point displacements q α . In the present invention, p α will be reformulated as two mappings from the coarse nodes to the boundary detail points and then to the internal detail points. Mathematically, p α can be expressed as the product of a coarse node - boundary mapping matrix Ψ b of size (2n α × 2r) and a boundary - internal mapping matrix, where r is the number of coarse nodes, is the number of detail points, and n is bis the number of boundary detail points,

[0071]

[0072] where the matrix Ψ α , M α maps the coarse node displacement Q α to the boundary detail point displacement q b , and then maps it to the internal detail point displacement q i by solving the local static equilibrium problem, that is,

[0073] q α = M α q b , q b = Ψ α Q α .

[0074] The construction of these two mappings will be introduced in detail below.

[0075] 1) Coarse node - boundary mapping Ψ α

[0076] The key to constructing the coarse node - boundary mapping Ψ α lies in the construction of the IIS that covers the boundary detail points of the coarse element. Based on a coarse element D α the IIS is constructed and defined as as follows,

[0077]

[0078] where Q I is a displacement component in the coarse node displacement vector Q I . ψ I (x) is a specific IIS basis function. Combining all the x and y components, the following matrix form is obtained,

[0079]

[0080] where the matrix ψ α (x), Q α are of sizes (2×2r) and (2r×1) respectively. At each boundary detail point x k , k = 1, 2,..., n b the matrix ψ α (x) is calculated and rearranged row - by - row to obtain the coarse node - boundary matrix Ψ α ,

[0081]

[0082] which is used to achieve the coarse node displacement Qα Mapping of the displacement q of the boundary detail points b . The different positions (i.e., quantity and position) of the coarse nodes determine different basis functions ψ α (x) and the corresponding IIS and matrix mapping Ψ α . This method proposes two strategies for constructing IIS, or two types of coarse elements: B-spline coarse elements and boundary B-spline coarse elements.

[0083] As Figure 1 shown, the IIS covers the boundary detail points on the coarse element D α . Therefore, the common boundary detail points of adjacent coarse elements have the same value, which ensures the C1 continuity of the numerical shape functions. Specifically, for adjacent coarse elements D α and D β and a common coarse node I of them, the numerical shape function of the coarse node I is defined as Φ α (x) and Φ β (x) on D α,I and D β,I respectively. Because the intermediate interpolation surfaces and of adjacent coarse elements have the same value at their common boundary detail points, according to the formula of B-spline surfaces, Φ α,I (x) and Φ β,I (x) have the same value and partial derivative at the boundary of the coarse elements. Therefore, the constructed shape functions have C1 continuity between the coarse elements. In addition, the IIS has the partition of unity property, which also ensures that the constructed numerical shape functions satisfy this property.

[0084] 2) Boundary-Interior Mapping M α

[0085] The boundary-interior mapping matrix M α maps the displacement q α of the boundary detail points to the displacement q b of the interior detail points by solving the static equilibrium equation of the boundary displacement constraints within the coarse element D i . Its knot vector D α =X×Y is the analysis grid, with the B-spline basis functions at the detail points as the shape functions and the values of the coarse node-boundary mapping matrix as the Dirichlet boundary conditions.

[0086] This method observes that instead of using the full Newton-Raphson method to solve the local problem, only using the first iteration of the Newton-Raphson method can greatly improve the efficiency while maintaining a very close accuracy to the former. The static equilibrium equation is:

[0087]

[0088] wherein is the tangential stiffness matrix, and are the external force and internal force vectors respectively. Considering the boundary detail points and the internal detail points are the retained nodes and truncated nodes respectively, the above equation can be rewritten as:

[0089]

[0090] where K b , K i , K bi , K ib are sub - matrices of b and f i and f i are sub - vectors of the load force. By solving q i in the second row of the above equation, and assuming f

[0091]

[0092] Therefore

[0093] wherein

[0094] where I is the identity matrix of size (2n b × 2n b ), and M α is the boundary - internal mapping matrix of size .

[0095] Based on the two constructed mapping matrices, a C1 - continuous numerical shape function is constructed.

[0096] Figure 2 is the schematic diagram of the construction of the C1 - continuous numerical shape function of the present invention. Among them, (a) is to introduce coarse nodes on the coarse element, (b) is to construct the intermediate interpolation surface IIS to obtain the coarse node - boundary mapping, (c) is to solve the local equilibrium equation to construct the boundary - internal mapping, (d) is to construct the B - spline surface / solid shape function, and (e) is to simulate to obtain a continuous stress field, and it can be seen that it satisfies C1 - continuity.

[0097] To test the performance of the proposed shape function, it is applied in the MPM simulation to obtain the coarsened MPM, abbreviated as cMPM. The following is a specific solution case using the method of the present invention:

[0098] Figure 3(a) is a three-dimensional very large scale heterogeneous curved beam experiment with a size of 20×2×2, containing 40 softer elliptic cylinders with various shapes. The Young's moduli of the elliptic cylinders and other parts are 1e 2 , 1e 5 respectively. The curved beam is fixed at both ends and subjected to the action of gravity. We conduct the following three groups of tests: cMPM with a coarse grid, B-spline MPM (BSMPM) with a fine grid, and BSMPM with a coarse grid. Figure 3 (b)(c)(d) show the grids of the three groups of tests. Among them, the fine-grid BSMPM is regarded as the reference solution. The simulation deformation results at t = 0.3 and 0.6 s are as Figure 4 shown. cMPM obtains a deformation very close to the reference solution, while the BSMPM on the coarse grid cannot correctly reflect the heterogeneous material distribution of the structure. The maximum displacement errors of the two also verify this, which are 0.016 and 0.095 respectively. On the other hand, compared with the BSMPM on the fine grid, cMPM achieves an acceleration ratio of 11 times, verifying the efficiency of the numerical shape function of the present invention.

[0099] The above are only some of the preferred solutions of the present invention, but they are not intended to limit the present invention. Those of ordinary skill in the relevant technical field can also make various changes and modifications, as well as expand to the corresponding thermal simulation, electromagnetic simulation, etc., without departing from the spirit and scope of the present invention. Therefore, all technical solutions obtained by means of equivalent replacement or equivalent transformation fall within the protection scope of the present invention.

Claims

1. A simulation method for ultra-large-scale heterogeneous structures based on constructing C1 continuous B-spline surface / body numerical shape functions, characterized in that: include: In the process of mechanical simulation of heterogeneous structures, the simulation area is discretized using a regular Cartesian coarse background grid, and coarse nodes are introduced on each coarse unit; An intermediate interpolation surface / body is constructed on the coarse unit to ensure C1 continuity between coarse units. The boundary of the coarse unit area covered by the intermediate interpolation surface / body is the boundary detail point, and the inner area of ​​the boundary is the inner detail point. The coarse node-boundary displacement transformation in matrix form is obtained by sampling the boundary detail points. By solving the static equilibrium equation in the coarse unit, the displacement mapping relationship between the boundary detail points and the internal detail points is established to obtain the boundary-internal displacement transformation; Based on the coarse node-boundary displacement transformation and boundary-internal displacement transformation, the displacement mapping relationship between coarse nodes and detail points is obtained. A B-spline surface / body shape function with C1 continuity is constructed for each coarse element to obtain a numerical shape function, which is applied to Galerkin simulation to obtain a continuous stress / strain field. Where: In each coarse unit D α Introduce r coarse nodes For each coarse unit D α Constructing intermediate interpolation surfaces / bodies Where Q I is the coarse node displacement vector Q I A displacement component of I (x) is The basis functions of , combining all x and y components, are obtained in matrix form: For each boundary detail point x k ,k=1,2,…,n b , calculate the matrix ψ α (x) and rearrange them row by row to obtain the coarse node-boundary displacement transformation Ψ α : It is used to realize the coarse node displacement Q α Displacement q to boundary detail point b Mapping; n b is the number of boundary detail points.

2. The simulation method for ultra-large-scale heterogeneous structures based on constructing C1 continuous B-spline surface / body numerical shape functions according to claim 1, characterized in that: There are two methods for introducing coarse nodes, corresponding to two coarse units: B-spline coarse unit and boundary B-spline coarse unit. In the B-spline coarse element D α , select the D α All corner points on its adjacent units are coarse nodes The number of coarse nodes is r = 16; B-spline coarse element D α , select the D α The endpoints of the edge and its adjacent edges are thick nodes The number of coarse nodes is r=12.

3. The simulation method for ultra-large-scale heterogeneous structures based on constructing C1 continuous B-spline surface / body numerical shape functions according to claim 1, characterized in that: Construct an intermediate interpolation surface / body for the B-spline coarse element and define a B-spline basis function B at each coarse node. I (x), then Expressed as, Since the coarse nodes select the corner points of adjacent coarse units, the above formula gives Will cross the coarse unit D α The boundary of the coarse element is covered, and the boundary detail points are covered, which ensures the C1 continuity of the constructed numerical shape function between the coarse elements.

4. The simulation method for ultra-large-scale heterogeneous structures based on constructing C1 continuous B-spline surface / body numerical shape functions according to claim 1, characterized in that: Construct an intermediate interpolation surface / body for the boundary B-spline coarse element, construct a B-spline curve on each edge, and stretch it along the direction of the adjacent edge to obtain a local coverage of the boundary detail points where x k is the boundary detail point, C 0 is from x k The nearest edge 0 B-spline curve on, X0 and X1 are edges e 0 The two endpoints, C 1 and C 2 is the adjacent edge e 1 and e 2 B-spline curve on x i is x k To the edge i The projection point, i = 0, 1, 2, weight a according to x k The relative distance ratio is calculated. Where H is the coarse element size and s is the fine element size. Through the above stretching operation, It will also cross the coarse unit D α The boundary of the coarse element is covered, and the boundary detail points are covered, which ensures the C1 continuity of the constructed numerical shape function between the coarse elements.

5. The simulation method for ultra-large-scale heterogeneous structures based on constructing C1 continuous B-spline surface / body numerical shape functions according to claim 1, characterized in that: By solving the coarse element D α The static equilibrium equation of the local boundary constraint within the boundary is displaced by q b Mapped to internal detail point displacement q i , which uses the coarse node-boundary displacement transformation as the displacement constraint, and the static equilibrium equation is: Where K b ,K i ,K bi ,K ib is a submatrix of the tangential stiffness matrix, f b and f i is a sub-vector of the load, and by solving the second line of the above equation, q i , assuming that f i =0, we get: in where q α is the detail point displacement vector, I is the identity matrix, M α is the computed boundary-interior mapping matrix.

6. The simulation method for ultra-large-scale heterogeneous structures based on constructing C1 continuous B-spline surface / body numerical shape functions according to claim 5, characterized in that: In solving the coarse element D α Only the first iteration of the Newton-Raphson method is used when solving the static equilibrium equations within local boundary constraints.

7. The simulation method for ultra-large-scale heterogeneous structures based on constructing C1 continuous B-spline surface / body numerical shape functions according to claim 1, characterized in that: In the coarse unit D α In the example, the numerical shape function is represented as a B-spline surface / body Φ α (x): Φ α (x)=B α (x)p α Among them B α (x) is the B-spline basis function matrix, p α is the controlling B-spline shape function Φ α (x) shaped matrix; by numerical shape function Φ α (x) and coarse node displacement Q α The linear combination of can get any point x∈D α Displacement u(x): u(x)=Φ α (x)Q α =B α (x)p α Q α ,x∈D α . In order to reflect the material distribution of the coarse unit, p α It will be expressed as a coarse node-detail point displacement mapping, that is, a coarse node-boundary mapping Ψ α and boundary-interior mapping M α The product of Where r is the number of coarse nodes, is the number of detail points, n b The number of boundary detail points.

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