Concrete structure damage variable-scale finite element simulation method based on two layers of grids
By establishing a finite element model with a two-layer mesh of structure and material in concrete structures and dynamically updating scale transformation and connection, the cross-scale problem of concrete structure damage simulation in existing technologies is solved, and high-precision and efficient damage evolution analysis is achieved.
Patent Information
- Application Number
- CN202511531385.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-10-24
- Publication Date
- 2026-02-06
AI Technical Summary
In existing technologies, the simulation analysis of damage evolution in concrete structures cannot accurately consider the random non-uniform characteristics and complex material composition of concrete, and cannot simulate damage evolution characteristics across scales.
A two-layer mesh-based method is used to establish finite element models at both the structural and material scales within the analysis domain of the concrete structure. By simulating scale transformation and dynamically updating the collaborative finite element model through structural-material scale connection, the cross-scale evolution simulation of concrete structure damage from the material scale to the structural scale is realized.
It achieves high-precision and high-efficiency simulation of the damage evolution process of concrete structures, breaking through the bottleneck of traditional single-structural-scale analysis, and enabling refined analysis of the damage process of concrete structures.
Smart Images
Figure CN121480144A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the field of concrete damage analysis, and particularly to a two-layer mesh-based concrete structure damage variable-scale finite element simulation method. BACKGROUND
[0002] Concrete structures play a significant role in national infrastructure systems such as water conservancy and hydropower, civil engineering, and transportation. Under various loads, the working behavior of concrete structures is complex, and local high-stress areas are usually in the nonlinear stress stage, making them prone to damage and even cracking and failure, which poses a significant challenge to long-term safe service. Therefore, accurately simulating the damage evolution process of concrete structures under loads is of great significance for objectively evaluating their structural safety.
[0003] Concrete is a typical multi-scale material, and at the material scale, it is usually considered as a three-phase heterogeneous composite material composed of aggregates, mortar, and the interface transition zone between them. Although in the elastic stress stage, concrete can be regarded as a homogeneous material and its stress and deformation behavior can be described by a linear elastic constitutive model at the structural scale, in the damage and cracking stage, the structural scale constitutive model based on the assumption of "homogeneity" cannot accurately describe the complex nonlinear mechanical behavior of concrete. The main reason is that the damage evolution of concrete is directly related to its material composition and structure, showing obvious random, heterogeneous, and cross-scale characteristics. Therefore, accurately simulating the damage evolution process of concrete structures under loads requires considering the material composition and structure of concrete.
[0004] At present, the simulation and analysis of the damage evolution process of concrete structures mainly uses numerical methods represented by the finite element method. Specifically, concrete is regarded as a homogeneous material at the structural scale, and a concrete structure damage finite element model is established to simulate the load process of concrete structures, thereby obtaining the damage state of concrete structures at different stress stages. The above technology has the following shortcomings:
[0005] (1) In the existing technology, concrete is regarded as a homogeneous material, which has the inherent defect of not considering the random and heterogeneous characteristics of concrete at the material scale.
[0006] (2) In the existing technology, a concrete damage constitutive model based on the structural scale is used, which cannot accurately reflect the damage mechanism resulting from the complex material composition and structure of concrete.
[0007] (3) In the existing technology, the simulation of the damage evolution process of concrete structures is carried out at a single structural scale, which cannot simulate the cross-scale evolution characteristics of concrete structure damage from the material scale to the structural scale. SUMMARY
[0008] The technical problem solved by the present application is to provide a two-layer grid-based concrete structure damage variable scale finite element simulation method aiming at the deficiencies of the prior art.
[0009] To solve the above technical problems, the technical scheme adopted by the present application is:
[0010] A two-layer grid-based concrete structure damage variable scale finite element simulation method comprises the following steps:
[0011] Step 1: The concrete structure is subjected to structure scale and material scale finite element grid division, respectively, to obtain structure-material two-layer grids covering all the concrete structure simulation regions.
[0012] Step 2: Material properties are assigned to the structure scale elements and the material scale elements to establish a structure scale finite element model and a material scale finite element model of the concrete structure, respectively.
[0013] Step 3: Based on the simulation scale transformation criterion and the structure scale-material scale connection technology, the structure scale-material scale collaborative finite element model is dynamically updated to simulate the concrete structure damage evolution process.
[0014] In step 3, the simulation scale transformation criterion is a judgment criterion for transforming the structure scale simulation region into the material scale simulation region, and the general expression is as follows:
[0015]
[0016] In the formula, is the stress of the integral point of the structure scale element located in the structure scale simulation region.
[0017] In step 3, when the concrete damage is tensile stress damage, the simulation scale transformation criterion is based on the maximum tensile stress criterion, and the tensile stress is positive, and the specific expression is:
[0018]
[0019] In the formula, is the first principal stress of the integral point of the structure scale element located in the structure scale simulation region.
[0020] The maximum tensile stress allowable value is the peak tensile strength of the concrete.
[0021] In step 3, the simulation method of the damage evolution process of the concrete structure comprises the following steps:
[0022] Step 3-1, a structure scale-material scale collaborative finite element model is established and set in the initial state to only include a structure scale simulation region.
[0023] Step 3-2, the loading process of the concrete structure is divided into K increments.
[0024] Step 3-3, for any increment, the structure scale-material scale collaborative finite element balance iteration solution of the damage evolution process of the concrete structure in the increment is carried out; when the iteration converges, the integral point stress of each structure scale unit in the structure scale simulation region in the structure scale-material scale collaborative finite element model is obtained.
[0025] Step 3-4, each integral point stress obtained in step 3-3 is judged by using the simulation scale transformation criterion of formula (1); when there is an integral point stress satisfying formula (1), step 3-5 is executed; otherwise, jump to step 3-8.
[0026] Step 3-5, the simulation scale of the simulation region occupied by all the structure scale units corresponding to all the integral point stresses satisfying formula (1) is transformed from the structure scale to the material scale.
[0027] Step 3-6, each material scale unit included in the structure scale-material scale collaborative finite element model is subjected to structure scale-material scale connection, so as to complete the dynamic update of the structure scale-material scale collaborative finite element model.
[0028] Step 3-7, for the current increment, the structure scale-material scale collaborative finite element balance iteration solution of the damage evolution process of the concrete structure is carried out again, and steps 3-3 to 3-6 are repeated until the integral point stress of each structure scale unit in the current increment does not satisfy formula (1).
[0029] Step 3-8, steps 3-3 to 3-7 are repeated to carry out the structure scale-material scale collaborative finite element balance iteration solution of the damage evolution process of the concrete structure in the next increment.
[0030] In step 3-2, the number K of increments is set according to the size of the load F to be applied and the simulation accuracy of the damage evolution process of the structure.
[0031] In steps 3-5, the method of transforming the structure scale to the material scale is as follows: all structure scale units corresponding to integral point stresses satisfying formula (1) are eliminated from the structure scale-material scale collaborative finite element model, and material scale units associated with the eliminated structure scale units are included in the structure scale-material scale collaborative finite element model.
[0032] In steps 3-5, the material scale units associated with the eliminated structure scale units refer to material scale units having at least one node located within the simulation region occupied by the eliminated structure scale units.
[0033] In step 3-6, the dynamically updated structure scale-material scale collaborative finite element model includes a structure scale simulation region and a material scale simulation region .
[0034] The structure scale simulation region refers to a simulation region in the concrete structure in the elastic stress stage, which is analyzed using structure scale units.
[0035] The material scale simulation region refers to a simulation region in the concrete structure in the non-elastic stress stage, which is analyzed using material scale units.
[0036] The structure scale simulation region and the material scale simulation region do not overlap and jointly constitute all simulation regions of the concrete structure , that is , and .
[0037] Macro material scale connection refers to establishing a material scale unit node displacement constraint equation based on the shape function of the structure scale unit to ensure the deformation coordination between the structure scale unit and the material scale unit in the structure scale-material scale collaborative finite element model.
[0038] The specific expression of the material scale unit node displacement constraint equation based on the shape function of the structure scale unit is as follows:
[0039] (3)
[0040] In the formula, is the number of structure scale unit nodes;
[0041] is the coordinate vector of the meso-unit node ;
[0042] is the function value of the shape function of the structure scale unit corresponding to the th node with respect to , ;
[0043] is the displacement vector of the structure scale unit node; is the displacement vector of the structure scale unit node;
[0044] is the displacement vector of the material scale unit node. is the displacement vector of the material scale unit node.
[0045] The present application has the following beneficial effects:
[0046] 1. The present application obtains the structure-material two-layer grid covering all simulation regions of the concrete structure by respectively performing structure scale and material scale finite element grid division on the concrete structure, and dynamically updates the structure scale-material scale collaborative finite element model of the concrete structure based on simulation scale transformation and structure scale-material scale connection.
[0047] 2. The present application can dynamically determine the simulation region requiring material scale simulation in the simulation process through simulation scale transformation, and can ensure the deformation coordination between the structure scale unit and the material scale unit in the structure scale-material scale collaborative finite element model through the establishment of the material scale unit node displacement constraint equation based on the structure scale unit shape function.
[0048] 3. The present application can simulate the cross-scale evolution process of the concrete structure damage from the material scale to the structure scale by considering the material composition structure of the concrete in the material scale simulation region, breaks through the bottleneck that the traditional single structure scale analysis method cannot accurately simulate the cross-scale evolution characteristics of the concrete damage, and has high precision and high efficiency, thereby providing an effective solution for the refined analysis of the concrete structure damage evolution process. DETAILED DESCRIPTION
[0049] Figure 1 is the flow chart of the concrete structure damage variable scale finite element simulation method based on the two-layer grid of the present application.
[0050] Figure 2 is the schematic diagram of the concrete structure geometry and size, boundary condition and load action of the embodiment of the present application.
[0051] Figure 3 is the schematic diagram of the three-phase material composition structure of the concrete structure of the embodiment of the present application.
[0052] Figure 4 is the schematic diagram of the structure-material two-layer grid of the concrete structure of the embodiment of the present application.
[0053] Figure 5 is the schematic diagram of the concrete structure damage evolution process of the embodiment of the present application.
[0054] Figure 6 This is a schematic diagram of the load-displacement curve of a concrete structure according to an embodiment of the present invention. Detailed Implementation
[0055] The present invention will now be described in further detail with reference to the accompanying drawings and specific preferred embodiments.
[0056] like Figure 1 As shown, the variable-scale finite element method for simulating concrete structure damage based on a two-layer mesh includes the following steps.
[0057] Step 1: Perform finite element mesh generation on the concrete structure at both the structural and material scales to obtain a two-layer mesh covering the entire simulated area of the concrete structure: structural and material meshes.
[0058] The aforementioned structural dimensions refer to the spatial dimensions of concrete as a homogeneous material.
[0059] The aforementioned material scale refers to the spatial scale that considers concrete as a three-phase non-homogeneous material mainly composed of aggregates, mortar, and the interfacial transition zone between the two.
[0060] The aforementioned structure-material two-layer mesh includes a structure-scale finite element mesh and a material-scale finite element mesh.
[0061] In this embodiment, as Figure 2 As shown, the tensile damage evolution process of a concrete L-shaped specimen is simulated. The specimen is 500 mm high and 500 mm wide. Displacement constraints are applied to the bottom of the specimen, and a vertical uniformly distributed displacement load u with a value of 0.8 mm is applied to the right side. In this embodiment, the preferred method for dividing the structure-material two-layer mesh includes the following steps.
[0062] Step 1-1: Preferably, three-node triangular elements are used to perform structural-scale finite element mesh generation on the L-shaped concrete structure to obtain a structural-scale finite element mesh covering the entire structural simulation area.
[0063] Step 12: The three-phase material composition structure of the concrete structure is generated by random sampling and placement.
[0064] The three-phase material composition structure of a concrete structure refers to the concrete material composition structure formed by aggregates, mortar, and the interfacial transition zone (ITZ) between them, which are randomly and non-uniformly distributed throughout the entire simulated area of the concrete structure.
[0065] In this embodiment, the concrete specimens were grade 1 concrete with aggregate particle sizes ranging from 5 to 20 mm and an aggregate content of 50%. The three-phase material composition structure of the L-shaped concrete specimens was generated using a random sampling method. Figure 3 As shown.
[0066] exist Figure 3In the embodiment, the concrete L-shaped specimen is meshed by three-node triangular elements and four-node quadrilateral elements. The three-node triangular elements are used for meshing the coarse aggregate and the mortar, and the four-node quadrilateral elements are used for meshing the interface transition zone.
[0067] Step 13, based on the generated three-phase material composition structure of the concrete structure, the concrete structure is meshed by material-scale finite element meshes to obtain the material-scale finite element meshes covering the entire structure simulation area as shown in FIG. 4. Figure 4
[0068] Step 2, material properties are assigned to the structure-scale elements and the material-scale elements to respectively establish the structure-scale finite element model and the material-scale finite element model of the concrete structure.
[0069] The structure-scale element refers to an element in the structure-scale finite element mesh. The structure-scale finite element model of the concrete structure refers to a finite element model of the concrete structure containing and only containing all the structure-scale elements.
[0070] The material-scale element refers to an element in the material-scale finite element mesh, including the aggregate element, the mortar element and the interface transition zone element. The material-scale finite element model of the concrete structure refers to a finite element model of the concrete structure containing and only containing all the material-scale elements.
[0071] The material parameters refer to the model parameters of the material constitutive model adopted by the structure-scale elements and the model parameters of the material constitutive model adopted by the aggregate elements, the mortar elements and the interface transition zone elements in the material-scale elements.
[0072] In the embodiment, the material constitutive model adopted by the structure-scale elements and the aggregate elements in the material-scale elements is a linear elastic constitutive model.
[0073] A, the material constitutive model adopted by the structure-scale elements and the aggregate elements in the material-scale elements is a linear elastic constitutive model.
[0074] B, the material constitutive model adopted by the mortar elements and the interface transition zone elements in the material-scale elements is an elastic-plastic damage constitutive model.
[0075] Table 1 below shows the material parameters of the structure-scale elements and the aggregate elements, the mortar elements and the interface transition zone elements in the material-scale elements.
[0076] Table 1 Material parameters of structure-scale elements, aggregate elements, mortar elements and interface transition zone elements
[0077]
[0078] Step 3, based on the simulation scale transformation criterion and the structure-scale-material-scale connection technology, the structure-scale-material-scale collaborative finite element model is dynamically updated to simulate the damage evolution process of the concrete structure.
[0079] The simulation scale includes a structure scale and a material scale.
[0080] The simulation scale transformation criterion refers to a judgment criterion adopted when transforming the structure scale simulation region into the material scale simulation region, and a general expression is as follows:
[0081]
[0082] In the formula, is the stress of the structure scale unit integral point located in the structure scale simulation region.
[0083] In this embodiment, the concrete damage is preferably tensile stress damage, and the simulation scale transformation criterion is based on the maximum tensile stress criterion, with tension as positive, and the specific expression of formula (1) is given as follows:
[0084]
[0085] In the formula, is the first principal stress of the structure scale unit integral point located in the structure scale simulation region. is the maximum tensile stress allowable value, which is taken as the peak tensile strength of the concrete, i.e., 1.45 MPa.
[0086] In step 3, the simulation method of the concrete structure damage evolution process includes the following steps.
[0087] Step 3-1, a structure scale-material scale collaborative finite element model is established, which only includes the structure scale simulation region in the initial state.
[0088] Step 3-2, the load action process of the concrete structure is divided into K incremental steps. Among them, the number K of incremental steps is set according to the size of the required load action F and the simulation accuracy of the structure damage evolution process.
[0089] In this embodiment, the load action process of the concrete L-shaped specimen is divided into 32 incremental steps, i.e., a displacement incremental load of 0.025 mm is applied in each incremental step.
[0090] Step 3-3, for any incremental step, the structure scale-material scale collaborative finite element balance iteration solution of the concrete structure damage evolution process in the incremental step is carried out; when the iteration converges, the integral point stress of each structure scale unit in the structure scale-material scale collaborative finite element model in the structure scale simulation region is obtained.
[0091] Step 3-4, the stress of each integral point obtained in step 3-3 is judged by using the simulation scale transformation criterion of formula (1); when there is an integral point stress satisfying formula (1), step 3-5 is executed; otherwise, jump to step 3-8.
[0092] In the embodiment, the stress of each integral point obtained in step 3-3 is judged by using the maximum tensile stress criterion based on formula (2).
[0093] Step 3-5, the simulation scale of the simulation region occupied by all structure scale units corresponding to the integral point stress satisfying formula (1) (preferably satisfying formula (2) in the embodiment) is transformed from structure scale to material scale.
[0094] The method of transforming the structure scale to the material scale is preferably: all structure scale units corresponding to the integral point stress satisfying formula (1) (preferably satisfying formula (2) in the embodiment) are eliminated from the structure scale-material scale collaborative finite element model, and the material scale units associated with the eliminated structure scale units are included in the structure scale-material scale collaborative finite element model.
[0095] The material scale unit associated with the eliminated structure scale unit refers to a material scale unit having at least one node located within the simulation region occupied by the eliminated structure scale unit.
[0096] Step 3-6, each material scale unit included in the structure scale-material scale collaborative finite element model is subjected to structure scale-material scale connection, thereby completing dynamic updating of the structure scale-material scale collaborative finite element model.
[0097] The dynamically updated structure scale-material scale collaborative finite element model includes a structure scale simulation region and a material scale simulation region .
[0098] The structure scale simulation region refers to a simulation region in the concrete structure in the elastic stress stage, which is analyzed by using structure scale units.
[0099] The material scale simulation region refers to a simulation region in the concrete structure in the non-elastic stress stage, which is analyzed by using material scale units.
[0100] The structure scale simulation region and the material scale simulation region do not overlap and jointly constitute the entire simulation region of the concrete structure , that is and .
[0101] Macro-material scale connection refers to ensuring deformation coordination between structural scale elements and material scale elements in a structural-material scale collaborative finite element model by establishing material scale element nodal displacement constraint equations based on the shape function of structural scale elements.
[0102] The specific expression for the material-scale element nodal displacement constraint equation based on the structural-scale element shape function is as follows:
[0103] (3)
[0104] In the formula, This represents the number of nodes in a structural unit.
[0105] To observe the unit nodes in detail The coordinate vector;
[0106] For structural scale units corresponding to the first The shape function of each node with respect to The function value;
[0107] For structural scale unit number The displacement vector of each node;
[0108] Material-scale element nodes The displacement vector.
[0109] Step 3-7: For the current increment step, restart the structural-material scale collaborative finite element equilibrium iteration solution of the concrete structure damage evolution process, and repeat steps 3-3 to 3-6 until the integral point stress of each structural scale element in the current increment step does not satisfy formula (1).
[0110] Step 3-8: Repeat steps 3-3 to 3-7 to carry out the structural-material scale co-finite element equilibrium iterative solution of the concrete structure damage evolution process in the next incremental step.
[0111] Figure 5 The finite element simulation results of the damage evolution process of the L-shaped concrete specimen are presented, that is, the evolution process of the damage variable d during the loading process. Among them, (a), (b), (c) and (d) show the finite element simulation results of the damage variable d at the 4th, 10th, 16th and 32nd increment steps, respectively.
[0112] Figure 6 The finite element simulation results of the load (reaction force)-displacement curve of the concrete L-shaped specimen are presented.
[0113] The preferred embodiments of the present application are described in detail above, but the present application is not limited to the specific details of the above-described embodiments, and various equivalent transformations of the technical solutions of the present application can be made within the technical concept of the present application, and these equivalent transformations all belong to the protection scope of the present application.
Claims
1. A finite element method for simulating damage in concrete structures based on a two-layer mesh, characterized in that: Includes the following steps: Step 1: Perform finite element mesh generation on the concrete structure at both the structural and material scales to obtain a two-layer mesh covering the entire simulated area of the concrete structure: structural and material meshes. Step 2: Assign material properties to structural-scale elements and material-scale elements to establish structural-scale finite element models and material-scale finite element models of the concrete structure, respectively; Step 3: Based on the simulation scale transformation criterion and the structural scale-material scale connection technology, dynamically update the structural scale-material scale collaborative finite element model to simulate the damage evolution process of concrete structures.
2. The finite element method for variably scaled damage simulation of concrete structures based on a two-layer mesh as described in claim 1, characterized in that: In step 3, the simulation scale transformation criterion refers to the judgment criterion used when transforming the structural scale simulation region into the material scale simulation region. The general expression is as follows: ; In the formula, The stress at the integral point of the structural-scale element located within the structural-scale simulation region is denoted as .
3. The finite element method for variably scaled damage simulation of concrete structures based on a two-layer mesh as described in claim 2, characterized in that: In step 3, when the concrete damage is tensile stress damage, the simulation scaling criterion is based on the maximum tensile stress criterion, with tension as positive. The specific expression is as follows: ; In the formula, The stress at the integral point of the structural-scale element located within the structural-scale simulation region. The first principal stress; The maximum allowable tensile stress is taken as the peak tensile strength of the concrete.
4. The finite element method for variably scaled damage simulation of concrete structures based on a two-layer mesh as described in claim 2, characterized in that: Step 3, the simulation method for the damage evolution process of concrete structures, includes the following steps: Step 3-1: Establish a structural-material scale co-simulation finite element model and set it in its initial state to only contain the structural scale simulation region; Step 3-2: Divide the load application process of the concrete structure into K incremental steps; Step 3-3: For any incremental step, perform a structural-material scale co-finite element equilibrium iterative solution for the damage evolution process of the concrete structure under that incremental step; when the iteration converges, obtain the integral point stress of each structural scale element in the structural scale simulation region in the structural-material scale co-finite element model; Step 3-4: For each integration point stress obtained in step 3-3, use the simulation scale transformation criterion of formula (1) to judge; if there is an integration point stress that satisfies formula (1), then execute step 3-5; otherwise, jump to step 3-8. Steps 3-5: Change the simulation scale of the simulation region occupied by all structural scale elements corresponding to all integral point stresses that satisfy formula (1) from the structural scale to the material scale. Steps 3-6: Connect each material-scale element included in the structural-material-scale co-factorial finite element model to the structural-material-scale model, thereby completing the dynamic update of the structural-material-scale co-factorial finite element model. Step 3-7: For the current increment step, re-run the structural-material scale collaborative finite element equilibrium iterative solution of the concrete structure damage evolution process, and repeat steps 3-3 to 3-6 until the integral point stress of each structural scale element in the current increment step does not satisfy formula (1). Step 3-8: Repeat steps 3-3 to 3-7 to carry out the structural-material scale co-finite element equilibrium iterative solution of the concrete structure damage evolution process in the next incremental step.
5. The finite element method for variable-scale damage simulation of concrete structures based on a two-layer mesh as described in claim 4, characterized in that: In step 3-2, the number of incremental steps K is set according to the magnitude of the load F to be applied and the simulation accuracy of the structural damage evolution process.
6. The finite element method for variably scaled damage simulation of concrete structures based on a two-layer mesh as described in claim 4, characterized in that: In steps 3-5, the method for transforming the structural scale to the material scale is as follows: all structural scale elements corresponding to the stresses at all integration points that satisfy formula (1) are eliminated from the structural scale-material scale co-finite element model, and the material scale elements associated with the eliminated structural scale elements are incorporated into the structural scale-material scale co-finite element model.
7. The finite element method for variable-scale damage simulation of concrete structures based on a two-layer mesh as described in claim 6, characterized in that: In steps 3-5, the material scale element associated with the eliminated structural scale element refers to a material scale element that has at least one node located within the simulation region occupied by the eliminated structural scale element.
8. The finite element method for variably scaled damage simulation of concrete structures based on a two-layer mesh as described in claim 4, characterized in that: In steps 3-6, the dynamically updated structural-material scale co-finite element model includes the structural-scale simulation region. and material-scale simulation region ; Structural Scale Simulation Region It refers to the simulated region in the elastic stress stage of a concrete structure, which is analyzed using structural scale elements; Material-scale simulation region This refers to the simulated region in a concrete structure that is in the inelastic stress stage, and is analyzed using material-scale elements. The structural-scale simulation region and the material-scale simulation region do not overlap and together constitute the entire simulation region of the concrete structure. ,Right now and .
9. The finite element method for variable-scale damage simulation of concrete structures based on a two-layer mesh as described in claim 4, characterized in that: Structural-material scale connection refers to ensuring deformation coordination between structural-scale elements and material-scale elements in a structural-material scale collaborative finite element model by establishing nodal displacement constraint equations for material-scale elements based on the shape functions of structural-scale elements.
10. The finite element method for variable-scale damage simulation of concrete structures based on a two-layer mesh as described in claim 9, characterized in that: The specific expression for the material-scale element nodal displacement constraint equation based on the structural-scale element shape function is as follows: ; In the formula, This represents the number of nodes in a structural unit. Material-scale element nodes The coordinate vector; For structural scale units corresponding to the first The shape function of each node with respect to The function value, ; For structural scale unit number The displacement vector of each node; Material-scale element nodes The displacement vector.