A Fast Numerical Simulation Modeling Method for Large Deformation Beam Structures Based on Absolute Nodal Coordinates

Abstract

This invention relates to a rapid numerical simulation modeling method for large deformation beam structures based on two-dimensional shear beam elements with absolute nodal coordinates. For widely used isotropic materials, the constitutive relation employed by the absolute nodal coordinate method is the isotropic expression from continuum mechanics. This precise expression results in multiple degrees of freedom for absolute nodal coordinate shear beam elements. However, existing two-dimensional shear beam elements still require a relatively large number of elements to obtain relatively accurate results, leading to large computational scale and long processing time in numerical simulations. This invention constructs four displacement modes by increasing the number of element nodes: three-node quadratic and cubic fully polynomial displacement modes, and four-node quadratic and cubic fully polynomial displacement modes. Compared to existing two-node fully polynomial displacement mode shear beam elements, these four displacement modes have faster convergence speeds and result in smaller computational scale and shorter processing time for numerical simulations of large deformation beam structures.

Description

Technical Field

[0001] This invention relates to numerical simulation modeling of large deformation beam structures, specifically to a rapid numerical simulation modeling method for large deformation beam structures using absolute nodal coordinate shear beam elements. Background Technology

[0002] The concept of modeling in absolute coordinates makes the absolute nodal coordinate method more suitable for modeling large deformation and large rotation multibody systems with significant rigid-flexible coupling characteristics. For widely used large deformation multibody systems using isotropic materials and containing beam structures, the constitutive relation expressed by this modeling approach is the isotropic expression in continuum mechanics, meaning that the longitudinal and transverse strains satisfy a strict coupling relationship with respect to Poisson's ratio. This is a precise expression and does not require special or simplified treatment due to differences in the dimensions and deformations between the longitudinal and transverse sides of the beam structure. This results in an increase in the number of polynomial terms in the transverse direction of the displacement mode of the absolute nodal coordinate shear beam element considering cross-sectional shear deformation, in addition to the longitudinal direction. Therefore, the absolute nodal coordinate shear beam element itself, established to satisfy this coupling relationship, has a relatively large number of degrees of freedom.

[0003] Currently developed two-dimensional shear beam elements with absolute nodal coordinates can yield accurate results as the number of elements increases. However, a relatively large number of elements are required to obtain high-accuracy results, and the convergence speed is slow. Thus, numerical simulation of large deformation beam structures suffers from problems of large computational scale, slow simulation speed, and long simulation time. Summary of the Invention

[0004] The purpose of this invention is to provide a fast numerical simulation modeling method for large deformation beam structures using two-dimensional shear beam elements with absolute node coordinates, which offers fast convergence and small computational scale, in order to solve the problem of numerical simulation of large deformation beam structures using two-dimensional shear beam elements with absolute node coordinates.

[0005] The objective of this invention can be achieved through the following technical solutions:

[0006] By increasing the number of element nodes to construct higher-order polynomial displacement modes for shear beam elements with absolute nodal coordinates, four displacement modes were obtained: a 3-node quadratic complete polynomial displacement mode, a cubic complete polynomial displacement mode, and a 4-node quadratic complete polynomial displacement mode and a cubic complete polynomial displacement mode, for a total of four displacement modes. These four displacement modes exhibit fast convergence speeds for shear beam elements, resulting in smaller computational scales, faster simulation speeds, and shorter simulation times for numerical simulations of large deformation beam structures.

[0007] Displacement Mode 1: 3-node quadratic complete polynomial displacement mode

[0008] The 3-node displacement mode includes the content of a complete polynomial of the highest second degree, namely (1 xy xyx 2y 2 ), and on this basis, some incomplete higher-order terms were added ( x 2 yxy 2 x 2 y 2 x 3 x 4 x 5 The displacement mode is specifically in the form of:

[0009]

[0010] The resulting element has 8 degrees of freedom per node, and a total of 24 degrees of freedom for 3 nodes. The 8 degrees of freedom for each node correspond to 8 node coordinates r, r... ,x ,r ,y ,r ,xy , where r=[ r 1, r [2] represents the position coordinates in two directions of the plane.

[0011] Displacement Mode 2: 3-node cubic complete polynomial displacement mode

[0012] The 3-node displacement mode includes the content of a complete polynomial of the highest degree, namely (1 xy xyx 2 y 2 x 3 x 2 yxy 2 y 3 ), and on this basis, some incomplete higher-order terms were added ( x 2 y 2 x 2 y 3 xy 3 x 4 x 5 The displacement mode is specifically in the form of:

[0013]

[0014] The resulting unit has 10 degrees of freedom per node, and 3 nodes for a total of 30 degrees of freedom. The 10 degrees of freedom per node, i.e., the 10 node coordinates, are denoted as r, r... ,x ,r ,y ,r ,yy ,r ,yyy .

[0015] Displacement Mode 3: 4-Node Quadratic Complete Polynomial Displacement Mode

[0016] The 4-node displacement mode includes the content of a complete polynomial of the highest second degree, namely (1 xy xyx 2 y 2 ), and on this basis, some incomplete higher-order terms were added ( x 3 x 2 yxy 2 x 3 yx 2 y 2 x 3 y 2 The displacement mode is specifically in the form of:

[0017]

[0018] The resulting element has 6 degrees of freedom per node, and 4 nodes for a total of 24 degrees of freedom. The 6 degrees of freedom for each node correspond to the coordinates of the 6 nodes, r, r. ,y ,r ,yy .

[0019] Displacement Mode 4: 4-node cubic complete polynomial displacement mode

[0020] The 4-node displacement mode includes the content of a complete polynomial of the highest degree, namely (1 xy xyx 2 y 2 x 3 x 2 yxy 2 y 3 ), and on this basis, some incomplete higher-order terms were added ( x 3 yxy 3 x2 y 2 x 2 y 3 x 3 y 2 x 3 y 3 The displacement mode is specifically in the form of:

[0021]

[0022] The resulting element has 8 degrees of freedom per node, and 32 degrees of freedom for 4 nodes. The 8 degrees of freedom for each node correspond to 8 node coordinates r, r... ,y ,r ,yy ,r ,yyy .

[0023] Compared with existing technologies, the absolute node coordinate shear beam element of this invention has a fast convergence speed, and the numerical simulation calculation of large deformation beam structures is small in scale, fast, and short in time. Attached Figure Description

[0024] Figure 1 This is a schematic diagram of a beam structure with large deformation. Detailed Implementation

[0025] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0026] Large deformation beam structures, such as Figure 1 As shown, the beam is 10 meters long, with a square cross-section and a side length of 0.1 meters. The elastic modulus of the beam material is E = 2.1 × 10. 11 Pa is an isotropic material with a density of 7800 kg / m³. 3 The Poisson's ratio is 0.3. The beam ends are subjected to concentrated moments. In order to comprehensively test the numerical simulation of large deformation of the beam structure by the models of various types of elements, the applied bending moments are increased from small to large, with ML / EIπ being 0.2, 0.6, 1, 1.6, and 2 respectively.

[0027] The two-dimensional shear beam element with absolute nodal coordinates was proposed by Mohil Patel in 2018. It features two nodes and a quadratic complete polynomial displacement mode. Its displacement mode is as follows:

[0028]

[0029] Each node in the corresponding unit has 8 degrees of freedom, and the two nodes together have 16 degrees of freedom.

[0030] Tables 1 and 2 show a comparison of the X and Y displacements at the ends of large deformation beam structures analyzed by the elements proposed by Mohil Patel and the elements corresponding to the four displacement modes provided in this invention. The number of elements for the beam structures shown is 8 in each case.

[0031] The comparison results show that by increasing the number of nodes, the four displacement modes obtained have yielded results that are close to the theoretical solution when the number of structural elements is only 8. However, the analysis results of the elements proposed by Mohil Patel are much worse than the theoretical solution. To obtain results with the same accuracy, the number of elements needs to be increased to at least 32.

[0032] Table 1 Comparison of X-direction displacement at the beam end

[0033]

[0034] Table 2 Comparison of Y-direction displacement at the beam end

[0035]

Claims

1. A rapid numerical simulation modeling method for large deformation beam structures based on the absolute nodal coordinate method, characterized in that... Including any one of the following absolute nodal coordinate shear beam element displacement modes: Displacement Mode 1: A 3-node quadratic complete polynomial displacement mode, specifically in the form of: The displacement mode includes the content of a complete polynomial of the highest second degree, namely (1 xy xyx 2 y 2 ), and on this basis, some incomplete higher-order terms were added ( x 2 yxy 2 x 2 y 2 x 3 x 4 x 5 ). Displacement mode 2: 3-node cubic complete polynomial displacement mode, specifically in the form of: The displacement mode includes the content of a complete polynomial of the highest degree, namely (1 xy xyx 2 y 2 x 3 x 2 yxy 2 y 3 ), and on this basis, some incomplete higher-order terms were added ( x 2 y 2 x 2 y 3 xy 3 x 4 x 5 ). Displacement mode 3: 4-node quadratic complete polynomial displacement mode, specifically in the form of: The displacement mode includes the content of a complete polynomial of the highest second degree, namely (1 xy xyx 2 y 2 ), and on this basis, some incomplete higher-order terms were added ( x 3 x 2 yxy 2 x 3 yx 2 y 2 x 3 y 2 ). Displacement mode 4: 4-node cubic complete polynomial displacement mode, specifically in the form of: The displacement mode includes the content of a complete polynomial of the highest degree, namely (1 xy xyx 2 y 2 x 3 x 2 yxy 2 y 3 ), and on this basis, some incomplete higher-order terms were added ( x 3 yxy 3 x 2 y 2 x 2 y 3 x 3 y 2 x 3 y 3 ).

2. The rapid numerical simulation modeling method for large deformation beam structures based on the absolute nodal coordinate method according to claim 1, characterized in that... Based on the absolute nodal coordinates obtained from displacement mode 1, each node of the shear beam element has 8 degrees of freedom, for a total of 24 degrees of freedom across 3 nodes. The 8 degrees of freedom for each node, i.e., the 8 nodal coordinates, are r, r... ,x ,r ,y ,r ,xy .

3. The rapid numerical simulation modeling method for large deformation beam structures based on the absolute nodal coordinate method according to claim 1, characterized in that... Based on the absolute nodal coordinates obtained from displacement mode 2, each node of the shear beam element has 10 degrees of freedom, for a total of 30 degrees of freedom across 3 nodes. Each node has 10 degrees of freedom, meaning each node has 10 nodal coordinates r, r... ,x ,r ,y ,r ,yy ,r ,yyy .

4. The rapid numerical simulation modeling method for large deformation beam structures based on the absolute nodal coordinate method according to claim 1, characterized in that... Based on the absolute nodal coordinates obtained from displacement mode 3, each node of the shear beam element has 6 degrees of freedom, for a total of 24 degrees of freedom across 4 nodes. Each node has 6 degrees of freedom, meaning each node has 6 nodal coordinates r, r... ,y ,r ,yy .

5. The rapid numerical simulation modeling method for large deformation beam structures based on the absolute nodal coordinate method according to claim 1, characterized in that... Based on the absolute nodal coordinates obtained from displacement mode 4, each node of the shear beam element has 8 degrees of freedom, for a total of 32 degrees of freedom across 4 nodes. Each node has 8 degrees of freedom, meaning each node has 8 nodal coordinates r, r... ,y ,r ,yy ,r ,yyy .

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