A finite element simulation method for describing the growth morphology of plant tendrils

Abstract

The invention discloses a finite element simulation method for describing the growth morphology of plant tendrils. The method comprises the following steps: step S1, constructing an initial morphological model of plant tendrils, setting simulation conditions for tendril growth, step S2, establishing a constitutive relationship for describing the growth of plant tendrils based on a hyperelastic Neo-Hookean model and a finite deformation theory, step S3, constructing a constitutive model for describing the growth of plant tendrils, and step S4, obtaining a simulation result of the growth morphology of plant tendrils based on the initial morphological model of the plant tendrils and an iterative result of step S3. The method introduces the finite deformation theory into nonlinear large deformation hyperelastic materials to establish different constitutive relationships, establishes constitutive relationships including isotropic and anisotropic free growth, accurately simulates the morphological changes of plant tendril growth by changing the growth function in the finite deformation theory, and adopts an implicit Newton-Raphson integral iteration method to improve the stability of the model algorithm and make the iteration more accurate and rapid.

Description

Technical Field

[0001] The invention relates to the technical field of nonlinear material mechanics and digital simulation, and in particular to a finite element simulation method for describing the growth morphology of plant tendrils. Background Art

[0002] The growth of biological tissue is a complex process that occurs at different time and space scales. This process relies on the stimulation of mechanical signals to adjust to various complex and changing natural environments. In-depth research on the growth and shrinkage of biological tissues and the expansion and contraction of biological tissues from a mechanical perspective is of great significance for understanding complex life processes, revealing the internal mechanisms of diseases, and designing bionic structures.

[0003] The growth process of plant tendrils is a nonlinear large deformation problem that changes with time, so a simple theoretical model obviously does not meet the requirements and needs to rely on a more complex growth model. The finite element method can process complex models through efficient numerical calculations, but the current general finite element simulation software does not have a module suitable for tissue growth. Therefore, developing a simulation method for the plant curling growth process based on the finite element method is of great significance to the exploration of plant growth mechanisms and life sciences. Summary of the invention

[0004] The object of the present invention is to provide a finite element simulation method for describing the growth morphology of plant tendrils, which can solve the above-mentioned technical problems and realize accurate simulation of the curling growth process of plants.

[0005] To this end, the technical solution of the present invention is as follows:

[0006] A finite element simulation method for describing the growth morphology of plant tendrils, the steps are as follows:

[0007] S1. Construct an initial morphological model of plant tendrils, match the initial morphological model with plant characteristic parameters consistent with the plant tendrils to be simulated, including density and mechanical constant composition; set simulation conditions for tendril growth;

[0008] S2. Based on the hyperelastic Neo-Hookean model and finite deformation theory, the constitutive relationship for describing the growth of plant tendrils is established, including the total deformation gradient F, the Jacobian determinant J of the deformation gradient, and the elastic left Cauchy-Green deformation tensor B, which are used to describe the growth process of the tissue. e , Cauchy stress σ and tangent stiffness D;

[0009] S3, based on the constitutive relationship established in step S2, construct a constitutive model for describing the growth of plant tendrils, and obtain the state variables at several moments in the tendril growth process and the final position coordinates of each growth point on the tendril at the end of the iteration through a finite element simulation method based on the plant characteristic parameters and simulation conditions set in step S1; the state variables include the tangent stiffness tensor D and the Cauchy stress σ at each iteration time t;

[0010] S4. Based on the initial morphological model of the plant tendril constructed in step S1, the state variables at several moments in the tendril growth process obtained by iterating the constitutive model in step S3, the tendril growth function, and the final position coordinates of each growth point on the tendril at the end of the iteration, the final growth morphological simulation result of the plant tendril is obtained.

[0011] Furthermore, step S1 is implemented based on ABAQUS software, and its specific implementation steps are:

[0012] S101. Construct a straight rod model as the initial morphological model of the plant tendril, set the density to 1E-15 in the density setting of the material module, and set the number of state variables that depend on the solution to 10 in the non-independent variable module; set the user material type to select material mechanics in the user material module, and set the mechanical constants based on the characteristics of the plant to be simulated, including the elastic modulus, Poisson's ratio, and the growth rate coefficient K of the plant tendril growth model. 1 , K 2 and K 3 ;

[0013] S102, setting simulation conditions for tendril growth includes sequentially performing assembly operations on the model constructed in step S101, setting analysis steps, setting model boundary conditions, and meshing the model.

[0014] Further, in step S102, the assembly operation method includes: assembling the straight rod model constructed in step S101, and setting the rotation angle of the straight rod model based on the bottom end of the initial morphological model of the plant tendril according to the growth characteristics of the plant tendril; the analysis step is set to: select dynamic implicit, turn on geometric nonlinearity and set the time length; the boundary condition is set to: set the bottom end of the model to ENCASTRE and the top end to free; the model meshing method is: in the unit type, set the unit library to standard, the family to three-dimensional stress, the geometric order to linear, the shape to hexahedron, the hybrid formula, and the viscosity to default.

[0015] Furthermore, in step S2, the constitutive relationship for describing the growth of plant tendrils includes:

[0016] The expression of the total deformation gradient F is:

[0017] F=Fe F g ,

[0018] Where A is the initial coordinate position vector of a growth point inside the tendril, and a is the new coordinate position vector of the growth point after the growth and deformation over a certain period of time; F e is the elastic deformation part of the total deformation gradient F, F g is the irreversible growth deformation part of the total deformation gradient F;

[0019] The expression of the Jacobian determinant J of the deformation gradient is:

[0020] J=J e J g =det(F),

[0021] In the formula, J e is the elastic variation part of the Jacobian determinant of the deformation gradient, Jg is the growth variation part of the Jacobian determinant of the deformation gradient; is the growth amount, is the total volume growth, V is the original volume, △V is the change in volume;

[0022] Elastic left Cauchy-Green deformation tensor B e The expression is:

[0023] B e =F e F eT ,

[0024] Where B is the left Cauchy-Green tensor representing the deformation in space, B = FF T ;

[0025] The expression of Cauchy stress σ is:

[0026]

[0027] Where I is the unit tensor;

[0028] The expression of tangent stiffness D is:

[0029]

[0030] In the formula, and Both are non-standard fourth-order tensors and satisfy the Einstein summation convention.

[0031] Furthermore, in step S3, the iterative calculation process of the constitutive model is implemented based on Microsoft Visual Studio software, and the specific iterative calculation steps are:

[0032] S301, obtaining the three growth rate coefficients and the time length T set in step S1;

[0033] S302, calculate the time t of the current iteration, t=t 0 +△t; where t 0 is the previous iteration time, △t is the time increment;

[0034] S303, according to the growth characteristics of the plant tendril, select a growth function that is suitable for the growth morphology of the plant tendril to calculate the growth amount of the plant tendril in three directions at time t in,

[0035] i) When the plant tendril growth morphology is an Archimedean tendril, its growth function adopts the growth function corresponding to the Archimedean tendril, and its expression is:

[0036]

[0037] ii) When the plant tendril growth morphology is an Euler tendril, its growth function adopts the growth function corresponding to the Euler tendril, and its expression is:

[0038]

[0039] iii) When the plant tendril growth form is a space circular tendril form, its growth function adopts the growth function corresponding to the space circular tendril, and its expression is:

[0040]

[0041] iv) When the plant tendril growth form is a spatial Euler tendril form, its growth function adopts the growth function corresponding to the spatial Euler tendril, and the growth function expression is:

[0042]

[0043] In the formula, X, Y and Z are the 0 The coordinate position vector of any growth point in the state is the coordinate in the X direction, Y direction and Z direction respectively; t is the current moment;

[0044] S304, calculating the reversible elastic deformation part F of the total deformation gradient e ;

[0045] S305, reversible elastic deformation part F based on total deformation gradient e, calculate the left Cauchy-Green tensor B e ;

[0046] S306, based on the left Cauchy-Green tensor B e , calculate the Cauchy stress σ;

[0047] S307, calculating the tangent stiffness matrix D based on the Cauchy stress σ;

[0048] S308. Repeat steps S301 to S307 until the current iteration time t> time length T, and the iteration ends.

[0049] Furthermore, the time length T is set to 1, and the iteration time increment △t is set to 0.0001.

[0050] Furthermore, the specific implementation steps of step S4 are:

[0051] S401, create a new job in the job module of ABAQUS software, select User subroutine file in General to associate the constitutive model constructed by the UMAT subroutine, so as to realize the continuous call of the iterative calculation results obtained by the constitutive model in step S3;

[0052] S402, ABAQUS software obtains the growth function determined in step S3, the tangent stiffness tensor D and Cauchy stress σ obtained at each iteration time t, and the final position coordinates of each growth point on the tendril at the end of the iteration, so that ABAQUS software continues to perform finite element simulation on the growth morphology of the plant tendril based on the initial morphological model of the plant tendril constructed in step S1, and finally obtains the simulation result of the growth morphology of the plant tendril.

[0053] Compared with the existing technology, the finite element simulation method for describing the growth morphology of plant tendrils introduces finite deformation theory for nonlinear large deformation hyperelastic materials and establishes different constitutive relations, including isotropic and anisotropic free growth constitutive relations. By changing the growth function in the finite deformation theory, the morphological changes of plant tendril growth can be accurately simulated; at the same time, by adopting the implicit Newton-Raphson integral iteration method, the stability of the model algorithm is improved, making the iteration more accurate and rapid, which provides support for nonlinear large deformation finite element simulation and plant morphology research. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 A flow chart of the finite element simulation method for describing the growth morphology of plant tendrils of the present invention;

[0055] FIG2( a ) is a schematic diagram of a model of the initial morphology of a plant tendril in step S1 of the finite element simulation method for describing the growth morphology of a plant tendril of the present invention;

[0056] FIG2( b ) is a schematic diagram of the coordinate position change of any growth point in the growth process of a plant tendril in the finite element simulation method for describing the growth morphology of a plant tendril of the present invention;

[0057] Figure 3 A visualized schematic diagram of volume growth in the growth gradient Jacobian determinant of the finite element simulation method for describing the growth morphology of plant tendrils of the present invention;

[0058] Figure 4 It is a flow chart of step S3 of the finite element simulation method for describing the growth morphology of plant tendrils of the present invention;

[0059] FIG5( a ) is a schematic diagram of the actual growth morphology of plant tendril A in an embodiment of the present invention;

[0060] FIG5( b ) is a schematic diagram of the simulated growth morphology of plant tendril A obtained by finite element simulation method in an embodiment of the present invention;

[0061] FIG6( a ) is a schematic diagram of the actual growth morphology of plant tendril B in an embodiment of the present invention;

[0062] FIG6( b ) is a schematic diagram of the simulated growth morphology of the plant tendril B obtained by the finite element simulation method in an embodiment of the present invention;

[0063] FIG. 7( a ) is a schematic diagram of an angle of a plant tendril C having a growth angle in an embodiment of the present invention;

[0064] FIG. 7( b ) is a schematic diagram of an initial morphological model of a plant tendril C having a growth angle constructed in step S1 in an embodiment of the present invention;

[0065] FIG8( a ) is a schematic diagram of the actual growth morphology of a plant tendril C in an embodiment of the present invention;

[0066] FIG8( b ) is a schematic diagram of the simulated growth morphology of a plant tendril C obtained by a finite element simulation method in an embodiment of the present invention;

[0067] FIG9( a ) is a schematic diagram of the actual growth morphology of a plant tendril D in an embodiment of the present invention;

[0068] FIG9( b ) is a schematic diagram of the simulated growth morphology of the plant tendril D obtained by the finite element simulation method in an embodiment of the present invention. DETAILED DESCRIPTION

[0069] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, but the following embodiments are by no means intended to limit the present invention in any way.

[0070] See also Figure 1 The specific implementation steps of the finite element simulation method for describing the growth morphology of plant tendrils are as follows:

[0071] S1. Constructing the initial morphological model of plant tendrils and setting the simulation conditions for tendril growth;

[0072] Specifically, referring to FIG. 2( a ), step S1 is implemented based on finite element simulation software ABAQUS, and the implementation steps are as follows:

[0073] S101. Create a straight rod model in ABAQUS software. In the Density setting of the material module (property), set the density to 1E-15; in the Depvar module, set the number of solution-dependent state variables to 10; in the UserMaterial module, set the user material type (Hybrid formulation) to select material mechanics (Incremental), and set the mechanical constants (Mechanical Constants) accordingly, including the elastic modulus, Poisson's ratio, and the growth rate coefficient K of the plant tendril growth model in three directions. 1 , K 2 and K 3 , the above mechanical constants are set based on the actual parameter values ​​given by the characteristics of the plant to be simulated; then, the initial morphological model of the plant tendril is constructed;

[0074] S102, in ABAQUS software, sequentially assemble the initial model constructed in step S101, set the analysis steps, set the model boundary conditions, and mesh the model; wherein,

[0075] i) Assembly operation method: assemble the straight rod model constructed in step S101; then, according to the growth characteristics of the plant tendril, the rotation angle β of the straight rod model is set with the bottom of the initial morphological model of the plant tendril as a reference; wherein, if there is no rotation angle required, the operation of setting the rotation angle in the assembly operation is skipped;

[0076] ii) The analysis step is set by selecting dynamic implicit, turning on geometric nonlinearity and setting the time length. The time length T is generally set to 1.

[0077] iii) The model boundary conditions are set as follows: the bottom of the model is set to ENCASTRE and the top of the model is set to FREE;

[0078] iv) The model is meshed in the following manner: in the unit type, the unit library is set to standard, the family is three-dimensional stress, the geometric order is linear, the shape is hexahedron, hybrid formula, and the viscosity is default; in the meshing process, the meshing method adopted by the present invention can avoid the hourglass phenomenon in the simulation process of the growth morphology of plant tendrils, that is, when using linear reduced integration units, the unit deformation is manifested as an alternating trapezoidal shape due to the use of linear reduced integration, similar to an hourglass icon; this mode belongs to the zero-energy mode, which means that no energy is consumed during the simulation process, which may lead to inaccurate calculation results.

[0079] S2. Using the hyperelastic Neo-Hookean model and introducing the finite deformation theory, a constitutive relationship is established to describe the growth of plant tendrils.

[0080] In order to describe the finite deformation motion of plant tendrils under the action of growth function, the finite deformation theory is introduced to describe the finite deformation motion. The finite deformation motion includes the growth deformation, elastic deformation and constitutive relationship of the tendril.

[0081] Tendril growth is based on the growth of several growth points inside it. The tendril growth model can be regarded as a spiral curve motion in a plane, that is, it gradually grows from an initial straight rod to a curved shape.

[0082] Assumption: If the position vector of any growth point inside the tendril at the initial moment is A, then after the growth deformation, the position vector of the growth point after deformation is a;

[0083] Then: after a growth period of t, its position relationship under the deformation field φ is x=φ(A,t);

[0084] In the above setting, the coordinate position of any growth point is determined based on the constructed tendril coordinate system. Specifically, see Figure 2(b), introducing the absolute coordinate system o-xy and the local coordinate system o i -x i y i The origin O of the absolute coordinate system is set at the bottom of the straight rod, and the x-axis and y-axis are set in the horizontal and vertical directions respectively; the far point O of the local coordinate system i Located on the inside of the bend of the rod, x i Axis and y i The axis has angles θ and γ with the x-axis and y-axis respectively; among them, θ is the rotation angle of the local coordinate system relative to the absolute coordinate system, and γ is the deflection angle of the local coordinate system, that is, the coordinate change of the growth point is used to describe the bending change of the tendril growth.

[0085] Based on this, in the above-mentioned finite deformation theory, the growth process of the tissue can be explained by the total deformation gradient F, that is, the total deformation gradient F is used to represent the initial After growth and deformation The mapping is represented as:

[0086]

[0087] The total deformation gradient F can be decomposed into the reversible elastic deformation part F e and the irreversible growth deformation part F g , whose expression is:

[0088] F=F e F g ;

[0089] Correspondingly, the Jacobian determinant of the deformation gradient can also be decomposed into the elastic change J e and growth changes g , whose expression is:

[0090] J=J e J g = det(F);

[0091] When the plant grows, the Jacobian of the growth gradient is expressed as:

[0092]

[0093] In the formula, is the growth amount, is the total volume growth, When plant tissues grow isotropically, When plant tissue grows anisotropically, it is three components with different values; V is the original volume, △V is the change in volume;

[0094] like Figure 3 As shown, V represents the initial volume. After a certain period of time, under the action of the growth function, the volume changes by △V, and then the initial volume changes to V+△V.

[0095] therefore,

[0096] At this time, the elastic left Cauchy-Green deformation tensor B e It is expressed as:

[0097] B e =F e F eT ,

[0098] Where B is the left Cauchy-Green tensor representing the deformation in space, B = FFT ;

[0099] Based on this, the potential energy function of the Neo-Hookean model expressed by the total deformation gradient and growth is:

[0100]

[0101] Where, λ is the first Lamé parameter, λ = νE / (1-2ν)(1+ν); G is the second Lamé parameter, G = E / 2(1+ν); E and ν are the elastic modulus and Poisson's ratio of the plant, respectively; I 1 is the first principal variable of B.

[0102] Furthermore, the expression of the first-type Piola-Kirchhoff stress P is:

[0103] And PF T = Jσ,

[0104] In the formula, F T is the transpose of the total deformation gradient F;

[0105] Then, the Cauchy stress σ at this time is obtained from the relationship between the Kirchhoff stress τ, the first Piola-Kirchhoff stress P and the Cauchy stress σ, and its expression is:

[0106]

[0107] Where I is the unit tensor;

[0108] Furthermore, the tangent stiffness D is obtained from the total derivative of the Cauchy stress σ with respect to the total deformation gradient F, and its expression is:

[0109]

[0110] In the formula, and Both are non-standard fourth-order tensors and satisfy the Einstein summation convention.

[0111] Step 3: Based on the constitutive relationship established in step S2, a constitutive model for describing the growth of plant tendrils is constructed;

[0112] The initial morphological model of the plant tendril constructed using ABAQUS software in step 1 is not sufficient to continue to describe its subsequent growth morphology. Therefore, it is necessary to customize the growth properties of the plant tendril based on the complex constitutive relationship constructed in step 2 to effectively simulate the growth morphology of different types of plant tendrils.

[0113] Specifically, in step 3, the constitutive model used to describe the growth of plant tendrils is implemented using Microsoft VisualStudio 2019 software, and the running process of the constitutive model is written in Fortran language to form a subroutine UMAT, so that the ABAQUS software in step 1 will pass the necessary mechanical constants and time length to the subroutine based on the initial settings, so as to update the stress, deformation state and tangent stiffness through the constitutive relationship contained in the UMAT subroutine to perform tendril growth simulation iterations.

[0114] See also Figure 4 ,The specific steps for obtaining the state variables generated by the morphological changes during the growth of plant tendrils are:

[0115] S301, obtain the three growth rate coefficients K set in step S1 1 , K 2 and K 3 , and time length T; wherein, time length T is the total solution time of the entire finite element simulation analysis;

[0116] S302, calculate the time t of the current iteration, t=t 0 +△t; where t 0 is the previous iteration time, △t is the time increment; in this step, the incremental step of the finite element simulation process is used to approximately replace the time increment △t of the plant tendril growth, so the growth time of each iteration is increased by △t on the basis of the previous iteration; among them, when the first iteration calculation is performed, t 0 =0; △t is set to 0.0001;

[0117] S303, according to the type of plant, select a growth function that is suitable for the growth form of its tendril to calculate the growth amount of the plant tendril in three directions at time t in,

[0118] i) When the plant tendril growth morphology is an Archimedean tendril, its growth function adopts the growth function corresponding to the Archimedean tendril, and the growth function expression is:

[0119]

[0120] ii) When the plant tendril growth morphology is an Euler tendril, its growth function adopts the growth function corresponding to the Euler tendril, and the growth function expression is:

[0121]

[0122] iii) When the plant tendril growth form is a space circular tendril form, its growth function adopts the growth function corresponding to the space circular tendril, and the growth function expression is:

[0123]

[0124] iv) When the plant tendril growth form is a spatial Euler tendril form, its growth function adopts the growth function corresponding to the spatial Euler tendril, and the growth function expression is:

[0125]

[0126] In the formula, X, Y and Z are the previous moment t 0 The coordinate position vector of any growth point in the state is the coordinate in the X direction, Y direction and Z direction respectively; t is the current moment;

[0127] Based on the growth function, the growth amount of each growth point of the tendril at each iteration moment can be calculated, and then after each iteration, the coordinate position of each growth point on the tendril is synchronously updated;

[0128] S304, calculating the reversible elastic deformation part F of the total deformation gradient e , and its calculation formula is:

[0129] F e =F / F g ,

[0130] Where F is the total deformation gradient, which is expressed by the formula: Calculated; F g is the irreversible growth deformation part of the total deformation gradient, and its calculation formula is: L is the initial length, △L is the length change;

[0131] S305. Calculate the left Cauchy-Green tensor B e , and its calculation formula is:

[0132] B e =F e (F e ) T ,

[0133] In the formula, (F e ) T F e The transpose of

[0134] S306. Calculate the Cauchy stress σ, the calculation formula is:

[0135]

[0136] Where J is the Jacobian of the deformation gradient, λ is the first Lamé parameter, and J eis the elastic variation of the Jacobian of the deformation gradient; G is the second Lamé parameter; I is the unit tensor;

[0137] S307, calculate the tangent stiffness matrix D, the calculation formula is:

[0138]

[0139] Where J is the Jacobian of the deformation gradient, B is the left Cauchy-Green tensor, σ is the Cauchy stress, λ is the first Lamé parameter, G is the second Lamé parameter, and J e is the elastic variation of the Jacobian of the deformation gradient, and is a nonstandard fourth-order tensor whose subscripts are dummy indices and satisfy the Einstein summation convention;

[0140] S308, repeating steps S301 to S307 until the current iteration time t> the time length T, and the iteration ends;

[0141] S4. During the iteration process of step S3, the initial morphological model of the plant tendril constructed in step S1 is based on the growth function determined in step S3, the tangent stiffness tensor D and the Cauchy stress σ obtained at each iteration time t, and the final position coordinates of each growth point on the plant tendril at the end of the iteration. The finite element simulation of the growth morphology of the plant tendril is continuously performed to finally obtain the simulation result of the growth morphology of the plant tendril.

[0142] Specifically, step S4 is also implemented based on the finite element simulation software ABAQUS, and its specific implementation steps are:

[0143] S401, create a new job in the job module of ABAQUS software, select User subroutine file in General to associate the constitutive model constructed in step S3 (specifically, the UMAT subroutine) to continuously call the iterative calculation results obtained by the constitutive model in step S3;

[0144] S402, ABAQUS software obtains the growth function determined in step S3, the tangent stiffness tensor D and Cauchy stress σ obtained at each iteration time t, and the final position coordinates of each growth point on the tendril at the end of the iteration, so that ABAQUS software continues to perform finite element simulation on the growth morphology of the plant tendril based on the initial morphological model of the plant tendril constructed in step S1, and finally obtains the simulation result of the growth morphology of the plant tendril.

[0145] In order to further demonstrate the accuracy and effectiveness of the finite element simulation method of the present invention for describing the growth morphology of plant tendrils, the growth morphology of different types of plant tendrils was simulated respectively. At the same time, as a control group, different types of plant tendrils were grown and cultured to compare the similarity between the simulation results and the actual growth results.

[0146] Embodiment 1:

[0147] A finite element description method for the growth morphology of a plant tendril A is specifically implemented using the above-mentioned implementation steps; wherein the growth morphology of the plant tendril A is a curling change caused by the asymmetric growth of its inner and outer sides, simulating that during the growth process of the plant tendril, the cross-sectional direction of the tendril grows and changes uniformly, while there are asynchronous growth changes on the inner and outer sides caused by the displacement difference, see Figure 5(a) for details.

[0148] Based on this, in step S1, among the mechanical constants, the elastic modulus, Poisson's ratio, and the growth rate coefficient K of the plant curling growth model are 1 , K 2 and K 3 are respectively set to 100, 0.4, 0.1, 0.04 and 0.001; in step S3, the growth function adopts the growth function corresponding to the Archimedean tendril; after the finite element simulation of the present invention, the simulation result shown in Figure 5(b) is obtained; it can be seen from the figure that the plant tendril growth form is fixed at one end and free at the other end, and lateral growth occurs in the cross-sectional direction. Specifically, the plant tendril growth form obtained by finite element simulation grows faster than the right side in the x-axis direction, and bends inward, which will produce a trend in which the curvature of the tendril gradually decreases from its base to the top. As time goes by, the curvature gradually decreases, and finally grows into an Archimedean curve shape, which is completely consistent with the actual tendril growth form of the plant as shown in Figure 5(a).

[0149] Embodiment 2:

[0150] A finite element description method for the growth morphology of a plant tendril B is specifically implemented using the above-mentioned implementation steps; wherein the growth morphology of the plant tendril B is affected by the combined effects of inner and outer asymmetric growth and apical dominance phenomenon similar to biology, see Figure 6(a) for details.

[0151] Based on this, in step S1, among the mechanical constants, the elastic modulus, Poisson's ratio, and the growth rate coefficient K of the plant tendril growth model are 1 , K 2 and K 3are set to 100, 0.4, 0.1, 0.02 and 0.0006 respectively; in step S3, the growth function adopts the growth function corresponding to the plant tendril in the shape of an Euler curve; after the finite element simulation of the present invention, the simulation result shown in Figure 6(b) is obtained; it can be seen from the figure that the initially straight tendril gradually grows into an Euler curve under the action of the growth function, and the ring distance of the tendril is uneven. The ring distance formed at the tip with a larger growth rate is smaller and the bending curvature is larger, while the ring distance at the bottom with a smaller growth rate is larger and the bending curvature is smaller; the tendril model has one end fixed and the other end free, and the cross-sectional direction grows horizontally, which is completely consistent with the actual tendril growth morphology of the plant as shown in Figure 6(a).

[0152] Embodiment 3:

[0153] A finite element description method for the growth morphology of a plant tendril C is specifically implemented using the above-mentioned implementation steps; wherein the plant tendril C grows at a certain angle during the growth process, see FIG8(a) for details.

[0154] Based on this, in step S1, see Figure 7(a) and Figure 7(b), in the assembly step, select the rotation example, take the bottom end as the reference, and then set the rotation angle to 10°, so that the initial model is changed from the straight rod in the space in the previous example to an inclined rod with a deflection angle of β degrees for the growth process, β = 10°; In addition, among the mechanical constants of step S1, the elastic modulus, Poisson's ratio, and the growth rate coefficient K of the plant tendril growth model are 1 , K 2 and K 3 They are set to 100, 0.4, 0.06, 0 and 0 respectively; in step S3, the growth function adopts the growth function corresponding to the spatial circular plant tendril; after the finite element simulation of the present invention, the simulation result shown in Figure 8(b) is obtained; it can be seen from the figure that the projection of the model on the xy plane is a rectangle. When there is uneven growth on the x-axis on the plane, the tendril will bend into a circular shape, that is, when this happens, the projection of the growth shape of the model in this example on the xy plane will be a circular shape. The tendril model has one end fixed and the other end free, and no lateral growth occurs in the cross-sectional direction; the simulation result is completely consistent with the actual tendril growth shape of the plant as shown in Figure 8(a).

[0155] Embodiment 4:

[0156] A finite element description method for the growth morphology of a plant tendril D is specifically implemented using the above-mentioned implementation steps; wherein the plant tendril D grows on a plane under the combined effects of asymmetric growth on both sides and apical dominance, and there is a certain angle in the growth process, see Figure 9(a) for details.

[0157] Based on this, in step S1, the initial model is changed from the straight rod in the space in the previous example to an inclined rod with a deflection angle of β degrees for the growth process, β = 20°; in addition, among the mechanical constants in step S1, the elastic modulus, Poisson's ratio, and the growth rate coefficient K of the plant tendril growth model are 1 , K 2 and K 3 are set to 100, 0.4, 0.0008, 0 and 0 respectively; in step S3, the growth function adopts the growth function corresponding to the spatial Euler-shaped plant tendril; after the finite element simulation of the present invention, the simulation result shown in Figure 9(b) is obtained; it can be seen from the figure that when the asymmetric growth on both sides and the apical dominance act together on the plane, the tendril will form an Euler curve shape, that is, when this happens, the projection of the growth shape of the model in this example on the xy plane will be an Euler curve shape. The tendril model has one end fixed and the other end free, and no lateral growth occurs in the cross-sectional direction; the simulation result is completely consistent with the actual tendril growth shape of the plant as shown in Figure 9(a).

[0158] In summary, the finite element simulation method for describing the growth morphology of plant tendrils of the present invention can accurately simulate the morphological changes of the curling growth of plants.

Claims

1. A finite element simulation method for describing the growth morphology of plant tendrils, characterized in that: Here are the steps: S1. Construct an initial morphological model of plant tendrils, match the initial morphological model with plant characteristic parameters consistent with the plant tendrils to be simulated, including density and mechanical constant composition; set simulation conditions for tendril growth; S2. Based on the hyperelastic Neo-Hookean model and finite deformation theory, the constitutive relations for describing the growth of plant tendrils are established, including the total deformation gradient, the Jacobian determinant J of the deformation gradient, the elastic left Cauchy-Green deformation tensor, the Cauchy stress and the tangent stiffness used to describe the growth process of the tissue; S3. Based on the constitutive relationship established in step S2, a constitutive model for describing the growth of plant tendrils is constructed, and based on the plant characteristic parameters and simulation conditions set in step S1, the state variables at several moments in the tendril growth process and the final position coordinates of each growth point on the tendril at the end of the iteration are obtained by finite element simulation; the state variables include the tangent stiffness tensor and Cauchy stress at each iteration time t; the iteration adopts the implicit Newton-Raphson integral iteration method; S4. Based on the initial morphological model of the plant tendril constructed in step S1, the Cauchy stress and tangent stiffness at each iteration moment in the tendril growth process, the tendril growth function, and the final position coordinates of each growth point on the tendril at the end of the iteration obtained by iterating the constitutive model in step S3, the growth morphology of the plant tendril is continuously simulated by finite element, and finally a simulation result of the growth morphology of the plant tendril is obtained.

2. The finite element simulation method for describing the growth morphology of plant tendrils according to claim 1, characterized in that: Step S1 is implemented based on ABAQUS software, and its specific implementation steps are as follows: S101. Construct a straight rod model as the initial morphological model of the plant tendril, set the density to 1E-15 in the density setting of the material module, and set the number of state variables that depend on the solution to 10 in the non-independent variable module; set the user material type to select material mechanics in the user material module, and set mechanical constants based on the characteristics of the plant to be simulated, including elastic modulus, Poisson's ratio, and growth rate coefficients K1, K2, and K3 of the plant tendril growth model; S102, setting simulation conditions for tendril growth includes sequentially performing assembly operations on the model constructed in step S101, setting analysis steps, setting model boundary conditions, and meshing the model.

3. The finite element simulation method for describing the growth morphology of plant tendrils according to claim 2, characterized in that: In step S102, the assembly operation method includes assembling the straight rod model constructed in step S101, and setting the rotation angle of the straight rod model based on the bottom end of the initial morphological model of the plant tendril according to the growth characteristics of the plant tendril; the analysis step is set as follows: selecting dynamic implicit, turning on geometric nonlinearity and setting the time length; the boundary condition is set as follows: setting the bottom end of the model to ENCASTRE and the top end to free; the model meshing method is as follows: in the unit type, setting the unit library to standard, the family to three-dimensional stress, the geometric order to linear, the shape to hexahedron, the hybrid formula, and the viscosity to default.

4. The finite element simulation method for describing the growth morphology of plant tendrils according to claim 1, characterized in that: In step S2, the constitutive relations used to describe the growth of plant tendrils include: The expression of the total deformation gradient F is: F=F e F g , Where A is the initial coordinate position vector of a growth point inside the tendril, and a is the new coordinate position vector of the growth point after the growth and deformation over a certain period of time; F e is the elastic deformation part of the total deformation gradient F, F g is the irreversible growth deformation part of the total deformation gradient F; The expression of the Jacobian determinant J of the deformation gradient is: J=J e J g =det(F),J g =θ 3 =(V+△V) / V,J e =J / θ 3 In the formula, J e is the elastic variation part of the Jacobian of the deformation gradient, J g is the growth change part of the Jacobian determinant of the deformation gradient; θ is the growth amount, θ 3 is the total volume growth, θ 3 =θ1×θ2×θ3; V is the original volume, △V is the change in volume; Elastic left Cauchy-Green deformation tensor B e The expression is: B e =F e F eT , Where B is the left Cauchy-Green tensor representing the deformation in space, B = FF T ; The expression of Cauchy stress σ is: Where I is the unit tensor; The expression of tangent stiffness D is: In the formula, and Both are non-standard fourth-order tensors and satisfy the Einstein summation convention.

5. The finite element simulation method for describing the growth morphology of plant tendrils according to claim 1, characterized in that: In step S3, the iterative calculation process of the constitutive model is implemented based on Microsoft Visual Studio software, and the specific iterative calculation steps are as follows: S301, obtaining the three growth rate coefficients and the time length T set in step S1; S302, calculating the time t of the current iteration, t=t0+Δt; where t0 is the time of the previous iteration, and Δt is the time increment; S303, according to the growth characteristics of the plant tendril, select a growth function that is suitable for the growth morphology of the plant tendril to calculate the growth amount θ of the plant tendril in three directions at time t i ;in, i) When the plant tendril growth morphology is an Archimedean tendril, its growth function adopts the growth function corresponding to the Archimedean tendril, and its expression is: θ1=1+K1t, θ2=1+K2Xt+K3Yt, θ3=1; ii) When the plant tendril growth morphology is an Euler tendril, its growth function adopts the growth function corresponding to the Euler tendril, and its expression is: θ1=1+K1t, θ2=1+K2Xt+K3XYt, θ3=1; iii) When the plant tendril growth form is a space circular tendril form, its growth function adopts the growth function corresponding to the space circular tendril, and its expression is: θ1=1, θ2=1, θ3=1+K1Xt; iv) When the plant tendril growth form is a spatial Euler tendril form, its growth function adopts the growth function corresponding to the spatial Euler tendril, and the growth function expression is: θ1=1, θ2=1, θ3=1+K1XZt; Where X, Y and Z are the coordinates of the position vector of any growth point in the previous moment t0 in the X, Y and Z directions respectively; t is the current moment; S304, calculating the reversible elastic deformation part F of the total deformation gradient e ; S305, reversible elastic deformation part F based on total deformation gradient e , calculate the left Cauchy-Green tensor B e ; S306, based on the left Cauchy-Green tensor B e , calculate the Cauchy stress σ; S307, calculating the tangent stiffness matrix D based on the Cauchy stress σ; S308. Repeat steps S301 to S307 until the current iteration time t> time length T, and the iteration ends.

6. The finite element simulation method for describing the growth morphology of plant tendrils according to claim 5, characterized in that: The time length T is set to 1, and the iteration time increment △t is set to 0.0001.

7. The finite element simulation method for describing the growth morphology of plant tendrils according to claim 5, characterized in that: The specific implementation steps of step S4 are: S401, create a new job in the job module of ABAQUS software, select User subroutinefile in General to associate the constitutive model constructed by the UMAT subroutine, so as to realize the continuous call of the iterative calculation results obtained by the constitutive model in step S3; S402, ABAQUS software obtains the growth function determined in step S3, the tangent stiffness tensor D and Cauchy stress σ obtained at each iteration time t, and the final position coordinates of each growth point on the tendril at the end of the iteration, so that ABAQUS software continues to perform finite element simulation on the growth morphology of the plant tendril based on the initial morphological model of the plant tendril constructed in step S1, and finally obtains the simulation result of the growth morphology of the plant tendril.

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