Numerical calculation method and device for super-elastic shell growth deformation control and medium

Abstract

This invention discloses a numerical calculation method, apparatus, and medium for controlling the growth and deformation of a hyperelastic shell. The method includes: determining the base surface of the initial configuration and the target surface, and parameterizing a defined discrete complex surface; calculating the fundamental geometric quantities at each node of the discrete complex surface; determining a growth function based on the calculated fundamental geometric quantities; inputting the growth function into a finite element model to simulate the growth process of the sample, and verifying the effectiveness of the obtained growth function. The proposed solution can achieve precise shape control of the hyperelastic shell during the growth and deformation process, thus having significant application value in the field of intelligent manufacturing. This invention can be widely applied in the technical field of soft intelligent device design and development.

Description

Technical Field

[0001] This invention relates to the technical field of soft intelligent device design and development, and in particular to a numerical calculation method, device and medium for controlling the growth and deformation of a hyperelastic shell. Background Technology

[0002] "Soft material growth" refers to the phenomenon where the mass or volume of a soft material sample changes under specific conditions. These changes can be caused by biological factors, such as swelling after tissue damage; or by physical factors, such as hydrogel swelling and rubber aerodynamic expansion. Although the factors inducing growth deformation vary, the resulting growth effect can be attributed to the pre-strain applied to local material points by the growth field (growth function) within the sample.

[0003] Since the function of soft material devices is closely related to their shape, researchers often seek to control the shape of samples to meet the needs of engineering applications. This requires careful design of the material composition or microstructure within the sample and control of external loading conditions to obtain the desired sample configuration; this method is called "shape control" or "shape programming." To achieve sample shape control, it is necessary to study the inverse problem of growth deformation (i.e., given the initial and target configurations of the sample, determining the distribution of the growth function within the sample based on the governing equations, considering factors such as material properties and growth patterns). The research results on the inverse problem can reveal the intrinsic mechanisms of the morphological evolution of biological soft tissues in nature and can also guide the development of novel soft material functional devices, thus possessing significant importance.

[0004] Although shell-shaped soft materials are relatively common in nature and engineering, theoretical research on the inverse problem of shell sample growth and deformation is lacking due to the difficulty in modeling shell structures (requiring the use of curvilinear coordinate systems and consideration of curvature effects). Furthermore, to control the shape of complex discrete surfaces (such as 3D laser-scanned surfaces) in engineering applications, corresponding numerical calculation schemes are needed. However, current research faces two major challenges: how to determine the relationship between the sample growth function and the geometry of the target surface, and how to use appropriate algorithms to extract the geometric features of discrete surfaces. Summary of the Invention

[0005] In order to at least partially solve one of the technical problems existing in the prior art, the purpose of this invention is to provide a numerical calculation method, device and medium for controlling the growth deformation of a hyperelastic shell for industrial applications.

[0006] The first technical solution adopted in this invention is:

[0007] A numerical calculation method for controlling the growth deformation of a hyperelastic shell includes the following steps:

[0008] Determine the base of the initial configuration and target surface The bottom surface and target surface All are discrete complex surfaces;

[0009] Parameterize a given discrete complex surface;

[0010] Calculate the fundamental geometric quantities at each node of a discrete complex surface;

[0011] The growth function is determined based on the calculated fundamental geometric quantities.

[0012] The growth function is input into the finite element model to simulate the growth process of the sample and verify the effectiveness of the obtained growth function.

[0013] Furthermore, the bottom surface and target surface All are composed of triangular units and nodes Represents; each triangular mesh f i Connect three nodes Each node has three-dimensional coordinates (x, y). j ,y j z j These triangular meshes are connected by common nodes, which defines the shape of the surface.

[0014] Furthermore, the parameterization of the determined discrete complex surface includes:

[0015] Using the parametric algorithm in the C++ geometry library libitl, a defined discrete complex surface is transformed. Mapped to the two-dimensional parametric plane Ω r The parameterization process preserves the surface. Each triangular unit Node connection information, two-dimensional parametric plane Ω r The corresponding elements and nodes are represented as follows: and

[0016] Furthermore, the calculation of the fundamental geometric quantities at each node of the discrete complex surface includes:

[0017] Calculation of the first fundamental quantity: To ensure the linear continuity of the first fundamental quantity within the element, a virtual node is introduced located at the centroid of the triangular mesh. and Assume the parametric equations of the triangular mesh (denoted as s). (i) It has the following bilinear distribution form:

[0018]

[0019] In the formula, (x,y,z) are the coordinates of the three-dimensional surface node, (X,Y) are the coordinates of the two-dimensional parametric plane node, and K 11 -K 34 These are the constant coefficients of the parametric equations;

[0020] Triangular mesh f i The first fundamental quantities {E,F,G} on are:

[0021]

[0022] Node Substituting the two-dimensional coordinates into equation (2), we obtain the coordinates of each node on the three-dimensional surface. {E} j ,F j G j};

[0023] Calculate the second fundamental quantities: The second fundamental quantities {L, M, N} satisfy the following system of equations:

[0024]

[0025] In the formula, {κ1,κ2} are the principal curvatures, and {p1,p2} represent the corresponding principal directions; and Let represent the values ​​of the first derivatives of the target surface parametric equation s with respect to X and Y at node i, respectively;

[0026] By solving equation (3), we obtain the expression for {L,M,N}:

[0027]

[0028] Combining the {E} of each node j ,F j G j} and equation (4), solve to obtain the {L} of each node j M j N j}

[0029] Furthermore, determining the growth function based on the calculated fundamental geometric quantities includes:

[0030] growth function It has the following four sets of solutions:

[0031] Groups 1 and 2:

[0032] Groups 3 and 4:

[0033] in:

[0034]

[0035] Q1 = E r F t +F r G t Q2 = E t F r +F t G r

[0036] Q3 = E t F t Q1Δ r +F r G r Q1Δ t Q4 = F t G t Q2Δ r +E r F r Q2Δ t

[0037]

[0038] In the formula, {E r ,F r G r} represents the initial surface The first fundamental coefficient; {E t ,F t G t} represents the target surface The first fundamental quantity coefficient;

[0039] Based on the obtained fundamental geometric quantities, the growth function of the discrete mesh is obtained. and

[0040] Furthermore, to ensure that the growth function conforms to physical meaning, the following criteria are used to select a suitable set of solutions:

[0041]

[0042] Furthermore, the step of inputting the growth function into the finite element model to simulate the sample growth process and verify the effectiveness of the obtained growth function includes:

[0043] Along the bottom surface The top nodes are stretched along their normal directions to grow a geometric model with a preset thickness, and the bottom surface is then stretched. As the bottom surface of the shell;

[0044] The entire shell is divided into multiple triangular elements, and the displacement degrees of freedom of the three nodes on the triangular elements are restricted.

[0045] growth tensor As a state variable in the UMAT subroutine, it is transferred from the unit tensor within a preset period. Gradually change to the target value.

[0046] The second technical solution adopted in this invention is:

[0047] A numerical calculation device for controlling the growth deformation of a hyperelastic shell, comprising:

[0048] The surface determination module is used to determine the bottom surface of the initial configuration. and target surface

[0049] The surface parameterization module is used to parameterize a given discrete complex surface.

[0050] The basic quantity calculation module is used to calculate the basic geometric quantities at each node of a discrete complex surface.

[0051] The growth function determination module is used to determine the growth function based on the calculated fundamental geometric quantities.

[0052] The growth verification module is used to input the growth function into the finite element model, simulate the growth process of the sample, and verify the effectiveness of the obtained growth function.

[0053] The third technical solution adopted in this invention is:

[0054] A numerical calculation device for controlling the growth deformation of a hyperelastic shell, comprising:

[0055] At least one processor;

[0056] At least one memory for storing at least one program;

[0057] When the at least one program is executed by the at least one processor, the at least one processor implements the method described above.

[0058] The fourth technical solution adopted in this invention is:

[0059] A computer-readable storage medium storing a processor-executable program, which, when executed by a processor, performs the method described above.

[0060] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0061] (1) Various geometric mappings can be achieved through non-uniform growth.

[0062] The numerical calculation scheme proposed in this invention establishes a direct link between solid mechanics and differential geometry. Through the non-uniform growth of the shell, a wide range of geometrical mappings, including topological transformations, conformal mappings, and isometric mappings, can be achieved. This demonstrates the flexibility and high adaptability of the scheme in implementing different types of geometrical transformations, providing a theoretical basis and computational methods for designing and manufacturing intelligent soft devices with specific geometric shapes.

[0063] (2) Applicable to discrete surfaces without analytical expressions.

[0064] The numerical calculation scheme proposed in this invention is applicable to discrete surfaces without analytical expressions; therefore, complex surfaces in typical engineering applications can also be considered within the scope of shape control schemes. Furthermore, this research can promote the development of mechanical metamaterial design and processing technology, especially in fields requiring precise control of material microstructure and macromorphology, such as biomimetic engineering, the development of smart structures, and micro / nano fabrication technologies.

[0065] (3) Simple and easy-to-use growth function calculation formula.

[0066] The growth function calculation formula established in this invention has an explicit analytical expression, offering the advantages of simplicity and ease of use. On one hand, it helps researchers efficiently calculate the desired growth function distribution, thereby controlling sample morphological changes. On the other hand, this calculation formula provides a powerful mathematical tool for describing the growth process of biological tissues, contributing to a deeper understanding of the mechanisms of biological morphogenesis. Attached Figure Description

[0067] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following description is provided with accompanying drawings of the relevant technical solutions in the embodiments of the present invention or the prior art. It should be understood that the accompanying drawings described below are only for the purpose of clearly illustrating some embodiments of the technical solutions of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0068] Figure 1 This is a flowchart illustrating the steps of a numerical calculation method for controlling the growth deformation of a hyperelastic shell in an embodiment of the present invention.

[0069] Figure 2 A schematic diagram of the kinematic model and bottom parameterization of the shell sample: (a) initial configuration of the shell; (b) initial reference plane; (c) parameter plane; (d) current configuration of the shell; (e) target surface.

[0070] Figure 3 This is a schematic diagram of discrete surface parameterization.

[0071] Figure 4A schematic diagram to realize changes in facial expressions: (a) parameterization of a three-dimensional surface; (b) numerical simulation process.

[0072] Figure 5 A schematic diagram for realizing changes in the shape of a car: (a) parameterization of the three-dimensional surface; (b) numerical simulation process.

[0073] Figure 6 A schematic diagram to illustrate different facial variations: (a) parameterization of the three-dimensional surface; (b) numerical simulation process. Detailed Implementation

[0074] The embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention. The step numbers in the following embodiments are set only for ease of explanation, and there is no limitation on the order between the steps. The execution order of each step in the embodiments can be adaptively adjusted according to the understanding of those skilled in the art.

[0075] In the description of this invention, it should be understood that the orientation descriptions, such as up, down, front, back, left, right, etc., are based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limiting this invention.

[0076] In the description of this invention, "several" means one or more, "multiple" means two or more, "greater than," "less than," and "exceeding" are understood to exclude the stated number, while "above," "below," and "within" are understood to include the stated number. If "first" or "second" is used, it is only for distinguishing technical features and should not be construed as indicating or implying relative importance, or implicitly indicating the number of indicated technical features, or implicitly indicating the order of the indicated technical features. Furthermore, "and / or" describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A alone, A and B simultaneously, or B alone. The character " / " generally indicates that the preceding and following related objects have an "or" relationship.

[0077] In the description of this invention, unless otherwise explicitly defined, terms such as "set up," "install," and "connect" should be interpreted broadly, and those skilled in the art can reasonably determine the specific meaning of the above terms in this invention in conjunction with the specific content of the technical solution.

[0078] To address the existing technical problems, this invention aims to propose a numerical calculation method for controlling the growth deformation of hyperelastic shells for industrial applications, such as... Figure 1 As shown, the method includes selecting the initial and target base surfaces, parameterizing the three-dimensional surface, calculating the fundamental geometric quantities of the surface, determining the growth function, and numerical simulation verification. This method can achieve precise shape control of the hyperelastic shell during growth and deformation, and therefore has significant application value in the field of intelligent manufacturing.

[0079] I. Theoretical Modeling of Growth and Deformation of Hyperelastic Shells

[0080] In the Einstein summation convention used below, the Greek letters (α, β, γ…) change from 1 to 2, and the Latin letters (i, j, k…) change from 1 to 3. Figure 2 Taking the shell-shaped sample shown as an example, it is assumed that the shell sample of uniform thickness has a reference configuration in three-dimensional space. The lower surface Let θ be the bottom surface of the shell, and 2h be the thickness. Introduce a curvilinear coordinate system {θ}. α} α＝1,2 Will Expressed as a parametric equation, r(θ) α )={X 1 (θ α ),X 2 (θ α ),X 3 (θ α )},θ α ∈Ω r , where Ω r Represents the parametric plane θ 1 θ 2 The parameter field. Therefore, The tangent vector of any point along the coordinate curve is obtained from Given, the normal vector is represented as n.

[0081] Due to the growth effect, the sample deformation reaches the current configuration. Write the new position vector as x(θ) α Z) = x i (θ α ,Z)e i , where e i Using Cartesian coordinates as the basis. To describe the deformation of the shell during growth, the total deformation gradient tensor is multiplied and decomposed into... That is, elastic deformation tensor With growth tensor Multiplication. By considering the incompressible constraint condition of elastic deformation and the surface force-free boundary condition, the three-dimensional governing equations satisfied by the freely grown sample are:

[0082]

[0083] Where Div(·) is the divergence operator, and Det(·) represents the determinant of the matrix. This is the nominal stress. To describe the elastic response of a material, commonly used hyperelastic models such as the neo-Hookean and Gent models can be used, and their elastic strain energy function is denoted as... Therefore, the nominal stress tensor For growth tensor The function is p, where p represents hydrostatic pressure.

[0084] II. Analytical Solution of the Inverse Problem of Growth and Deformation

[0085] To facilitate the analytical solution of the inverse problem, the unknown functions x and p are expanded into Taylor series along the sample thickness direction:

[0086]

[0087] Deformation gradient tensor Elastic deformation tensor and nominal stress tensor The corresponding series expansion is also performed.

[0088]

[0089] The growth tensor considered in this application has a symmetric form:

[0090]

[0091] Where g α For covariant basis vector g α The corresponding contravariant basis vector. Assume the growth function {λ1, λ2, λ...} s Linear distribution along the shell thickness direction:

[0092]

[0093] Based on the above preparations, the main steps for analytically solving the growth function are as follows:

[0094] (1) Given the initial bottom surface and target surface Given the parametric equations, calculate the fundamental geometric quantities of the surface. Assume... and The parametric equations are given as follows:

[0095]

[0096] Then the curved surface The fundamental geometric quantities {E} r ,Fr G r ,L r M r N r It can be calculated according to the following formula:

[0097]

[0098] Similarly, curved surfaces can be obtained. The fundamental geometric quantities {E} t ,F t G t ,L t M t N t}

[0099] (2) Assume the shell is in a stress-free state in its current configuration (i.e. Analyze the sufficient conditions satisfied by the three-dimensional control equations. By neglecting O(Z) in the three-dimensional control equations (1) 2 From the terms, we can see that the sufficient condition for the system of equations (1) to be satisfied is: in It is a unit tensor.

[0100] (3) Based on the above sufficient conditions, derive the explicit relationship between the growth function and the fundamental geometric quantities of the surface. According to the sufficient conditions... The growth function is derived. It has the following four sets of solutions:

[0101] Groups 1 and 2:

[0102] Groups 3 and 4:

[0103] in:

[0104]

[0105] Q1 = E r F t +F r G t Q2 = E t F r +F t G r ,

[0106] Q3 = E t F t Q1Δ r +F r G r Q1Δ t Q4 = F t G t Q2Δr +E r F r Q2Δ t ,

[0107]

[0108] To ensure that the growth function conforms to physical meaning, the following criteria should be used to select a suitable set of solutions:

[0109]

[0110] Based on sufficient conditions The growth function is derived. It has the following unique solution:

[0111]

[0112]

[0113]

[0114] in When the initial bottom surface and target surface After the parametric equations are given, the fundamental geometric quantities of the surface need to be calculated according to equation (7). Substituting the obtained fundamental quantities into equations (8) and (11), the geometric quantities of the surface can be determined. arrive Shape-controlled growth function.

[0115] III. Numerical Calculation Scheme for Shell Growth and Deformation Control

[0116] The analytical solution to the aforementioned inverse problem only applies to cases where both the initial and target surfaces have analytical expressions. However, in engineering, some complex surfaces have discrete forms (such as three-dimensional laser scanning surfaces), and their shapes are typically represented by interconnected meshes. To extend the applicability of equations (8) and (11) to discrete complex surfaces without analytical expressions, this invention proposes a numerical calculation method for shell growth deformation control, see [link to relevant documentation]. Figure 1 The main steps include:

[0117] S1, the base of the given initial configuration and target surface

[0118] These surfaces are typically composed of triangular units. and nodes Represented. Each triangular mesh f i Connect three nodes Each node has three-dimensional coordinates (x, y). j ,y j zj These triangular meshes are connected by common nodes, which defines the shape of the surface.

[0119] S2. Parameterize the three-dimensional surface.

[0120] Using parametric algorithms from the C++ geometry computation library, a given discrete complex surface (denoted as...) is transformed... Mapped to the two-dimensional parametric plane Ω r This parameterization process preserves Each unit Node connection information, Ω r The corresponding elements and nodes are represented as follows: and

[0121] S3. Calculate the fundamental geometric quantities at each node of a discrete complex surface.

[0122] S31. Calculate the first fundamental quantity. For example... Figure 3 As shown, to ensure the linear continuity of the first fundamental quantity within the element, a virtual node is introduced at the centroid of the triangular mesh. and And assume the parametric equations of the mesh (denoted as s). (i) It has the following bilinear distribution form:

[0123]

[0124] Where (x, y, z) are the coordinates of the 3D surface nodes, (X, Y) are the coordinates of the 2D parametric plane nodes, and K... 11 -K 34 These are the constant coefficients of the parametric equations. (This is achieved by...) and Substituting the coordinates into equation (12), the coefficient K can be derived. 11 -K 34 expression

[0125]

[0126] Based on equations (12) and (13), the parametric equation s (i) Regarding (θ) 1 ,θ 2 The first-order partial derivative of ) is calculated by the following formula.

[0127]

[0128] Therefore, the triangular mesh f i The first fundamental quantity {E,F,G} on is

[0129]

[0130] Will Substituting the two-dimensional coordinates into equation (15), the nodes of the three-dimensional surface can be determined. {E} j ,F j G j When multiple units share a node, the {E} at that node... j ,F j G j It is represented by the arithmetic mean calculated from each unit.

[0131] S32. Calculate the second fundamental quantity. The second fundamental quantity {L, M, N} should satisfy the following system of equations:

[0132]

[0133] Where {κ1,κ2} are the principal curvatures, and {p1,p2} represent the corresponding principal directions. By solving equation (16), the expression for {L,M,N} can be obtained.

[0134]

[0135] Each node The principal curvatures {κ1,κ2} and principal directions {p1,p2} at the point can be calculated using a C++ program. Combined with the first fundamental quantity {E} calculated in step S31... j ,F j G j}, {L} at each node j M j N j It can be obtained through equation (17).

[0136] S4. Determine the growth function.

[0137] Substituting the obtained geometric fundamental quantities into equations (8) and (11), the growth function of the discrete mesh can be obtained. and In order for the growth function to conform to the physical meaning, a suitable set of growth functions should be selected using the criteria shown in Equation (10).

[0138] S5. Input the growth function into the finite element model to simulate the growth process of the sample and verify the effectiveness of the obtained growth function.

[0139] The method of the present invention will be further described below with reference to specific embodiments.

[0140] Example 1: Achieving changes in facial expressions

[0141] Example 1 specifically includes the following steps:

[0142] S101, The base of the given initial configuration and target surface The chosen face is a male with a calm expression. The chosen face is a male face with a shocked expression. and Each has 7963 connected triangular units.

[0143] S102. Parameterize the 3D surface. Utilize the parameterization algorithms in the geometry calculation library to parameterize the given discrete complex surface. and Mapped to the two-dimensional parametric plane Ω r This parameterization process preserves the individual elements of the surface. The node connection information.

[0144] S103. Calculate the fundamental geometric quantities at each node of a discrete complex surface.

[0145] S1031, Calculate the first fundamental quantity. By connecting the nodes... and Substituting the coordinates into equation (12), the coefficient K can be derived. 11 -K 34 The expression. Substituting the two-dimensional coordinates into equation (15), the nodes of the three-dimensional surface can be determined. {E} j ,F j G j When multiple units share a node, the {E} at that node... j ,F j G j It is represented by the arithmetic mean calculated from each unit.

[0146] S1032. Calculate the second fundamental quantity. Calculate the value for each node using a C++ program. The principal curvatures {k1,k2} and principal directions {p1,p2} at the location. Combined with the first fundamental quantity {E} calculated in step (3.1) j ,F j G j}, {L} at each node j M j N j It can be obtained through equation (17).

[0147] S104. Determine the growth function. Substituting the obtained geometric fundamental quantities into equations (8) and (11), the growth function of the discrete mesh can be obtained. and In order to make the growth function conform to the physical meaning, a suitable set of growth functions is selected using the criteria shown in Equation (10).

[0148] S105. Input the growth function into the finite element model to simulate the sample growth process and verify the effectiveness of the obtained growth function. First, along... The normal direction of each node is stretched to grow a geometric model with a certain thickness and then... The bottom surface of the shell was selected. The material model was chosen as neo-Hookean hyperelastic material with a Poisson's ratio μ = 0.4995, indicating near-incompressibility. The entire shell was divided into 15926 C3D6H (6-node linear triangular prism, hybrid with constant pressure) elements. To eliminate rigid body motion, the displacement degrees of freedom at the three nodes were restricted. The loading step time increment was set to 10. -6 Automatically adjusts between 0.1 and 0.1. Growth tensor As a state variable in the UMAT subroutine, it changes during time t∈[0,10]. Gradually change to the target value.

[0149] Numerical simulation results of shell growth and deformation are as follows: Figure 4 As shown, the shell grows to fit the target surface very well, thus verifying the correctness of the growth function settings.

[0150] Example 2: Achieving changes in the shape of a car

[0151] Example 2 specifically includes the following steps:

[0152] S201, Given the base of the initial configuration and target surface The Beetle was chosen. Selected as a taxi. and Each has 64,070 connected triangular units.

[0153] S202. Parameterize the 3D surface. Utilize the parameterization algorithms in the geometry calculation library to parameterize the given discrete complex surface. and Mapped to the two-dimensional parametric plane Ω r This parameterization process preserves the individual elements of the surface. The node connection information.

[0154] S203. Calculate the fundamental geometric quantities at each node of a discrete complex surface.

[0155] S2031, Calculate the first fundamental quantity. By connecting the nodes... and Substituting the coordinates into equation (12), the coefficient K can be derived. 11 -K 34 The expression. Substituting the two-dimensional coordinates into equation (15), the nodes of the three-dimensional surface can be determined. {E} j ,F j G j When multiple units share a node, the {E} at that node... j ,F j G j It is represented by the arithmetic mean calculated from each unit.

[0156] S2032. Calculate the second fundamental quantity. Calculate the value for each node using a C++ program. The principal curvatures {κ1,κ2} and principal directions {p1,p2} at the point. Combined with the first fundamental quantity {E} calculated in step (3.1) j ,F j G j}, {L} at each node j M j N j It can be obtained through equation (17).

[0157] S204. Determine the growth function. Substituting the obtained geometric fundamental quantities into equations (8) and (11), the growth function of the discrete mesh can be obtained. and In order to make the growth function conform to the physical meaning, a suitable set of growth functions is selected using the criteria shown in Equation (10).

[0158] S205. Input the growth function into the finite element model to simulate the sample growth process and verify the effectiveness of the obtained growth function. First, along... The normal direction of each node is stretched to grow a geometric model with a certain thickness and then... The bottom surface of the shell is used. The material model is selected as neo-Hookean hyperelastic material with a Poisson's ratio μ = 0.4995, indicating near-incompressibility. The entire shell is divided into 128,140 C3D6H elements. To eliminate rigid body motion, the displacement degrees of freedom at three nodes are restricted. The loading step time increment is set to 10... -6 Automatically adjusts between 0.1 and 0.1. Growth tensor As a state variable in the UMAT subroutine, it changes during time t∈[0,10]. Gradually change to the target value.

[0159] Numerical simulation results of shell growth and deformation are as follows: Figure 5As shown, the shell grows to fit the target surface very well, thus verifying the correctness of the growth function settings.

[0160] Example 3: Achieving different facial transformations

[0161] Example 3 specifically includes the following steps:

[0162] S301, The base of the given initial configuration and target surface Selected as the robot, the first target surface The selected target surface is a female face. The face was selected as male. and Each has 16,030 connected triangular units.

[0163] S302. Parameterize the 3D surface. Utilize the parameterization algorithms in the geometry calculation library to parameterize the given discrete complex surface. and Mapped to the two-dimensional parametric plane Ω r (A disk with a diameter of 1). This parameterization process preserves the individual elements of the surface. The node connection information.

[0164] S303. Calculate the fundamental geometric quantities at each node of a discrete complex surface.

[0165] S3031, Calculate the first fundamental quantity. By connecting the nodes... and Substituting the coordinates into equation (12), the coefficient K can be derived. 11 -K 34 The expression. Substituting the two-dimensional coordinates into equation (15), the nodes of the three-dimensional surface can be determined. {E} j E j G j When multiple units share a node, the {F} at that node... j ,F j G j It is represented by the arithmetic mean calculated from each unit.

[0166] S3032. Calculate the second fundamental quantity. Calculate the value for each node using a C++ program. The principal curvatures {κ1,κ2} and principal directions {p1,p2} at the point. Combined with the first fundamental quantity {E} calculated in step (3.1) j ,F j G j}, {L} at each node j Mj N j It can be obtained through equation (17).

[0167] S304. Determine the growth function. Substituting the obtained geometric fundamental quantities into equations (8) and (11), the growth function of the discrete mesh can be obtained. and In order to make the growth function conform to the physical meaning, a suitable set of growth functions is selected using the criteria shown in Equation (10).

[0168] S305. Input the growth function into the finite element model to simulate the sample growth process and verify the effectiveness of the obtained growth function. First, along... The normal direction of each node is stretched to grow a geometric model with a certain thickness and then... The bottom surface of the shell is used. The material model is selected as neo-Hookean hyperelastic material with a Poisson's ratio μ = 0.4995, indicating near-incompressibility. The entire shell is divided into 32060 C3D6H elements. To eliminate rigid body motion, the displacement degrees of freedom at three nodes are restricted. The loading step time increment is set to 10... -6 Automatically adjusts between 0.1 and 0.1. Growth tensor State in UMAT subroutines

[0169] The variable, during time t∈[0,5] (first phase), from Gradually changed to The corresponding target value gradually changes during the time period t∈[5,10] (second stage) to The corresponding target value.

[0170] Numerical simulation results of shell growth and deformation are as follows: Figure 6 As shown, the shell grows to fit the target surface very well, thus verifying the correctness of the growth function settings.

[0171] The above-described embodiments are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Therefore, any changes made in accordance with the shape and principle of the present invention should be covered within the protection scope of the present invention.

[0172] This embodiment also provides a numerical calculation device for controlling the growth deformation of a hyperelastic shell, including:

[0173] The surface determination module is used to determine the bottom surface of the initial configuration. and target surface

[0174] The surface parameterization module is used to parameterize a given discrete complex surface.

[0175] The basic quantity calculation module is used to calculate the basic geometric quantities at each node of a discrete complex surface.

[0176] The growth function determination module is used to determine the growth function based on the calculated fundamental geometric quantities.

[0177] The growth verification module is used to input the growth function into the finite element model, simulate the growth process of the sample, and verify the effectiveness of the obtained growth function.

[0178] This embodiment provides a numerical calculation device for controlling the growth and deformation of a hyperelastic shell. It can execute the numerical calculation method for controlling the growth and deformation of a hyperelastic shell provided in the method embodiment of the present invention, and can execute any combination of the implementation steps of the method embodiment. It has the corresponding functions and beneficial effects of the method.

[0179] This embodiment also provides a numerical calculation device for controlling the growth deformation of a hyperelastic shell, including:

[0180] At least one processor;

[0181] At least one memory for storing at least one program;

[0182] When the at least one program is executed by the at least one processor, the at least one processor implements Figure 1 The method shown.

[0183] This embodiment provides a numerical calculation device for controlling the growth and deformation of a hyperelastic shell. It can execute the numerical calculation method for controlling the growth and deformation of a hyperelastic shell provided in the method embodiment of the present invention, and can execute any combination of the implementation steps of the method embodiment. It has the corresponding functions and beneficial effects of the method.

[0184] This application also discloses a computer program product or computer program, which includes computer instructions stored in a computer-readable storage medium. A processor of a computer device can read the computer instructions from the computer-readable storage medium and execute the computer instructions, causing the computer device to perform... Figure 1 The method shown.

[0185] This embodiment also provides a storage medium storing instructions or programs that can execute the numerical calculation method for controlling the growth deformation of a hyperelastic shell provided in the method embodiment of the present invention. When the instructions or programs are run, any combination of implementation steps of the method embodiment can be executed, and the method has the corresponding functions and beneficial effects.

[0186] In some alternative embodiments, the functions / operations mentioned in the block diagrams may not occur in the order shown in the operation diagrams. For example, depending on the functions / operations involved, two consecutively shown blocks may actually be executed substantially simultaneously, or the blocks may sometimes be executed in reverse order. Furthermore, the embodiments presented and described in the flowcharts of this invention are provided by way of example to provide a more comprehensive understanding of the technology. The disclosed methods are not limited to the operations and logic flows presented herein. Alternative embodiments are contemplated in which the order of various operations is altered and sub-operations described as part of a larger operation are executed independently.

[0187] Furthermore, although the invention has been described in the context of functional modules, it should be understood that, unless otherwise stated, one or more of the described functions and / or features may be integrated into a single physical device and / or software module, or one or more functions and / or features may be implemented in a separate physical device or software module. It is also understood that a detailed discussion of the actual implementation of each module is unnecessary for understanding the invention. Rather, given the properties, functions, and internal relationships of the various functional modules in the apparatus disclosed herein, the actual implementation of the module will be understood within the scope of conventional skill of an engineer. Therefore, those skilled in the art can implement the invention as set forth in the claims using ordinary techniques without excessive experimentation. It is also understood that the specific concepts disclosed are merely illustrative and not intended to limit the scope of the invention, which is determined by the full scope of the appended claims and their equivalents.

[0188] If the aforementioned functions are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, essentially, or the part that contributes to the prior art, or a portion of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0189] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-included system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transmit programs for use by, or in conjunction with, an instruction execution system, apparatus, or device.

[0190] More specific examples of computer-readable media (a non-exhaustive list) include: electrical connections (electronic devices) having one or more wires, portable computer disk drives (magnetic devices), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic devices, and portable optical disc read-only memory (CDROM). Furthermore, computer-readable media can even be paper or other suitable media on which the program can be printed, since the program can be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, interpreting, or otherwise processing as necessary, and then stored in computer memory.

[0191] It should be understood that various parts of the present invention can be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0192] In the foregoing description of this specification, references to terms such as "one embodiment," "another embodiment," or "some embodiments" indicate that a specific feature, structure, material, or characteristic described in connection with an embodiment or example is included in at least one embodiment or example of the present invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0193] Although embodiments of the invention have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

[0194] The above is a detailed description of the preferred embodiments of the present invention. However, the present invention is not limited to the above embodiments. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the present invention. All such equivalent modifications or substitutions are included within the scope defined by the claims of this application.

Claims

1. A numerical method for controlling the growth deformation of a super-elastic shell, characterized in that, Includes the following steps: determining the base surface of the initial configuration and the target surface ; Parameterize a given discrete complex surface; Calculate the fundamental geometric quantities at each node of a discrete complex surface; The growth function is determined based on the calculated fundamental geometric quantities. The growth function is input into the finite element model to simulate the growth process of the sample and verify the effectiveness of the obtained growth function. The parameterization of the determined discrete complex surface includes: Using the parameterization algorithm from the C++ geometry computation library libigl, the determined discrete complex surface is mapped to a two-dimensional parameter plane ; wherein the parameterization process preserves the node connection information of the individual triangular elements of the surface , the corresponding elements and nodes on the two-dimensional parameter plane are represented as and ; and ; The calculation of the fundamental geometric quantities at each node of the discrete complex surface includes: Calculate the first fundamental quantity: Assume the parametric equations of the triangular mesh have the following bilinear distribution form: （1） In the formula, These are the coordinates of the nodes on the three-dimensional surface. These are the coordinates of the nodes in the two-dimensional parametric plane. These are the constant coefficients of the parametric equations; Triangular mesh The first fundamental quantity for: （2） Node Substituting the two-dimensional coordinates into equation (2), we obtain the coordinates of each node on the three-dimensional surface. of ; Calculate the second fundamental quantity: the second fundamental quantity The following system of equations must be satisfied: （3） In the formula, It is the principal curvature. Indicates the corresponding main direction; and These represent the nodes respectively. Parametric equations of the target surface right and The value of the first derivative; By solving equation (3), we obtain The expression: （4） Combining the nodes By solving equation (4), the values ​​of each node can be obtained. ; The determination of the growth function based on the calculated geometric fundamental quantities includes: growth function It has the following four sets of solutions: in: In the formula, Indicates the bottom surface The first fundamental coefficient; Representing the target surface The first fundamental coefficient; Based on the obtained fundamental geometric quantities, the growth function of the discrete mesh is obtained. and ; To ensure that the growth function conforms to physical meaning, the following criteria are used to select a suitable set of solutions: 。 2. The numerical calculation method for controlling the growth deformation of a hyperelastic shell according to claim 1, characterized in that, The bottom surface and target surface All are composed of triangular units and nodes Representation; each triangular grid Connect three nodes Each node has three-dimensional coordinates. These triangular meshes are connected by common nodes, which defines the shape of the surface.

3. The numerical calculation method for controlling the growth deformation of a hyperelastic shell according to claim 1, characterized in that, The step of inputting the growth function into the finite element model to simulate the sample growth process and verify the effectiveness of the obtained growth function includes: Along the bottom surface The top nodes are stretched along their normal directions to grow a geometric model with a preset thickness, and the bottom surface is then stretched. As the bottom surface of the shell; The entire shell is divided into multiple triangular elements, and the displacement degrees of freedom of the three nodes on the triangular elements are restricted. growth tensor As a state variable in the UMAT subroutine, it is transferred from the unit tensor within a preset period. Gradually change to the target value.

4. A numerical calculation device for controlling the growth deformation of a hyperelastic shell, used to implement the method described in any one of claims 1-3, characterized in that, include: The surface determination module is used to determine the bottom surface of the initial configuration. and target surface ; The surface parameterization module is used to parameterize a given discrete complex surface. The basic quantity calculation module is used to calculate the basic geometric quantities at each node of a discrete complex surface. The growth function determination module is used to determine the growth function based on the calculated fundamental geometric quantities. The growth verification module is used to input the growth function into the finite element model, simulate the growth process of the sample, and verify the effectiveness of the obtained growth function.

5. A numerical calculation device for controlling the growth deformation of a hyperelastic shell, characterized in that, include: At least one processor; At least one memory for storing at least one program; When the at least one program is executed by the at least one processor, the at least one processor implements the method of any one of claims 1-3.

6. A computer-readable storage medium storing a processor-executable program, characterized in that, The processor-executable program, when executed by the processor, is used to perform the method as described in any one of claims 1-3.

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