CN114239334A - Method, system and medium for controlling configuration evolution of superelastic thin plate based on growth deformation - Google Patents

Method, system and medium for controlling configuration evolution of superelastic thin plate based on growth deformation Download PDF

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CN114239334A
CN114239334A CN202111324510.0A CN202111324510A CN114239334A CN 114239334 A CN114239334 A CN 114239334A CN 202111324510 A CN202111324510 A CN 202111324510A CN 114239334 A CN114239334 A CN 114239334A
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curved surface
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王炯
李展锋
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South China University of Technology SCUT
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Abstract

The invention discloses a method, a system and a medium for controlling the configuration evolution of a super-elastic sheet based on growth deformation, wherein the method comprises the following steps: s1, calculating a first type basic quantity and a second type basic quantity of the target curved surface according to a parameter equation of the target curved surface; s2, judging whether the coordinate curve forms an orthogonal curvature net according to the first type basic quantity and the second type basic quantity, and if so, executing a step S4; if not, go to step S3; s3, carrying out independent variable transformation on the parameter equation of the target curved surface so that the new coordinate curve forms an orthogonal curvature net on the curved surface; and S4, determining the initial plane configuration of the plate-shaped sample according to the first basic quantity and the second basic quantity, and the distribution of the growth function in the plate-shaped sample. The invention considers two conditions of whether the parameter curved surface coordinate net is an orthogonal curvature net or not, and can calculate the growth function of any parameter curved surface. The invention can be widely applied to the technical field of design and development of software intelligent devices.

Description

Method, system and medium for controlling configuration evolution of superelastic thin plate based on growth deformation
Technical Field
The invention relates to the technical field of design and development of software intelligent devices, in particular to a method, a system and a medium for controlling the configuration evolution of a superelasticity thin plate based on growth deformation.
Background
The growing (or swelling, etc.) deformation of soft material samples (e.g., soft biological tissue, polymer gels, etc.) is ubiquitous in nature and in the engineering arts. Due to genetic, biochemical, external environmental and mechanical loading effects, the growth field inside the soft material sample often has non-uniform or incompatible properties, commonly referred to as differential growth. Under differential growth conditions, soft material samples can exhibit a wide variety of geometric configurations and surface topography evolution. In the field of industrial application, by setting the material composition or microstructure in a soft material sample, the distribution of the growth field inside the sample can be controlled, so as to realize the required evolution of the sample configuration or other functions, and the technology is called 'shape programming' or 'shape control'. The technology has wide application prospect in the design and manufacture of novel soft intelligent devices (such as drivers, sensors, soft robots and the like).
For the growth deformation of soft material samples, most of the existing theoretical and experimental research work focuses on the positive problem, namely what mechanical behavior characteristics the sample will have given the initial configuration, growth function, external load and constraint conditions of the soft material sample. In theoretical modeling, the mechanical behavior of a soft material sample is mostly studied based on the superelastic material constitutive. The prediction results of the model are generally good at describing growth distortions and instabilities of the sample. However, in order to meet the development requirements of the soft intelligent device in the industrial application field, the inverse problem of the growth deformation of the soft material sample must be studied, i.e. how to set the growth field (growth function) inside the soft material sample to realize the specific configuration evolution of the soft material sample through differential growth?
The current research situation aiming at the inverse problem of the growth deformation of soft materials is far from being satisfactory. In particular, a widely applicable scheme is lacked, and the growth function distribution in the plate-shaped soft material sample can be determined according to any three-dimensional curved surface configuration (meeting the necessary smoothness requirement). The solution of the problem has important significance for the design and development of new software intelligent devices in the industrial field.
Disclosure of Invention
To solve at least one of the technical problems in the prior art to some extent, the present invention provides a method, a system and a medium for controlling the evolution of a superelastic sheet configuration based on growth deformation.
The technical scheme adopted by the invention is as follows:
a method for controlling the structural evolution of a superelasticity thin plate based on growth deformation comprises the following steps:
s1, calculating a first type basic quantity and a second type basic quantity of the target curved surface according to a parameter equation of the target curved surface;
s2, judging whether the coordinate curve forms an orthogonal curvature net according to the first type basic quantity and the second type basic quantity, and if so, executing a step S4; if not, go to step S3;
s3, carrying out independent variable transformation on the parameter equation of the target curved surface so that the new coordinate curve forms an orthogonal curvature net on the curved surface;
and S4, determining the initial plane configuration of the plate-shaped sample according to the first basic quantity and the second basic quantity, and the distribution of the growth function in the plate-shaped sample.
Further, the first-type basic quantity { E, F, G } and the second-type basic quantity { L, M, N } are calculated by the following formula:
E＝r,X·r,X,F＝r,X·r,Y,G＝r,Y·r,Y,
L＝r,XX·n,M＝r,XY·n,N＝r,YY·n,
wherein n is r,X×r,Y/|r,X×r,YI is a unit normal vector of the curved surface; r is,XAs the first derivative of X, r,YIs the first derivative of Y, r,XYFirst derivative to X, Y, r,XXIs the second derivative of X, r,YYThe second derivative to Y.
Further, the determining whether the coordinate curve forms an orthogonal curvature net according to the first type basic quantity and the second type basic quantity includes:
judging whether the condition F is equal to M and is equal to 0, and if so, judging that an orthogonal curvature net is formed; otherwise, it is determined that the orthogonal curvature net cannot be formed.
Further, the performing independent variable transformation on the parameter equation of the target curved surface includes:
establishing a mapping from { S, T } to { X, Y } including transforming X and Y as follows:
X＝X(S,T),Y＝Y(S,T),
wherein X (S, T) and Y (S, T) are sufficiently smooth and Jacobi determinant of the mapping from { S, T } to { X, Y }, is
Figure BDA0003346488430000021
Establishing a { X, Y } plane definition domain omegarAnd { S, T } plane domain
Figure BDA0003346488430000022
Bijective relationship of (1), object curved surface
Figure BDA0003346488430000023
The new parameter equation is r*(S, T) ═ r (X (S, T), Y (S, T)), and the curved surface r is calculated*Natural coordinate base of (S, T)
Figure BDA0003346488430000024
Figure BDA0003346488430000025
Figure BDA0003346488430000026
Wherein Θ is1And Θ2Are respectively the included angles between the two main directions and the X axis;
Figure BDA0003346488430000027
Figure BDA0003346488430000028
obtaining expressions of S (X, Y) and T (X, Y), according to the targetMarked curved surface
Figure BDA0003346488430000031
Equation of parameters r*(S, T), and recalculating the first type basic quantity { E, F, G } and the second type basic quantity { L, M, N } of the target curved surface.
Further, the growth function is calculated by the following formula:
Figure BDA0003346488430000032
wherein λ is1And λ2Respectively radial and circumferential growth functions, and Z is a thickness direction coordinate of the plate.
Further, before step S1, the method further includes the following steps:
determining the initial configuration, growth conditions and elastic constitutive relation of the super-elastic sheet sample.
Further, the following steps are also included after step S4:
and verifying the effectiveness and the precision of the obtained growth function by adopting a finite element simulation method.
The other technical scheme adopted by the invention is as follows:
a system for controlling evolution of a configuration of a superelastic sheet based on growth deformation, comprising:
the basic quantity calculating module is used for calculating a first basic quantity and a second basic quantity of the target curved surface according to a parameter equation of the target curved surface;
the conversion judging module is used for judging whether the coordinate curve forms an orthogonal curvature net according to the first type basic quantity and the second type basic quantity, and if so, the conversion judging module is switched to the function determining module; if not, turning to a basic quantity adjusting module;
the basic quantity adjusting module is used for carrying out independent variable transformation on a parameter equation of the target curved surface so that a new coordinate curve forms an orthogonal curvature net on the curved surface;
a function determination module for determining an initial planar configuration of the plate-like sample and a distribution of the growth function inside the plate-like sample from the first basis quantity and the second basis quantity.
The other technical scheme adopted by the invention is as follows:
a system for controlling evolution of a configuration of a superelastic sheet based on growth deformation, comprising:
at least one processor;
at least one memory for storing at least one program;
when executed by the at least one processor, cause the at least one processor to implement the method described above.
The other technical scheme adopted by the invention is as follows:
a computer readable storage medium in which a processor executable program is stored, which when executed by a processor is for performing the method as described above.
The invention has the beneficial effects that: the invention considers two conditions whether the parameter curved surface coordinate network is an orthogonal curvature network, the growth function can be directly calculated for the curved surface of which the coordinate network is the orthogonal curvature network, and for other curved surfaces, the coordinate network is the orthogonal curvature network through coordinate conversion, so that the growth function is calculated.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following description is made on the drawings of the embodiments of the present invention or the related technical solutions in the prior art, and it should be understood that the drawings in the following description are only for convenience and clarity of describing some embodiments in the technical solutions of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flow chart of a method for programming the differential growth shape of a superelastic sheet in an embodiment of the present invention;
FIG. 2 is a diagram of coordinate transformation between domain from { X, Y } to { S, T } and planar region in an embodiment of the present invention
Figure BDA0003346488430000042
Projected to a curved surface
Figure BDA0003346488430000041
Is mapped r*A schematic diagram of (a);
FIG. 3 is a diagram showing the results of numerical simulation of the growth process of a plate according to an embodiment of the present invention;
FIG. 4 is a diagram illustrating the coordinate transformation from { X, Y } to { S, T } of the helicoid domain and the numerical simulation results of the helicoid growth process in an embodiment of the present invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention. The step numbers in the following embodiments are provided only for convenience of illustration, the order between the steps is not limited at all, and the execution order of each step in the embodiments can be adapted according to the understanding of those skilled in the art.
In the description of the present invention, it should be understood that the orientation or positional relationship referred to in the description of the orientation, such as the upper, lower, front, rear, left, right, etc., is based on the orientation or positional relationship shown in the drawings, and is only for convenience of description and simplification of description, and does not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus, should not be construed as limiting the present invention.
In the description of the present invention, the meaning of a plurality of means is one or more, the meaning of a plurality of means is two or more, and larger, smaller, larger, etc. are understood as excluding the number, and larger, smaller, inner, etc. are understood as including the number. If the first and second are described for the purpose of distinguishing technical features, they are not to be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated or implicitly indicating the precedence of the technical features indicated.
In the description of the present invention, unless otherwise explicitly limited, terms such as arrangement, installation, connection and the like should be understood in a broad sense, and those skilled in the art can reasonably determine the specific meanings of the above terms in the present invention in combination with the specific contents of the technical solutions.
The embodiment aims to provide a method for controlling the evolution of the configuration of a superelastic thin plate based on growth (or swelling, expansion and the like) deformation, and the flow of the method is shown in fig. 1. The method can realize the accurate control of the three-dimensional configuration evolution of the super-elastic sheet in the growth deformation process, thereby having important application value in the field of manufacturing soft material intelligent devices.
The study object of the present embodiment is a three-dimensional Euclidean space
Figure BDA0003346488430000051
One of which has a uniform thickness. Setting the initial configuration of the superelastic sheet by appropriate selection of Cartesian coordinates
Figure BDA0003346488430000052
Region K in (1)r＝Ωr×[0,h]Wherein the thickness h of the plate is much smaller than the plane area omegarThe size of (c). The unit vector along the coordinate axis is denoted as { e1,e2K }. Considering the coordinates of any material point in the super-elastic sheet as (X, Y, Z), the position vector of the point is R ═ Xe1+Ye2+Zk。
Consider the case where the superelastic sheet is biaxially grown along the X and Y axes. In this growth case, the in-plane growth field can be given a diagonal tensor
Figure BDA0003346488430000053
Is shown, wherein λ1(X, Y, Z) and λ2(X, Y, Z) are growth functions in the X-axis and Y-axis directions, respectively. The growth field being linearly distributed along the thickness of the plate, i.e.
Figure BDA0003346488430000054
Since the growth fields in the plates may be incompatible, residual stresses may develop in the plates, which in turn lead to elastic deformation of the plates. After deformation, the plate reaches the current configuration κtAssume the plate is at κtThe material point in (1) has a new position vector r ═ xe1+ye2+ zk, where the current coordinate (x, y, z) depends on κrReference coordinates (X, Y, Z) in (a). Total deformation gradient tensor
Figure BDA0003346488430000055
Can be calculated by
Figure BDA0003346488430000056
Wherein
Figure BDA0003346488430000057
Is an in-plane two-dimensional gradient operator, subscript ",X”，“,Y”，“,X"is taken as the derivative along the coordinate axis. Following the method proposed by Rodriguez et al, the total deformation gradient tensor can be decomposed into
Figure BDA0003346488430000058
Wherein
Figure BDA0003346488430000059
Is the elastic strain tensor. Since the elastic response of soft materials is generally isochoric (e.g., biological soft tissue, polymer gel), the elastic strain tensor should satisfy the following constraint equation
Figure BDA00033464884300000510
Where Det (-) denotes the determinant of the matrix.
Let the elastic response of the sheet material be described by the neo-Hookean material constitutive, i.e. the elastic strain energy function of the material is
Figure BDA00033464884300000511
Wherein
Figure BDA00033464884300000512
C0Is the material constant. From elastic strain energy function, nominal stress tensor
Figure BDA00033464884300000513
Given by:
Figure BDA00033464884300000514
where p (X, Y, Z) is the lagrangian multiplier associated with constraint equation (3).
Regarding the initial configuration, growth conditions and elastic constitutive relation of the superelastic sheet sample, as shown in fig. 1, the configuration evolution control scheme proposed in this embodiment includes the following steps:
s101, according to the target curved surface
Figure BDA00033464884300000515
To calculate the target surface
Figure BDA00033464884300000516
And a second type of basis weights { L, M, N }, wherein { E, F, G } and { L, M, N } are calculated according to the following equations:
Figure BDA0003346488430000061
wherein n is r,X×r,Y/|r,X×r,YAnd | is a unit normal vector of the curved surface.
S102, determine whether the initial coordinate curve forms an orthogonal curvature net, that is, whether the condition F-M-0 is satisfied. If yes, jumping to step 104, otherwise, proceeding to step 103.
S103, carrying out independent variable transformation on the parameter equation of the curved surface, so that the new coordinate curve forms an orthogonal curvature net on the curved surface. Establishing a mapping from { S, T } to { X, Y } including transforming X and Y as follows:
X＝X(S,T),Y＝Y(S,T), (6)
wherein X (S, T) and Y (S, T) are sufficiently smooth and their Jacobi determinant
Figure BDA0003346488430000062
Based on the above transformation, a { X, Y } plane definition domain Ω can be establishedrAnd { S, T } plane domain
Figure BDA0003346488430000063
As shown in fig. 2. Curved surface
Figure BDA0003346488430000064
The new parameter equation is r*(S, T) ═ r (X (S, T), Y (S, T)), thereby calculating curved surface r*Natural coordinate base of (S, T)
Figure BDA0003346488430000065
Figure BDA0003346488430000066
Wherein Θ is1And Θ2Respectively the included angles between the two main directions and the X axis,
Figure BDA0003346488430000067
to ensure r*The coordinate curves of (S, T) can form an orthogonal curvature net, a curved surface
Figure BDA0003346488430000068
At any point r*(S0,T0) Is/are as follows
Figure BDA0003346488430000069
And
Figure BDA00033464884300000610
alignment with the main direction is required. Accordingly, Θ1And Θ2Needs to be determined according to the following equation:
(LF-ME)cos2Θ+(LG-NE)cosΘsinΘ+(MG-NF)sin2Θ＝0, (9)
wherein, the basic quantities { E, F, G } and { L, M, N } of the curved surface need to be calculated according to the initial curved surface r (X, Y). Using the bijective relationship between { S, T } and { X, Y }, an expression for dS and dT is given. The method comprises the following steps of utilizing a bijective relation between { S, T } and { X, Y } to derive first derivatives of S (X, Y) and T (X, Y):
Figure BDA00033464884300000611
wherein the content of the first and second substances,
Figure BDA00033464884300000612
and
Figure BDA00033464884300000613
is an integral factor
Figure BDA00033464884300000614
Thus, an expression for dS and dT is written:
Figure BDA0003346488430000071
finding suitable integral factors
Figure BDA0003346488430000072
And
Figure BDA0003346488430000073
dS and dT are made integrable, and a single integration of dS and dT, respectively, provides an explicit expression for S (X, Y) and T (X, Y). If { sin Θi,cosΘi}i＝1,2Are continuously differentiable, and they are at the point (X)0,Y0)∈ΩrNot simultaneously zero, then in (X)0,Y0) Must have an integral factor in the neighborhood of
Figure BDA0003346488430000074
After obtaining the expressions of S (X, Y) and T (X, Y), it is necessary to obtain the target surface
Figure BDA0003346488430000077
Equation of parameters r*(S, T), and recalculating the first basic quantity { E, F, G } and the second basic quantity { L, M, N } of the curved surface.
And S104, determining the initial plane configuration of the plate-shaped sample and the distribution of the internal growth function field of the plate-shaped sample according to the first and second basic quantities of the target curved surface. Wherein the growth function field is calculated according to the following formula:
Figure BDA0003346488430000075
in the formula, λ1And λ2Respectively radial and circumferential growth functions, and Z is a thickness direction coordinate of the plate. Since the basic quantities { E, F, G } and { L, M, N } of the curved surface can uniquely determine the curved surface, the target growth configuration can be generated based on the formula (13).
And S105, verifying the effectiveness and the precision of the obtained growth function by adopting numerical methods such as finite element simulation.
The above method is explained in detail below with reference to the drawings and specific embodiments.
This example provides a theoretical method of shape programming for growth deformation of superelastic sheets. The method comprises the steps of calculating basic quantities { E, F, G } and { L, M, N } of the target curved surface, judging whether a coordinate curve forms an orthogonal curvature net or not, performing variable conversion of a defined domain, calculating a growth function, performing numerical verification and the like.
Firstly, an example that the coordinate curve can form an orthogonal curvature net is shown, here we select a rotating surface as a parametric surface, which specifically includes the following steps:
s201, according to the target curved surface
Figure BDA0003346488430000078
The first type basic quantity { E, F, G } and the second type basic quantity { L, M, N } of the curved surface are calculated. For a rotating surface, the parametric equation satisfies the following relation:
r(X,Y)＝(f(X)cos(2πY),f(X)sin(2πY),g(X)), (14)
wherein f (X) and g (X) are arbitrary smoothing functions. Corresponding to the parameter equation, the surface basic quantities { E, F, G } and { L, M, N } can be calculated as
Figure BDA0003346488430000076
S202, the initial coordinate curve forms an orthogonal curvature wire mesh on the curved surface, and the process may jump to step S203, depending on whether F is 0 or M.
And S203, determining the distribution of the internal growth function field of the plate-shaped sample according to the first and second basic quantities of the target curved surface. Substituting the surface basic quantities { E, F, G } and { L, M, N } of the rotating surface into the formula (13) to obtain a growth function as follows:
Figure BDA0003346488430000081
an expected rotation plane can be generated according to the growth function. To achieve more specific results, four rotational surfaces, ellipsoid, conical, catenary, and torus, are selected for illustration below. Their growth functions are shown below:
1) ellipsoid (X is more than or equal to 0 and less than or equal to 1, Y is more than or equal to 0 and less than or equal to 1)
Figure BDA0003346488430000082
2) Conical surface (X is more than or equal to 0 and less than or equal to 1, Y is more than or equal to 0 and less than or equal to 1)
Figure BDA0003346488430000083
3) Catenary surface (X is more than or equal to 0 and less than or equal to 1, Y is more than or equal to 0 and less than or equal to 1)
Figure BDA0003346488430000084
4) Toroid (X is more than or equal to 0 and less than or equal to 1, Y is more than or equal to 0 and less than or equal to 1)
Figure BDA0003346488430000085
And S204, carrying out numerical verification on the effectiveness and the precision of the obtained growth function field. And simulating the growth deformation of the thin plate through numerical calculation, wherein the growth function is given by the shape design formula, and verifying whether the setting of the growth function is correct or not according to the numerical result. In the numerical calculation, the initial configuration of the plate was chosen to be Ωr＝[0,1]×[0,1]The numerical calculation steps are as follows:
s2041, writing a constitutive model of the Neo-Hookeank compressible super-elastic material by using a subprogram UMAT of commercial finite element software ABAQUS, wherein lambda is1And λ2Set to the state variable. During the numerical calculation, UMAT will be invoked at the integration point of each cell. The material constants are chosen to ensure that the poisson ratio μ is 0.4995, i.e., close to the incompressible state. The thickness of the plate is set to 0.01 and the plate is divided into 20000C 3D8IH (an 8-node linear crack, hybrid, linear pressure, compatible modes) units. To simulate the growth process of a thin plate, the growth function λ1And λ2Linearly from 1 to the specified value.
S2042, applying initial displacement disturbance to the geometric model to further induce out-of-plane deformation. Before the growth is carried out, the plate is subjected to linear buckling analysis to obtain instability modes, wherein one linear buckling mode is multiplied by a dissipation coefficient and then is applied to the sample as an initial defect.
S2043, deformation gradient tensor based on input displacement data and state variables
Figure BDA0003346488430000091
And growth tensor
Figure BDA0003346488430000092
The value of (a) is determined. Subprogram passing
Figure BDA0003346488430000093
And (4) calculating elastic strain, further calculating Cauchy stress and consistent tangent operator, and transmitting the Cauchy stress and consistent tangent operator to a finite element main program as output data.
The results of numerical simulation of the sheet growth deformation are shown in fig. 3, and it can be seen that, among the four curved surfaces, the sheet after growth closely matches the target curved surface, and therefore the correctness of the setting of the growth function can be verified. Wherein, FIG. 3(a) is a schematic diagram of a simulation of an ellipsoid; FIG. 3(b) is a schematic diagram of a simulation of a conical surface; FIG. 3(c) is a schematic simulation of the catenary profile; FIG. 3(d) is a schematic diagram of a simulation of a torus.
To further illustrate the applicability of the method of this embodiment, an example is shown next, in which the initial coordinate curve cannot directly form an orthogonal curvature net, and here we select a helicoid as a parametric surface, which specifically includes the following steps:
s301, according to the target curved surface
Figure BDA0003346488430000097
The first type basic quantity { E, F, G } and the second type basic quantity { L, M, N } of the curved surface are calculated. For the helicoid, the parametric equation satisfies the following relation:
r(X,Y)＝(X sin(2πY),X cos(4πY),2Y). (21)
corresponding to the parameter equation, the curved surface basic quantity F is 0,
Figure BDA0003346488430000094
s302, the initial coordinate curve cannot form an orthogonal curvature net on the curved surface, and the process proceeds to step S303 according to the condition that M ≠ 0.
And S303, carrying out independent variable transformation on the parameter equation of the curved surface, so that the new coordinate curve forms an orthogonal curvature net on the curved surface. Establishing a mapping from { S, T } to { X, Y }:
X＝X(S,T),Y＝Y(S,T). (22)
by the above transformation, a curved surface
Figure BDA0003346488430000098
The new parameter equation is r*(S, T) ═ r (X (S, T), Y (S, T)), thereby calculating curved surface r*Natural coordinate base of (S, T)
Figure BDA0003346488430000095
Figure BDA0003346488430000096
Wherein { sin Θi,cosΘi}i＝1,2Comprises the following steps:
Figure BDA0003346488430000101
using the bijective relationship between { S, T } and { X, Y }, an expression for dS and dT is given. Wherein the integral factor
Figure BDA0003346488430000102
And
Figure BDA0003346488430000103
comprises the following steps:
Figure BDA0003346488430000104
the expression for dS and dT is:
Figure BDA0003346488430000105
integral factor of the above
Figure BDA0003346488430000106
And
Figure BDA0003346488430000107
so that dS and dT can be multiplied, after the integration constant is selected, the explicit expression of S (X, Y) and T (X, Y) is:
Figure BDA0003346488430000108
s (X, Y) and T (X, Y) will define the domain ΩrTransition to a New Domain
Figure BDA0003346488430000109
According to the target curved surface
Figure BDA00033464884300001010
Equation of parameters r*(S, T), recalculating the basic quantities { E, F, G } and { L, M, N } of the curved surface:
Figure BDA00033464884300001011
and S304, determining the distribution of the internal growth function field of the plate-shaped sample according to the first and second basic quantities of the target curved surface. Substituting the surface basic quantities { E, F, G } and { L, M, N } of the helicoids into the formula (13) to obtain a growth function:
Figure BDA00033464884300001012
s305, simulating the growth deformation of the thin plate through numerical calculation, wherein the growth function is given by the shape design formula, and verifying whether the setting of the growth function is correct according to the numerical result. In the numerical calculation, the initial configuration of the plate is selected as
Figure BDA00033464884300001013
Numerical calculation divisionThe same is true for the case where the coordinate curves can form orthogonal curvature nets, except for the initial configuration.
The results of numerical simulation of the deformation of the growth of the sheet are shown in fig. 4, and it can be seen that the sheet after growth closely matches the helicoid, and therefore the correctness of the setting of the above-mentioned growth function can be verified.
In summary, compared with the prior art, the method of the embodiment has the following beneficial effects:
(1) the theoretical model established in this embodiment considers two situations whether the parametric curved surface coordinate network is an orthogonal curvature network, the growth function can be directly calculated for a curved surface of which the coordinate network is the orthogonal curvature network, and for other curved surfaces, the coordinate network needs to be the orthogonal curvature network through coordinate conversion so as to calculate the growth function. Theoretically, the growth field of any parametric surface that satisfies the necessary smoothness conditions can be determined by this theoretical approach.
(2) The technical scheme provided by the embodiment can accurately control the configuration evolution of the thin plate from two dimensions to three dimensions. Fig. 3-4 show numerically calculated growth configurations in which the growth function is determined using the present approach. The numerical calculation result shows that the method of the embodiment can accurately control the growth configuration of the super-elastic sheet.
(3) The growth function calculation formula (13) established by the embodiment has a very simple form, on one hand, the growth function calculation formula is beneficial to popularization and application, and convenience is provided for efficient design and manufacture of intelligent soft instruments. On the other hand, the analysis result can also provide deeper insight for the growth behavior of certain soft biological tissues in nature.
The embodiment also provides a system for controlling the evolution of the configuration of the superelastic thin plate based on growth deformation, which comprises:
the basic quantity calculating module is used for calculating a first basic quantity and a second basic quantity of the target curved surface according to a parameter equation of the target curved surface;
the conversion judging module is used for judging whether the coordinate curve forms an orthogonal curvature net according to the first type basic quantity and the second type basic quantity, and if so, the conversion judging module is switched to the function determining module; if not, turning to a basic quantity adjusting module;
the basic quantity adjusting module is used for carrying out independent variable transformation on a parameter equation of the target curved surface so that a new coordinate curve forms an orthogonal curvature net on the curved surface;
a function determination module for determining an initial planar configuration of the plate-like sample and a distribution of the growth function inside the plate-like sample from the first basis quantity and the second basis quantity.
The structural evolution control system for the superelastic sheet based on growth deformation can execute the structural evolution control method for the superelastic sheet based on growth deformation provided by the method embodiment of the invention, can execute any combination of the implementation steps of the method embodiment, and has corresponding functions and beneficial effects of the method.
The embodiment also provides a system for controlling the evolution of the configuration of the superelastic thin plate based on growth deformation, which comprises:
at least one processor;
at least one memory for storing at least one program;
when executed by the at least one processor, cause the at least one processor to implement the method as shown in fig. 1.
The structural evolution control system for the superelastic sheet based on growth deformation can execute the structural evolution control method for the superelastic sheet based on growth deformation provided by the method embodiment of the invention, can execute any combination of the implementation steps of the method embodiment, and has corresponding functions and beneficial effects of the method.
The embodiment of the application also discloses a computer program product or a computer program, which comprises computer instructions, and the computer instructions are stored in a computer readable storage medium. The computer instructions may be read by a processor of a computer device from a computer-readable storage medium, and executed by the processor to cause the computer device to perform the method illustrated in fig. 1.
The embodiment also provides a storage medium, which stores instructions or a program capable of executing the method for controlling evolution of configuration of a superelastic sheet based on growth deformation provided by the embodiment of the method, and when the instructions or the program are executed, the steps can be executed in any combination of the embodiment of the method, and the corresponding functions and advantages of the method are achieved.
In alternative embodiments, the functions/acts noted in the block diagrams may occur out of the order noted in the operational illustrations. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality/acts involved. Furthermore, the embodiments presented and described in the flow charts of the present invention are provided by way of example in order to provide a more thorough 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 changed and in which sub-operations described as part of larger operations are performed independently.
Furthermore, although the present invention is described in the context of functional modules, it should be understood that, unless otherwise stated to the contrary, one or more of the described functions and/or features may be integrated in 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 will also be appreciated that a detailed discussion of the actual implementation of each module is not necessary for an understanding of the present invention. Rather, the actual implementation of the various functional modules in the apparatus disclosed herein will be understood within the ordinary skill of an engineer, given the nature, function, and internal relationship of the modules. Accordingly, those skilled in the art can, using ordinary skill, practice the invention as set forth in the claims without undue experimentation. It is also to be understood that the specific concepts disclosed are merely illustrative of and not intended to limit the scope of the invention, which is defined by the appended claims and their full scope of equivalents.
The functions, if implemented in the form of software functional units and sold or used as a stand-alone product, may be stored in a computer readable storage medium. Based on such understanding, the technical solution of the present invention may be embodied in the form of a software product, which is stored in a storage medium and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device) to execute all or part of the steps of the method according to the embodiments of the present invention. And the aforementioned storage medium includes: a U-disk, a removable hard disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a magnetic disk or an optical disk, and other various media capable of storing program codes.
The logic and/or steps represented in the flowcharts or otherwise described herein, e.g., an ordered listing of executable instructions that can be considered to implement logical functions, can be embodied in any computer-readable medium for use by or in connection with an instruction execution system, apparatus, or device, such as a computer-based system, processor-containing system, or other system that can fetch the instructions from the instruction execution system, apparatus, or device and execute the instructions. For the purposes of this description, a "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device.
More specific examples (a non-exhaustive list) of the computer-readable medium would include the following: an electrical connection (electronic device) having one or more wires, a portable computer diskette (magnetic device), a Random Access Memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber device, and a portable compact disc read-only memory (CDROM). Additionally, the computer-readable medium could even be paper or another suitable medium upon which the program is printed, as the program can be electronically captured, via for instance optical scanning of the paper or other medium, then compiled, interpreted or otherwise processed in a suitable manner if necessary, and then stored in a computer memory.
It should be understood that portions of the present invention may be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, the various steps or methods may 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, any one or combination of the following techniques, which are known in the art, may be used: a discrete logic circuit having a logic gate circuit for implementing a logic function on a data signal, an application specific integrated circuit having an appropriate combinational logic gate circuit, a Programmable Gate Array (PGA), a Field Programmable Gate Array (FPGA), or the like.
In the foregoing description of the specification, reference to the description of "one embodiment/example," "another embodiment/example," or "certain embodiments/examples," etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the present invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.
While the preferred embodiments of the present invention have been illustrated and described, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (10)

1. A method for controlling the structural evolution of a superelasticity thin plate based on growth deformation is characterized by comprising the following steps:
s1, calculating a first type basic quantity and a second type basic quantity of the target curved surface according to a parameter equation of the target curved surface;
s2, judging whether the coordinate curve forms an orthogonal curvature net according to the first type basic quantity and the second type basic quantity, and if so, executing a step S4; if not, go to step S3;
s3, carrying out independent variable transformation on the parameter equation of the target curved surface so that the new coordinate curve forms an orthogonal curvature net on the curved surface;
and S4, determining the initial plane configuration of the plate-shaped sample according to the first basic quantity and the second basic quantity, and the distribution of the growth function in the plate-shaped sample.
2. The evolution control method for the configuration of a superelastic thin plate according to claim 1, wherein the first type basic quantities { E, F, G } and the second type basic quantities { L, M, M } are obtained by the following formula:
E＝r，X·r，X，F＝r，X·r，Y，G＝r，Y·r，Y，
L＝r，XX·n，M＝r，XY·n，N＝r，YY·n，
wherein n is r, and r is,X×r，Y/|r，X×r，Yi is a unit normal vector of the curved surface; r is the sum of the total number of the carbon atoms,Xthe first derivative to X, r,Yis the first derivative to Y, r,XYthe first derivative to X, Y, r,XXis the second derivative to X, r,YYthe second derivative to Y.
3. The evolution control method for controlling the configuration of a superelastic thin plate based on growth deformation according to claim 2, wherein the step of judging whether the coordinate curves form orthogonal curvature net according to the first basic quantity and the second basic quantity comprises:
judging whether the condition F is equal to M and is equal to 0, and if so, judging that an orthogonal curvature net is formed; otherwise, it is determined that the orthogonal curvature net cannot be formed.
4. The evolution control method for the configuration of the superelastic thin plate based on growth deformation according to claim 1, wherein the performing independent variable transformation on the parameter equation of the target curved surface comprises:
establishing a mapping from { S, T } to { X, Y } including transforming X and Y as follows:
X＝X(S，T)，Y＝Y(S，T)，
wherein Jacobi determinant of mapping from { S, T } to { X, Y }
Figure FDA0003346488420000011
Establishing a { X, Y } plane definition domain omegarAnd { S, T } plane domain
Figure FDA0003346488420000012
Bijective relationship of (1), object curved surface
Figure FDA0003346488420000016
The new parameter equation is r*(S, T) ═ r (X (S, T), Y (S, T)), and the curved surface r is calculated*Natural coordinate base of (S, T)
Figure FDA0003346488420000013
Figure FDA0003346488420000014
Figure FDA0003346488420000015
Wherein Θ is1And Θ2Are respectively the included angles between the two main directions and the X axis;
Figure FDA0003346488420000021
Figure FDA0003346488420000022
obtaining expressions of S (X, Y) and T (X, Y), and obtaining the expression according to the target curved surface
Figure FDA0003346488420000024
Equation of parameters r*(S, T), and recalculating the first type basic quantity { E, F, G } and the second type basic quantity { L, M, N } of the target curved surface.
5. The evolution control method for the configuration of a superelastic thin plate based on growth deformation according to claim 2, wherein the growth function is obtained by calculating according to the following formula:
Figure FDA0003346488420000023
wherein λ is1And λ2Respectively radial and circumferential growth functions, and z is a thickness direction coordinate of the plate.
6. The evolution control method for configuration of a superelastic thin plate according to claim 1, further comprising the following steps before step S1:
determining the initial configuration, growth conditions and elastic constitutive relation of the super-elastic sheet sample.
7. The evolution control method for configuration of a superelastic thin plate according to claim 1, further comprising the following steps after step S4:
and verifying the effectiveness and the precision of the obtained growth function by adopting a finite element simulation method.
8. A system for controlling evolution of configuration of a superelastic thin plate based on growth deformation, comprising:
the basic quantity calculating module is used for calculating a first basic quantity and a second basic quantity of the target curved surface according to a parameter equation of the target curved surface;
the conversion judging module is used for judging whether the coordinate curve forms an orthogonal curvature net according to the first type basic quantity and the second type basic quantity, and if so, the conversion judging module is switched to the function determining module; if not, turning to a basic quantity adjusting module;
the basic quantity adjusting module is used for carrying out independent variable transformation on a parameter equation of the target curved surface so that a new coordinate curve forms an orthogonal curvature net on the curved surface;
a function determination module for determining an initial planar configuration of the plate-like sample and a distribution of the growth function inside the plate-like sample from the first basis quantity and the second basis quantity.
9. A system for controlling evolution of configuration of a superelastic thin plate based on growth deformation, comprising:
at least one processor;
at least one memory for storing at least one program;
when executed by the at least one processor, cause the at least one processor to implement the method of any one of claims 1-7.
10. A computer-readable storage medium, in which a program executable by a processor is stored, wherein the program executable by the processor is adapted to perform the method according to any one of claims 1 to 7 when executed by the processor.
CN202111324510.0A 2021-11-10 2021-11-10 Method, system and medium for controlling configuration evolution of superelastic thin plate based on growth deformation Pending CN114239334A (en)

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