WO2022116372A1

WO2022116372A1 - Method and apparatus for unified isotropic and anisotropic virtual material energy modelling

Info

Publication number: WO2022116372A1
Application number: PCT/CN2021/072432
Authority: WO
Inventors: 刘学慧; 何浩; 吴笛
Original assignee: 中国科学院软件研究所
Priority date: 2020-12-01
Filing date: 2021-01-18
Publication date: 2022-06-09
Also published as: CN112487645A; CN112487645B

Abstract

A method and an apparatus for unified isotropic and anisotropic virtual material energy modelling. In the definition of energy, a set of directions and functions are given in which the defined energy function may not only be isotropic but the material also covers linear and non-linear. The provided intuitive isotropic and anisotropic unified modelling method enables models to cover a wider range of materials. The method provides the user with the choice of direction and the definition of tensile energy in the corresponding direction, and provides the user with the choice of interaction between directions, thereby implementing the modelling of elastic materials with different elastic moduli.

Description

A unified energy modeling method and device for isotropic and anisotropic virtual materials

technical field

The invention belongs to the technical field of computer graphics, relates to how to model materials of dynamic objects, and in particular relates to a method and device for energy modeling of isotropic and anisotropic virtual materials of dynamic objects.

Background technique

Deformation body deformation simulation is commonly used in film special effects, simulation, virtual reality and other technologies. As one of the commonly used materials, superelastic materials are widely used in all walks of life, especially in the application of complex geometric features and different material properties, which can often obtain distinctive and indescribable deformation effects. These deformation effects are largely dependent on the constitutive model of the material, which is the function of stress and strain used to describe the deformable material. With the continuous expansion of the field of simulation, the study of constitutive model has become a multidisciplinary central subject, which has attracted great attention from researchers in mechanics, materials science, physics, applied mathematics, and graphics. Common material constitutive models in graphics modeling include co-rotational linear model, St.Venant-Kirchhoff model, Neohookean model, etc. Based on these models, different materials can be designed by adjusting material parameters such as Young's modulus and Poisson coefficient, and even more realistic material parameters can be obtained by obtaining dynamic data of the object. However, it is very limited to obtain various deformation effects only by adjusting the material parameters. In fact, the common standard constitutive model is only a small subset of the entire material space. With the continuous expansion of virtual reality applications, especially the demand for simulation of medical and polymer materials, the traditional isotropic material model, that is, the elasticity of the deformable body in all directions, is no longer suitable. More object materials in the real world exhibit different elastic characteristics in different directions, that is, anisotropy. Considering anisotropic materials, the entire material space is even more huge.

The world is full of strangeness. In order to describe the properties of these rich materials, researchers from all walks of life have proposed many material models. In the traditional models based on the continuum assumption, most of the elastic materials are regarded as isotropic hyperelastic bodies, and the strain energy-strain energy density function Ψ per unit volume of the hyperelastic materials is regarded as the first, The functions of the second and third invariants Ι ₁ , Ι ₂ , and Ι ₃ to describe the deformation of the material under load. in,

I ₁ =tr(F ^T F),I ₂ =tr[(F ^T F) ² ],I ₃ =det(F ^T F)

F is the deformation gradient. and have

Among these models, the most commonly used models are Mooney model, Mooney-Rivlin model, Hart-Smith model, Yeoh-Fleming model, Gent model and so on. According to the continuous discovery of experiments, the researchers extended the strain energy density function Ψ represented by invariants Ι ₁ , Ι ₂ , Ι ₃ from linear to polynomial, to exponential, logarithmic, etc., to obtain A more precise description of elastic material properties.

In 1967, Valansi and Landel believed that the previous strain energy density function expressed in invariants was due to

and

The mutual implication makes the experimental design very difficult to obtain the correct strain energy density function. They proposed a strain energy function in the form of principal elongations λ ₁ , λ ₂ , λ ₃ , and expressed the strain energy density function as the following separation mode:

The Valansi-Landel model, also known as the Valansi-Landel hypothesis, considers that when the elastic material is isotropic, the strain energy function is interchangeable for the three principal elongations, so that the strain energy function can be expressed as in the above formula Separate symmetrical functions of the three principal elongations, and since the functions of the different invariants are the same, this will simplify the theoretical and experimental work of the entire model.

Since the Valansi-Landel model has strict mathematical derivation and a more intuitive physical explanation, researchers have further given different functional forms based on the principal elongation based on the work of Valansi-Landel, such as the high-order polynomial representation of the Ogden model, etc. , to more realistically predict the deformation characteristics of the material. In 2015, the graphics researcher Xu et al. further expanded the strain energy density function based on the principal elongation into the functional form of the following modes:

ψ(λ ₁ ,λ ₂ ,λ ₃ )=f(λ ₁ )+f(λ ₂ )+f(λ ₃ )+g(λ ₁ λ ₂ )+g(λ ₁ λ ₃ )+g(λ ₂ λ ₃ )+h(λ ₁ λ ₂ λ ₃ ),

And provides an interface for users to fit data with spline functions. Because the Valansi-Landel assumption no longer holds in the case of material anisotropy, Xu et al. added a direction-defined orthotropic term to the above model to simulate orthotropic materials. Shariff has done a lot of work in the study of the anisotropy model in terms of principal elongation. In his 2016 work, Shariff reduced his constitutive model for representing anisotropy to a function of the principal elongation and the corresponding principal direction. In order to deal with the characteristics of a specific direction a, Shariff introduces the fourth and fifth invariants:

And the following constitutive model including principal elongation and anisotropy invariants is given:

Although Shariff's work incorporates constitutive models based on invariants and principal elongations, in practice the anisotropic term must be much smaller than the isotropic term in model design, which is reported by graphics researchers Xu et al. Also mentioned at work. In the modeling work of anisotropy, another work worth mentioning is the direction-based modeling method proposed by Diani in 2004. According to various molecular structure models in rubber materials, Diani proposed direction-based anisotropy and isotropy models based on tetrahedron, hexahedron, octahedron and other structures. The direction of the model is defined by the center of the polyhedron to the vertex of the polyhedron [Diani2004] (Diani, J., Brieu, M., Vacherand, J.M., Rezgui, A., 2004. Directional model for isotropic and anisotropic rubber-like materials. Mech.Mater .36, 313–321.). But as pointed out in his work, the constitutive model that defines the material based on the tensile energy in the direction is actually an anisotropic model. In order to obtain the isotropic constitutive model based on the direction, Diani et al. gave a set of models that they thought was the closest to the isotropy from the fitting effect of the constitutive model based on the direction of each polyhedron, but theoretically they What is proposed is still an anisotropic model.

SUMMARY OF THE INVENTION

In order to solve the problem of unified editing and modeling of isotropic and anisotropic materials in computer simulation, the present invention designs a modeling method of orientation-based constitutive model. The constitutive model defined in the present invention not only constructs the function of the material elongation in the direction, but also constructs the function of the material interaction between the directions. The user can choose the direction by himself (see Figure 1), or can define the material according to a set of directions given by the present invention. In the given set of material directions, not only isotropic and anisotropic linear and nonlinear functions can be defined, but also the Poisson's ratio of the model can be adjusted.

The present invention proposes a method for modeling a virtual material that is isotropic and anisotropic in direction, but is different from the existing method in which the existing polyhedron model is used as the direction selection. The direction selection of the present invention is determined by the user according to the The material required is selected, and the constitutive model of the material is not only a function of the directional elongation, but also a function of the interaction between the directions. According to the stiffness calculation in the direction, the present invention theoretically provides a set of directions, in which not only the linear model and the nonlinear model of the isotropic material can be defined, but also the Poisson's ratio of the material can be defined.

For realizing the purpose of the present invention, the technical scheme adopted in the present invention is as follows:

A unified energy modeling method for isotropic and anisotropic virtual materials, it is defined that the material constitutive model Ψ consists of a set of tensile energy Ψ _Traction and rotation energy Ψ _Rotation in the direction ξ _k . Steps include:

Calculate the elastic matrix of the corresponding linear material according to the user-defined or pre-given directions to determine the influence of the selected direction on the defined material;

Define the tensile energy function and its weight of the selected direction according to the required material, thereby obtaining the tensile energy Ψ _Traction corresponding to the current direction;

According to the user's material requirements, consider whether to add the rotation energy function of the current direction and its weight to obtain the rotation energy Ψ _Rotation corresponding to the current direction;

The user-defined material constitutive model Ψ is composed of the tensile energy Ψ _Traction and the rotational energy Ψ _Rotation .

Further, Ψ is defined as follows:

Ψ=Ψ _Traction + Ψ _Rotation (1)

Among them, the calculation formula of stretching energy Ψ _Traction is:

KT _k is the weight of the stretch energy of the material in the direction ξ _k .

n is the number of user-defined material directions, and F is the deformation gradient.

The stretch energy function in this direction selected by the user.

is the stretch measure of material deformation in the direction ξ _k .

Among them, the calculation formula of the corner energy Ψ _Rotation is:

KR _k is the weight of the corner energy of the material in the direction ξ _k ,

is the corner energy function.

is the normalized mode of the direction ξ _k , expressed as

The definitions of θ _k and φ _k are shown in Figure 1. θ _k represents the angle between the projection of the direction ξ _k on the X ₁ X ₃ plane and the coordinate axis X ₁ , and φ _k represents the angle between the direction ξ _k and the coordinate axis X ₂ . η' _k , η″ _k , η″′ _k are three local coordinate axes based on material orientation constructed by the direction ξ _k and the world coordinate axes _xi (x, y and z axes),

i=1,2,3, respectively:

Under the constitutive model defined in the present invention, Ψ _Traction is in the following set of directions {m _i , 1≤i≤9}:

and in the weight

Below, the functions f, g, h:

is isotropic. Therefore, the relevant energy function in the direction {m _i , 1≤i≤9} can be defined by the combination of f, g and h functions, including basic mathematical calculations (four operations), polynomial combinations, logarithms and other calculations, so as to construct Isotropic linear and nonlinear constitutive models of materials.

Using this set of directions, according to the user-defined Poisson's ratio ν and elastic modulus E, the following formulas can be used to calculate the tensile energy weight KT and the corner energy weight KR in the direction, and assemble a constitutive model that meets the user's requirements.

In the present invention, although the material constitutive model in any direction is anisotropic from the beginning, the superposition of the directions {m _i , 1≤i≤9} and the definition of the functions f, g, h can guarantee the defined The constitutive model of is isotropic and can be either linear or nonlinear.

In the present invention, the Poisson's ratio of the material is adjusted by the superposition of the corner energy, so as to cover a wider range of applied materials.

Based on the same inventive concept, the present invention also provides a unified isotropic and anisotropic virtual material energy modeling device, which is an electronic device (computer, server, smart phone, etc.), which includes a memory and a processor, The memory stores a computer program configured to be executed by the processor, the computer program comprising instructions for performing the steps in the method of the present invention.

Beneficial effects of the present invention:

The purpose of the present invention is to provide a unified isotropic, anisotropic linear and nonlinear constitutive model modeling method of virtual material, and provide a more convenient modeling tool for simulation simulation, animation production and the like. The method proposed by the present invention unifies the isotropic and anisotropic linear and nonlinear models in one model, which breaks away from the limitation of the traditional model based on strain invariants or principal elongation, through the stretching in the direction And the interaction between the directions achieves the modeling of the model and the modification of the model parameters. This model is simple and intuitive, easy to simulate, animation design. The model of the invention can ensure the symmetry of the stiffness matrix calculation, and ensure the stability and robustness of the entire simulation and animation design.

Description of drawings

Fig. 1 is the direction design of the present invention.

FIG. 2 is the design flow of the orientation-based constitutive model of the present invention.

3 is a deformation diagram and a volume change diagram of a conventional Neo-Hookean material and an isotropic and anisotropic material constructed by the method of the present invention under normal load.

Fig. 4 is a graph of different deformations produced by different Poisson's ratio models constructed by the method of the present invention under the same stretching length.

Fig. 5 is a deformation diagram of a hollow thin tube of an orthotropic material constructed by the method of the present invention.

Figure 6 is a deformation diagram of the linear and nonlinear models constructed by the method of the present invention.

Detailed ways

The present invention will be described in detail below through specific embodiments and in conjunction with the accompanying drawings.

The direction-based constitutive model modeling method of hyperelastic materials proposed by the present invention simultaneously considers the stretching in the defined direction and the energy acting between directions, and establishes a unified isotropic and anisotropic linear and nonlinear Modeling methods for virtual materials. This model can provide not only anisotropic direction selection, but also different selection of energy functions in the direction.

Fig. 1 is the direction design of the present invention. Among them, X ₁ , X ₂ , and X ₃ represent the world coordinate axes, θ _k represents the angle between the projection of the direction ξ _k on the X ₁ X ₃ plane and the coordinate axis X ₁ , and φ _k represents the direction ξ _k and the coordinate axis X ₂ the included angle. ξ _k represents the selected direction.

Referring to Fig. 2, the modeling method provided by the present invention includes the following steps:

1) Select the direction according to whether the required material is isotropic or anisotropic

in

Represents the coordinate component of the direction ξ _k in the world coordinate system, or formula (6) of the present invention can also be selected to give a set of directions.

2) Assume that the material is stretched in the selected direction ξ _k as

of linear materials. Calculate the elasticity matrix according to formula (9):

Among them, Ψ is the material constitutive model defined in the present invention, and ε _ij and ε _mn represent the components of Green's strain tensor.

Then, the influence of the energy function of the current direction ξ _k on the elastic matrix of the material is obtained as follows:

in:

A _ij represents the j-th element of the i-th row of the matrix A, and A _mn represents the n-th element of the m-th row of the matrix A.

According to the calculation result of the above formula, the user can determine whether the currently selected direction ξ _k needs to be further modified, or whether more directions need to be added, so as to decide whether to go back to step 1) or enter step 3). Elasticity matrices reflect the properties of linear elastic materials, and they usually have the following form:

For isotropic materials, it can be further expressed as:

where E is Young's modulus and ν is Poisson's ratio. Specifically, according to the currently set direction, the influence of the set direction on the elastic matrix (such as (11) or (12)) of the required material can be obtained from the formula (10), and the formula of all the set directions can be obtained by the formula (10). The synthesis of (10) determines whether the current set direction needs to be added or modified.

3) Set the stretching energy function of the current direction ξ _k according to the requirements of the user's material isotropy or anisotropy

and its weight KT _k . The stretching energy function in the current direction ξ _k can choose the combination of the weights given by the formula (7) in the present invention. Thus, the stretching energy corresponding to the current direction ξ _k is obtained:

Changing the weight in a certain direction is equivalent to adjusting the stiffness (or elastic modulus) in a certain direction. Different weights are assigned to different directions, so that the mechanical properties of the material in different directions are different, and different material constitutives are constructed.

For example, under the weight given by the formula (7) of the present invention, the linear stretching energy has the following elastic matrix in the group of directions given by the formula (6):

This is a linear isotropic material. Among them, the dimensionless Young's modulus of elasticity E=5/6, the shear modulus G=25/12, and the Poisson's ratio ν=1/4, which satisfies the isotropy. But the linear stretch energy in the same direction is weighted as follows

, we get different elasticity matrices:

It can be seen that (15) is a different linear material than that shown by equation (14). Due to the material

Not isotropic, this is the elastic matrix of an anisotropic material. So not only the choice of orientation, but also the weight of the orientation will affect the constructed material.

According to the above calculation, the user can determine whether the selected direction needs further modification or addition or deletion, so as to decide whether to go back to step 1) or to go to step 4).

4) According to the user's material requirements, consider whether to add the corner energy function of the current direction

and its weight KR _k to obtain the corner energy corresponding to the current direction ξ _k :

Likewise, by calculating the effect of the rotational energy on the elastic matrix of the linear material, D _ijmn , the user can determine whether the rotational energy setting in the current direction is appropriate, where

Add the rotation energy of the corresponding direction equal weight to (11), namely

, the following elastic matrix is obtained:

It can be seen that (14) is an elastic matrix of an isotropic material different from (11). Among them E=187/45, G=11/6, ν=2/15, satisfy isotropy. The Poisson's ratio ν can be adjusted by constructing a superposition model of the stretching and turning energies.

According to whether the user needs to increase or decrease the direction of rotation, go to step 1) or go to step 5).

5) According to all n directions selected by the user, obtain the total energy function of the constitutive model of the material defined by the user:

Ψ=Ψ _Traction + Ψ _Rotation (1)

in:

is the sum of the stretch energies in all directions selected.

is the sum of the directions chosen considering their corner energy.

By setting the boundary conditions of the simulated environment and applying the load, according to the constitutive structure of the material model set by the user, we obtain the element stiffness matrix of the simulated object as follows:

The dynamic changes of the simulation can be calculated by bringing in the balance equation, and the entire modeling process can be concluded.

Fig. 3 shows the conventional Neo-Hookean materials and isotropic and anisotropic materials constructed with the direction-based material energy construction method of the present invention and the nine directions given in Equation (6) under normal loads. The deformation diagram of and its volume change diagram. 9-ax-iso represents the isotropic material constructed using the nine directions given by (6); 9-ax-aniso represents the anisotropic material constructed using the same nine orientations. During the simulation, the model is first given a positive load, and then the load is removed. The right side corresponds to the screenshot during the simulation process, and the left side is the volume change diagram of the entire simulation process. As can be seen from the figure, whether it is isotropic or anisotropic, the deformation model can finally return to the original state like the traditional model, and the volume change is uniform throughout the simulation process. The volume change of anisotropic materials is slightly jittery, which is also in line with its anisotropic properties. In addition, the anisotropic material also exhibits its forward and shear changes under forward loading.

Fig. 4 shows the different deformations produced by different Poisson's ratio models constructed with the direction-based material energy construction method of the present invention and the nine directions given in formula (6) under the same tensile length.

FIG. 5 shows the deformation diagram of the hollow thin tube of the orthotropic material constructed by the direction-based material energy construction method of the present invention and the nine directions given in the formula (6). Among them, the first column represents the view of the rightward pulling force given by the model to the middle part, and the second column shows the elastic strength that the model gives 100 times of other directions in the longitudinal, radial and tangential directions respectively (with "100x stiffer" ” indicates the direction of the schematic diagram, and the third to fifth columns indicate the middle section view of the model in the simulation process of the respective model. Since they give 100 times the elastic strength of other directions in the longitudinal, radial and tangential directions, respectively, different models have different effects when a rightward pulling force is given to the middle part. Among them, for the tangentially reinforced model, due to its greater load required for tangential stretching, the model maintains its round walls throughout the simulation process; and for the longitudinal model, due to its greater longitudinal stretching requirement For the radial model, the whole model shows the characteristic of maintaining the original wall thickness.

FIG. 6 shows the deformation diagrams of the linear and nonlinear models constructed with the direction-based material energy construction method of the present invention and the nine directions given in formula (6). Since this method can construct a nonlinear model, it can give the model of incompressible material that generates huge internal force as the model is continuously compressed under load. Both models in this image are given a continuous downward pull on the mandible. Due to the constructed incompressible material, the nonlinear model on the right will show more deformation details and normal deformation effects; while the linear model on the left, which cannot produce normal incompressible properties, pierces itself.

Based on the same inventive concept, another embodiment of the present invention provides a unified energy modeling device for isotropic and anisotropic virtual materials, which is an electronic device (computer, server, smart phone, etc.), which includes a memory and a processor, the memory stores a computer program configured to be executed by the processor, the computer program comprising instructions for performing the steps in the method of the present invention.

Based on the same inventive concept, another embodiment of the present invention provides a computer-readable storage medium (eg, ROM/RAM, magnetic disk, optical disk), where the computer-readable storage medium stores a computer program, and when the computer program is executed by a computer , realize each step of the method of the present invention.

The specific embodiments of the present invention disclosed above and the accompanying drawings are intended to help understand the content of the present invention and implement them accordingly. Those skilled in the art can understand that various replacements can be made without departing from the spirit and scope of the present invention. , variations and modifications are possible. The present invention should not be limited to the contents disclosed in the embodiments of the present specification and the accompanying drawings, and the protection scope of the present invention is subject to the scope defined by the claims.

Claims

A unified energy modeling method for isotropic and anisotropic virtual materials, characterized by comprising the following steps:

Calculate the elastic matrix of the corresponding linear material according to the user-defined or pre-given directions to determine the influence of the selected direction on the defined material;

Define the tensile energy function and its weight of the selected direction according to the required material, thereby obtaining the tensile energy Ψ Traction corresponding to the current direction;

According to the user's material requirements, add the rotation energy function of the current direction and its weight to obtain the rotation angle energy Ψ Rotation corresponding to the current direction;

The user-defined material constitutive model Ψ is composed of the tensile energy Ψ Traction and the rotational energy Ψ Rotation .
The method of claim 1, wherein the calculation formula of Ψ, Ψ Traction and Ψ Rotation is:

Ψ = Ψ Traction + Ψ Rotation ,

in,
is the stretching energy function in the selected direction ξ k ; KT k is the weight of the stretching energy in the selected direction ξ k ;

n is the number of user-defined material directions; KR k is the weight of the rotational energy in the selected direction ξ k ;
is the rotation energy function, F is the deformation gradient;
is the stretch measure of material deformation in the direction ξ k .
3. The method of claim 2, wherein isotropic, anisotropic linear and nonlinear constitutive models of the desired material are defined by the selection of directions.
The method according to claim 2 or 3, wherein the predetermined direction is {m i , 1≤i≤9}, comprising:
The method according to claim 3, characterized in that, for isotropic materials, the correlation energy function in the direction {m i , 1≤i≤9} is defined according to the functions f, g, h, and the correlation energy function is a function The combination of f, g, and h functions, including the basic four operations, polynomial combination, logarithmic calculation; among them, the calculation formula of f, g, h is:
The method according to claim 5, characterized in that, although the material constitutive model in any direction is anisotropic from the beginning, but through the superposition of the directions {m i , 1≤i≤9} and the functions f, g The definition of , h can ensure that the defined constitutive model is isotropic, and can be both linear and nonlinear.
7. The method of claim 6, wherein the Poisson's ratio of the material is adjusted by the superposition of the turning energy to cover a wider range of applied materials.
The method according to claim 1, wherein, according to the Poisson's ratio ν and the elastic modulus E defined by the user, the following formula is used to calculate the tensile energy weight KT and the corner energy weight KR in the direction, and then assemble the User-requested constitutive model:
A unified energy modeling device for isotropic and anisotropic virtual materials, characterized by comprising a memory and a processor, wherein the memory stores a computer program, and the computer program is configured to be executed by the processor, The computer program comprises instructions for carrying out the method of any of claims 1-8.
A computer-readable storage medium, characterized in that the computer-readable storage medium stores a computer program, and when the computer program is executed by a computer, the method described in any one of claims 1 to 8 is implemented.