CN112487645A - Energy modeling method and device for unified isotropic and anisotropic virtual materials - Google Patents

Energy modeling method and device for unified isotropic and anisotropic virtual materials Download PDF

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CN112487645A
CN112487645A CN202011384893.6A CN202011384893A CN112487645A CN 112487645 A CN112487645 A CN 112487645A CN 202011384893 A CN202011384893 A CN 202011384893A CN 112487645 A CN112487645 A CN 112487645A
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刘学慧
何浩
吴笛
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Abstract

The invention belongs to the field of computer graphics, and relates to a method and a device for energy modeling of a unified isotropic and anisotropic virtual material. Under the energy definition of the present invention, the present invention presents a set of directions and functions in which the energy function defined by the present invention can be not only isotropic, but also materials that encompass both linearity and nonlinearity. The invention aims to provide an intuitive isotropic and anisotropic unified modeling method, and enables a model to cover a wider material range. The method can not only provide the selection of the direction and the definition of the stretching energy in the corresponding direction for the user, but also provide the selection of the action among the directions of the user, thereby realizing the modeling of the elastic materials with different elastic moduli.

Description

Energy modeling method and device for unified isotropic and anisotropic virtual materials
Technical Field
The invention belongs to the technical field of computer graphics, and relates to material modeling for a dynamic object, in particular to an energy modeling method and device for isotropic and anisotropic virtual materials of the dynamic object.
Background
Deformation body deformation simulation is commonly used in technologies such as movie special effects, simulation, virtual reality and the like. The super-elastic material is one of the common materials, is widely applied to various industries, and especially can obtain distinctive and self-evident deformation effects on the application of representing complex geometric characteristics and different material characteristics. These deformation effects depend to a large extent on the constitutive model of the material, i.e. the functional relationship of stress to strain that is used to describe the deformation material. With the continuous expansion of the simulation field, the research of the constitutive model has become a central subject of multidisciplinary intersection, and has attracted great attention of researchers in mechanics, materials science, physics, applied mathematics and graphics. Common material constitutive models in graphical modeling include co-rotation linear models, st.venant-Kirchhoff models, Neohookean models, and the like. Based on the models, different materials can be designed by adjusting material parameters such as Young modulus and Poisson coefficient, and even more real material parameters can be obtained by acquiring dynamic data of a real object. However, obtaining a variety of deformation effects by merely adjusting material parameters is very limited. 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 requirements of medical and polymer material simulation, the traditional isotropic material model, i.e. the deformation body, has the same elasticity in all directions and is no longer suitable. More object materials in the real world exhibit different elastic characteristics, i.e. anisotropy, in different directions. The overall material space is even larger in view of the anisotropic material.
The world is very strange. To characterize these abundant materials, various researchers have proposed a number of material models. In the traditional model based on continuous medium hypothesis, most elastic materials are regarded as isotropic hyperelastomers, and the strain energy-strain energy density function psi of the hyperelastic materials per unit volume is regarded as the first, second and third invariant I of Green strain1、Ι2、Ι3To describe the deformation of the material under load. Wherein the content of the first and second substances,
I1＝tr(FTF),I2＝tr[(FTF)2],I3＝det(FTF)
f is the deformation gradient. And is provided with
Figure BDA0002809372070000011
Figure BDA0002809372070000012
Figure BDA0002809372070000013
Among these models, the most commonly used model is the Mooney model, the Mooney-Rivlin model, the Hart-Smith model, the Yeoh-Fleming model, the Gent model, and the like. Researchers find that the constant I is used according to the continuous experiment1、Ι2、Ι3The expressed strain energy density function Ψ extends from a linear representation to a polynomial representation to an exponential function, logarithmic function representation, and the like, to obtain a more accurate description of the elastic material properties.
In 1967, Valansi and Landel considered that the conventional invariant strain energy density function was due to
Figure BDA0002809372070000021
And
Figure BDA0002809372070000022
the mutual implication makes the experimental design very difficult, and the correct strain energy density function can not be obtained. They propose a main elongation lambda1、λ2、λ3Expressed in terms of a strain energy function, and the strain energy density function is expressed as a separate mode as follows:
Figure BDA0002809372070000023
the vallisi-Landel model, also known as the vallisi-Landel hypothesis, assumes that the strain energy function is interchangeable for the three principal elongations when the elastic material is isotropic, so that the strain energy function can be expressed as a function of the separation symmetry of the three principal elongations in the above equation, and that this simplifies the theoretical and experimental work of the whole model, since the functions of the different invariants are identical.
Due to the strict mathematical derivation and the more intuitive physical explanation of the vallisi-Landel model, researchers further provide different functional forms based on main elongation on the vallisi-Landel work, such as a high-order polynomial representation of an Ogden model and the like, so as to predict the deformation characteristics of the material more realistically. In 2015, the graphical investigator Xu et al further extended the strain energy density function based on the amount of main elongation to a functional form of the following pattern:
ψ(λ1,λ2,λ3)＝f(λ1)+f(λ2)+f(λ3)+g(λ1λ2)+g(λ1λ3)+g(λ2λ3)+h(λ1λ2λ3)，
and provides an interface for the user to fit the data with a spline function. Since Valansi-Landel assumes that it is no longer true in the case of material anisotropy, Xu et al add a orthotropic term defined in one direction to the above model to simulate orthotropic materials. In the study of anisotropy models expressed in principal elongations, Shariff made a lot of work. In 2016, Shariff ascribed its constitutive model representing anisotropy as a function of principal elongation and corresponding principal direction. To handle the characteristics of a particular direction a, Shariff introduces the 4 th, 5 th invariants:
Figure BDA0002809372070000024
and giving a constitutive model containing principal elongations, anisotropy invariants as follows:
Figure BDA0002809372070000025
while the work of Shariff combines constitutive models based on invariant and principal elongations, in practice its anisotropic term must be much smaller than the isotropic term in model design, which is also mentioned in the work of graphical researchers Xu et al. Another worth mentioning work in the anisotropic modeling work was the direction-based modeling method proposed by Diani in 2004. Diani proposes a direction-based anisotropic, isotropic model based on tetrahedral, hexahedral, octahedral, etc. structures, based on various molecular structure models in rubber materials. The orientation of the model is defined from the center of the polyhedron to the vertex of the polyhedron [ Diani2004] (Diani, J., Brieu, M., Vacherand, J.M., Rezgui, A.,2004. directive model for isotropic and isotropic rubber-like materials. Material.36, 313-321.). But as noted in its work, the constitutive model that defines a material based on tensile energy in a direction is actually an anisotropic model. To obtain a directionally-based isotropic constitutive model, Diani et al give a set of models that they consider to be closest to isotropy from the fitting effect of the constitutive model based on the directions of the polyhedrons, but theoretically they propose an anisotropic model.
Disclosure of Invention
In order to solve the problems of uniform editing and modeling of isotropic and anisotropic materials in computer simulation, the invention designs a modeling method of a direction-based constitutive model. The constitutive model defined by the invention not only constructs a function of the elongation of the material in the direction, but also constructs a function of the interaction of the material between the directions. The user may select the direction himself (see fig. 1) or may define the material according to a set of directions given by the invention. Not only can isotropic, anisotropic linear and nonlinear functions be defined in a given set of material directions, but also the poisson's ratio of the model can be adjusted.
The invention provides a method for modeling a virtual material based on isotropy and anisotropy of directions, but different from the existing polyhedral model as the direction selection in the prior method, the direction selection of the invention is selected by a user according to the required material, and the constitutive model of the material is not only a function of the direction elongation but also a function of the direction interaction. From the stiffness calculation in the direction, the present invention theoretically gives a set of directions in which not only linear and non-linear models of isotropic materials can be defined, but also the poisson's ratio of the material can be defined.
In order to realize the purpose of the invention, the technical scheme adopted by the invention is as follows:
a unified energy modeling method for isotropic and anisotropic virtual materials is used for defining a material constitutive model psi from a group of directions xikTensile energy ofTractionAnd the angle of rotation psiRotationThe method comprises the following steps:
calculating an elasticity matrix corresponding to the linear material according to a user-defined or pre-given direction to determine the influence of the selected direction on the defined material;
defining the stretching energy function of the selected direction and the weight thereof according to the required material, thereby obtaining the stretching energy psi corresponding to the current directionTraction；
According to the material requirement of a user, whether a corner energy function and the weight thereof in the current direction are added or not are considered, and the corner energy psi corresponding to the current direction is obtainedRotation；
By stretching energy ΨTractionAnd the angle of rotation psiRotationA user-defined material constitutive model Ψ is constructed.
Further, Ψ is defined as follows:
Ψ＝ΨTraction+ΨRotation (1)
wherein the stretching energy ΨTractionThe calculation formula of (2) is as follows:
Figure BDA0002809372070000041
KTkis the direction xi of the materialkWeight of upper stretching energy.
Figure BDA0002809372070000042
n is the user-defined number of material directions and F is the deformation gradient.
Figure BDA0002809372070000043
A function of the stretch energy in that direction selected for the user.
Figure BDA0002809372070000044
Is the direction xi of material deformationkThe amount of stretch of (a).
Wherein the angle of rotation energy ΨRotationThe calculation formula of (2) is as follows:
Figure BDA0002809372070000045
KRkis the direction xi of the materialkThe weight of the energy of the upper rotation angle,
Figure BDA0002809372070000046
as a function of the rotational angle energy.
Figure BDA0002809372070000047
Figure BDA0002809372070000048
Is direction xikIs expressed as
Figure BDA0002809372070000049
θkAnd phikSee FIG. 1, definition of θkDirection of presentation xikAt X1 X3Projection on plane and coordinate axis X1Angle of (d), phikDirection of presentation xikAnd the coordinate axis X2The angle of (d). Eta 'of'k,η”k,η”'kIs formed by direction xikAnd world coordinate axis xiThree local coordinate axes (x-axis, y-axis and z-axis) based on material direction,
Figure BDA00028093720700000410
the method comprises the following steps:
Figure BDA00028093720700000411
psi under the constitutive model defined in the present inventionTractionIn the following set of directions mi,1≤i≤9}：
Figure BDA0002809372070000051
And in the weight
Figure BDA0002809372070000052
Functions f, g, h:
Figure BDA0002809372070000053
is isotropic. Thus, the direction { m } can be defined by a combination of f, g, h functionsiAnd i is more than or equal to 1 and less than or equal to 9), including basic mathematical calculation (four arithmetic operations), polynomial combination, logarithm calculation and the like, so as to construct isotropic linear and nonlinear constitutive models of the material.
By utilizing the group of directions, the tensile energy weight KT and the corner energy weight KR in the direction can be calculated by utilizing the following formulas according to the Poisson ratio v and the elastic modulus E defined by the user, and the constitutive model meeting the user requirements is assembled.
Figure BDA0002809372070000054
Figure BDA0002809372070000055
In the present invention, the constitutive model of the material in any direction is anisotropic from the beginning, but passes through the direction { m }iAnd i is more than or equal to 1 and less than or equal to 9, and the definition of the functions f, g and h can ensure that the defined constitutive model is isotropic and can be linear or nonlinear.
In the invention, the Poisson ratio of the material is adjusted by superposing corner energy so as to cover more widely applied materials.
Based on the same inventive concept, the present invention also provides an energy modeling apparatus for a unified isotropic, anisotropic virtual material, which is an electronic apparatus (computer, server, smartphone, etc.) comprising a memory and a processor, the memory storing a computer program configured to be executed by the processor, the computer program comprising instructions for performing the steps of the method of the present invention.
The invention has the beneficial effects that:
the invention aims to provide a modeling method of a unified isotropic, anisotropic linear and nonlinear constitutive model of a virtual material, and provides a more convenient modeling tool for simulation, animation production and the like. The method provided by the invention unifies isotropic, anisotropic linear and nonlinear models into one model, the model breaks away from the limitation of the traditional model based on strain invariants or main elongations, and the modeling of the model and the modification of model parameters are achieved through the stretching in the direction and the action between the directions. The model is simple and intuitive, and is easy to simulate and design animation. The model of the invention can ensure the symmetry of the rigidity matrix calculation and ensure the stability and robustness of the whole simulation and animation design.
Drawings
FIG. 1 is a design of the present invention.
FIG. 2 is a design flow of the orientation-based constitutive model of the present invention.
Fig. 3 is a deformation diagram under forward load and a volume change diagram of a conventional Neo-Hookean material and isotropic and anisotropic materials constructed using the method of the present invention.
FIG. 4 is a graph of the different deformations produced by different Poisson's ratio models constructed using the method of the present invention at the same stretch length.
FIG. 5 is a deformation diagram of a thin hollow tube of orthotropic material constructed using the method of the present invention.
FIG. 6 is a deformation diagram of linear and non-linear models constructed using the method of the present invention.
Detailed Description
The present invention will be described in detail below with reference to specific embodiments and accompanying drawings.
The modeling method of the constitutive model of the superelasticity material based on the direction provided by the invention simultaneously considers the stretching in the defined direction and the energy acting between the directions, and establishes a unified modeling method of linear and nonlinear virtual materials with isotropy and anisotropy. The model can provide not only anisotropic direction selection, but also different selection of the energy function in the direction.
FIG. 1 is a directional design of the present invention. Wherein, X1、X2、X3Representing world coordinate axis, thetakDirection of presentation xikAt X1 X3Projection on plane and coordinate axis X1Angle of (d), phikDirection of presentation xikAnd the coordinate axis X2The angle of (d). XikIndicating the selected direction.
Referring to fig. 2, the modeling method provided by the present invention includes the following steps:
1) the direction is selected according to whether the desired material is isotropic or anisotropic
Figure BDA0002809372070000061
Wherein
Figure BDA0002809372070000062
Direction of presentation xikIn the world coordinate systemThe following coordinate components, or formula (6) of the present invention may be selected to give a set of directions.
2) Assuming that the material is in a selected direction xikThe tensile measurement of
Figure BDA0002809372070000063
A linear material of (2). The elasticity matrix is calculated according to equation (9):
Figure BDA0002809372070000071
where Ψ is the material constitutive model defined in the present invention, ∈ij、εmnRepresenting the components of the Greens strain tensor.
Further obtain the current direction xikThe influence of the energy function of (a) on the elastic matrix of the material is as follows:
Figure BDA0002809372070000072
wherein:
Figure BDA0002809372070000073
Aijthe ith row and jth element of the matrix A, AmnRepresenting the nth element of the mth row of matrix a.
According to the formula, the user can determine the current selection direction xikWhether further modification is required or whether more directions need to be added to decide whether to go back to step 1) or to step 3). The elastic matrix reflects the properties of the linear elastic material, which generally has the following form:
Figure BDA0002809372070000074
and for isotropic materials, it can be further specified as:
Figure BDA0002809372070000075
where E is the Young's modulus and ν is the Poisson's ratio. Specifically, the influence of the set direction on the elasticity matrix (e.g., (11) or (12)) of the desired material can be derived from equation (10) based on the currently set direction, and whether the currently set direction needs to be added or modified is determined by the integration of equations (10) for all set directions.
3) Setting the current direction xi according to the isotropic or anisotropic requirement of the user materialkStretching energy function of
Figure BDA0002809372070000081
And weight KT thereofk. Current direction xikThe stretching energy function of (3) may be selected as a combination of the weights given by equation (7) in the present invention. From which the current direction xi is obtainedkThe corresponding stretching energy is as follows:
Figure BDA0002809372070000082
changing the weight in a certain direction is equivalent to adjusting the stiffness (or elastic modulus) in a certain direction. Different weights are configured in different directions, so that the mechanical properties of the material in different directions are different, and different material constitutive is constructed.
For example, under the weight given by equation (7) of the present invention, the linear stretching energy has the following elastic matrix under the set of directions given by equation (6):
Figure BDA0002809372070000083
this is a linear isotropic material. Wherein, dimensionless Young's elastic modulus E is 5/6, shear modulus G is 25/12, Poisson's ratio v is 1/4, and isotropy is satisfied. But the linear stretching energy in the same direction is weighted as follows
Figure BDA0002809372070000084
Next, different elastic matrices are obtained:
Figure BDA0002809372070000085
it can be seen that (15) is a linear material different from that shown in equation (14). Due to the material
Figure BDA0002809372070000091
Isotropy is not satisfied, which is an elastic matrix of anisotropic material. Not only the choice of direction, but also the weight of the direction will influence the material built.
Based on the above calculations, the user can determine whether the selected direction needs further modification or addition or deletion to decide whether to go back to step 1) or to step 4).
4) According to the material requirement of a user, whether a corner energy function of the current direction is added or not is considered
Figure BDA0002809372070000092
And its weight KRkObtaining the current direction xikThe corresponding corner energy is as follows:
Figure BDA0002809372070000093
also, the influence D of the corner energy on the elastic matrix of the linear material is calculatedijmnThe user can decide whether the setting of the rotational energy of the current direction is appropriate, wherein
Figure BDA0002809372070000094
Adding a corresponding rotation energy of equal weight to the direction to (11), i.e.
Figure BDA0002809372070000095
Next, the following elastic matrix is obtained:
Figure BDA0002809372070000096
it can be seen that (14) is an elastic matrix of isotropic material different from (11). Wherein E is 187/45, G is 11/6, and ν is 2/15, and isotropy is satisfied. The Poisson ratio v can be adjusted by constructing a superposition model of stretching and corner energy.
Turning to step 1) or entering step 5) according to whether the user needs to increase or decrease the rotation of the direction.
5) Obtaining a total energy function of the constitutive model of the user-defined material according to all n directions selected by the user:
Ψ＝ΨTraction+ΨRotation (1)
wherein:
Figure BDA0002809372070000101
is the sum of the selected stretching energies in all directions.
Figure BDA0002809372070000102
Is the sum of the directions chosen to take into account their corner energies.
By setting the boundary conditions of the simulation environment and applying the load, according to the structure of the material model set by the user, the unit stiffness matrix of the simulated object is obtained as follows:
Figure BDA0002809372070000103
the dynamic change of the simulation can be calculated by introducing a balance equation, and the whole modeling process is finished.
Fig. 3 shows deformation diagrams under forward load and volume change diagrams of the conventional Neo-Hookean material and the isotropic and anisotropic materials constructed using the direction-based material energy construction method of the present invention and the nine directions given in equation (6). Wherein 9-ax-iso represents an isotropic material constructed using the nine directions given in (6); 9-ax-aniso denotes an anisotropic material built with the same nine directions. The forward load is firstly given to the model in the simulation process, and then the load is removed. The right side corresponds to a screen shot in the simulation process, and the left side is a volume change diagram of the whole simulation process. It can be seen from the figure that the deformation model can finally return to the original shape as the traditional model, and the change of the whole simulation process volume is uniform, no matter the isotropy or the anisotropy of the construction is carried out. The volume change of the anisotropic material is slightly jittered, which also conforms to the anisotropic nature thereof. In addition, anisotropic materials also exhibit positive and shear changes that occur under positive loading.
Fig. 4 shows the different deformations generated by the same stretching length of the different poisson ratio models constructed by the direction-based material energy construction method of the present invention and the nine directions given in equation (6).
Fig. 5 shows a hollow thin tube deformation diagram of orthotropic material constructed using the direction-based material energy construction method of the present invention and the nine directions given in equation (6). Wherein the first column shows a view of the right-hand pulling force exerted by the model on the middle, the second column gives a schematic directional diagram of the elastic strength (denoted by "100 x stiffer") of the model in the longitudinal, radial and tangential directions, respectively, giving the model a factor of 100 in the other directions, and the third to fifth columns show a view of the middle section of the model during the simulation of the respective model. Since they impart elastic strength in multiples of 100 in the other directions, respectively, in the longitudinal, radial and tangential directions, different models produce different effects when a rightward pulling force is imparted to the middle. Wherein, for the tangentially reinforced model, because the tangential stretching of the model requires a larger load, the model can be more consistent with the circular wall in the whole simulation process; for the longitudinal model, because the longitudinal stretching needs larger load, the model shows smaller deformation; and for the radial model, the whole model shows the characteristic of keeping the original wall thickness.
Fig. 6 shows a deformation diagram of linear and nonlinear models constructed using the direction-based material energy construction method of the present invention and the nine directions given in equation (6). Because the method can construct a nonlinear model, huge internal force generated by continuous compression of the model under load, namely the model of the incompressible material can be provided. Both models in this figure give a continuous downward pull on the mandible. The right non-linear model will show more deformation details and normal deformation effects due to the built non-compressible material; whereas the linear model of the left figure, by itself, punctures itself due to the inability to produce the normal incompressible property.
Based on the same inventive concept, another embodiment of the present invention provides an energy modeling apparatus for a unified isotropic, anisotropic virtual material, which is an electronic apparatus (computer, server, smartphone, etc.) comprising a memory and a processor, the memory storing a computer program configured to be executed by the processor, the computer program comprising instructions for performing the steps of the inventive method.
Based on the same inventive concept, another embodiment of the present invention provides a computer-readable storage medium (e.g., ROM/RAM, magnetic disk, optical disk) storing a computer program, which when executed by a computer, performs the steps of the inventive method.
The foregoing disclosure of the specific embodiments of the present invention and the accompanying drawings is directed to an understanding of the present invention and its implementation, and it will be appreciated by those skilled in the art that various alternatives, modifications, and variations may be made without departing from the spirit and scope of the invention. The present invention should not be limited to the disclosure of the embodiments and drawings in the specification, and the scope of the present invention is defined by the scope of the claims.

Claims (10)

1. A unified isotropic, anisotropic virtual material energy modeling method is characterized by comprising the following steps:
calculating an elasticity matrix corresponding to the linear material according to a user-defined or pre-given direction to determine the influence of the selected direction on the defined material;
defining the stretching energy function of the selected direction and the weight thereof according to the required material, thereby obtaining the stretching energy psi corresponding to the current directionTraction；
Adding a corner energy function and weight thereof in the current direction according to the material requirement of a user to obtain the corner energy psi corresponding to the current directionRotation；
By stretching energy ΨTractionAnd the angle of rotation psiRotationA user-defined material constitutive model Ψ is constructed.
2. The method of claim 1, wherein Ψ, ΨTractionAnd ΨRotationThe calculation formula of (2) is as follows:
Ψ＝ΨTraction+ΨRotation，
Figure FDA0002809372060000011
Figure FDA0002809372060000012
Figure FDA0002809372060000013
wherein the content of the first and second substances,
Figure FDA0002809372060000014
for a selected direction xikA stretching energy function of (a); KT (karat)kIs the selected direction xikWeight of upper stretching energy; n is the number of material directions defined by the user; KR (Kr)kIs the selected direction xikUpper rotary energy rightWeighing;
Figure FDA0002809372060000015
f is the deformation gradient as a function of rotational energy;
Figure FDA0002809372060000016
is the direction xi of material deformationkThe amount of stretch of (a).
3. The method of claim 2, wherein the isotropic, anisotropic linear and nonlinear constitutive models of the desired material are defined by the selection of the orientation.
4. A method according to claim 2 or 3, wherein said predetermined direction is { m }iAnd i is more than or equal to 1 and less than or equal to 9, comprising:
Figure FDA0002809372060000017
5. a method according to claim 3, characterized in that for isotropic materials the direction { m } is defined according to the functions f, g, hiI is more than or equal to 1 and less than or equal to 9, and the related energy function is the combination of functions f, g and h and comprises the basic four arithmetic operation, polynomial combination and logarithmic calculation; wherein the calculation formula of f, g and h is as follows:
Figure FDA0002809372060000021
6. the method of claim 5, wherein the constitutive model of the material in any direction, though anisotropic from the beginning, passes through the direction { m }iI is more than or equal to 1 and less than or equal to 9, and the definition of the functions f, g and h can ensure that the defined constitutive model is isotropic and can be linear,can also be non-linear.
7. The method of claim 6, wherein the poisson's ratio of the material is adjusted by superposition of corner energies to cover a wider range of materials.
8. The method of claim 1, wherein the directional stretching energy weight KT and the corner energy weight KR are calculated according to the poisson's ratio v and the elastic modulus E defined by the user by using the following formulas, and then the constitutive model meeting the user requirement is assembled:
Figure FDA0002809372060000022
9. an apparatus for energy modeling of a unified isotropic, anisotropic virtual material, comprising a memory and a processor, the memory storing a computer program configured to be executed by the processor, the computer program comprising instructions for performing the method of any of claims 1-8.
10. A computer-readable storage medium, characterized in that the computer-readable storage medium stores a computer program which, when executed by a computer, implements the method of any one of claims 1 to 8.
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