CN108877944B - Virtual cutting method based on grid model including Kelvin viscoelastic model - Google Patents

Virtual cutting method based on grid model including Kelvin viscoelastic model Download PDF

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CN108877944B
CN108877944B CN201810665525.5A CN201810665525A CN108877944B CN 108877944 B CN108877944 B CN 108877944B CN 201810665525 A CN201810665525 A CN 201810665525A CN 108877944 B CN108877944 B CN 108877944B
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张小瑞
占怡
孙伟
刘佳
朱利丰
宋爱国
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Nanjing University of Information Science and Technology
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Abstract

The invention discloses a virtual cutting method of a grid model based on a Kelvin viscoelastic model, which comprises the steps of constructing the Kelvin viscoelastic model, solving a displacement increment in time by using parameters of a virtual cutting material and cutting time, calculating new displacement, strain and stress of each node, generating a simulated cut, incorporating the Kelvin viscoelastic model into a grid, replacing approximate calculation with stress deformation, and solving the problems of grid distortion and continuous cutting in a finite element model and the problems of different deformation forms in a non-grid model. And the calculation amount is greatly reduced, and the performance is optimized. By the model, only the position of the force application time point is needed to be calculated for deformation, and after the force is removed, the grid automatically recovers due to the balance of the force is broken, so that the calculation amount is reduced; and the grid lines with viscoelasticity among the openings are removed aiming at cutting, so that the grid is automatically deformed, and compared with the traditional shape and size similar to the openings, the method is more practical and is simple to operate.

Description

Virtual cutting method based on grid model incorporating Kelvin viscoelastic model
Technical Field
The present invention relates to virtual cutting, and in particular to a virtual cutting method based on a mesh model incorporating a kelvin viscoelastic model.
Background
In recent years, with the development of virtual reality technology, it has become possible to perform simulated surgery by virtual reality. Virtual surgery gives medical personnel great convenience, and they can use this technique to carry out repeated simulation exercise, promote own technique. The current virtual surgery presents the process of the simulated surgery on a platform through modeling, rendering and calculation, and uses a finite element model, a non-grid model and the like. However, both finite element models and meshless models have some disadvantages. For finite element models, it depends deeply on the mesh, and a distorted or low quality mesh can cause large errors. During the re-engagement process, the resulting distortion elements may even cause model instability. Compared to classical mesh model based infrastructure, it is not suitable for cutting that simulates cut mesh structure and continuity. The purpose of the meshless model is to overcome the problems associated with finite element models. In contrast to finite element models, meshless models reconstruct virtual soft tissue on the basis of discrete and separate point elements, and the relationship between each point element is not associated with a mesh. Therefore, the point elements are random and not constrained by the grid, suitable for discontinuous scenes. Although the above mesh-free model is promising in simulating soft tissue cutting procedures, the interaction between the virtual surgical instrument and the soft tissue becomes a problem. To simplify the simulation process, most methods consider that as long as the soft tissue is swept by the virtual scalpel, the tissue is separated. However, the results are not as simple as expected. During the cutting process, it was observed that significant deformation may occur prior to soft tissue dissection. As can be appreciated, different instruments interacting with soft tissue often result in different forms of deformation. In addition, the methods corresponding to the finite element model and the meshless model are relatively large in calculation amount and relatively complex.
Disclosure of Invention
The invention aims to: in view of the above-mentioned drawbacks of the prior art, the present invention aims to provide a virtual cutting method based on a mesh model incorporating a kelvin viscoelastic model, which improves the efficiency of virtual deformation and cutting by using a mesh incorporating the kelvin viscoelastic model.
The technical scheme is as follows: a Kelvin viscoelastic model is constructed based on a virtual cutting method of a grid model including the Kelvin viscoelastic model, and a general discrete control equation of the Kelvin viscoelastic model is as follows:
Figure BDA0001707562230000011
wherein, K n Is a global stiffness matrix that is a function of,
Figure BDA0001707562230000021
is a global hysteresis stiffness matrix;
Figure BDA0001707562230000022
Figure BDA0001707562230000023
global hysteresis stiffness matrix applying the hysteresis function:
Figure BDA0001707562230000024
wherein B, φ, Δ ∈ n The elements in (A) are all constants; delta is a parameter of the virtual cut material, b is a constant,
Figure BDA0001707562230000025
represents time;
solving [ t ] from parameters of the virtual cut material and the cut time n ，t n+1 ]Increment of displacement in time
Figure BDA0001707562230000026
And calculating new displacement, strain and stress of each node to generate a simulated cut.
Further, the calculation of the new displacement, strain and stress of each node is specifically based on [ t [ [ t ] n ，t n+1 ]Increment of displacement in time
Figure BDA0001707562230000027
And (3) calculating:
at t n+1 In time, the increments of displacement, stress, strain are in turn:
Figure BDA0001707562230000028
σ n+1 ＝σ n +Δσ n ；
ε n+1 ＝ε n +Δε n ；
wherein the content of the first and second substances,
Figure BDA0001707562230000029
Δε＝∑ n B i ΔU i ，B i phi and phi i Is the strain matrix:
Figure BDA00017075622300000210
wherein L is a constant;
wherein the Kelvin viscoelasticity model is [ t ] n ，t n+1 ]The stress increment in (a) is: delta sigma n ＝Δε n Ε k +σ 0，n
(ii) a Wherein, the relation of stress and strain is as follows:
Figure BDA00017075622300000211
therein, e k Is a linear relaxation coefficient, representing the relaxation time in the time interval t n ，t n+1 ]In the stress change caused by a unit step strain increment, c 0 ，c 1 ，τ 1 Is a material parameter;
t n+1 the initial stresses at time were:
Figure BDA0001707562230000031
further, the constitutive equation of the kelvin viscoelastic model is:
Figure BDA0001707562230000032
wherein σ 1 Expressing stress, eta is damping coefficient of damper, sigma 2 The time derivatives representing stress, E 2 E 1 Respectively representing the stiffness of two springs, ∈ 1 Represents strain,. epsilon 2 Representing the time derivative of the strain.
Further, the constitutive relation of strain and stress in the kelvin viscoelastic model is:
Figure BDA0001707562230000033
where σ denotes stress, Ε denotes elastic modulus (young's model), epsilon denotes strain, c 0 And c 1 Is a material parameter, t represents time, τ 1 Is a time constant.
Further, before the kelvin viscoelastic model is constructed, a stress judgment step is further included: let the collision region be A, and set the threshold value f 1 ，d 1 And d 2 If the force is less than f 1 Or the width of A is greater than or equal to d 2 Then the mesh model is only deformed; if the force is greater than or equal to f 1 And the width of A is less than d 1 If the cutting speed is more than 0, the cutting condition is a first cutting condition; if the force is greater than or equal to f 1 And the width of A is greater than or equal to d 1 And is less than d 2 If so, judging the cutting condition II;
the first cutting condition is specifically as follows: consider A as a straight line L 1 Reading the position of the end points, setting the two end points as rigid cores, fixing, and copying L 1 To obtain L 1 、L 2 Before copying, is linked to L 1 The intersection point of the upper grid line and the left side of A is connected with L 1 Upper, connecting the right intersection point to L 2 Upper, L 1 、L 2 Because the stress imbalance generates elastic deformation according to Hooke's law F ═ k Δ x, wherein k is the stiffness coefficient of the spring, and only the stress in the horizontal direction is considered;
the second cutting condition is specifically as follows: and (3) regarding A as a rectangular area, fixing the width, deleting grid lines in A, generating elastic deformation on two long sides due to unbalanced stress according to Hooke's law F-k delta x, and only considering the stress in the horizontal direction.
Has the advantages that: the Kelvin viscoelastic model is brought into the grid, the viscoelasticity is the basic characteristic of biological soft tissues, so that the soft tissues have viscoelasticity, the simulated cutting is more practical, and the feedback setting of subsequent force is more facilitated. The method of the invention is based on a high-quality grid model, replaces approximate calculation with stress deformation, and solves the problems of grid distortion and continuous cutting in a finite element model and the problems of different deformation forms in a non-grid model. And the grid model provided by the invention replaces the calculation of the positions of partial points with the balance of force, thereby greatly reducing the calculation amount and optimizing the performance. By the model, only the position of the force application time point is needed to be calculated for deformation, and after the force is removed, the grid automatically recovers due to the balance of the force is broken, so that the calculation amount is reduced; and removing the viscoelastic grid lines between the openings aiming at cutting, breaking the equilibrium state of force, and enabling the grid to deform automatically. Because deformation and cutting accord with reality more, the model effect that the later stage was rendered and is obtained is splendid.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a Kelvin visco-elastic model;
FIG. 3 is a mesh model;
FIG. 4 is a schematic view of a first cutting scenario;
fig. 5 is a schematic diagram of the second cutting case.
Detailed Description
The technical solution is described in detail below with reference to a preferred embodiment and the accompanying drawings.
As shown in fig. 1, a virtual cutting method based on a mesh model incorporating a kelvin viscoelastic model mainly includes the steps of incorporating the kelvin viscoelastic model into a mesh, combining viscoelasticity, applying force, judging relevant conditions, and cutting, and specifically includes the following steps:
step 1: kelvin viscoelastic model was incorporated for the mesh.
Real life soft tissue has properties of hysteresis, relaxation and creep, collectively known as viscoelasticity. The viscoelastic mechanism model can describe the viscoelasticity of soft tissue, which is very important in biological properties. Biomechanical characteristics of different soft tissues can be described by modifying relevant parameters that can be obtained in vivo experiments. The kelvin viscoelastic model used herein is a standard linear model, the structure of which is shown in fig. 2;
the springs in the model represent the linear elastic characteristics of the soft tissue, and the dampers represent the damping characteristics of the soft tissue as it changes. The geometric and kinematic equations in viscoelasticity are the same as those in elasticity. The solution of the viscoelastic boundary value can be obtained by solving a constitutive equation of the motion equation, the geometric equation, the boundary condition and the initial condition;
the constitutive equation of the kelvin viscoelasticity model is:
Figure BDA0001707562230000051
wherein σ 1 Expressing stress, eta is damping coefficient of damper, sigma 2 The time derivatives representing stress, E 2 E 1 Respectively representing the stiffness of two springs, ∈ 1 Represents strain,. epsilon 2 Representing the time derivative of the strain;
the constitutive relationship of strain and stress is:
Figure BDA0001707562230000052
where σ denotes stress, Ε denotes elastic modulus (young's model), epsilon denotes strain, c 0 And c 1 Is a material parameter, t represents time, τ 1 Is a time constant;
and 2, step: a combination of viscoelasticity.
To incorporate viscoelasticity into the deformation model, an incremental version of the viscoelasticity model is used. First, the deformation simulation time T is divided into n time slices T 1 ，t 2 ，...，t n . Each time interval
Figure BDA0001707562230000053
Referred to as increments. Stress, strain and displacement at each moment are respectively sigma 1 ，σ 2 ，...，σ n ，ε 1 ，ε 2 ，...，ε n ，
Figure BDA0001707562230000054
From t n To t n+1 The increments of displacement, stress and strain are respectively
Figure BDA0001707562230000055
Δσ n And Δ ε n . In the deformation simulation process, the volume force generated by the acceleration is not considered, the soft tissue is not compressible, and the volume is not changed. The volume force b will therefore not change, assuming an external force on the boundary Γ
Figure BDA0001707562230000059
Is constant;
in the kelvin viscoelasticity model, the relaxation of soft tissue under external force can be represented by the relaxation constitutive relation:
Figure BDA0001707562230000056
when at t n And t n+1 And Δ t → 0, the stress increment is:
Figure BDA0001707562230000057
wherein e denotes the stiffness of the spring, τ denotes the time constant;
the stress-strain relationship of the kelvin viscoelastic model is represented by (2) in combination with (4):
Figure BDA0001707562230000058
wherein E k Is a linear relaxation coefficient, representing the relaxation time in the time interval t n ，t n+1 ]Of the strain induced by a unit step strain increment. c. C 0 ，c 1 ，τ 1 As a parameter of the material. Obtaining t by the following formula n+1 Stress of time σ 0，n ：
Figure BDA0001707562230000061
The Kelvin viscoelasticity model at [ t ] was obtained by the following formula n ，t n+1 ]Stress increment in (2):
Δσ n ＝Δε n Ε k +σ 0，n (7)
at t n+1 The increment of displacement, stress and strain is respectively as follows:
Figure BDA0001707562230000062
σ n+1 ＝σ n +Δσ n (9)
ε n+1 ＝ε n +Δε n (10)
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0001707562230000063
Δε＝∑ n B i ΔU i ，B i phi (phi) and phi (phi) i Is a strain matrix of the form:
Figure BDA0001707562230000064
the stress and strain can be calculated according to (8):
Figure BDA0001707562230000065
wherein L is a constant. In the soft tissue deformation process, the stress, the physical strength and the external force meet the balance condition, namely according to the virtual working principle of deformation, the total virtual operation is zero, and a general discrete control equation of the viscoelastic model can be obtained:
Figure BDA0001707562230000066
wherein, K n Is a global stiffness matrix that is a function of,
Figure BDA0001707562230000067
is a global hysteresis stiffness matrix, similar to the combined form of the global stiffness matrix. The global stiffness matrix and the global hysteresis stiffness matrix applying the hysteresis function may be expressed as:
Figure BDA0001707562230000068
Figure BDA0001707562230000069
Figure BDA00017075622300000610
wherein the material is homogeneous, B, phi, delta epsilon n The elements in (A) are all constant. Delta is a material parameter, b is a constant,
Figure BDA0001707562230000071
representing time. Given the corresponding material parameters and time, according to (12) [ t ] can be solved n ，t n+1 ]Increment of displacement in
Figure BDA0001707562230000072
Finally, calculating new displacement, strain and stress of each node;
and 3, step 3: and (5) stress and judging related conditions.
The mesh model is shown in FIG. 3, where the collision region is denoted as A, and a threshold value f is set 1 ，d 1 And d 2 If the force is less than f 1 Or the width of A is greater than or equal to d 2 Then the mesh model is only deformed; if the force is greater than or equal to f 1 And the width of A is less than d 1 If the cutting speed is greater than 0, the cutting condition is a first cutting condition (as shown in FIG. 4); if the force is greater than or equal to f 1 And A isIs greater than or equal to d 1 And is less than d 2 If so, the method belongs to the second cutting condition (as shown in FIG. 5);
and 4, step 4: and (5) deforming.
According to the stress condition, completing an algorithm of the viscoelastic coupling part, and calculating displacement;
and 5: and (6) cutting.
When the case belongs to case 1 or case 2, the mesh model will generate a cut;
5-1: regarding A as a straight line L in the first cutting condition 1 Reading the position of the end points, setting the two end points as rigid cores, fixing, and copying L 1 To obtain L 1 、L 2 Before copying, is linked to L 1 The intersection point of the upper grid line and the left side of A is connected with L 1 Upper, connecting the right intersection point to L 2 Thus, L 1 、L 2 Because the stress imbalance is generated according to Hooke's law F-k delta x (wherein k is the stiffness coefficient of the spring), elastic deformation is generated, only the stress in the horizontal direction is considered, and finally the stress balance state is achieved, so that a notch is generated;
5-2: and in the second cutting condition, the A is regarded as a rectangular area, the width is fixed, and grid lines in the A are deleted, so that two strips of the A are elastically deformed due to unbalanced stress according to the Hooke's law, only the stress in the horizontal direction is considered, and finally, a stress balance state is achieved, and a notch is formed.

Claims (4)

1. A virtual cutting method based on a grid model including a Kelvin viscoelastic model is characterized in that the Kelvin viscoelastic model is constructed, and a general discrete control equation of the Kelvin viscoelastic model is as follows:
Figure FDA0003779157960000011
wherein, K n Is a global stiffness matrix that is a function of,
Figure FDA0003779157960000012
is a global hysteresis stiffness matrix;
Figure FDA0003779157960000013
Figure FDA0003779157960000014
global hysteresis stiffness matrix applying the hysteresis function:
Figure FDA0003779157960000015
wherein BETA, φ, Δ ∈ n The elements in (A) are all constants; delta is a parameter of the virtual cut material, b is a constant,
Figure FDA0003779157960000016
represents time;
solving [ t ] from parameters of the virtual cut material and the cut time n ，t n+1 ]Increment of displacement in time
Figure FDA0003779157960000017
Calculating new displacement, strain and stress of each node to generate a simulated cut;
the calculation of the new displacement, strain and stress of each node is specifically based on [ t [ ] n ，t n+1 ]Increment of displacement in time
Figure FDA0003779157960000018
And (3) calculating:
at t n+1 In time, the increments of displacement, stress, strain are in turn:
Figure FDA0003779157960000019
σ n+1 ＝σ n +Δσ n ；
ε n+1 ＝ε n +Δε n ；
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA00037791579600000110
Δε＝Σ n B i ΔU i ，Β i phi and phi i Is the strain matrix:
Figure FDA00037791579600000111
wherein L is a constant;
wherein the Kelvin viscoelasticity model is [ t ] n ，t n+1 ]The stress increment in (a) is: delta sigma n ＝Δε n Ε k +σ 0，n (ii) a Wherein, the relation between the stress and the strain is as follows:
Figure FDA0003779157960000021
wherein, Ε k Is a linear relaxation coefficient, representing the relaxation time in the time interval t n ，t n+1 ]In the stress change caused by a unit step strain increment, c 0 ，c 1 ，τ 1 Is a material parameter;
t n+1 the initial stresses at time were:
Figure FDA0003779157960000022
2. the virtual cutting method based on a mesh model incorporating a kelvin viscoelastic model according to claim 1, wherein the constitutive equation of the kelvin viscoelastic model is:
Figure FDA0003779157960000023
wherein σ 1 Expressing stress, eta is damping coefficient of damper, sigma 2 The time derivatives representing stress, E 2 And e 1 Respectively representing the stiffness of two springs, ∈ 1 Represents strain,. epsilon 2 Representing the time derivative of the strain.
3. The virtual cutting method based on a mesh model incorporating a kelvin viscoelastic model according to claim 1, characterized in that the constitutive relation of strain and stress in the kelvin viscoelastic model is:
Figure FDA0003779157960000024
where σ denotes stress, Ε denotes the elastic modulus, e denotes strain, c 0 And c 1 Is a material parameter, t represents time, τ 1 Is a time constant.
4. The method of claim 1, wherein the step of determining the stress further comprises the steps of: let the collision region be A, and set a threshold value f 1 ，d 1 And d 2 If the force is less than f 1 Or the width of A is greater than or equal to d 2 Then the mesh model is only deformed; if the force is greater than or equal to f 1 And the width of A is less than d 1 If the cutting speed is more than 0, the cutting condition is a first cutting condition; if the force is greater than or equal to f 1 And the width of A is greater than or equal to d 1 And is less than d 2 If so, the cutting condition is a second cutting condition;
the first cutting condition is specifically as follows: consider A as a straight line L 1 Reading the position of the end points, setting the two end points as rigid cores, fixing, and copying L 1 To obtain L 1 、L 2 Before copying, is linked to L 1 The grid line on the grid line is connected with the left intersection point on A of AL 1 Upper, connecting the right intersection point to L 2 Upper, L 1 、L 2 Because the stress imbalance generates elastic deformation according to Hooke's law F-k delta x, wherein k is the stiffness coefficient of the spring, and only the stress in the horizontal direction is considered;
the second cutting condition is specifically as follows: and (3) regarding A as a rectangular area, fixing the width, deleting grid lines in A, generating elastic deformation on two long sides due to unbalanced stress according to Hooke's law F-k delta x, and only considering the stress in the horizontal direction.
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