CN113470165A - Soft tissue modeling method based on radial basis point interpolation method and mass point spring method - Google Patents

Soft tissue modeling method based on radial basis point interpolation method and mass point spring method Download PDF

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CN113470165A
CN113470165A CN202110670815.0A CN202110670815A CN113470165A CN 113470165 A CN113470165 A CN 113470165A CN 202110670815 A CN202110670815 A CN 202110670815A CN 113470165 A CN113470165 A CN 113470165A
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CN113470165B (en
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李春泉
彭佳琦
潘雨晨
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Nanchang University
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Abstract

The invention discloses a soft tissue modeling method based on a radial basis point interpolation method and a mass point spring method, which comprises the following steps: constructing a soft tissue model for the acquired medical image information by using a three-dimensional modeling technology, wherein the soft tissue model is divided into an operation area and a non-operation area according to a function area, and establishing a transition area based on an edge extraction strategy of a support area; preprocessing model parameters and parameters in a modeling method; the operating area simulates the deformation behavior of the soft tissue model, and the force is transmitted to the non-operating area through the transition area; and ending the single-step iteration of calculating the deformation, and entering the next round of circulation. The soft tissue model is divided into an operation area and a non-operation area, the operation area adopts high-precision RPIM which is convenient for applying boundary conditions, an implicit integration method Newmark method is introduced to dynamically analyze the change of displacement and ensure the stability of the system; the MSM is simple to use and realize in a non-operation area and high in calculation efficiency.

Description

Soft tissue modeling method based on radial basis point interpolation method and mass point spring method
Technical Field
The invention belongs to the technical field of soft tissue deformation model modeling, and particularly relates to a soft tissue modeling method based on a radial basis point interpolation method and a mass point spring method, which is used for solving the problem of soft tissue modeling in a virtual surgery experiment.
Background
The virtual operation provides ideal operation simulation effect and teaching environment, replaces the traditional operation skill training, greatly reduces the training cost, is beneficial to preoperative plan making and operation practice, and simultaneously provides new change for the traditional operation teaching mode. The rapid development of computer technology and the continuous progress of modern medicine promote the powerful role of the advanced field of multidisciplinary intersection in medical training. Therefore, research related to virtual surgery, such as soft tissue modeling method research, has important practical significance.
The establishment of the soft tissue model is the basis in the virtual surgery simulation system, and the simulation of deformation and cutting is the key technology of the virtual surgery simulation system. In the early stage of researching the deformation of an object, the method has the advantages of simple realization, easy operation and the like based on a non-physical geometric algorithm, so that the method occupies the mainstream position of related research. The geometric model obtains the basic unit information of the surface model from the CT tomography image of the source data without a large amount of complex physical calculation, but lacks material information, is difficult to carry out deformation simulation on the model with a complex topological structure, and influences the fidelity.
Then, modeling methods based on physics are continuously proposed and matured, and the kinematics equation including physical characteristics such as mass force and rotational inertia is developed from an initial kinematics equation including only velocity and angular velocity. Compared with a non-physical model, the physical model can reflect the deformation process in the virtual surgery process more truly. At present, physical models such as Finite Element Method (FEM), Mass-Spring Method (MSM), and Meshless (mesless) are the mainstream of soft tissue modeling methods.
The finite element method is a numerical solution, and seeks an approximate solution of the field variable distribution to the problem that it is difficult to obtain an analysis result. In the finite element method, a continuum of complex shapes is divided into different cells, known as finite cells. The cells are connected together by a topology called a mesh. The shapes of unit bodies of all finite units are different, the connection modes of the unit bodies are different, the finite element method can be applied to the solution domain of a complex model for modular processing, and the soft tissue model based on the FEM has higher accuracy. Therefore, the finite element method is widely applied to soft tissue modeling. However, the limitations of finite elements are quite evident: the calculation cost when forming the FEM network is high; the stress precision is low; the self-adaptive information analysis is difficult, and when the problem of super-large deformation is solved, the distorted grid can be generated, so that the grid needs to be reconstructed, and the solution precision is seriously influenced. The root of the method is that the finite element method system equation stage utilizes unit or grid information.
The mass-spring model was favored by researchers since the beginning of its introduction. The mass-spring model is an elastic deformation model based on hooke's law. In the particle spring model, the model is discretized into a number of particles connected by constraints, and such constraints with certain physical characteristics are called springs. When a single mass point or a plurality of mass points are subjected to external force, the displacement changes of the mass points are mutually influenced and transmitted, so that the purpose of deformation is achieved, the realization is simple, and the calculated amount is small. Since the kinetic equation for the particle spring has no physical parameters of the integrated material, the parameters required in the structural equation for the model are empirically derived. In addition to the spring tensor between the particles, the force vector of the particles is also related to the compensation of the force and displacement changes of the particles in different directions when the grid is changed. Such as spring torsion, changes in confinement, and changes in volume that occur in the model.
The meshless method is a system algebraic equation established in the whole problem domain and is a method for carrying out dispersion without utilizing predefined mesh information. The meshless approach utilizes a set of interspersed nodes, referred to as field nodes, that do not form a mesh and do not require any pre-defined nodal connectivity information for constructing interpolation or approximation expressions for variable unknown functions. The mesh-free method is based on point approximation, can completely or partially eliminate meshes, does not need initial division and reconstruction of the meshes, can ensure the calculation precision, and can reduce the calculation difficulty compared with a finite element method.
Slow-fire Element Method (DEM) was proposed by Nayroles et al in 1992 to introduce the moving least squares approximation into the galileon (Galerkin) Method, after which some scholars improved DEM, kept the neglect terms and introduced the lagrange multiplier Method into the essential boundary conditions, and the Element-Free Galerkin Method (EFG) was proposed, which caused the research heat tide of the meshless Method. Pauly realized a soft tissue cutting simulation in [69] in 2006, in which visibility criteria were introduced and succeeded in simulating a hysteromyomectomy surgery simulation system. Zou et al propose a mesh-free deformation model based on a Radial base Point Interpolation Method (RPIM) for interactive simulation applications, and experimental results in the article show higher accuracy. Cheng et al propose a novel mesh-free soft tissue interactive cutting simulation model. Unlike most existing methods that consider soft tissue cutting procedures, the proposed model is able to simulate a complete cutting procedure, including three phases: deformation before cutting, deformation after cutting and cutting, and the model is applied to a liver cutting simulation system and obtains satisfactory visual effect and tactile feedback. However, none of their solutions completely solves the problem of the traditional model that cannot balance real-time performance and accuracy.
In summary, due to the complexity of the soft tissue structure of the human body, the modeling method of the soft tissue model is always a key and difficult part of the virtual surgery system, and there are few soft tissue modeling methods that can meet the requirements of the virtual surgery reality.
Disclosure of Invention
Aiming at the defects and problems in the prior art, the invention aims to provide a soft tissue modeling method based on a radial basis point interpolation method and a mass point spring method. The invention is suitable for tissues and organs with various shapes and sizes, and can enable the model to better meet the practical requirements of virtual surgery.
The invention is realized by the following technical scheme:
a soft tissue modeling method based on a radial basis point interpolation method and a mass point spring method is characterized in that the model is divided into an operation (pathological change) area and a non-operation (health) area according to a functional area so as to balance the real-time performance and the accuracy of the system; the stability and high precision of a system operation area are ensured by introducing RPIM of a hidden integration method Newmark into the operation area; the MSM model with small calculated amount is adopted in the non-operation area to reduce the calculation complexity, and meanwhile, the mass point spring method based on biomechanics can effectively express the nonlinearity and viscoelasticity of healthy biological tissues; in order to relieve the splitting problem of the model functional area, an edge extraction strategy based on a support domain is proposed to establish a transition area.
The method comprises the following steps:
step 1: and constructing a soft tissue model for the acquired medical image information by utilizing a three-dimensional modeling technology.
Step 2: preprocessing model parameters and parameters in a modeling method.
And step 3: the operating region simulates the deformation behavior of the soft tissue model, and the force is transmitted to the non-operating region through the transition region.
And 4, step 4: and ending the single-step iteration of calculating the deformation, and entering the next round of circulation.
Further, in the deformation calculation process of the step 2, in the operation region, according to a radial base point difference method, an implicit integration method (Newmark) is used for calculating an equivalent stiffness matrix, and a kinetic equation is solved. The nonlinear spring, the damping spring and the virtual body spring in the non-operation area can simulate the biological characteristic effect of soft tissues. The mixed model is divided according to the functional areas, so that the calculated amount during deformation is reduced, the accuracy of the deformation area is ensured, and the accuracy and the real-time performance are balanced finally.
The specific steps of the step 2 are as follows:
2.1 systematic equation for Soft tissue model
Figure BDA0003118665040000041
Here, M represents a mass matrix of the node, K represents a stiffness matrix of the node, C represents a damping matrix of the node, and F represents an overall force vector;
for the operating field of the model, the nodes of the problem domain are numbered 1 to n, and for the nodes i, j there are
Figure BDA0003118665040000042
∫ΩρΦTΦdΩ＝Mij
∫ΩcΦTΦdΩ＝Cij
B is a strain matrix, phi is a shape function, rho is density, c is a damping coefficient, and D is a material constant matrix.
The radial basis function is:
Figure BDA0003118665040000043
wherein in the formula
Figure BDA0003118665040000044
Representing a calculation point X (X, y, z) and a node Xi(xi,yi,zi) A distance between, αcIs more than or equal to 0, and r is a constant.
2.2 for the non-operation area of the model, setting any adjacent points as i, j, kijThe elastic coefficient is expressed into two different stages according to the nonlinear characteristics of soft tissues, the nonlinear state in small deformation is expressed as follows, the elastic coefficient is not constant any more, but is replaced by a cubic polynomial related to the deformation degree to present a nonlinear relation
Figure BDA0003118665040000045
The linear state expression of non-small deformation is as follows, instead of being a first order polynomial on the degree of deformation, exhibiting a linear relationship
Figure BDA0003118665040000046
In the formula k1、k2Is a constant, Δ uijThe distance between adjacent nodes i, j,
Figure BDA0003118665040000047
is the initial distance.
Damping spring, virtual spring simulation viscoelasticity are added in traditional mass point spring, and virtual spring, damping spring can simulate the influence each other of each mass point when the soft tissue atress, and the damping spring specific expression between the mass point is as follows, and the damping coefficient is the first order polynomial about deformation degree:
Figure BDA0003118665040000048
in the formula b1、b2Is a constant number of times, and is,
Figure BDA0003118665040000049
is the initial position.
The virtual spring is connected with the gravity center and the mass point, so that the situation that the deformation is not controlled due to the continuous increase of the load is relieved. When soft tissue begins to change after load is applied, the virtual spring generates force acting on the node, so that excessive change of the volume can be avoided, and the expression of the coefficient of the virtual spring is as follows:
Figure BDA0003118665040000051
hiocoefficient of a virtual spring from node i to center of gravity O, h is constant, Oi、
Figure BDA0003118665040000052
The volume of the current cell and the initial volume, respectively.
And 2.3, processing the transition region. And constructing a boundary of the hybrid model by using a support domain, wherein a cuboid support domain is used for judging whether the node is located in the support domain, if the node is located in the support domain or the boundary, the node belongs to the edges of two regions, otherwise, the node belongs to a non-operation region:
Figure BDA0003118665040000053
dsx、dsx、dsxthe dimensions of the support field in the x, y, z directions, respectively, (x)g,yg,zg) Is to calculate the coordinates of the points,node coordinates (x, y, z). According to the conditions that the resultant force of acting forces of a radial base point interpolation method region and a spring particle model region of a transition node is 0 and the displacements are equal, two different approximate displacements of the same transition node are obtained, all the transition nodes are traversed, an approximate displacement function constructed by an RPIM model and an MSM model is obtained, a functional relation between the two is established, and the function is used as the approximate displacement function of the transition node, so that smooth transition between the two regions is realized.
2.4 solving the kinetic equation. The Newmark method calculates the structural dynamic response according to the assumed acceleration change rule of the time increment, namely the acceleration is linearly distributed or is constant in the time interval delta t, and the acceleration expression is as follows:
Figure BDA0003118665040000054
the displacement, the speed and the acceleration have the following relations:
Figure BDA0003118665040000055
Figure BDA0003118665040000056
Figure BDA0003118665040000057
representing acceleration at time t + deltat and time t respectively,
Figure BDA0003118665040000058
representing the speed at time t + Δ t and t, u, respectivelyt+Δt，utRespectively representing the displacement at time t + Δ t and at time t, δ, β being parameters controlling the accuracy and stability of the value.
The acceleration and the speed are converted into displacement expression, and the expression is as follows
Figure BDA0003118665040000061
Figure BDA0003118665040000062
Converting the acceleration and the speed into displacement expressions and substituting the displacement expressions into an integral discrete equation to obtain:
Figure BDA0003118665040000063
Figure BDA0003118665040000064
Figure BDA0003118665040000065
equivalent stiffness matrix in dynamic equation of operation region expressed by the above formula
Figure BDA0003118665040000066
And an equivalence matrix Q.
Integration constants are respectively
Figure BDA0003118665040000067
Figure BDA0003118665040000068
a6＝Δt(1-δ)；a7＝δΔt
Compared with the prior art, the invention has the beneficial effects that:
(1) the invention simulates real soft tissue based on a RPIM-MSM mixed soft tissue model, and divides the soft tissue model into two regions according to the operation requirement: the operating area and the non-operating area, namely the lesion area, adopt high precision, facilitate to exert RPIM of the boundary condition, introduce the hidden integral method Newmark method to carry on the dynamic analysis to the change of the displacement and guarantee the stability of the system; the MSM is simple to use and realize in a non-operation area and high in calculation efficiency.
(2) The invention provides an MSM based on biomechanics, which is used for showing nonlinearity and viscoelasticity, showing biological characteristics of healthy soft tissues and improving the authenticity of a model.
Drawings
FIG. 1 is a flow chart of a hybrid model deformation calculation according to the present invention;
FIG. 2 is a structural diagram of a biomechanically based mass spring according to the present invention;
FIG. 3 is a schematic diagram of the transition region of the present invention.
Detailed Description
The invention will be further described with reference to the accompanying drawings.
A soft tissue modeling method based on a radial basis point interpolation method and a mass point spring method is disclosed, and the flow of the soft tissue modeling method is shown in figure 1 and comprises the following steps:
step 1: constructing a soft tissue model for the acquired medical image information by utilizing a three-dimensional modeling technology;
step 2: preprocessing model parameters and parameters in a modeling method;
and step 3: the operating area simulates the deformation behavior of the soft tissue model, and the force is transmitted to the non-operating area through the transition area;
and 4, step 4: and ending the single-step iteration of calculating the deformation, and entering the next round of circulation.
Further, in the deformation calculation process in the step 2, in the operation area, according to a radial base point difference method, a Newmark is used for calculating an equivalent stiffness matrix, and a kinetic equation is solved. The nonlinear spring, the damping spring and the virtual body spring in the non-operation area can simulate the biological characteristic effect of soft tissues. The mixed model is divided according to the functional areas, so that the calculated amount during deformation is reduced, the accuracy of the deformation area is ensured, and the accuracy and the real-time performance are balanced finally.
The specific steps of the step 2 are as follows:
2.1 systematic equation for Soft tissue model
Figure BDA0003118665040000071
Here, M represents the mass matrix of the node, K represents the stiffness matrix of the node, C represents the damping matrix of the node, and F represents the overall force vector.
For the operating field of the model, the nodes of the problem domain are numbered 1 to n, and for the nodes i, j there are
Figure BDA0003118665040000072
∫ΩρΦTΦdΩ＝Mij
∫ΩxΦTΦdΩ＝Cij
B is a strain matrix, phi is a shape function, rho is density, c is a damping coefficient, and D is a material constant matrix.
The radial basis function is:
Figure BDA0003118665040000073
in the formula (I), the compound is shown in the specification,
Figure BDA0003118665040000074
representing a calculation point X (X, y, z) and a node Xi(xi,yi，zi) A distance between, αcIs more than or equal to 0, and r is a constant.
2.2 for the non-operation area of the model, setting any adjacent points as i, j, kijThe elastic coefficient is expressed into two different stages according to the nonlinear characteristics of soft tissues, the nonlinear state in small deformation is expressed as follows, the elastic coefficient is not constant any more, and is replaced by a cubic polynomial about the deformation degree to present a nonlinear relation:
Figure BDA0003118665040000081
the linear state expression for non-small deformations, instead of being a first order polynomial on the degree of deformation, exhibits a linear relationship as follows:
Figure BDA0003118665040000082
in the formula k1、k2Is a constant, Δ uijThe distance between adjacent nodes i, j,
Figure BDA0003118665040000083
is the initial distance.
As shown in fig. 2, a damping spring and a virtual spring are added in a traditional mass point spring to simulate viscoelasticity, the virtual spring and the damping spring can simulate the mutual influence of mass points when soft tissues are stressed, the specific expression of the damping spring among the mass points is as follows, and the damping coefficient is a first-order polynomial about the deformation degree:
Figure BDA0003118665040000084
in the formula b1、b2Is a constant number of times, and is,
Figure BDA0003118665040000085
is the initial position.
The virtual spring is connected with the gravity center and the mass point, so that the situation that the deformation is not controlled due to the continuous increase of the load is relieved. When soft tissue begins to change after load is applied, the virtual spring generates force acting on the node, so that excessive change of the volume can be avoided, and the expression of the coefficient of the virtual spring is as follows:
Figure BDA0003118665040000086
hiocoefficient of a virtual spring from node i to center of gravity O, h is constant, Oi、
Figure BDA0003118665040000087
The volume of the current cell and the initial volume, respectively.
2.3 treatment of the transition zone (as shown in FIG. 3). And constructing a boundary of the hybrid model by using a support domain, wherein a cuboid support domain is used for judging whether the node is located in the support domain, if the node is located in the support domain or the boundary, the node belongs to the edges of two regions, otherwise, the node belongs to a non-operation region:
Figure BDA0003118665040000088
dsx、dsx、dsxthe dimensions of the support field in the x, y, z directions, respectively, (x)g，yg，zg) Is the calculated point coordinates, node coordinates (x, y, z). According to the conditions that the resultant force of acting forces of a radial base point interpolation method region and a spring particle model region of a transition node is 0 and the displacements are equal, two different approximate displacements of the same transition node are obtained, all the transition nodes are traversed, an approximate displacement function constructed by an RPIM model and an MSM model is obtained, a functional relation between the two is established, and the function is used as the approximate displacement function of the transition node, so that smooth transition between the two regions is realized.
2.4 solving the kinetic equation. The Newmark method calculates the structural dynamic response according to the assumed acceleration change rule of the time increment, and the Newmark method calculates the structural dynamic response according to the assumed acceleration change rule of the time increment, namely the acceleration is linearly distributed or is constant in the time interval delta t, and the acceleration expression is as follows:
Figure BDA0003118665040000091
the displacement, the speed and the acceleration have the following relations:
Figure BDA0003118665040000092
Figure BDA0003118665040000093
Figure BDA0003118665040000094
representing acceleration at time t + deltat and time t respectively,
Figure BDA0003118665040000095
representing the speed at time t + Δ t and t, u, respectivelyt+Δt，utThe displacement at time t + Δ t and the displacement at time t are represented, δ and β are parameters for controlling the accuracy and stability of the numerical values, and the Newmark method is unconditionally stable when the following formula is set.
Figure BDA0003118665040000096
Converting acceleration and velocity into a displacement representation
Figure BDA0003118665040000097
Figure BDA0003118665040000098
Converting acceleration and speed into displacement expression and substituting the displacement expression into an integral discrete equation
Figure BDA0003118665040000099
Figure BDA00031186650400000910
Figure BDA00031186650400000911
Equivalent stiffness matrix in dynamic equation of operation region expressed by the above formula
Figure BDA00031186650400000912
And an equivalence matrix Q.
Integration constants are respectively
Figure BDA0003118665040000101
Figure BDA0003118665040000102
a6＝Δt(1-δ)；a7＝δΔt。
The foregoing merely represents preferred embodiments of the invention, which are described in some detail and detail, and therefore should not be construed as limiting the scope of the invention. It should be noted that, for those skilled in the art, various changes, modifications and substitutions can be made without departing from the spirit of the present invention, and these are all within the scope of the present invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.

Claims (3)

1. A soft tissue modeling method based on a radial basis point interpolation method and a mass point spring method is characterized by comprising the following steps:
s1: constructing a soft tissue model for the acquired medical image information by using a three-dimensional modeling technology, wherein the soft tissue model is divided into an operation area and a non-operation area according to a function area, and establishing a transition area based on an edge extraction strategy of a support area;
s2: preprocessing model parameters and parameters in a modeling method;
in the deformation calculation process of pretreatment, calculating an equivalent stiffness matrix by using an implicit integration method according to a radial base point difference method in an operation area, and solving a kinetic equation; the nonlinear spring, the damping spring and the virtual body spring in the non-operation area simulate the biological characteristic effect of soft tissues;
s3: the operating area simulates the deformation behavior of the soft tissue model, and the force is transmitted to the non-operating area through the transition area;
s4: and ending the single-step iteration of calculating the deformation, and entering the next round of circulation.
2. The soft tissue modeling method based on radial basis point interpolation and particle spring method as claimed in claim 1, wherein the system equation of the soft tissue model in step S1 is:
Figure FDA0003118665030000011
in the formula, M represents a mass matrix of a node, K represents a stiffness matrix of the node, C represents a damping matrix of the node, and F represents an overall force vector.
3. The soft tissue modeling method based on radial basis point interpolation and particle spring method as claimed in claim 2, wherein the respective regional parameter processing of the model in step S2 includes:
s21, numbering the nodes of the problem domain as 1 to n for the operation region of the model, and having the nodes i and j
Figure FDA0003118665030000012
∫ΩρΦTΦdΩ＝Mij
∫ΩcΦTΦdΩ＝Cij
In the formula, B is a strain matrix, phi is a shape function, rho is density, c is a damping coefficient, and D is a material constant matrix; kijIs the coefficient of elasticity, MijIs a mass coefficient, CijIs a damping coefficient;
the radial basis function is:
Figure FDA0003118665030000021
in the formula
Figure FDA0003118665030000022
It represents the calculation point X (X, y, z) and the node Xi(xi,yi,zi) A distance between, αcR is a constant and is more than or equal to 0;
s22 model non-operation area
S221, setting any adjacent point as i, j, kijFor the elastic coefficient, the elastic coefficient expression is divided into two different stages according to the non-linear characteristics of the soft tissue:
(1) the non-linear state at small deformations is expressed as follows:
Figure FDA0003118665030000023
the elastic coefficient is not constant any more, but is replaced by a cubic polynomial about the deformation degree, and a nonlinear relation is presented;
(2) the linear state expression for non-small deformations is as follows:
Figure FDA0003118665030000024
the elastic coefficient is replaced by a first order polynomial on the degree of deformation, exhibiting a linear relationship, where k1、k2Is a constant, Δ uijThe distance between adjacent nodes i, j,
Figure FDA0003118665030000025
is the initial distance;
s222, the specific expression of the damping coefficient of the damping spring between the mass points is as follows:
Figure FDA0003118665030000026
the damping coefficient is a polynomial of degree of deformation of the order in which b1、b2Is a constant number of times, and is,
Figure FDA0003118665030000027
is an initial position;
s223, virtual spring has connected focus and mass point, connects and alleviates the uncontrolled condition of deformation because of the load lasts to increase, and soft tissue begins to change after applying the load, and virtual spring produces the power that acts on the node and can avoid the excessive change of volume, and virtual spring coefficient expression is as follows:
Figure FDA0003118665030000028
hiocoefficient of a virtual spring from node i to center of gravity O, h is constant, Oi、
Figure FDA0003118665030000029
Respectively the volume of the current unit and the initial volume;
s23, processing for transition area
Utilizing a cuboid support domain to construct a boundary of a hybrid model, judging whether a node is located in the support domain, if the node is located in the support domain or the boundary, the node belongs to the edges of two regions, otherwise, the node belongs to a non-operation region:
Figure FDA0003118665030000031
in the formula (d)sx、dsx、dsxThe dimensions of the support field in the x, y, z directions, respectively, (x)g,yg,zg) Is the calculated point coordinates, node coordinates (x, y, z);
obtaining two different approximate displacements of the same transition node according to the condition that the resultant force of acting forces of a radial base point interpolation method region and a spring mass point model region, which are applied to the transition node, is 0 and the displacements are equal, traversing all the transition nodes to obtain an approximate displacement function constructed by a radial base point interpolation model and a mass point spring model, then establishing a functional relationship between the two approximate displacement functions, and taking the approximate displacement function as the transition node to realize smooth transition between the two regions;
s24 solving kinetic equation
The Newmark method calculates the structural dynamic response according to the assumed acceleration change rule of the time increment, namely the acceleration is linearly distributed or is constant in the time interval delta t, and the acceleration expression is as follows:
Figure FDA0003118665030000032
the displacement, the speed and the acceleration have the following relations:
Figure FDA0003118665030000033
Figure FDA0003118665030000034
in the formula (I), the compound is shown in the specification,
Figure FDA0003118665030000035
representing acceleration at time t + deltat and time t respectively,
Figure FDA0003118665030000036
representing the speed at time t + Δ t and t, u, respectivelyt+Δt、utRespectively representing the displacement at the moment t + delta t and the displacement at the moment t, wherein delta and beta are parameters for controlling the accuracy and the stability of the numerical value;
convert acceleration and velocity into displacement expressions:
Figure FDA0003118665030000037
Figure FDA0003118665030000038
converting the acceleration and the speed into displacement expressions and substituting the displacement expressions into an integral discrete equation to obtain:
Figure FDA0003118665030000039
Figure FDA00031186650300000310
Figure FDA0003118665030000041
in the formula, an equivalent stiffness matrix in a dynamic equation of an operation region is expressed
Figure FDA0003118665030000042
And an equivalence matrix Q;
the integration constants are:
Figure FDA0003118665030000043
Figure FDA0003118665030000044
a6＝Δt(1-δ)；a7＝δΔt。
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