CN113139310A - Method, simulator and system for simulating focused ultrasound temperature field based on FDTD - Google Patents

Method, simulator and system for simulating focused ultrasound temperature field based on FDTD Download PDF

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CN113139310A
CN113139310A CN202110333517.2A CN202110333517A CN113139310A CN 113139310 A CN113139310 A CN 113139310A CN 202110333517 A CN202110333517 A CN 202110333517A CN 113139310 A CN113139310 A CN 113139310A
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王祥达
林伟军
滕世国
苏畅
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Shaoguan East Sunshine Automation Equipment Co Ltd
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Abstract

The invention provides a method, a simulator and a system for simulating a focused ultrasound temperature field based on FDTD, wherein the method comprises the following steps: s1: obtaining a numerical simulation equation of a focused ultrasound temperature field; wherein, the numerical simulation equation of the focused ultrasound temperature field is expressed by a Pennes biological heat conduction equation; s2: and solving a numerical simulation equation in S1 by adopting a time domain finite difference method with spatial second-order precision and temporal first-order precision to realize the temperature rise effect caused by simulated focused ultrasound. The invention solves the Pennes biological heat conduction equation by using a finite difference time domain algorithm with spatial second-order precision and temporal first-order precision, thereby establishing a method for simulating a focused ultrasound temperature field based on FDTD. The numerical simulation precision of the focused ultrasonic temperature field really corresponds to a time domain finite difference method with spatial second-order precision and temporal first-order precision, and the effectiveness and the accuracy of simulation are ensured.

Description

Method, simulator and system for simulating focused ultrasound temperature field based on FDTD
Technical Field
The invention relates to the technical field of focused ultrasound temperature field simulation, in particular to a method, a simulator and a system for simulating a focused ultrasound temperature field based on FDTD.
Background
High Intensity Focused Ultrasound (HIFU) tumor ablation is a new tumor treatment that has become popular in recent years. The ultrasonic generator is characterized in that high-intensity ultrasonic waves emitted by an external ultrasonic transducer pass through the skin noninvasively and form an ultrasonic focus in a tumor in the body. The temperature at the focus is raised instantly, so that the surrounding tumor tissue is coagulated and necrosed, and finally scars are formed or the scar is absorbed by metabolism. As disclosed in chinese patent publication No.: CN101108268A, published: 2008-01-23, discloses a multi-mode thermal field forming method of phased array focused ultrasound in the field of biomedical engineering, comprising the following steps: (1) adopting a multi-element array spherical surface open pore unequal-interval annular array treatment head; (2) adopting each sound field mode to correspond to the phase and amplitude of a group of driving signals; (3) determining the available high heat mode sound intensity in the target tissue by using a heuristic method; (4) and regulating the sound intensity of the thermal field in the heating mode to ensure that the temperature rising rate of the cold point of the heating area is kept at a preset value, and finally achieving the purpose of forming the thermal field in any expected uniform heating mode.
However, in the prior art, the residual heat of the focus or the focus position of the focus is inaccurate, and the surrounding healthy tissues are easily killed, so that unnecessary damage is caused. Therefore, the temperature field caused by the focused ultrasound is researched through simulation, and the method has important significance for the design of focused ultrasound equipment, the guarantee of the safety and the effectiveness of focused ultrasound therapy, the formulation of a proper strategy of the focused ultrasound therapy and the like.
Disclosure of Invention
The invention provides a method, a simulator and a system for simulating a focused ultrasound temperature field based on FDTD (finite time division) for overcoming the problem of inaccurate temperature field caused by focused ultrasound simulation in the prior art, which can effectively and accurately simulate the focused ultrasound temperature field.
In order to solve the technical problems, the technical scheme of the invention is as follows: a method for simulating a focused ultrasound temperature field based on FDTD, the method comprising the following steps:
s1: obtaining a numerical simulation equation of a focused ultrasound temperature field; wherein, the numerical simulation equation of the focused ultrasound temperature field is expressed by a Pennes biological heat conduction equation;
s2: and solving a numerical simulation equation in S1 by adopting a time domain finite difference method with spatial second-order precision and temporal first-order precision to realize the temperature rise effect caused by simulated focused ultrasound.
Preferably, the Pennes biological heat transfer equation:
Figure BDA0002997259830000021
wherein T represents a biological tissue temperature; kappatRepresents a thermal conductivity coefficient; ctRepresents the specific heat of the biological tissue; cbRepresents the vascular specific heat; wbRepresenting the blood perfusion rate of capillaries within the biological tissue; t isaRepresents the initial temperature; qvRepresents the amount of heat absorbed by the tissue per unit volume per unit time; qmRepresenting the rate of heat generation by biological metabolism.
Further, taking
Figure BDA0002997259830000022
Indicating the rise in temperature, ignoring QmGenerally, the starting temperature is a constant temperature, and the Pennes biological heat transfer equation is expressed as:
Figure BDA0002997259830000023
for focused ultrasound temperature field simulation, QvNamely external heat source input brought to biological tissues by focused ultrasound; for a single frequency stationary focused ultrasound field, an ultrasound heat source QvIs represented as follows:
Qv＝αIav＝αρ0c0<V2>＝1/2×αρ0c0V0 2
wherein α represents a sound absorption coefficient; i isavRepresents the average sound intensity; c. C0Represents the wave velocity; v represents the particle vibration velocity;<*>the periodic average of the variables in the device is shown; v0Representing the particle vibration velocity magnitude.
Further, the time-domain finite difference method with spatial second-order precision specifically adopts spatial grid points to divide the analog simulation area, and the distance between adjacent spatial grid points is 0.1 λ, wherein λ represents the wavelength of the ultrasonic wave.
Further, the time domain finite difference method of the spatial second-order precision specifically solves the following:
for a unary function f (x), where x is an argument, the second-order center-of-precision difference format for its second derivative is:
Figure BDA0002997259830000024
wherein f is(2)|iThe superscript (2) in (1) denotes the second derivative of the unary function f (x), f $iThe index i in (a) indicates the value (x) of the argument corresponding to the ith grid point0+iΔx)，x0Is the initial value of the argument, Δ x is the difference of the arguments between adjacent grid points;
the temperature rises at different time and different space grid point positions are respectively defined as
Figure BDA0002997259830000031
Superscript n denotes the time coordinate (t)0+ n Δ t), where t is0Denotes the starting time, Δ t denotes the time step of the numerical simulation; the indices i, j, k denote the cartesian space coordinates (x)0+iΔx,y0+jΔy,z0+ k Δ z) where (x)0,y0,z0) Denotes the starting position, and Δ x, Δ y, Δ z denote the spatial step in the x, y, z directions, respectively.
Further, the time-domain finite difference method with the first-order time precision specifically solves the following problems:
the explicit time-discrete format of equation (2) is:
Figure BDA0002997259830000032
second spatial derivative in equation (4)
Figure BDA0002997259830000033
The difference format of (3) uses equation (3) such that there is:
Figure BDA0002997259830000034
a simulator comprises an acquisition module, a time domain finite difference method solving module with spatial second-order precision and a time domain finite difference method solving module with temporal first-order precision;
the acquisition module is used for acquiring a Pennes biological heat conduction equation;
the module for solving the Pennes biological heat conduction equation is respectively solved by the module for solving the time domain finite difference method with the spatial second-order precision and the module for solving the time domain finite difference method with the temporal first-order precision, so that the temperature rise effect caused by the simulated focused ultrasound is realized.
Preferably, the solution module of finite difference method of time domain with second order precision in space realizes the following solution:
for a unary function f (x), where x is an argument, the second-order center-of-precision difference format for its second derivative is:
Figure BDA0002997259830000041
wherein f is(2)|iThe superscript (2) in (1) denotes the second derivative of the unary function f (x), f $iThe index i in (a) indicates the value (x) of the argument corresponding to the ith grid point0+iΔx)，x0Is the initial value of the argument, Δ x is the difference of the arguments between adjacent grid points;
the temperature rises at different time and different space grid point positions are respectively defined as
Figure BDA0002997259830000042
Superscript n denotes the time coordinate (t)0+ n Δ t), where t is0Denotes the starting time, Δ t denotes the time step of the numerical simulation; the indices i, j, k denote the cartesian space coordinates (x)0+iΔx,y0+jΔy,z0+ k Δ z) where (x)0,y0,z0) Denotes the starting position, and Δ x, Δ y, Δ z denote the spatial step in the x, y, z directions, respectively.
Further, the time domain finite difference method solving module with the first-order precision realizes the following solving:
the explicit time-discrete format of equation (2) is:
Figure BDA0002997259830000043
second spatial derivative in equation (4)
Figure BDA0002997259830000044
The difference format of (3) uses equation (3) such that there is:
Figure BDA0002997259830000045
a computer system comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the method as claimed in any one of claims 1 to 6 when executing the computer program.
Compared with the prior art, the technical scheme of the invention has the beneficial effects that:
the invention solves the Pennes biological heat conduction equation by using a finite difference time domain algorithm with spatial second-order precision and temporal first-order precision, thereby establishing a method for simulating a focused ultrasound temperature field based on FDTD. The numerical simulation precision of the focused ultrasonic temperature field is really corresponding to a time domain finite difference method with spatial second-order precision and temporal first-order precision, and the effectiveness and the accuracy of analog simulation are improved.
Drawings
Fig. 1 is a flow chart of the steps of the method according to the present embodiment.
FIG. 2 is the variation of the 2-norm and ∞ -norm of the error in the normalized temperature rise of the present embodiment with space step.
FIG. 3 is a graph of the 2-norm and ∞ -norm of the error in the normalized temperature rise of the present embodiment as a function of step size in time.
Fig. 4 is a graph showing the change in the temperature rise at the center of the region according to the present embodiment with time.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and are used for illustration only, and should not be construed as limiting the patent. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
Example 1
As shown in fig. 1, a method for simulating a focused ultrasound temperature field based on FDTD simulation, the method comprises the following steps:
s1: obtaining a numerical simulation equation of a focused ultrasound temperature field; wherein, the numerical simulation equation of the focused ultrasound temperature field is expressed by a Pennes biological heat conduction equation;
s2: and solving a numerical simulation equation in S1 by adopting a time domain finite difference method with spatial second-order precision and temporal first-order precision to realize the temperature rise effect caused by simulated focused ultrasound.
In a specific embodiment, the Pennes biological heat transfer equation:
Figure BDA0002997259830000051
wherein T represents a biological tissue temperature; kappatRepresents a thermal conductivity coefficient; ctRepresents the specific heat of the biological tissue; cbRepresents the vascular specific heat; wbRepresenting the blood perfusion rate of capillaries within the biological tissue; t isaRepresents the initial temperature; qvRepresents the amount of heat absorbed by the tissue per unit volume per unit time; qmRepresenting the heat generation rate of the biological metabolism, Q when calculatedmAnd is generally negligible.
Get
Figure BDA0002997259830000052
Indicating the rise in temperature, ignoring QmGenerally, the starting temperature is a constant temperature, and the Pennes biological heat transfer equation is expressed as:
Figure BDA0002997259830000061
for focused ultrasound temperature field simulation, QvNamely external heat source input brought to biological tissues by focused ultrasound; for a single frequency stationary focused ultrasound field, an ultrasound heat source QvIs represented as follows:
Qv＝αIav＝αρ0c0<V2>＝1/2×αρ0c0V0 2
wherein α represents a sound absorption coefficient; i isavRepresents the average sound intensity; c. C0Represents the wave velocity; v represents the particle vibration velocity;<*>the periodic average of the variables in the device is shown; v0Representing the particle vibration velocity magnitude.
In a specific embodiment, the time-domain finite difference method with spatial second-order precision specifically uses spatial grid points to divide the analog simulation area, and the distance between adjacent spatial grid points is 0.1 λ, where λ represents the wavelength of the ultrasonic wave.
Further, the time domain finite difference method of the spatial second-order precision specifically solves the following:
for a unary function f (x), the second-order center-of-precision difference format for its second derivative is:
Figure BDA0002997259830000062
wherein f is(2)|iThe superscript (2) in (1) denotes the second derivative of the unary function f (x), f $iThe index i in (a) indicates the value (x) of the argument corresponding to the ith grid point0+iΔx)，x0Is the initial value of the argument, Δ x is the difference of the arguments between adjacent grid points;
the temperature rises at different time and different space grid point positions are respectively defined as
Figure BDA0002997259830000063
Superscript n denotes the time coordinate (t)0+ n Δ t), where t is0Denotes the starting time, Δ t denotes the time step of the numerical simulation; the indices i, j, k denote the cartesian space coordinates (x)0+iΔx,y0+jΔy,z0+ k Δ z) where (x)0,y0,z0) Denotes the starting position, and Δ x, Δ y, Δ z denote the spatial step in the x, y, z directions, respectively.
In a specific embodiment, the time-domain finite difference method with first-order precision specifically solves the following:
the explicit time-discrete format of equation (2) is:
Figure BDA0002997259830000064
second spatial derivative in equation (4)
Figure BDA0002997259830000065
The difference format of (3) uses equation (3) such that there is:
Figure BDA0002997259830000071
to confirm the reliability of the method described in this example, the following verification was performed:
1. initial value of temperature rise in limited area
Heat source term Q without considering the right side of biological Heat transfer equation (2)vAnd items relating to blood vessels
Figure BDA0002997259830000072
Then equation (2) has the form:
Figure BDA0002997259830000073
for a finite region of a cuboid with sides a, b, c, respectively, it is assumed that the surface temperature rise of this region remains unchanged:
Figure BDA0002997259830000074
the initial temperature rise distribution is:
Figure BDA0002997259830000075
the temperature rise changes with time and space as follows:[9]
Figure BDA0002997259830000076
the numerical simulation was set as follows. The rectangular parallels have equal side lengths, a, b, c, 10cm, and a density rho0＝1000kg/m3Coefficient of thermal conductivity κt0.6W/(m.K), specific heat Ct＝4180J/(kg·K)，
Figure BDA0002997259830000077
2. The numerical calculation error is characterized by using the 2-norm and the infinity norm as follows:
Figure BDA0002997259830000081
wherein the content of the first and second substances,
Figure BDA0002997259830000082
the numerical solution of the temperature rise is shown,
Figure BDA0002997259830000083
represents an analytic solution to the temperature rise of equation (9).
And setting the time step length to satisfy that the delta t is 0.0001s and keeping the time step length unchanged, and analyzing the influence of the space step length on the numerical calculation result of the temperature rise. The 2-norm and ∞ -norm of the error in temperature rise as a function of the spatial step are shown in fig. 1, with the error in the numerical calculation of the temperature rise increasing with increasing spatial step. The fitted curve of the error characterized by the norm as a function of the spatial step distribution is also plotted in fig. 1, and the results of the fitted curve show that the numerical calculation results of the temperature rise are indeed second-order accurate in space.
And similarly, setting the space step length as delta x being 0.001m and keeping the space step length unchanged, and analyzing the influence of the time step length on the numerical calculation result of the temperature rise. The variation of the 2-norm and the ∞ -norm of the error of the normalized temperature rise with the time step is shown in fig. 2, the error of the numerical calculation of the temperature rise likewise increases with the increase of the time step, and the results of fitting the curve also confirm that the results of the numerical calculation of the temperature rise are indeed of first order accuracy in time.
The space step length and the time step length are set to be Δ x-0.005 m and Δ t-0.1 s, respectively, and the total calculation time is 2500 s. Fig. 3 shows a curve of the temperature rise of the area center over time obtained by numerical simulation calculation and a curve of the temperature rise of the area center over time obtained by theoretical calculation using equation (9), from which it can be clearly seen that the two curves have a very high goodness of fit, and the numerical simulation result and the analysis result have very good consistency.
Example 2
A simulator comprises an acquisition module, a time domain finite difference method solving module with spatial second-order precision and a time domain finite difference method solving module with temporal first-order precision;
the acquisition module is used for acquiring a Pennes biological heat conduction equation;
the Pennes biological heat conduction equation:
Figure BDA0002997259830000084
wherein T represents a biological tissue temperature; kappatRepresents a thermal conductivity coefficient; ctRepresents the specific heat of the biological tissue; cbRepresents the vascular specific heat; wbRepresenting the blood perfusion rate of capillaries within the biological tissue; t isaRepresents the initial temperature; qvRepresents the amount of heat absorbed by the tissue per unit volume per unit time; qmRepresenting the heat generation rate of the biological metabolism, Q when calculatedmAnd is generally negligible.
Get
Figure BDA0002997259830000091
Indicating the rise in temperature, ignoring QmGenerally, the starting temperature is a constant temperature, and the Pennes biological heat transfer equation is expressed as:
Figure BDA0002997259830000092
for focused ultrasound temperature field simulation, QvNamely external heat source input brought to biological tissues by focused ultrasound; for a single frequency stationary focused ultrasound field, an ultrasound heat source QvIs represented as follows:
Qv＝αIav＝αρ0c0<V2>＝1/2×αρ0c0V0 2
wherein α represents a sound absorption coefficient; i isavRepresents the average sound intensity; c. C0Represents the wave velocity; v represents the particle vibration velocity;<*>the periodic average of the variables in the device is shown; v0Representing the velocity of particle vibrationThe magnitude of the degree.
The solution module of the finite difference method of the time domain with the spatial second-order precision and the solution module of the finite difference method of the time first-order precision respectively carry out solution coupling on the Pennes biological heat conduction equation, and the temperature rise effect caused by the simulated focused ultrasound is realized.
The time domain finite difference method solving module with the spatial second-order precision realizes the following solving:
for a unary function f (x), where x is an argument, the second-order center-of-precision difference format for its second derivative is:
Figure BDA0002997259830000093
wherein f is(2)|iThe superscript (2) in (1) denotes the second derivative of the unary function f (x), f $iThe index i in (a) indicates the value (x) of the argument corresponding to the ith grid point0+iΔx)，x0Is the initial value of the argument, Δ x is the difference of the arguments between adjacent grid points;
the temperature rises at different time and different space grid point positions are respectively defined as
Figure BDA0002997259830000094
Superscript n denotes the time coordinate (t)0+ n Δ t), where t is0Denotes the starting time, Δ t denotes the time step of the numerical simulation; the indices i, j, k denote the cartesian space coordinates (x)0+iΔx,y0+jΔy,z0+ k Δ z) where (x)0,y0,z0) Denotes the starting position, and Δ x, Δ y, Δ z denote the spatial step in the x, y, z directions, respectively.
The time domain finite difference method solving module with the time first-order precision realizes the following solving:
the explicit time-discrete format of equation (2) is:
Figure BDA0002997259830000101
second spatial derivative in equation (4)
Figure BDA0002997259830000102
The difference format of (3) uses equation (3) such that there is:
Figure BDA0002997259830000103
example 3
A computer system comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the method when executing the computer program as follows:
s1: obtaining a numerical simulation equation of a focused ultrasound temperature field; wherein, the numerical simulation equation of the focused ultrasound temperature field is expressed by a Pennes biological heat conduction equation;
s2: and solving a numerical simulation equation in S1 by adopting a time domain finite difference method with spatial second-order precision and temporal first-order precision to realize the temperature rise effect caused by simulated focused ultrasound.
It should be understood that the above-described embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the claims of the present invention.

Claims (10)

1. A method for simulating a focused ultrasound temperature field based on FDTD is characterized in that: the method comprises the following steps:
s1: obtaining a numerical simulation equation of a focused ultrasound temperature field; wherein, the numerical simulation equation of the focused ultrasound temperature field is expressed by a Pennes biological heat conduction equation;
s2: and solving a numerical simulation equation in S1 by adopting a time domain finite difference method with spatial second-order precision and temporal first-order precision to realize the temperature rise effect caused by simulated focused ultrasound.
2. The FDTD-based simulation focused ultrasound temperature field method of claim 1, wherein: the Pennes biological heat conduction equation:
Figure FDA0002997259820000011
wherein T represents a biological tissue temperature; kappatRepresents a thermal conductivity coefficient; ctRepresents the specific heat of the biological tissue; cbRepresents the vascular specific heat; wbRepresenting the blood perfusion rate of capillaries within the biological tissue; t isaRepresents the initial temperature; qvRepresents the amount of heat absorbed by the tissue per unit volume per unit time; qmRepresenting the rate of heat generation by biological metabolism.
3. The FDTD-based simulation focused ultrasound temperature field method of claim 2, wherein: get
Figure FDA0002997259820000012
Indicating the rise in temperature, ignoring QmGenerally, the starting temperature is a constant temperature, and the Pennes biological heat transfer equation is expressed as:
Figure FDA0002997259820000013
for focused ultrasound temperature field simulation, QvNamely external heat source input brought to biological tissues by focused ultrasound; for a single frequency stationary focused ultrasound field, an ultrasound heat source QvIs represented as follows:
Figure FDA0002997259820000014
wherein α represents a sound absorption coefficient; i isavRepresents the average sound intensity; c. C0Represents the wave velocity; v represents the particle vibration velocity;<*>the periodic average of the variables in the device is shown; v0Representing the particle vibration velocity magnitude.
4. The FDTD-based simulation focused ultrasound temperature field method according to claim 3, wherein: the time domain finite difference method of the spatial second-order precision specifically adopts spatial grid points to divide an analog simulation area, and the distance between the adjacent spatial grid points is 0.1 lambda, wherein lambda represents the wavelength of ultrasonic waves.
5. The FDTD-based simulation focused ultrasound temperature field method according to claim 4, wherein: the time domain finite difference method of the spatial second-order precision concretely solves the following steps:
for a unary function f (x), where x is an argument, the second-order center-of-precision difference format for its second derivative is:
Figure FDA0002997259820000021
wherein f is(2)|iThe superscript (2) in (1) denotes the second derivative of the unary function f (x), f $iThe index i in (a) indicates the value (x) of the argument corresponding to the ith grid point0+iΔx)，x0Is the initial value of the argument, Δ x is the difference of the arguments between adjacent grid points;
the temperature rises at different time and different space grid point positions are respectively defined as
Figure FDA0002997259820000022
Superscript n denotes the time coordinate (t)0+ n Δ t), where t is0Denotes the starting time, Δ t denotes the time step of the numerical simulation; the indices i, j, k denote the cartesian space coordinates (x)0+iΔx,y0+jΔy,z0+ k Δ z) where (x)0,y0,z0) Denotes the starting position, and Δ x, Δ y, Δ z denote the spatial step in the x, y, z directions, respectively.
6. The FDTD-based simulation focused ultrasound temperature field method according to claim 4, wherein: the time domain finite difference method with the time first-order precision concretely solves the following steps:
the explicit time-discrete format of equation (2) is:
Figure FDA0002997259820000023
second spatial derivative in equation (4)
Figure FDA0002997259820000024
The difference format of (3) uses equation (3) such that there is:
Figure FDA0002997259820000025
7. a simulator based on the method of any one of claims 1 to 6, wherein: the simulator comprises an acquisition module, a time domain finite difference method solving module with spatial second-order precision and a time domain finite difference method solving module with temporal first-order precision;
the acquisition module is used for acquiring a Pennes biological heat conduction equation;
the module for solving the Pennes biological heat conduction equation is respectively solved by the module for solving the time domain finite difference method with the spatial second-order precision and the module for solving the time domain finite difference method with the temporal first-order precision, so that the temperature rise effect caused by the simulated focused ultrasound is realized.
8. The simulator of claim 7, wherein: the time domain finite difference method solving module with the spatial second-order precision realizes the following solving:
for a unary function f (x), where x is an argument, the second-order center-of-precision difference format for its second derivative is:
Figure FDA0002997259820000031
wherein f is(2)|iThe superscript (2) in (1) denotes the second derivative of the unary function f (x), f $iThe index i in (a) indicates the value (x) of the argument corresponding to the ith grid point0+iΔx)，x0Is the initial value of the argument, Δ x is the difference of the arguments between adjacent grid points;
the temperature rises at different time and different space grid point positions are respectively defined as
Figure FDA0002997259820000032
Superscript n denotes the time coordinate (t)0+ n Δ t), where t is0Denotes the starting time, Δ t denotes the time step of the numerical simulation; the indices i, j, k denote the cartesian space coordinates (x)0+iΔx,y0+jΔy,z0+ k Δ z) where (x)0,y0,z0) Denotes the starting position, and Δ x, Δ y, Δ z denote the spatial step in the x, y, z directions, respectively.
9. The simulator of claim 8, wherein: the time domain finite difference method solving module with the first-order time precision realizes the following solving:
the explicit time-discrete format of equation (2) is:
Figure FDA0002997259820000033
second spatial derivative in equation (4)
Figure FDA0002997259820000034
The difference format of (3) uses equation (3) such that there is:
Figure FDA0002997259820000041
10. a computer system comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, wherein: the processor, when executing the computer program, performs the steps of the method according to any of claims 1 to 6.
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