CN112765802A - Method for evolving water wave waveform based on high-order water wave model - Google Patents
Method for evolving water wave waveform based on high-order water wave model Download PDFInfo
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Abstract
A method for evolving a water wave waveform based on a high-order water wave model comprises the steps of constructing the water wave model, carrying out logarithmic transformation, constructing a test function f, evolving the water wave model and determining parameters. The invention considers a complex actual water wave model, combines the waveform characteristics of periodic waves and solitary waves, evolves a high-order high-dimensional water wave model, obtains corresponding parameters and a corresponding symbolic representation form, and shows the physical characteristics of a new waveform corresponding to the model in the motion process and the propagation process. Through simulation experiments, the invention realizes the combination of theoretical analysis and practical problems, enriches the display of the interaction of long waves corresponding to the models, can be used for analyzing the physical characteristics of long-wave models in practical problems and expands the application range of the long waves.
Description
Technical collar city
The invention belongs to the technical field of waveform processing, and particularly relates to determination of an evolution water wave waveform of a high-order water wave model.
Background
In real life, research on a nonlinear high-dimensional high-order water wave model relates to a plurality of natural disciplines such as biology, hydraulics, physics, mechanics, computer disciplines and the like. In the research process, students find that a high-dimensional water wave model can be simulated according to the natural state of water waves, parameters of a high-order high-dimensional water wave model are determined through the evolution of a known wave pattern, a mathematical model of the water waves can be obtained according to parameter analysis, and actual problems are abstracted into the mathematical model which is favorable for research and analysis. The energy change and the propagation state of the high-dimensional high-order water wave can be better observed through the research on the parameters of the mathematical model. Therefore, the research of high-dimensional high-order water wave models is gradually valued by scientists. Previous studies on this type of model were limited to periodic or soliton wave patterns alone. The periodic wave graph can well show the propagation process and physical properties of the periodic wave, but only can show the properties of the periodic wave, and the analysis of multiple angles cannot be achieved. Similarly, the graph of the solitary wavelets can only show the wave change and the propagation process when the solitary wavelets collide, has certain limitation, cannot analyze more complex waveforms, and cannot adapt to more complex natural conditions. For the wave diagram of the low-dimensional water wave model, the process of wave propagation is difficult to show due to low dimensionality, which is not beneficial to the research and analysis of waves.
Disclosure of Invention
The technical problem to be solved by the invention is to overcome the defects of the prior art and provide a method for evolving a water wave waveform based on a high-order water wave model, which is simple, high in operation speed and high in accuracy.
The technical scheme adopted for solving the technical problems comprises the following steps:
(1) construction of water wave model
Constructing a 2+ 1-dimensional 4-order nonlinear water wave model according to the formula (1):
αuyt+βuxxxy+γuxuxy+δuyuxx＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and a space variable x and a space variable yxIs the first order partial derivative of u with respect to x, uyIs the first order partial derivative of u with respect to y, utIs the first order partial derivative of u with respect to t, uxxIs the second order partial derivative of u with respect to x, uyyIs u with respect to yOrder partial derivative uttIs the second order partial derivative of u with respect to t, uxxxxIs the fourth order partial derivative of u with respect to x, uyyyyIs the fourth order partial derivative, u, of yttttIs the fourth order partial derivative of u with respect to t, α, β, γ, ε are rational numbers, the ratio of α to β, γ, ε is 1: 1-5: -5 to-1: -5 to-1.
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf)x (2)
f＝f(x,y,t)
f is a differentiable function with respect to the variables x, y, t, and N is an even number.
(3) Construction of a test function f
The test function f is constructed according to equation (3):
ξi＝kix+wiy+cit，
wherein a isiIs amplitude, kiIs the wave velocity in the x direction, wiWave velocity in the y direction, ciAs the frequency of the wave, i e [1, 2, 3 ]]，ai,ki,wi,ciThe value range of (A): a isi∈[-10,10],ki、wi、ci∈[-10,0)∪(0,10]。
(4) Evolution of water wave models
Obtaining an arbitrary function u with respect to the parameter a according to equation (2)i,ci,wi,kiThe symbol expression of (a):
the formula (4) is to simulate a water wave model by the determined interaction form of the periodic wave and the solitary wave, and find out when a non-blasting waveform exists by combining with the specific waveform display of the actual water wave generation and transmission.
(5) Determining parameters
Step (4), when the long-wave waveform appears in the non-blasting model, finding out the relation existing among partial parameters, assigning the parameters without the relation in a determined range, and obtaining a when the parameters are determined to be consistent with the actual long-wave waveformi,ki,wi,ciSpecific values of the parameters and the final form of equation (4).
In formula (1) of step (1) of constructing a water wave model of the present invention, the optimum ratio of α to β, γ, and ∈ is 1: 1: -3: -3.
In the formula (2) of the logarithmic transformation step (2), N is an even number of-10 to 14.
In the formula (2) of the logarithmic transformation step (2) of the present invention, the value of N is preferably-2.
In formula (3) of step (3) of constructing the test function of the present invention, ciA value of 1, a1A value of 0, a2A value of 1, a3The value is-6.25, k1A value of 1, k2The value is 0.5, k3A value of 0.1, w1Value of 1, w2A value of 0.2, w3The value was 0.04.
The invention combines the period method and the soliton method to generate a new waveform, can better adapt to variable and complex practical application conditions, and can well research the high-dimensional long-wave model by a wide-field method and a direct planning method.
The invention provides a simulation method based on a high-dimensional water wave model, namely a field method and a direct fitting method, which are simpler and can be completed with the assistance of special simulation software aiming at analyzing various existing waveforms of the existing ships, oceans, climates and the like. Through the value taking of the discrete parameters, the physical characteristics of the new waveform corresponding to the model in the motion process and the propagation process are shown, and the combination of theoretical analysis and practical problems is realized. The invention enriches the display of the interaction of the long waves corresponding to the model and expands the application range of the long waves.
Drawings
FIG. 1 is a flowchart of example 1 of the present invention.
FIG. 2 is a three-dimensional view of a 2+1 dimensional 4-order water wave model modeled with a first set of parameters at t-4.
FIG. 3 is a three-dimensional view of a 2+1 dimensional 4-order water wave model modeled with a first set of parameters when t is 0.
FIG. 4 is a three-dimensional view of a 2+1 dimensional 4-order water wave model modeled with a first set of parameters at t 4.
Detailed Description
The present invention will be described in further detail with reference to the following drawings and examples, but the present invention is not limited to these examples.
Example 1
Taking a known BLMP long-wave model as an example, the method for evolving a water wave waveform based on a high-order water wave model of the present embodiment comprises the following steps (see fig. 1):
(1) construction of water wave model
Constructing a 2+ 1-dimensional 4-order nonlinear water wave model according to the formula (1):
αuyt+βuxxxy+γuxuxy+δuyuxx＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and space variables x and yxIs the first order partial derivative of u with respect to x, uyIs the first order partial derivative of u with respect to y, utIs the first order partial derivative of u with respect to t, uxxIs the second order partial derivative of u with respect to x, uyyIs the second order partial derivative of u with respect to y, uttIs the second order partial derivative of u with respect to t, uxxxxIs the fourth order partial derivative of u with respect to x, uxyyyIs the fourth order partial derivative, u, of yttttIs the fourth order partial derivative of u with respect to t, alpha, beta, gamma, epsilon are rational numbers, the ratio of alpha to beta, gamma, epsilon is 1: 1-5: -5 to-1: -5 to-1, the ratio of α to β, γ, ε being 1: 1: -3: -3.
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf)x (2)
f＝f(x,y,t)
wherein f is a differentiable function related to variables x, y, and t, N is an even number, and N takes an even number of-10 to 14, and the value of N in this embodiment is-2.
(3) Construction of a test function f
The test function f is constructed according to equation (3):
ξi＝kix+wiy+cit，
wherein a isiIs amplitude, kiIs the wave velocity in the x direction, wiWave velocity in the y direction, ciAs the frequency of the wave, i e [1, 2, 3 ]]，ai∈[-10,10],ki、wi、ci∈[-10,0)∪(0,10]The values of this embodiment are: c. CiTake 1, a1Take 0, a2Take 1, a3Take-6.25, k1Take 1, k2Take 0.5, k3Take 0.1, w1Taking 1, w2Take 0.2, w30.04 is taken.
(4) Evolution of water wave models
Obtaining a long wave u with respect to the parameter a according to equation (2)i,ci,wi,kiThe symbol expression of (a):
the formula (4) is to simulate a water wave model by the determined interaction form of the periodic wave and the solitary wave, and find out when a non-blasting waveform exists by combining with the specific waveform display of the actual water wave generation and transmission.
(5) Determining parameters
Finding out the relation between partial parameters when the long wave form appears in the non-blasting model, assigning the parameters without relation in the determined range, and determining the long wave formWhen the shapes are consistent, a is obtainedi,ki,wi,ciThe specific values of the parameters and the final form of equation (4) (see fig. 2, 3, 4).
Example 2
Taking a known long-wave model as an example, the method for evolving a water wave waveform based on a high-order water wave model of the embodiment comprises the following steps:
(1) construction of water wave model
Constructing a 2+ 1-dimensional 4-order nonlinear water wave model according to the formula (1):
αuyt+βuxxxy+γuxuxy+δuyuxx＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and a space variable x and a space variable yxIs the first order partial derivative of u with respect to x, uyIs the first order partial derivative of u with respect to y, utIs the first order partial derivative of u with respect to t, uxxIs the second order partial derivative of u with respect to x, uyyIs the second order partial derivative of u with respect to y, uttIs the second order partial derivative of u with respect to t, uxxxxIs the fourth order partial derivative of u with respect to x, uyyyyIs the fourth order partial derivative, u, of yttttIs the fourth order partial derivative of u with respect to t, alpha, beta, gamma, epsilon are rational numbers, the ratio of alpha to beta, gamma, epsilon is 1: 1-5: -5 to-1: -5 to-1, the ratio of α to β, γ, ε being 1: 1: -5: -5.
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf)x (2)
f＝f(x,y,t)
wherein f is a differentiable function related to variables x, y, and t, N is an even number, and N takes an even number of-10 to 14, and the value of N in this embodiment is-10.
(3) Construction of a test function f
The test function f is constructed according to equation (3):
ξi＝kix+wiy+cit，
wherein a isiIs amplitude, kiIs the wave velocity in the x direction, wiWave velocity in the y direction, ciAs the frequency of the wave, i e [1, 2, 3 ]]，ai,ki,wi,ciThe value range of (A): a isi∈[-10,10],ki、wi、ci∈[-10,0)∪(0,10]A of the present embodimentiThe value is 0.1, ki、wi、ciIs 10.
The other steps were the same as in example 1. To obtain ai,ki,wi,ciSpecific values of the parameters and the final form of equation (4).
Example 3
Taking a known long-wave model as an example, the method for evolving a water wave waveform based on a high-order water wave model of the embodiment comprises the following steps:
(1) construction of water wave model
Constructing a 2+ 1-dimensional 4-order nonlinear water wave model according to the formula (1):
αuyt+βuxxxy+γuxuxy+δuyuxx＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and space variables x and yxIs the first order partial derivative of u with respect to x, uyIs the first order partial derivative of u with respect to y, utIs the first order partial derivative of u with respect to t, uxxIs the second order partial derivative of u with respect to x, uyyIs the second order partial derivative of u with respect to y, uttIs the second order partial derivative of u with respect to t, uxxxxIs the fourth order partial derivative of u with respect to x, uyyyyIs the fourth order partial derivative, u, of yttttIs the fourth order partial derivative of u with respect to t, alpha, beta, gamma, epsilon are rational numbers, the ratio of alpha to beta, gamma, epsilon is 1: 1-5: -5 to-1: -5 to-1, the ratio of α to β, γ, ε being 1: 5: -1: -1.
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf)x (2)
f＝f(x,y,t)
wherein f is a differentiable function related to variables x, y, and t, N is an even number, N takes the value of-10 to 14 even numbers, and N takes the value of 14 in this embodiment.
(3) Construction of a test function f
The test function f is constructed according to equation (3):
ξi＝kix+wiy+cit, wherein aiIs amplitude, kiIs the wave velocity in the x direction, wiWave velocity in the y direction, ciAs the frequency of the wave, i e [1, 2, 3 ]]，ai,ki,wi,ciThe value range of (A): a isi∈[-10,10],ki、wi、ci∈[-10,0)∪(0,10]A of the present embodimentiHas a value of 10, ki、wi、ciIs 0.1.
The other steps were the same as in example 1. To obtain ai,ki,wi,ciSpecific values of the parameters and the final form of equation (4).
In order to verify the beneficial effects of the invention, the inventor uses the method for determining the optimal evolution waveform of the high-order water wave model in embodiment 1 of the invention to perform simulation experiments, and the experimental conditions are as follows:
the results of simulation experiments in example 1 are shown in fig. 2, 3, and 4, where fig. 2 is a three-dimensional view of u when the spatial variable t is-4, fig. 3 is a three-dimensional view of u when the spatial variable t is 0, and fig. 4 is a three-dimensional view of u when the spatial variable t is 4. As can be seen from fig. 2, when the space variable t takes-4, the two waves collide with each other, the amplitude decreases, and the energy of the waves decreases. As can be seen from fig. 3, when the spatial variable t takes 0, the two waves still collide, and the propagation direction does not change. As can be seen from fig. 4, when the spatial variable t takes 4, the wave continues to propagate from the original position along the direction, and the energy does not change. Test results show that the long wave propagation process can be completely shown in the embodiment 1 of the invention, and the method in the embodiment 1 is proved to be good in simulation of the model.
Claims (5)
1. A method for evolving water wave waveform based on a high-order water wave model is characterized by comprising the following steps:
(1) construction of water wave model
Constructing a 2+ 1-dimensional 4-order nonlinear water wave model according to the formula (1):
αuyt+βuxxxy+γuxuxy+δuyuxx＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and two space variables x and yxIs the first order partial derivative of u with respect to x, uyIs the first order partial derivative of u with respect to y, utIs the first order partial derivative of u with respect to t, uxxIs the second order partial derivative of u with respect to x, uyyIs the second order partial derivative of u with respect to y, uttIs the second order partial derivative of u with respect to t, uxxxxIs the fourth order partial derivative of u with respect to x, uyyyyIs the fourth order partial derivative, u, of yttttIs a fourth-order partial derivative of u relative to t, alpha, beta, gamma and epsilon are rational numbers, and the ratio of alpha to beta, gamma and epsilon is 1: 1-5: 5-1;
(2) logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf)x (2)
f＝f(x，y，t)
where f is a differentiable function with respect to the variables x, y, t, and N is an even number;
(3) construction of a test function f
The test function f is constructed according to equation (3):
ξi＝kix+wiy+cit，
wherein a isiIs amplitude, kiIs the wave velocity in the x direction, wiWave velocity in the y direction, ciIs the frequency of the wave, i takes the values 1, 2, 3, ai，ki，wi，ciThe value range of (A): a isi∈[0，10]，ki、wi、ci∈(0，10]；
(4) Evolution of water wave models
Obtaining an arbitrary function u with respect to the parameter a according to equation (2)i，ci，wi，kiThe symbol expression of (a):
the formula (4) is that a water wave model is simulated by the determined interaction form of periodic waves and solitary waves, and when non-blasting waveforms exist is found out by combining with the specific waveform display of actual water wave generation and transmission;
(5) determining parameters
Step (4), when the long-wave waveform appears in the non-blasting model, finding out the relation existing among partial parameters, assigning the parameters without the relation in a determined range, and obtaining a when the parameters are determined to be consistent with the actual long-wave waveformi，ki，wi，ciSpecific values of the parameters and the final form of equation (4).
2. The method for evolving water wave waveforms based on higher-order water wave models of claim 1, wherein: in the formula (1) of the step (1) of constructing the water wave model, the ratio of alpha to beta, gamma and epsilon is 1: 3.
3. The method for evolving water wave waveforms based on higher-order water wave models of claim 1, wherein: in the formula (2) in the logarithmic transformation step (2), N is an even number of-10 to 14.
4. The method for evolving water wave waveform based on higher-order water wave model according to claim 1 or 3, wherein: in the formula (2) of the logarithmic transformation step (2), N takes the value of-2.
5. The method for evolving water wave waveforms based on higher-order water wave models of claim 1, wherein: in formula (3) in the step (3) of constructing the test function, ciA value of 1, a1A value of 0, a2A value of 1, a3The value is-6.25, k1A value of 1, k2The value is 0.5, k3A value of 0.1, w1Value of 1, w2A value of 0.2, w3The value was 0.04.
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