JPH1144560A - Method for processing open boundary in non-linear wave propagation simulation - Google Patents

Abstract

PROBLEM TO BE SOLVED: To express calculations at an open boundary with accuracy of an equal level to that at the interior by applying a unidirectional propagation equation at the open boundary equivalent to that of an internal point and separating the equation discretely with the use of a difference equation of higher accuracy than at the internal point. SOLUTION: In steps S1-S3 of an internal area process, a bilateral wave equation is selected, then a difference equation at an internal point is applied, and a calculation formula at an internal area is obtained. In steps S4-S6 of an open boundary process, a unidirectional propagation equation equivalent to the wave equation of the internal area is obtained, and a calculation formula at the open boundary is obtained by applying a highly accurate difference equation at an end point. Then in step S7, a calculation model for a seabed configuration, a wave-making boundary, an open boundary, etc., is set. A simulation condition, e.g. a waveform, a lattice interval, a time interval, etc., at the wave- making boundary is input in step S8. A nonlinear wave propagation simulation is executed in step S9.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention belongs to the technical field of non-linear wave propagation simulation in various coastal engineering problems such as port design, tsunami propagation prediction, and coastal wave prediction.

[0002]

2. Description of the Related Art When performing a numerical wave simulation for a sea wave or the like, usually, an open boundary is always provided at a part of a boundary of a calculation region, that is, a numerical value which can be performed only in a finite region where there is no boundary in reality. A virtual boundary arises from applying the simulation.

Conventionally, in the open boundary processing of a linear wave (sine wave), a wave speed obtained by linear theory is applied at an open boundary, assuming that a sine wave propagates at a constant wave speed, and the wave speed is not changed and the wave speed is not changed. Open boundary processing was performed using the relational expression (radiation condition) obtained as the propagation at. In the case of a linear wave, the wave speed is determined by the depth of the water and the cycle of the wave without depending on the amplitude of the wave, and can be calculated in advance.
However, in nonlinear waves, the wave speed depends on the amplitude,
Since the waveform is not a sine wave, such a method cannot be applied. That is, although the amplitude of the wave at the open boundary is obtained as a calculation result for the first time, the wave velocity at the open boundary needs to be accurately known for the calculation, which is not easy. There is also an example in which the radiation condition is applied as it is using the wave velocity according to the linear theory as an approximate value (conventional method 1). There is a problem that can not be.

For this reason, a sponge boundary is used as an open boundary processing method for nonlinear waves (conventional method 2). This incorporates the image of the model aquarium into numerical calculations,
A layer that absorbs and dissipates energy is added to the open boundary.

[0005]

However, in the above-mentioned conventional method 2, it is complicated to incorporate a sponge boundary into a general-purpose differential code, and it is necessary to find a solution within a sponge boundary that is not originally required. A significant increase in computation time for large objects is also a major problem. Further, since energy is absorbed in the sponge layer, there is a problem that if the calculation time is long, internal waves are affected by a kind of diffraction effect.

[0006] In particular, as a major practical problem common to these, when an irregular wave such as a sea wave is targeted,
In order to obtain statistically meaningful results in consideration of irregularities, it is necessary to perform calculations for a considerably long time, but the current processing method cannot perform a sufficient amount of calculations continuously. There is a problem that.

SUMMARY OF THE INVENTION The present invention solves the above-mentioned conventional problems. The principle and numerical values related to an efficient open boundary processing method for smoothly letting a wave out of a calculation region at an open boundary in a nonlinear wave propagation simulation are disclosed. It aims to provide a calculation algorithm.

[0008]

An open boundary is provided for convenience of analysis in a place where there is essentially no boundary. Although waves actually propagate freely, conventional open boundary processing is used. Is forcibly attenuated there or forced out of the analysis area at a predetermined wave speed, which is not natural. That is, unlike a wall or the like, the open boundary is not actually a physical boundary, but it can be said that the physical boundary condition is applied to the open boundary in the conventional open boundary processing. So what are the conditions imposed on the open boundary? Returning to the origin, it is necessary and sufficient to satisfy only the basic equation system, just like the interior points. That is, "at the open boundary, the motion of the water particles follows exactly the same physical laws as the interior points."
This means that the existence of an open boundary cannot be recognized by a wave, so the conditions imposed on the open boundary must be equivalent to the basic equation system describing the physical laws of the interior points.

The feature of the method proposed by the present invention is that the physical quantity at the open boundary also follows exactly the same physical law as that of the interior point, and therefore, an equation equivalent to the interior point, more specifically, at the open boundary, Since only waves traveling from the outside to the outside are allowed and no information from outside is required, use the equation of a wave propagating in one direction corresponding to the wave equation showing a two-way wave, It is to express with the same degree of accuracy as a point.

[0010]

Embodiments of the present invention will be described below with reference to the drawings. FIG. 1 is a diagram for explaining a processing procedure of a nonlinear wave propagation simulation according to the present invention.

First, in steps S1 to S3, a bidirectional wave equation is selected as internal area processing, and then a difference equation at an internal point is applied to obtain a calculation equation in the internal area. Next, in steps S4 to S6, as an open boundary process, a one-way propagation equation equivalent to the wave equation in the internal region was obtained, and a high-precision difference equation at an end point was applied to obtain a calculation expression at the open boundary. Thereafter, in step S7, a calculation model for the seafloor topography, wave-making boundary, open boundary, and the like is set, and in step S8, simulation conditions such as the waveform at the wave-making boundary, lattice spacing, and time interval are input. Perform a propagation simulation.

A specific example of the present invention will be described. Boussine, famous for weakly dispersive and nonlinear wave theory
A procedure for deriving the above-described evaluation formula at the open boundary using the sq equation as an example will be described. The Boussinesq equation is represented by the following (Equation 1).

[0013]

(Equation 1)

Here, u is the vertical average velocity vector, η is the amount of water surface fluctuation from the average water surface, g is the gravitational acceleration, z = −h
(X, y) is the sea floor, and t is time. Where xy
A coordinate system is defined with the plane on the average water surface and the z-axis pointing vertically upward.

Conventionally, a difference equation obtained by discretizing equation (1) is used at an internal point, and the values of u and η are separately combined with equation (1) at the open boundary by the above-described conventional method 1 or conventional method 2. Let it be solved. On the other hand, in the present invention, the internal point is exactly the same as the conventional one, but is equivalent to the equation (1) at the open boundary, but K is the equation of a wave propagating in only one direction.
dV equation, that is,

[0016]

(Equation 2)

It is assumed that the following holds. Note that the subscripts indicate partial derivatives with respect to the variables (time and space).

At the open boundary, equation (1) is equivalent (has the same nonlinearity and dispersibility), but the equation of a wave propagating outside the open boundary is applied. It is a feature of the present invention to use a difference formula that provides higher accuracy.

The difference will be specifically described as follows. First, it is assumed that the first and second derivatives of the function f at the internal points are evaluated using a normal central difference, that is, two adjacent points as follows.

[0020]

(Equation 3)

Where f (I), f _x (I), f
_xx (I) and the like are respective function values at the I-th grid point, and Δx is a grid width. Here, the difference equation regarding x has been described, but the same applies to y.

On the other hand, when N is a node number at an end, a high-precision difference formula for the first and second derivatives of the function f at the end (open boundary) is as follows. The first derivative uses a backward difference using the immediately preceding two points, and the second derivative uses a backward difference using the immediately preceding three points.

[0023]

(Equation 4)

At the open boundary, the KdV equation (Equation 2) is applied, and the differentiation in the equation is discretized using the above high-precision difference equation, which is the calculation equation at the open boundary. When the space derivative is replaced with the time derivative using the relationship u _t = gη _x, the calculation formula is as follows. Note that this equation is a technique for lowering the order of spatial differentiation and is not essential to the present invention.

[0025]

(Equation 5)

The spatial derivative is given by equation (4).
f is u, η, etc., and the value of the corresponding time step is substituted. Δt is a time step.

As described above, when a specific procedure is performed in accordance with the above procedure, a calculation formula at the open boundary is obtained as shown in Expression (5). Then, when the value at the open boundary is calculated by such a calculation formula, the calculation can be performed more stably than in the conventional method. Here, a specific method has been shown using the KdV equation of equation (2) with respect to the Boussinesq equation of equation (1). However, the present invention is not limited to this, and other equation systems may be used. It is clear that the calculation procedure at the open boundary can be obtained by a completely similar procedure. The same physical laws as the interior points should be applied at the open boundary and therefore the same basic equations should be applied, except that at the open boundary only waves that go out can be applied, so the one-way propagation equation is applied And
Further, the originality of the present invention resides in that a high-precision difference equation is adopted so that the same degree of accuracy as the internal points can be obtained in the discretization.

Next, the results and effects of the simulation according to the present invention will be described. As an application example, consider a case where a wave propagates from left to right on a uniform slope having a 1/100 gradient as shown in FIG. In this example, since the water depth becomes shallower with the propagation of the wave, strong nonlinearity appears.

First, as an example of calculation by the conventional open boundary processing, the state of propagation of a wave every 10 seconds is shown in FIG.
It is. The left end generates a regular wave with an amplitude of 1 cm and a period of 2 seconds at the wave-making boundary. The right end is the open boundary. In this calculation, the lattice interval is set to 10 cm, and the time interval is set to 0.05 second. The wave becomes a non-linear wave that is vertically asymmetric on the slope as it propagates, and it passes through the open boundary as it reaches the open boundary at the right end, but the turbulence generated from the right end over time It can be seen that it has propagated to the left end. FIG. 4 is an enlarged view of the waveform near the right end at the time of 60 seconds (third graph from the bottom in FIG. 3). The turbulence generated at the open boundary has already propagated to the left. I understand. Then, this disturbance spreads over the entire calculation area with the passage of time. In the case of calculation for regular waves, in this case, almost steady state was reached in 50 seconds, and there is no problem if this is adopted as a final result. Proves impossible. In fact, when the calculation was continued after this, the waveform was greatly disturbed as shown in FIG. 5 and diverged immediately thereafter.

FIG. 6 shows a comparison of calculation results obtained by using the conventional method and the method of the present invention. Up to 40 seconds, when the wave has not yet reached the open boundary (x = 50 m), there is no notable difference between the two, but when the wave reaches the open boundary and comes out of it, the calculation result by the conventional method Shows that the waveform is considerably disturbed by the error. On the other hand, the calculation result according to the present invention is a relatively well-ordered cunoid wave, and the wave is clearly removed at the open boundary, confirming the effectiveness of the present invention.

FIG. 7 shows a case where an irregular wave having a Bled Schneider / Easy spectrum having a significant wave period T _1/3 = 3S and a significant wave height H _1/3 = 2 cm is input to the terrain of FIG. The calculation results are shown for each time. In this case,
Even when the time step is 120,000 steps, the wave smoothly escapes from the open boundary and is calculated stably. The wave number is about 400, and if such a wave passes, there will be no practical problem. As described above, the open boundary processing method according to the present invention is extremely excellent in practical use, and as a result, it can be said that the simulation of nonlinear irregular waves has come to practical use.

[0032]

As is apparent from the above description, according to the present invention, first, a one-way propagation equation equivalent to the wave equation in the internal region at the open boundary is applied. For example, Boussi inside
In the case of the nesq equation, the KdV equation (one-dimensional) or KP equation (two-dimensional) is used at the open boundary, and in the case of other nonlinear wave equations, the Boussinesq equation is used to calculate the KdV equation or KP equation.
One-way propagation equations can be derived in the same way as the equations are derived. This makes it possible to give a wave to the open boundary by exactly the same mechanism as the internal point. In addition, it is possible to represent only the waves that escape from the calculation area to the outside, and it is possible to satisfy the original function required for the open boundary.

Next, the one-way propagation equation is discretized from the internal points by using a differential equation with higher precision. Since only one side (inside) information is available at the end of the open boundary, the accuracy is lower than that of the inner point. Therefore, at the open boundary, an evaluation formula with higher accuracy than the internal point is used. As a result, the calculation at the end can be performed with the same accuracy as that of the internal point, and the calculation at the open boundary is physically performed in combination with the application of the one-way propagation equation equivalent to the wave equation of the internal region at the open boundary. In addition, the calculation can be performed almost indistinguishably from the inside in terms of calculation accuracy.

[Brief description of the drawings]

FIG. 1 is a diagram for explaining a processing procedure of a nonlinear wave propagation simulation according to the present invention.

FIG. 2 is a diagram showing a calculation model of a seabed topography.

FIG. 3 is a diagram showing a result of a calculation example by the conventional open boundary processing.

FIG. 4 is an enlarged view of a waveform near the right end at 60 seconds in FIG. 3;

FIG. 5 is a diagram showing occurrence of strong turbulence leading to divergence according to a conventional method.

FIG. 6 is a diagram showing a comparison of calculation results using a conventional method and the method of the present invention.

FIG. 7 is a diagram showing a calculation result according to the present invention for a certain irregular wave.

Claims (3)

[Claims] In the processing of an open boundary which is a boundary of a wave propagation calculation area, a one-way propagation equation equivalent to a wave equation of an inner area is applied, and then the one-way propagation equation is applied with higher precision than an inner point. An open boundary processing method in a nonlinear wave propagation simulation, wherein a calculation expression at an open boundary is obtained by discretization using a difference expression. 2. The method according to claim 1, wherein the high-precision difference equation uses a plurality of internal points immediately before the open boundary. 3. The method according to claim 1, wherein the wave equation in the internal region is a Boussinesq equation, and the one-way propagation equation is a KdV equation.