CN109741428B - Three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation - Google Patents

Three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation Download PDF

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CN109741428B
CN109741428B CN201910035897.4A CN201910035897A CN109741428B CN 109741428 B CN109741428 B CN 109741428B CN 201910035897 A CN201910035897 A CN 201910035897A CN 109741428 B CN109741428 B CN 109741428B
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朱鉴
李泓澍
刘培钰
张浩晨
陈炳丰
蔡瑞初
郝志峰
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Guangdong University of Technology
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Abstract

The invention provides a three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation. Based on a constrained interpolation profile method, when the convection term of a two-dimensional fluid motion control equation is solved and calculated by using a semi-Lagrange method, a high-precision interpolation method is invented aiming at the calculation of physical quantity at a backspacing point, and the convection precision is improved. In addition, in order to reduce memory consumption, only the value of the physical field and the first derivative thereof are stored as calculation variables, and the high-order derivative is calculated approximately based on the derived Taylor expansion under the premise of ensuring that the calculation precision is not damaged. The invention can keep the third-order high precision on the premise of less time and memory consumption and has the characteristic of tight template. Compared with the existing method, the method has obvious improvement on the aspects of visual quality, speed, memory consumption and the like, and can effectively improve the convection accuracy and speed of fluid simulation. Besides fluid simulation, the invention can be used in other fields requiring high interpolation precision, such as image/video super-resolution and the like.

Description

Three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation
Technical Field
The invention relates to the technical field of computer animation, in particular to a three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation, which can effectively reduce convection numerical dissipation, reduce memory overhead and enhance the sense of reality of fluid animation.
Background
In the fields of computer graphics and virtual reality, physical-based computer animation technology is always the research focus of researchers, and physical-based fluid dynamic simulation technology is a hot problem. Fluid simulation techniques are also widely used in many fields, and the development of engineering fields such as aerospace, aviation, navigation and the like is not independent of the support of fluid techniques.
In fluid simulation, the numerical dissipation increases the viscosity of the fluid, making it more viscous than expected, and smears detail such that visual quality is compromised. Among factors causing numerical dissipation, accuracy of convection has a great influence on visual quality of fluid simulation, and related scholars make various attempts to develop an accurate convection solver. Among them, the high-order interpolation format is widely applied to the field of Computational Fluid Dynamics (CFD), including intrinsic non-oscillation (ENO), weighted ENO (weno), etc., however, the amount of computation is too large to be applied to the field of graphics, and these interpolation algorithms are performed on a wide template, and are not suitable for simulation on a non-uniform grid. The Linear method (Linear), the forward and backward error compensation correction method (BFECC), the MacCormack method and other convection methods are widely used in graphics, but the calculation accuracy is limited, and the highest accuracy can only reach second-order accuracy.
The Constrained Interpolation Profile (CIP) method only constructs an interpolation function in a single grid unit, has a tight template characteristic and has third-order precision, but expanding the CIP to solve a high-dimensional convection equation is a difficult task and needs a large amount of computing time and memory consumption. In addition, the multi-dimensional convection solvers based on CIP in the prior art, such as monotonic CIP (mcip), non-split CIP (uscip), etc., can only partially solve these problems, some at the cost of losing numerical accuracy or causing instability. How to develop an efficient and high-precision high-dimensional CIP scheme is significant for improving the computational efficiency and precision of the convection method, but is still a challenging problem.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation, which comprises the following steps: two-dimensional Taylor Expansion based Constrained Interpolation Profile (2d Taylor Expansion based Constrained Interpolation Profile, 2 dTECIP). When the existing CIP-based method is applied to high dimension, the problems of high calculation overhead, large memory occupation, low precision or instability and the like generally exist. The method only takes the physical quantity and the first derivative thereof as unknown variables, and the high-order derivative is not saved as variables, so that a large amount of memory overhead can be reduced. Secondly, the high-order derivative is calculated according to the requirement by using local Taylor expansion, the calculation speed is high, the third-order precision is kept, and abundant fluid details can be effectively kept. Finally, the method does not destroy the tight template property of the original CIP method, and is also suitable for non-uniform grids except for regular grids.
The technical scheme of the invention is as follows: the invention discloses a three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation, which is mainly used for improving the precision of solving a convection item in a fluid motion control equation. When the convection term of the fluid motion control equation is solved by using a semi-Lagrange method, high-precision interpolation is adopted for calculation of physical quantity at a backspacing point, numerical value dissipation is reduced, and meanwhile, in order to reduce memory consumption, only the value of a physical field and the first derivative thereof are stored as calculation variables.
For a two-dimensional convection equation, let φ be a convection physical quantity, 3 variables are stored at each grid point i, i.e. the physical quantity and its first-order partial derivative: phi is ai、
Figure BDA0001945895740000021
And the corresponding value at the backspacing point is obtained by high-precision interpolation. The method comprises the following steps:
s1), setting the side length of a grid unit as h, setting the four grid points as A, B, C, D, setting all sides of the grid unit to be parallel to the coordinate axis of the two-dimensional space, setting the distance between the side AB and the side CD to be parallel to the X axis and the distance between the side AB and the side CD to be h along the positive direction of the Y axis, and setting the distance between the side AD and the side BC to be parallel to the Y axis and the distance between the side AD and the side BC to be h along the positive direction of the X axis. Setting points E and G as projection points of the point P on the sides AB and AD respectively;
s2), calculating second-order partial derivatives of phi at the point A, the point B and the point D
Figure BDA0001945895740000022
The value of (c). The specific calculation is as follows:
based on the following third order precision taylor expansion,
Figure BDA0001945895740000023
let Δ x be Δ y be h, calculate the second order partial derivative at grid point a
Figure BDA0001945895740000024
The values, one can obtain:
Figure BDA0001945895740000025
similarly, let Δ x be-h and Δ y be h, calculate the second-order partial derivative at grid point B
Figure BDA0001945895740000026
The values, one can obtain:
Figure BDA0001945895740000027
similarly, let Δ x be h and Δ y be-h, calculate the second-order partial derivative at grid point D
Figure BDA0001945895740000028
The values, one can obtain:
Figure BDA0001945895740000029
s3), by phi at grid points a and B based on one-dimensional CIP interpolation algorithmA、φB、
Figure BDA00019458957400000210
Calculate phi of point EE、
Figure BDA00019458957400000211
Passing through points A and B
Figure BDA00019458957400000212
Calculating point E
Figure BDA00019458957400000213
By phi at grid points A and DA、φD、
Figure BDA00019458957400000214
Calculate phi at point GG、
Figure BDA00019458957400000215
Through grid points A and D
Figure BDA0001945895740000031
Calculating at point G
Figure BDA0001945895740000032
S4), calculating phi of the P point according to the following formulaP、
Figure BDA0001945895740000033
Figure BDA0001945895740000034
Figure BDA0001945895740000035
Figure BDA0001945895740000036
Where ξ and η are the lengths of AE and AG, respectively.
Further, based on the one-dimensional CIP interpolation algorithm, pass phi at the grid points a and B in step S3)A、φB、
Figure BDA0001945895740000037
Figure BDA0001945895740000038
Calculate phi at point EE、
Figure BDA0001945895740000039
The concrete formula is as follows:
φE＝C3(xE-xA)3+C2(xE-xA)2+C1(xE-xA)+C0；
Figure BDA00019458957400000310
wherein, C0＝φA，
Figure BDA00019458957400000311
xEDenotes the abscissa, x, of point EAAnd (4) representing the abscissa of the point A, and h is the side length of a grid unit.
Further, based on the one-dimensional CIP interpolation algorithm, through A and B in step S3)
Figure BDA00019458957400000312
Calculating at point E
Figure BDA00019458957400000313
The concrete formula is as follows:
Figure BDA00019458957400000314
Figure BDA00019458957400000315
wherein the content of the first and second substances,
Figure BDA00019458957400000316
Figure BDA00019458957400000317
xEdenotes the abscissa, x, of point EAAnd (4) representing the abscissa of the point A, and h is the side length of a grid unit.
Further, based on the one-dimensional CIP interpolation algorithm, pass phi at the grid points a and D in step S3)A、φD、
Figure BDA00019458957400000318
Figure BDA00019458957400000319
Calculate phi at point GG、
Figure BDA00019458957400000320
The concrete formula is as follows:
φG＝C3(yG-yA)3+C2(yG-yA)2+C1(yG-yA)+C0；
Figure BDA0001945895740000041
wherein, C0＝φA，
Figure BDA0001945895740000042
yGDenotes the ordinate, y, of the G pointAAnd (4) representing the ordinate of the point A, and h is the side length of the grid unit.
Further, in step S3), based on the one-dimensional CIP interpolation algorithm, the grid points a and D are passed
Figure BDA0001945895740000043
Figure BDA0001945895740000044
Calculating at point G
Figure BDA0001945895740000045
The specific formula is as follows:
Figure BDA0001945895740000046
Figure BDA0001945895740000047
wherein the content of the first and second substances,
Figure BDA0001945895740000048
Figure BDA0001945895740000049
yGdenotes the ordinate, y, of the G pointAAnd (4) representing the ordinate of the point A, and h is the side length of the grid unit.
The invention has the beneficial effects that:
the invention can keep the third-order high precision on the premise of less time and memory consumption and has the characteristic of tight template. Compared with the existing method, the method has obvious improvement on the aspects of visual quality, speed, memory consumption and the like, and can effectively improve the convection accuracy and speed of fluid simulation. Furthermore, the present invention can be used in other fields requiring high interpolation accuracy, such as image/video super-resolution, etc., besides fluid simulation.
Drawings
FIG. 1 is a schematic illustration of interpolation for two-dimensional fluid simulation according to the present invention;
FIG. 2 is a flow chart of the calculation of the method of the present invention applied to a fluid simulation;
fig. 3 is a comparison of rendering graphs of two-dimensional fluid simulation results, in which (a), (b), (c), (d), and (e) are rendering graphs of fluid simulation results of a Linear and BFECC algorithm, an MCIP and USCIP algorithm, and the method (2dTECIP) of the present invention, respectively, using low-order precision.
Detailed Description
The following further describes embodiments of the present invention with reference to the accompanying drawings:
the invention provides a three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation, which is mainly used for improving the precision of calculating and solving convection terms in a fluid motion control equation so as to improve the precision of fluid simulation. When the convection term of the fluid motion control equation is solved by using a semi-Lagrange method, high-precision interpolation is adopted for calculation of physical quantity at a backspacing point, numerical value dissipation is reduced, and meanwhile, in order to reduce memory consumption, only the value of a physical field and a first derivative thereof are stored as calculation variables, and a high-order derivative is calculated approximately based on a derived Taylor expansion on the premise of ensuring that calculation precision is not damaged.
For a two-dimensional convection equation, let φ be a convection physical quantity, 3 variables are stored at each grid point i, i.e. the physical quantity and its first-order partial derivative: phi is ai、
Figure BDA0001945895740000051
And the corresponding value at the backspacing point is obtained by high-precision interpolation. The method comprises the following steps:
s1), for the taylor expansion:
Figure BDA0001945895740000052
when the value of Δ x is 0,
Figure BDA0001945895740000053
Figure BDA0001945895740000054
when the value of y is equal to 0,
Figure BDA0001945895740000055
Figure BDA0001945895740000056
by combining the above 5 equations and neglecting the remainder of the third order or more, the following taylor expansion equation of the third order precision is obtained:
Figure BDA0001945895740000057
after finishing, obtaining:
Figure BDA0001945895740000058
s2), as shown in fig. 1, for the two-dimensional space grid cell where the back-off point P is located, let the side length of the grid cell be h, and the point E, G be the projection points of P on the sides AB and AD, respectively. Value phi of P point positionP、
Figure BDA0001945895740000059
The corresponding values of the four grid points A, B, C, D of the grid cell are obtained through the invented high-precision interpolation;
s3), calculating second-order partial derivatives of phi at the point A, the point B and the point D
Figure BDA00019458957400000510
The value of (c) is specifically calculated as follows:
based on the taylor expansion equation obtained in step S1), let Δ x be Δ y be h, the second-order partial derivative at grid point a is calculated
Figure BDA0001945895740000061
The values, one can obtain:
Figure BDA0001945895740000062
similarly, let Δ x be-h and Δ y be h, calculate the second-order partial derivative at grid point B
Figure BDA0001945895740000063
The values, one can obtain:
Figure BDA0001945895740000064
similarly, let Δ x be h and Δ y be-h, calculate the second-order partial derivative at grid point D
Figure BDA0001945895740000065
The values, one can obtain:
Figure BDA0001945895740000066
s4), by phi at grid points a and B based on one-dimensional CIP interpolation algorithmA、φB、
Figure BDA0001945895740000067
Calculate phi at point EE、
Figure BDA0001945895740000068
The calculation formula is as follows:
φE＝C3(xE-xA)3+C2(xE-xA)2+C1(xE-xA)+C0；
Figure BDA0001945895740000069
wherein, C0＝φA，
Figure BDA00019458957400000610
xEDenotes the abscissa, x, of point EAAnd (4) representing the abscissa of the point A, and h is the side length of a grid unit.
Passing through points A and B
Figure BDA00019458957400000611
Calculating at point E
Figure BDA00019458957400000612
The calculation formula is as follows:
Figure BDA00019458957400000613
Figure BDA00019458957400000614
wherein the content of the first and second substances,
Figure BDA00019458957400000615
Figure BDA00019458957400000616
xEis the abscissa of point E, xAThe abscissa of the point A is, and h is the side length of the grid unit.
By phi at grid points A and DA、φD、
Figure BDA00019458957400000617
Calculate phi at point GG、
Figure BDA00019458957400000618
The calculation formula is as follows:
φG＝C3(yG-yA)3+C2(yG-yA)2+C1(yG-yA)+C0；
Figure BDA00019458957400000619
wherein, C0＝φA，
Figure BDA0001945895740000071
yGDenotes the ordinate, y, of the G pointAAnd (4) representing the ordinate of the point A, and h is the side length of the grid unit.
Passing through grid pointsAt A and D
Figure BDA0001945895740000072
Calculating point G
Figure BDA0001945895740000073
The specific calculation formula is as follows:
Figure BDA0001945895740000074
Figure BDA0001945895740000075
wherein the content of the first and second substances,
Figure BDA0001945895740000076
Figure BDA0001945895740000077
yGdenotes the ordinate, y, of the G pointAAnd (4) representing the ordinate of the point A, and h is the side length of the grid unit.
S5), calculating phi at the point P,
Figure BDA0001945895740000078
the values are specifically calculated as follows:
based on the taylor expansion equation obtained in step S1),
Figure BDA0001945895740000079
let Δ x be ξ and Δ y be η, the value at point a can be found
Figure BDA00019458957400000710
The value:
Figure BDA00019458957400000711
let Δ x ═ ξ, Δ y ═ η, one can obtain the value at point E
Figure BDA00019458957400000712
The value:
Figure BDA00019458957400000713
let Δ x be ξ and Δ y be η, the value at point G being obtained
Figure BDA00019458957400000714
The value:
Figure BDA00019458957400000715
rearranging the above three formulas to obtain the following formula to calculate phi at the point P,
Figure BDA00019458957400000716
Figure BDA0001945895740000081
Figure BDA0001945895740000082
Figure BDA0001945895740000083
referring to fig. 2, the flow chart of the fluid simulation is shown, and the convection interpolation method provided by the present invention is mainly used for calculating the convection term of the fluid simulation. Fig. 3 is a comparison of rendering graphs of two-dimensional fluid simulation results, fig. 3(e) is a rendering graph of results of the method (2dTECIP) of the present invention for two-dimensional fluid simulation, fig. 3(a), fig. 3(b), fig. 3(c), and fig. 3(d) are rendering graphs of fluid simulation results of Linear and BFECC algorithms, MCIP and USCIP algorithms, respectively, using low-order precision, and it can be seen from the graphs that the simulation effect of three methods based on CIP is better than that of Linear and BFECC algorithms with low-order precision. Due to the defects of the MCIP and USCIP methods, the method achieves better effect in two-dimensional fluid simulation than the MCIP and USCIP methods, and the simulated smoke is richer in detail.
The foregoing embodiments and description have been presented only to illustrate the principles and preferred embodiments of the invention, and various changes and modifications may be made therein without departing from the spirit and scope of the invention as hereinafter claimed.

Claims (5)

1. A three-order high-precision convection interpolation algorithm suitable for two-dimensional fluid simulation is characterized in that: when the convection term of the fluid motion control equation is solved by using a semi-Lagrange method, high-precision interpolation is adopted for calculation of physical quantity at a backspacing point, numerical value dissipation is reduced, and meanwhile, in order to reduce memory consumption, only the value of a physical field and the first derivative thereof are stored as calculation variables, and the high-order derivative is approximately calculated based on a derived Taylor expansion formula on the premise of ensuring that the calculation precision is not damaged;
for a two-dimensional convection equation, let φ be a convection physical quantity, 3 variables are stored at each grid point i, i.e. the physical quantity and its first-order partial derivative: phi is ai、
Figure FDA0001945895730000011
The corresponding value on the backspacing point is obtained through high-precision interpolation; the method comprises the following steps:
s1), setting the side length of a grid unit as h, setting the four grid points as A, B, C, D, wherein the side length of the grid unit is h, all sides of the grid unit are parallel to the coordinate axis of a two-dimensional space, the side AB and the side CD are parallel to the X axis, the distance between the side AB and the side CD along the positive direction of the Y axis is h, and the side AD and the side BC are parallel to the Y axis, and the distance between the side AD and the side BC along the positive direction of the X axis is h; setting points E and G as projection points of the point P on the sides AB and AD respectively;
s2), calculating second-order partial derivatives of phi at the point A, the point B and the point D
Figure FDA0001945895730000012
The value of (c) is specifically calculated as follows:
based on the following third order precision taylor expansion,
Figure FDA0001945895730000013
let Δ x be Δ y be h, calculate the second order partial derivative at grid point a
Figure FDA0001945895730000014
The values, one can obtain:
Figure FDA0001945895730000015
similarly, let Δ x be-h and Δ y be h, calculate the second-order partial derivative at grid point B
Figure FDA0001945895730000016
The value of (c) can be given as:
Figure FDA0001945895730000017
similarly, let Δ x be h and Δ y be-h, calculate the second-order partial derivative at grid point D
Figure FDA0001945895730000018
The value of (c) can be given as:
Figure FDA0001945895730000019
s3), based on one-dimensional Constrained Interpolation Profile (CIP) interpolation algorithm, through phi at grid points A and BA、φB、
Figure FDA00019458957300000110
Figure FDA00019458957300000111
Calculate phi of point EE、
Figure FDA00019458957300000112
Passing through grid points A and B
Figure FDA00019458957300000113
Calculating point E
Figure FDA00019458957300000114
Figure FDA00019458957300000115
By phi at grid points A and DA、φD、
Figure FDA0001945895730000021
Calculate phi at point GG、
Figure FDA0001945895730000022
Through grid points A and D
Figure FDA0001945895730000023
Calculating at point G
Figure FDA0001945895730000024
S4), calculating phi of the P point according to the following formulaP、
Figure FDA0001945895730000025
Figure FDA0001945895730000026
Figure FDA0001945895730000027
Figure FDA0001945895730000028
Where ξ and η are the lengths of AE and AG, respectively.
2. The three-order high-precision convective interpolation algorithm suitable for two-dimensional fluid simulation according to claim 1, wherein: step S3), based on the one-dimensional CIP interpolation algorithm, pass phi at the grid points a and BA、φB、
Figure FDA0001945895730000029
Calculate phi at point EE、
Figure FDA00019458957300000210
The concrete formula is as follows:
φE＝C3(xE-xA)3+C2(xE-xA)2+C1(xE-xA)+C0；
Figure FDA00019458957300000211
wherein, C0＝φA，
Figure FDA00019458957300000212
xEDenotes the abscissa, x, of point EAAnd (4) representing the abscissa of the point A, and h is the side length of a grid unit.
3. The three-order high-precision convection current plug suitable for two-dimensional fluid simulation of claim 1Value algorithm, characterized by: step S3), based on the one-dimensional CIP interpolation algorithm, passing through the grid points a and B
Figure FDA00019458957300000213
Calculating at point E
Figure FDA00019458957300000214
The concrete formula is as follows:
Figure FDA00019458957300000215
Figure FDA00019458957300000216
wherein the content of the first and second substances,
Figure FDA00019458957300000217
Figure FDA00019458957300000218
xEdenotes the abscissa, x, of point EAAnd (4) representing the abscissa of the point A, and h is the side length of a grid unit.
4. The three-order high-precision convective interpolation algorithm suitable for two-dimensional fluid simulation according to claim 1, wherein: step S3), based on the one-dimensional CIP interpolation algorithm, pass phi at the grid points a and DA、φD、
Figure FDA0001945895730000031
Calculate phi at point GG、
Figure FDA0001945895730000032
The concrete formula is as follows:
φG＝C3(yG-yA)3+C2(yG-yA)2+C1(yG-yA)+C0；
Figure FDA0001945895730000033
wherein, C0＝φA，
Figure FDA0001945895730000034
yGDenotes the ordinate, y, of the G pointAAnd (4) representing the ordinate of the point A, and h is the side length of the grid unit.
5. The three-order high-precision convective interpolation algorithm suitable for two-dimensional fluid simulation according to claim 1, wherein: step S3), based on the one-dimensional CIP interpolation algorithm, pass through the grid points a and D
Figure FDA0001945895730000035
Calculating point G
Figure FDA0001945895730000036
The specific formula is as follows:
Figure FDA0001945895730000037
Figure FDA0001945895730000038
wherein the content of the first and second substances,
Figure FDA0001945895730000039
Figure FDA00019458957300000310
yGrepresenting the ordinate of the G point，yAAnd (4) representing the ordinate of the point A, and h is the side length of the grid unit.
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