CN109741428A - A kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation - Google Patents
A kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation Download PDFInfo
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Abstract
The present invention provides a kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation.Based on constraint Interpolation Profile method, when solving the convective term for calculating two dimensional fluid motion control equation using Semi -Lagrangian method, a kind of high-precision interpolation method has been invented in the calculating of physical quantity at rollback point, improves convection current precision.In addition, the value and its first derivative of storage physical field are as calculating variable, the higher derivative then Taylor expansion approximate calculation based on derivation under the premise of guaranteeing that computational accuracy is not impaired to reduce memory consumption.The present invention can keep three ranks high-precision under the premise of time and less memory consumption, and have tight template properties.Compare existing method, this method visual quality, speed, in terms of be significantly improved, be capable of effectively lifting fluid simulation convection current accuracy and speed.Other than fluid simulation, the present invention can also be used to require other field, such as image/video super-resolution of high interpolation precision etc..
Description
Technical field
The present invention relates to Computer Animated Graph fields, are a kind of three rank high-precision convection current suitable for two dimensional fluid simulation
Convection current numerical dissipation can be effectively reduced in interpolation algorithm, reduce memory overhead, enhance the sense of reality of fluid animation.
Background technique
In computer graphics and field of virtual reality, the Computer Animated Graph based on physics is researcher always
Research emphasis, and the fluid dynamic simulation technology based on physics is then one of hot issue.Fluid simulation technology exists
Multiple fields are also widely used, and the development of engineering field such as space flight, aviation, navigation etc. also be unable to do without the support of fluid technique.
In fluid simulation, numerical dissipation increases the viscosity of fluid, keeps it more viscous than expected, and smear details with
Cause visual quality impaired.Cause in the factor of numerical dissipation, the precision of convection current has very big shadow to the visual quality of fluid simulation
It rings, related scholar is to develop accurate convection current solver to have carried out various trials.Wherein, high-order interpolation format is widely used in counting
The field fluid operator mechanics (CFD), including essential non-oscillatory (ENO), weighting ENO (WENO) etc., however too very much not due to calculation amount
Suitable for graphics field, and these interpolation algorithms carry out in wide template, are not suitable for being imitated on non-uniform grid
Very.Linear method (Linear), front and back are extensive to convective methods such as error compensation antidote (BFECC), MacCormack methods
For in graphics, but computational accuracy is limited, and highest only reaches second order accuracy.
Interpolation Profile (CIP) method of constraint only constructs interpolating function in single grid cell, has tight template properties, and
With third-order, but extending CIP to solve higher-dimension convection equation is a difficult task, needs largely to calculate time and interior
Deposit consumption.In addition, the multidimensional convection current solver based on CIP in the prior art, such as dullness CIP (MCIP), nondividing type CIP
(USCIP) etc., it can only partly solve these problems, some are even to lose numerical precision or shakiness is caused to be set to cost.Such as
It is quite significant for the computational efficiency and precision that promote convective methods what develops a kind of efficient, high-precision higher-dimension CIP scheme, but
It is still a challenging problem.
Summary of the invention
In view of the deficiencies of the prior art, it is slotting to provide a kind of three rank high-precision convection current suitable for two dimensional fluid simulation by the present invention
Value-based algorithm: constraint Interpolation Profile method (2d Taylor Expansion based Constrained of the two dimension based on Taylor expansion
Interpolation Profile, 2dTECIP).When the existing method based on CIP is applied to higher-dimension, usually there is computing cost
The problems such as height, EMS memory occupation are big, precision reduces or is unstable.The method of the present invention is only using physical quantity and its first derivative as not
Know variable, higher derivative is not intended as variable save, it is possible to reduce a large amount of memory overheads.Secondly, the present invention uses local Taylor
On-demand computing higher derivative is unfolded, calculating speed is very fast and keeps third-order, can more effectively keep fluid details abundant.
Finally, method does not destroy the tight template properties of former CIP method, non-uniform grid is equally applicable in addition to regular grid.
The technical solution of the present invention is as follows: a kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation, this
Invention is mainly used for improving the precision for solving convective term in fluid motion governing equation.By solving stream using Semi -Lagrangian method
When the convective term of body motion control equation, the calculating of physical quantity uses high-precision interpolation at rollback point, reduces numerical value
It dissipates, while to reduce memory consumption, the value and its first derivative of storage physical field are as calculating variable.
3 variables, the i.e. object are then stored on each mesh point i if φ is convection current physical quantity for two-dimensional convection equation
Reason amount and its single order local derviation: φi、Analog value is acquired by high-precision interpolation in rollback point.Including following step
It is rapid:
S1), for two-dimensional space grid cell locating for rollback point P, if grid cell side length is h, four mesh points point
Not Wei A, B, C, D, all sides of grid cell are parallel with the reference axis of two-dimentional dimension space, while AB and while CD be parallel to X-axis and
The two along Y-axis forward direction spacing be h, while AD and while BC be parallel to Y-axis and the two along X-axis forward direction spacing be h.The E and G that sets up an office points
It Wei not subpoint of the point P on side AB and AD；
S2), the second order local derviation of φ at point A, point B and point D is calculatedValue.Specific calculating is as follows:
Based on following third-order Taylor expansion,
Δ x=Δ y=h is enabled, the second order local derviation at mesh point A is calculatedValue, can obtain:
Similarly, Δ x=-h, Δ y=h are enabled, the second order local derviation at mesh point B is calculatedValue, can obtain:
Similarly, Δ x=h, Δ y=-h are enabled, the second order local derviation at mesh point D is calculatedValue, can obtain:
S3), it is based on one-dimensional CIP interpolation algorithm, passes through the φ at mesh point A and BA、φB、Calculate point E's
φE、At point A and BCalculate point E's
Pass through the φ at mesh point A and DA、φD、Calculate the φ at point GG、Pass through mesh point A and D
PlaceIt calculates at point G
S4), the φ of P point is calculated according to the following formulaP、
Wherein, ξ and η is respectively the length of AE and AG.
Further, step S3) in be based on one-dimensional CIP interpolation algorithm, pass through the φ at mesh point A and BA、φB、 Calculate the φ at point EE、Specific formula are as follows:
φE=C3(xE-xA)3+C2(xE-xA)2+C1(xE-xA)+C0；
Wherein, C0=φA,xEIndicate E point abscissa,
xAIndicate that A point abscissa, h are grid cell side length.
Further, step S3) in be based on one-dimensional CIP interpolation algorithm, pass through A and point BIt calculates at point ESpecific formula are as follows:
Wherein,
xEIndicate E point abscissa, xAIndicate that A point abscissa, h are grid list
First side length.
Further, step S3) in be based on one-dimensional CIP interpolation algorithm, pass through the φ at mesh point A and DA、φD、 Calculate the φ at point GG、Specific formula are as follows:
φG=C3(yG-yA)3+C2(yG-yA)2+C1(yG-yA)+C0；
Wherein, C0=φA,yGIt indicates that G point is vertical to sit
Mark, yAIndicate that A point ordinate, h are grid cell side length.
Further, step S3) in be based on one-dimensional CIP interpolation algorithm, at mesh point A and D It calculates at point GSpecific formula are as follows:
Wherein,
yGIndicate G point ordinate, yAIndicate that A point ordinate, h are grid list
First side length.
The invention has the benefit that
The present invention can keep three ranks high-precision under the premise of time and less memory consumption, and have tight template special
Property.Compare existing method, this method visual quality, speed and in terms of be significantly improved, can be effective
The convection current accuracy and speed of lifting fluid simulation.In addition, the present invention can also be used to require high interpolation essence other than fluid simulation
Other field of degree, such as image/video super-resolution etc..
Detailed description of the invention
Fig. 1 is interpolation schematic diagram when the method for the present invention is simulated for two dimensional fluid；
Fig. 2 is calculation flow chart when the method for the present invention is applied to fluid simulation；
Fig. 3 is the rendering figure comparison of two dimensional fluid analog result, and (a), (b), (c), (d), (e) are respectively to use low order in figure
The fluid simulation result of the Linear and BFECC algorithm of precision, MCIP and USCIP algorithm and the method for the present invention (2dTECIP)
Rendering figure.
Specific embodiment
Specific embodiments of the present invention will be further explained with reference to the accompanying drawing:
The present invention provides a kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation, and the present invention mainly uses
The precision for solving convective term in fluid motion governing equation is calculated in improving, to improve the precision of fluid simulation.By utilizing half
When Lagrangian method solves the convective term of fluid motion governing equation, the calculating of physical quantity uses high-precision at rollback point
Interpolation reduces numerical dissipation, while to reduce memory consumption, the value and its first derivative of storage physical field become as calculating
Amount, the higher derivative then Taylor expansion approximate calculation based on derivation under the premise of guaranteeing that computational accuracy is not impaired.
3 variables, the i.e. object are then stored on each mesh point i if φ is convection current physical quantity for two-dimensional convection equation
Reason amount and its single order local derviation: φi、Analog value is acquired by high-precision interpolation in rollback point.Including following step
It is rapid:
S1), it is directed to Taylor expansion:
As Δ x=0,
As Δ y=0,
In summary 5 formula, and ignore the remainder of three ranks or more, obtain the Taylor expansion equation of following third-order:
It is obtained after arrangement:
S2), as shown in Figure 1, for two-dimensional space grid cell locating for rollback point P, if grid cell side length is h, point
E, G is respectively subpoint of the P on side AB, AD.The value φ of P point positionP、By four grids of grid cell
The analog value of point A, B, C, D are acquired by the high-precision interpolation invented；
S3), the second order local derviation of φ at point A, point B and point D is calculatedValue, it is specific calculate it is as follows:
Based on the Taylor expansion equation that step S1) is obtained, Δ x=Δ y=h is enabled, calculates the second order local derviation at mesh point AValue, can obtain:
Similarly, Δ x=-h, Δ y=h are enabled, the second order local derviation at mesh point B is calculatedValue, can obtain:
Similarly, Δ x=h, Δ y=-h are enabled, the second order local derviation at mesh point D is calculatedValue, can obtain:
S4), it is based on one-dimensional CIP interpolation algorithm, passes through the φ at mesh point A and BA、φB、It calculates at point E
φE、Its calculating formula is as follows:
φE=C3(xE-xA)3+C2(xE-xA)2+C1(xE-xA)+C0；
Wherein, C0=φA,xEIndicate E point abscissa,
xAIndicate that A point abscissa, h are grid cell side length.
At point A and BIt calculates at point EIts calculating formula is such as
Under:
Wherein,
xEFor E point abscissa, xAFor A point abscissa, h is grid cell side
It is long.
Pass through the φ at mesh point A and DA、φD、Calculate the φ at point GG、Its calculating formula is as follows:
φG=C3(yG-yA)3+C2(yG-yA)2+C1(yG-yA)+C0；
Wherein, C0=φA,yGIt indicates that G point is vertical to sit
Mark, yAIndicate that A point ordinate, h are grid cell side length.
At mesh point A and DCalculate point G'sSpecific meter
Formula is as follows:
Wherein,
yGIndicate G point ordinate, yAIndicate that A point ordinate, h are grid list
First side length.
S5), the φ at point P is calculated,Value, specific calculating are as follows:
Based on the Taylor expansion equation that step S1) is obtained,
Δ x=ξ, Δ y=η are enabled, it can be at invocation point AValue:
Δ x=- ξ, Δ y=η are enabled, it can be at invocation point EValue:
Δ x=ξ, Δ y=- η are enabled, it can be at invocation point GValue:
Three formula above of rearrangement, obtains following formula to calculate the φ at point P,
Fluid simulation flow chart can be found in Fig. 2, and convection current interpolation method provided by the invention is mainly used for calculating fluid simulation
Convective term.Fig. 3 is the rendering figure comparison of two dimensional fluid analog result, and Fig. 3 (e) is that the method for the present invention (2dTECIP) is used for two-dimensional flow
Body simulation result render figure, Fig. 3 (a), Fig. 3 (b), Fig. 3 (c), Fig. 3 (d) be respectively using low order precision Linear and
The fluid simulation result rendering figure of BFECC algorithm, MCIP and USCIP algorithm, it can be seen from the figure that three based on CIP kinds of sides
Method simulates Linear the and BFECC algorithm that effect is better than low order precision.Due to the defect of MCIP and USCIP method above-mentioned, originally
For the effect that inventive method obtains in two dimensional fluid simulation better than MCIP and USCIP, the smog details of simulation is richer.
The above embodiments and description only illustrate the principle of the present invention and most preferred embodiment, is not departing from this
Under the premise of spirit and range, various changes and improvements may be made to the invention, these changes and improvements both fall within requirement and protect
In the scope of the invention of shield.
Claims (5)
1. a kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation, it is characterised in that: by utilizing half
When Ge Langfa solves the convective term of fluid motion governing equation, the calculating of physical quantity uses high-precision and inserts at rollback point
Value reduces numerical dissipation, while to reduce memory consumption, the value and its first derivative of storage physical field become as calculating
Amount, the higher derivative then Taylor expansion approximate calculation based on derivation under the premise of guaranteeing that computational accuracy is not impaired；
3 variables, the i.e. physical quantity are then stored on each mesh point i if φ is convection current physical quantity for two-dimensional convection equation
And its single order local derviation: φi、Analog value is acquired by high-precision interpolation in rollback point；The following steps are included:
S1), for two-dimensional space grid cell locating for rollback point P, if grid cell side length is h, four mesh points are respectively
A, all sides of B, C, D, grid cell are parallel with the reference axis of two-dimentional dimension space, while AB and while CD be parallel to X-axis and the two
Along Y-axis forward direction spacing be h, while AD and while BC be parallel to Y-axis and the two along X-axis forward direction spacing be h；The E and G that sets up an office is respectively
Subpoint of the point P on side AB and AD；
S2), the second order local derviation of φ at point A, point B and point D is calculatedValue, it is specific calculate it is as follows:
Based on following third-order Taylor expansion,
Δ x=Δ y=h is enabled, the second order local derviation at mesh point A is calculatedValue, can obtain:
Similarly, Δ x=-h, Δ y=h are enabled, the second order local derviation at mesh point B is calculatedValue, can obtain:
Similarly, Δ x=h, Δ y=-h are enabled, the second order local derviation at mesh point D is calculatedValue, can obtain:
S3), it is based on one-dimensional constraint Interpolation Profile method (CIP) interpolation algorithm, passes through the φ at mesh point A and BA、φB、 Calculate the φ of point EE、At mesh point A and BCalculate point E's
Pass through the φ at mesh point A and DA、φD、Calculate the φ at point GG、At mesh point A and DIt calculates at point G
S4), the φ of P point is calculated according to the following formulaP、
Wherein, ξ and η is respectively the length of AE and AG.
2. a kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation according to claim 1, special
Sign is: in step S3), being based on one-dimensional CIP interpolation algorithm, passes through the φ at mesh point A and BA、φB、It calculates
φ at point EE、Specific formula are as follows:
φE=C3(xE-xA)3+C2(xE-xA)2+C1(xE-xA)+C0；
Wherein, C0=φA,
xEIndicate E point abscissa, xAIndicate that A point abscissa, h are grid cell side length.
3. a kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation according to claim 1, special
Sign is: in step S3), one-dimensional CIP interpolation algorithm is based on, at mesh point A and B
It calculates at point ESpecific formula are as follows:
Wherein, xEIndicate E point abscissa, xAIndicate that A point abscissa, h are grid cell side length.
4. a kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation according to claim 1, special
Sign is: in step S3), being based on one-dimensional CIP interpolation algorithm, passes through the φ at mesh point A and DA、φD、It calculates
φ at point GG、Specific formula are as follows:
φG=C3(yG-yA)3+C2(yG-yA)2+C1(yG-yA)+C0；
Wherein, C0=φA,yGIt indicates that G point is vertical to sit
Mark, yAIndicate that A point ordinate, h are grid cell side length.
5. a kind of three rank high-precision convection current interpolation algorithms suitable for two dimensional fluid simulation according to claim 1, special
Sign is: in step S3), one-dimensional CIP interpolation algorithm is based on, at mesh point A and D
Calculate point G'sSpecific formula are as follows:
Wherein,
yGIndicate G point ordinate, yAIndicate that A point ordinate, h are grid cell side
It is long.
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| US20090040220A1 (en) * | 2007-02-05 | 2009-02-12 | Jonathan Gibbs | Hybrid volume rendering in computer implemented animation |
| US20090171596A1 (en) * | 2007-12-31 | 2009-07-02 | Houston Benjamin Barrie | Fast characterization of fluid dynamics |
| US20130027407A1 (en) * | 2011-07-27 | 2013-01-31 | Dreamworks Animation Llc | Fluid dynamics framework for animated special effects |
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2019
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Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6266071B1 (en) * | 1998-11-20 | 2001-07-24 | Silicon Graphics, Inc. | Method of producing fluid-like animations using a rapid and stable solver for the Navier-Stokes equations |
| US20090040220A1 (en) * | 2007-02-05 | 2009-02-12 | Jonathan Gibbs | Hybrid volume rendering in computer implemented animation |
| US20090171596A1 (en) * | 2007-12-31 | 2009-07-02 | Houston Benjamin Barrie | Fast characterization of fluid dynamics |
| US20130027407A1 (en) * | 2011-07-27 | 2013-01-31 | Dreamworks Animation Llc | Fluid dynamics framework for animated special effects |
Non-Patent Citations (3)
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| 傅德月等: "高阶CIP数值方法及其在相关物理问题中的应用", 《计算物理》 * |
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