CN102298794A - Real-time water drop simulation method based on surface grids - Google Patents
Real-time water drop simulation method based on surface grids Download PDFInfo
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Abstract
The invention discloses a real-time water drop simulation method based on surface grids. In the method, a three-dimensional fluid movement model is simplified into a surface grid model; the average curvature flow is calculated by implicit iteration in each step to produce surface tension; and the shape of a water drop when the water drop contacts a solid surface is changed by contact angle operation. In the invention, grid fusion and split are realized by a series of grid optimization and connectivity tests, the quality of the grids is improved, and water drop motion effects (including a wavelet during collision of the water drop, water column cutoff and flow of the water drop on the solid surface) which approach to those of various real physical experiments are achieved in real time.
Description
Technical field
The present invention relates to the physical simulation technical field, relate in particular to a kind of real-time water droplet emulation mode based on the veil lattice.
Background technology
Water droplet often appears in the scene of daily life, such as the glass in bathroom, and the vehicle window of rainy day etc.Fluid with large volume is compared, and water droplet has bigger viscosity and surface tension, so its emulation difficulty is bigger.When water droplet was represented with three-dimensional voxel, so big viscosity and surface tension needed very little step-length could guarantee precision and stability.Three-dimensional voxel represent need be very big memory cost could recording surface details, make when making the general scene that comprises a lot of water droplets very difficulty.Therefore general fluid emulation mode is not suitable for the emulation of water droplet.
Wang has proposed a kind of fluid emulation mode (H. Wang based on the diving equation, G. Miller, and G. Turk, " Solving general shallow wave equations on surfaces; " in Proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on Computer animation (SCA 2007), 2007, pp. 229 – 238), the grid that this employing height field is represented Simulated Water very easily drops in flowing of solid surface, but can not handle the fracture of the solid surface and the water column of depression.Th ü rey has then adopted the veil lattice to represent the details on water droplet surface (N. Th ü rey, C. Wojtan, M. Gross, and G. Turk, " A multiscale approach to mesh-based surface tension flows; " ACM Transactions on Graphics (SIGGRAPH 2010), vol. 29, and no. 4, pp. 48:1 – 48:10, July 2010), but their method calculated amount is very big, needs calculated off-line.
The implicit expression solution of mean curvature flow can be with reference to paper (the M. Desbrun of Desbrun, M. Meyer, P. Schr der, and A. H. Barr, " Implicit fairing of irregular meshes using diffusion and curvature flow; " in Proceedings of the 26th annual conference on Computer graphics and interactive techniques (SIGGRAPH ' 99), 1999, pp. 317 – 324; D. Mathieu, M. Mark, S. Peter, and A. H. Barr, " Discrete differential geometry operators in nD. " Springer-Verlag, 2000, pp. 35 – 57), the Laplace-Beltrami operator is used in the sparse linear space, therefore can carries out iteration, but the method that they propose can cause VOLUME LOSS with big step-length.To revise volume, can adopt (I. Eckstein, J.-P. Pons, Y. Tong, C.-C. J. Kuo, and M. Desbrun, " Generalized surface flows for mesh processing; " in Proceedings of the fifth Eurographics symposium on Geometry processing (SGP 2007), 2007, pp. 183 – 192) the volume modification method that proposes.
Because the three-dimensional voxel expression is not suitable for representing water droplet, the veil lattice represent then to have a lot of advantages.Big viscosity makes that the velocity field trend is smooth, therefore can ignore the motion of water droplet inside, and only pay close attention to its surface.Simultaneously, surface tension can directly be calculated by the mean curvature on surface.These phenomenons make that the surface trend is smooth, and make the easier realization of change in topology of grid.
Summary of the invention
The present invention is directed to the water droplet that occurs in film and the recreation, the big and problem of unstable of its motion calculation amount has proposed a kind of real-time water droplet emulation mode based on the veil lattice.
The objective of the invention is to be achieved through the following technical solutions: a kind of real-time water droplet emulation mode based on grid, this method adopt the gore grid to represent water droplet; Contact angle when contacting with surface tension and with solid by deformation operation analog stream bulk viscosity; Realize the fusion and the division of grid then by grid operations, and grid is optimized.
Further, described by the contact angle of deformation operation analog stream bulk viscosity when contacting with surface tension and with solid, comprise following substep:
(1) each summit in the grid is applied external force, comprise gravity, solid surface absorption affinity, friction force and a part of viscosity resistance, these external force are used to revise the speed and the position on summit;
(2) mean curvature flow of implicit expression computing grid;
(3) explicit formation contact angle, i.e. the angle of water droplet edge and solid surface formation; Specific as follows: as at first to search for the summit that all contact with solid surface, therefrom extract osculatory, and calculate every bit and solid surface angulation on the osculatory; If angle greater than maximum angular, then applies outside external force to this summit; If angle less than minimum angle, then applies inside external force to this summit;
(4) grid is carried out the volume correction, the VOLUME LOSS that computation process causes is revised by local and two kinds of methods of the overall situation.
Further, describedly realize the fusion and the division of grid, and grid is optimized comprises following substep by grid operations:
(1) grid merges: specific as follows: for the grid that two its bounding boxs intersect, find out the intersection that two grids intersect, former grid is cut along this intersection, and trigonometric ratio again.To the triangle on each grid, calculate its inside that is positioned at the another one grid or outside, will be labeled as inner triangle deletion, and all will be labeled as an outside new grid of triangle merging becoming;
(2) grid division: in the process of carrying out the limit deletion action, if after deleting a limit, grid just no longer is a stream shape, then herein grid is divided into two.It is specific as follows: after the limit that will delete is determined, search for a ring neighborhood on two summit, if find public summit, and this summit does not belong to the adjacent triangle in limit therewith, so just insert new summit and triangle, make former grid be split into two new grids at this place, three summits.If the grid at the place, limit of deletion has been a tetrahedron, then with this tetrahedron deletion;
(3) grid optimization: comprise edge contraction, limit division, limit upset;
The invention has the beneficial effects as follows, the degree of freedom of fluid emulation is reduced to the summit of two-dimensional grid from three-dimensional voxel, thereby significantly reduced the room and time expense of calculating.Whole model was divided into for two steps: deformation operation makes the water droplet shape change; Grid operations makes grid produce and merges and splitting effect, and grid is optimized.The present invention can combine with existing fluid policy technology easily, is used for making high-precision fluid animation.
Description of drawings
Fig. 1 is the algorithm flow chart that the present invention proposes;
Fig. 2 is the formation contact angle computing synoptic diagram that the present invention proposes;
Fig. 3 is the grid splitting-up method synoptic diagram that the present invention proposes, and Fig. 3 (a) has shown by adding new summit and triangle grid is split into tetrahedral process that Fig. 3 (b) has shown a kind of situation that needs special processing;
Fig. 4 is two water droplets of the present invention collisions, merges and the design sketch of concussion, and wherein Fig. 4 (a) is a ground water droplet before the collision, Fig. 4 (b) and (c) show two states in the fusion concussion process;
Fig. 5 is that a water of the present invention is dropped in solid surface, and by the design sketch of solid surface absorption formation osculatory, wherein Fig. 5 (a) is a ground water droplet before the drippage, and Fig. 5 (b) is the water droplet in the adsorption process, and Fig. 5 (c) is the water droplet that finally forms osculatory;
Fig. 6 is that the fluid of the different surfaces coefficient of tension of the present invention shows difform design sketch on the surface of same material;
Fig. 7 is the fluid Rayleigh-Plateau wild effect design sketch of the different surfaces coefficient of tension of the present invention, wherein Fig. 7 (a) is a coefficient of tension effect hour, Fig. 7 (b) is the effect of moderate tension coefficient, and Fig. 7 (c) is the effect of coefficient of tension when big.
Embodiment
All water droplets are all represented by the gore grid.The data structure of veil lattice comprises apex coordinate, the summit that each triangle comprised, and on the veil lattice one the ring neighborhood syntople.Grid can adopt half of structure, and perhaps other the structure that can directly obtain a ring neighborhood on the grid is represented.
Because big relatively viscosity and strong surface tension under the small scale, the motion of water droplet mainly is to be subjected to the surface influence.Fluid emulation mode based on the veil lattice mainly is divided into two steps, and the first step has comprised the veil lattice distortion computing based on physics law, and external force, friction force, surface tension are considered in this computing and other are with the relevant physical phenomenon of water droplet.Second step comprised the grid operations of a series of change mesh topology, comprised that grid merges, grid division and grid optimization.
The total system streamline as shown in Figure 1.We realize the deformation operation of grid by 4 sub-steps, upgrade the speed and the position on each summit of grid.Each summit in the grid is applied external force, comprise gravity and contact force, at first act on the shape and speed of grid.Subsequently, the mean curvature flow of implicit expression computing grid is simulated capillary effect.When water droplet contacted with solid surface, in order to produce different effect of impregnation may, according to the situation of contact angle, explicitly formed contact angle on osculatory again.Finally revise the volumetric wear that produces in the whole process by volume correction computing.The distortion computing is carried out the fusion of grid after finishing again, division and Optimizing operation.
Separate between these operations, and carry out in order, therefore can be considered as operator to grid vertex speed and position.After having passed through previous operator computing, the speed on summit and position are just as the input of next operator.In order to simplify expression, we use
With
Represent speed and the position of summit i before the operator computing, use
With
Represent speed and position after the operator computing.Specifically be described below:
One, each summit in the grid is applied external force
The external force that needs to add comprises gravity, solid surface absorption affinity, friction force and a part of viscosity resistance.With
The speed of expression summit i before this step, adding gravity can be expressed as
, wherein
Be the speed after the computing,
Be acceleration of gravity,
It is time step.The position of summit i
The forward direction Euler method is calculated, and formula is
After vertex position upgraded, we detected this summit and whether have penetrated solid surface.If penetrated solid surface really, nearest subpoint is as the reposition on this summit on solid surface just to find out this summit, and speed also is changed to:
Wherein,
Be that solid surface exists
The speed of point,
Be that solid surface exists
The normal direction of point.This formula has been eliminated the relative velocity of this summit in the solid surface normal direction.All found out in case penetrate the summit of solid surface, we apply friction force to these summits, and formula is as follows:
Wherein,
It is friction factor.We use then
With
Difference recomputate the position on summit
Although viscosity resistance is not considered to external force in the emulation of voxelization fluid, here our opposite grid applies damping and obtains the viscosity resistance similar effects.Our damping is divided into two:
Wherein, first is that common damping produces, and effect is to make speed be in the ratio that per step iteration reduces
, second is that explicit Laplacian-Beltrami operator is with ratio of damping
Effect produces.Second closely similar with the damping in the Navier-Stokes equation, and just it is defined in surface mesh fully.
Usually span is 0.3 to 0.5,
Usually between 0 to 0.1.Use very large
Value can be avoided a lot of grid defectives, therefore when the very poor and king-sized damping of needs of grid character, can with
Value is established very greatly.
Two, the mean curvature flow of implicit expression computing grid
We produce capillary effect by mean curvature flow.If
Be the surface
Last any point,
Be the coefficient of curvature flow,
Be
The Laplace-Beltrami operator.The continuous representation form of mean curvature flow is:
，
Wherein,
It is normal
The mean curvature of direction definition.Method (the M. Desbrun that we propose with Desbrun, M. Meyer, P. Schr der, and A. H. Barr, " Implicit fairing of irregular meshes using diffusion and curvature flow; " in Proceedings of the 26th annual conference on Computer graphics and interactive techniques (SIGGRAPH ' 99), 1999, pp. 317 – 324.) it is carried out discretize:
Wherein,
Be the vector of vertex position,
Be time step,
Be the concentrated area mass matrix,
It is the symmetrical matrix that has comprised the cotangent coefficient that discrete Laplace-Beltrami operator calculates.With
A ring neighborhood of representing summit i
Wherein,
With
Be two the relative angles of limit that constitute with i and j.Because
With
All be symmetrical matrix, therefore discrete mean curvature flow formula can all multiply by by both sides
Become the positive definite symmetrical matrix.Method of conjugate gradient (Golub, Gene H. that we decompose with non-complete pre-service Cholesky; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins) find the solution this linear system, consequently a new vertex position vector.After the position that calculating makes new advances, calculate new vertex velocity:
Three, explicit formation contact angle
When water droplet when solid surface flows, osculatory can be subjected to the infiltrating influence of solid material.Different infiltration character causes water droplet different shapes to occur, and contact angle is used for describing this character.We calculate contact angle by explicitly, and according to the size correction osculatory of contact angle.
The first step that forms contact angle is to find out osculatory on the veil lattice.Add external force one again in the step, we have marked all and have penetrated the summit of solid surface, the contact summit is defined as with non-penetrates the summit that penetrates that the summit links to each other, and osculatory just is defined as the set that connects the limit that contacts the summit, as shown in Figure 2.
After we found out osculatory, we made it reach stable contact angle just with mobile osculatory.As shown in Figure 2, establish O and be a contact summit, A, B, C are the proximal surface of water droplet at gas-liquid surface side grid, and we are with the normal area weighted sum of A, B, three faces of the C normal as an O.
With
Angle just can be used as a little
The estimation of place's contact angle.
Poor according to current contact angle and stable contact angle judged the moving direction on contact summit, realizes by the method that applies external force to the summit:
Wherein,
With
Be respectively minimal contact angle and maximum contact angle,
Be the coefficient of boundary force,
It is the liquid level normal
In the projection of solid surface, as shown in Figure 2.This formulate contact summit can not change the position easily, and makes that border external force trend is minimum.
Four, grid is carried out the volume correction
Our volume modification method is divided into local correction and the overall situation revised for two steps.
At first calculate the general speed of each water droplet, comprise point-to-point speed and velocity of rotation, then it is removed from the velocity field of grid, then Sheng Xia speed
Make water droplet produce deformation and volume change.If
Be the volume change of unit area, its computing method are:
Wherein
It is i place, summit water droplet normal to a surface.Calculate by a ring neighborhood then
Local mean value:
，
Wherein,
It is the total area of summit i.Then will
From
In deduct:
?。
Overall situation correction also can be used for volume is revised.At first calculate and work as the poor of front volume and initial volume, then grid is carried out translation, distance is
, wherein
Be volume change,
It is total surface area.Translation is moved the summit by the normal along each summit and is realized:
When viewpoint was far away apart from water droplet, the method that adopts overall situation correction was to improve computing velocity; When needing the details of water droplet, we select for use local correction to keep the small echo on water droplet surface, and carry out overall situation correction as required again.When
After the correction, the original velocity field of the general speed add-back of whole water droplet, then present velocity field has just had the character of volume conservation again.
Finish after the above-mentioned steps, just realized the deformation operation of single water droplet.Next step is exactly the grid operations that will carry out a plurality of merge water droplets, division and optimization.Specifically be described below:
One, grid merges
What grid merged simulation is the process of the afterwards synthetic big water droplet of water droplet collision.We make up a bounding box for each grid, and the grid for two its bounding boxs intersect quickens to find out the intersection point of all limits and face by the AABB tree, if there is intersection point, then two grids are carried out logic and operation.Concrete grammar is as follows: method (the D. Baraff that adopts Baraff to propose, A. Witkin, and M. Kass, " Untangling cloth, " ACM Transactions on Graphics (SIGGRAPH 2003), vol. 22, no. 3, pp. 862 – 870, July 2003) then connect these intersection points successively and form intersection, with former grid along this intersection trigonometric ratio again.To the triangle on each grid, calculate still outside, its inside that is positioned at the another one grid, the triangle that is labeled as in inside is deleted, and all are labeled as the grid that outside triangles merge becomes a new closure.This method can be used in the self intersection computing of grid equally.
Two, grid division
What the grid division was simulated is the process that a big water droplet is drawn as two little water droplets.In the process of carrying out the limit deletion action, if after deleting a limit, grid just no longer is a stream shape, then herein grid is divided into two, as shown in Figure 3.Specific as follows: the limit of AB for deleting among Fig. 3, search for a ring neighborhood on two summit, if find public adjacent vertex, and this summit does not belong to the adjacent triangle in limit therewith, promptly put E, we just add new summit and triangle at A, B, E place, the grid branch are split, shown in Fig. 3 (a).If find that whole connection grid is a tetrahedron after determining to delete limit AB, so just directly with this tetrahedron deletion, shown in Fig. 3 (b).
Three, grid optimization
Each step iteration is all carried out local optimum one time to grid, comprise edge contraction, the limit division, limit upset (Hugues Hoppe, Tony DeRose, Tom Duchamp, John McDonald, Werner Stuetzle. " Mesh Optimization " ACM SIGGRAPH 1993 Proceedings, 19-26).When the length on limit during, just this edge branch is split greater than threshold value; When the length on limit during less than threshold value, just that this edge is folding, its medium and small threshold value is made as 0.04, and big threshold value is made as 3 times of little threshold value.In order to quicken optimizing process and improve to optimize quality, we make up two priority queries by raft and rickle, and the limit divides and begins to carry out successively from the longest limit, and edge contraction is from the shortest limit.The limit upset is carried out according to the sequence number on limit, if the cosine weight on a limit is a negative value, then this edge is overturn.
We compose a weight for simultaneously every limit, make that the grid length of side can self-adaptation.Such as, we compose with less weight for the limit on the fluid-solid surface of contact, even the physical length on limit also can be folded greater than threshold value like this, so just can only keep less summit on surface of contact.Because osculatory is very responsive for change in topology, if therefore certain bar limit belongs to osculatory, then it is not carried out the limit turning operation.
By above-mentioned steps, the inventor has realized some embodiments of the present invention at 8 nuclear Xeon CPU that have been equipped with two 2.4GHz on the computing machine of 16G internal memory.
Embodiment one, the inventor tested the effect of two water droplet collision and fusion, as shown in Figure 4.Because the collision of water droplet forms small echo on the water droplet surface, these small echos are propagated along the surface of water droplet, and the water droplet vibrations finally stop vibration for several times under the effect of damping.
Embodiment two, the inventor tested the effect that a water is dropped in solid surface, as shown in Figure 5.The surface is a water wetted material, and behind water droplet contact solid surface, osculatory is expansion rapidly under the strong absorption affinity effect on surface.Final integral drip water is spread out in the plane, reaches steady state (SS).
Embodiment three, the inventor tested the effect of the liquid of the different surfaces coefficient of tension at identical solid surface, as shown in Figure 6.Three formed contact angles of water droplet are identical, but the big more liquid of surface tension coefficient, its shape trends towards sphere more; And the little liquid of surface tension coefficient then more trends towards spreading out from the teeth outwards.
Embodiment four, inventor's Rayleigh-Plateau wild effect of having tested the liquid of the different surfaces coefficient of tension.When water flows out from water swivel, because the Rayleigh-Plateau wild effect, water column can be blocked is a lot of little water droplets.Surface tension coefficient is big more, and water column is easy more being blocked just.The arithmetic speed of this example is 50FPS.
Claims (3)
1. the real-time water droplet emulation mode based on grid is characterized in that, this method adopts the gore grid to represent water droplet; Contact angle when contacting with surface tension and with solid by deformation operation analog stream bulk viscosity; Realize the fusion and the division of grid then by grid operations, and grid is optimized.
2. according to the described real-time water droplet emulation mode of claim 1, it is characterized in that based on grid, described by the contact angle of deformation operation analog stream bulk viscosity when contacting with surface tension and with solid, comprise following substep:
(1) each summit in the grid is applied external force, comprise gravity, solid surface absorption affinity, friction force and a part of viscosity resistance, these external force are used to revise the speed and the position on summit;
(2) mean curvature flow of implicit expression computing grid;
(3) explicit formation contact angle, i.e. the angle of water droplet edge and solid surface formation; Specific as follows: as at first to search for the summit that all contact with solid surface, therefrom extract osculatory, and calculate every bit and solid surface angulation on the osculatory; If angle greater than maximum angular, then applies outside external force to this summit; If angle less than minimum angle, then applies inside external force to this summit;
(4) grid is carried out the volume correction, the VOLUME LOSS that computation process causes is revised by local and two kinds of methods of the overall situation.
3. according to the described real-time water droplet emulation mode of claim 1, it is characterized in that, describedly realize the fusion and the division of grid, and grid is optimized comprises following substep by grid operations based on grid:
(1) grid merges: specific as follows: for the grid that two its bounding boxs intersect, find out the intersection that two grids intersect, former grid is cut along this intersection, and trigonometric ratio again; To the triangle on each grid, calculate its inside that is positioned at the another one grid or outside, will be labeled as inner triangle deletion, and all will be labeled as an outside new grid of triangle merging becoming;
(2) grid division: in the process of carrying out the limit deletion action, if after deleting a limit, grid just no longer is a stream shape, then herein grid is divided into two; It is specific as follows: after the limit that will delete is determined, search for a ring neighborhood on two summit, if find public summit, and this summit does not belong to the adjacent triangle in limit therewith, so just insert new summit and triangle, make former grid be split into two new grids at this place, three summits; If the grid at the place, limit of deletion has been a tetrahedron, then with this tetrahedron deletion;
(3) grid optimization: comprise edge contraction, limit division, limit upset.
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Application publication date: 20111228 |