High-speed Stereolithography 3D Printing Simulation Method Based on Fluid-Structure Interaction

Abstract

The present invention provides a high-speed photocuring 3D printing simulation method based on fluid-structure interaction, which comprises the following steps: S1: Establish a model of the photocuring 3D printing process, and the model includes: photocuring reaction, mass transfer process of reactants, light intensity distribution, phase distribution, resin flow and phase change process, and elastic deformation of the resin tank bottom; S2: Construct a geometric model of the resin tank, and determine the initial conditions and boundary conditions of the geometric model; S3: Perform mesh division on the geometric model of the resin tank; S4: Determine the simulation parameters during the simulation process; S5: Select different strategies according to the printing mode for numerical solution. The above numerical simulation method for photocuring 3D printing can use the same set of models to simulate and analyze the two printing modes of continuous printing and layer-by-layer printing, and can be used to deeply analyze the photocuring forming mechanism under the coupling action of multiple physical fields, improve the process through virtual printing, optimize process parameters, and thus improve the printing speed and the forming quality of parts.

Description

Technical Field

[0001] The present invention belongs to the technical field of photocuring 3D printing, and relates to a high-speed photocuring 3D printing simulation method based on fluid-structure interaction. Background Art

[0002] Photocuring 3D printing is one of the earliest developed additive manufacturing technologies. This technology utilizes the principle of free radical photopolymerization, and through ultraviolet light irradiation, the liquid resin crosslinks and polymerizes to transform into a solid state, ultimately forming a three-dimensional entity. It has the advantages of fast printing speed and high precision, and has extensive applications in the manufacturing of small and medium-sized complex structural parts. With the rapid development of photocuring 3D printing in fields such as biomedicine, microfluidics, and mechanical metamaterials, higher requirements have been put forward for the forming accuracy and printing speed of printed parts. More and more key problems affecting forming have received extensive attention, such as the difficulty in separating the formed part from the elastic substrate during the printing process, the reduction in accuracy caused by ultraviolet light scattering, and the continuous scouring effect of the reflux resin on the semi-cured section during continuous printing.

[0003] Since the forming process of photocuring 3D printing involves complex physical and chemical phenomena, including multiple factors such as free radical photopolymerization reaction, curing phase change, resin flow and mass transfer, and deformation of the elastic substrate, the improvements that can be made solely by experimental means are relatively limited, and it is difficult to comprehensively study the adsorption force generation mechanism and the evolution process of the solid-liquid interface morphology during the printing process. Therefore, it is necessary to combine multi-physical field coupling models for simulation to deeply analyze the key physical phenomena and forming mechanisms of photocuring 3D printing. At present, most of the simulation schemes only consider single factors such as free radical photopolymerization reaction or resin flow, ignoring the coupling effect of multi-physical fields, and it is difficult to have a comprehensive and profound understanding of the entire printing process. On the other hand, for continuous photocuring forming, there is currently no effective simulation model for simulation analysis. Summary of the Invention

[0004] In view of the above problems, the present invention provides a high-speed photocuring 3D printing simulation method based on fluid-structure interaction, which is characterized in that:

[0005] S1: Establish a model of the photocuring 3D printing process, and the model includes: photocuring reaction, mass transfer process of reactants, light intensity distribution, phase distribution, resin flow and phase change process, and elastic deformation of the resin tank bottom;

[0006] S2: Construct a geometric model of the resin tank, and determine the initial conditions and boundary conditions of the geometric model;

[0007] S3: Perform mesh division on the geometric model of the resin tank;

[0008] S4: Determine the simulation parameters during the simulation process;

[0009] S5: numerically solve the model of the photocuring 3D printing process according to the printing mode. When the printing mode is continuous printing, the complete model is solved. When the printing mode is layered printing, the solution includes two stages: stage one corresponds to the static curing process, solving the photocuring reaction, the mass transfer process of the reactants, the light intensity distribution, and the phase distribution; stage two corresponds to the separation and reset process, solving the transport process of the reactants, the phase distribution, the resin flow and phase change process, and the elastic deformation of the bottom of the resin tank. After completing the solution of stage two, return to stage one to enter a new cycle until the printing is completed.

[0010] Optionally, the photocuring reaction at least includes photoinitiation, free radical generation, chain growth, chain termination, and oxygen inhibition, which are specifically described as follows:

[0011]

[0012] Among them, PI, M. RM n M m 、O2、RM n OO is photoinitiator, primary free radical, monomer, monomer free radical, chain free radical, polymer chain, oxygen, oxidation product, hv is light quantum, k is p , k t , k O are the reaction rates of chain growth, chain termination and oxygen inhibition, respectively.

[0013] Optionally, the mass transfer process of the reactants is described by the convection-diffusion equation of the reactant concentration, which is specifically expressed as follows:

[0014]

[0015]

[0016]

[0017]

[0018] Free radical substances Unified Indicates that [PI], [M] and [O2] are PI, M, O2 molar concentration, t is time, v is the resin velocity vector, D O is the oxygen diffusion coefficient, R i is the primary free radical generation rate, which is specifically expressed as follows:

[0019]

[0020] in where Φ is the quantum yield, ε is the molar extinction coefficient, and I(x, y, z) is the spatial distribution of the light intensity.

[0021] Optionally, the light intensity distribution is obtained by integrating the Gaussian distribution of the point light source intensity within the illumination domain at the focal plane and decays in the depth direction according to the Beer-Lambert law, which is specifically expressed as follows:

[0022]

[0023] I(x,y,z) = ∫ Ω G(X - x, Y - y)dXdY exp(-ε[PI]z) (8)

[0024] where G(x, y) is the light intensity distribution of the point light source at the focal plane, I0 is the peak light intensity of the point light source, w0 is the Gaussian radius, and Ω is the illumination area.

[0025] Optionally, the phase distribution is determined by the degree of double bond conversion, which is specifically expressed as follows:

[0026]

[0027] θ s = H(α - α c ) (10)

[0028] λ = 1 - θ s (11)

[0029] where α and α c are the degree of double bond conversion and the critical value of the degree of conversion, respectively, [M] and [M]0 are the monomer concentration and the initial monomer concentration, respectively, θ s is the phase fraction of the solid phase, H(·) is the unit step function, and λ is the porosity.

[0030] Optionally, the flow process of the resin is controlled by the incompressible NS equation, and the phase change process of the resin is realized by adding a Carman-Kozeny type penalty term to the original NS equation, which is specifically expressed as follows:

[0031]

[0032]

[0033] where v is the resin velocity vector, ρ f is the resin density, p is the resin pressure, μ is the resin viscosity, S is the penalty term, and the definition of the penalty term is as follows:

[0034] S = A(v - V0) (14)

[0035]

[0036] Among them, A is the penalty coefficient, V0 is the platform lifting speed, C is the paste zone constant, λ is the porosity, and q is a small constant to avoid a zero denominator.

[0037] Optionally, the elastic deformation of the bottom of the resin tank follows the large deformation elastic theory, and the specific description is as follows:

[0038]

[0039]

[0040] Among them, all physical quantities are for the elastic body at the bottom of the resin tank, ρ s is the density, u is the displacement, t is the time, F is the deformation gradient tensor, Σ is the second Piola-Kirchhoff stress tensor, is the elastic constitutive relation, and E is the Green strain tensor.

[0041] Optionally, the flow process of the resin and the elastic deformation of the bottom of the resin tank generate fluid-structure interaction, and the specific description of the coupling condition is as follows:

[0042] u = u mesh (18)

[0043]

[0044] σ s ·n s +σ f ·n f = 0 (20)

[0045] Among them, u mesh is the moving mesh displacement, σ s is the Cauchy stress tensor of the elastic body at the bottom of the resin tank, σ f is the Cauchy stress tensor of the resin liquid, n s is the outer normal direction vector of the elastic body at the bottom of the resin tank on the fluid-structure interaction interface, n f is the outer normal direction vector of the resin liquid on the fluid-structure interaction interface.

[0046] The fluid-structure interaction is described by the Arbitrary Lagrangian-Eulerian method / moving mesh method. In the moving mesh method, the deformation of the mesh is determined by the modified Laplace equation, and the specific description is as follows:

[0047]

[0048] Among them, δ is the area of the mesh element.

[0049] Advantages of the present invention: The method comprehensively considers the coupled effects of multiple factors such as free radical photopolymerization reaction, curing phase change, resin flow and mass transfer, and deformation of the elastic substrate. At the same time, different solution strategies can be selected according to the printing mode, and it is applicable to both layer-by-layer printing and continuous printing modes. It can be used to predict the change of adsorption force, the evolution mode of solid-liquid interface, and the influence of substrate elastic deformation on the separation of printed parts during the forming process, which helps to deeply analyze the forming mechanism of stereolithography 3D printing. Description of the Drawings

[0050] Figure 1 It is a schematic diagram of an axisymmetric geometric model.

[0051] Figure 2 It is an adaptive mesh diagram.

[0052] Figure 3 It is a velocity vector diagram of the flow field within a thin film deformation cycle.

[0053] Figure 4 It is the flow field particle vector diagram observed in the experiment and the corresponding solid-liquid morphology. Detailed Embodiment

[0054] The following further describes the embodiments of the present invention in detail with reference to the drawings.

[0055] A numerical simulation method for stereolithography 3D printing includes:

[0056] S1: Establish a model of the stereolithography 3D printing process, and the model includes: photopolymerization reaction, mass transfer process of reactants, light intensity distribution, phase distribution, resin flow and phase change process, and elastic deformation of the resin tank bottom;

[0057] In this embodiment, based on the FreeFem++ open-source finite element program library, a transient solver for the stereolithography 3D printing process model is developed, and the diffusion-convection equation of reactant concentration with reaction source term, the incompressible NS equation with additional Carman-Kozeny type penalty term, and the large deformation membrane element control equation considering that the resin tank bottom is only an elastic thin film are solved by coupling.

[0058] The photopolymerization reaction includes photoinitiation, free radical generation, chain growth, chain termination, and oxygen inhibition, and the specific description is as follows:

[0059]

[0060] Among them, PI, M, RM n M m 、O2、RM nOO are respectively photoinitiator, primary radical, monomer, monomer radical, chain radical, polymerized polymer chain, oxygen, oxidation product, hv is a light quantum, k p , k t , k O are respectively the reaction rates of chain growth, chain termination, and oxygen inhibition polymerization reactions.

[0061] The mass transfer process of the reactants is described by the convective diffusion equation of the reactant concentration, and the specific expression is as follows:

[0062]

[0063]

[0064]

[0065]

[0066] Among them, the radical substances are uniformly represented by , [PI], [M], [O2] are respectively the molar concentrations of PI, M, O2, t is time, v is the resin velocity vector, D O is the oxygen diffusion coefficient, R i is the primary radical generation rate, and the specific expression is as follows:

[0067]

[0068] Among them is the quantum yield, ε is the molar extinction coefficient, and I(x, y, z) is the spatial distribution of the light intensity.

[0069] The light intensity distribution is obtained by integrating the Gaussian distribution of the point light source intensity in the illumination area at the focal plane and attenuates according to the Beer-Lambert law in the depth direction. The specific expression is as follows:

[0070]

[0071] I(x, y, z) = ∫ Ω G(X - x, Y - y)dXdYexp(-ε[PI]z) (8)

[0072] Among them, G(x, y) is the light intensity distribution of the point light source at the focal plane, I0 is the peak light intensity of the point light source, w0 is the Gaussian radius, and Ω is the illumination area.

[0073] The phase distribution is determined by the degree of double bond conversion, and the specific expression is as follows:

[0074]

[0075] θ s =H(α - α c ) (10)

[0076] λ = 1 - θ s (11)

[0077] where α and α c are the degree of double - bond conversion and the critical value of conversion respectively, [M] and [M]0 are the monomer concentration and the initial monomer concentration respectively, θ s is the phase fraction of the solid phase, H(·) is the unit step function, and λ is the porosity.

[0078] The flow process of the resin is controlled by the incompressible NS equations. The phase - change process of the resin is realized by adding a Carman - Kozeny - type penalty term to the original NS equations, and the specific expression is as follows:

[0079]

[0080]

[0081] where v is the resin velocity vector, ρ f is the resin density, p is the resin pressure, μ is the resin viscosity, S is the penalty term, and the definition of the penalty term is as follows:

[0082] S = A(v - V0) (14)

[0083]

[0084] where A is the penalty coefficient, V0 is the platform - lifting velocity, C is the paste - zone constant, λ is the porosity, and q is a small constant to avoid the denominator being zero.

[0085] The elastic deformation of the resin - tank bottom follows the large - deformation elastic theory, and the specific expression is as follows:

[0086]

[0087]

[0088] where all physical quantities are for the elastomer at the resin - tank bottom, ρ s is the density, u is the displacement vector, t is the time, F is the deformation - gradient tensor, Σ is the second - Piola - Kirchhoff stress tensor, is the elastic constitutive relation, and E is the Green's strain tensor.

[0089] The flow process of the resin and the elastic deformation of the resin - tank bottom produce fluid - solid coupling, and the specific expression of the coupling condition is as follows:

[0090] u = u mesh (18)

[0091]

[0092] σ s ·n s + σ f ·n f =0 (20)

[0093] where u mesh is the moving mesh displacement, σ s is the Cauchy stress tensor of the elastomer at the bottom of the resin tank, σ f is the Cauchy stress tensor of the resin liquid, n s is the unit outer normal vector of the elastomer at the bottom of the resin tank on the fluid-structure interface, n f is the unit outer normal vector of the resin liquid on the fluid-structure interface.

[0094] The fluid-structure interaction is described by the Arbitrary Lagrangian-Eulerian method / moving mesh method. In the moving mesh method, the deformation of the mesh is determined by the modified Laplace equation, which is specifically expressed as follows:

[0095]

[0096] where δ is the area of the mesh element. The added correction factor can distribute the deformation of the deformed domain to each element according to the element size, thus avoiding too large deformation of small elements resulting in element inversion.

[0097] To improve the convergence of the model, the fully coupled formulation is adopted for the moving mesh method. There is only one set of displacement and velocity variables in the entire computational domain. In the solid domain, the displacement is the solid displacement and the velocity is the solid velocity. In the fluid domain, the displacement is the mesh and the velocity is the fluid velocity. On the fluid-structure coupling boundary, the fluid and the solid share the nodes and the unknowns at the nodes, so that the fluid-structure coupling conditions are automatically satisfied.

[0098] Considering the periodicity of the deformation of the elastic substrate, in order to avoid geometric errors caused by frequent updating of the deformed mesh frame, the initial mesh is used as the reference frame, and the control equations of the fluid are established on it, which are specifically expressed as follows:

[0099]

[0100]

[0101] where F is the deformation gradient tensor of the mesh, $v$ is the convective velocity on the moving mesh, and $J$ is the determinant of the deformation gradient tensor of the mesh. It should be noted that, due to the different meanings of displacement variables in different domains under the fully coupled formulation, the meanings of corresponding displacement-derived variables such as the deformation gradient tensor also change.

[0102] S2: Construct the geometric model of the resin tank and determine the initial conditions and boundary conditions of the geometric model.

[0103] Establish the Figure 1 axisymmetric model shown (coordinates are $r$, $\theta$, $z$) for subsequent simulations. The entire model mainly targets the liquid film and a part of the cured resin area immediately above it. The central axis of the cylindrical sample coincides with the $z$-axis, and the solid phase area is lifted upward at a speed of $V_0$. The height of the entire model is $H_0$, the radius is $R_0$, and it is irradiated by a constant ultraviolet light from bottom to top. The focal plane of the light coincides with the bottom of the resin tank, and the bottom of the resin tank is an elastic film, which is simulated using membrane elements. In the axisymmetric coordinate system, the Green's strain tensor $E$ of the film is defined as follows:

[0104]

[0105]

[0106] where $u$ and $w$ are the components of the film displacement vector $u$ in the $r$ and $z$ directions respectively. The elastic constitutive relation is defined as follows:

[0107]

[0108] where $E$ Y is the Young's modulus of the film, $h$ is the film thickness, and $\nu$ is the Poisson's ratio of the film.

[0109] The initial conditions and boundary conditions of the geometric model include: The external liquid communicates with the atmosphere, the relative pressure $p$ at the boundary is set to zero, the velocity at the initial moment is 0, and it is in a hydrostatic state. The specific description is as follows:

[0110]

[0111] The lower boundary is in contact with the external oxygen through an oxygen-permeable membrane to maintain a constant oxygen concentration. The right boundary communicates with the unexposed resin in the resin tank, and the concentrations of all substances remain in balance with the initial values. The upper boundary adopts a no-flux boundary condition:

[0112]

[0113]

[0114] where $[O_2]_0$ and $[PI]_0$ are the initial concentration values of oxygen and photoinitiator respectively, and the initial concentration of free radical substances is 0.

[0115] The elastic film is fixed at the lower right corner of the model, and the deformation domain is fixed at the upper and right boundaries:

[0116]

[0117] The general axisymmetric conditions are satisfied on the central axis, which will not be elaborated here.

[0118] S3: Mesh the geometric model of the resin tank.

[0119] The initial mesh is a triangular mesh obtained by bisecting a uniform quadrilateral mesh element. The mesh is re-divided at each subsequent time step, and the new mesh is adaptively generated according to the velocity vector and reactant concentration. This adaptive process is implemented by calling the adaptmesh method in FreeFem++. The values of the unknowns on the new and old meshes are copied through the finite element interpolation provided by the software.

[0120] S4: Determine the simulation parameters during the simulation process.

[0121] The simulation parameters used in the simulation are shown in the following table:

[0122] Table 1

[0123]

[0124]

[0125] S5: Numerically solve the model of the stereolithography 3D printing process according to the printing mode. When the printing mode is continuous printing, solve the complete model. When the printing mode is layer-by-layer printing, solve the model including two stages: Stage 1 corresponds to the static curing process, and solves the photocuring reaction, the mass transfer process of reactants, the light intensity distribution, and the phase distribution. Stage 2 corresponds to the separation and reset process, and solves the transport process of reactants, the phase distribution, the resin flow and phase change process, and the elastic deformation of the bottom of the resin tank. After completing the solution of Stage 2, return to Stage 1 and enter a new loop until the printing ends.

[0126] In this embodiment, the continuous printing process is solved. The simulation results show that when the forming process tends to the stable stage, the elastic film serving as the bottom of the resin tank will undergo periodic adsorption and detachment. Figure 2 It is an adaptive mesh diagram. Figure 3It is a local flow field vector diagram within a cycle. Due to the boundary condition of a constant oxygen concentration at the bottom, affected by the principle of oxygen inhibition of polymerization, a liquid thin layer that is difficult to solidify will form between the bottom of the tank and the forming section, which is called the dead zone. The area where the concentration drops sharply in the monomer cloud map and the corresponding grid is very dense is the solid-liquid boundary between the solidified resin and the liquid resin. It can be observed that the solid-liquid interface presents a non-uniformly decaying wave morphology, and there is a local narrow area where the flow velocity of the reflux resin is relatively large, forming a large pressure gradient, thereby increasing the fluid negative pressure near the axis. The negative pressure acts on the elastic film at the bottom of the resin tank, causing it to undergo elastic deformation and be sucked up. When the distance between the film and the forming section gradually decreases, the overall thickness of the dead zone becomes thinner, resulting in a slowdown in the flow velocity of the reflux resin, and the fluid negative pressure adsorbing the film also decreases accordingly, eventually causing the film to fall off. In this way, a cycle of film deformation process is completed.

[0127] In the experiment, by performing particle image velocimetry on the existing experimental data, the velocity vector diagram near the dead zone during the printing process can be obtained. Then, based on the position where the velocity suddenly changes, the morphology of the solid-liquid interface can be roughly determined, such as Figure 4 shown. The solid-liquid morphology obtained in the experiment is in good agreement with the simulation results.

[0128] The above-described embodiments only represent several implementation manners of the present invention. The description is relatively specific and detailed, but it cannot be construed as a limitation on the scope of the invention patent. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several modifications and improvements can still be made, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention patent shall be subject to the appended claims.

Claims

1. A high-speed photocuring 3D printing simulation method based on fluid-structure interaction, characterized in that: S1: Establish a model of the photocuring 3D printing process, the model including: photocuring reaction, mass transfer process of reactants, light intensity distribution, phase distribution, resin flow and phase change process, elastic deformation of the resin tank bottom; S2: Construct a geometric model of the resin tank and determine the initial conditions and boundary conditions of the geometric model; S3: Perform mesh division on the geometric model of the resin tank; S4: Determine the simulation parameters during the simulation process; S5: Numerically solve the model of the photocuring 3D printing process according to the printing mode. When the printing mode is continuous printing, solve the complete model. When the printing mode is layer-by-layer printing, solve including two stages: Stage 1 corresponds to the static curing process, solving the photocuring reaction, mass transfer process of reactants, light intensity distribution, phase distribution. Stage 2 corresponds to the separation and reset process, solving the transport process of reactants, phase distribution, resin flow and phase change process, elastic deformation of the resin tank bottom. After completing the solution of Stage 2, return to Stage 1 to enter a new cycle until the printing ends.

2. The high-speed photocuring 3D printing simulation method based on fluid-structure interaction according to claim 1, characterized in that: The photocuring reaction at least includes photoinitiation, free radical generation, chain growth, chain termination, and oxygen inhibition polymerization, and is specifically described as follows: ， Among them 、 、 、 、 、 、 、 are photoinitiator, primary radical, monomer, monomer radical, chain radical, polymerized polymer chain, oxygen, and oxidation product, respectively is a light quantum 、 、 are the reaction rates of chain growth, chain termination, and oxygen inhibition polymerization reactions, respectively 3. A method for simulating high-speed photocuring 3D printing based on fluid-structure interaction according to claim 2, characterized in that: The mass transfer process of the reactants is described by the convective diffusion equation of the reactant concentration, and is specifically described as follows: ， Among them, free radical substances , , are uniformly represented by . , , , are respectively , , , in molar concentration, is time, is the resin velocity vector, is the oxygen diffusion coefficient, is the primary free radical generation rate, and the specific expression is as follows: ， wherein is the quantum yield, is the molar extinction coefficient, is the spatial distribution of the light intensity.

4. A method for simulating high-speed photocuring 3D printing based on fluid-structure interaction according to claim 3, characterized in that: The light intensity distribution is obtained by integrating the Gaussian distribution of the point light source intensity in the illumination domain at the focal plane and attenuates in the depth direction according to the Beer-Lambert law, and is specifically described as follows: ， wherein is the light intensity distribution of the point light source at the focal plane, is the peak light intensity of the point light source, is the Gaussian radius, is the illumination area.

5. A high-speed photocuring 3D printing simulation method based on fluid-structure interaction according to claim 3, characterized in that: The phase distribution is determined by the degree of double bond conversion, and is specifically described as follows: ， Among them and are the double bond conversion degree and the critical value of the conversion degree respectively, and are the monomer concentration and the initial monomer concentration respectively, is the phase fraction of the solid phase, is the unit step function, is the porosity.

6. A simulation method for high-speed photocuring 3D printing based on fluid-structure interaction according to claim 5, characterized in that: The flow process of the resin is controlled by the incompressible NS equation, and the phase change process of the resin is realized by adding a Carman-Kozeny type penalty term on the basis of the original NS equation, and is specifically described as follows: ， Among them is the resin velocity vector, and is the resin density,[[]] is the resin pressure,[[]] is the resin viscosity,[[]] is the penalty term, and the definition of the penalty term is as follows: ， wherein is the penalty coefficient, is the platform lifting speed, is the paste zone constant, is the porosity, is a small constant to avoid a zero denominator.

7. A method for simulating high-speed photocuring 3D printing based on fluid-structure interaction according to claim 1, characterized in that: The elastic deformation of the resin tank bottom follows the large deformation elastic theory, and is specifically described as follows: ， where all physical quantities are for the elastomer at the bottom of the resin tank, is the density, is the displacement, is the time, is the deformation gradient tensor, is the second Piola-Kirchhoff stress tensor, is the elastic constitutive relation, is the Green strain tensor.

8. A simulation method for high-speed photocuring 3D printing based on fluid-structure interaction according to claim 5 or 6, characterized in that: The flow process of the resin and the elastic deformation of the resin tank bottom generate fluid-structure interaction, and the specific description of the coupling conditions is as follows: ， Among them is the moving mesh displacement, is the Cauchy stress tensor of the elastomer at the bottom of the resin tank, is the Cauchy stress tensor of the resin liquid, is the unit normal vector of the elastomer at the bottom of the resin tank on the fluid-structure interface, is the unit normal vector of the resin liquid on the fluid-structure interface; The fluid-structure interaction is described by the arbitrary Lagrangian-Eulerian method / dynamic mesh method. In the dynamic mesh method, the deformation of the mesh is determined by the modified Laplace equation, and is specifically described as follows: ， Among them is the grid cell area.

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