CN107220399A - Weight the whole flow field analogy method of non-oscillatory scheme substantially based on Hermite interpolation - Google Patents

Weight the whole flow field analogy method of non-oscillatory scheme substantially based on Hermite interpolation Download PDF

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CN107220399A
CN107220399A CN201710176365.3A CN201710176365A CN107220399A CN 107220399 A CN107220399 A CN 107220399A CN 201710176365 A CN201710176365 A CN 201710176365A CN 107220399 A CN107220399 A CN 107220399A
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王镇明
朱君
赵宁
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Nanjing University of Aeronautics and Astronautics
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Abstract

Weight the whole flow field analogy method of non-oscillatory scheme substantially based on Hermite interpolation, limited bulk Robust HWENO forms are constructed under cartesian coordinate and immersed Boundary Method is combined and calculate compressible circumferential motion problem, Robust HWENO forms are better than HWENO forms robustness, it is easy to be generalized in higher-dimension or movement and adaptive mesh.Limited bulk Robust HWENO forms under structured grid need high-quality grid when directly simulating above mentioned problem, and high-quality mess generation is extremely complex, immersed Boundary Method is a kind of method that can handle object plane border very well and can effectively used on all kinds of calculating grids.Two methods are effectively combined for this present invention, can on better simply cartesian grid the effectively compressible circumferential motion problem of numerical simulation, the numerical simulation result of several classical permanent and Non―steady Problems fully demonstrates effectiveness of the invention.

Description

Total flow field simulation method based on Hermite interpolation basic weighting non-oscillation format
Technical Field
The invention discloses a full flow field simulation method based on a Hermite interpolation basic weighting non-oscillation format, and relates to the technical field of computational fluid mechanics engineering.
Background
In computational fluid mechanics numerical simulation, the structure and the application of a high-precision format are always the key contents of research, because the high-precision format can accurately simulate the whole flow field, the solved structure can be well simulated, and the shock wave position can be accurately captured. In 1983, Harten first proposed a tvd (total Variation distinguishing) format, and in 1987, with Osher, proposed an ENO (essential Non-oscillotory) high-precision format, the main idea of the ENO format is to select the smoothest template from the successively expanded templates to construct a polynomial to find the value at the cell boundary, thereby achieving the effects of high precision, high resolution and substantially no oscillation at the discontinuity. However, in the implementation process of the method, the ENO format may cause waste of the calculation result, resulting in low calculation efficiency, and therefore Liu, Osher, Chan, and the like propose a WENO (Weighted essential Non-oscillotory) format, which improves the utilization rate of the calculation result and improves the ENO format of the r-order precision to the r + 1-order precision. In 1996, Jiang and Shu proposed a new WENO format that enabled numerical accuracy to be improved to the order of 2 r-1. Subsequently, Qiu and Shu developed such a method, named HWENO format and used it as a restriction for the RKDG (Runge-Kutta Disconnectious Galerkin) method. In the next year, they generalized this approach to two-dimensional situations. In 2008 Zhu and Qiu constructed the finite volume HWENO format with fourth order accuracy. The classical numerical calculation methods have high requirements on grid quality and cannot be directly applied to the problem of compressible streaming including complex object plane boundaries in a calculation flow field, and on the other hand, the high-precision format on the unstructured grid is complex in structure, so that the high-precision format is necessary to be used in a Cartesian grid to process the problem of simulation of the full flow field including the complex object plane. For numerical simulations of these flow fields involving complex geometric objects, spurious oscillations of the solution can occur at object plane boundaries due to the arbitrary intersection of the object plane boundaries with the cartesian grid. In view of this problem, Dadone et al systematically studied object plane boundary processing methods of complex geometric shapes in a patch grid, named st (symmetry technique) Method and ccst (capacitor corrected symmetry technique) Method, and generalized these immersion boundary methods to non-patch cartesian grids, named GBCM (Ghost-Cell Method) Method. In addition, Forrer et al 1998 proposed a dipping boundary Method, named FGCM (Forrer's Ghost Cell Method). The methods can effectively process the boundary conditions of the complex object plane and can effectively reduce the generation difficulty of the used computational grid.
In addition, the numerical method with the first-order precision does not generate non-physical numerical oscillation but excessively smoothes the strong discontinuity when capturing the shock wave, and the strong discontinuity has important significance on the follow-up research of the problem, so that the introduction of a high-precision numerical calculation format for carrying out related numerical simulation is necessary. When complex surfaces appear in a calculation area, many high-precision calculation formats designed based on structural grids cannot be directly applied to relevant numerical calculation, complex skin grids need to be generated first, and then high-precision calculation formats based on the skin grids are constructed, so that the steps are complex and tedious, and the popularization is difficult.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a full flow field simulation method based on a Hermite interpolation basic weighting non-oscillation format, which can carry out numerical simulation on the full flow field by uniformly adopting a Cartesian grid and a high-precision calculation format aiming at the problem of flow bypassing of various complex objects. The invention firstly provides a structure of a finite volume Robust HWENO high-precision numerical calculation format under a Cartesian grid, the format can be better popularized to a high-dimensional situation and a self-adaptive and mobile grid situation, and then aiming at the non-body-sticking characteristic of the Cartesian grid, an appropriate boundary processing method is adopted to process the boundary condition of an object surface, and the two can be effectively combined and applied to numerical simulation of the problem of non-stick compressible streaming of the object surface with a complex geometric shape. Compared with the WENO format, the HWENO format constructs a polynomial through a Hermite interpolation method, and both a function value and a derivative value are needed, so that the relation among grid units is strengthened, the format is more compact and stable, and compared with the HWENO format, the Robust of the Robust HWENO format is enhanced, and the method can be popularized better.
In order to achieve the purpose, the invention adopts the following technical scheme:
in a Cartesian coordinate, a flow field calculation area contains an object, grid points in the object are assigned to form a full flow field by a virtual unit immersion boundary method, and then the full flow field is numerically simulated by using a Robust HWENO format, and the method specifically comprises the following steps:
1) assigning values to grid unit points in the object by using a virtual unit immersion boundary method, so that a flow field containing the object forms a single-medium full flow field;
2) establishing a control equation of the viscous-free fluid, wherein the control equation is an equation system related to a time variable t and related to space variables x and y, and dispersing the space variables of the control equation to obtain a semi-discrete finite volume format, and the method comprises the following specific steps of:
1.1 respectively deriving spatial variables x and y to obtain an equation set, and increasing the number of control equations;
1.2, integrating two sides of a control equation on a target unit simultaneously to obtain a finite volume format, wherein the finite volume format comprises a time derivative term and a space integral term;
1.3 solving the spatial integral term by using a Gauss product-solving formula so as to obtain a semi-discrete finite volume format related to a time derivative term;
3) using a third-order TVD Runge-Kutta discrete formula on a time variable so as to change a semi-discrete finite volume format into a space-time full-discrete finite volume format;
4) and obtaining the flow field value on the next time layer according to a space-time full-discrete finite volume format, and sequentially iterating to obtain the numerical simulation when the full flow field is stable.
In order to optimize the technical scheme, the specific measures adopted further comprise:
in the step 1), two virtual unit methods of ST and FGCM are used for processing the boundary condition of the object plane, so that the flow field is approximate to the full flow field of a single medium, and the method specifically comprises the following steps:
a. classifying points in a calculation area into virtual unit points inside an object and points outside the object;
b. finding out the symmetrical points of all the virtual unit points relative to the object plane, and determining the nearest grid unit I in which the symmetrical points fall1And grid cells I on the upper, lower, left and right sides thereof2，I3，I4，I5；
c. Using grid cells I1，I2，I3，I4，I5Determining the flow field value of the symmetrical point through an interpolation formula;
d. and obtaining the physical quantity relation between the virtual unit point and the symmetrical point by an ST and FGCM virtual unit method, thereby obtaining the flow field value of the virtual unit point and forming the flow field containing the object into a full flow field of a single medium.
In step 1.1, the control equation is:
where U ═ p, ρ U, ρ v, E)TDenotes a conservation variable, f (u) ═ p u, p u2+p，ρuv，u(E+p))T，g(U)＝(ρV，ρuv，ρv2+p，V(E+p))TF (U), g (U) denotes flux, UtMeaning U is derived from t, f (U)xDenotes f (U) derivative of x, g (U)yDenotes g (U) derivative on y, ρ, p, U, v, E denote fluid density, pressure, horizontal velocity, vertical velocity and energy, respectively, T denotes transposition, U0Represents an initial state value;
and respectively deriving the space variables x and y for the control equation to obtain an equation:
wherein,meaning that U is derived over x,meaning that U derives from y, h (U, V)xAnd q (U, W)xDenotes h (U, V) q (U, W) respectively deriving x, r (U, V)yAnd s (U, W)yIt is shown that r (U, V) s (U, W) is derived for y, h (U, V) is f '(U) V, r (U, V) is g' (U) V, q (U, W) is f '(U) W, s (U, W) is g' (U) W, f '(U), g' (U) is f (U), and g (U) is derived for U.
In step 1.2, the control equation is set in the target unit IijThe internal integral has:
wherein,meaning that U is derived over x,meaning that U derives from y, h (U, V)xAnd q (U, W)xDenotes h (U, V) q (U, W) respectively deriving x, r (U, V)yAnd s (U, W)yIt is shown that r (U, V) s (U, W) is derived for y, h (U, V) is f '(U) V, r (U, V) is g' (U) V, q (U, W) is f '(U) W, s (U, W) is g' (U) W, f '(U), g' (U) is f (U), and g (U) is derived for U.
In step 1.3, solving by using a Gauss product formula, wherein the specific process is as follows:
a. taking 8 Gauss points on the boundary of the target unit, wherein the points are respectivelyAndand the target unit and the surrounding 9 units form a mother template;
b. selecting 8 sub-templates from the mother template, wherein each sub-template must comprise a target unit;
c. solving the reconstruction polynomial p of each Gauss point of the target unit in 8 sub-templates by using a Hermite interpolation methodl(x, y), where l is 1, …, 8, where l represents the corresponding sub-template number, then:
wherein,respectively represent U in grid unit Ii-1，j-1、Ii，j-1、Ii+1，j-1、Ii-1，j、Iij、Ii+1，j、Ii-1，j+1、Ii，j+1、Ii+1，j+1Is determined by the average value of (a) of (b),respectively represent V in grid cell Ii-1，j、Ii+1，jIs determined by the average value of (a) of (b),respectively represent W in grid cell Ii，j-1、Ii，j+1H is the grid step length;
d. taking the linear weight gammalCalculating smoothness indicator βlFor evaluating the reconstruction polynomial pl(x, y), l ═ 1, …, 8 smoothness on the target cell, and the formula is calculated as:
wherein α is a multiple index, D is a derivative operator, and r is a polynomial plThe highest degree of (x, y);
e. calculating the non-linear weight omegalThe calculation formula is as follows:
wherein,to calculate the transition values in the process, βlIs a smooth indicator, 10 ═ c-6；
f. Find U at each Gauss point GkThe approximate expression of (a) is:
wherein G iskDenotes the kth Gauss point, k ═ 1, …, 8;
g. repeating the steps c to f, and solving V and W at each Gauss point GkAn approximate expression of (c);
h. solving by using a Gauss product formula:
wherein,representing the boundary length of the target cell, akFor the weighting coefficients of the Gauss product formula,is an external normal vector of the unit,andreplacing with numerical flux;
i. and substituting the calculation result into a semi-discrete finite volume format containing a time derivative term to obtain an ordinary differential equation related to the time derivative.
In step 3), the three-order TVD Runge-Kutta discrete formula is:
wherein,for intermediate transition values, Δ t is the time step, and the superscript n denotes the nth time layer，Is an integral approximation.
In step 4), the space-time full-discrete finite volume format is an iterative formula about a time layer, an initial state value is known, a flow field value of a next time layer is obtained through the iterative formula, and full-flow-field numerical simulation in stable time is sequentially obtained.
The invention has the beneficial effects that: the limited volume Robust HWENO format under the Cartesian grid is combined with the method of immersing the boundary virtual unit, and the method is applied to the numerical simulation without the viscous flow problem with complex geometric shapes, so that the generation of complex grids and the construction of high-precision formats under unstructured grids are avoided, and compared with the WENO format and the HWENO format, the Robust HWENO format enhances the compactness and the robustness of the formats and can be well popularized to the numerical simulation under higher-dimensional and moving grids. The traditional numerical method needs to do certain equivalent transformation or even cannot solve the problems of double Mach reflection with a real inclined plane, cylindrical flow around and airfoil flow around, but the method can capture strong shock waves and contact discontinuity basically without oscillation and simultaneously keep the high-order precision of space-time consistency in a smooth region of solution.
Drawings
FIG. 1 is a schematic diagram of a large template T.
Fig. 2 is a schematic diagram of grid cells intersecting an object plane boundary in a cartesian grid.
Fig. 3a-3b are density contour plots of the pre-stage problem using two boundary processing methods, ST and FGCM, respectively, in the first embodiment.
Fig. 4a-4b are pressure contour plots of the first embodiment, in which the pre-step problem employs two boundary processing methods, ST and FGCM, respectively.
Fig. 5a-5b are density contour plots of the two mach-zehnder problem using the two boundary processing methods ST and FGCM, respectively, in example two.
Fig. 6a-6b are pressure contour plots of the two mach-zehnder problem using the two boundary processing methods ST and FGCM, respectively, in example two.
Fig. 7a-7b are density contour plots of the cylindrical streaming problem using two boundary processing methods, ST and FGCM, respectively, in the third embodiment.
Fig. 8a-8b are contour graphs of mach numbers for the cylindrical winding problem using two boundary processing methods, ST and FGCM, respectively, in the third embodiment.
9a-9b are pressure contour plots of the NACA0012 airfoil flow problem using the ST and FGCM boundary processing methods, respectively, in the fourth embodiment.
10a-10b are graphs of airfoil surface pressure coefficients for the NACA0012 airfoil flow-around problem using two boundary processing methods, ST and FGCM, respectively, in the fourth embodiment.
Detailed Description
The present invention will now be described in further detail with reference to the accompanying drawings.
One, limited volume RonustHWENO format construction
Consider the two-dimensional Euler equation:
where t denotes a time variable, x, y denotes a space variable, and U ═ p, ρ U, ρ v, E)TDenotes a conservation variable, f (u) ═ p u, p u2+p，ρuv，u(E+p))T，g(U)＝(ρv，ρuv，ρv2+p，v(E+p))TF (U), g (U) denotes flux, f (U)xDenotes f (U) derivative of x, g (U)yDenotes g (U) derivation of y, ρ, p, u, v, ERepresenting fluid density, pressure, horizontal velocity, vertical velocity, and energy, T representing transpose, U0Indicating the initial state value.
Assuming the grid cell step size is fixed, xi+1/2-xi-1/2＝Δx，yj+1/2-yj-1/2Δ y, grid center isGrid cell is Iij＝[xi-1/2，xi+1/2]×[yj-1/2，yj+1/2]And subscripts i and j are coordinate serial numbers.
The derivation of equation (1) is:
wherein,meaning that U is derived over x,meaning that U derives from y, h (U, V)xAnd q (U, W)xDenotes h (U, V) q (U, W) respectively deriving x, r (U, V)yAnd s (U, W)yIt is shown that r (U, V) s (U, W) derives y, wherein h (U, V) is f '(U) V, r (U, V) is g' (U) V, q (U, W) is f '(U) W, s (U, W) is g' (U) W, f '(U), g' (U) is f (U), and g (U) derives U.
Definition ofIndicating U in grid cell IijThe average value of the values of (a) to (b),indicating V in grid cell IijThe average value of the values of (a) to (b),denotes W in grid cell IijInner average value, i.e.
First, find each Gauss point GkIs positioned at U (G)k，t)，V(Gk，t)，W(GkAnd t) is as follows:
step 1, as shown in FIG. 1, target Unit IijAnd a mother template is composed of 9 grid units around, and the grid step length is assumed to be h, as shown in FIG. 1, wherein I5Is a target unit IijTo and fromRespectively represent U, V and W in grid unit ImAverage value of (m — 1, …, 9).
And 2, selecting 8 sub-templates from the mother template, wherein the 8 sub-templates are respectively as follows: s1＝{I1，I2，I4，I5}，S2＝{I2，I3，I5，I6}，S3＝{I4，I5，I7，I8}，S4＝{I5，I6，I7，I9}，S5＝{I1，I2，I3，I4，I5，I7}，S6＝{I1，I2，I3，I5，I6，I9}，S7＝{I1，I4，I5，I7，I8，I9}，S8＝{I3，I5，I6，I7，I8，I9In which IiIs the grid cell of the corresponding serial number.
Step 3, respectively calculating each Gauss point G in each sub-templatekReconstruction polynomial p of U, V, Wn(x，y)，n＝1，…，8。
Step 3.1, reconstruction polynomial p of Un(x, y), n ═ 1, …, 8, the solution is as follows:
step 3.1.1 at sub-template S1，S2，S3，S4Upper structure polynomial pn(x, y), n is 1, …, 4, such that:
wherein
n＝1，k＝1，2，4，5，kx＝4，ky＝2；n＝2，k＝2，3，5，6，kx＝6，ky＝2；
n＝3，k＝4，5，7，8，kx＝4，ky＝8；n＝4，k＝5，6，8，9，kx＝6，ky＝8；
Step 3.1.2, in the sub-template S5，S6，S7，S8Upper structure polynomial pn(x, y), n is 5, …, 8, such that it satisfies:
wherein
n＝5，k＝1，2，3，4，5，7；n＝6，k＝1，2，3，5，6，9；
n＝7，k＝1，4，5，7，8，9；n＝8，k＝3，5，6，7，8，9；
Step 3.1.3 obtaining each seedTemplate at each Gauss point GkInterpolation polynomial p of (A)n(x, y), n ═ 1, …, 8, as follows:
step 3.2, reconstruction polynomial p of Vn(x, y), n ═ 1, …, 8, the solution is as follows:
step 3.2.1 at sub-template S1，S2，S3，S4Upper structure polynomial pn(x, y), n is 1, …, 4, such that:
n＝1，k＝1，2，4，5，kx＝1，4，5，ky＝1，2，5；n＝2，k＝2，3，5，6，kx＝3，5，6，ky＝2，3，5；
n＝3，k＝4，5，7，8，kx＝4，5，7，ky＝5，7，8；n＝4，k＝5，6，8，9，kx＝5，6，9，ky＝5，8，9；
step 3.2.2 at sub-template S5，S6，S7，S8Upper structure polynomial pn(x, y), n is 5, …, 8, such that it satisfies:
wherein
n＝5，kx＝1，2，3，4，5，7；n＝6，kx＝1，2，3，5，6，9；
n＝7，kx＝1，4，5，7，8，9；n＝8，kx＝3，5，6，7，8，9；
Step 3.2.3, obtaining the G point of each sub-template at each Gauss pointkInterpolation polynomial p of (A)n(x, y), n ═ 1, …, 8, as follows:
wherein,represents pn(x, y) deriving x with a bias (n is 1, …, 4), h is the step size;
step 3.3, reconstruction polynomial p of Wn(x, y), n ═ 1, …, 8, the solution is as follows:
step 3.3.1 at sub-template S1，S2，S3，S4Upper structure polynomial pn(x, y), n is 1, …, 4, such that:
wherein
n＝1，k＝1，2，4，5，kx＝1，4，5，ky＝1，2，5；n＝2，k＝2，3，5，6，kx＝3，5，6，ky＝2，3，5；
n＝3，k＝4，5，7，8，kx＝4，5，7，ky＝5，7，8；n＝4，k＝5，6，8，9，kx＝5，6，9，ky＝5，8，9；
Step 3.3.2, in the sub-template S5，S6，S7，S8Upper structure polynomial pn(x, y), n is 5, …, 8, such that it satisfies:
wherein
n＝5，ky＝1，2，3，4，5，7；n＝6，ky＝1，2，3，5，6，9；
n＝7，ky＝1，4，5，7，8，9；n＝8，ky＝3，5，6，7，8，9；
Step 3.3.3, obtaining the G point of each sub-template at each Gauss pointkInterpolation polynomial p of (A)n(x, y), n ═ 1, …, 8, as follows:
wherein,represents pn(x, y) the partial derivatives are calculated for y (n is 1, …, 4), and h is the step size.
Step 4, taking each Gauss point GkLinear weight of (A) is
Step 5, calculate smooth indicator βl1, …, 8, the calculation formula is as follows:
step 5.1, the smooth indicator calculation formula of the reconstructed U is as follows:
step 5.2, the smooth indicator calculation formula of the reconstructed V is as follows:
step 5.3, the smooth indicator calculation formula of the reconstructed W is as follows:
wherein a is a multiple index, | I5And | is the area of the target unit, D is the derivation operator, and the subscript l is the serial number corresponding to the sub-template.
Step 6, passing linear weight glAnd a smoothness indicator βlCalculating the non-linear weight omegal：
Wherein,for the transition value in the calculation, βlIs a smooth indicator, and takes a value of 10-6The denominator is prevented from being zero.
Step 7, further obtaining U, V and W at each Gauss point GkApproximate values of (a):
wherein,represents pl(Gk) The partial derivatives of the x are calculated,represents pl(Gk) Partial derivatives of y, pl(Gk) Coordinates G representing Gauss pointskSubstituting the corresponding polynomial plTaking values after (x, y);
secondly, the equations (1), (2) and (3) are compared with the target unit IijThe internal integration yields a semi-discrete finite volume format as follows:
wherein,indicating U in grid cell IijThe average value of the values of (a) to (b),indicating V in grid cell IijThe average value of the values of (a) to (b),denotes W in grid cell IijInner average value, i.e.F＝(f(U)，g(U))T，H＝(h(U，V)，r(U，V))T，Q＝(q(U，W)，s(U，W))TAnd T represents a transpose,for the outer normal vector, the integral term on the right side of the equation can be obtained by using a q-point Gauss product formula as follows:
wherein,representing the boundary length of the target cell, akFor the weighting coefficients of the Gauss product formula,is an external normal vector of the unit,replacing with numerical flux;
finally, utilizing a three-order TVD Runge-Kutta discrete formula:
obtaining a space-time fully discrete finite volume format, whereinFor intermediate transition values, Δ t is the time step, the superscript n denotes the nth temporal layer,is an integral approximation.
Second, virtual unit method
In practical calculations, due to the non-conformality of the cartesian grid, the limited volume Robust HWENO format is very difficult to deal with the problem that an object plane with a complex geometric shape can constrict the flow, and non-physical oscillations can be generated at the intersection of the grid and the object plane. To compensate for this deficiency, the present invention treats the boundary of the object plane using the immersion boundary method. As shown in fig. 2, solid triangles represent virtual grid cell points inside the object plane, and solid squares represent points of symmetry of the virtual grid cell points with respect to the boundary of the object plane.
The value of a point A in an object is determined by utilizing the value of a point B in a fluid through the boundary condition of an object plane, the momentum relationship and the like, and then the numerical simulation of the full flow field is carried out. Two different methods are introduced below to determine the relationship between the flow field value at the point A and the flow field value at the point B, and a numerical experiment is performed to numerically simulate the full flow field containing the object surface with the complex geometric shape in the calculation region.
2.1 ST (symmetry technique) method
The method is the simplest and most direct method for determining the flow field values of the point A and the point B, and the calculation formula is as follows:
in the above formula, r is density, p is pressure,in the form of a velocity vector, the velocity vector,the unit external normal vector at the object plane is represented, so that the given boundary condition can satisfy the wall surface non-penetration condition.
2.2 FGCM (Forrer's Ghost Cell Method) Method
Forrer et al proposed a virtual cell immersion boundary method, the idea being that the velocity at point A still uses the general symmetric reflection, and the density and pressure are found by the following equations:
wherein x iswIs the point where the normal vector passing through point a intersects the object plane boundary, is the unit external normal vector of the object plane, and similarly, the boundary condition given in this way also satisfies the wall surface non-penetration condition.
Third, the combination of the limited volume Robust HWENO format and the virtual unit method
From the above steps, the limited volume Robust HWENO format is a limited volume high-precision numerical calculation format. The finite volume method starts with a fluid equation in the form of integral conservation, and constructs an integral discrete equation by discretizing the equation on grid cells. Compared with a finite difference method, the finite volume method is more flexible in grid processing and good in conservative property. However, for the finite volume Robust HWENO format constructed on a cartesian structural grid, it is difficult to solve complex interface streaming problems such as double mach reflection, cylindrical streaming, etc. The problem is particularly prominent because the finite volume weighted substantially free-oscillation format requires a total of 9 surrounding (including itself) grid cells for polynomial reconstruction of the field values of the grid cell streams, especially where the grid intersects the boundary of the object plane, where the grid intersects the object plane in an arbitrary manner. In view of these problems, in the double mach-zehnder reflectometry numerical simulation, the conventional solution is to tilt the incident shock wave, so that the slope coincides with the grid lines, and the numerical simulation becomes clear and simple. Although the reflection of the laser wave can be captured well, the method is different from the experimental simulation in form or nature. In the cylindrical flow-around problem or the airfoil flow-around problem, the traditional solution is to use a mixed mesh or an unstructured mesh, which undoubtedly increases the difficulty of mesh generation and the difficulty of format construction on the corresponding mesh. The invention combines the limited volume Robust HWENO format of the structural grid with the method of immersing the boundary virtual unit, effectively simulates the problem of double Mach reflection with a real inclined plane, and accurately captures the reflection condition after the action of the shock wave and the inclined plane. Similarly, better numerical simulation results are obtained in high mach number cylindrical flows and airfoils.
Four examples are given below as specific examples of the disclosed method.
Embodiment one, step problem. The problem is a classical example proposed by Emery in 1968 for testing the nonlinear hyperbolic conservation law format. The initial data is a mach number of 3 for the horizontal incoming stream, a density of 1.4, a pressure of 1, a duct region of 0, 3' [0, 1], a step of 0.2 height at 0.6 from the left boundary, and the step extending to the end of the duct. The upper and lower boundaries are reflection boundaries, the left boundary is an incoming flow boundary, and the right boundary is an outgoing flow boundary. Fig. 3a-3b and fig. 4a-4b show contour plots of density and pressure at time t-4, respectively.
Example two, the problem of beveled double mach-zehnder reflections. The problem is to describe the change of a strong shock wave incident on a slope at an angle of 30 ° to the plane, the incoming flow being a strong shock wave with mach number 10. The calculation region is taken to be [0, 3]′[0，2]Starting fromShock was perpendicular to the x-axis and the initial data were:
wherein the left and right states are UL＝(8，66.009，0，563.544)T，UR＝(1.4，0，0，2.5)TThe boundary condition is processed by taking left and right state values for the left and right boundaries, respectively. Lower bound: when in useWhen the reflection boundary is set, setting the reflection boundary as a reflection boundary; when in useWhen it is, it is set to the left state. An upper boundary: when x is more than g (t), taking a right state; when in useWhen it is in the left state, whereinFig. 5a-5b and fig. 6a-6b show contour plots of density and pressure at time t 0.2, respectively.
Example three, cylindrical streaming problem. Considering a supersonic, non-viscous fluid flowing toward a cylinder with a diameter of 1, the initial data is a horizontal incoming flow mach number of 3, a density of 1, a pressure of 1, and a calculated region of [ -10, 10 ]? [10, 10], the left boundary is the incoming flow boundary and the right boundary is the supersonic outflow boundary. Figures 7a-7b and 8a-8b show the density and mach number contours obtained by the two methods in the calculation region, respectively.
Fourth embodiment, NACA0012 airfoil shape transonic velocity flow problem. The incoming flow mach number is 0.8, and the attack angle a is 1.25 degrees. Fig. 9a-9b and fig. 10a-10b respectively show a pressure contour map, an airfoil surface pressure coefficient map and a residual map of the ST method and the FGCM method, and it can be seen from the maps that the finite volume Robust HWENO format combined with the two immersed boundary methods can capture the position of the shock wave and the intensity of the shock wave well, a strong shock wave is formed on the upper surface, and a weak shock wave is formed on the lower surface.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.

Claims (7)

1. A full flow field simulation method based on a Hermite interpolation basic weighting non-oscillation format is characterized in that a flow field calculation area contains an object under Cartesian coordinates, grid points in the object are assigned through a virtual unit immersion boundary method to form a full flow field, and then the full flow field is numerically simulated by using a Robust HWENO format, and the method comprises the following specific steps:
1) assigning values to grid unit points in the object by using a virtual unit immersion boundary method, so that a flow field containing the object forms a single-medium full flow field;
2) establishing a control equation of the viscous-free fluid, wherein the control equation is an equation system related to a time variable t and related to space variables x and y, and dispersing the space variables of the control equation to obtain a semi-discrete finite volume format, and the method comprises the following specific steps of:
1.1 respectively deriving spatial variables x and y to obtain an equation set, and increasing the number of control equations;
1.2, integrating two sides of a control equation on a target unit simultaneously to obtain a finite volume format, wherein the finite volume format comprises a time derivative term and a space integral term;
1.3 solving the spatial integral term by using a Gauss product-solving formula so as to obtain a semi-discrete finite volume format related to a time derivative term;
3) using a third-order TVD Runge-Kutta discrete formula on a time variable so as to change a semi-discrete finite volume format into a space-time full-discrete finite volume format;
4) and obtaining the flow field value on the next time layer according to a space-time full-discrete finite volume format, and sequentially iterating to obtain the numerical simulation when the full flow field is stable.
2. The full-flow-field simulation method based on the hermitian interpolation basic weighting non-oscillation format, as claimed in claim 1, wherein: in the step 1), two virtual unit methods of ST and FGCM are used for processing the boundary condition of the object plane, so that the flow field is approximate to the full flow field of a single medium, and the method specifically comprises the following steps:
a. classifying points in a calculation area into virtual unit points inside an object and points outside the object;
b. finding out the symmetrical points of all the virtual unit points relative to the object plane, and determining the nearest grid unit I in which the symmetrical points fall1And grid cells I on the upper, lower, left and right sides thereof2，I3，I4，I5；
c. Using grid cells I1，I2，I3，I4，I5Determining the flow field value of the symmetrical point through an interpolation formula;
d. and obtaining the physical quantity relation between the virtual unit point and the symmetrical point by an ST and FGCM virtual unit method, thereby obtaining the flow field value of the virtual unit point and forming the flow field containing the object into a full flow field of a single medium.
3. The full-flow-field simulation method based on the hermitian interpolation basic weighting non-oscillation format as claimed in claim 2, characterized in that: in step 1.1, the control equation is:
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>f</mi> <msub> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <mi>x</mi> </msub> <mo>+</mo> <mi>g</mi> <msub> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <mi>y</mi> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>U</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
where U ═ p, ρ U, ρ v, E)TDenotes a conservation variable, f (u) ═ p u, p u2+p，ρuv，u(E+p))T，g(U)＝(ρv，ρuv，ρv2+p，v(E+p))TF (U), g (U) denotes flux, UtMeaning U is derived from t, f (U)xDenotes f (U) derivative of x, g (U)yDenotes g (U) derivative of y, p, u, v, ERespectively representing fluid density, pressure, horizontal velocity, vertical velocity and energy, T representing transposition, U0Represents an initial state value;
and respectively deriving the space variables x and y for the control equation to obtain an equation:
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>V</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>h</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>,</mo> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </msub> <mo>+</mo> <mi>r</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>,</mo> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mi>y</mi> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>V</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>x</mi> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced>1
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>W</mi> <mi>t</mi> </msub> <mo>+</mo> <mi>q</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>,</mo> <mi>W</mi> </mrow> <mo>)</mo> </mrow> <mi>x</mi> </msub> <mo>+</mo> <mi>s</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>,</mo> <mi>W</mi> </mrow> <mo>)</mo> </mrow> <mi>y</mi> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>W</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>y</mi> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
wherein,meaning that U is derived over x,meaning that U derives from y, h (U, V)xAnd q (U, W)xDenotes h (U, V) q (U, W) respectively deriving x, r (U, V)yAnd s (U, W)yDenotes that r (U, V) s (U, W) is respectively derived for y, h (U,v ═ f '(U) V, r (U, V) ═ g' (U) V, q (U, W) ═ f '(U) W, s (U, W) ═ g' (U) W, f '(U), g' (U) respectively denote f (U), and g (U) is derived from U.
4. The full-flow-field simulation method based on the hermitian interpolation basic weighting non-oscillation format, as claimed in claim 3, wherein: in step 1.2, the control equation is set in the target unit IijThe internal integral has:
<mrow> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mi>&amp;Delta;</mi> <mi>x</mi> <mi>&amp;Delta;</mi> <mi>y</mi> </mrow> </mfrac> <msub> <mo>&amp;Integral;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mi>F</mi> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>d</mi> <mi>s</mi> </mrow>
<mrow> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mi>&amp;Delta;</mi> <mi>y</mi> </mrow> </mfrac> <msub> <mo>&amp;Integral;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mi>H</mi> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>d</mi> <mi>s</mi> </mrow>
<mrow> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mi>&amp;Delta;</mi> <mi>x</mi> </mrow> </mfrac> <msub> <mo>&amp;Integral;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mi>Q</mi> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>d</mi> <mi>s</mi> <mo>;</mo> </mrow>
wherein,indicating U in grid cell IijThe average value of the values of (a) to (b),indicating V in grid cell IijThe average value of the values of (a) to (b),denotes W in grid cell IijInner average value, i.e.F＝(f(U)，g(U))T，H＝(h(U，V)，r(U，V))T，Q＝(q(U，W)，s(U，W))TThe indices i and j are coordinate numbers, Δ x denotes the step size in the x-direction, Δ y denotes the step size in the y-direction,is the outer normal vector.
5. The full-flow-field simulation method based on the hermitian interpolation basic weighting non-oscillation format, as claimed in claim 4, wherein: in step 1.3, solving by using a Gauss product formula, wherein the specific process is as follows:
a. taking 8 Gauss points on the boundary of the target unit, wherein the points are respectivelyAndand the target unit and the surrounding 9 units form a mother template;
b. selecting 8 sub-templates from the mother template, wherein each sub-template must comprise a target unit;
c. solving the reconstruction polynomial p of each Gauss point of the target unit in 8 sub-templates by using a Hermite interpolation methodl(x, y), where l is 1, …, 8, where l represents the corresponding sub-template number, then:
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>-</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>x</mi> <mi>y</mi> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> </mrow> <mi>h</mi> </mfrac> <mi>x</mi> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> </mrow> <mi>h</mi> </mfrac> <mi>y</mi> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>10</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>-</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>x</mi> <mi>y</mi> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> </mrow> <mi>h</mi> </mfrac> <mi>x</mi> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> </mrow> <mi>h</mi> </mfrac> <mi>y</mi> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <mn>10</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>-</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>2
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>x</mi> <mi>y</mi> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> </mrow> <mi>h</mi> </mfrac> <mi>x</mi> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <mi>h</mi> </mfrac> <mi>y</mi> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>10</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>-</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>x</mi> <mi>y</mi> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> </mrow> <mi>h</mi> </mfrac> <mi>x</mi> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <mi>h</mi> </mfrac> <mi>y</mi> <mo>+</mo> <mfrac> <mrow> <mn>10</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>-</mo> <msub> <mover> <mi>V</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>-</mo> <msub> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <mn>12</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>x</mi> <mi>y</mi> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> <mi>y</mi> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mrow> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>24</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> </mrow> <mn>24</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>x</mi> <mi>y</mi> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> <mi>x</mi> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> <mi>y</mi> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mn>24</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mn>24</mn> </mfrac> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mn>7</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>x</mi> <mi>y</mi> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> <mi>x</mi> <mo>-</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> <mi>y</mi> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>24</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mn>24</mn> </mfrac> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mn>8</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> </mfrac> <mi>x</mi> <mi>y</mi> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> <mi>x</mi> <mo>-</mo> <mfrac> <mrow> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mrow> <mn>2</mn> <mi>h</mi> </mrow> </mfrac> <mi>y</mi> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mn>24</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>5</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>6</mn> </msub> <mo>-</mo> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>7</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>8</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mn>9</mn> </msub> </mrow> <mn>24</mn> </mfrac> </mtd> </mtr> </mtable> <mo>;</mo> </mrow>
wherein,respectively represent U in grid unit Ii-1，j-1、Ii，j-1、Ii+1，j-1、Ii-1，j、Iij、Ii+1，j、Ii-1，j+1、Ii，j+1、Ii+1，j+1Is determined by the average value of (a) of (b),respectively represent V in grid cell Ii-1，j、Ii+1，jIs determined by the average value of (a) of (b),respectively represent W in grid cell Ii，j-1、Ii，j+1H is the grid step length;
d. taking the linear weight gammalCalculating smoothness indicator βlFor evaluating the reconstruction polynomial pl(x, y), l ═ 1, …, 8 smoothness on the target cell, and the formula is calculated as:
<mrow> <msub> <mi>&amp;beta;</mi> <mi>l</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mo>|</mo> <mi>&amp;alpha;</mi> <mo>|</mo> <mo>=</mo> <mn>1</mn> </mrow> <mi>r</mi> </munderover> <mrow> <mrow> <msub> <mo>&amp;Integral;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </msub> <mrow> <mo>|</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mo>|</mo> <mrow> <mo>|</mo> <mi>&amp;alpha;</mi> <mo>|</mo> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>D</mi> <mi>&amp;alpha;</mi> </msup> <msub> <mi>p</mi> <mi>l</mi> </msub> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> </mrow> <mi>d</mi> <mi>y</mi> <mo>;</mo> </mrow>
wherein α is a multiple index, D is a derivative operator, and r is a polynomial plThe highest degree of (x, y);
e. calculating the non-linear weight omegalThe calculation formula is as follows:
<mrow> <msub> <mover> <mi>&amp;omega;</mi> <mo>~</mo> </mover> <mi>l</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>&amp;gamma;</mi> <mi>l</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>&amp;epsiv;</mi> <mo>+</mo> <msub> <mi>&amp;beta;</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>,</mo> <msub> <mi>&amp;omega;</mi> <mi>l</mi> </msub> <mo>=</mo> <mfrac> <msub> <mover> <mi>&amp;omega;</mi> <mo>~</mo> </mover> <mi>l</mi> </msub> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>8</mn> </munderover> <msub> <mover> <mi>&amp;omega;</mi> <mo>~</mo> </mover> <mi>k</mi> </msub> </mrow> </mfrac> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mn>8</mn> <mo>;</mo> </mrow>
wherein,to calculate the transition values in the process, βlIs a smooth indicator, 10 ═ c-6；
f. Find U at each Gauss point GkThe approximate expression of (a) is:
<mrow> <mi>U</mi> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;ap;</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>8</mn> </munderover> <msub> <mi>&amp;omega;</mi> <mi>l</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>p</mi> <mi>l</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow>
wherein G iskDenotes the kth Gauss point, k ═ 1, …, 8;
g. repeating the steps c to f, and solving V and W at each Gauss point GkAn approximate expression of (c);
h. solving by using a Gauss product formula:
<mrow> <msub> <mo>&amp;Integral;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mi>F</mi> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>d</mi> <mi>s</mi> <mo>&amp;ap;</mo> <mo>|</mo> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>8</mn> </munderover> <msub> <mi>a</mi> <mi>k</mi> </msub> <mi>F</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> </mrow>
<mrow> <msub> <mo>&amp;Integral;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mi>H</mi> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>d</mi> <mi>s</mi> <mo>&amp;ap;</mo> <mo>|</mo> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>8</mn> </munderover> <msub> <mi>a</mi> <mi>k</mi> </msub> <mi>H</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>,</mo> <mi>V</mi> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> </mrow>
<mrow> <msub> <mo>&amp;Integral;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mi>H</mi> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>d</mi> <mi>s</mi> <mo>&amp;ap;</mo> <mo>|</mo> <mo>&amp;part;</mo> <msub> <mi>I</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>8</mn> </munderover> <msub> <mi>a</mi> <mi>k</mi> </msub> <mi>H</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>,</mo> <mi>V</mi> <mo>(</mo> <mrow> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mover> <mi>n</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>;</mo> </mrow>
wherein,representing the boundary length of the target cell, akFor the weighting coefficients of the Gauss product formula,is an external normal vector of the unit,andreplacing with numerical flux;
i. and substituting the calculation result into a semi-discrete finite volume format containing a time derivative term to obtain an ordinary differential equation related to the time derivative.
6. The full-flow-field simulation method based on the hermitian interpolation basic weighting non-oscillation format, as claimed in claim 5, wherein: in step 3), the three-order TVD Runge-Kutta discrete formula is:
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mi>n</mi> </msup> <mo>+</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mi>L</mi> <mo>(</mo> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mi>n</mi> </msup> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mi>n</mi> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mi>&amp;Delta;</mi> <mi>t</mi> <mi>L</mi> <mo>(</mo> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mi>n</mi> </msup> <mo>+</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msup> <mo>+</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mi>&amp;Delta;</mi> <mi>t</mi> <mi>L</mi> <mo>(</mo> <msup> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msup> <mo>)</mo> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
wherein,for intermediate transition values, Δ t is the time step, the superscript n denotes the nth temporal layer,is an integral approximation.
7. The full-flow-field simulation method based on the hermitian interpolation basic weighting non-oscillation format as claimed in claim 6, characterized in that: in step 4), the space-time full-discrete finite volume format is an iterative formula about a time layer, an initial state value is known, a flow field value of a next time layer is obtained through the iterative formula, and full-flow-field numerical simulation in stable time is sequentially obtained.
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