A method for constructing monomial basis functions of discontinuous finite elements applicable to arbitrary meshes

Abstract

The present invention provides a method for constructing a discontinuous finite element monomial basis function applicable to any grid, belongs to the field of computer-aided engineering, and is used for hybrid grid calculations. The present invention divides the calculation area into a plurality of unstructured grids to maximize the filling of the calculation area. The expressions of the basis functions are consistent for different units and can be directly used for the discontinuous finite element method on the unstructured hybrid grid. The diagonal part of the mass matrix of the basis function space is 1, and the absolute value of the non-diagonal part is less than 1, which satisfies the main diagonal dominance and makes the calculation more stable; the basis function is defined in global coordinates, and when calculating the gradient, it is not necessary to use the Jacobian matrix of the unit isoparametric transformation, which is more direct and efficient.

Description

Technical Field

[0001] The invention belongs to the field of computer-aided engineering, and in particular relates to a discontinuous finite element monomial basis function construction method suitable for arbitrary grids. Background Art

[0002] In the field of computer-aided engineering, high-order precision formats have smaller numerical dispersion and dissipation, and are suitable for solving multi-scale flow problems, such as turbulence, aeroacoustics, etc. Among the many high-order precision calculation methods, the discontinuous finite element method has attracted much attention due to its good mathematical properties and its convenience for expansion to high dimensions and high orders. In the discontinuous finite element method, discontinuities in solutions are allowed between units, that is, the gradient of the solution may be discontinuous on the unit interface. This discontinuity makes the method more suitable for dealing with problems with discontinuous solutions or gradients. For each unit, a local discrete equation is established, and the weighted residual method (Galerkin method) is usually used to obtain the discrete equation.

[0003] The selection of basis functions is particularly important for the discontinuous finite element method. In the traditional discontinuous finite element method, the Lagrangian basis function is used, and the position of the Lagrangian point varies with the shape of the mesh. For mixed meshes containing triangles, quadrilaterals, tetrahedrons, hexahedrons, triangular prisms, pyramids and other shapes, multiple basis functions appear in the algorithm, which makes the construction of limiters, implicit time advancement and boundary condition processing inconvenient. Summary of the invention

[0004] In order to solve the above technical problems, the present invention constructs a basis function suitable for different grid shapes for hybrid grid calculation, and the specific steps are as follows:

[0005] Step 1: Divide the computational area of ​​the problem to be solved using an unstructured grid unit generation algorithm. Since the geometric shape of the computational area may be complex, a single grid shape may not be able to fill it with high quality. Dividing it into multiple unstructured grids can maximize the filling of the computational area. Among them, the three-dimensional units of the unstructured grid include: tetrahedron, hexahedron, triangular prism, pyramid; the two-dimensional units include: triangle, quadrilateral; the one-dimensional units include: line segment.

[0006] Step 2: Define the local coordinates (ξ,η) of the element:

[0007]

[0008]

[0009] Among them, (x, y) is the global coordinate of the grid, (x c ,y c) is the unit center defined by node average, h is the average distance from the node to the center; ξ represents the local coordinate of the x component, and η represents the local coordinate of the y component.

[0010] Step 3: According to the local coordinates, define the polynomial space composed of monomials, expressed as the basis function φ:

[0011] φ={1,ξ,η,ξ 2 ,ξη,η 2}

[0012] Here, {·} represents the polynomial space composed of basis functions.

[0013] Step 4: Calculate the inner product m of the basis function φ ij ,

[0014] m ij =<φ i φ j >=∫ e φ i φ j dΩ

[0015] Among them, φ i ,φ j represents the i-th and j-th basis functions in the polynomial space, <> represents the inner product, and dΩ represents the volume integral differential element.

[0016] Renormalize the basis functions of the polynomial space

[0017]

[0018] To satisfy,

[0019]

[0020]

[0021] Get the renormalized basis function space That is, we get the monomial basis function:

[0022]

[0023] in, represents the i-th basis function.

[0024] Step 5: Construct the compressible Euler equations:

[0025]

[0026] in, denote the convection term, represents the time partial derivative, represents the spatial gradient operator, and U is the conserved variable of the compressible fluid. Multiply both sides of the Euler equation by the monomial basis function constructed in step 4 As a test function, we have

[0027]

[0028] Among them, dV represents the volume differential element, dS represents the surface differential element, represents the inviscid flux, represents the convection term;

[0029] Viscous flux Calculation using Riemann flux

[0030]

[0031] Where n represents the external normal vector of the unit surface, λ max is the maximum eigenvalue, which can be calculated using the following formula:

[0032] λ max ={‖u·n‖+c}

[0033] u and c represent the local fluid velocity and sound speed, respectively, and ‖·‖ indicates the absolute value.

[0034] Step 6: According to step 5, we can get the ordinary differential equation about time

[0035]

[0036] in, is the time derivative of the variable. R(U) is a spatial discrete operator and can be calculated using the following formula:

[0037]

[0038] Step 7: Use the three-step, third-order Runge-Kutta method to advance the time. n Solve for U at the next moment n+1

[0039] U (1) =U n +ΔtR(U n )

[0040]

[0041]

[0042] Δt is the time step length, U (1) ,U (2) Temporary variables for intermediate steps.

[0043] The beneficial effects of the present invention are as follows:

[0044] (1) The expressions of the basis functions are consistent for different elements and can be directly used in the discontinuous finite element method on unstructured hybrid grids.

[0045] (2) The diagonal part of the mass matrix of the basis function space is 1, and the absolute value of the non-diagonal part is less than 1, which satisfies the main diagonal dominance and makes the calculation more stable.

[0046] (3) The basis functions are defined in global coordinates. When calculating the gradient, there is no need to use the Jacobian matrix of the unit isoparametric transformation, which is more direct and efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 Schematic diagram of a unit cell of an unstructured network.

[0048] Figure 2 It is the density cloud map of the initial time position and the 2-second time position in Example 2.

[0049] Figure 3 Schematic diagram of two sets of grid units: regular triangle and irregular triangle. DETAILED DESCRIPTION

[0050] In order to better understand the purpose, structure and function of the present invention, a method for constructing a discontinuous finite element monomial basis function applicable to any grid is further described in detail below in conjunction with the accompanying drawings.

[0051] Example 1

[0052] The specific steps of the present invention are as follows:

[0053] Step 1: Divide the computational domain of the problem to be solved using an unstructured grid generation algorithm. Since the geometric shape of the computational domain may be complex, a single grid shape may not be able to fill it with high quality. Dividing it into multiple unstructured grids can maximize the filling of the computational domain. Among them, the three-dimensional units of the unstructured grid include: tetrahedron, hexahedron, triangular prism, pyramid; the two-dimensional units include: triangle, quadrilateral; the one-dimensional units include: line segment.

[0054] Step 2: Define the local coordinates (ξ,η) of the element:

[0055]

[0056]

[0057] Among them, (x, y) is the global coordinate of the grid, (x c ,y c) is the unit center defined by node average, h is the average distance from the node to the center; ξ represents the local coordinate of the x component, and η represents the local coordinate of the y component.

[0058] Step 3: According to the local coordinates, define the polynomial space composed of monomials, expressed as the basis function φ:

[0059] φ={1,ξ,η,ξ 2 ,ξη,η 2}

[0060] Here, {·} represents the polynomial space spanned by basis functions.

[0061] Step 4: Calculate the inner product m of the basis function φ ij ,

[0062] m ij =<φ i φ j >=∫ e φ i φ j dΩ

[0063] Among them, φ i ,φ j represents the i-th and j-th basis functions in the polynomial space, <> represents the inner product, and dΩ represents the volume integral differential element.

[0064] Renormalize the basis functions of the polynomial space

[0065]

[0066] To satisfy,

[0067]

[0068]

[0069] Get the renormalized basis function space That is, we get the monomial basis function:

[0070]

[0071] in, represents the i-th basis function, represents the basis function space.

[0072] Step 5: Construct the compressible Euler equations:

[0073]

[0074] in, denote the convection term, represents the time partial derivative, represents the spatial gradient operator, and U is the conserved variable of the compressible fluid. Multiply both sides of the Euler equation by the monomial basis function constructed in step 5 As a test function, we have

[0075]

[0076] Among them, dV represents the volume differential element, dS represents the surface differential element, represents the inviscid flux, represents the convection term.

[0077] Viscous flux Calculation using Riemann flux

[0078]

[0079] Where n represents the external normal vector of the unit surface, λ max is the maximum eigenvalue, which can be calculated using the following formula:

[0080] λ max ={‖u·n‖+c}

[0081] u and c represent the local fluid velocity and sound speed, respectively, and ‖·‖ indicates the absolute value.

[0082] Step 6: According to step 5, we can get the ordinary differential equation about time

[0083]

[0084] in, is the time derivative of the variable. R(U) is the spatial discretization operator in step 6 and can be calculated using the following formula:

[0085]

[0086] Step 7: Use the three-step, third-order Runge-Kutta method to advance the time. n Solve for U at the next moment n+1

[0087] U (1) =U n +ΔtR(U n )

[0088]

[0089]

[0090] Δt is the time step length, U (1) ,U (2) Temporary variables for intermediate steps.

[0091] The isentropic vortex problem has an exact solution. The accuracy of this method is verified by using the isentropic vortex problem. An isentropic vortex is added to the initial mean flow {ρ,u,v,p}={1,1,1,1}. The size of the vortex is given by the following formula:

[0092]

[0093] Here r 2 =(x-5) 2 +(y-5) 2 , ε=5.0; the calculation area is [0,10]×[0,10], the isentropic vortex problem has an exact solution, the boundary conditions are given according to the exact solution, the calculation area is divided into unstructured grids, the monomial basis function is constructed, and then the third-order step 7 Runge-Kutta method is used to advance to 2 seconds in time. Figure 2 The density cloud diagram at the initial moment and calculated to 2 seconds is used to verify the accuracy of the format for two-dimensional problems.

[0094] In order to verify the adaptability of this method to the grid, the calculation grid uses two sets of grids: regular triangles and irregular triangles. The grid units are as follows: Figure 3 The accuracy analysis results are shown in the following table:

[0095]

[0096]

[0097] It can be seen that the position of the vortex and the exact solution are calculated by precise error comparison, achieving the ideal second-order and third-order accuracy.

[0098] It is to be understood that the present invention is described by some embodiments, and it is known to those skilled in the art that various changes or equivalent substitutions may be made to these features and embodiments without departing from the spirit and scope of the present invention. In addition, under the teachings of the present invention, these features and embodiments may be modified to adapt to specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the scope of protection of the present invention.

Claims

1. A method for constructing discontinuous finite element monomial basis functions applicable to arbitrary grids, characterized in that: The following steps are involved: Step 1: Divide the computational domain of the problem to be solved using unstructured grid elements; Step 2: Define the local coordinates (ξ,η) of the unstructured grid cell: Among them, (x, y) is the global coordinate of the grid, (x c ,y c ) is the unit center defined by node average, h is the average distance from the node to the center; ξ represents the local coordinate of the x component, and η represents the local coordinate of the y component; Step 3: According to the local coordinates, define the polynomial space composed of monomials, expressed as the basis function φ: φ={1,ξ,η,ξ 2 ,xii,h 2 } Where {·} represents the polynomial space composed of basis functions; Step 4: Calculate the inner product m of the basis function φ ij : m ij =<φ i f j >=∫ e f i f j dΩ Among them, φ i ,φ j represents the i-th and j-th basis functions in the polynomial space, <> represents the inner product, and dΩ represents the volume differential element; Renormalize the basis functions of the polynomial space: To meet Get the renormalized basis function space That is, we get the monomial basis function: in, represents the i-th basis function; The monomial basis function Applied to the Euler equation for compressible fluid, the specific steps are as follows: Formulate the compressible Euler equations: in, denote the convection terms of the conservation variables of the compressible fluid, represents the time partial derivative, represents the spatial gradient operator, U is the conserved variable of the compressible fluid; multiply the monomial basis function on both sides of the compressible Euler equation As a test function, we have Among them, dV represents the volume differential element, dS represents the surface differential element, represents the inviscid flux, represents the convection term; Viscous flux Calculation using Riemann flux Where n represents the external normal vector of the unit surface, λ max is the maximum eigenvalue, which can be calculated using the following formula: λ max ={‖u·n‖+c} u and c represent the local fluid velocity and sound speed, respectively, and ‖·‖ indicates the absolute value; Then we get the ordinary differential equation for time in, is the time derivative of the variable; R(U) is the spatial discrete operator and can be calculated using the following formula: Finally, the three-step three-order Runge-Kutta method is used to advance the time. n Solve for U at the next moment n+1 IN (1) =U n +ΔtR(U n ) Δt is the time step length, U (1) ,U (2) Temporary variables for intermediate steps.

2. The method for constructing a discontinuous finite element monomial basis function applicable to any grid according to claim 1, characterized in that: The unstructured grid units include three-dimensional units: tetrahedron, hexahedron, triangular prism, pyramid; two-dimensional units: triangle, quadrilateral; one-dimensional units: line segment.

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