WO2019137236A1 - Method and system for constructing non-coordinated interpolation function for optimizing calculation precision of finite element software and storage medium - Google Patents

Abstract

A method and system for constructing a non-coordinated interpolation function for optimizing calculation precision of finite element software and a storage medium, used to improve the calculation precision of finite element software. The method comprises: constructing a non-coordinated interpolation function using a single linear transformation coordinate system (S1); determining the number of coordinate elements, the number of terms, and the degree of the equation of the non-coordinated interpolation function according to features of a target unit; at the same time, the equation of the constructed non-coordinated interpolation function comprises a first partial polynomial having a complete degree and a second partial polynomial having an incomplete degree; the first partial polynomial increases from low to high after covering each coordinate element combination; the second partial polynomial is symmetrically distributed, the total degree of each term is greater than or equal to the highest degree of the first partial polynomial, and the degree of each coordinate element does not exceed the highest degree of the first partial polynomial (S2).

Description

Non-coordinating interpolation function construction method, system and storage medium for optimizing finite element software calculation precision Technical field

The invention relates to the field of simulation technology, in particular to a method, a system and a storage medium for constructing a non-coordinating interpolation function for optimizing the calculation precision of a finite element software.

Background technique

Currently, finite element is an indispensable part of engineering analysis and design. Finite element calculation software is now widely used in various fields of structural, solid and fluid analysis engineering. In fact, finite elements are used in almost every field of engineering analysis.

After the mathematical model of engineering or physical problems (basic variables, basic equations, solution domain and boundary conditions, etc.) is determined, the finite element method as a numerical method for analyzing it can be summarized into the following three parts:

(1) A solution domain representing a structure or a continuum is discretized into a plurality of sub-domains (units), and interconnected by nodes on their boundaries to form a combination. This part is the pre-processing part of the finite element software, that is, the unit division part, and the technology of this part is very mature.

(2) The approximate field function assumed in each unit is used to slice the unknown field variables to be solved in the full solution domain. The approximation function in each cell is expressed by the value of the unknown field function and its derivative at each node of the cell and its corresponding interpolation function. This part is the finite element interpolation function forming part in the finite element software. The finite element interpolation function is difficult to construct. There are many problems that can not construct the finite element interpolation function that satisfies the basic convergence requirements. It has always been a difficult problem in the field of finite element research.

The construction result of the finite element interpolation function of the same problem is not unique. The selection of the finite element interpolation function has great influence on the calculation and analysis precision of the finite element software, which is directly related to the success or failure of the finite element software calculation result. There are three key conditions for constructing high-precision finite element interpolation functions: First, the higher (complete) order (number of times) of the polynomial used in the finite element interpolation function is higher, and the higher the accuracy is. Second, the finite element interpolation function is in the common boundary of adjacent units. The displacement (the derivative with displacement), that is, the displacement from the adjacent unit to the same common boundary should be consistent; otherwise, the displacement conflict causes energy loss and reduces the calculation accuracy; the third is that the finite element interpolation function should be suitable for the surface (curve) boundary. At present, it is impossible to construct a finite element interpolation function that can satisfy the above conditions at the same time. It is very difficult to construct a high-order complete finite element interpolation function.

The construction of the existing interpolation function generally adopts the isoparametric coordinate method. Whether it is for a planar solid element, a three-dimensional solid element, a flat thin plate unit or a space curved thin shell element, the calculation accuracy is low, the scope of application is limited, and coordination cannot be achieved. / or not suitable for problems such as curve boundaries. E.g:

1) A 4-node quadrilateral unit that has been constructed based on the isoparametric coordinate method. The unit has only one complete coordination, which can only meet the basic convergence requirements of finite element calculation, and the calculation accuracy is low.

2) An 8-node curved quadrilateral unit constructed based on the isoparametric coordinate method. The number of nodes in the unit is doubled, but only one-order complete coordination is available, which can only meet the basic convergence requirements of finite element calculation, and the calculation accuracy is low. When the unit is rectangular, the unit can be fully coordinated twice, but it is not suitable for the curve boundary, and the scope of use is very limited.

3) A 12-node curved quadrilateral unit constructed based on the isoparametric coordinate method. The unit interpolation function has only 2 complete coordinations and the calculation accuracy is low. When the unit is rectangular, the unit interpolation function can be fully coordinated three times, but it is not suitable for the curve boundary, and the scope of use is very limited.

4) An 8-node arbitrary hexahedral element constructed based on the isoparametric coordinate method. When the unit is an arbitrary hexahedral element, it is suitable for the line-shaped boundary, but the element interpolation function has only 1 order completeness, which can only satisfy the basics of finite element calculation. Convergence requirements, low calculation accuracy. When the unit is a rectangular parallelepiped, the unit interpolation function can be fully coordinated twice, but it is not suitable for the line boundary, and the scope of use is very limited.

5) The 20-node surface hexahedral element constructed based on the isoparametric coordinate method. Regardless of whether the unit is an arbitrary hexahedral element or a rectangular parallelepiped, the finite element interpolation function has only two complete coordinations, and the calculation accuracy is limited.

6) A 32-node surface hexahedral element constructed based on the isoparametric coordinate method. Regardless of whether the unit is an arbitrary hexahedral element or a rectangular parallelepiped, the finite element interpolation function has only two complete coordinations, and the calculation accuracy is low.

7), only four nodes constructed based on the isoparametric coordinate method (w related three-node parameters w, θ _x , θ _y and four-node parameters w, θ _x , θ _y , θ _xy ) three complete rectangular thin plate element displacement non- Coordinating the interpolation function, although the complete order of the unit displacement non-coordinating interpolation function is higher, the unit boundary normal angular displacement is not coordinated, and is not suitable for any polyline boundary, and the applicable range is very limited.

8) Based on the isoparametric coordinate method, it is not possible to construct a quadratic complete 4-node arbitrary quadrilateral thin-plate unit, and it is impossible to solve the unit coordination problem.

9) Based on the isoparametric coordinate method, the planar 4-node non-coordinating rectangle and the triangular thin plate element displacement non-coordinating interpolation function can only be used for the space thin shell structure by the coordinate transformation method. The applicable range is very limited and uncoordinated.

10), based on the isoparametric coordinate method, the three-dimensional 8-node coordinated low-order complete quadrilateral hyperparameter surface element displacement non-coordinating interpolation function and three-dimensional 8-node low-order complete coordination surface quadrilateral relative degree of freedom shell element displacement non-coordinating interpolation function. The two unit displacement interpolation functions are coordinated, and the thick and thin shell structure is common, but only has 1 order completeness, and the calculation precision is low. When the thickness of the shell tends to a thin curved shell, there are shear "locking" and film "locking", etc. problem.

In summary, the current structural finite element software is based on a single isoparametric coordinate method (or area coordinate method) construction unit to solve the physical quantity (displacement, temperature, fluid and electromagnetic, etc.) interpolation function, the constructed unit interpolation function can not be high The order is complete and coordinated, even if it is complete, it is only low-order complete, and the calculation accuracy is low. For the structural problem, a high-order complete and coordinated finite element interpolation function that satisfies the basic convergence requirements of finite elements has not been constructed.

On the other hand, the high-order completeness of the unit is more important than its coordination. In some cases, high-order complete and coordinated finite element interpolation functions cannot be constructed. Some finite element coordinated interpolation functions appear to be too rigid and the convergence effect is not good. In fact, as long as the unit can guarantee the coordination and convergence when the unit is subdivided, it is also necessary to construct a high-order complete non-coordinating finite element interpolation function to provide a comparison and verification method for the finite element analysis results.

Summary of the invention

The object of the invention is to disclose a non-coordinating interpolation function construction method, system and storage medium for optimizing the calculation precision of the finite element software, so as to improve the calculation precision of the finite element software.

To achieve the above object, the present invention discloses a non-coordinating interpolation function construction method for optimizing the calculation precision of a finite element software, including:

Constructing a non-coordinating interpolation function in a single linear transformation coordinate system;

Determining the number of coordinate elements, the number of items, and the number of times of the non-coordinating interpolation function equation according to the characteristics of the target unit; meanwhile, the constructed non-coordinating interpolation function equation includes a first partial polynomial with complete completeness and a second partial polynomial with less complete number; The first part of the polynomial increases from low to high after covering each coordinate element combination; the second partial polynomial is symmetrically distributed, and the total number of times is greater than the highest number of times of the first partial polynomial and the number of each coordinate element does not exceed The first part of the polynomial is the highest number of times.

Corresponding to the above method, the present invention also discloses a non-coordinating interpolation function construction system for optimizing the calculation precision of the finite element software, comprising a memory, a processor, and a computer program stored on the memory and operable on the processor, The steps of the above method are implemented when the processor executes the computer program.

Corresponding to the above method, the present invention also discloses a computer readable storage medium having stored thereon a computer program, the program being implemented by a processor to implement the steps of the above method.

Based on the non-coordinating interpolation function constructed by the present invention, the finite element is solved by the variational principle or the weighted residual method which is equivalent to the original problem mathematical model (basic equation, boundary condition), and the base unknown value (the node value of the field function is established). The algebraic equations or the ordinary differential equations can solve the equations to solve the problem. This part is assembled and solved for algebraic equations or ordinary differential equations. The technology of this part is also very mature, and there are standard fixed solution modules. Thereby, the present invention has the following beneficial effects:

Based on a single linear transformation coordinate system to construct a non-coordinating interpolation function for solving physical quantities, the first part of the polynomial has high-order completeness, which can greatly improve the calculation accuracy of the finite element analysis software, improve the safety and reliability of the structural design, and optimize the structural design. It is more adaptable to various curved (curve) boundaries, which brings huge economic benefits for engineering, aviation and aerospace construction.

On the other hand, the present invention achieves an unexpected technical effect by constructing a high-order complete non-coordinated finite element interpolation function by using a single linear coordinate transformation. When establishing a unit stiffness matrix and solving a system of displacement equations, it is not necessary to solve the whole solution. Coordinates of the isoparametric local coordinate (ξ, η) transformed Jacobian matrix [J] inverse matrix, when the element is distorted, the inverse matrix of the Jacobian matrix [J] may tend to infinity and cause computational errors, while the unit comparable matrix The inverse matrix of [J] is a non-matrix matrix in the unit. It is not possible to avoid this problem by controlling the shape of the element. However, using a single linear coordinate transformation to construct a high-order complete non-coordinated finite element interpolation function has no such problem, so a single The high-order complete non-coordinating finite element interpolation function constructed by linear coordinate transformation has better anti-distortion than the high-order complete coordination finite element interpolation function constructed by mixed linear coordinate transformation.

The invention will now be described in further detail with reference to the accompanying drawings.

DRAWINGS

The accompanying drawings, which are incorporated in the claims In the drawing:

1 is a flow chart of a method for constructing a finite element non-coordinating interpolation function disclosed in a preferred embodiment of the present invention;

2(a) is a schematic diagram of a global coordinate system in a planar linear transformation coordinate system; and FIG. 2(b) is a schematic diagram of linear transformation coordinates in a planar linear transformation coordinate system;

3(a) is a schematic diagram of a global coordinate system in a spatial linear transformation coordinate system; FIG. 3(b) is a schematic diagram of linear transformation coordinates in a spatial linear transformation coordinate system;

Figure 4 is a schematic diagram of an 8-node curved quadrilateral unit;

Figure 5 is a schematic diagram of a 12-node curved quadrilateral unit;

Figure 6 is a schematic diagram of a 20-node curved quadrilateral unit;

Figure 7 is a schematic diagram of a 32-node curved quadrilateral unit;

Figure 8 is a schematic view of a 4-node curved quadrilateral thin plate unit;

Figure 9 is a schematic view of an 8-node curved quadrilateral thin plate unit;

Figure 10 is a schematic view of a space 4-node quadrilateral flat shell unit;

Figure 11 is a schematic diagram of a spatial 8-node quadrilateral flat shell unit;

Figure 12 is a schematic view of a space 4-node quadrilateral curved thin shell unit;

Figure 13 is a schematic diagram of a spatial 8-node quadrilateral curved thin shell unit.

Detailed ways

The embodiments of the present invention are described in detail below with reference to the accompanying drawings.

Example 1

As shown in FIG. 1, the finite element non-coordinating interpolation function construction method disclosed in this embodiment includes:

Step S1 constructs a non-coordinating interpolation function with a single linear transformation coordinate system.

The linear coordinate transformation system, that is, the transformation relationship of the two orthogonal coordinate systems is linear. The orthogonal surface coordinate transformation system existing on the curved thin shell element is equivalent to the global coordinate system of the general structural unit, and can also be transformed into a linear transformation coordinate system.

The contribution of the linear transformation coordinate transformation system: First, the unit can be changed into a rectangular (line) shape unit, so that the coordinate values of some unit nodes become simple 0 and 1, thereby reducing the difficulty of constructing the finite element interpolation function and improving the calculation. The purpose of accuracy is to make the complete order of the polynomial not increase by coordinate transformation. The coordinate system of the finite element interpolation function polynomial is several complete polynomials. There are several complete polynomials in the whole coordinate system, which makes the structure high. A well-ordered finite element non-coordinating interpolation function is possible.

The linear transformation coordinate system is divided into a plane linear transformation coordinate system and a spatial linear transformation coordinate system.

(a) In the case of two dimensions, the transformation formula of the global coordinate system and the linear coordinate system transformation is:

The shape of the unit after coordinate transformation is shown in Fig. 2(a) and Fig. 2(b). One corner of the quadrilateral element is at the origin of the coordinate, and the two corner points are located on the coordinate axis. There are six undetermined coefficients A _i , B _i , C _i , (i=1, 2) in the coordinate transformation relationship, and six linear transformation coordinate values of 0 or 1 can be transformed.

(b) In the case of three dimensions, the transformation formula of the global coordinate system and the linear coordinate system transformation is:

The shape of the unit after the coordinate change is shown in Fig. 3(a) and Fig. 3(b). One corner of the hexahedral element is at the origin of the coordinate, and the three corner points are located on the coordinate axis. The coordinate change relationship has 12 undetermined coefficients A _i , B _i , C _i , D _i , (i=1, 2, 3), and 12 linear transformation coordinate values of 0 or 1 can be transformed.

Similarly, the transformation formula of the two-dimensional orthogonal curvilinear coordinate system with respect to the subsequent unit surface is transformed with the linear coordinate system:

There are six undetermined coefficients A _i , B _i , C _i , (i=1, 2) in the coordinate transformation relationship determined by a system of equations that transforms six linear transformation coordinate values of 0 or 1.

Step S2: determining a coordinate element number, a number of items, and a number of times of the non-coordinating interpolation function equation according to characteristics of the target unit; meanwhile, the constructed non-coordinating interpolation function equation includes a high-order complete first partial polynomial and a high-order incomplete second part a polynomial; the first partial polynomial increases from low to high after the combination of the coordinate elements; the second partial polynomial is symmetrically distributed, and the total number of times is greater than the highest number of times of the first partial polynomial and each coordinate element The number of times does not exceed the maximum number of times of the first partial polynomial.

In this step, for the two-dimensional linear transformation coordinate system, the first partial polynomial is superimposed from low to high and covers a Pascal triangle distribution after covering each coordinate element combination; the second partial polynomial is selected from the first partial Pascal triangle. Mirrored in an inverted triangle.

Thereby, the present embodiment constructs a finite element non-coordinating interpolation function based on a single linear transformation coordinate system. In the selection of finite element interpolation polynomials, all items of the finite element interpolation polynomial use linear transformation coordinates. As long as the finite element interpolation polynomial does not appear incomplete terms, and the element boundary is a straight line or a plane, and the middle node is divided into boundaries, the finite element non-coordinating interpolation function naturally tends to coordinate on the element boundary, thereby solving the structure very ingeniously and simply. Order-complete finite element interpolation functions lead to inconsistencies.

The following is an analysis of the constructor for each target unit as follows:

1) As shown in Fig. 4, when the target unit is a two-dimensional 8-node high-order complete quadrilateral curved side unit, the constructed unit displacement non-coordinating interpolation function is:

In the following equations, T ₁ , T ₂ , and T ₃ are the coordinates in the unit linear transformation coordinate system; u, v, and w respectively correspond to the displacements in the three coordinate directions in the unit, and θ _x and θ _y are respectively w pairs. The partial derivatives of the coordinates x and y in the unit; a _i , b _i , c _i , i=1, 2, 3, ... are the undetermined coefficients of the unit displacement non-coordinating interpolation function, which will not be described later.

In the present embodiment, corresponding to the above step S2, when determining the number of items of the non-coordinating interpolation function equation according to the characteristics of the target unit, corresponding to u, v (or u', v in each of the following 1 to 10 cases) The total number of interpolation-related node displacements of ') is equal to the number of known nodes, and the total number of interpolation-related node displacements corresponding to w (or w') is the product of the number of known nodes and the associated displacement component.

The unit displacement non-coordinating interpolation function can establish a system of equations according to the displacement of the unit nodes, and the simultaneous solution determines the undetermined coefficients a ₁ ～ a ₈ and b ₁ ～ b ₈ , which have 2 order completeness, and the 8-node quadrilateral unit can only have 2 The order completeness is 1 order higher than the completeness of the traditional unit displacement interpolation function. The completeness of the unit displacement non-coordinating interpolation function is improved once, and its convergence performance and anti-distortion performance will be greatly improved.

When the common boundary of two adjacent elements is a straight line and the nodes in the boundary are evenly divided, as long as the unit interpolation polynomial does not have incomplete terms, the common boundary is coordinated, and is independent of other boundary shapes of the unit. Therefore, when dividing the unit, Keep the common boundary of two adjacent units as a straight line and the nodes in the boundary are divided into boundaries, and the free outer boundary of the unit has no coordination requirement, which can be a curve. As long as the unit interpolation polynomial does not appear incomplete, the two-dimensional 8-node unit at this time The displacement non-coordinating interpolation function has high-order complete coordination and is suitable for the curve boundary, and does not increase the difficulty of unit division. This two-dimensional 8-node quadrilateral curved edge unit can degenerate a 6-node triangular curved edge unit.

2) As shown in Fig. 5, when the target unit is a two-dimensional 12-node high-order complete coordinated quadrilateral curved side unit, the constructed unit displacement non-coordinating interpolation function is:

The unit displacement non-coordinating interpolation function can establish a system of equations according to the displacement of the unit nodes, and the simultaneous solution determines the undetermined coefficients a ₁ ～ a ₁₂ and b ₁ ～ b ₁₂ , which have 3 order completeness, which is more complete than the traditional unit displacement interpolation function. High 2nd order, its convergence performance and anti-distortion performance is very good. When the common boundary of two adjacent elements is a straight line and the nodes in the boundary are evenly divided, as long as the unit interpolation polynomial does not have incomplete terms, the common boundary is coordinated, and is independent of other boundary shapes of the unit. Therefore, when dividing the unit, Keep the common boundary of two adjacent units as a straight line and the nodes in the boundary are divided into boundaries, and the free boundary of the unit has no coordination requirement, which can be a curve. As long as the unit interpolation polynomial does not appear incomplete, the two-dimensional 12-node unit displacement at this time The non-coordinating interpolation function has high-order complete coordination and can be adapted to arbitrary curve boundaries without the difficulty of adding units. This two-dimensional 12-node quadrilateral curved edge unit can degenerate a 9-node triangular curved edge unit.

3) As shown in Fig. 6, when the target unit is a three-dimensional 20-node high-order complete coordination surface hexahedral element, the constructed unit displacement non-coordinating interpolation function is:

The unit displacement non-coordinating interpolation function can establish a system of equations according to the displacement of the unit nodes, and the simultaneous solution determines the undetermined coefficients a ₁ ～ a ₂₀ and b ₁ ～ b ₂₀ , which have 2 order completeness, which is more complete than the traditional unit displacement interpolation function. High 1st order. The curved hexahedral element can be automatically degenerated into a curved pentahedron unit and a curved side tetrahedral unit; the six quadrilateral side surfaces can be automatically degenerated into a triangular side surface.

The coordination problem of the unit displacement non-coordinating interpolation function at the cell boundary is divided into three cases:

(a) Common angular line coordination of hexahedrons

Similar to the plane problem, the common angular line of the unit is required to be a straight line and the middle node is divided into its common angular line. As long as the unit interpolation polynomial does not have incomplete terms, the unit displacement non-coordinating interpolation function is coordinated on the common angular line, and other parts of the unit. The shape is irrelevant.

(b) Hexahedral quadrilateral lateral surface coordination

It is required that the quadrilateral side surface common to the unit is a plane, the four sides are straight lines, and the nodes in the edge are divided into their boundaries. As long as the unit interpolation polynomial does not have incomplete terms, the unit displacement non-coordinating interpolation function is on the common quadrilateral side plane. That is, coordination, regardless of the shape of other faces of the unit.

(c) Triangular triangular side surface coordination

Only the triangular side surface common to the unit is required to be a plane, and the edge of the triangle can be a curve. As long as the unit interpolation polynomial does not have incomplete terms, the unit displacement non-coordinating interpolation function is coordinated on the common triangle side plane, regardless of the shape of other planes of the unit. . The sides of the triangle can be curved conditions, making it possible to construct degenerate tetrahedral elements suitable for any curved boundary.

In the unit division, the plane inside the structure uses plane hexahedral elements, and the degenerate tetrahedral unit, pentahedral element and hexahedral element exposed surface can be used to simulate the outer surface boundary of the structure. As long as the unit interpolation polynomial does not appear incomplete, it can be guaranteed. The coordination of the unit displacement non-coordinating interpolation function can also simulate the surface boundary of the structure well. Consistent with the conventional unit division method, it does not increase the difficulty of unit division.

4) As shown in FIG. 7, when the target unit is a three-dimensional 32-node high-order complete coordination surface hexahedral element, the constructed unit displacement non-coordinating interpolation function is:

The unit displacement non-coordinating interpolation function can establish a system of equations according to the displacement of the unit nodes, and the simultaneous solution determines the undetermined coefficients a ₁ ～ a ₃₂ and b ₁ ～ b ₃₂ , which have 3 order completeness, which is more complete than the traditional unit displacement interpolation function. High 2nd order. The curved hexahedral element can also be automatically degenerated into a curved pentahedral unit and a curved side tetrahedral unit; the six quadrilateral side surfaces can be automatically degenerated into triangular side surfaces to fit the curved structural boundary.

The coordination problem of the unit displacement non-coordinating interpolation function at the cell boundary is divided into three cases:

(a) Common angular line coordination of hexahedrons

Similar to the plane problem, the common angular line of the unit is required to be a straight line and the middle node is divided into its common angular line. As long as the unit interpolation polynomial does not have incomplete terms, the unit displacement non-coordinating interpolation function is coordinated on its common angular line, and other units The shape of the part is irrelevant.

(b) Hexahedral quadrilateral lateral surface coordination

It is required that the common quadrilateral side surface of the unit is a plane, the four sides are straight lines, and the nodes in the side are divided into their sides. As long as the unit interpolation polynomial does not have incomplete terms, the unit displacement non-coordinating interpolation function is on the common quadrilateral side plane. Coordinated, regardless of the shape of the other faces of the unit.

(c) Triangular triangular side surface coordination

Only the triangular side surface common to the unit is required to be a plane, and the edge of the triangle can be a curve. As long as the unit interpolation polynomial does not have incomplete terms, the unit displacement non-coordinating interpolation function is coordinated on the common triangle side plane, regardless of the shape of other planes of the unit. . The sides of the triangle can be curved conditions, making it possible to construct degenerate tetrahedral elements suitable for any curved boundary.

In the unit division, the plane inside the structure uses plane hexahedral elements, and the degenerate tetrahedral unit, pentahedral element and hexahedral element exposed surface can be used to simulate the outer surface boundary of the structure. As long as the unit interpolation polynomial does not appear incomplete, it can be guaranteed. The coordination of the unit displacement non-coordinating interpolation function can also simulate the surface boundary of the structure well. Consistent with the conventional unit division method, it does not increase the difficulty of unit division.

5), as shown in Figure 8, when the target unit is a two-dimensional 4-node and each node has three correlation displacement components (three-node parameters are w, θ _x , θ _y , w, θ _x , θ _y related, this case When the high-order complete arbitrary quadrilateral thin plate element with the total number of displacements is the product of the number of known nodes and the number of node displacement components, the constructed unit displacement non-coordinating interpolation function is:

w=c ₁ +c ₂ T ₁ +c ₃ T ₂ +c ₄ T ₁ ² +c ₅ T ₁ T ₂ +c ₆ T ₂ ² +c ₇ T ₁ ³ +c ₈ T ₁ ² T ₂ +c ₉ T ₁ T ₂ ² +c ₁₀ T ₂ ³ +c ₁₁ T ₁ ³ T ₂ +c ₁₂ T ₁ T ₂ ³ .

The unit displacement non-coordinating interpolation function can establish a system of equations according to the displacement of the unit nodes, and the simultaneous solution determines the undetermined coefficients a ₁ to a ₁₂ , which have 3 order completeness, and the complete order is higher. The unit displacement non-coordinating interpolation function is The deflection and tangential corners on the common boundary of adjacent units can be coordinated. However, the conventional quadrilateral thin plate unit has only a rectangular unit and a triangular unit, and is not suitable for any polygonal line boundary, and the application range is very limited.

The unit displacement non-coordinating interpolation function coordinates the deflection and tangential rotation angle on the element boundary, but the normal rotation angle is not coordinated, that is, the C ¹ order non-coordination problem. This problem has been considered as an unsolvable problem. As long as the unit interpolation polynomial only appears as a quadratic complete polynomial, at this time, the normal corner of the element boundary is coordinated.

6) As shown in Fig. 9, when the target unit is a two-dimensional 8-node and each node has three high-order complete curved quadrilateral thin plate elements with relevant displacement components, the constructed unit displacement non-coordinating interpolation function is:

The unit displacement non-coordinating interpolation function can establish a system of equations according to the displacement of the unit nodes, and the simultaneous solution determines the undetermined coefficients a ₁ to a ₂₄ , which have 5 order completeness, and the complete order is high, which is suitable for any curved boundary.

When the common boundary of two adjacent elements is a straight line and the nodes in the boundary are equally divided, the unit displacement non-coordinating interpolation function coordinates the deflection and the tangential rotation angle on the element boundary. As long as the cell interpolation polynomial only appears three complete polynomials, the normal corners are also coordinated, so that the C ¹ order non-coordination problem is better solved.

For the thin shells commonly used in engineering, the appropriate orthogonal curve coordinates and corresponding geometric equations are used to construct high-order complete curved thin shell elements directly in the spatial orthogonal curvilinear coordinate system according to the principle of the above-mentioned unit, like the plane problem. , calculate the unit stiffness matrix, and then perform space coordinate conversion. It mainly includes the following four cases of 7-10. In the interpolation function equation constructed in the following four cases, Z ₁ and Z ₂ are the coordinates in the linear transformation coordinate system of the two-dimensional orthogonal curve or the linear coordinate in the curved thin shell element respectively; In the shell, the two-dimensional orthogonal curve coordinates in the curved thin shell element become two-dimensional orthogonal linear coordinates; u', v' respectively correspond to the two-dimensional orthogonal curve in the unit curved surface or the displacement in the direction of the linear coordinate, w' corresponds The displacement in the normal direction of the unit surface, θ _x' and θ _y' are the partial derivatives of the two-dimensional orthogonal curve or the linear coordinate x', y' in the unit curved surface; a _i , b _i , c _i , respectively i=1, 2, 3, ... is the undetermined coefficient of the unit displacement non-coordinating interpolation function, which will not be described later.

7) When the target unit is a three-dimensional 4-node and each node has three high-order complete arbitrary quadrilateral flat shell elements with relevant displacement components, the relevant displacement components are respectively w′, θ _x′ , θ _y′ , ie for 4 Any quadrilateral thin shell element of the node plate, as shown in Fig. 10, can transform the displacement non-coordinating interpolation function of any quadrilateral flat sheet element into a non-coordinating interpolation function of the space thin shell element by coordinate transformation, wherein the node displacement vector is in the two coordinate system The conversion relationship is:

a' _i =La;a _i =L ^T a' _i a _i =[u _i v _i w _i θ _xi θ _yi θ _zi ];a' _i =[u' _i v' _i w' _i θ' _xi θ ' _yi θ' _zi ]

Where a' _i , a _i is the node displacement; T, L, λ are the transformation matrices.

The conversion relationship between the element stiffness matrix and the load column vector is:

K' ^e =TK ^e T;Q' ^e =TQ

Other conversion relationships are implemented in a conventional manner.

The key is the construction of the displacement non-coordinating interpolation function of the arbitrary quadrilateral flat shell element. Based on the single linear transformation coordinate method of the present invention, the unit displacement non-coordinating interpolation function can be assumed as:

The interpolation function can establish a system of equations according to the displacement of the unit nodes, and determine the undetermined coefficients by simultaneous solution; have 3 order completeness, complete order is high, and the thin-shell element displacement non-coordinating interpolation function is on the common boundary of adjacent units The deflection and tangential corners can be coordinated. The unit displacement non-coordinating interpolation function coordinates the deflection and tangential rotation angle on the element boundary, but the normal rotation angle is not coordinated, that is, the C ¹ order non-coordination problem. This problem has been considered as an unsolvable problem.

As long as the unit interpolation polynomial only appears as a quadratic complete polynomial, at this time, the normal corner of the element boundary is coordinated, so that the C ¹ order non-coordination problem is better solved.

8) When the target unit is a three-dimensional 8-node and each node has three high-order complete curved quadrilateral flat shell elements with relevant displacement components, the relevant displacement components are respectively w′, θ _x′ , θ _y′ , ie for 8 Arbitrary quadrilateral thin shell element of the node plane, as shown in Fig. 11, by coordinate transformation, the displacement non-coordinating interpolation function of any quadrilateral flat sheet element can be converted into a spatial thin shell element displacement non-coordinating interpolation function. The node displacement vector is transformed between two coordinate systems:

a' _i =La _i ;a _i =L ^T a' _i

a _i =[u _i v _i w _i θ _xi θ _yi θ _zi ]; a' _i =[u' _i v' _i w' _i θ' _xi θ' _yi θ' _zi ]

Where a' _i , a _i is the node displacement; T, L, λ are the transformation matrices.

The conversion relationship between the element stiffness matrix and the load column vector is:

K' ^e =TK ^e T;Q' ^e =TQ

Other conversion relationships are implemented in a conventional manner.

The key is the construction of the displacement non-coordinating interpolation function of the arbitrary quadrilateral flat shell element. Based on the single linear transformation coordinate method of the present invention, the unit displacement non-coordinating interpolation function can be assumed as:

The above interpolation function can establish a system of equations according to the displacement of the unit nodes, and determine the undetermined coefficients by a simultaneous solution; it has a 5th order completeness, and the complete order is high, which is suitable for the curved boundary.

When the common boundary of two adjacent elements is a straight line and the nodes in the boundary are equally divided, the unit displacement non-coordinating interpolation function coordinates the deflection and the tangential rotation angle on the element boundary. As long as the cell interpolation polynomial only appears three complete polynomials, the normal corners are also coordinated, so that the C ¹ order non-coordination problem is better solved.

9) When the target unit is a three-dimensional 4-node and each node has three high-order complete quadrilateral curved thin-shell elements with relevant displacement components, the relevant displacement components are w', θ _x' , and θ _y' , respectively. It can be assumed that the overall coordinates of any point inside the curved thin shell element are:

Where N _i (ξ, η) is a conventional shape function.

The rigid body displacement {δ' _Ri } of the node in the global coordinate system is given by the rigid body motion of the unit microbody, and the motion of the microcentre centroid includes rotation about three coordinate axes and translation along the coordinate axis. The rigid body motion of the curved shell element is:

{V' _R }={u' ₀ v' ₀ w' ₀ θ' _x0 θ' _y0 θ' _z0 } ^T

In the global coordinate system, the rigid body displacement of the rigid body motion obtained by the dynamic method is:

{δ' _Ri }={u' _Ri v' _Ri w' _Ri θ' _xRi θ' _yRi θ' _zRi } ^T =[T _Ri ]{V' _R }

{δ' _R ^e }={δ' _R1 δ' _R2 δ' _R3 δ' _R4 } ^T =[T _R ]{V' _R }

Wherein, the submatrix of the transformation matrix T _R :

[T _R ]={T _R1 T _R2 T _R3 T _R4 } ^T

Where x ₀ , y ₀ , z ₀ are the global coordinates of the centroid of the unit micro-body.

In the curvilinear coordinate system, the rigid body displacement of the joint generated by the rigid body motion is:

{δ _Ri }={u _Ri v _Ri w _Ri θ _xRi θ _yRi θ _zRi } ^T =[L _i ] ^T [T _Ri ][L ₀ ] ^T {V _R }

Where L _i is a transformation matrix.

In the formula, λ _x'αi =cos(x', α) and the like are cosines in the respective directions of the x, y, and z axes in the orthogonal main curve coordinate system α, β, δ. In the curvilinear coordinate system, the rigid body motion of the centroid of the curved thin shell element is:

{V _R }={u ₀ v ₀ w ₀ θ _x0 θ _y0 θ _z0 } ^T

The unit node displacement vector converts the relationship between the global coordinates and the curve coordinates as:

In the curvilinear coordinate system, the total displacement field of the unit after the displacement of the rigid body is:

Where I is a unit matrix of 20 x 20.

The rigid body displacement does not produce strain, so the strain matrix in the orthogonal main curve coordinate system is:

Where B is the strain matrix of the elements in the orthogonal main curve coordinate system.

The rigid body displacement does not generate nodal forces, for static condensation, and the finite element equation is established according to the principle of virtual work:

among them,

Is the node load of the element in the orthogonal main curve coordinate system,

Is the original stiffness matrix of the element in the orthogonal main curve coordinate system

Obtained by displacement, ie

among them,

The conversion relationship between the stiffness matrix and the load vector in the global coordinate system is:

By integrating the element stiffness matrix and the load vector of the overall coordinate system, the node displacement solution equation of the global coordinate system can be obtained.

In the orthogonal main curve coordinate system (α, β, δ), α, β, δ are the arc lengths of the curve coordinates. For the 4-node arbitrary quadrilateral elements, as shown in Fig. 11, based on the original single linear transformation coordinate method, Similar to the planar thin plate problem, linear coordinate transformation of the arc length coordinates of the curve can be assumed that the unit displacement non-coordinating interpolation function is:

The equations are established according to the displacement of the unit nodes, and the simultaneous solution determines the undetermined coefficients. The unit displacement non-coordinating interpolation function has 3 order completeness, and the complete order is higher. The unit displacement non-coordinating interpolation function can coordinate the deflection and tangential rotation angle on the common boundary of adjacent units. As long as the unit non-coordinating interpolation polynomial only appears as a quadratic complete polynomial, the normal corners of the element boundary are coordinated, so that the C ¹ order non-coordination problem is better solved.

10), when the target unit is a three-dimensional 8-node and each node has three high-order complete quadrilateral curved thin shell elements with relevant displacement components, the relevant displacement components are respectively w′, θ _x′ , θ _y′ , ie The 8-node curved thin shell element, as shown in Figure 13, assumes that the overall coordinates of any point inside the curved shell element are:

Where N _i (ξ, η) is a conventional shape function.

The rigid body displacement {δ' _Ri } of the node in the global coordinate system is given by the rigid body motion of the unit microbody, and the motion of the microcentre centroid includes rotation about three coordinate axes and translation along the coordinate axis. The rigid body motion of the shell element is:

{V' _R }={u' ₀ v' ₀ w' ₀ θ' _x0 θ' _y0 θ' _z0 } ^T

In the global coordinate system, the rigid body displacement of the rigid body motion obtained by the dynamic method is:

{δ' _Ri }={u' _Ri v' _Ri w' _Ri θ' _xRi θ' _yRi θ' _zRi } ^T =[T _Ri ]{V' _R }

{δ' _R ^e }={δ' _R1 δ' _R2 δ' _R3 δ' _R4 δ' _R5 δ' _R6 δ' _R7 δ' _R8 } ^T = [T _R ]{V' _R }

Wherein, the submatrix of the transformation matrix T _R :

[T _R ]={T _R1 T _R2 T _R3 T _R4 T _R5 T _R6 T _R7 T _R8 } ^T

Where x ₀ , y ₀ , z ₀ are the global coordinates of the centroid of the unit micro-body.

In the curvilinear coordinate system, the rigid body displacement of the joint generated by the rigid body motion is:

{δ _Ri }={u _Ri v _Ri w _Ri θ _xRi θ _yRi θ _zRi } ^T =[L _i ] ^T [T _Ri ][L ₀ ] ^T {V _R }

Where L _i is a transformation matrix.

In the formula, λ _x'αi =cos(x', α) and the like are cosines in the respective directions of the x, y, and z axes in the orthogonal main curve coordinate system α, β, δ.

In the curvilinear coordinate system, the rigid body motion of the centroid of the curved thin shell element is:

{V _R }={u ₀ v ₀ w ₀ θ _x0 θ _y0 θ _z0 } ^T

The unit node displacement vector converts the relationship between the global coordinates and the curve coordinates as:

In the curvilinear coordinate system, the total displacement field of the unit after the displacement of the rigid body is:

Where I is a unit matrix of 40 x 40.

The rigid body displacement does not produce strain, so the strain matrix in the orthogonal main curve coordinate system is:

Where B is the strain matrix of the elements in the orthogonal main curve coordinate system.

The rigid body displacement does not generate nodal forces, for static condensation, and the finite element equation is established according to the principle of virtual work:

among them,

Is the node load of the element in the orthogonal main curve coordinate system,

Is the original stiffness matrix of the element in the orthogonal main curve coordinate system

Obtained by displacement, ie

among them,

The conversion relationship between the stiffness matrix and the load vector in the global coordinate system is:

By integrating the element stiffness matrix and the load vector of the overall coordinate system, the node displacement solution equation of the global coordinate system can be obtained.

In the orthogonal main curve coordinate system (α, β, δ), α, β, δ are the arc lengths of the curve coordinates. For the 8-node arbitrary quadrilateral element, based on the original single linear transformation coordinate method, it is similar to the planar thin plate problem. Perform a linear coordinate transformation on the arc length coordinates of the curve, and assume that the unit displacement non-coordinating interpolation function is:

The equations can be established according to the displacement of the unit nodes, and the undetermined coefficients are determined by the simultaneous solution. The unit displacement non-coordinating interpolation function has 5 order completeness, and the complete order is high, which is suitable for any curved boundary.

When dividing the unit, as long as the common boundary of the two adjacent units is a straight line in the curvilinear coordinate system and the nodes in the boundary are evenly divided, and the free outer boundary of the unit has no coordination requirement, it can be a curve to fit the curve boundary. The unit displacement non-coordinating interpolation function can coordinate the deflection and tangential rotation angle on the common boundary of the unit, but the normal rotation angle cannot be coordinated.

When the common boundary of two adjacent elements is a straight line in the curvilinear coordinate system and the nodes in the boundary are equally divided, the unit displacement non-coordinating interpolation function coordinates the deflection and the tangential rotation angle at the element boundary. As long as the cell interpolation polynomial only appears three complete polynomials, the normal corners are also coordinated, so that the C ¹ order non-coordination problem is better solved.

Further, taking the eight-node quadrilateral element as an example, the eight-node quadrilateral element model constructed by the existing method is mostly a straight-sided unit. In solving the curve boundary problem, it is limited by the position of the interpolation point, and the curve boundary cannot be well fitted. reality. In this embodiment, a planar high-order non-coordinated eight-node curved quadrilateral element can be constructed by numerically integrating the element stiffness matrix in the area coordinate system. The interpolation function used is:

The unit has a general interpolation point, which can fit the curve boundary well. The structure of the shape function is closer to the real situation, and a single linear coordinate transformation has better anti-distortion. And through the rectangular plate subjected to uniform unidirectional tension, grid distortion test, bending of thick curved beam subjected to bending shear force, bending of curved thin curved beam, MacNeal cantilever beam, with four irregular quadrangles The cantilever beam of the unit and the cantilever beam are verified by pure bending, the cantilever is subjected to linear bending and shearing force, and the Cook oblique beam. The performance and convergence effect of the embodiment in the case of sparse mesh and the lateral linear bending load The performance under the action is better than the existing method, which provides higher precision in simulating linear elements and solving linear boundary problems, and works well in fitting linear boundaries. In short, this embodiment is proved by the unit completeness proof and the example test. The constructed unit not only has the unit's secondary completeness, has strong anti-distortion ability and anti-distortion, but also solves the curve boundary problem.

Example 2

The embodiment discloses a non-coordinating interpolation function construction system for optimizing the calculation precision of the finite element software, comprising a memory, a processor, and a computer program stored on the memory and operable on the processor, the processor executing the computer program The steps of the above method embodiments are implemented.

Example 3

The embodiment discloses a computer readable storage medium on which a computer program is stored, and when the program is executed by the processor, the steps of the foregoing method embodiments are implemented.

In summary, the high-order completeness of the unit is more important than its coordination. In some cases, it is impossible to construct a high-order complete and coordinated finite element interpolation function. Some finite element coordinated interpolation functions are too rigid and have poor convergence. In fact, as long as the unit can guarantee the coordination and convergence when the unit is subdivided, it is also necessary to construct a high-order complete non-coordinating finite element interpolation function to provide a comparison and verification method for the finite element analysis results. Therefore, the non-coordinating interpolation function construction method, system and storage medium for optimizing the calculation precision of the finite element software disclosed in the above embodiments of the present invention have the following beneficial effects:

Based on a single linear transformation coordinate system to construct a non-coordinating interpolation function for solving physical quantities, the first part of the polynomial has high-order completeness, which can greatly improve the calculation accuracy of the finite element analysis software, improve the safety and reliability of the structural design, and optimize the structural design. It is more adaptable to various curved (curve) boundaries, which brings huge economic benefits for engineering, aviation and aerospace construction.

On the other hand, when the present invention constructs a high-order complete non-coordinating finite element interpolation function by using a single linear coordinate transformation, an unexpected technical effect is obtained, which does not need to be solved when establishing a unit stiffness matrix and solving a system of displacement equations. The inverse coordinate matrix of the Jacobian matrix [J] transformed by the global coordinates of the isoparametric local coordinates (ξ, η). When the element is distorted, the inverse matrix of the Jacobian matrix [J] may tend to infinity and cause computational errors, while the unit is comparable. The inverse matrix of matrix [J] is a non-matrix matrix in the unit. It can't avoid the problem by controlling the shape of the element. However, using a single linear coordinate transformation to construct a high-order complete non-coordinated finite element interpolation function has no such problem. The high-order complete non-coordinating finite element interpolation function constructed by single linear coordinate transformation has better anti-distortion than the high-order complete coordination finite element interpolation function constructed by mixed linear coordinate transformation.

The above description is only the preferred embodiment of the present invention, and is not intended to limit the present invention, and various modifications and changes can be made to the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and scope of the present invention are intended to be included within the scope of the present invention.

Claims (8)

1. A non-coordinating interpolation function construction method for optimizing the calculation precision of finite element software, characterized in that it comprises:

   Constructing a non-coordinating interpolation function in a single linear transformation coordinate system;

   The coordinate element number, the number of items and the number of times of the non-coordinating interpolation function equation are determined according to the characteristics of the target unit; meanwhile, the constructed non-coordinating interpolation function equation comprises a high-order complete first partial polynomial and a high-order incomplete second partial polynomial; The first partial polynomial is incremented from low to high after covering each coordinate element; the second partial polynomial is symmetrically distributed, and the total number of times is above the first partial polynomial and the number of times of each coordinate element Not exceeding the highest number of times of the first partial polynomial.
2. The non-coordinating interpolation function construction method for optimizing the calculation precision of the finite element software according to claim 1, wherein the constructed non-coordinating interpolation function comprises any one or any combination of the following:

   1) When the target unit is a two-dimensional 8-node high-order complete quadrilateral curved side unit, the constructed unit displacement non-coordinating interpolation function is:

   

   2) When the target unit is a two-dimensional 12-node high-order complete quadrilateral curved side unit, the constructed unit displacement non-coordinating interpolation function is:

   

   3) When the target unit is a three-dimensional 20-node high-order complete surface hexahedral element, the constructed unit displacement non-coordinating interpolation function is:

   

   4) When the target unit is a three-dimensional 32-node high-order complete surface hexahedral element, the constructed unit displacement non-coordinating interpolation function is:

   

   5) When the target unit is a two-dimensional 4-node and each node has three high-order complete arbitrary quadrilateral thin plate elements with relevant displacement components, the relevant displacement components are respectively w, θ _x , θ _y , and the constructed unit displacement The non-coordinating interpolation function is:

   

   6) When the target unit is a two-dimensional 8-node and each node has three high-order complete curved quadrilateral thin plate elements with relevant displacement components, the relevant displacement components are respectively w, θ _x , θ _y , and the constructed unit The displacement non-coordinating interpolation function is:

   

   Wherein, in the above-mentioned various programs, T ₁ , T ₂ , and T ₃ are respectively coordinates in a unit linear transformation coordinate system; u, v, and w respectively correspond to displacements in three coordinate directions in the unit, and θ _x and θ _y are respectively w The partial derivative of the coordinates x, y in the element; a _i , b _i , c _i , i = 1, 2, 3, ... is the undetermined coefficient of the unit displacement non-coordinating interpolation function.
3. The non-coordinating interpolation function construction method for optimizing the calculation precision of the finite element software according to claim 2, wherein the transformation formula of the global coordinate system and the linear coordinate system transformation is:

   In the case of two dimensions,

   

   There are six undetermined coefficients A _i , B _i , C _i , (i=1, 2) in the coordinate transformation relationship determined by transforming the equations of six linear transformation coordinate values of 0 or 1.

   In the case of 3D,

   

   The coordinate change relationship has 12 undetermined coefficients A _i , B _i , C _i , D _i , (i = 1, 2, 3) determined by a system of equations that transforms 12 linear transformed coordinate values of 0 or 1.
4. The method for constructing a non-coordinating interpolation function for optimizing the calculation precision of the finite element software according to claim 1, characterized in that, for a space thin shell generally used in engineering, suitable orthogonal curve coordinates and corresponding geometric equations are used, according to the above The principle of the unit, like the plane problem, directly constructs the high-order complete curved thin shell element in the spatial orthogonal curvilinear coordinate system, calculates the element stiffness matrix, and then performs the space coordinate transformation; the specific includes:

   1) When the target unit is a three-dimensional 4-node and each node has three high-order complete arbitrary quadrilateral flat shell elements with relevant displacement components, the relevant displacement components are respectively w′, θ _x′ , θ _y′ The constructed unit displacement non-coordinating interpolation function is:

   

   2) When the target unit is a three-dimensional 8-node and each node has three high-order complete curved quadrilateral flat shell elements with relevant displacement components, the relevant displacement components are respectively w′, θ _x′ , θ _y′ The constructed unit displacement non-coordinating interpolation function is:

   

   3) When the target unit is a three-dimensional 4-node and each node has three high-order complete quadrilateral curved thin shell elements with relevant displacement components, the relevant displacement components are respectively w′, θ _x′ , θ _y′ , constructed The unit displacement non-coordinating interpolation function is:

   

   4) When the target unit is a three-dimensional 8-node and each node has three high-order complete quadrilateral curved thin shell elements with relevant displacement components, the relevant displacement components are respectively w′, θ _x′ , θ _y′ , constructed The unit displacement non-coordinating interpolation function is:

   

   Wherein, in the above-mentioned various programs, Z ₁ and Z ₂ are respectively coordinates in a linear transformation coordinate system of a two-dimensional orthogonal curve or a linear coordinate in a curved thin shell unit; when the three-dimensional curved thin shell is a flat thin shell, the curved thin shell unit The inner two-dimensional orthogonal curve coordinates become two-dimensional orthogonal linear coordinates; u', v' respectively correspond to the two-dimensional orthogonal curve in the unit curved surface or the displacement in the direction of the linear coordinate, and w' corresponds to the normal direction of the unit surface The displacement, θ _x' and θ _y' are the partial derivatives of w' to the two-dimensional orthogonal curve or the linear coordinates x', y' in the unit surface; a _i , b _i , c _i , i = 1, 2, 3 , ... is the undetermined coefficient of the unit displacement non-coordinating interpolation function.
5. The method for constructing a non-coordinating interpolation function for optimizing the calculation precision of the finite element software according to claim 4, further comprising:

   For the curved shell element on the curvilinear coordinate system, the complete rigid body displacement is supplemented in the displacement mode.
6. The non-coordinating interpolation function construction method for optimizing the calculation precision of the finite element software according to claim 4 or 5, wherein the transformation formula of the two-dimensional orthogonal curve coordinate system of the unit curved surface and the linear coordinate system transformation is:

   

   There are six undetermined coefficients A _i , B _i , C _i , (i=1, 2) in the coordinate transformation relationship determined by a system of equations that transforms six linear transformation coordinate values of 0 or 1.
7. A non-coordinating interpolation function construction system for optimizing the calculation precision of a finite element software, comprising: a memory, a processor, and a computer program stored on the memory and operable on the processor, wherein the processor executes The computer program implements the steps of the method of any of the preceding claims 1 to 6.
8. A computer readable storage medium having stored thereon a computer program, wherein the program is executed by a processor to perform the steps of any of the methods of any of claims 1 to 6.

Images