CN112632825B - Electrostatic field smooth finite element numerical algorithm based on finite element super-convergence - Google Patents
Electrostatic field smooth finite element numerical algorithm based on finite element super-convergence Download PDFInfo
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Abstract
An electrostatic field smooth finite element numerical algorithm based on finite element super-convergence. Inputting data, wherein the data is numerical data; according to the node numbers and the node coordinates, all the nodes are circulated according to the node sequence numbers, and whether the nodes are internal nodes is judged; when the node is an internal node, determining a unit set near a reconstruction point, and calculating a super-convergence point coordinate and a super-convergence point original gradient near the reconstruction point so as to calculate a polynomial expansion and reconstruction gradient, wherein the internal node is a common node of at least a preset number of units; when the node is not an internal node, determining a unit set near the reconstruction point, and calculating gradients of the reconstruction point in different units and a gradient average value of the reconstruction point. The invention provides a high-precision and high-efficiency electrostatic field smooth finite element numerical calculation method, which greatly improves the precision and convergence speed of electrostatic field gradient calculation on the premise of ensuring the calculation efficiency.
Description
Technical Field
The invention belongs to the technical field of numerical algorithms, and relates to an electrostatic field smooth finite element numerical algorithm based on finite element super-convergence.
Background
Finite Element Method (FEM) is a numerical computation method based on variational principle and sliced interpolation. Since the 20 th century and the 50 th century, under the promotion of computer technology, FEM has been expanded from the initial field of aeronautical technology to other fields such as civil construction, mechanical manufacturing, hydraulic engineering, shipbuilding, electronic technology, etc., and has been developed from simple static analysis to the solution of complex problems such as dynamic, nonlinear, multi-physics coupling, etc., and the application range and depth thereof have been greatly expanded. Due to high calculation efficiency and more unit division modes, the finite element method can be used for simulating complex geometric bodies in various forms, and becomes an important analysis method in engineering and scientific numerical simulation.
With the continuous research, the problems of the finite element method are revealed. For some problems, researchers may be more concerned about the gradient of the solution, such as the gradient of electric potential-electric field strength, and the gradient of temperature-heat flux, while the gradient of the finite element solution obtained by the lagrange unit in the conventional finite element method is discontinuous at the boundary of the unit, has larger difference with the actual continuous gradient and has lower precision. To solve this problem, the related scholars propose a gradient reconstruction method in which the gradient approximation of each node is constructed by averaging the contributions of surrounding cells, thereby improving the resolution accuracy of the gradient. The earliest gradient reconstruction methods were simple averaging, approximating the continuous gradient by directly averaging the gradient of the nodal finite element solution. On the basis, the related scholars propose a weighted average method with different weighting factors, which can be divided into area average, distance average, angle average and area adjustment average, and the gradient approximation effect obtained by the methods when the unit grids are not uniform is poor. Thom é introduces an average kernel to the gradient of the finite element solution by using a spline function, and can obtain gradient approximation with high precision by convolution to reconstruct the gradient, but the method requires uniform grid division and lower boundary precision. Louis introduces a large-range integral average method, which has no limitation on grid division, but has large calculation amount and strong dependence on basic solution.
Disclosure of Invention
In order to solve the problems existing in the prior art, namely the problem that the boundary gradient of the traditional finite element method is discontinuous and the gradient approximation effect of the weighted average method is poor, the invention aims to provide an electrostatic field smooth finite element numerical algorithm based on finite element super-convergence.
The invention adopts the following technical scheme:
an electrostatic field smooth finite element numerical algorithm based on finite element super convergence, the smooth finite element numerical algorithm comprising the steps of:
step 3, when the node is an internal node, determining a unit set near the reconstruction point, and calculating the coordinate of the super-convergence point near the reconstruction point and the original gradient of the super-convergence point so as to calculate the polynomial expansion and reconstruction gradient;
and 4, when the node is not an internal node, determining a unit set near the reconstruction point, and calculating the gradient of the reconstruction point in different units and the gradient average value of the reconstruction point.
In step 2, the preset number is 4 for the bilinear quadrilateral cells.
In step 3, the reconstruction gradient of each node in the definition domain is determined
So as to be composed of
The gradient determined from the shape function N (x, y) is continuous and is expressed as follows:
wherein σ * For continuous gradient approximation, N (x, y) is a shape function, and x and y in N (x, y) are node coordinates,
to reconstruct the gradient.
The reconstructed gradient
By a polynomial σ of the same order as the shape function N (x, y) p And (3) expansion determination, wherein the expression is as follows:
σ p ＝Pa
wherein σ p Is a polynomial, P is a polynomial vector, and a is a coefficient vector to be solved.
For the one-dimensional case, the expression of the polynomial vector P and the coefficient vector a to be solved is:
P＝[1 x x 2 … x n ]
a＝[a 1 a 2 a 3 … a n+1 ] T
wherein x is a coordinate value, a 1 ，a 2 ，…，a n+1 Is 1,x, …, x in the polynomial vector P n The coefficient of (c).
For the two-dimensional case, in a bilinear quadrilateral cell, the expression of the polynomial vector P is:
P＝[1 x y xy]
wherein x is an abscissa value and y is an ordinate value.
And (3) performing least square fitting on the considered unit cell by using a group of super-convergence points around the reconstruction node to solve the coefficient vector a to be solved:
wherein z is i ＝(x i ,y i ) I is more than or equal to 1 and less than or equal to n is a selected super convergence point,(x i ,y i ) Is the coordinate of the hyper-convergence point, n is the number of hyper-convergence points, σ h (x i ,y i ) Is a point (x) i ,y i ) Gradient of (a), P (x) i ,y i ) Is a point coordinate z i ＝(x i ,y i ) The value of 1 is less than or equal to i and less than or equal to n is substituted into P = [1 x y xy =]And obtaining a numerical value vector.
According to the G (a) minimizing condition, it is possible to obtain:
wherein, P T (x i ,y i ) Is P (x) i ,y i ) The transposed matrix of (2).
The expression of the coefficient vector a is:
a＝A -1 B
after determining the polynomial vector P and the coefficient vector a, substituting and obtaining the reconstruction point coordinates
And obtaining continuous gradient approximation, namely obtaining continuous electric field approximation.
Compared with the prior art, the invention has the beneficial effects that:
the method for calculating the smooth finite element numerical value of the electrostatic field has high precision and high efficiency, and greatly improves the precision and convergence rate of electrostatic field gradient calculation on the premise of ensuring the calculation efficiency.
Drawings
FIG. 1 is a schematic flow chart of a smooth finite element numerical algorithm according to the present invention;
FIG. 2 is a schematic diagram showing the comparison of gradient calculation results between the method of the present invention and other methods;
FIG. 3 is a schematic diagram showing the comparison of the error and the operation time of the method of the present invention with other methods.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention clearer, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. The embodiments described herein are only some embodiments of the invention, and not all embodiments. All other embodiments obtained by a person skilled in the art without making any inventive step on the basis of the spirit of the present invention are within the scope of protection of the present invention.
The smooth finite element numerical algorithm comprises the following steps:
for a bilinear quadrilateral cell, the preset number is 4.
Step 3, when the node is an internal node, determining a unit set near the reconstruction point, and calculating the coordinate of the super-convergence point near the reconstruction point and the original gradient of the super-convergence point so as to calculate the polynomial expansion and reconstruction gradient;
and 4, when the node is not an internal node, determining a unit set near the reconstruction point, and calculating the gradient of the reconstruction point in different units and the gradient average value of the reconstruction point.
To illustrate the smooth recovery method of the gradient, we now assume that the electrostatic field problem defined at Ω is as follows:
u| Γ ＝g (2)
where u is the potential, Γ is the boundary defining the domain Ω, and g is the value on the boundary Γ.
Approximate solution u obtained by traditional finite element method h (x, y) is as shown in formula (3) and the gradient σ h (x, y) is as shown in formula (4), wherein N i (x, y) is (x) i ,y i ) Shape function, u i For node parameters, N (x, y) is a shape function matrix, U is a node parameter vector, and D is a differential operator.
σ h (x,y)＝Du h (x,y) (4)
The gradient of the finite element solution calculated by the formula (4) is not continuous at the node and the unit boundary, and the accuracy is low. The smooth finite element numerical algorithm aims to determine the reconstruction gradient of each node in the definition domain
So as to be composed of
The gradient determined with N (x, y) is continuous, as shown in equation (5), where σ * Is a continuous gradient approximation.
In the electrostatic field smoothing finite element method in the present invention,
is expanded by a polynomial of the same order as the shape function N (x, y) p Determining x and y in N (x and y) are node coordinates,
to reconstruct the gradient.
This polynomial is valid on the set of units containing the reconstruction point under consideration, as shown in equation (6).
σ p ＝Pa (6)
Wherein, P is a polynomial vector, and a is a coefficient vector to be solved. For a one-dimensional problem, P and a can be written as:
P＝[1 x x 2 … x n ] (7)
a＝[a 1 a 2 a 3 … a n+1 ] T (8)
wherein x is a coordinate value, a 1 ，a 2 ，…，a n+1 1,x, …, x in the polynomial vector P n The coefficient of (a).
For two-dimensional problems, using the same terms as those appearing in N (x, y) may improve the computation result slightly, so for a bilinear quad cell, P may be written as:
P＝[1 x y xy] (9)
wherein x is an abscissa value and y is an ordinate value.
The method for determining the coefficient vector a to be solved in the formula (6) is to carry out least square fitting on a considered unit chip by utilizing a group of super-convergence points around a reconstruction node
Wherein z is i ＝(x i ,y i ) I is more than or equal to 1 and less than or equal to n is a selected super-convergence point, (x) i ,y i ) Is the coordinate of the hyper-convergence point, n is the number of hyper-convergence points, σ h (x i ,y i ) Is a point (x) i ,y i ) Gradient of (a), P (x) i ,y i ) Is a point coordinate z i ＝(x i ,y i ) The value of 1 is less than or equal to i and less than or equal to n is substituted into P = [1 x y xy =]And obtaining a numerical value vector.
According to the G (a) minimizing conditions, it is possible to obtain:
wherein, P T (x i ,y i ) Is P (x) i ,y i ) The transposed matrix of (2).
Can be solved by the formula (11):
a＝A -1 B (12)
after the polynomial vector P and the coefficient vector a are determined, the reconstructed point coordinates can be taken into formula (6) to obtain
And further obtaining continuous gradient approximation, namely obtaining continuous electric field approximation. The algorithm flow is shown in fig. 1.
Example analysis:
the electrostatic field problem is defined in the rectangular area Ω: x is more than or equal to 0 and less than or equal to 1,0 and less than or equal to 1, the differential equation is shown as a formula (14), and the boundary conditions are shown as a formula (15) and a formula (16).
T(0,y)＝-y 3 T(1,y)＝3y-y 3 0≤y≤1 (15)
T(x,0)＝0T(x,1)＝3x 2 -1 0≤x≤1 (16)
The gradient calculation results of the method and the traditional finite element method, the simple average method and the weighted average method in a non-uniform grid division mode are shown in figure 2, and the gradient approximation effect of the method is obviously superior to that of other methods.
The error and run time ratio for the method of the invention and other methods are shown in figure 3.
The method greatly improves the precision and the convergence speed of finite element gradients on the premise of ensuring the calculation efficiency.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting the same, and although the present invention is described in detail with reference to the above embodiments, those of ordinary skill in the art should understand that: modifications and equivalents may be made to the embodiments of the invention without departing from the spirit and scope of the invention, which is to be covered by the claims.
Claims (4)
1. An electrostatic field smooth finite element numerical algorithm based on finite element super convergence, which is characterized by comprising the following steps:
step 1, inputting data, wherein the data is numerical data and comprises a unit number, a unit node number, a node coordinate, a node potential value and a node original electric field, the node potential value is a node function value, and the node original electric field is a node original gradient value;
step 2, according to the unit node numbers and the node coordinates in the step 1, all the nodes are circulated according to the unit node number sequence, and whether the nodes are internal nodes or not is judged, wherein the internal nodes are common nodes of at least a preset number of units;
step 3, when the node is an internal node, determining a unit set near the reconstruction point, and calculating the coordinate of the super-convergence point near the reconstruction point and the original gradient of the super-convergence point so as to calculate the polynomial expansion and reconstruction gradient;
determining a reconstructed gradient for each node within a domain of definition
So as to be composed of
The gradient determined with the shape function N (x, y) is continuous and is expressed as follows:
wherein σ * For continuous gradient approximation, N (x, y) is a shape function, and x and y in N (x, y) are node coordinates，
To reconstruct the gradient;
reconstructing a gradient
By a polynomial σ of the same order as the shape function N (x, y) p And (4) performing expansion determination, wherein the expression is as follows:
σ p ＝Pa
wherein σ p Is a polynomial, P is a polynomial vector, and a is a coefficient vector to be solved;
and (3) performing least square fitting on the considered unit cell by using a group of super-convergence points around the reconstruction node to solve the coefficient vector a to be solved:
wherein z is i ＝(x i ，y i ) I is more than or equal to 1 and less than or equal to n is a selected super-convergence point, (x) i ，y i ) Is the coordinate of the hyper-convergence point, n is the number of hyper-convergence points, σ h (x i ，y i ) Is a point (x) i ，y i ) Gradient of (a) p (x i ，y i ) Is a polynomial, P (x) i ，y i ) Is a point coordinate z i ＝(x i ，y i ) The value of 1 is less than or equal to i and less than or equal to n is substituted into P = [1 x y xy =]Obtaining a numerical vector;
according to the G (a) minimizing condition, it is possible to obtain:
wherein, P T (x i ，y i ) Is P (x) i ，y i ) The transposed matrix of (2);
the expression of the coefficient vector a is:
a＝A -1 B
after determining the polynomial vector P and the coefficient vector a, substituting the coordinates of the reconstruction point into the polynomial vector P and obtaining the reconstructed point coordinates
Obtaining continuous gradient approximation, namely obtaining continuous electric field approximation;
and 4, when the node is not an internal node, determining a unit set near the reconstruction point, and calculating the gradient of the reconstruction point in different units and the gradient average value of the reconstruction point.
2. The electrostatic field smoothing finite element numerical algorithm based on finite element super convergence of claim 1, wherein:
in step 2, the preset number is 4 for bilinear quadrilateral cells.
3. The electrostatic field smooth finite element numerical algorithm based on finite element super-convergence of claim 1, wherein:
for the one-dimensional case, the expression of the polynomial vector P and the coefficient vector a to be solved is:
P＝[1 x x 2 …x n ]
a＝[a 1 a 2 a 3 …a n+1 ] T
wherein x is a coordinate value, a 1 ，a 2 ，...，a n+1 Are 1,x, a., x, respectively, in a polynomial vector P n The coefficient of (a).
4. The electrostatic field smooth finite element numerical algorithm based on finite element super-convergence of claim 1, wherein:
for the two-dimensional case, in bilinear quadrilateral cells, the expression of the polynomial vector P is:
P＝[1 x y xy]
wherein x is an abscissa value and y is an ordinate value.
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