CN110046400B - High-precision numerical simulation calculation method for static characteristic analysis of concrete gravity dam based on h-p type finite element method - Google Patents
High-precision numerical simulation calculation method for static characteristic analysis of concrete gravity dam based on h-p type finite element method Download PDFInfo
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Abstract
The invention relates to a high-precision numerical simulation calculation method for static characteristic analysis of a concrete gravity dam based on an h-p type finite element method, and belongs to the technical field of simulation. The high-precision numerical simulation calculation method for static characteristic analysis of the concrete gravity dam based on the h-p type finite element method comprises the following steps: constructing a calculation model of the hydraulic concrete gravity dam, and calculating a displacement field and a stress field of the hydraulic concrete gravity dam model by adopting an h-p type finite element method; and judging whether the obtained displacement field and the stress field meet the precision requirement, if not, encrypting the grid again and improving the order of the interpolation polynomial, and repeating the steps until a satisfactory result is obtained. The invention provides a novel high-precision finite element calculation analysis method for a concrete gravity dam, aiming at the problems of overlong calculation time, overlarge error and lower precision of the existing concrete gravity dam model based on the traditional finite element static analysis.
Description
Technical Field
The invention relates to a high-precision numerical simulation calculation method for static characteristic analysis of a concrete gravity dam based on an h-p type finite element method, and belongs to the technical field of simulation.
Background
The gravity dam is one of the most important dam types used by human beings at first, and is widely applied due to the advantages of simple design and construction, strong reliability, good adaptability to terrains, high durability and the like. The effective safety evaluation standard of the hydraulic structure is generally directed at the displacement and stress conditions of each part of the dam body, and the dam body is damaged when the displacement and stress exceed the safety standard, so that the static analysis (including stress analysis and displacement analysis) is an inevitable and significant topic in the research of the hydraulic structure. At present, the calculation method of static analysis in hydraulic structure safety analysis is mainly a finite element method, a material mechanics method and an elastic theory analysis method, the material mechanics method and the elastic theory analysis method are applied in the early stage, and the analysis method also makes a contribution to the analysis of some hydraulic structure problems, but because of the inherent limitations of the calculation method, the time consumption in the calculation process is long, so that the development of a novel numerical simulation calculation method is particularly important in the hydraulic structure analysis. The conventional finite element method (h-type finite element method) continuously improves the precision of a finite element solution mainly in a mode of gradually reducing the element grid size h (commonly called subdivision grid), and is widely applied to the aspects of design calculation and safety evaluation of hydraulic buildings, such as various earth and rockfill dams, panel rock-fill dams, concrete gravity dams, arch dams, flood discharge ports and other hydraulic buildings.
In the related research of hydraulic buildings, the traditional finite element method plays a non-negligible role, and solves the problems in the aspect of a large number of hydraulic structures, but due to the limitation of the calculation method, the calculation result cannot be completely satisfied, the convergence rate of numerical simulation calculation is slow, the error is large, and the precision is low, so that the research of a novel finite element method for solving the problem which cannot be well solved by the traditional finite element method is very important. Most of researches on problems using the finite element method are the conventional h-type finite element method or the development based on the conventional finite element, so that the following problems exist:
(1) The control of the traditional h-type finite element calculation precision depends on the experience of researchers, and in order to obtain better calculation precision, grids need to be divided again for many times;
(2) The traditional h-type finite element is based on a low-order interpolation function, inherent defects exist in numerical simulation, a large number of encryption grids are needed for obtaining high precision, and pretreatment and calculation cost are high.
The traditional finite element method (h-type finite element method) and the p-type finite element method have been successfully applied to various practical engineering fields (including hydraulic structure analysis), and in recent years, the h-p type finite element method has been rapidly developed, but in the hydraulic structure analysis, particularly in the static force analysis of the concrete gravity dam, no relevant application technology of the h-p type finite element method exists at home and abroad.
The invention provides an application of an h-p type finite element method to the field of hydraulic and hydroelectric engineering based on national natural science foundation (subsidy number: 51769011).
Disclosure of Invention
Aiming at the problems and the defects in the prior art, the invention provides a high-precision numerical simulation calculation method for analyzing the static characteristics of a concrete gravity dam based on an h-p type finite element method. The technical problem solved by the invention is as follows: the existing simulation calculation technology applied to hydraulic structure analysis based on the traditional finite element method has the defects that due to the limitation of the calculation method, the calculation result cannot meet the actual application requirements of engineering, for example, when numerical simulation calculation is carried out on actual engineering problems, the convergence rate is low, the error is large, and the precision is low.
The h-p type finite element method has the core idea that the advantages of the h type finite element method and the p type finite element method are combined together, namely the cell grid size h (subdivision grid) is reduced and the order p of an interpolation polynomial on a cell is improved, so that the convergence speed and precision of a finite element solution can be improved more quickly, and the h-p type finite element method has the advantages of higher convergence speed, higher calculation result precision and smaller error in the solving process and has unique advantages. The conventional low-order finite element method is a method of constructing a finite element solution using a low-order interpolation polynomial, while the h-p type finite element method is a method of constructing a finite element solution using a high-order (nonlinear) interpolation polynomial, and is a high-order nonlinear finite element method, and the h-p type finite element method is a new development of the finite element method, which can achieve higher precision and faster convergence rate.
The invention is realized by the following technical scheme.
A high-precision numerical simulation calculation method for static characteristic analysis of a concrete gravity dam based on an h-p type finite element method comprises the following steps:
step 1.1 geometric model establishment: establishing a two-dimensional structure model by contrasting an actual engineering drawing, wherein the two-dimensional structure model structure consists of a dam body part and a dam foundation part;
step 1.2, grid division: finite element meshing is carried out by adopting quadrilateral meshes, so that slender quadrilateral meshes are avoided, and meshes are divided in a boundary region in an encrypted manner so as to be better simulated;
step 1.3 setting material properties: simulating the operating conditions with a linear elastic material, setting corresponding material properties for the grid cells, wherein the settings for the elastic modulus are: setting the values of the elastic moduli of the main body part and the dam foundation part of the concrete gravity dam as the static elastic modulus values corresponding to the actually used materials of the main body part of the concrete gravity dam and the actual rock mass of the dam foundation part;
step 1.4 setting boundary conditions: setting vertical constraints on the bottom of the dam foundation, and respectively setting normal constraints on the two side surfaces of the dam foundation part along the width direction of the river channel;
step 1.5, setting load: simulating hydrostatic pressure, silt pressure and uplift pressure of water body quality;
step 2.1, solving to obtain a structural displacement array a
According to the equation: ka = F (1), where K = ∑ ΣeGTKeG, structural Overall stiffness matrix, KeIs a cell stiffness matrix; f = ∑ ΣeGTfeStructure node load array; g is a conversion matrix of the degree of freedom of the unit node and the degree of freedom of the structure node, GTFor the transposition of G, superscript T represents the transposition of matrix; e represents a single unit; a is a structure displacement array;
then according to formula unit rigidity matrix Ke＝∫ΩBTDBdΩ (2)，
Unit equivalent node load array
Nodal force within a cell
External nodal force
In the above formulas (2) to (5), Ω is represented inside the cell; b = LNIL is a differential operator, NIIs an interpolation function matrix or a shape function matrix, I is the index of the cell, ΓtIs the outer boundary of the cell; d is a stress matrix, b is a physical force,
determining the boundary conditions of the load according to the material properties and the boundary conditions of the load and the displacement in the step 1 respectively;
the formulas (2) to (5) and NISubstituting an interpolation polynomial which is an h-p type finite element method into the linear equation set (1), and solving the linear equation set (1) to obtain a structure displacement array a;
NIby using interpolation polynomial of h-p finite element method, a typical two-dimensional h-p finite element function is based on Legendre orthogonal polynomial, taking a two-dimensional quadrilateral unit as an example (shown in figure 1), and under a coordinate system (eta, xi), a point p is1、p2、p3And p4Four vertices, Γ, divided into cells1、Γ2、Γ3And Γ4The four sides of the cell are divided, and the structural form of the shape function is as follows:
p is more than or equal to 1, the point mode basis function is consistent with the conventional Lagrange basis function, and a quadrilateral four-node unit is used as an example and is expanded as follows:
p is more than or equal to 2, and the side mode basis function is as follows:
in the formula:
here, Pn(t) is a Legendre polynomial with an order n ≧ 0
Similarly, the edge Γ may be defined as followsk(2 ≦ k ≦ 4) associated shape function:
internal mode basis function when p is greater than or equal to 4:
the h-p type finite element method adopted by the invention has the advantages that the order of the interpolation polynomial can be gradually improved from p =1 in sequence, the rigidity matrix of the low order can be continuously used after the order of the interpolation polynomial is improved, only the high order part needs to be calculated, the repeated calculation of the low order part of the rigidity matrix is avoided, the method has good inheritability, and the calculation cost is saved;
step 2.2, according to formula aI= Ga solving unit displacement array aI；
Step 2.3, according to a displacement field formula u = NIaIStress field formula σ = D ∈ = DBaISolving to obtain a displacement field u and a stress field sigma;
and 3, judging whether the displacement field and the stress field obtained in the step 2 meet the precision requirement, if not, continuing to encrypt the grid and improve the order of the interpolation polynomial at the same time, and returning to the step 2.
In the step 1.2, the grid division adopts quadrilateral grids: the apex angle of the quadrilateral elements is from 5 to 175, preferably close to 90, of the apex angle of the quadrilateral elements. As the shape function adopts a high-order interpolation function, compared with the traditional finite element method, the method can effectively accelerate the convergence rate and improve the calculation precision to obtain a high-precision finite element solution.
And (3) for the non-standard model, under the condition that no theoretical solution or experimental result is compared, simultaneously encrypting the grids, sequentially increasing the order of the interpolation polynomial, and then obtaining the displacement field by static calculation. Whether the results obtained after observation deviate from the results calculated previously by within an acceptable range.
The accuracy does not meet the requirements or the direct deviation of the obtained results is not within the acceptable range. And (3) returning to the step 2 again, and recalculating the displacement field and the stress field until the numerical calculation result meets the precision requirement.
Reference numerals not specifically explained in the above formulas are those whose meanings are well known to those skilled in the art.
The invention has the beneficial effects that:
the invention provides a novel high-precision finite element calculation analysis method for a concrete gravity dam, aiming at the problems of low convergence rate, overlarge error and low precision of the existing concrete gravity dam model calculation based on the traditional finite element static characteristic analysis, which comprises the following steps: the method has the characteristics of less grid division number, high calculation efficiency, high convergence rate and smaller error.
Drawings
FIG. 1 is a schematic diagram of a standard quadrilateral parent cell of the present invention;
FIG. 2 is a schematic view of a concrete gravity dam according to an embodiment of the present invention 1;
FIG. 3 is a cloud image of a displacement field obtained by calculating a finite element model by an h-type finite element method after step 1.5 in example 1 of the present invention;
FIG. 4 is a cloud image of a displacement field obtained by computing a finite element model by an h-p type finite element method in example 1 of the present invention.
Detailed Description
The invention is further described with reference to the following drawings and detailed description.
Examples 1
A certain hydropower station is developed in a dam mode, and a central hub consists of a barrage, a flood discharge and sand discharge building, a water-approaching power generation system, a vertical lifting ship machine, an alternating power station and the like. The river retaining dam is made of a rolled concrete gravity dam, the elevation of the whole dam top is as high as 622m, and the whole maximum dam height is 108m. The normal water level of the reservoir is 612m. The rock performance parameters used in the calculation were as follows:
mass density: 2.8t/m3And the Poisson ratio: 0.25, elastic modulus: 10000MPa; the concrete parameters used by the gravity dam are as follows: mass density: 2.4, poisson ratio: 0.2, elastic modulus: 22000MPa; the parameters used in the plant are as follows: mass density: 2.50t/m3And the Poisson ratio: 0.167, elastic modulus: 25500MPa.
The main loads used in the concrete gravity dam design calculations are as follows.
The various static loading values are as follows:
(1) Dead weight of dam body
The self weight of the gravity dam body concrete is applied according to one-time construction. And in the calculation modeling range, the dead weight of the bedrock is converted into a bedrock dead weight stress field and then is applied to the calculation model according to the initial dead weight stress field.
(2) Hydrostatic pressure
Due to the water pressure generated by the water depth of the upstream and downstream. Water volume weight: 9.81kN/m3, upstream water level: 602.00m, downstream water level: 535.735m.
(3) Pressure of silt
Silt floating volume weight: 8.1kN/m3, internal friction angle: 24 deg.. Elevation of silt in front of a dam: el560.00m (fouling age 40 years).
(4) Uplift pressure
The uplift pressure of the base surface of the concrete dam is designed and calculated according to the pumping and pressure reduction effect. And determining the positions of the upstream curtain, the downstream curtain and the dam foundation drain hole. The calculation graph and coefficient of the uplift pressure are selected according to the design standard of hydraulic building load. The internal osmotic pressure strength coefficient of the dam body of the concrete gravity dam is 0.2, and the uplift pressure strength coefficient in front of the main drainage hole is 0.2.
The concrete gravity dam adopting the parameters is subjected to high-precision numerical simulation calculation method for static analysis based on an h-p type finite element method, and the method comprises the following steps:
step 1.1 geometric model establishment: establishing a two-dimensional structure model by contrasting an actual engineering drawing, wherein the two-dimensional structure model structure consists of a dam body part and a dam foundation part, and a structure model drawing obtained by adopting CAD is shown in figure 2;
step 1.2, grid division: finite element meshing is carried out by adopting quadrilateral meshes, so that slender quadrilateral meshes are avoided, and meshes are divided in a boundary region in an encrypted manner so as to be better simulated; the grid division in the step 1.2 adopts quadrilateral grids: the vertex angle of the quadrilateral unit is 5-175 degrees, preferably close to 90 degrees; the grid is divided into 356, 1424, 3204, 5696, 8900, 12816 cells;
step 1.3 setting material properties: simulating the operating conditions with a linear elastic material, setting corresponding material properties for the grid cells, wherein the settings for the elastic modulus are: setting the values of the elastic moduli of the main body part and the dam foundation part of the concrete gravity dam as the static elastic modulus values corresponding to the actual materials of the main body part of the concrete gravity dam and the actual rock mass of the dam foundation part;
step 1.4 setting boundary conditions: setting vertical constraint on the bottom of the dam foundation, and respectively setting normal constraint on the two side surfaces of the dam foundation part in the width direction of the river channel;
step 1.5, setting load: simulating hydrostatic pressure, silt pressure and uplift pressure of water body quality;
the cloud picture of the displacement field obtained by calculating the finite element model by the h-p type finite element method in combination with the step 1 and the step 2 is shown in fig. 4, and the displacement field U is along the river channel direction (X direction)xThe calculation results are shown in table 1:
TABLE 1
As can be seen from table 1, the calculation result of the maximum displacement Ux in the X direction of the two-dimensional concrete gravity dam is obtained when the two-dimensional concrete gravity dam is subjected to the basic load combination of the dead weight, the water pressure before the dam, the silt pressure, the water pressure after the dam, the uplift pressure and the like. When a finite element calculation model of a two-dimensional concrete gravity dam is solved based on an h-type finite element method, a specific calculation model is shown in fig. 3, and when 20023 grids are divided, the obtained calculation result of the maximum displacement Ux in the X direction is as follows: the X-direction maximum displacement Ux =7.012mm, when the concrete gravity dam model is calculated by using the H-p type finite element method, the X-direction maximum displacement which can be achieved when 1424 grids are divided, namely the X-direction maximum displacement Ux =7.015mm, results of the X-direction maximum displacement and the X-direction maximum displacement are similar, but the number of the grids divided by the h-p type finite element calculation analysis is much smaller than that of the grids divided by the h type finite element calculation analysis, which shows that the h-p type finite element method can achieve the precision of the h type finite element method when the grids are divided more by increasing the order of the polynomial and simultaneously encrypting the grids under the condition that the grids are divided relatively less.
As can be seen from FIG. 3, the displacement U in the X direction of the dam is realized by using the h-type finite element method under the condition of 20023 gridsxThe size is 7.012mm; displacement U of 1424 meshes compared to the h-p type finite element in FIG. 4xThe sizes of 7.015mm are close, which shows that the h-p type finite element has more accurate result under the condition of less grid number than the h type finite element under the condition of more grid number.
Step 3, judging whether the displacement field and the stress field obtained in the step 2 meet the precision requirement, if not, encrypting the grid and improving the order of the interpolation polynomial, and returning to the step 2;
the above description describes the embodiments of the present invention in detail, but the present invention is not limited to concrete gravity dams, and within the knowledge of those skilled in the art, the h-p type finite element method can be applied to numerical simulation calculations of other hydraulic structures without departing from the spirit of the present invention.
Claims (1)
1. A high-precision numerical simulation calculation method for static analysis of a concrete gravity dam based on an h-p type finite element method is characterized by comprising the following steps of: the method comprises the following steps:
step 1, constructing a calculation model of a hydraulic concrete gravity dam, comprising the following steps of:
step 1.1 geometric model establishment: establishing a two-dimensional structure model by contrasting an actual engineering drawing, wherein the two-dimensional structure model structure consists of a dam body part and a dam foundation part;
step 1.2, grid division: carrying out finite element meshing by adopting a quadrilateral mesh;
step 1.3 setting material properties: simulating the working condition by using a linear elastic material;
step 1.4 setting boundary conditions: setting vertical constraints on the bottom of the dam foundation, and respectively setting normal constraints on the two side surfaces of the dam foundation part along the width direction of the river channel;
step 1.5, setting load: simulating hydrostatic pressure, silt pressure and uplift pressure of the water mass;
step 2, calculating a displacement field and a stress field of the finite element model by adopting an h-p type finite element method, and comprising the following steps:
step 2.1, solving to obtain a structure displacement array a
According to the equation: ka = F (1), where K = ΣeGTKeG is a structural global stiffness matrix, KeIs a cell stiffness matrix; f = ΣeGTfeLoading arrays for structural nodes(ii) a G is a conversion matrix of the degree of freedom of the unit node and the degree of freedom of the structure node, GT is the transposition of G, and superscript T represents the transposition of the matrix; e represents a single unit; a is a structure displacement array;
recalculating cell stiffness matrix Ke＝∫ΩBTDBdΩ (2)，
Unit equivalent node load array fe＝fe int+fe ext (3)，
Nodal force within a cell
External nodal force
In the above formulas (2) to (5), Ω is represented inside the cell; b = LNIL is a differential operator, NIIs an interpolation function matrix or a shape function matrix, I is the label of a unit, gammatIs the outer boundary of the cell; d is a stress matrix, b is a physical strength,
determining the boundary conditions of the load according to the material properties and the boundary conditions of the load and the displacement in the step 1 respectively;
the formulas (2) to (5) and NISubstituting interpolation polynomial into the (1) for a finite element method, and solving a linear equation set (1) to obtain a structural displacement array a;
step 2.2, according to formula aI= Ga solving to obtain unit displacement array aI；
Step 2.3, according to a displacement field formula u = NIaIStress field formula σ = D ∈ = DBaISolving to obtain a displacement field u and a stress field sigma;
step 3, judging whether the displacement field and the stress field obtained in the step 2 meet the precision requirement, if not, simultaneously encrypting the grid and increasing the order of the interpolation polynomial, and returning to the step 2;
in the step 1.2, the grid division adopts quadrilateral grids: the vertex angle of the quadrilateral unit is 5-175 degrees.
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