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Efficient multiscale finite element method for simulating two-dimensional water flow in porous media

A porous medium and finite element technology, applied in the field of hydraulics, can solve the problem of high consumption of basis function construction, achieve the effects of reduced calculation time, high element flexibility, and strong anti-deformation ability

Active Publication Date: 2019-05-07
NANJING UNIV
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  • Summary
  • Abstract
  • Description
  • Claims
  • Application Information

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Problems solved by technology

[0004] Aiming at the deficiencies of the above-mentioned prior art, the object of the present invention is to provide an efficient multi-scale finite element method for simulating two-dimensional water flow motion in porous media, which uses domain decomposition technology to improve the construction algorithm and subdivision method of basis functions , decomposing the basis function construction problem into several sub-problems, and solving unknown items in batches, can greatly reduce the calculation consumption required to construct the basis function, so as to solve the problem that the basis function construction consumes too much when solving complex groundwater problems in the prior art

Method used

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  • Efficient multiscale finite element method for simulating two-dimensional water flow in porous media
  • Efficient multiscale finite element method for simulating two-dimensional water flow in porous media
  • Efficient multiscale finite element method for simulating two-dimensional water flow in porous media

Examples

Experimental program
Comparison scheme
Effect test

Embodiment 1

[0055] Example 1: Continuum model of two-dimensional steady flow

[0056] The research area is a square area: Ω=[50m, 150m]×[50m, 150m], permeability coefficient K(x,y)=x 2 m / d, the research equation is the steady flow equation:

[0057]

[0058] Boundary condition is constant head boundary condition The source-sink item is 0, and this model has an analytical solution: H=x 2 -3y 2 .

[0059] Sub-example 1.1: Solved using LFEM, LFEM-F, MSFEM-L, MSFEM-O, EMSFEM-L and EMSFEM-O. LFEM-F divides the study area into 88200 units, other methods divide the study area into 1800 units; MSFEM divides each coarse grid unit into 49 fine grid units (7×7), EMSFEM divides each A coarse grid is divided into 8 medium grid units, each medium grid unit is divided into 6 fine grid units, a total of 48 fine grid units.

[0060] image 3 is the absolute error value of the water head at the y=100m section of the above numerical method, it can be seen that the error of LFEM is the largest, th...

Embodiment 2

[0066] Example 2: Gradient medium model of two-dimensional unsteady flow

[0067] The research area is a square area: Ω=[0,10km]×[0m,10km], the research equation is:

[0068]

[0069] The thickness of the aquifer in the study area is 10m, and the left and right sides are the boundaries of constant water head, the water heads are 10m and 0m respectively, and the upper and lower sides are separated from each other. The permeability coefficient increases from 1m / d to 250m / d from the left boundary to the right boundary, that is, K(x,y)=1+x / 40m / d, and the water storage coefficient S=0.00001-0.000009x / 1000 / m, at coordinates ( There is a pumping well at 5200m, 5200m) with a flow rate of 1000m 3 / d. The water head at the initial moment changes linearly from left to right: H 0 (x, y) = 10-x / 1000m.

[0070] Sub-example 2.1: In this example, the pumping time is 5 days and the time step is 1 day. LFEM, LFEM-F, MSFEM-L, MSFEM-O, EMSFEM-L and EMSFEM-O are used to solve the problem. ...

Embodiment 3

[0078] Embodiment 3: two-dimensional submerged flow model (non-linear model)

[0079] The research equation is the Boussinesq equation:

[0080] -▽·K(x,y,H)▽H=W,

[0081] All parameters in this example have been dimensionless and have no units; the study area is: Ω=[0,1]×[0,1], the boundary water head is the boundary of constant head and both are 0, the base level b=-4, the permeability coefficient for:

[0082]

[0083] Where T=(1+x)(1+y), the initial water head is 0, this model has an analytical solution: H=xy(1-x)(1-y), and the source-sink term W is given according to the analytical solution.

[0084] The Boussinesq equation is a nonlinear equation that can be solved iteratively:

[0085] -▽·K(x,y,H (n-1) )▽H (n) =W,

[0086] The set iteration error is η=10 -4 , that is, iterate until |H (n) -H (n-1) |<η.

[0087] Using MSFEM-O and EMSFEM-O to solve this example, the study area is divided into 1800 parts. MSFEM divides the coarse grid unit into 49 fine grid un...

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Abstract

The invention discloses an efficient multi-scale finite element method for simulating two-dimension water flow movement in porous media. The method comprises the steps that a problem needing to be solved is converted into a variation form; boundary conditions of a research area are determined, coarse grid unit dimension is set, the research area is divided, and coarse grid units are obtained; middle grid unit dimension is set, and each coarse grid unit is divided into middle grid units; fine grid unit dimension is set, and each middle grid unit is divided into fine grid units; the degenerate ellipse type problem on the coarse grid units is converted into subproblems of the number of the middle grid units through the area decomposition technique, and values of a multi-scale primary function on all nodes of the middle grid units are obtained by solving the subproblems; a total stiffness matrix can be obtained through the variation form, and a system of simultaneous equations of a water head total stiffness matrix and a right end term is solved through an effective calculation method; water heads of all nodes on the research area are obtained. Compared with a traditional finite element method and a multi-scale finite element method, the calculation efficiency is higher.

Description

technical field [0001] The invention belongs to the technical field of hydraulics, and in particular relates to an efficient multi-scale finite element method for simulating two-dimensional water flow motion in porous media. Background technique [0002] Groundwater resources are an important part of water resources and one of the important sources of water for industry, agriculture and cities. In hydrogeology, groundwater level can reflect the mechanical energy of groundwater. The distribution of groundwater is closely related to the project implementation plan, construction method, construction time, project funds and other factors; therefore, the study of numerical calculation methods related to groundwater level is very necessary for the analysis of groundwater distribution and movement, and has important research value . [0003] The traditional finite element method is one of the commonly used numerical calculation methods for groundwater, and it is widely used in th...

Claims

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Application Information

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Patent Type & Authority Patents(China)
IPC IPC(8): G06F17/50
CPCG06F30/23
Inventor 谢一凡吴吉春薛禹群常勇谢春红
Owner NANJING UNIV
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