Limited unit method


In a finite element method, divide the calculation domain discrirting into a limited number of unloadable and interconnected units, select the base function within each unit, with The line-shaped combination of the unit base function is approximated the true solution in the unit, and the overall base function of the entire calculation domain can be constructed by each unit base function, and the solution within the entire calculation domain can be regarded as approximation from all units. Solution. In river numerical simulation, common finite element calculation methods are the development of Ritzfa and Garaijinfa, minimum multiplier, etc., which are developed by variational method and weighted balance. The finite element method is also divided into a variety of calculation formats depending on the power function and interpolation function employed. For the choice of power functions, there is a configuration method, a torque method, a minimum multiplier, and a Garakin method, from the shape of the computing unit mesh, a triangular mesh, a quadrilateral mesh and a polygon mesh, from interpolation The accuracy of the function is divided into linear interpolation functions and highly interpolated functions. Different combinations also constitute different finite element calculation formats. For the power function, the Galerkin method is the basis function of the power function as the approximation function; the least squares method is the power function is equal to the margin itself, and the total minimum value of the volume is for the consolidation coefficient. The square error is minimal; in the configuration method, first select N configuration points in the calculation domain. The approximate solution is strictly satisfied with the differential equation in the selected N configuration points, that is, the balance of the equation is 0 on the configuration point. Interpolation functions are generally consisting of different power polynomials, but there are also product representations with triangular functions or index functions, but most commonly used polynomial interpolation functions. The finite element interpolation function is divided into two categories. One type only requires interpolation polynomial itself in the interpolated point to take a known value, called the Lagrange (lagrange) polynomial interpolation; the other requires not only the interpolation polynomial itself, but also requires it The derivative value is known in the interpolated point, called the Hermite polynomial interpolation. The unit coordinates have a Cartesian right-angle coordinate system and no introspective natural coordinates, symmetrical and asymmetrical. The commonly used unusual coordinates are a partial coordinate system. Its definition depends on the geometry of the unit, one-dimensional viewing length ratio, two-dimensional viewing area ratio, three-dimensional consideration as a volume ratio. In two-dimensional finite element, the application of the triangular unit application is also more widely used. For the two-dimensional triangular and quad-shaped power unit, the commonly used interpolation function is a linear interpolation function and a second order or higher order function in the LagRange interpolation coordinate system, the linear interpolation function in the area coordinate system, second order or higher Order insertion function, etc.

Solution Step

Establishing the Integral Equation

According to the principle of orthogonalization of the variational principle or equation balance and the power function, Establish integral expression of the equivalent of the initial boundary value problem of differential equations, which is the starting point of the finite element method.

Regional unit score

According to the physical characteristics of the shape and practical problem of the area, the region is connected to each other, the unit is connected to each other. . The regional unit division is a pre-preparation for finite element methods. This part of the workload is relatively large, in addition to numbered the computing unit and node and determines the relationship between each other, but also indicates the position coordinates of the node, but also needs columns The node serial number and the corresponding boundary value of the natural boundary and the nature boundary.

Determine the unit base function

Select the interpolation function that satisfies a certain interpolation condition according to the number of nodes in the unit and the approximate solution, select the interpolation function that meets a certain interpolation condition as a unit base function. . The base function in the finite element method is selected in the unit, and since each unit has a rule geometry, a certain method can be followed when selecting a base function.

unit analysis

Approxes the linear combination expression of the solving function in each unit by the linear combination expression of the cell-based function; the approximate function is substituted into the integration equation. And the unit area is integrated, and an algebraic equation group containing the tolerance (the parameter value of each node in the unit) is obtained, referred to as a unit finite element equation.

Overall Synthesis

After the unit finite element equation is obtained, all unit finite element equations in the region are accumulated according to certain orders, forming an overall finite element. equation.

The processing of boundary conditions

There are three forms of general boundary conditions, divided into nature boundary conditions (Diilikle boundary conditions), natural border conditions ( Limann boundary conditions), mixed boundary conditions (Keti boundary conditions). For natural boundary conditions, it is generally available in integral expressions. For nature boundary conditions and mixed boundary conditions, the overall finite element equation is required to be corrected according to certain orders.

Designing Finite Element Equation

According to the overall finite element equation group of boundary conditions, it is a closed equation group that contains all unknown amounts, adopting appropriate The numerical calculation method is solved, and the function value of each node can be obtained.

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