The methods are presented on the basis of the results presented in the textbooks by Zenkevich, Morgan and Rumyantsev.
Weighted residual methods
A large group of methods for the approximate solution of differential equations is based on a mathematical formulation associated with the integral representation of the weighted residual. This group of methods is called weighted residual methods.
Let there be a differential equation and a boundary condition to it:
Here L is a differential operator; x i? spatial coordinates; V and S? volume and outer border of the study area; u 0 is the exact solution.
while the coefficients? unknown quantities to be determined using some mathematical procedure.
In residual methods, this procedure consists of two sequential stages. At the first stage, by substituting the approximate solution (3) into equation (1), the error function, or residual, is found, which characterizes the degree of difference from the exact solution:
As a result, an algebraic equation is obtained containing the current coordinates and M, as before, unknown coefficients.
At the second stage, requirements are imposed on the residual function (4) that minimize either the residual itself (collocation method) or the weighted residual (least squares method and Galerkin's method).
In the collocation method, it is believed that the differential equation is satisfied only at some selected (arbitrarily) points? collocation points, the number of which is equal to the number of unknown coefficients. At these M points, the residual must be zero, which leads to a system of M algebraic equations for M coefficients:
In the methods of weighted residual, the weighted residual is first formed by multiplying it by some weight functions, and then it is minimized on average:
In the least squares method? the Rayleigh-Ritz method? the error itself is selected as the weighting function, i.e. , and it is required that the value (functional) obtained in this way is minimal:
For this, the following condition must be met:
leading to a system of algebraic equations for unknown coefficients.
In the Galerkin method, the functions themselves, called basic ones, are taken as weight functions, and their orthogonality is required to the residual:
If? is a linear operator, then system (9) becomes a system of algebraic equations for the coefficients.
Basic concept of the finite element method
The main difficulty in the direct application of the classical methods of weighted residuals is associated with the choice of basis functions for the domain as a whole. These functions must not only satisfy the boundary conditions, but also adequately describe the geometry and other characteristics of the problem. All these conditions are usually difficult to fulfill, especially for objects (structures) of complex geometry in the presence of complex heat transfer, and therefore the capabilities of the methods in their classical sense are very limited.
With the advent of high-speed computers, the idea of localizing approximating functions in small regions (subdomains), called finite elements
An important feature of the FEM is that initially, with local approximation of a function on finite elements, they can be considered independently of each other. This means that each element can be considered isolated from the entire set and the function on this element can be approximated using its values at its nodes, regardless of what place the considered element will take in the related model, and on the behavior of the function on other finite elements. From a mathematical point of view, this means the following. For each element, a local (element) approximating function is written:
where? the number of nodes belonging to the th element; ? the values of the required function in its nodes; ? basic function; ? the volume of the element.
Since each element is considered separately, its properties are studied independently of other elements, i.e. the differential equation with the corresponding boundary conditions is solved for each th element, for example, by the Galerkin method:
The matrices obtained on the basis of (2.2) for individual elements, which contain the nodal values of the function as an unknown, are formed into global matrices for the entire domain of definition. Solving the system of algebraic equations obtained in this way, the values of the desired function at the nodes are determined, which makes it possible to find an approximate solution to the problem for the entire region as a whole:
where is the number of elements, the aggregate of which approximates the region as a whole.
The implementation in the framework of the FEM of the representation of the domain of definition by a set of finite elements determines the following important advantages of the FEM, providing its wide application for solving problems of field theory:
* local approximation at each element is uniquely determined by the values of the required function at the nodal points;
* provides a wide variation in the setting of boundary conditions in individual sections of the border (external and internal) area;
* curved sections of the boundaries of the region can be approximated by straight lines;
* the size and geometric shape of the elements may be different;
* the interconnection of elements does not have to follow any regular structure;
* material properties of each element can be individual and, moreover, anisotropic;
* it is possible to improve the accuracy of solving the problem by increasing the number of elements, limited only by the power of the computer used;
* due to the presence of common nodal points, global matrices are banded, i.e. contain a large number of zeros.
In accordance with the concept of FEM, the main stages of its application to solving boundary value problems of field theory are as follows:
* construction of a mesh from finite elements interconnected at nodal points.
In this case, the boundaries of the external elements approximate the boundary of the region as a whole;
* getting basic functions of elements;
* building a matrix representation for each element on the basis;
* unification of all elements into an ensemble by means of matrix transformations;
* setting boundary conditions for elements;
* solution of the resulting system of equations: ordinary differential first order (non-stationary process) or algebraic (stationary process);
* conclusion and evaluation of results; calculation of any other function depending on the values at the nodes of the found solution to the problem.
The first stage of the finite element procedure - the decomposition of the object under study (structure or its parts) into finite elements interconnected at the nodal points - includes the following operations:
* selection of types of elements, the totality of which approximates the object;
* setting the size and, thus, the number of elements;
* numbering of elements and nodes, and indexing of the latter.
Lecture 6
Weighted residuals method
Weighted residuals method
The least squares method is pretty simple in concept. However, the so-called weighted residual method... In this method, the system of equations for determining the unknown coefficients is constructed as follows:
Here 
- some system of "weight" functions. Hence, by the way, the name "method of weighted residuals".
The mathematical meaning of this approach is as follows. Note that the integrals in (28) are the scalar products of the residual function by the weight functions. If we use a geometric analogy, then we can say that the integrals in (28) are the projections of the residual function onto the weight functions.
If it were possible to use the complete system of functions as weighting functions, then the obtained solution would be exact. However, for obvious reasons, it is necessary to use a finite number of weight functions.
Let us write system (28) in relation to the considered example (1):
That is, again, as in the least squares method, the problem is reduced to solving a system of linear equations. But the elements of the matrix and vector have a different form:
The weighting function system can be selected in various ways. Let's try the simplest option first: the first three power series functions:
Recall that we must restrict ourselves to only three weight functions, since in this example we are looking for an approximate solution in the form of a linear combination of three functions (18), and the approximate solution (17) contains three unknown coefficients:.
Substituting (18) and (31) into (30), we obtain
,
and system solution:
Substituting the found values of the coefficients in (17), we obtain
Table 3
| x | Exact solution | Weighted residuals method (weight functions: 1, x,x 2) |
| 0.25 | -0.0716449 | -0.0611209 |
| 0.5 | -0.1013212 | -0.0780438 |
| 0.75 | -0.0716449 | -0.0565199 |
 |
The corresponding graph is in Figure 9.
Fig. 9
As you can see, the results turned out to be worse than when using both the finite difference method and the least squares method. The reason for this trouble is not that the method of weighted residuals is bad. The point is that the system of weighting functions was chosen unsuccessfully. As already mentioned, in "Mathematical digression" (the second paragraph of this section) these functions are neither normalized nor orthogonal. In the same place, an orthonormal system of functions equivalent to (31) was obtained by the Gram-Schmidt method. Now let's try to use the functions of this system as weight functions:
In this case, the matrix and vector are:
and the solution of the system:
As a result of substitution of these values in (17):
Table 4
| x | Exact solution | Weighted residual method (orthonormal system of power functions) |
| 0.25 | -0.0716449 | -0.0717608 |
| 0.5 | -0.1013212 | -0.1010489 |
| 0.75 | -0.0716449 | -0.717608 |
It can be seen here that a seemingly insignificant improvement in the choice of weighting functions led to a significant increase in the accuracy of the approximate solution. By the way, note that although the matrices and obtained by the least squares method are different in the latter case, the solutions of these linear systems practically coincided. The graph of the approximate solution is therefore not shown. It would look like an exact repetition of Figure 8.
A large group of methods for the approximate solution of differential
equations is based on a mathematical formulation associated with
integral representation of the weighted residual. This group of methods is called weighted residuals .
Let there be a differential equation and a boundary condition to it:
Here L−differential operator; x i- spatial coordinates; V and S- volume and outer border of the investigated area; u 0- exact solution.
in this case, the coefficients are unknown quantities to be determined using some mathematical procedure.
In residual methods, this procedure consists of two sequential stages. At the first stage, by substituting the approximate solution (2.1.3) into equation (2.1.1), we find the function mistake, or discrepancy which characterizes degree of distinction from accurate solutions :
As a result, an algebraic equation is obtained containing the current coordinates and M still unknown coefficients.
At the second stage, requirements are imposed on the residual function (2.1.4) that minimize either the residual itself (collocation method) or the weighted residual (least squares method and Galerkin's method).
In the collocation method, it is assumed that the differential equation is satisfied only at some selected (arbitrarily) points - collocation points, the number of which is equal to the number of unknown coefficients. In these M points, the residual must be zero, which leads to the system M algebraic equations for M coefficients:
In the methods of weighted residual, the weighted residual is first formed by multiplying it by some weight functions, and then it is minimized on average:
In the least squares method - the Rayleigh-Ritz method - the error itself is chosen as the weighting function, i.e. , and it is required that the value (functional) obtained in this way is minimal:
For this, the following condition must be met:
leading to a system of algebraic equations for unknown coefficients.
In the Galerkin method, the functions themselves are taken as weight functions, called basic, and you need them orthogonality to residual :
If is a linear operator, then system (2.1.9) turns into a system of algebraic equations for the coefficients.
Let's consider the Galerkin method using a specific example. An equation is given on the interval:
Comparison of the approximate results obtained by various methods with the exact solution is given in Table 1.
150. EXPRESS AND IMPLICABLE DIFFERENCE SCHEMES. METHOD OF WEIGHED MISTAKES. BUBNOV-GALERKIN METHOD.
Difference scheme is a finite system of algebraic equations assigned to a differential problem containing a differential equation and additional conditions (for example, boundary conditions and / or initial distribution). Thus, difference schemes are used to reduce a differential problem of a continuous nature to a finite system of equations, the numerical solution of which is in principle possible on computers. Algebraic equations associated with a differential equation are obtained using the difference method, which distinguishes the theory of difference schemes from other numerical methods for solving differential problems (for example, projection methods, such as the Galerkin method).
A solution to a difference scheme is called an approximate solution to a differential problem.
Although the formal definition does not impose significant restrictions on the form of algebraic equations, in practice it makes sense to consider only those schemes that in some way correspond to the differential problem. Important concepts in the theory of difference schemes are the concepts of convergence, approximation, stability, and conservatism.
Explicit schemas
Explicit schemas compute the value of the result across multiple adjacent data points. An example of an explicit scheme for differentiation: (2nd order of approximation). Explicit schemas are often unstable.
Here V *
- an approximate solution,
F- a function that satisfies the boundary conditions,
N m - trial functions, which should be equal to zero at the boundary of the region,
A m - unknown coefficients that must be found from the condition of the best satisfaction of the differential operator,
M- the number of trial functions.
If you substitute V* into the original differential operator, then we get a residual that takes different values at different points of the region.
R = LV * + P
Here W n- some weighting functions, depending on the choice of which the variants of the method of weighted residuals are distinguished,
S- the area of space in which the solution is sought.
When choosing delta-functions as weight functions, we will have a method that is called the method of pointwise collocation, for piecewise-constant functions - the method of collocation by subdomains, but the most common is the Galerkin method, in which trial functions are chosen as weight functions N... In this case, if the number of test functions is equal to the number of weight functions, after expanding certain integrals, we arrive at a closed system of algebraic equations for the coefficients A.
KA + Q = 0
Where the coefficients of the matrix K and the vector Q are calculated by the formulas:
After finding the coefficients A and substituting them in (1), we obtain a solution to the original problem.
The disadvantages of the method of weighted residuals are obvious: since the solution is sought over the entire region at once, the number of trial and weight functions must be significant to ensure acceptable accuracy, but at the same time difficulties arise in calculating the coefficients K ij and Q i, especially when solving plane and volumetric problems, when it is required to calculate double and triple integrals over regions with curvilinear boundaries. Therefore, in practice, this method was not used until the finite element method (FEM) was invented.
Weighted residuals method
FEM is based on the method of weighted residuals, the essence of which is as follows: a function is selected that satisfies differential equations and boundary conditions, but it is not chosen arbitrarily, since such a selection is hardly possible already in two-dimensional space, but using special methods.
Let the state of some medium be described by the following differential operator, with a given boundary condition:
Here L is a differential operator (for example, the Laplace operator),
V - phase variable - unknown function to be found,
P is a value independent of V,
V (G) = V g is a boundary condition of the first kind (Dirichlet), that is, the value of the phase variable is set on the boundary.
We will look for a solution using a function of the following form:
Here V * is an approximate solution,
F is a function that satisfies the boundary conditions,
N m - trial functions, which on the boundary of the region must be equal to zero,
A m - unknown coefficients that must be found from the condition of the best satisfaction of the differential operator,
M is the number of trial functions.
If we substitute V * into the original differential operator, then we get a residual that takes different values at different points of the region:
It is necessary to formulate a condition to minimize this discrepancy over the entire region. One of the options for such a condition may be the following equation:
Here W n are some weight functions, depending on the choice of which the variants of the method of weighted residuals are distinguished,
S is the region of space in which the solution is sought.
When choosing delta-functions as weight functions, we will have a method that is called the method of pointwise collocation, for piecewise-constant functions - the method of collocation by subdomains, but the most common is the Galerkin method, in which trial functions N are chosen as weight functions. In this case, if the number of test functions is equal to the number of weight functions, after expanding certain integrals, we arrive at a closed system of algebraic equations for the coefficients A.
where the coefficients of the matrix K and the vector Q are calculated by the formulas:
After finding the coefficients A and substituting them in (1), we obtain a solution to the original problem.
The disadvantages of the method of weighted residuals are obvious: since the solution is sought over the entire region at once, the number of trial and weight functions must be significant to ensure acceptable accuracy, but at the same time, difficulties arise in calculating the coefficients Kij and Qi, especially when solving plane and volumetric problems when it is required calculation of double and triple integrals over areas with curvilinear boundaries. Therefore, in practice, this method was not used until the finite element method was invented.
The idea of FEM is to use simple trial and weight functions in the method of weighted residuals, but not in the entire domain S, but in its individual subdomains (finite elements). The accuracy of the solution of the problem must be ensured by using a large number of finite elements (FE), while FE can be of a simple form and the calculation of integrals over them should not cause any particular difficulties. Mathematically, the transition from the method of weighted residuals to the FEM is carried out using special test functions, which are also called global basis functions, which have the following properties:
1) at the approximation node, the functions have a value equal to one;
2) the functions are nonzero only in FE containing this approximation node, in the rest of the region they are equal to zero.
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